Einstein at Work on Unified Field Theory : The Five-Dimensional Einstein-Bergmann Approach 9783031521263, 9783031521270

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Einstein Studies 17

Tobias Schütz

Einstein at Work on Unified Field Theory The Five-Dimensional Einstein-Bergmann Approach

Einstein Studies Volume 17

Series Editors Don Howard, Notre Dame, IN, USA Diana L. Kormos-Buchwald, Dept. of Humanities and Social Sciences, California Institute of Technology, Pasadena, CA, USA Editorial Board Members Giovanni Amelino-Camelia, Dipartimento di Fisica “Ettore Pancini”, University of Naples Federico II, Napoli, Napoli, Italy Alisa Bokulich, Ctr for Philosophy & History of Science, Boston University, Boston, MA, USA Alessandra Buonanno, Max Planck Institute for Gravitational Physics, Potsdam, Brandenburg, Germany Danian Hu, History, The City College of New York, GLEN ROCK, NJ, USA Michel Janssen, Tate Laboratory of Physics, University of Minnesota, Minneapolis, MN, USA Dennis Lehmkuhl, Institut für Philosophie, Universität Bonn, Bonn, NordrheinWestfalen, Germany John D. Norton, Dept of History & Philosophy of Sci, University of Pittsburgh, Pittsburgh, PA, USA Jurgen Renn, MPI for the History of Science, Berlin, Berlin, Germany Carlo Rovelli, Université de la Méditerranée, Centre de Physique Théoretique, Marseille, France Sahotra Sarkar, Dept of Philosophy, Rm 415, University of Austin, Austin, TX, USA Tilman Sauer, Institut für Mathematik, 5. OG, Johannes Gutenberg-Universität, Mainz, Rheinland-Pfalz, Germany Rainer Weiss, Massachusetts Inst of Technology, Cambridge, MA, USA

Einstein Studies is a multidisciplinary series, reflecting the wide variety of Einstein’s own contributions to our century. The series emphasizes the history of twentiethcentury science, but will also welcome works in the philosophy of science, as well as social, cultural, and political history. Original research in physics, mathematics, and philosophy will also be published in cases where such research explores the implications of Einstein’s work for contemporary issues.

Tobias Schütz

Einstein at Work on Unified Field Theory The Five-Dimensional Einstein-Bergmann Approach

Tobias Schütz Lahnstein, Germany

ISSN 2381-5833 ISSN 2381-5841 (electronic) Einstein Studies ISBN 978-3-031-52126-3 ISBN 978-3-031-52127-0 (eBook) https://doi.org/10.1007/978-3-031-52127-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Für Leon

Acknowledgments

First and foremost, I thank Prof. Dr. Tilman Sauer for supervising my thesis, which gave rise to this book. It all started with some very interesting seminars on Einstein and his theories of relativity, which were the motivation for my later projects. I am also deeply thankful to Prof. Dr. Diana Kormos Buchwald for her continuous support and for inviting me to the Einstein Papers Project (EPP) at Caltech. I am grateful to the entire EPP team for always supporting me and my research. I am also thankful to the Albert Einstein Archives (AEA) at The Hebrew University of Jerusalem as well as to the Library of the Eidgenössische Technische Hochschule Zürich (ETH) for letting me use their extensive archival material. I wish to thank my loving wife, Ilka, for her unfailing support. Lahnstein November 2023

vii

Contents

1

2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation and Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 The Collected Papers and the Princeton Manuscripts . . . . . . . . . 1.1.2 Prague Notebook. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Overview of Research Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Einstein’s Education and His Way to General Relativity with a Special Focus on Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Primary and Secondary School in Munich, Milano, and Aarau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Einstein’s Studies at ETH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Summary on Einstein’s Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Einstein’s Way to General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Einstein’s Unified Field Theory Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Outline of Einstein’s 1938 Program Based on Subsequent Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Einstein’s Princeton Assistants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Peter Gabriel Bergmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Valentine Bargmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Structure of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Einstein and Projective Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 The Prague Notebook and the Princeton Manuscripts . . . . . . . . 2.1.2 Projective Relativity and the Pasadena Diary . . . . . . . . . . . . . . . . . 2.1.3 Bergmann Correspondence and the Point at Infinity . . . . . . . . . . 2.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Pascal Line and Involution on a Conic. . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Analysis of Princeton Manuscripts and Further Working Sheets . . . . . 2.3.1 AEA 62-785r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 6 7 8 11 11 15 21 21 51 56 61 62 63 63 67 70 70 75 76 77 77 80 80 82 85 ix

x

3

4

Contents

2.3.2 AEA 62-787r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 AEA 62-789. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 AEA 62-789r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 AEA 124-446 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.6 Further Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Summary of the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Prague Notebook and Its Connection with Princeton Manuscripts . . . 2.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Conjecture on the Purpose of Einstein’s Ideas About Projective Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95 106 111 118 123 123 125 139

Different Pathways to the Generalization of Kaluza’s Theory. . . . . . . . . . 3.1 Einstein’s Field Equations of General Relativity . . . . . . . . . . . . . . . . . . . . . . 3.2 Einstein and Bergmann’s Generalized Kaluza Theory . . . . . . . . . . . . . . . . 3.2.1 Space Structure and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Field Equations of General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Field Equations in Generalized Kaluza Theory . . . . . . . . . . . . . . . 3.3 Einstein’s Opinion on the New Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Einstein’s Washington Manuscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Einstein and Bergmann Correspondence . . . . . . . . . . . . . . . . . . . . . . 3.4.2 The Donation to the Library of Congress. . . . . . . . . . . . . . . . . . . . . . 3.4.3 Differences Between the Two Versions of the Manuscript . . . 3.4.4 Axiomatic Structure of the Manuscript and its Differences to the Publication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.5 A Correction to the Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Concluding Remarks on the Washington Manuscript . . . . . . . . . . . . . . . . . 3.6 Manuscript Pages Dealing with the 1938 Theory . . . . . . . . . . . . . . . . . . . . . 3.6.1 AEA 62-789. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 AEA 62-798. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 AEA 62-802 and AEA 62-794 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.4 AEA 62-785. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143 146 148 149 160 164 176 179 181 182 189

Einstein’s Further Considerations on the Generalized Kaluza Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Einstein and Bergmann’s Correspondence: Dating the Letters . . . . . . . 4.2 Context of Einstein’s Further Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Most Important Relations of the Generalized Kaluza Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Wave Equations and the Stromgleichung . . . . . . . . . . . . . . . . . . . . . . 4.2.3 The Usage of Different Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Einstein’s Ansatz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Manuscript Pages Related to Publication, Correspondence, and Further Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Sheets Containing Sketches on Projective Geometry . . . . . . . . . 4.3.2 Power Series Expansions of β and η at Zero and Infinity . . . .

141

190 197 200 201 202 205 206 215 221 223 227 227 228 231 234 235 254 255 266

Contents

xi

4.3.3 4.3.4 4.3.5

Calculations on F1 to F4 and on the Laplace Operator . . . . . . . Rewriting the Equations in Terms of ρ . . . . . . . . . . . . . . . . . . . . . . . . Sheets Containing Idiosyncratic Relations and Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.6 Considerations on Different Expressions for ρ . . . . . . . . . . . . . . . . 4.3.7 AEA 62-791: An Example for Both Mistakes and Skills . . . . . 4.3.8 Further Sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

266 267

5

Considerations on Delta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Analysis of Related Manuscript Pages and Correspondence . . . . . . . . . 5.2.1 AEA 62-783. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Related Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 An Expression for the Functions α, β, δ, and η . . . . . . . . . . . . . . . 5.2.4 AEA 62-795. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 AEA 62-795r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.6 An Expression Similar to δ-Function from 1938 . . . . . . . . . . . . . . 5.2.7 Further Manuscript Pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.8 AEA 63-058r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

279 281 283 283 292 295 298 308 315 317 320 322

6

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

267 268 269 275 276

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Index of Research Notes, Letters, and Further Documents . . . . . . . . . . . . . . . . . 351

List of Figures

Fig. 1.1 Fig. Fig. Fig. Fig. Fig. Fig. Fig.

2.1 2.2 2.3 2.4 2.5 2.6 2.7

Fig. Fig. Fig. Fig.

2.8 2.9 2.10 2.11

Fig. 2.12 Fig. 2.13 Fig. 2.14 Fig. Fig. Fig. Fig.

2.15 2.16 2.17 2.18

Fig. Fig. Fig. Fig. Fig.

2.19 2.20 2.21 2.22 2.23

Fig. 2.24

Grossmann’s notes of Geiser’s lecture on infinitesimal geometry from 1898 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The double page of Einstein’s Prague notebook. . . . . . . . . . . . . . . . . . . . . Manuscript pages AEA 62-785r, 62-787r, 62-789, and 62-789r . . . Manuscript page AEA 124-446 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The back of the letter AEA 6-250. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . First entry in Einstein’s Pasadena diary from 1931/32 . . . . . . . . . . . . . . Involution on a conic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From the inscribed hexagon to the quadrangle and to the triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Manuscript page AEA 62-785r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reconstruction of Einstein’s first sketch on AEA 62-785r . . . . . . . . . Reconstruction of Einstein’s second sketch on AEA 62-785r . . . . . . Grossmann’s transcription of the first lecture on imaginary elements in 1897 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Manuscript page AEA 62-787r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reconstruction of Einstein’s top left sketch on AEA 62-787r . . . . . . Reconstruction of Einstein’s bottom left sketch on AEA 62-787r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reconstruction of Einstein’s middle sketch on AEA 62-787r . . . . . . Reconstruction of Einstein’s right sketch on AEA 62-787r . . . . . . . . Connection between two sketches on AEA 62-787r . . . . . . . . . . . . . . . . Transition from the elliptic to the hyperbolic and parabolic involutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Manuscript page AEA 62-789 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reconstruction of the right sketch on AEA 62-789 . . . . . . . . . . . . . . . . . Reconstruction of two of Einstein’s sketches on AEA 62-789 . . . . . Interpretation of Einstein’s top left sketch on AEA 62-789 . . . . . . . . Transition from Einstein’s left to the right sketch on AEA 62-789 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Manuscript page AEA 62-789r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 70 72 73 74 76 81 82 86 90 91 94 96 97 98 99 101 103 104 107 108 109 110 111 112 xiii

xiv

List of Figures

Fig. 2.25 Reconstruction of Einstein’s sketch at the bottom left on AEA 62-789r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 2.26 Reconstruction of Einstein’s sketch at the top of AEA 62-789r . . . . Fig. 2.27 Reconstruction of Einstein’s bottom right sketch on AEA 62-789r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 2.28 Sketches on AEA 62-789r and Grossmann’s lecture notes from 1897 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 2.29 Manuscript page AEA 124-446 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 2.30 Reconstruction of Einstein’s sketch on the left of AEA 124-446 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 2.31 Three sketches from AEA 124-446, 62-789, and Grossmann’s lecture notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 2.32 Reconstruction of Einstein’s sketch in the middle of AEA 124-446 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 2.33 Double page from the Prague notebook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 2.34 Reconstruction of Einstein’s first sketch on the double page . . . . . . . Fig. 2.35 Reconstruction of Einstein’s second sketch on the double page. . . . Fig. 2.36 Reconstruction of Einstein’s third sketch on the double page . . . . . . Fig. 2.37 Transition from Einstein’s hexagon to the quadrangle on the double page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 2.38 Similarities between two hexagons on AEA 62-787r and the double page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 2.39 Similarities between two quadrangles on AEA 62-787r and the double page. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 2.40 Einstein’s first sketch on page 50 of his Prague notebook . . . . . . . . . . Fig. 2.41 Reconstruction of Einstein’s fifth sketch on the double page . . . . . . . Fig. 2.42 Similarities between sketches on AEA 62-789, 124-446, and the double page. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 2.43 Similarities between two sketches on AEA 62-789 and the double page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 5.1 The manuscript page AEA 62-783. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 5.2 Plots of .gσ,τ (x) = i(x) · j (x) with .i(x) := x σ and −τ x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .j (x) := e

113 114 115 117 118 119 121 122 126 127 129 130 131 132 133 133 136 138 138 284 285

2

Fig. 5.3 Fig. 5.4 Fig. 5.5 Fig. 5.6 Fig. 5.7 Fig. 5.8 Fig. 5.9

−1σ ξ · a σ e−τ a e 2 a2 .. . . . . . . . . . .

Plots of .g(x) = x σ e−τ x and .h(ξ ) = B 2 2 Graph of the Gauss function .lr (α) = e−α r for different values of r. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The manuscript page AEA 62-795. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sketch drawn by Einstein on the manuscript page AEA 62-795. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The manuscript page AEA 62-795r. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plots of .A(α) for two different cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The graph of .f (r) = r −1 arctan(r/a) for different values of a.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

288 292 299 301 309 313 316

List of Tables

Table 1.1 Documents that will be discussed in the book . . . . . . . . . . . . . . . . . . . . . . 9 Table 4.1 Sequence of letters of Einstein and Bergmann’s correspondence from 1938. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

xv

Chapter 1

Introduction

The overall goal of this book is to date, categorize, contextualize, reconstruct, comment on, and interpret unidentified research notes by Albert Einstein (1879– 1955). The Albert Einstein Archives (AEA) at The Hebrew University of Jerusalem and the Einstein Papers Project at the California Institute of Technology (Caltech) have in their database and in their holdings more than 2000 manuscript pages with research notes written in Einstein’s hand that are for the most part not yet understood. These working sheets provide primary source material on Einstein’s way of theorizing, thinking, and calculating as well as on the heuristics of Einstein’s modus operandi, which cannot be captured by merely analyzing Einstein’s publications. In the following chapters, we will analyze research notes, scratch notebooks, unpublished manuscripts, letters, and publications. In particular, we will investigate Einstein’s program of finding a five-dimensional unified field theory for gravitation and electromagnetism in the context of the Einstein–Bergmann approach from 1938. In addition, the analysis of research notes allows us to draw conclusions about Einstein’s mathematical skills, particularly in the field of projective geometry. The structure of the book is based on the contents of the research notes that will be analyzed. In Chap. 2, we will look at several manuscript pages and a scratch notebook that were written some quarter century apart but contain strikingly similar considerations on projective geometry. This alone is an intriguing fact, especially as Einstein apparently did not use these ideas in his publications. Moreover, the analysis of the manuscript pages allows us to investigate Einstein’s knowledge and interest in projective geometry in due consideration of his mathematical education and how he put it to use for his research. The same manuscript pages also contain considerations that we will connect with a publication on the five-dimensional unified field theory from 1938 by Einstein and his assistant Peter Bergmann. We will discuss both the publication and the respective research notes in Chap. 3, where we will encounter further documents like an unpublished manuscript, which is related to the publication regarding its content but substantially differs in its internal structure. We will see that Einstein composed this manuscript alone without Bergmann being © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Schütz, Einstein at Work on Unified Field Theory, Einstein Studies 17, https://doi.org/10.1007/978-3-031-52127-0_1

1

2

1 Introduction

co-author and that he preferred his new manuscript over the publication. Parts of the research notes containing sketches on projective geometry are also related to an extant correspondence between Einstein and Bergmann from 1938. By browsing through all unidentified research notes of the database, we identified several further manuscript pages that document how Einstein tested and further developed his new theory and how he tried to find particle-like solutions. In Chap. 4, we will analyze Einstein and Bergmann’s correspondence as well as the related research notes and connect them with each other and with the publication. For this purpose, we will also use another correspondence between Einstein and his assistant Valentine Bargmann from 1939. The analysis of all those documents will allow us to identify specific requirements that Einstein imposed on the new theory. We also found further manuscript pages that are directly related to the 1939 correspondence when Einstein considered a certain integral form as well as .δ-distributions that he apparently used in order to search for particle solutions. We will discuss the respective research notes in Chap. 5.1

1.1 Motivation and Sources After establishing the theory of general relativity, Einstein wrote more than 40 technical papers on unified field theory with the aim to unify the theories of the electromagnetic and gravitational field (Sauer, 2014, p. 281). Pursuing this goal, he chose, rejected, and came back to several approaches over the years, however, he did not succeed (Sauer, 2019). Questions as for why Einstein rejected these theories, how he investigated the respective field equations, and what difficulties he encountered cannot be answered by merely looking at his publications (Sauer, 2019) as they usually do not reflect the process of how the results have been achieved and do not show false, unessential, or aborted approaches (Holmes et al., 2003a, p. vii). A hint on questions as posed above can be found in correspondence, where Einstein frequently wrote about new ideas, calculations, or suggestions on how to proceed further (Sauer, 2019, p. 378). A better understanding to such questions, however, can only be received by studying not only Einstein’s publications and correspondence but also his unpublished calculations on various manuscripts. Sauer concludes that this “is a considerable challenge, but it also offers the perspective of being able to understand Einstein’s thinking and, in fact, physical theorizing” (Sauer, 2019, p. 378). This is one of the goals of the present book, in which we will analyze publications, correspondence, and several working sheets containing research notes by Albert Einstein from around 1938.

1 The

present book originated from a Ph.D. project (Schütz, 2021). Previous results have been published in Sauer and Schütz (2019), Sauer and Schütz (2020), Schütz (2020), Sauer and Schütz (2021), and Sauer and Schütz (2022).

1.1 Motivation and Sources

3

Throughout the following chapters, we will encounter various kinds of research notes. Some of them have already been analyzed and others will be analyzed in the present book. Here, we will give a brief overview of some of Einstein’s extant notebooks and working sheets. We will also refer to further sections where we will discuss these documents in more detail. We start with probably the best known example of studied research notes of Albert Einstein, the so-called Zurich Notebook.2 This notebook consists of 84 pages, carefully analyzed by the Einstein scholars Janssen, Norton, Renn, Sauer, and Stachel (Janssen et al., 2007). Before parts of the first results of the analysis were published in Norton (1984), the question remained unclear of how Einstein suddenly became aware of several mathematical concepts and the importance of the metric tensor in the process of developing the theory of general relativity. For instance, this is documented in Stachel’s well-known paper on “the genesis of general relativity”: An attempt to understand how Einstein arrived at this truly revolutionary viewpoint is complicated by the fact that there is not a hint of it in his 1912 papers on gravitation; while the very next paper in 1913 starts off from this viewpoint. Therefore, one can only speculate on exactly what happened in the half-year between. (Stachel, 1979, p. 434)

The analysis of the Zurich notebook gave answers not only to this question such that the history of general relativity needed to be rewritten. As pointed out in Renn and Sauer (2003b, pp. 253–260), it is also a powerful example that the investigation of only correspondence and publications can mislead to erroneous conclusions: While a paper in 1913 suggests that Einstein was not familiar with generally covariant equations, the Zurich notebook clearly contradicts this view. We will further discuss the Zurich notebook several times throughout the book and particularly in Sect. 1.3.4.5. Another example of important research notes of Albert Einstein is the so-called Einstein Besso manuscript.3 As we will see in Sect. 1.3.4.7, shortly before finding the field equations in November 1915, Einstein struggled with three problems. Two of them were the wrong prediction of Mercury’s perihelion advance and the noncovariance of Einstein’s previous field equations with respect to rotating coordinate systems. The Einstein Besso manuscript written by both Einstein and his friend Michele Besso4 shows that they considered the derivation of Mercury’s perihelion advance that was finally published by Einstein in Einstein (1915c) already 2 years before in 1913, see Klein et al. (1995, pp. 344–359) and Earman and Janssen (1993, pp. 135–136). By an analysis of two pages of this manuscript, Janssen (Janssen, 1999, pp. 144–148) showed how Einstein missed the opportunity to detect the rotation problem at that time. In fact, further documents like four pages of Besso’s

2 For

facsimiles of this notebook, see Einstein (2007) and Klein et al. (1995, Doc. 10). facsimiles of the Einstein Besso manuscript, see Klein et al. (1995, Doc. 14). 4 Besso was a close and lifelong friend of Einstein, see Calaprice et al. (2015, pp. 65–66). 3 For

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1 Introduction

Nachlass5 suggest that Einstein might have considered this problem already in 1913, which gave new insights into Einstein’s modus operandi, see Janssen (2007). Notes on the rotation problem can also be found in the so-called Prague notebook6 from the time period 1909–1914, see Janssen (1999, pp. 137–138) or Klein et al. (1995, p. 345). This notebook also contains calculations and considerations on gravitational lensing from 1912, see Renn et al. (1997) and Renn and Sauer (2003a). Einstein published on this phenomenon only some 20 years later in 1936 (Einstein, 1936). The phenomenon was observed for the first time in 1979 (Walsh et al., 1979) and has in the meantime developed into a powerful tool in astrophysics and cosmology, see, for instance, (Huang et al., 2021; Bartelmann & Schneider, 2001; Wambsganss, 2006; Gaudi, 2012). By the analysis of the Prague notebook, the history of this important phenomenon has been rewritten. In fact, by further analysis of the notebook and related correspondence, it turned out that Einstein came back to this phenomenon in fall 1915, i.e., in the decisive time period when formulating his theory of general relativity (Sauer, 2008).7 The Prague notebook will play an important role in the book as we find a double page in it containing sketches on projective geometry. We will look at this notebook and in particular at the double page in several sections throughout the book. Below in Sect. 1.1.2, we will introduce it more comprehensively. As a last example of historical research using working sheets by Albert Einstein,8 we look at the Einstein–Podolsky–Rosen (EPR) paradox published in Einstein et al. (1935).9 As discussed in Sauer (2007a), the manuscript page AEA 62-575r dates from late 1954 or early 1955 and shows the only known version of Einstein discussing the argument for spin observables. At the same time, it shows one of the latest considerations by Einstein with quantum theory in general.10 The manuscript page belongs to a batch of sheets with mainly unidentified calculations. This batch will play an important role in the present book as we will analyze further pages throughout the following chapters. Hence, we will introduce these manuscripts in Sect. 1.1.1 in more detail. In fact, on some of the pages dated back to 1936 containing to this batch, Einstein considered gravitational lensing (Schütz, 2017) that made possible a characterization of Einstein’s different pathways of exploring what can be called space of implications (Sauer & Schütz, 2019).

5 For

facsimiles, see Renn (2005b, pp 127–130) and Janssen (2007). notebook is also called Scratch Notebook, see, for instance, Janssen (1999), Klein et al. (1993, Appendix A), Renn et al. (1997), or Sauer and Schütz (2021). 7 For a historical overview of gravitational lensing, see Trimble (2001), Barnothy (1989), and VallsGabaud (2006). 8 For further examples, see Sauer (2019). 9 In fact, Einstein did not compose this paper (Podolsky did) and he also did not fully agree with the content (Fine, 1996, p. 35). See also Howard (1985, 1990). 10 See also Sauer (2017). 6 This

1.1 Motivation and Sources

5

As already mentioned, we will discuss the Prague notebook as well as the batch of manuscripts in more detail. In addition, we also have a third main source, the socalled Washington Manuscript that was composed by Einstein in 1938 in the context of his publication on the generalization of Kaluza’s theory (Einstein & Bergmann, 1938).11 We will discuss both the publication and the manuscript in Chap. 3. An overview of all sources that will be used in the following chapters can be found in Sect. 1.2. Before discussing these sources in more detail, we finally note that analyzing research notes has not only become an important tool for Einstein scholars but also generally for historical research in both theoretical and practical fields of research.12 Such notebooks, manuscripts, laboratory notes, or research notes exist, to a differing extent, for several scientists as for André-Marie Ampère (1775– 1836) and Michael Faraday (1791–1867),13 Leonhard Euler (1707–1783),14 Galileo Galilei (1564–1642),15 Carl Friedrich Gauß (1777–1855),16 Kurt Gödel (1906– 1978),17 Felix Hausdorff (1868–1942),18 Heinrich Hertz (1857–1894),19 David Hilbert (1862–1943),20 Christiaan Huygens (1629–1695),21 Sofja Kowalewskaja (1850–1891),22 Samuel Pierpont Langley (1834–1906),23 Gottfried Wilhelm Leibniz (1646–1716),24 Ernst Mach (1838–1916),25 Isaac Newton (1643–1727),26 or Mary Sommerville (1780–1872).27.,28 In the case of Albert Einstein, a vast amount of publications, correspondence, manuscripts, and research notes is extant, which results in extensive archival and editorial work as we will see in the following.

11 See

also Sauer and Schütz (2020). research notes as a tool is not only used in the field of history of mathematics and physics but also in the field of history of chemistry, biology, or medicine, see Holmes et al. (2003b). 13 See Steinle (2003). 14 See Mikhailov (1959), Matvievskaja and Ožigova (1983), and Juškeviˇ c (1983). 15 See Büttner et al. (2001) and Renn and Damerow (2003). 16 See Reich and Roussanova (2012). 17 See Engelen (2019b,c,a). 18 See Purkert and Scholz (2010). 19 See Buchwald (2003). 20 See Majer (2010). 21 See Yoder (2013). 22 See Cooke (1984). 23 See Loettgers (2003). 24 See Leibniz Archiv/Leibniz Forschungsstelle Hannover (1990–2020). 25 See Hoffmann and Berz (2001, pp. 49–141) and Hoffmann (2003). 26 See Iliffe (2004) and Shapiro (2003). 27 See Stenhouse (2020). 28 This list shows only examples and does not claim to be complete. 12 Studying

6

1 Introduction

1.1.1 The Collected Papers and the Princeton Manuscripts After Einstein’s death in Princeton in 1955, Helen Dukas, who had been his secretary since 1928, and Otto Nathan, a close friend of Einstein, were named trustees of all of Einstein’s letters, manuscripts, and copyrights (Pais, 1982, p. 476). Until her death in 1982, Helen Dukas carefully organized and cataloged Einstein’s papers (Mendelsson et al., 2014). In 1970, Nathan discussed the idea of publishing a comprehensive edition with Princeton University Press (Stachel, 1987a, p. xii) resulting in The Collected Papers of Albert Einstein, with the first volume being published by John Stachel (Stachel, 1987a). To date, the Einstein Papers Project has published 16 volumes, most recently volume 16 in 2021 (Kormos Buchwald et al., 2021) covering Einstein’s writings until May 1929. The editors are now working on volume 17 (Kormos Buchwald et al., 2018a). Overall, the Einstein Papers Project aims to publish over 14,000 scientific and non-scientific documents including publications, correspondence, research notes, and further writings in some 30 volumes.29 For that purpose, Einstein’s documents in Princeton, which also contain material from the early years in Berlin, Switzerland, and Prague, were cataloged and microfilmed by the editorial team in Princeton, which is now located in Pasadena (Sauer, 2019). After Dukas’ death in 1982, the Hebrew University became the owner of the archives according to Einstein’s will (Stachel, 1987a, Foreword). Thus, all papers and books that had belonged to Einstein were shipped from Princeton to Jerusalem and the Albert Einstein Archives were constituted (Sauer, 2019). The archive now has an archival database of over 80,000 records which were made accessible due to digitization in 2012 (Mendelsson et al., 2014, p. 325). While preparing the Einstein estate to be shipped to Jerusalem, a large batch of some 1750 unidentified manuscript pages suddenly appeared behind a filing cabinet (Sauer, 2004, p. 13). Before sending them to Jerusalem, the editorial team in Princeton made Xerox copies of these documents (Sauer, 2019). In Jerusalem, the batch was microfilmed and labeled with the reel numbers 62 and 63,30 which is why we have now two versions of the manuscript pages (Sauer, 2004, 2019).31 The versions of copies are indeed copies of the same batch of manuscripts but differ in the physical sequence of sheets (Sauer, 2019). Probably, the Xerox copies reflect the original physical sequence more precisely since they were made before shipping. Due to this amount of manuscript pages, it will be a major challenge for the Einstein Papers Project to publish the documents. Most of the pages are undated

29 See

the homepage of the Einstein Papers Project: http://www.einstein.caltech.edu (visited on 06/10/2021). 30 The numbering is in such a way that pages marked by “r” are the reverse sides of the pages with the same archival number without the “r.” We will refer to the former by back page and call the latter the front of the page. 31 The manuscripts belonging to reels 62 and 63 are frequently called Princeton manuscripts in the literature as they stem from Einstein’s time in Princeton.

1.1 Motivation and Sources

7

and unidentified and contain research calculations in Einstein’s hand, without any remarks or explanations about the context, since they were made for Einstein’s own use (Sauer, 2004, pp. 13, 14). A tentative analysis of the pages came to the result that, in all probability, the manuscript pages were written after 1928 and deal mostly with the search of a unified field theory (Sauer, 2004, pp. 13, 16). As we already argued above,32 the Princeton manuscript pages are of special interest to us as they contain information about Einstein’s thinking, ideas, and paths to solve his upcoming questions at the time. In addition to reels 62 and 63, the database also contains further unidentified sheets written in Einstein’s hand. In the preparation of the book in hand, all sheets of reels 62 and 63 as well as further sheets containing unidentified calculations were studied carefully. In doing so, some pages with an archival number not belonging to reels 62 and 63 have also been identified and contextualized. For instance, we will discuss the page AEA 124-446 in Chap. 2 as it contains interesting sketches on projective geometry. A full list will be given in Sect. 1.2.

1.1.2 Prague Notebook Among Einstein’s early research notes which are extant belongs a scratch notebook written by Einstein probably between 1909 and 1915 (Klein et al., 1993, Appendix A). We will refer to this notebook as Prague notebook as Einstein wrote parts of it during his stay in Prague.33 In October 1909, Einstein became an associate professor of theoretical physics at the University of Zurich (Pais, 1982, p. 523). Only 2 years later, in March 1911, Einstein moved to Prague (Pais, 1982, p. 523). There, he published his famous paper on light bending (Einstein, 1911a), where he predicted that light is bent by heavy masses like the Sun. One year later in August 1912, Einstein moved back to Zurich, but this time he became a professor at ETH (Pais, 1982, p. 523) where he started his collaboration with Grossmann, culminating in their joint Entwurf theory, a precursor of Einstein’s general theory of relativity (Sauer, 2015b; Einstein & Grossmann, 1913). At the end of 1913, Einstein accepted a position in Berlin and moved there in 1914 (Pais, 1982, p. 523), where he accomplished his greatest achievement by completing the general theory of relativity in November 1915 (Einstein, 1915a).34 As professor in Prague, Einstein visited Berlin in April 1912 meeting the astronomer Erwin Freundlich (Renn et al., 1997). Probably on this occasion, they

32 See

the introduction of Sect. 1.1. fact, as we will see below, he spent most of the time during 1909 and 1915 in Zurich, and however, the notion of Zurich Notebook is already assigned as explained in the introduction of Sect. 1.1. The notebook here is frequently simply called Scratch Notebook, see Janssen (1999), Klein et al. (1993, Appendix A), Renn et al. (1997), or Sauer and Schütz (2021). 34 We will look at this time period in more detail in Sect. 1.3.4. 33 In

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1 Introduction

discussed the possibility of gravitational lensing (Renn & Sauer, 2003a). Notes regarding this were found in his Prague notebook on pages 43 to 48 (Renn et al., 1997). Due to an investigation of an unknown letter at the time, Sauer argued that additional notes of gravitational lensing on pages 51 and 52 of his notebook were written in early October 1915, thus shortly before his completion of general relativity (Sauer, 2008, p. 6). Einstein published his results not until 1936 (Einstein, 1936) and then only by instigation of the amateur scientist Rudi W. Mandl.35 From this time period in 1936, we recently reconstructed four pages of gravitational lensing which are part of the Princeton manuscripts (Schütz, 2017; Sauer & Schütz, 2019). It is not known whether Einstein recalled his early calculations (Renn et al., 1997). As explained above, pages 43 to 48 and 51 to 52 are about gravitational lensing. The two pages 49 and 50 between them do not deal with gravitational lensing. As we will see in Sect. 2.3, they deal with projective geometry. In addition, the sketches are very similar to sketches that we found on manuscript pages which we date to summer 1938. Thus, equally to the context of gravitational lensing, we have a second example in which Einstein considered a subject in his Prague notebook that appears more than 20 years later in his manuscript pages. The notebook also contains further interesting pages as, for instance, pages 41, 42, and even the top of page 43, directly preceding the pages of gravitational lensing. These pages are about mathematical recreations where, for example, Einstein showed that all triangles are isosceles or a proof that .64 = 65.36 Rowe (Rowe, 2011) argued that Einstein probably read about these recreations in Schubert’s “Mathematische Mußestunden” from 1898 (Schubert, 1900).37 As shown in Kormos Buchwald et al. (2013), it also contains entries on chemical subjects and a short note on his daughter Lieserl. As already mentioned above, it also contains considerations on Mercury’s perihelion advance (Janssen, 1999).

1.2 Overview of Research Notes Table 1.1 gives an overview of the most important sheets containing research notes that will be discussed in the following chapters. The table also lists correspondence between Einstein and his assistants P. Bergmann and V. Bargmann since we will discuss them comprehensively. The first entries of the table consist of several pages which might have different archival numbers, while the rest of the documents usually consists of only one sheet of paper. The second column classifies the documents roughly into the categories projective geometry, generalization of Kaluza’s theory

35 For further literature about the history of gravitational lensing and the calculations in the Prague notebook, see also Sauer (2015c, 2016) in addition to the already mentioned literature. 36 See also Sauer and Schütz (2022). 37 Rowe cited Schubert’s edition (Schubert, 1898) from 1898, see Rowe (2011, p. 57).

1.2 Overview of Research Notes

9

Table 1.1 Documents that will be discussed in the book Document/AEA Prague Notebookb Washington Msc Corr. AE PBd Corr. AE VBe AEA 1-133 AEA 1-136 AEA 2-119 AEA 5-242.1 AEA 5-258f AEA 6-250g

Categorization Projective geometry Washington Ms Further development Further development delta function Generalization Generalization Washington Ms Further development Projective geometry Projective geometry further development

AEA 6-259 AEA 36-778 AEA 52-574 AEA 62-065 AEA 62-223 AEA 62-373r AEA 62-783 AEA 62-785 AEA 62-785r AEA 62-786 AEA 62-786r AEA 62-787 AEA 62-787r AEA 62-788 AEA 62-789

Further development Further development Further development Further development Further development Generalization Delta function Generalization Projective geometry further development Further development Further development Further development Projective geometry further development Further development Projective geometry Generalization

AEA 62-789r AEA 62-790 AEA 62-791 AEA 62-792 AEA 62-792r AEA 62-794 AEA 62-795 AEA 62-795r AEA 62-796 AEA 62-797 AEA 62-798

Projective geometry further development Further development Further notes Further development Further development Generalization Delta function Delta function Further development Further development Generalization further development

Main sectionsa Sections 2.1.1 and 2.5 Section 3.4 Section 4.1 Sections 4.2 and 5.2.2 Section 3.4.5 Section 4.2.4 Section 3.4.2 Chapter 4 Section 2.1.2 Sections 2.3.6, 4.3.1.3, and Chap. 4 Section 4.2.2 Chapter 4 Section 5.2.7 Section 4.3.8 Section 5.2.7 Section 4.3.8 Section 5.2.1 Section 3.6.4 Sections 2.3.1 and 4.3.1.1 Chapter 4 Chapter 4 Chapter 4 Sections 2.3.2 and 4.3.1.2 Chapter 4 Sections 2.3.3, 3.6.1, and 4.3.1.3 Sections 2.3.4 and 4.3.1.3 Chapter 4 Section 4.3.7 Chapter 4 Chapter 4 Section 3.6.3 Section 5.2.4 Section 5.2.5 Chapter 4 Chapter 4 Section 3.6.2 and Chap. 4 (continued)

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1 Introduction

Table 1.1 (continued) Document/AEA AEA 62-799 AEA 62-800 AEA 62-802 AEA 62-803r AEA 62-805 AEA 62-807 AEA 63-026, 63-026r AEA 63-325 AEA 124-446 AEA 22-290.1, 28-434, 28-435, 28-512, 62-495r, 63-030, 63-030r AEA 26-417, 28-458, 28-459, 31-219, 36-765, 62-300, 62-317 to 62-322r AEA 31-224, 80-914 AEA 62-373, 63-042, 63-042r, 63-050, 63-058r a b c d

e

f g

Categorization Further development Generalization Generalization Further development Further development Generalization Generalization Generalization Projective geometry Further notes

Main sectionsa Chapter 4 Section 4.3.8 Section 3.6.3 Chapter 4 Chapter 4 Section 4.3.8 Section 4.3.8 Section 4.3.8 Section 2.3.5 Section 4.3.8

Further notes

Section 5.2.7

Further notes Further notes

Section 2.3.6 Section 5.2.8

Many documents will be discussed in several different sections. We give here only a selection. For a full list, see the index that is included in the appendix We will mainly look at pages 49 and 50, see Klein et al. (1993, Appendix A). See also AEA 3-013 The Washington manuscript has several archival numbers as AEA 2-121, 5-008, and 97-487 The correspondence between Einstein (AE) and P. Bergmann (PB) consists of the documents with archival numbers AEA 6-240, 6-262, 6-245, 6-246, 6-247, 6-249, 6-250, 6-251, 6-252, 6253, 6-254, 6-256, 6-242, 6-271, 6-258, 6-263, 6-264, 6-248, 6-265, 6-266, 6-261, 6-267, 6-268, 6-269, 6-270, 6-260, and 6-272. We will date them in Sect. 4.1, which also explains the preceding sequence The correspondence between Einstein (AE) and V. Bargmann (VB) consists of the documents with archival numbers AEA 6-273, 6-206, 6-274, 6-275, and 6-208. Some of them were addressed or forwarded to P. Bergmann See also the documents AEA 29-136 to 29-141 This document is a letter and belongs to the correspondence between Einstein and Bergmann. We will also discuss the back of the letter AEA 6-250.1

of electricity, further development of Einstein and Bergmann’s publication, delta functions, and further notes. The third column refers to the sections that mainly discuss the respective document. Some of them will be briefly discussed or even only mentioned in the book without a comprehensive analysis only to point out similarities to other research notes.38 We find these documents in the category “further notes.” In order to detect connections of a specific manuscript page or letter more easily to other documents, an index of all research notes, letters, and further documents that are mentioned or discussed in the book is included in the appendix.

38 For

instance, these sheets contain expressions, equations, functions, or sketches that remind us of considerations from other manuscript pages that will be discussed more comprehensively. See Sects. 2.3.6, 4.3.8, and 5.2.7.

1.3 Einstein’s Education and His Way to General Relativity with a Special. . .

11

A comprehensive and detailed analysis of most of the documents listed in Table 1.1 can be found in Schütz (2021).

1.3 Einstein’s Education and His Way to General Relativity with a Special Focus on Mathematics In order to assess Einstein’s heuristics, creativity, and way of theorizing as a theoretical physicist, we need to take into account Einstein’s education. With respect to our discussion of Einstein’s mathematical skills throughout the following chapters and especially in Chap. 2, we will here focus on the mathematical part and in particular on his education in geometry. We will also discuss Einstein’s road to the theory of general relativity and look at how Einstein handled mathematics in his publications and collaborations. Our aim is not to claim exclusivity or to give a comprehensive biography of Einstein’s education. Instead, we refer to Pais (1982), Isaacson (2007), Fölsing (1994), Frank (1947), Seelig (1954), and Neffe (2005).

1.3.1 Primary and Secondary School in Munich, Milano, and Aarau The first volume of the Collected Papers of Albert Einstein provides the most important primary documents about Einstein’s education in secondary school and university (Stachel, 1987a, appendixes A to E). A first detailed analysis and commentary of Einstein’s mathematical education was made by Pyenson, who concluded that “Einstein received excellent preparation for his future career” (Pyenson, 1980, p. 399) as he “was taught a great deal of mathematics [. . . ] at the hands of sympathetic teachers” (Pyenson, 1980, p. 425). Einstein’s education began with private instructions at home in 1884 (Stachel, 1987a, p. 370) and the Catholic primary school Peterschule in Munich which he entered in 1885 (Isaacson, 2007, p. 15).39 In 1888, he then entered secondary school at Luitpold-Gymnasium (Pais, 1982, p. 520).40 An important role in Einstein’s education also played Max Talmey41 who met Einstein in 1889, also played an important role in Einstein’s life. He was Einstein’s tutor, they became friends, and Talmey exposed him to a variety of advanced texts in mathematics, science, and philosophy (Stachel, 1987a, p. 371). On Talmey’s advice, Einstein studied one of the many editions of Theodor Spieker’s geometry books “Lehrbuch der 39 For

the curriculum, see Stachel (1987a, Appendix A). the curriculum and Einstein’s courses, see Stachel (1987a, Appendix B). 41 Talmey changed his name from Talmud to Talmey when he immigrated to the United States (Isaacson, 2007, p. 18). 40 For

12

1 Introduction

ebenen Geometrie” (Spieker, 1980), see Talmey (1932, p. 163). Among the physical textbooks that Talmey gave to Einstein belong an edition of Bernstein (1853–1857) and Büchner (1855), see Talmey (1932, p. 162) and Stachel (1987a, p. lxii). According to Einstein’s sister Maja Winteler-Einstein, he also worked through Violle (1892), Violle (1893a), and Violle (1893b), see Winteler-Einstein (1987, p. lxiv) and Stachel (1987a, p. lxiv).42 According to Talmey (Talmey, 1932, pp. 162– 163) and Einstein’s own recollections (Einstein, 1949b, p. 14), some of these books influenced Einstein for his entire life. Einstein recalled in his autobiographical notes that one particular book on geometry played an important role in his youth by emphasizing the lucidity and certainty of mathematical proofs:43 Im Alter von 12 Jahren erlebte ich ein zweites Wunder ganz verschiedener Art: An einem Büchlein über Euklidische Geometrie der Ebene [. . . ]. Da waren Aussagen wie z.B. das Sich-Schneiden der drei Höhen eines Dreiecks in einem Punkt, die—obwohl an sich keineswegs evident—doch mit solcher Sicherheit bewiesen werden konnten, dass ein Zweifel ausgeschlossen zu sein schien. Diese Klarheit und Sicherheit machte einen unbeschreiblichen Eindruck auf mich. (Einstein, 1949b, p. 8)

This book that Einstein called “heilige[s] Geometrie-Büchlein”44 (Einstein, 1949b, p. 10) probably was an edition of Sickenberger (1888) as suggested by Pyenson (1980, p. 405).45 As pointed out in Stachel (1987a, p. lxi), the second part of Sickenberger’s Leitfaden der elementaren Mathematik entitled Planimetrie fits best to Einstein’s description in his autobiographical notes as well as to Kayser (1930, pp. 35–37).46 The latter is a biography written by Einstein’s step son Rudolf Kayser who stated that Einstein’s “preoccupation with mathematics became the most beautiful adventure of his youth” (Kayser, 1930, p. 36).47

42 We will not discuss Einstein’s physical education here in more detail. For that purpose, we refer to Einstein’s biographies as well as to Cahan (2000). 43 “At the age of 12, I experienced a second wonder of a totally different nature: in a little book dealing with Euclidean plane geometry, which came into my hands at the beginning of a schoolyear. Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which—though by no means evident—could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty made an indescribable impression upon me.” (Einstein, 1949a, p. 10). 44 “Holy geometry booklet” (Einstein, 1949a, p. 11). 45 Without giving further reasons, Banesh Hoffmann and Helen Dukas who were Einstein’s assistants and secretary suggested that Heis and Eschweiler (1881) was this book Hoffmann and Dukas (1972, pp. 22–23). 46 In fact, Adolf Sickenberger was a mathematics teacher at Luitpold-Gymnasium at that time (Pyenson, 1980, p. 403) and his several book editions were used in Einstein’s class, see Stachel (1987a, Appendic B). The subject Planimetrie was taught during the fifth to seventh class from 1892 to 1894 (Stachel, 1987a, pp. 350–352). 47 Kayser wrote this biography under the pseudonym Anton Reiser. Einstein very much liked this biography as he expressed in a letter to E. F. Magnin on February 25, 1931: “Das Reiser’sche Buch ist nach meiner Ansicht das beste biographische Buch, das über mich geschrieben worden ist. Es stammt aus der Feder eines Mannes, der mich persönlich gut kennt [. . . ]” (AEA 47-569). “The

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13

By the fact that Einstein studied several geometry books, it comes about that one of Einstein’s earliest known writings can be found in such a book next to a proof showing that a cylinder can be developed onto a plane. Einstein criticized this proof in Heis and Eschweiler (1881, p. 76) by commenting:48 Der Beweis hat keinen Sinn, denn so gut wir annehmen können, daß sich diese prismatischen Räume entrollen lassen dürfen wir dies vom Cylinder sagen! (Stachel, 1987a, p. 3)

Of course, the young Einstein did not only study geometry. He recalled studying calculus eagerly as well. However, by describing this in his recollections, he immediately connected the respective subjects with geometry:49 Im Alter von 12-16 machte ich mich mit den Elementen der Mathematik vertraut inklusive der Prinzipien der Differential- und Integral-Rechnung. [. . . ] Diese Beschäftigung war im Ganzen wahrhaft fascinierend; es gab darin Höhepunkte, deren Eindruck sich mit dem der elementaren Geometrie sehr wohl messen konnte—der Grundgedanke der analytischen Geometrie, die unendlichen Reihen, der Differential- und Integral-Begriff. (Einstein, 1949b, p. 14)

This was the time when Einstein visited Luitpold-Gymnasium, and indeed, he apparently did very well in mathematics as it was confirmed by a mathematics teacher of Einstein who “so highly rated the boy’s mathematical knowledge and abilities that he recommended him for matriculation at the university” (Kayser, 1930, pp. 42–43). Philipp Frank who knew Einstein personally and who was his successor50 as a professor of theoretical physics in Prague from 1912 (Holton et al., 1968, p. 2) concluded accordingly:51 Einstein war seinen Mitschülern auf dem Gebiet der Mathematik sehr weit voraus [. . . ]. (Frank, 1949, p. 30)

At the end of 1894, Einstein withdrew from Luitpolt-Gymnasium without completing his schooling52 and spent most of 1895 in Milano with his family in

book by Reiser is, in my opinion, the best biography which has been written about me. It comes from the pen of a man who knows me well personally” (Pais, 1982, p. 48). 48 This is the book Hoffmann and Dukas identified with the holy geometry booklet, see footnote 45. In Hoffmann and Dukas (1972, p. 23), we find an image of Einstein’s comment in the margin of Heis and Eschweiler (1881, p. 76). In English, it says “The proof makes no sense, because if we can assume that these prismatic spaces are capable of being flattened out, we could just as well say it of the cylinder” (Hoffmann & Dukas, 1972, p. 23). 49 “At the age of 12–16, I familiarized myself with the elements of mathematics together with the principles of differential and integral calculus. [. . . ] This occupation was, on the whole, truly fascinating; climaxes were reached whose impression could easily compete with that of elementary geometry—the basic idea of analytic geometry, the infinite series, the concepts of differential and integral.” (Einstein, 1949a, p. 15). 50 Among Frank’s Ph.D. students was Peter Bergmann who became Einstein’s assistant upon Frank’s recommendation in Princeton in 1936 (Schmutzer, 2003, p. 411). Bergmann will play an important role in the following chapters, especially his joint publication in 1938 and related correspondence. 51 “In the field of mathematics Einstein was far ahead of his fellow students [. . . ]” (Frank, 1947, p. 16).

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1 Introduction

order to prepare for the entrance examinations at Eidgenössische Polytechnische Schule53 (now ETH) in self-study (Isaacson, 2007, pp. 23–25). In the same year, Einstein was allowed to take the entrance examinations at ETH although he was 2 years younger than the regular age of admission (Isaacson, 2007, p. 25). He passed the tests in mathematics and physics and was far ahead of most of the other candidates in mathematics; however, he failed the test in modern languages, zoology, and botany (Frank, 1947, p. 18). Among the required topics of the entrance examinations were Arithmetik und Algebra, Geometrie,54 and Darstellende Geometrie including projections, see Stachel (1987a, Appendix C).55 Due to his good mathematical performance, the director of ETH, Albin Herzog, advised him to earn a diploma at the cantonal school in Aarau and to come back thereafter (Frank (1947, p. 18) & Einstein (1955, pp.145–146)). Einstein followed this recommendation in October 1895 and enrolled in the Gewerbeabteilung56 of the Aargau Kontonsschule57 in Aarau58 (Pais, 1982, p. 521), which was “one of Switzerland’s finest secondary schools” (Pyenson, 1980, p. 418).59 His entrance report attests Einstein good grades in mathematics (2) and average grades in descriptive geometry (3) (Stachel, 1987a, Doc. 8).60 In his first year 1895/1896, Einstein attended the third class and learned analytic and descriptive geometry (Stachel, 1987a, pp. 359–361). The former was taught by Heinrich Ganter following his own textbook (Ganter & Rudio, 1894) that was co-authored with Ferdinand Rudio who was a professor at ETH (Pyenson, 1980, p. 414). This book had a special focus on conic sections.61 At the end of the school year, Einstein received the best possible grades (1) in arithmetic, algebra, and geometry, while he received grade 2 in descriptive geometry.62 Apparently, Einstein was not eager to learn descriptive geometry as his teacher gave him only grade 3–4 in diligence. In his final year, the grade system changed such that 6 became

52 Einstein

neither liked the atmosphere at Luitpold-Gymnasium nor at primary school, see Moszkowski (1921, p. 221). 53 “Federal Polytechnic School,” see Sect. 1.3.2. 54 Geometry was divided into Planimetrie, Stereometrie, Trigonometrie, and Analytische Geometrie. 55 In Stachel (1987a, Appendix C), we find a detailed list of topics. 56 Technical Commercial School (Stachel, 1987a, p. 11). 57 Aargau cantonal school. 58 Aarau is the principal town in the canton of Aargau, 50 kilometers away from Zurich (Stachel, 1987a, p. 11). 59 As Einstein described in Einstein (1956, pp. 9–10), he liked this school and its “liberal spirit” in contrast to the Luitpold-Gymnasium. For the curriculum at the cantonal school, see Stachel (1987a, Appendix D). 60 In the school year 1895/1896, 1 was the highest and 6 was the lowest grade (Stachel, 1987a, p. 14). 61 This becomes clear alone by the titles of the chapters as they are named after the conic sections: Der Punkt, Die gerade Linie, Der Kreis, Die Ellipse, Die Hyperbel, and Die Parabel (Ganter & Rudio, 1894, pp. iv–vi). 62 In the third quarter of the school year, he received grade 3, in the fourth quarter grade 2.

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the best and 1 the lowest grade.63 Again, he received the best possible grades (6) in arithmetic, algebra, and geometry. He also improved his descriptive geometry skills and received grade 4–5, while he apparently had not changed his eagerness in learning descriptive geometry as he received grade 4 in diligence (Stachel, 1987a, Doc. 10). Finally, in September 1896, Einstein took the secondary school leaving examination called Maturitätsprüfung. The examination questions have been published in Stachel (1987a, Doc. 21–Doc. 27).64 For instance, in geometry where Einstein received the best grade 6 (Stachel, 1987a, p. 29), a sequence of circles was given that touch an enveloped ellipse at a maximum that needed to be determined by Einstein.65 In algebra, Einstein also received grade 6 (Stachel, 1987a, p. 39)66 as it was also the case in descriptive geometry which was held as an oral examination (Stachel, 1987a, p. 25). Thus, he improved his skills in descriptive geometry over the year at Aarau from average to best grades. In total, Einstein left the school with excellent grades in algebra (6), geometry (6), and descriptive geometry (5). In physics, for comparison, Einstein received the finale grade 5–6 (Stachel, 1987a, Doc. 19). Even though we are not interested in Einstein’s French skills here, his written examination in this subject was on his “future plans” (Stachel, 1987a, Doc. 22) and reveals some enlightening insights of how he assessed his mathematical skills at that time. There, he stressed his mathematical abilities and even made them his motivation for enrolling at ETH:67 If I am lucky and pass my exams, I will enroll in the Zurich Polytechnic. I will stay there four years to study mathematics and physics. [. . . ] Here are the reasons that have led me to this plan. They are, most of all, my personal talent for abstract and mathematical thinking [. . . ]. As we will see in Sect. 1.3.2, his wish became true. (Isaacson, 2007, p. 31)

1.3.2 Einstein’s Studies at ETH After graduating from Aargau cantonal school, Einstein started his undergraduate studies in Switzerland at Zurich’s Eidgenössische Polytechnische Schule68 in

63 This is a fact that frequently causes confusion about Einstein’s mathematical skills as a student. For instance, see Pyenson (1980, p. 410). 64 For a more detailed discussion of Einstein’s mathematics and physics Matura examinations, see Hunziker (2005) and Pfeifer (2005). 65 For a short discussion and illustration of this question, see Sauer and Schütz (2021). 66 Einstein’s grade average for the written examination was the best in his class, namely 5 .1/3 (Stachel, 1987a, p. 25). 67 This quote was originally written in French by Einstein for his French examinations, where he also made several mistakes (Stachel, 1987a, Doc. 22). We used here Isaacson’s translation (Isaacson, 2007, p. 31). 68 Federal Polytechnic School.

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1 Introduction

October 1896, which has been named Eidgenössische Technische Hochschule69 (ETH) since 1911 (Isaacson, 2007, p. 32). He was one of the eleven students initially enrolled in department VI A: “Schule für Fachlehrer in mathematischer und naturwissenschaftlicher Richtung. Mathematische Sektion.”70 In 1899, this section was renamed in “Mathematisch-physikalische Sektion”71 (Stachel, 1987a, p. 43). From Einstein’s time at ETH, many documents are preserved such as Einstein’s Matrikel, Diploma, or ETH Programme and Reglemente.72 With the use of these and additional documents as students’ and professors’ notes, the Collected Papers of Albert Einstein published a comprehensive overview of the curriculum of courses which Einstein attended between winter term 1896/97 and summer term 1900 (Stachel, 1987a, Appendix E) as well as his ETH record and grade transcript. As stated in Muheim (1975), during that time, ETH became famous for mathematics mostly because of its teachers.73 However, Minkowski apparently was not satisfied with the interest of students in mathematics as he expressed in a letter to Hilbert in 1897: “Es giebt hier nur einen mathematischen Studenten, der mehr als 3 Semester hat. Das Colloquium wird hauptsächlich durch die Assistenten gestützt”74 (Rüdenberg & Zassenhaus, 1973, p. 105). Einstein as a student apparently was well aware of the good conditions at ETH, but in analogy to Minkowski’s quote, he missed the opportunity to gain a deep understanding in mathematics as he recalled in 1949:75 Dort hatte ich vortreffliche Lehrer (z.B. Hurwitz, Minkowski), so dass ich eigentlich eine tiefe mathematische Ausbildung hätte erlangen können. Ich aber arbeitete die meiste Zeit im physikalischen Laboratorium [. . . ]. (Einstein, 1949b, p. 14)

69 Swiss

Federal Institute of Technology. for Mathematics and Science Teachers. Mathematical Section.” Translation inspired by Stachel (1987a, p. 43). 71 “Mathematical-physical section.” Translation by the author. See also page 2 of ETH Reglement (1899), AEA 74-593. 72 The documents are accessible online on the ETH library platform or in the Albert Einstein Archives at The Hebrew University of Jerusalem. 73 Among the mathematicians at ETH at the end of nineteenth century were Carl Friedrich Geiser (1863–1919), Otto Wilhelm Fiedler (1867–1907), Johannes Rebstein (1873–1907), Albin Herzog (1875–1899), Ferdinand Rudio (1881–1920), Adolf Hurwitz (1892–1919), Arthur Hirsch (1893– 1936), Hermann Minkowski (1896–1902), and Alexander Beck (1899–1901), see Muheim (1975), Kirschmer (1961), and Frei and Stammbach (1994). Among Einstein’s mathematics teacher were Fiedler, Geiser, Hirsch, Hurwitz, Minkowski, Rebstein, and Rudio, see Einstein’s record and grade transcript (Stachel, 1987a, Doc. 28). For a general discussion of mathematics at ETH during Einstein’s time, see McCormmach (1976, pp. xi–xxxv). For general history of ETH from 1855 to 1905, see also Oechsli (1905). 74 “There is only one mathematics student who has more than 3 semesters. The colloquium is mainly supported by the assistants.” Translation by the author. 75 “There I had excellent teachers (for example, Hurwitz, Minkowski), so that I really could have gotten a sound mathematical education. However, I worked most of the time in the physical laboratory [. . . ]” (Einstein, 1949a, p. 15). 70 “School

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In fact, Einstein took several courses in mathematics, in particular courses held by Minkowski where he registered for in total nine courses, see also Pyenson (1980, pp. 418–419).76 In his later years, Einstein recalled that he did not attend his courses frequently:77 Einigen Vorlesungen folgte ich mit gespanntem Interesse. Sonst aber «schwänzte» ich viel und studierte zu Hause die Meister der theoretischen Physik mit heiligem Eifer. [. . . ] Sonst aber interessierte mich in den Studienjahren die höhere Mathematik wenig. (Einstein, 1955, p. 146)

This is also supported by Kayser: In contrast to the enthusiasm for mathematics of Einstein’s school years, his interest in this subject slackened considerably. [. . . ] No one could stir him to visit the mathematical seminars. His entire concern was physics. He did not yet see the possibility of seizing that formative power resident in mathematics, which later became the guide of his work. (Kayser, 1930, p. 51)

Minkowski’s courses might have been among the courses that Einstein did not attend. According to anecdotes, Minkowski expressed his astonishment when Einstein later established the special theory of relativity and recalled that Einstein was not eager in learning mathematics: “[. . . ] [F]rüher war Einstein ein richtiger Faulpelz. Um die Mathematik hat er sich überhaupt nicht gekümmert”78 (Seelig, 1954, p. 33). However, when Einstein recalled that he enjoyed working in Weber’s Laboratorium,79 he also mentioned one exception concerning his interest in mathematics:80 Auch faszinierten mich Professor Geisers Vorlesungen über Infinitesimalgeometrie, die wahre Meisterstücke pädagogischer Kunst waren und mir später beim Ringen um die allgemeine Relativitätstheorie sehr halfen. (Einstein, 1955, p. 146)

76 He registered for Geometrie der Zahlen, Funktionentheorie, Potentialtheorie, Elliptische Funktionen, Analytische Mechanik, Variationsrechnung, Algebra, Partielle Differentialgleichungen, and Anwendungen der analytischen Mechanik, see Stachel (1987a, Doc. 28) or Einstein’s Matrikel AEA 71-539. 77 “I followed some lectures with eager interest. Otherwise, however, I «cut» a lot and studied at home the masters of theoretical physics with holy fervor. [. . . ] Otherwise, however, I was little interested in mathematics during student years.” Translation by the author. 78 “[. . . ] Einstein used to be a real slacker. He did not care about mathematics at all.” Translation by the author. 79 As we learn from Einstein’s Matrikel, this does not mean that Einstein participated eagerly in every physical laboratory. In 1899, Einstein got a “director’s reprimand for nondiligence in physics practicum” (Stachel, 1987b, p. 27). 80 “I was also fascinated by professor Geiser’s lectures on infinitesimal geometry, which were true masterpieces of pedagogical art and which later helped me a lot when struggling with the theory of general relativity.” Translation by the author. We note that infinitesimal geometry is another name for differential geometry (Stachel, 1979, p. 435).

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1 Introduction

The fact that Einstein mentioned Geiser’s lectures and particularly his didactic abilities is intriguing as Geiser also taught projective geometry. We will come back to this in Sects. 1.3.4 and 2.3.2.81 In his recollections, Einstein also qualified his statements on his interest in mathematics when he made lack of time the reason for his neglect of mathematics. He used the metaphor of finding himself in the situation of Buridans Esel:82 Dass ich die Mathematik bis zu einem gewissen Grade vernachlässigte, hatte nicht nur den Grund, dass das naturwissenschaftliche Interesse stärker war als das mathematische, sondern das folgende eigentümliche Erlebnis. Ich sah, dass die Mathematik in viele Spezialgebiete gespalten war, deren jedes diese kurze uns vergönnte Lebenszeit wegnehmen konnte. So sah ich mich in der Lage von Buridans Esel, der sich nicht für ein besonderes Bündel Heu entschliessen konnte. Dies lag offenbar daran, dass meine Intuition auf mathematischem Gebiete nicht stark genug war, um das Fundamental-Wichtige, Grundlegende sicher von dem Rest der mehr oder weniger entbehrlichen Gelehrsamkeit zu unterscheiden. (Einstein, 1949b, p. 14)

In the following, we will concentrate on Einstein’s lectures on projective geometry as they will play an important role in Chap. 2. In his first year 1896– 1897 at ETH, Einstein took courses by Otto Wilhelm Fiedler on Darstellende Geometrie83 as well as on Projektivische Geometrie I 84 and received grades .4.5 and 4 in descriptive geometry85 and .4.5 in projective geometry86 (Stachel, 1987a, Doc. 28 and Appendix E).87 These were his worst grades in the first year (Stachel, 1987a, p. 46).88 For comparison, his good friend Grossmann received grades .5.5, .5.5, and 5, respectively, see AEA 70-755. In his second year 1897–1898, Einstein took the second part Projektivische Geometrie II held by Fiedler and received grade 4 (Stachel, 1987a, Doc. 28 and Appendix E). Again, this was his worst grade and Grossmann received grade .5.5, see AEA 70-755.89 This was also the year when

81 For

instance, see his lecture on Geometrische Theorie der Invarianten I in Grossmann (1898a, p. 49) attended by Einstein in 1898 (Stachel, 1987a, p. 366). 82 “The fact that I neglected mathematics to a certain extent had its cause not merely in my stronger interest in the natural sciences than in mathematics but also in the following strange experience. I saw that mathematics was split up into numerous specialities, each of which could easily absorb the short lifetime granted to us. Consequently I saw myself in the position of Buridan’s ass which was unable to decide upon any specific bundle of hay. This was obviously due to the fact that my intuition was not strong enough in the field of mathematics in order to differentiate clearly the fundamentally important, that which is really basic, from the rest of the more or less dispensable erudition” (Einstein, 1949a, p. 15). 83 Descriptive geometry. 84 Projective Geometry I. 85 In his first and second semesters. 86 In his second semester. 87 The best grade is 6, and the worst grade is 1. 88 He received grades .4.5 and 5 in differential and integral calculus, 5 in analytic geometry, and 5 in mechanics, see Stachel (1987a, p. 46) or AEA 71-539. 89 Einstein’s further grades were 5 in differential equations, .5.5 and 5 in physics, and .5.5 in mechanics (Stachel, 1987a, Doc. 28).

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Einstein attended Geiser’s lecture on Geometrische Theorie der Invariante without grades (Stachel, 1987a, Doc. 28 and Appendix E). In his leaving certificate from August 1900, he got grade .4 1/4 in both descriptive and projective geometry, see Stachel (1987a, Doc. 28) and AEA 29-234. About Fiedler’s lectures, Einstein once wrote to Mileva Mari´c90 in 1898: “Fiedler liest projektivische Geometrie, derselbe undelikate, rohe Mensch wie früher & dabei manchmal undurchsichtig, doch immer geistvoll & tief – kurz ein Meister aber leider auch ein arger Schulmeister”91 (Stachel, 1987a, Doc. 39). From the first lecture on projective geometry in summer 1897, a transcript is extant, written by Marcel Grossmann (Grossmann, 1897). Grossmann was a good friend and classmate of Einstein and earned his diploma together with Einstein in 1900. After this, he became an assistant to Fiedler, obtained his Ph.D. in 1902 under Fiedler’s supervision, and was from 1907 on his successor for “Darstellende Geometrie” and “Geometrie der Lage” (Graf-Grossmann, 2015, pp. 100–119).92 We know from Einstein’s recollections that Einstein frequently borrowed Grossmann’s notes for test preparation:93 Zur Vorbereitung für die Examina lieh er mir diese Hefte, die für mich einen Rettungsanker bedeuteten; wie es mir ohne sie ergangen wäre, darüber will ich lieber nicht spekulieren. (Einstein, 1955, p. 147)

In a letter to Grossmann from 1924, he called himself “ein Schlamper [. . . ], der ohne Hilfe von Grossmanns Heften nicht einmal seine Examina hätte machen können”94 (Kormos Buchwald et al., 2015b, Doc. 226). In fact, in one of Grossmann’s lecture transcripts of infinitesimal geometry (Grossmann, 1898b, p. 105), there is a note not made by Grossmann that can most probably be traced back to Einstein (Sauer, 2015b). As we already mentioned above, in Grossmann’s lecture notes on Geiser’s lecture Geometrische Theorie der Invarianten I that was attended by Einstein and Grossmann in 1898 (Stachel, 1987a, p. 366), we find considerations on projective geometry as well, see Grossmann’s transcription in Grossmann (1898a, p. 49).95 It is safe to assume that Einstein carefully studied Grossmann’s lecture notes on Fiedler’s descriptive and projective geometry and on Geiser’s geometric theory

90 She

was a fellow student who became his wife in 1903 and his ex-wife in 1919 (Pais, 1982, p. 47,300). 91 “Fiedler lectures on projective geometry, he is the same indelicate, rude man as before & in addition sometimes opaque, but always witty & profound—in brief, a master but, unfortunately, also a terrible schoolmaster” (Stachel, 1987b, Doc. 39). 92 Grossmann is well known among historians because of his mostly mathematical contributions to the general theory of relativity. For more details about his contributions and collaboration with Einstein, see Sauer (2015b) and Sect. 1.3.4.5. 93 “For test preparation, he lent me these notebooks, which were a lifeline for me; I would rather not speculate on how I would have been without them.” Translation by the author. 94 “[A] slovenly guy who wouldn’t even have been able to pass his examinations without the help of Grossmann’s notebooks” (Kormos Buchwald et al., 2015a, p. 226). 95 We mentioned this lecture above when quoting Einstein on his comments about Geiser’s lectures. We will come back to it in Sect. 2.3.2.

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1 Introduction

of invariants, as well. Grossmann not only composed notes in his student years but also when he himself was a teacher at ETH as Fiedler’s successor, see Grossmann (1907). Although we have no evidence for this, it is possible that Einstein read these notes as well. We will come back to this in Sect. 2.3 when discussing Einstein’s research notes. By Grossmann’s transcript (Grossmann, 1897), we know that topics of the first lecture course in projective geometry included, among others, the cross ratio of four distinct points, projective coordinates, duality, conics, collineations, harmonic relations, and imaginary elements. In addition, they discussed involutions and Pascal’s theorem. Fiedler’s textbooks on projective and descriptive geometry are also of our interest as he was Einstein’s teacher. His book on descriptive geometry (Fiedler, 1871) already included many elements of what later became projective geometry, which was not a clearly defined field at that time. At the end of nineteenth century, Fiedler provided an “organische Verbindung”96 between descriptive and projective geometry which is expressed in the title of his textbooks (Fiedler, 1883, 1885, 1888), see also Volkert (2019) and Kitz (2015, pp. 26–28). Finally, at the latest by Veblen and Young in 1910, the axiomatic foundation of projective geometry was established (Veblen and Young, 1910, 1918), see also Coolidge (1934, p. 227). Maurice Solovine, a good friend of Einstein who met him after graduating and was one of the members of Akademie Olympia97 (Pais, 1982, pp. 46–47), summarized Einstein’s handling of mathematics and its connection to physics as follows:98 Einstein, der sein mathematisches Rüstzeug mit unvergleichlicher Meisterschaft handhabte, hat sich oft gegen den Mißbrauch der Mathematik auf physikalischem Gebiete ausgesprochen. „Die Physik“, so sagte er, „ist ihrem Wesen nach eine konkrete und anschauliche Wissenschaft. Die Mathematik gibt uns nur die Mittel in die Hand, um die Gesetze auszudrücken, wonach die Erscheinungen sich vollziehen.“ (Solovine, 1960, p. xviii)

While this quote is in accordance with Einstein’s statement about his lack of intuition in mathematics, it does not make a statement of the importance of mathematics in Einstein’s physical ideas. However, Einstein himself recalled:99

96 “Organic

connection.” Translation by the author. Einstein, and their friend Conrad Habicht formed together a discussion group which they called Akademie Olympia in 1902/03, see Stachel (1987a, p. 382), Klein et al. (1993, Doc. 191), and Solovine (1959). Einstein called them “regelmässige philosophische Lese- und Diskussionsabende” (Stachel, 1989b, p. xxiv). For a list of texts discussed and read by the group, see Stachel (1989b, pp. xxiv–xxv). 98 “Einstein, who handled the mathematical instrument with incomparable dexterity, often spoke against the abusive use of mathematics in physics. Physics, he would say, is essentially a concrete and intuitive science. Mathematics serves only as a means of expressing the laws that govern phenomena” (Pyenson, 1980, p. 419). 99 “[I]t was not clear to me as a student that the approach to a more profound knowledge of the basic principles of physics is tied up with the most intricate mathematical methods” (Einstein, 1949a, pp. 15–16). 97 Solovine,

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[. . . ] [E]s wurde mir als Student nicht klar, dass der Zugang zu den tieferen prinzipiellen Erkentnissen in der Physik an die feinsten mathematischen Methoden gebunden war. (Einstein, 1949b, p. 16)

1.3.3 Summary on Einstein’s Education We conclude that Einstein had an education in mathematics and physics of high standards as a student of Aargau cantonal school and ETH. We also saw that he showed interest in specific mathematical subdisciplines. Furthermore, throughout his education, he always received good grades in physics and mathematics including geometry. Even though Einstein apparently was more interested in physics than in descriptive and projective geometry, he had an influential teacher with Fiedler and studied his lectures at least through the carefully worked out lecture notes by Grossmann who was a good friend of Einstein and became himself Fiedler’s successor in 1907. It seems likely that Einstein talked with his friend Grossmann about projective geometry even after his graduation from ETH. We also note that there is a tension between Einstein’s comments from later years and the contemporary sources. The later sources tend to downplay Einstein’s engagement with mathematics, while the contemporary sources show that he was a good and engaged student, who liked mathematics and thoroughly studied it. The fact that Einstein had an excellent mathematical education does not imply that he retained the mathematical tools and concepts he was exposed to during his studies throughout his academic career. In order to assess Einstein’s use of his mathematical knowledge, we will analyze his research notes that, by their very nature, reveal Einstein’s thinking and process of exploration of ideas.

1.3.4 Einstein’s Way to General Relativity The importance of mathematics in Einstein’s theories can be best shown by discussing Einstein’s way to his theory of general relativity. This section gives an overview of Einstein’s pathway toward his theory; however, we will not be able to discuss each step in detail. For this, we will refer to further literature.100 We will rather focus on the steps that are important for the following discussion. In particular, we will concentrate on statements, publications, and actions that enable us to discuss Einstein’s relationship with mathematics. Another focus will be put on preliminary works on the five-dimensional approaches. Finally, we will also

100 We

generally refer to Norton (2020), Renn and Sauer (1999), Stachel (1979), Stachel (1995), Norton (1984), Gutfreund and Renn (2015), Sauer (2015b), Stachel (2007), and Norton (2000). All of them provide a detailed discussion of how Einstein derived the field equations of general relativity, while (Norton, 2000) particularly focuses on Einstein’s mathematical skills.

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1 Introduction

briefly discuss situations where the analysis of Einstein’s research notes contributed to establishing the history of general relativity, see also Sect. 1.1. We will not derive the field equations of the theory of gravitation in this section but only discuss its history with the focus on mathematics. In doing so, we will also use the notation used in the original publications. For a derivation of the field equations with modern notation, see Chap. 3.

1.3.4.1

Annus Mirabilis 1905

We start with Einstein’s annus mirabilis 1905.101 In this year, he worked as a technical expert at a patent office in Bern (Pais, 1982) and published five groundbreaking papers “deren jede [. . . ] nobelpreiswürdig ist”102 (Weizsäcker, 2002, p. 256). He started with “Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt”103 on March 17, where Einstein, based on Planck’s theory of black body radiation, asserted that the energy of a light ray is not distributed continuously but consists of “einer endlichen Zahl von in Raumpunkten lokalisierten Energiequanten”104 (Einstein, 1905d, p. 133).105 For this work, Einstein was awarded the Nobel Prize in 1921 (Nobel Media, 2021a).106 His second paper on April 30 was his Dissertation107 with the title “Eine neue Bestimunng der Moleküldimensionen”108 (Einstein, 1905a) on a new method to determine the size of solute molecules.109 On May 11, Einstein then published “über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in

101 We will here only give a brief overview. Many books have been written on Einstein’s annus mirabilis as, for instance (Rigden, 2005; Stachel, 2001). For a connection between Einstein’s individual papers, see Renn and Hoffmann (2005). 102 “Each of which is worth a Nobel Prize.” Translation by the author. Weizsäcker only wrote about four papers as he put together two papers on special relativity. Concerning Einstein’s annus mirabilis, he also chose the words “1905 eine Explosion von Genie” (Weizsäcker, 2002, p. 256). 103 “On a Heuristic Point of View Concerning the Production and Transformation of Light” (Stachel, 1989c, Doc. 14). 104 “[A] finite number of energy quanta that are localized in points in space” (Stachel, 1989c, Doc. 14). 105 For a brief historical discussion of this work, see Stachel (1989b, pp. 134–148). 106 He received his Nobel Prize in 1922. Einstein was often nominated before. He never received a Nobel Prize for his contributions to the theories of special and general relativity. It is all the more astonishing that he was awarded for his work on light quanta in 1921 as his theory of general relativity was proved correctly in 1919 by the British solar eclipse expeditions, see our discussion below. For the circumstances around Einstein’s Nobel Prize, we here refer to Elzinga (2006), Friedman (2005), Friedman (1981), Pais (1982), and Fölsing (1994). For Einstein’s generally difficult political situation, see Rowe (2007) and Hentschel (1990, pp. 55–195). 107 He dedicated his dissertation to his friend Marcel Grossmann (Einstein, 1905a, Preface). 108 “A new determination of molecular dimensions” (Stachel, 1989c, Doc. 15). A slightly revised version was published in Einstein (1906). 109 For a brief historical discussion of this paper, see Stachel (1989b, pp. 170–182).

1.3 Einstein’s Education and His Way to General Relativity with a Special. . .

23

ruhenden Flüssigkeiten suspendierten Teilchen”110 (Einstein, 1905c) on Brownian motion showing that “in Flüssigkeiten suspendierte Körper von mikroskopisch sichtbarer Größe infolge der Molekularbewegung der Wärme Bewegungen von solcher Größe ausführen müssen, daß diese Bewegungen leicht mit dem Mikroskop nachgewiesen werden können”111 (Einstein, 1905c, p. 549).112 Finally, on June 30, Einstein published his first paper on special relativity113 carrying the title “Zur Elektrodynamik bewegter Körper”114 (Einstein, 1905e). In this paper, Einstein asserted that “für alle Koordinatensysteme, für welche die mechanischen Gleichungen gelten, auch die gleichen elektrodynamischen und optischen Gesetze gelten”115 (Einstein, 1905e, p. 891) and made this his first postulate of his theory calling it Prinzip der Relativität.116 More precisely, this requirement postulates that the laws of physics are the same in every inertial frame of reference.117 His second postulate is that “das Licht im leeren Raume stets mit einer bestimmten, vom Bewegungszustande des emittierenden Körpers unabhängigen Geschwindigkeit V fortpflanze”118 (Einstein, 1905e, p. 892) that we will refer to as the constancy of speed of light in the following. By these two postulates and by defining a new concept of simultaneity of distant events, Einstein then derived the Lorentz transformations between the spatial and time coordinates of two inertial frames of reference (Einstein, 1905e, §3).119 By interpreting these 110 “On the movement of small particles suspended in stationary liquids required by the molecularkinetic theory of heat” (Stachel, 1989c, Doc. 16). 111 “Bodies of microscopically visible size suspended in liquids must, as a result of thermal molecular motions, perform motions of such magnitude that these motions can easily be detected by a microscope.” (Stachel, 1989c, Doc. 16). 112 For a brief historical discussion of this paper, see Stachel (1989b, pp. 206–222). 113 Einstein did not use the term Relativitätstheorie in his paper from 1905. The first time Einstein used this term in a title was Einstein (1911b), see Stachel (1989b, p. 254). For a historical discussion of both papers on special relativity from 1905, see Norton (2014). 114 “On the electrodynamics of moving bodies” (Stachel, 1989c, Doc. 140). 115 “[I]n all coordinate systems in which the mechanical equations are valid, also the same electrodynamic and optical laws are valid” (Stachel, 1989c, p. 140). 116 “The principle of relativity” (Stachel, 1989c, p. 140). 117 An inertial frame of reference moves uniformly and rectilinearly with respect to any other inertial frame of reference. Inertial frames of reference are non-accelerated frames of reference. In his follow-up paper (Einstein, 1905b), Einstein formulated this postulate as follows: “Die Gesetze, nach denen sich die Zustände der physikalischen Systeme ändern, sind unabhängig davon, auf welches von zwei relativ zueinander in gleichförmiger Parallel-Translationsbewegung befindlichen Koordinatensystemen diese Zustandsänderungen bezogen werden” (Einstein, 1905b, p. 639). “The laws governing the changes of state of physical systems do not depend on which one of two coordinate systems moving in uniform parallel translation relative to each other these changes of state are referred to” (Stachel, 1989c, p. 172). 118 “[I]n empty space light is always propagated with a definite velocity V which is independent of the state of motion of the emitting body” (Stachel, 1989c, p. 140). 119 Lorentz derived these transformations in Lorentz (1904). Poincaré called them later in 1905 Lorentz transformations (Poincaré, 1905). The term is also frequently used in Lorentz covariance for gravitation theories that are compatible with special relativity (Norton, 2007a, p. 413). For

24

1 Introduction

transformations, he explained phenomena as length contraction120 or time dilation.121 The second part of his paper is then on electrodynamics where he assigned the transformation laws to Maxwell’s equations (Einstein, 1905e, §6,§9).122 In Einstein’s second paper on special relativity (Einstein, 1905b) on September 27, he derived further consequences: “Gibt ein Körper die Energie L in Form von Strahlung ab, so verkleiner sich sein Masse um .L/V 2 ”123 (Einstein, 1905b, p. 641). Here, V denotes the speed of light such that this is the first appearance of the nowadays famous energy mass equality.124 An analysis of the pre-general relativity papers concerning the mathematical ideas reveals that they have a “strikingly nonmathematical character” and “contain relatively few equations” (Bernstein, 1973, p. 62).125 Overbye supports this with respect to Einstein’s first paper on special relativity stating “Most of his argument is presented verbally; what little math is involved is algebra that any high school student could follow. Indeed, the whole paper is a testament to the power of simple language to convey deep and powerfully disturbing ideas” (Overbye, 2000, p. 135). Nevertheless, the mathematics behind these papers is a much-debated issue with respect to the question about the mathematical influence of Einstein’s wife Mileva Mari´c.126 While some authors state that Mari´c solved Einstein’s mathematical problems for these papers,127 others claim that she was rather the one who had

a historical discussion of how Lorentz derived his transformation laws, see Janssen and Stachel (2004, pp. 22–32). For a discussion on Poincaré’s contribution, see Damour (2017). For general historical overviews, see McCormmach (1970), and Miller (1973). 120 “Ein starrer Körper, welcher in ruhendem Zustand ausgemessen die Gestalt einer Kugel hat, hat also in bewegtem Zustande — vom ruhenden System aus betrachtet — die Gestalt eines Rotationsellipsoides” (Einstein, 1905e, p. 908). “A rigid body that has a spherical shape when measured in the state of rest thus in the state of motion—observed from a system at rest—has the shape of an ellipsoid of revolution” (Stachel, 1989c, p. 152). He then specified that if the body moves along the x-axis, the length of the body along the x-axis appears contracted. 121 “Sind in den Punkten A und B [. . . ] ruhende, im ruhenden System betrachtet, synchron gehende Uhren vorhanden, und bewegt man die Uhr in A mit der Geschwindigkeit v auf der Verbindungslinie nach B, so gehen nach Ankunft dieser Uhr in B die beiden Uhren nicht mehr synchron, sondern die von A nach B bewegte Uhr geht gegenüber der von Anfang an in B befindlichen [. . . ] nach” (Einstein, 1905e, p. 904). “If at the points A and B [. . . ] there are located clocks at rest which, observed in a system at rest, are synchronized, and if the clock in A is transported to B along the connecting line with velocity v, then upon arrival of this clock at B the two clocks will no longer be synchronized; instead, the clock that has been transported from A to B will lag [. . . ] behind” (Stachel, 1989c, p. 153). 122 For a brief historical discussion of Einstein’s paper on special relativity, see Stachel (1989b, pp. 253–274). 123 “If a body releases the energy L in the form of radiation, its mass decreases by .L/V 2 ” (Stachel, 1989c, p. 174). 124 For a historical discussion of the formula .E = mc2 , see Bodanis (2000). 125 Here we used (McCormmach, 1976, pp. xxvi–xxvii) who quoted (Bernstein, 1973, p. 62). 126 Einstein and Mari´ c started their studies at ETH together in 1896, married in 1903, and got divorced in 1919 (Pais, 1982, pp. 521–525). 127 See Trbuhoci´ c-Gjuri´c (1983), Troemel-Ploetz (1990), and Asmodelle (2015).

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the fundamental physical ideas.128 However, as shown by Stachel (1996), Stachel (1989a), and Weinstein (2012), historical sources do not support these claims. One important source in this discussion is the correspondence between Einstein and his wife during the years 1897 and 1903, which was published in Renn and Schulmann (1992). In their introduction, the editors Renn and Schulmann comment the question whether the letters offer new insights on the intellectual influences on Einstein with respect to his development of special relativity as follows: [W]e must conclude that the letters do not enable us to reconstruct in detail the development from some of Einstein’s well-known youthful speculations to his later contributions. But the letters do show how the interest in electrodynamics—Einstein’s avocation since the age of sixteen—has now become more specific, and that it is already focused on the topics that would mark his later fundamental contribution[.] [. . . ] The earliest letters in this volume already show him studying the contemporary literature. [. . . ] For almost 2 years the topic of electrodynamics of moving bodies surfaces only in passing references in Albert’s letter—and does not appear in Mileva’s at all—but his unceasing interest in the subject as well as the fact that he continues to discuss it with Mileva and with his friends Michele Besso and Marcel Grossmann can be taken for granted. In the spring of 1901, he discussed key problems [. . . ] with Besso; a month earlier, he had even written to Mileva about “our work on relative motion[.]” [. . . ] We do not know the content of the planned treatise, just as we do not know Mileva’s possible contribution to it; but whatever it was, all available evidence indicates that the planned paper was a far cry from the relativity paper of 1905. The passing references to the relativity theme do not constitute a Rosetta stone for deciphering its history. (Renn and Schulmann, 1992, pp. xix–xx)

1.3.4.2

The Equivalence Principle and Its Consequences

Let us now come back to Einstein’s road to general relativity. As we saw, the theory of special relativity distinguishes inertial frames of reference. Einstein’s next step was to formulate a theory of gravitation, which had been excluded in his theory of special relativity. Einstein’s “breakthrough came suddenly one day” in 1907 when he “was sitting on a chair in [his] patent office in Bern” (Ono, 1982, p. 47). The crucial idea was the equivalence principle: “If a man falls freely, he would not feel his weight. [. . . ] This simple thought experiment made a deep impression on me. This led me to the theory of gravity” (Ono, 1982, p. 47).129

128 See

Walker (1989, 1991). here cited from a lecture that Einstein gave in Kyoto on December 14, 1922, which was translated by Ono in 1982 (Ono, 1982). The Collected Papers provide an alternative translation of this passage in Kormos Buchwald et al. (2012a, p. 638): “I was sitting in a chair in the Patent Office in Bern when all of a sudden I was struck by a thought: ‘If a person falls freely, he will certainly not feel his own weight.’ [. . . ] This simple thought made a really deep impression on me. My excitement motivated me to develop a new theory of gravitation.” For a discussion on the available translations of this text, see Kormos Buchwald et al. (2012a, pp. 624–628). The original was published in Japanese in Kaizo 5.2 (1923), pp. 2–7, which has been reprinted in Kormos Buchwald et al. (2012a, Doc. 399). 129 We

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1 Introduction

This thought experiment was, as he recalled later in an unpublished manuscript, “der glücklichste Gedanke meines Lebens”130 (Janssen et al., 2002, p. 265). The equivalence of uniform acceleration and a homogenous gravitational field was first explored by Einstein in Einstein (1907). The term equivalence principle was then used first in Einstein (1912b), see Janssen et al. (2007, p. 495). The physics behind his thought experiment is straightforward (Miller, 1992, pp. 324–327). Einstein’s thought experiment says that an observer cannot ascertain whether he falls freely in a gravitational field with an acceleration .−g or whether he is accelerated by an acceleration .a = g.131 The reason for this is that all objects in a gravitational field experience the same acceleration; in other words, the inertial mass is equal to the gravitational mass. Einstein concluded:132 [Wir] wollen daher im folgenden die völlige physikalische Gleichwertigkeit von Gravitationsfeld und entsprechender Beschleunigung des Bezugssystems annehmen. (Einstein, 1907, p. 454)

Einstein immediately derived consequences from the equivalence principle as the bending of light (Einstein, 1907, pp. 459–462), the gravitational redshift (Einstein, 1907, pp. 458–459), or the anomalous precession of the perihelion of Mercury as it was expressed in a letter to Habicht from 1907, see Renn and Sauer (1999, p. 90) and Stachel (1979, p. 430). In 1911, Einstein then published specific calculations on the light bending in Einstein (1911a): By the equivalence principle, Einstein implied that a light ray is being bent in a gravitational field meaning that the velocity c of light depends on the gravitational field.133 This consideration is crucial as it implied that the general postulation of the constancy of the speed of light in the framework of the special relativity needed to be modified for gravitational theories (Klein et al., 1995, p. 123). Let .c0 be the velocity of light at the origin, let .Ф be the gravitational potential

130 “The happiest thought of my life.” Translation by the author. For a brief discussion of how Einstein’s happiest thought might have been influenced by Mach’s book “The Science of Mechanics” (Mach, 1919), see Heller (1991). 131 For a comprehensive discussion of the role of the equivalence principle in Einstein’s further publications on the road to general relativity, see Norton (1985). 132 “[W]e shall therefore assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system” (Stachel, 1989c, p. 302). 133 The corresponding formula in Einstein (1911a) was corrected by a factor 2 in the theory of general relativity, see Einstein (1915d, p. 834). Johann Georg von Soldner (Soldner, 1801) already considered the bending of light by using Newton mechanics and the corpuscular theory, see also Ginoux (2021), Sauer (2021), and Jaki (1978).

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at a point x relative to the origin, and the velocity of light at x is then ⎞ ⎛ Ф c = c0 1 + 2 , c

.

(1.1)

according to Einstein (1911a, p. 906).134 Various calculations on the bending of light rays can also be found in Einstein’s Prague notebook (Klein et al., 1995, p. 122). As already mentioned in Sect. 1.1.2, the analysis of certain pages of this notebook rewrote the history of gravitational lensing as respective considerations, calculations, and formulas can be found in it (Renn & Sauer, 2003a; Renn et al., 1997). These pages can be dated to the years 1912 and 1915 (Sauer, 2008). Further calculations on gravitational lensing could also be found on the Princeton manuscript pages dated to 1936 that are related to Einstein’s publication (Einstein, 1936), see Sauer and Schütz (2019), and Schütz (2017). Let us finally briefly mention two statements from Einstein showing his relation to mathematics around that time. In summer 1911, the first Solvay conference took place in Brussels, where Frederik Lindemann as scientific secretary of Nernst met Einstein. He wrote later to his father: “Einstein [. . . ] says he knows very little mathematics, but he seems to have had a great success with them” (Klein, 1965, p. 179).135 Another statement can be found in Seelig (1954): In 1909/10, Einstein met his students from the University of Zurich showing them a paper of Planck containing a mistake. Upon the student’s suggestion to write Planck, Einstein answered them by a slightly sarcastic comment about mathematics: “Das Resultat stimmt [. . . ], nur der Beweis ist falsch. Wir schreiben ihm einfach, wie der richtige Beweis lauten könnte. Die Hauptsache ist doch der Inhalt, nicht die Mathematik. Mit der Mathematik kann man nämlich alles beweisen”136 (Seelig, 1954, p. 122).

134 We note here that both observers would measure the same velocity of light with respect to their own reference system as a clock at x measured at the origin runs .1 + Ф/c2 times slower than the clock at the origin. 135 Klein (1965, p. 179) quoted from Smith (1962, p. 43). 136 “The result is correct [. . . ], only the proof is wrong. We simply write him how the correct proof might be. The essential is the content, not the mathematics. With mathematics, namely, one can prove anything.” Translation by the author.

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1.3.4.3

1 Introduction

Einstein’s First Gravitational Theories and Comments on Abraham’s Mathematical Approach

Before discussing Einstein’s first theories including gravitation, we will briefly mention Minkowski’s137 lecture on “Raum und Zeit” (Minkowski, 1909b)138 and his paper “Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern”139 both from 1908 (Minkowski, 1908). There, he introduced the four-dimensional space-time with the fourth imaginary coordinate .x4 = it as well as Raum-Zeit-Vektoren I. und II. Art140 corresponding to four-vectors and tensors of second rank in modern terms141 which are invariant under Lorentz transformations.142 In 1912, Max Abraham used Minkowski’s four-dimensional approach and tried to combine it with Einstein’s idea of allowing the speed of light to be variable (Abraham, 1912d,a). Upon Einstein’s objection, he corrected his first paper (Abraham, 1912d) in a short note (Abraham, 1912a) by restricting covariance with respect to Lorentz transformations only for infinitesimal regions.143 As we will see below, Einstein disliked this approach as he was convinced that Minkowski’s formalism and the covariance under the Lorentz group cannot be upheld if the speed of light is variable. However, he was apparently pushed by Abraham’s papers and, based on the equivalence principle, published a theory of gravity restricted to static fields, which did not make use of Minkowski’s four-dimensional formalism, see Einstein (1912b) and Einstein (1912d).144 Einstein considered an accelerated reference system with the coordinates x, y, z, and t that is accelerated with respect to the x-axis relatively to another reference system. He then derived that the velocity of light c depends on x by c = c0 + ax,

.

137 We

(1.2)

saw in Sect. 1.3.2 that Minkowski was one of Einstein’s teachers at ETH. In 1902, Minkowski moved to the University of Göttingen, where he died in 1909. 138 “Space and Time” (Rowe, 2009, p. 30). Minkowski held this lecture in Cologne on September 21, 1908. For a discussion of the lecture, see Rowe (2009). He presented first results in Göttingen’s colloquium in 1907, see Minkowski (1915). 139 “[The] fundamental equations for the electrodynamics of moving bodies” (Rowe, 2009, p. 36). 140 Space-time vectors of type I and type II. 141 Minkowski did not use the term tensor. 142 For a detailed discussion of Minkowski’s new formalism, see Corry (1997). 143 As pointed out by Renn, by this correction, “Abraham had effectively introduced the mathematical representation of the gravitational potential that was to be at the core of later general relativity, the general four-dimensional line element involving a variable metric tensor” (Renn, 2007, p. 311). 144 It was probably around this time in spring 1912 when Einstein predicted gravitational lensing (Renn et al., 1997), a fact that is only known by the analysis of research notes.

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where .c0 and a are constants (Einstein, 1912b, p. 359).145 By the equivalence principle, this relation also holds for a non-accelerated reference system inside a gravitational field. Einstein then concluded by Eq. (1.1) that the gravitational field is determined by c and proposed to generalize Poisson’s equation by the differential equation Δc = kcρ,

.

(1.3)

where k is constant and .ρ is the mass density (Einstein, 1912b, p. 135). Already in this paper (Einstein, 1912b, footnote 1), Einstein stated that Eq. (1.3) is not correct referring to his follow-up paper (Einstein, 1912d). There, he implied that Eq. (1.3) violates “das Prinzip der Gleichheit von actio und reactio”146 (Einstein, 1912d, p. 452) and suggested new differential equations147 ⎧

⎫ 1 grad2 c .Δc = k cσ + , 2k c

(1.4)

interpreting the additional term as “die Energiedichte des Gravitationsfeldes”148 (Einstein, 1912d, p. 457). We see that in both cases, Einstein’s theory of gravity was a scalar theory, where c represented both the velocity of light and the gravitational potential. In a supplement to (Einstein, 1912d), Einstein stated that the equations of motion can be derived by a variational principle δ

.

⎫ ⎧⎛ / c2 dt 2 − dx 2 − dy 2 − dz2 = 0.

(1.5)

Even though Einstein did not use the notation . ds 2 for the argument inside the square root, he considered the infinitesimal line element here. As stated in Stachel (2007, p. 102), this might have been the hint for Einstein for interpreting . ds as the distance of two neighboring points (Stachel, 2007, p. 102). By preserved lecture notes on mechanics held by Einstein at the University of Zurich or at Prague University (Klein et al., 1993, p. 4), we know that he was familiar with the variational principle in order to derive the equations of motion, see Stachel (2007, p. 100).149

= 0, .c0 is the velocity of light at the origin. principle of equality of action and reaction” (Klein et al., 1996, p. 115). 147 In contrast to his previous paper, Einstein used here the notation .σ instead of .ρ. 148 “[T]he energy density of the gravitational field” (Klein et al., 1995, p. 118). 149 For the lecture notes, see Klein et al. (1993, Doc. 1) and in particular pages 91–95 and 116–117. Einstein also had one course on mechanics at ETH, see Klein et al. (1993, Appendix B). Some of Einstein’s early ideas on a theory of gravitation can also be found in the Prague notebook (Klein et al., 1995, p. 122). 145 By .x

146 “[T]he

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1 Introduction

Let us briefly come back to Abraham’s works (Abraham, 1912d,a) who used Minkowski’s four-dimensional formulation by introducing an imaginary time coordinate .u = ict or . du = ic dt and applied it to Einstein’s result of variable speed of light (Abraham, 1912d, pp. 1–2).150 Einstein criticized Abraham’s approach both in his own publication (Einstein, 1912b, p. 355) and probably also in a private correspondence (Renn & Sauer, 1999, p. 91). Both Abraham and Einstein publicly disputed on the question of whether “das Aufgeben des Postulates von der Konstanz der Lichtgeschwindigkeit”151 demanded “Verzicht auf die Invarianz der Gleichungssysteme gegenüber Lorentztransformationen”152 (Einstein, 1912c, p. 1059), as it was stated by Einstein in Einstein (1912b, p. 368).153 They published short notes on this dispute in Annalen der Physik, see Abraham (1912c), Einstein (1912c), Abraham (1912b), and Einstein (1912a).154 In two letters from 1912, Einstein commented on Abraham’s work revealing his point of view regarding mathematics and physics. In his first letter to Heinrich Zangger155 on January 27, 1912, Einstein criticized Abraham’s formal approach supporting himself a physical one:156 Abraham hat [. . . ] bedenkliche Denkfehler [. . . ] gemacht, sodass die Sache wohl unrichtig ist. Das kommt davon, wenn man formal operiert, ohne dabei physikalisch zu denken! (Klein et al., 1993, Doc. 344)

In a second letter to Besso on March 26, 1912, Einstein was even more explicit:157 Abrahams Theorie ist aus dem hohlen Bauche, d. h. aus blossen mathematischen Schönheitserwägungen geschöpft und vollständig unhaltbar. Ich kann gar nicht begreifen, wie sich der intelligente Mann zu solcher Oberflächlichkeit hat hinreissen lassen können. (Klein et al., 1993, Doc. 377)

150 See

also Renn and Sauer (1999, p. 91) and Klein et al. (1995, pp. 122–128).

151 “[A]bandoning the postulate of the constancy of the velocity of light” (Schulmann et al., 1998a,

p. 130). 152 “[R]elinquishment

of the invariance of the systems of equations with respect to the Lorentz transformations” (Schulmann et al., 1998a, p. 130). 153 “Damit ist also erwiesen, daß man auch für unendlich kleine Raum-Zeitgebiete nicht an der Lorentztransformation festhalten kann, sobald man die universelle Konstanz von c aufgibt” (Einstein, 1912b, p. 368). “This proves, therefore, that the Lorentz transformation also cannot be considered valid for small spacetime regions as soon as one gives up the universal constancy of c” (Klein et al., 1996, p. 106). 154 For a discussion of this dispute, see also Norton (2007a, pp. 422–426) and Klein et al. (1995, pp. 125–126). 155 Zangger was a forensic scientist and trusted friend of Einstein, see Calaprice et al. (2015, pp. 79–80). 156 “Abraham [. . . ] made some serious mistakes in reasoning so that the thing is probably wrong. This is what happens when one operates formally, without thinking physically“ (Klein et al., 1995, p. 250). 157 “Abraham’s theory has been created out of thin air, i.e., out of nothing but considerations of mathematical beauty, and is completely untenable. How this intelligent man could let himself be carried away with such superficiality is beyond me” (Klein et al., 1995, p. 278).

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However, he continued and confessed:158 Im ersten Augenblick (14 Tage lang!) war ich allerdings auch ganz „geblüfft“ durch die Schönheit und Einfachheit seiner Formeln. (Klein et al., 1993, Doc. 377)

Renn suggests that after all, Einstein might have been influenced by Abraham to have a closer look at his formalism with respect to the generalization of the line element (Renn, 2007, p. 316).159 Before this happened, however, Einstein was rather skeptical about Minkowski’s four-dimensional formulation of special relativity (Stachel, 1979, p. 434). About Laue’s book (Laue, 1911) on this subject, Einstein is reported to have said:160 Das Buch von Laue verstehe ich selbst kaum. (Frank, 1949, p. 334)

With respect to Minkowski’s formulation of special relativity and his colleagues in Göttingen, he apparently said playfully:161 Die Leute in Göttingen kommen mir manchmal vor, als wollten sie einem nicht helfen, etwas übersichtlich zu formulieren, sondern als wollten sie uns Physikern nur beweisen, um wieviel gescheiter sie sind als wir. (Frank, 1949, p. 335)

According to Pais (1982, p. 152), Einstein even denoted Minkowski’s formalism by “überflüssige Gelehrsamkeit.”162 As it is rumored, he also said that these mathematicians made his relativity so hard that the physicists do not understand it anymore (Stachel, 1979, p. 434), fitting to Einstein’s quote from above. Hilbert somehow agreed with this, but he made Einstein’s low mathematical skills responsible for this, and he acknowledged at the same time that it was Einstein who knew how to handle mathematics:163 Jeder Straßenjunge in unserem mathematischen Göttingen versteht mehr von vierdimensionaler Geometrie als Einstein. Aber trotzdem hat Einstein die Sache gemacht und nicht die großen Mathematiker. (Frank, 1949, p. 335)

Clearly, in the process of developing the theory of general relativity, Einstein’s view on mathematics and especially the usage of mathematics changed dramatically.

158 “To be sure, at the first moment (for 14 days!) I too was totally “bluffed” by the beauty and simplicity of his formulas” (Klein et al., 1995, p. 278). 159 For a comprehensive discussion of Abraham’s work and the controversy with Einstein, we here refer to Renn (2007). 160 “I myself can hardly understand Laue’s book” (Frank, 1947, p. 206). 161 “The people in Göttingen sometimes strike me, not as if they wanted to help one formulate something clearly, but instead as if they wanted only to show us physicists how much brighter they are than we” (Frank, 1947, p. 206). 162 “Superfluous learnedness.” According to Pais (1982, p. 152), Einstein told this V. Bargmann, who in turn related it to Pais. 163 “Every boy in the streets of our mathematical Göttingen understands more about fourdimensional geometry than Einstein. Yet, despite that, Einstein did the work and not the mathematicians” (Frank, 1947, p. 206).

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1 Introduction

He expressed this in his review paper from 1916, where he as first reference mentioned and acknowledged Minkowski’s work:164 Die Verallgemeinerung der Relativitätstheorie wurde sehr erleichtert durch die Gestalt, welche der speziellen Relativitätstheorie durch Minkowski gegeben wurde, welcher Mathematiker zuerst die formale Gleichwertigkeit der räumlichen Koordinaten und der Zeitkoordinate klar erkannte und für den Aufbau der Theorie nutzbar machte.” (Einstein, 1916a, p. 769)

1.3.4.4

Nordström’s First Theory on Gravitation

The dispute between Einstein and Abraham was followed by the Finnish theoretical physicist Gunnar Nordström165 as he pointed out in the introduction of his article “Relativitätsprinzip und Gravitation”166 from October 1912 (Nordström, 1912).167 As Nordström had an idea that was taken up later by Einstein and Bergmann in their generalization of Kaluza’s theory,168 we will here and in the following discuss Nordström’s works briefly.169 Nordström tried to resolve the problem concerning the variable speed of light and its non-compatibility with special relativity and Minkowski’s formalism by letting the speed of light being constant. In doing so, he could start from Minkowski’s theory without getting the problems Abraham had. Hence, he suggested to adapt the theory of gravity such that inertial and gravitational mass became equal (Nordström, 1912, p. 1126). He suggested two options: First, one could consider only the part of the four force that is orthogonal to the four velocity or let the mass of a body depend on the gravitational potential.170

164 “The

generalization of the theory of relativity has been facilitated considerably by Minkowski, a mathematician who was the first one to recognize the formal equivalence of space coordinates and the time coordinate, and utilized this in the construction of the theory” (Kox et al., 1997, p. 146). 165 For a short biography, see Isaksson (1985). 166 “The Principle of Relativity and Gravitation” (Nordström, 2007a). 167 Nordström himself got in a dispute with Abraham earlier in 1909–1910 as it is documented in their publications (Nordström, 1909, 1910; Abraham, 1909, 1910). For a brief discussion of their dispute on relativistic electrodynamics and for further literature, see Norton (2007a, pp. 431–432). 168 Einstein and Bergmann did not mention Nordström’s work. 169 For a detailed discussion, see Norton (2007a), Isaksson (1985), and Nugayev (2017). For a more modern point of view of Nordström’s influences, see Ravndal (2004). 170 “Entweder faßt man nicht .F , sondern nur den auf den Bewegungsvektor senkrechten Teil desselben als bewegende Kraft auf, oder man nimmt die Masse eines Massenpunktes nicht als konstant, sondern als von dem Gravitationspotential abhängig” (Nordström, 1912, p. 1126). “Either one takes not .F itself but only its component perpendicular to the velocity vector as the accelerating force, or one takes the mass of a mass point to be not constant but dependent on the gravitational potential” (Nordström, 2007a, p. 490). We suppressed a footnote in this quote, which referred to Minkowski’s way to treat the electrodynamic force.

1.3 Einstein’s Education and His Way to General Relativity with a Special. . .

33

Einstein read Nordström’s paper and communicated to him that in his theory, a rotating system will acquire a smaller acceleration in a gravitational field than a non-rotating system. Nordstörm agreed and commented in the addendum to the proofs of his paper “daß meine Theorie mit der Einsteinschen Äquivalenzhypothese nicht vereinbar ist, nach welcher ein unbeschleunigtes Bezugssystem im homogenen Gravitationsfelde einem beschleunigten Bezugssystem im gravitationsfreien Raume äquivalent wäre”171 (Nordström, 1912, p. 1129). However, Nordström did not see a problem in this. Instead, he published another comprehensive article on his theory in 1913 (Nordström, 1913a). This time, he finished with the equation ⎛ ⎞ v2 ∂ϕ dvz =− 1− 2 g . dt ∂z c

(1.6)

of a body with velocity v and its components .vx , .vy , and .vz in a gravitational field with potential .

∂ϕ ∂ϕ ∂ϕ ∂ϕ = = = 0, = const. and ∂x ∂y ∂t ∂z

(1.7)

where g is a gravitational factor. This time, he himself concluded:172 Die Bewegungskomponente senkrecht zur Feldrichtung ist gleichförmig. Die Fallbeschleunigung ist um so kleiner, je größer die Geschwindigkeit ist, aber unabhängig von der Richtung der Geschwindigkeit. Ein in horizontaler Richtung geworfener Körper fällt langsamer als ein ohne Anfangsgeschwindigkeit vertikal fallender. (Nordström, 1913a, p. 878)

Nordström published further articles on his theory of gravitation and Einstein carefully read them. We will come back to this below.

1.3.4.5

Einstein’s Turn to Mathematics, the Zurich Notebook, and the Entwurf Theory

In a guest lecture at Kyoto in 1922, Einstein specified how he realized that he would need sophisticated mathematical methods in order to formulate his theory of gravity.

171 “[T]hat my theory is not compatible with Einstein’s hypothesis of equivalence, according to which an unaccelerated reference system in a homogenous gravitational field is equivalent to an accelerated reference system in a gravitation free speace” (Nordström, 2007a, p. 497). 172 “The velocity component perpendicular to the field direction is uniform. Gravitational acceleration becomes smaller as the velocity increases, but this is independent of the direction of the velocity. A body projected horizontally falls slower than one without initial velocity falling vertically” (Nordström, 2007b, p. 520). In both the German and English quotes, we suppressed italics.

34

1 Introduction

In particular, he needed non-Euclidean geometry and differential geometry:173 Denn, wenn alle solche Systeme[, die Beschleunigung aufweisen] als möglich zugelassen würden, gälte in jedem einzelnen die EUKLIDische Geometrie nicht mehr. [. . . ] Was also müssen wir hier suchen, wonach forschen? Diese Frage blieb mir bis 1912 unbeantwortet, bis mir einfiel, daß die GAUSSsche Oberflächentheorie den Schlüssel zu diesem Geheimnis darstellen könnte. [. . . ] Bis dahin wußte ich nichts von den tiefgehenden Abhandlungen RIEMANNS über die Grundlagen der Geometrie. Ich erinnerte mich an die GAUSSsche Theorie, die wir während unserer Studienzeit in der Geometrie-Vorlesung des MathematikDozenten GEISER gehört hatten. So kam ich auf diese Idee und dachte weiter daran, daß die Grundlagen der Geometrie auch für die Physik Bedeutung haben müssen. (Haubold and Yasui, 1986, pp. 276–277)

This idea did not come before March or April174 1912 to Einstein as we know from a letter to Besso from March 26, 1912:175 In der letzten Zeit arbeitete ich rasend am Gravitationsproblem. Nun ist es soweit, dass ich mit der Statik fertig bin. Von dem dynamischen Feld weiss ich noch gar nichts, das soll erst jetzt folgen. (Klein et al., 1993, Doc. 377)

Stachel stated that Einstein in 1912 generalized the scalar c to a ten-component gμν and that he in mid-1912 realized that these components form the coefficients of the square of the invariant line element

.

.

ds 2 =

Σ

gik dxi dxk

(1.8)

of a four-dimensional space-time manifold (Stachel, 2007, p. 106).176 Stachel suggests that it was this time when he turned to Grossmann (Stachel, 2007, pp. 106–107). Also around that time in August 1912, Einstein left Prague and returned to Zurich (Pais, 1982, p. 523).

173 We suppressed two footnotes in this quote. “If all [accelerated] systems are equivalent, then Euclidean geometry cannot hold in all of them. [. . . ] What must we search for at this point? This problem remained insoluble to me until 1912, when I suddenly realized that Gauss’s surface coordinates had a profound significance. However, I did not know at that time that Riemann had studied the foundations of geometry in an even more profound way. I suddenly remembered that Gauss’s theory was contained in the geometry course given by Geiser when I was a student. . . . I realized that the foundations of geometry have physical significance” (Pais, 1982, pp. 211–212). We note that this is not a word for word translation of the German quote. 174 Einstein submitted (Einstein, 1912b) on February 26 and (Einstein, 1912d) on March 23. 175 “Lately I have been working like mad on the gravitation problem. Now I have gotten to the stage where I am finished with the statics. I do not know anything yet about the dynamic field, that will come only now” (Klein et al., 1995, p. 276). 176 We used here subscripts for coordinate differentials according to the original notation. When we derive the theory of gravitation in Chap. 3, we will use modern notation.

1.3 Einstein’s Education and His Way to General Relativity with a Special. . .

35

Stachel’s suggestion is also supported by Einstein’s recollection in 1955 where he stressed that the generalized line element reduced the problem to a purely mathematical one:177 Das Problem der Gravitation war damit reduziert auf ein rein mathematisches. Gibt es Differentialgleichungen für die .gik welche invariant sind gegenüber nicht-linearen Koordinaten-Transformationen? Solche Differentialgleichungen und nur solche kamen als Feldgleichungen des Gravitationsfeldes in Betracht. Das Bewegungsgesetz materieller Punkte war dann durch die Gleichung der geodätischen Linie gegeben. Mit dieser Aufgabe im Kopf suchte ich 1912 meinen alten Studienfreund Marcel Großmann auf, der unterdessen Professor der Mathematik am Eidgenössischen Polytechnikum geworden war. Er fing sofort Feuer, obwohl er der Physik gegenüber als echter Mathematiker eine etwas skeptische Einstellung hatte. (Einstein, 1955, p. 151)

Einstein realized that he could formulate the theory of gravity only with knowing sophisticated mathematical methods, which was only possible for him with Grossmann’s help:178 Grossmann, Du mußt mir helfen, sonst werd’ ich verrückt! (Kollros, 1956, p. 278)

In the foreword of the Czech translation179 of Einstein’s “Über die spezielle und die allgemeine Relativitätstheorie (Gemeinverständlich)”180 (Einstein, 1917), Einstein recalled:181 Den entscheidenden Gedanken von der Analogie des mit der Theorie verbundenen mathematischen Problems mit der Gaußschen Flächentheorie hatte ich allerdings erst 1912 nach meiner Rückkehr nach Zürich, ohne zunächst Riemanns und Riccis, sowie LeviCivitas Forschungen zu kennen. Auf diese wurde ich erst durch meinen Freund Großmann in Zürich aufmerksam, als ich ihm das Problem stellte, allgemein kovariante Tensoren aufzusuchen, deren Komponenten nur von Ableitungen der Koeffizienten der quadratischen Fundamentalinvariante abhängen. (Einstein, 1923c, Preface)

177 “The

problem of gravitation was thus reduced to a purely mathematical one. Do differential equations exist for the .gik , which are invariant under non-linear coordinate transformations? Differential equations of this kind and only of this kind were to be considered as field equations of the gravitational field. The law of motion of material points was then given by the equation of the geodesic line. With this problem in mind I visited my old friend Grossmann who in the meantime had become professor of mathematics at the Swiss polytechnic. He at once caught fire, although as a mathematician he had a somewhat skeptical stance towards physics” (Sauer, 2015b, p. 471). 178 “Grossmann, you have to help me, or else I’ll go crazy” (Sauer, 2015b, p. 468). 179 The Czech version was published in 1923 (Einstein, 1923c), see Kox et al. (1996, p. 535). 180 “On the special and the general theory of relativity (a popular account)” (Kox et al., 1997, Doc. 42). 181 Here we used the German version published in Kox et al. (1996, p. 535). “However, only after my return in 1912 to Zurich did I hit upon the decisive idea about the analogy between the mathematical problem connected with my theory and the theory of surfaces by Gauss-originally without knowledge of the research by Riemann, Ricci, and Levi-Civita. The latter research came to my attention only through my friend Grossmann in Zurich when I posed the problem to him only to find generally covariant tensors whose components depend only upon the derivatives of the coefficients of the quadratic fundamental invariant” (Kox et al., 1997, p. 418).

36

1 Introduction

Einstein explicitly mentioned the Gaussian surface theory that he knew from Geiser’s lecture. Indeed, Geiser treated among others the line element, Gaussian measure of curvature, and geodesics in his lecture, see also Reich (1994, pp. 160– 166). In particular, on June 10, 1898, Geiser considered a surface and a curve .ϕ(p, q) between two points .P0 and .P1 on the surface. By a variation of ⎛P1 s=

ds =

.

P0

⎛P1 /

E dp2 + 2F dp dq + G dq 2 ,

(1.9)

P0

he then determined its minimum. The respective passages of the lecture notes are shown in Fig. 1.1.182 Stachel in Stachel (2007, p. 104) pointed out that comparing these considerations and Einstein’s variational principle in Eq. (1.5) could have been Einstein’s thoughts on the analogy quoted above. Einstein started working with Grossmann shortly after arriving in Zurich (Sauer, 2015b, p. 468). Already in October 1912, he was very optimistic and held mathematics in high esteem as he expressed in a letter to Arnold Sommerfeld183 on October 29, 1912:184 Ich [. . . ] glaube nun mit Hilfe eines hiesigen befreundeten Mathematikers aller Schwierigkeiten Herr zu werden. Aber das eine ist sicher, dass ich mich im Leben noch nicht annähernd so geplag[t] habe, und dass ich grosse Hochachtung für die Mathematik eingeflösst bekommen habe, die ich bis jetzt in ihren subtileren Teilen in meiner Einfalt für puren Luxus ansah! (Klein et al., 1993, Doc. 421)

It is interesting to see how fast Einstein’s view on mathematics and the use of it for his theory of general relativity changed during the year 1912. From the time period when Einstein and Grossmann worked together, research notes are preserved in the Zurich notebook, see Einstein (2007) and Klein et al. (1995, Doc. 10). This notebook has been analyzed in great detail by the Einstein scholars Janssen, Renn, Sauer, Norton, and Stachel, who published their results in several papers as in Norton (1984), Castagnetti et al. (1994), Renn and Sauer (1996), Renn and Sauer (1999), Janssen (1999), Norton (2000), Renn and Sauer (2003b), and Renn (2005c,a) and finally in Janssen et al. (2007) and particularly in Janssen et al. (2007).185 The notes can be dated between summer 1912 and spring 1913 (Renn and Sauer, 1996, p. 865) with the most entries from winter 1912–1913 (Janssen et al., 2007, p. 492). It shows the first considerations Einstein’s in order to find a theory

182 We know by a pencil comment probably made by Einstein that he read Grossmann’s lecture notes on Geiser’s infinitesimal geometry, see Sauer (2015b, p. 458). 183 For a short biography, see Calaprice et al. (2015, pp. 158–159). 184 “I [. . . ] believe that I can overcome all difficulties with the help of a mathematician friend of mine here. But one thing is certain: never before in my life have I troubled myself over anything so much, and I have gained enormous respect for mathematics, whose more subtle parts I considered until now, in my ignorance, as pure luxury!” (Klein et al., 1995, p. 324). 185 See also its further volumes (Renn & Schemmel, 2007).

1.3 Einstein’s Education and His Way to General Relativity with a Special. . .

37

Fig. 1.1 Grossmann’s notes of Geiser’s lecture on infinitesimal geometry from 1898. The passages are from Grossmann (1898b, pp. 57–59) and show Geiser’s treatment of Gaussian surface theory. Original located at ETH-Bibliothek Zürich, Hs 421:16

of gravitation by using mathematical concepts as, for example, the general line element and Gauss’s theory of surfaces, Minkowski’s four-dimensional space-time geometry with imaginary time coordinate, theory of invariants, Riemann geometry, and differential and tensor calculus (Renn & Sauer, 1996, 1999). In particular, the turn to mathematics and Einstein’s growing familiarity with mathematics and the corresponding literature can be observed in Einstein’s notes in the Zurich notebook (Renn & Sauer, 1996, p. 869). The underlying question in the Zurich notebook was always to find the field equations connecting the energy momentum tensor with a tensor consisting of the metric and its first and second derivatives (Renn & Sauer, 1996, p. 868). These equations, however, should satisfy four heuristic requirements: First, the field equations should have the broadest possible covariance and should be covariant under a transformation group larger than the Lorentz group (Janssen et al., 2007, p. 494). This requirement, for instance, led Einstein to the Riemann tensor that was probably pointed out to him by Grossmann (Janssen et al., 2007, p. 604,610). Second, the equivalence principle should hold implying that any solution of the field equations should hold for observers both in an accelerated coordinate system and in

38

1 Introduction

a coordinate system at rest in a gravitational field (Janssen et al., 2007, pp. 494– 496). Third, in the case of weak static fields, the field equations needed to reduce to Poisson’s equation according to the Newton theory (Janssen et al., 2007, pp. 496– 498). Fourth, the energy–momentum conversation law should hold (Janssen et al., 2007, pp. 498–500). The analysis of the Zurich notebook shows how Einstein tried to find candidates for the field equations satisfying these four requirements. The Einstein scholars identified two different strategies (Janssen et al., 2007, pp. 500–501): The mathematical strategy started from quantities that are generally covariant as the Riemann tensor in order to build up field equations connecting the metric and its first and second derivatives with the energy momentum tensor. These field equations are then needed to satisfy the rest of the requirements. We will see later that Einstein struggled with finding such field equations that reduce to the classical theory in case of the Newtonian limit. The second and physical strategy started from the weak static fields where Einstein tried to generalize the laws of special theory of relativity and the gravitation theory of static fields. The problem occurring here was to find equations that are covariant under the largest possible group of transformations. The analysis of the Zurich notebook also showed how Einstein had improved his mathematical skills during that time period as he, for instance, at some point became more and more familiar with tensor calculus (Janssen et al., 2007, p. 523). At this time, Einstein switched to the mathematical strategy finally leading him to the Riemann tensor and to the Ricci tensor where he also considered the so-called November tensor (Janssen et al., 2007, pp. 603,606–610). Einstein, however, discarded this approach and switched back to the physical strategy. We will see later that Einstein came back to the November tensor in his first November publication in 1915 that then led him to the final field equations of general relativity, see Eq. (1.18).186 The analysis of the Zurich notebook revealed that he not only discovered the November tensor by his mathematical strategy, but he already derived the linearized weak field equations in the Zurich notebook on its page 20L as well (Janssen et al., 2007, pp. 603,606,632–636). Nevertheless, Einstein abandoned the mathematical strategy at the end, came back to the physical approach, and derived the so-called Entwurf theory equations (Janssen et al., 2007, pp. 681–683,699– 704,706–712).

186 Much has been written on the question why Einstein rejected the November tensor in 1912– 1913. For a discussion of this question, see Janssen and Renn (2007), Janssen and Renn (2015a), Norton (2007b), and Renn and Sauer (2007). For an overview of the different perspectives, see Räz (2016) and Weinstein (2018).

1.3 Einstein’s Education and His Way to General Relativity with a Special. . .

39

This theory was the result of Einstein and Grossmann’s collaboration, which they published in their joint paper “Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation”187 (Einstein & Grossmann, 1913).188 The idiosyncrasy of this paper is the stringent separation between physical and mathematical part. While Einstein was the author of the physical part (first part), Grossmann was the author of the mathematical part (second part). The latter discussed the subjects Allgemeine Tensoren, Differentialoperationen an Tensoren, Spezielle Tensoren (Vektoren), and Mathematische Ergänzungen zum physikalischen Teil.189 Einstein and Grosmmann’s Entwurf theory “contained virtually all the essential features of the final general theory of relativity” (Norton, 1985, p. 219), where all equations were written in a generally covariant form—except for the most important field equations (Norton, 2000, p. 144). In fact, Einstein formulated the field equations to be190 κ · Θμν = ┌μν ,

.

(1.10)

where .κ is a constant, .Θμν is the energy–momentum tensor describing the matter, and .┌μν denotes an unknown tensor of second rank built up by the metric components .gμν (Einstein & Grossmann, 1913, p. 233). This time, he denoted the variational principle in order to derive the equations of motion by ⎧⎛ δ

.

⎫ ds = 0,

(1.11)

where it is191 .

ds 2 =

Σ

gμν dxμ dxν .

(1.12)

μν

He also explicitly considered the special case .gμν = 0 for .μ /= ν and .g11 = g22 = g33 = −1, .g44 = c2 according to special relativity, and Minkowski’s spacetime metric (Einstein & Grossmann, 1913, pp. 228–229). Furthermore, Einstein

187 “Outline

of a Generalized Theory of Relativity and of a Theory of Gravitation” (Klein et al., 1996, Doc. 13). 188 The offprint was available in 1913 (Einstein & Grossmann, 1913). It was then reprinted in Zeitschrift für Mathematik und Physik in 1914 (Einstein & Grossmann, 1914a). For Grossmann’s role in developing the Entwurf theory as well as for his academic career, see Sauer (2015b). 189 General Tensors, Differential Operations on Tensors, Special Tensors (Vectors), and Mathematical Supplements to the Physical Part (Klein et al., 1996, Doc. 13). 190 In Einstein and Grossmann’s notation, this is a contravariant equation as Einstein used Greek letters, see Einstein and Grossmann (1913, p. 246). 191 Again, we use Einstein’s notation with the subscripts for the coordinates.

40

1 Introduction

interpreted the line element “ds als invariantes Maß für den Abstand zweier unendlich benachbarter Raumzeitpunkte”192 (Einstein & Grossmann, 1913, p. 230). However, Einstein did not find the correct expression for .┌μν such that it is a tensor with respect to arbitrary transformations and such that it reduces to the Laplacian of the gravitational potential .Δϕ in the Newtonian limit according to Poisson’s equation. This means that Einstein was not able to find what is nowadays called the Einstein tensor .Gμν = Rμν − 1/2gμν R, where .Rμν is the Ricci tensor and R the Ricci scalar.193 In particular, Einstein concluded that “[e]s [. . . ] aber hervorgehoben werden [muss], daß es sich als unmöglich erweist, unter dieser Voraussetzung einen Differentialausdruck .┌μν zu finden, der eine Verallgemeinerung von .Δϕ ist, und sich beliebigen Transformationen gegnüber als Tensor erweist”194 (Einstein & Grossmann, 1913, p. 233). In fact, Einstein referred to Grossmann’s part, where Grossmann considered what is nowadays called the Ricci tensor and concluded that “sich dieser Tensor im Spezialfall des unendlich schwachen statischen Schwerefeldes nicht auf den Ausdruck .Δϕ reduziert”195 (Einstein & Grossmann, 1913, p. 257). This shows us how close Einstein and Grossmann had been to the final field equations.196 As pointed out in Renn and Sauer (2003b, pp. 253–260), Einstein and Grossmann’s wrong conclusions frequently had been attributed to their nonfamiliarity with generally covariant equations, a fact that was rectified by the analysis of the Zurich notebook. Let us finally briefly summarize the mathematical concepts and ideas that Grossmann probably pointed out to Einstein. We already saw in the Zurich notebook that this was the case with the Riemann tensor. In particular, Grossmann drew Einstein’s attention to the mathematical tools of Riemannian geometry (Riemann, 1867) on n-dimensional manifolds where the Gaussian theory of surfaces only was a special case, usually using the notations E, F , and G for the three metric components .g11 , .g12 , and .g21 of the two-dimensional line element, see Sauer (2015b, pp. 472–473), Sauer (2005a, p. 806), and Janssen et al. (2007, pp. 610– 611).197 Furthermore, as also indicated by Einstein’s quote from above appearing in the foreword of his popular book on general relativity, due to Grossmann, Einstein

192 “We note in this connection that ds is to be conceived as the invariant measure of the distance between two infinitely close space-time points” (Klein et al., 1996, p. 157). 193 We will discuss the derivation of Einstein’s general theory of gravity as well as its generalization in the five-dimensional case comprehensively in Chap. 3. There, we will also give the definitions for the tensors mentioned here. 194 “But it must be stressed that, given this assumption, it proves impossible to find a differential expression .┌μν that is a generalization of .Δϕ and that proves to be a tensor with respect to arbitrary transformation” (Klein et al., 1996, p. 160). 195 “It turns out, however, that in the special case of the infinitely weak, static gravitational field this tensor does not reduce to the expression .Δϕ” (Klein et al., 1996, p. 185). 196 For a detailed analysis of why Einstein and Grossmann rejected the Ricci tensor, see Norton (1984). 197 See, for instance, the first equation at the bottom picture in Fig. 1.1.

1.3 Einstein’s Education and His Way to General Relativity with a Special. . .

41

got to know the works of Gregorio Ricci-Curbastro and Tullio Levi-Civita on differential calculus.198 He probably also pointed to Christoffel’s works on quadratic differential forms in Christoffel (1869), see Janssen et al. (2007, p. 610). For further contributions by Grossmann, we here refer to Sauer (2015b, pp. 468–488).199

1.3.4.6

Nordström’s Further Works and His Five-Dimensional Approach

In summer 1913, Nordström probably visited Einstein in Zurich where they discussed their theories (Isaksson, 1985, p. 38).200 In July 1913, Nordström then published his second theory of gravitation. This time, he did not set the gravitational source density (“Ruhdichte”) arbitrarily but, inspired by von Laue, let it be determined by the trace T of the energy momentum tensor (Nordström, 1913b, pp. 533–534). He then developed a scalar field theory which was “the first logically consistent relativistic field theory of gravitation ever formulated” (Pais, 1982, p. 233). By starting from the field equations from his previous theory, he let the gravitational factor201 g depend on the potential and derived the new field equations202 ⎧ Ф

.

∂ 2Ф ∂ 2Ф ∂ 2Ф ∂ 2Ф + + + ∂x 2 ∂y 2 ∂z2 ∂(ict)2

⎫ = −T ,

(1.13)

where .Ф is the gravitational potential, see Nordström (1913b, pp. 537–538,550). In September 1913, Einstein spoke “zum gegenwärtigen Stande des Gravitatonsproblems”203 at the 85. Naturforscherversammlung zu Wien204 and devoted a large part of his lecture to Nordström’s theory.205 After deriving his field equations by a variational principle (Einstein, 1913, p. 1254), Einstein showed that Nordström’s scalar theory satisfied four requirements that were posed by Einstein at the beginning.206 His only critique was that in Nordström’s theory, the inertia of a

198 See

Ricci and Levi-Civita (1900) and Ricci and Levi-Civita (1901). For a discussion and translation, see Hermann (1975). 199 For a brief comment on contributions from Grossmann as well as from Besso to general relativity, see also Janssen and Renn (2015b). 200 See also Norton (2007a, p. 455). 201 We already encountered this gravitational factor in Eq. (1.6). 202 We note that the left hand side is equal to .−Ф□Ф. 203 “On the current state of the problem of gravitation” (Norton, 2007a, p. 465). 204 85th Congress of the German Natural Scientists and Physicians in Vienna. Translation inspired by Norton (2007a, p. 465). 205 He only briefly mentioned Abraham’s work as it did not confirm with the laws of special relativity. 206 These four requirements were satisfaction of the conservation law of momentum and energy, equality of inertial and gravitational masses of closed systems, reduction to special relativity as a limiting case, and independence of observable natural laws from the absolute value of the

42

1 Introduction

body was not caused by the remaining bodies.207 Einstein then continued presenting their Entwurf theory.208 In 1914, Einstein published a paper on Nordström’s theory (Einstein & Fokker, 1914) co-authored by Adriaan Daniel Fokker, who just recently in 1913 obtained his Ph.D. under Lorentz and spent one semester with Einstein in Zurich (Pais, 1982, p. 236). Einstein and Fokker applied, similar to the Entwurf theory, Ricci’s and Levi Civita’s differential calculus on Nordström’s theory. By these methods, they derived Nordström’s theory by allowing the possibility of choosing a coordinate system such that the speed of light is constant (Einstein & Fokker, 1914, p. 322).209 They then derived the field equations Σ .

iklm

1 Σ Tτ τ , γim γkl (ik, lm) = κ √ −g τ

(1.14)

which yield Nordström’s field equations if one chooses a coordinate system where the speed of light is constant (Einstein & Fokker, 1914, pp. 326–327). By .Riklm = (ik, lm) and by the notation of covariant, contravariant, and mixed tensors with Latin, Greek (.γ ), and curly (.T ) letters, the left hand side of Eq. (1.14) is equal to the Ricci scalar R such that we can write210 κ R = √ T. −g

.

(1.15)

Thus, Einstein and Fokker derived equations that are generally covariant in contrast to Nordström’s theory which only was Lorentz covariant (Pais, 1982, p. 236). Einstein and Fokker were well aware that using the Riemann tensor would be an important step by developing general relativity as they point out at the end of their paper:211

gravitational potential, see Einstein (1913, p. 1250). We used here an abbreviated English version of these requirements from Norton (2007a, p. 465). 207 As pointed out by Ravndal (2004, p. 7), Nordström’s theory does not predict light deflection nor the correct perihelion shift of Mercury. 208 While both Einstein and Grossmann presented their theory together at the Schweizerische Naturforschende Gesellschaft at the beginning of September, he alone presented it 2 weeks later in Vienna (Sauer, 2015b, p. 481). 209 In other words, as Einstein and Fokker stated in Einstein and Fokker (1914, p. 324), Nordström’s theory is a special case of the Entwurf theory, see also Pais (1982, pp. 236–237). 210 See Einstein and Grossmann (1913, p. 246,256), Einstein and Fokker (1914, p. 325), and Fliesßach (2016, p. 97). The notation R is a modern notation. We will come back to this later when discussing Einstein’s final field equations, see Eq. (1.22). 211 “Finally, it is plausible that the role which the Riemann-Christoffel tensor plays in the present investigation would also open a way for a derivation of the Einstein-Grossmann gravitation equations in a way independent of physical assumptions. The proof of the existence or nonexistence of such a connection would be an important theoretical advance.” Translation inspired by Pais (1982, pp. 236–237).

1.3 Einstein’s Education and His Way to General Relativity with a Special. . .

43

Endlich legt die Rolle, welche bei der vorliegenden Untersuchung der RiemannChristoffelsche Differentialtensor spielt, den Gedanken nahe, daß er auch für eine von physkalischen Annahmen unabhängige Ableitung der Einstein-Großmannschen Gravitationsgleichungen einen Weg öffnen würde. Der Beweis der Existenz oder Nichtexistenz eines derartigen Zusammenhanges würde einen wichtigen theoretischen Fortschritt bedeuten.212 (Einstein & Fokker, 1914, p. 328)

In 1914, Nordström published “über die Möglichkeit, das elektromagnetische Feld und das Gravitationsfeld zu vereinigen.”213 In this paper, he introduced a fifth dimension in order to unify the theories of gravitation and electromagnetism. He concluded:214 Es wird gezeigt, daß eine einheitliche Behandlung des elektromagnetischen Feldes und des Gravitationsfeldes möglich ist, wenn man die vierdimensionale Raumzeitwelt als eine durch eine fünfdimensionale Welt gelegte Fläche auffaßt. (Nordström, 1914c, p. 506)

Interestingly, in this paper, he did not mention Einstein’s works, in particular the work with Fokker (Einstein & Fokker, 1914), but mentioned Mie’s theory (Mie, 1913).215 He elaborated his theory further in Nordström (1914d) and Nordström (1915).216 Theodor Kaluza in 1919 had a similar idea of introducing a fifth dimension in order to unify the theories of electromagnetism and gravitation (Kaluza, 1921). This theory became famous and was further elaborated by Oskar Klein (Klein, 1926a,b).217 We will come back to this in Sects. 1.4 and 1.5 as well

212 Einstein and Fokker also pointed out that Einstein and Grossmann excluded such a connection in their Entwurf theory in Einstein and Grossmann (1914a, §4.2), see also Norton (2007a, p. 473). 213 “On the possibility of unifying the electromagnetic field and the gravitational field.” Translation by the author. 214 “It is shown that a unified treatment of the electromagnetic field and the gravitational field is possible if one conceives the four-dimensional space-time world as a plain laid through the fivedimensional world.” Translation by the author. 215 Mie and Einstein had a discussion on Mie’s theory (Mie, 1913) after his Vienna lecture, where Mie expressed his dissatisfaction that Einstein did not mention Mie’s work, see Einstein et al. (1913). He also emphasized Abraham’s contribution that, in Mie’s opinion, was not enough appreciated by Einstein’s talk (Einstein et al., 1913, p. 1262). Einstein answered that he had roughly read Mie’s work but did not speak about it as the equality of inertia and gravitational masses is not satisfied in Mie’s work (Einstein et al., 1913, p. 1263). In Mie (1914a,b), Mie then criticized both Einstein’s and Nordström’s theories. Einstein (Einstein, 1914b) as well as Nordström (Nordström, 1914b) responded in the same volume of Physikalische Zeitschrift. For further discussion, see Norton (2007a, p. 477) and Smeenk and Martin (2007). 216 In 1914, Nordström also published one further article on the laws of free fall and the movement of planets in his gravitation theory (Nordström, 1914a). 217 For a detailed discussion, see Wünsch (2003). While (Wünsch, 2003, p. 526) states that Einstein “seemingly” had not known Nordström’s theory, we rather argue that he had read at least (Nordström, 1914c), which was published in Physikalische Zeitschrift. Einstein himself published in Physikalische Zeitschrift, see, for instance, (Einstein, 1914b). As we already saw, he also referred to many articles that were published there as, for instance, Abraham’s and Nordström’s papers. We also saw that Einstein knew Nordström’s gravitation theory very well and himself elaborated it. Hence, it seems unlikely to us that Einstein had not read Nordström’s publication.

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1 Introduction

as in Chap. 3 when discussing Einstein and Bergmann’s generalization of Kaluza’s theory.218

1.3.4.7

Einstein’s Final Steps to the Theory of General Relativity

In April 1914, Einstein moved to Berlin (Pais, 1982, p. 524).219 In late June to early July 1915, Einstein gave six lectures in Göttingen on general relativity trying to convince the mathematicians there, not only Klein and Hilbert but also Emmy Noether and others (Pais, 1982, p. 524). In mid-October, however, Einstein was no longer satisfied with his theory as first, it predicted the wrong precession of Mercury’s perihelion, second, his equations were not covariant under transformations that rotate the coordinate system, and third, the uniqueness of the gravitational Lagrangian that he proved in Einstein (1914a) was incorrect. He expressed this in several letters to Arnold Sommerfeld (Schulmann et al., 1998b, Doc. 153) as well as to Hendrik Lorentz (Schulmann et al., 1998b, Docs. 129, 177).220 These letters as well as manuscripts as the Einstein Besso manuscript (Klein et al., 1995, Doc. 14), the Prague Notebook (Klein et al., 1993, Appendix B),221 pages of Besso’s

218 There, we will also look at the time periods when Einstein considered such an approach in order to find a unified field theory, see also Sauer (2014). 219 This ended Einstein and Grossmann’s collaboration (Sauer, 2015b, p. 483). They published one further joint paper (Einstein & Grossmann, 1914b) on the hole argument as already done by Einstein in his addendum to the Entwurf theory (Einstein & Grossmann, 1914a, pp. 260–261), see Sauer (2015b, pp. 483–485). Einstein then published a comprehensive paper on Einstein and Grossmann’s theory (Einstein, 1914a) using the term “allgemeine Relativitätstheorie” (“general theory of relativity”) for the first time in the title (Sauer, 2015b, p. 486). By the hole argument, Einstein thought that physically meaningful and generally covariant equations for the gravitational field cannot exist. For a discussion of the hole argument, see Renn and Sauer (2007), Janssen (2014), Janssen (2005), Janssen (2007), and Stachel (2014). 220 In particular, he recapitulated this time period when he wrote to Lorentz on January 1, 1916: “Die allmählich aufdämmemde Erkenntnis von der Unrichtigkeit der alten GravitationsFeldgleichungen hat mir letzten Herbst böse Zeiten bereitet. Ich hatte schon früher gefunden, dass die Perihelbewegung des Merkur sich zu klein ergab. Dazu fand ich, dass die Gleichungen nicht kovariant waren für Substitutionen, die einer gleichförmigen Rotation des (neuen) Bezugssystems entsprachen. Endlich fand ich, dass meine letztes Jahr angestellte Betrachtung zur Bestimmung der Lagrange’schen Funktion H des Gravitationsfeldes durchaus illusorisch war, indem sie leicht so modifiziert werden konnte, dass man H überhaupt keiner einschrän- kenden Bedingung zu unterwerfen brauchte, sodass es ganz frei gewählt werden konnte” (Schulmann et al., 1998b, Doc. 177). “Trying times awaited me last fall as the inaccuracy of the older gravitational field equations gradually dawned on me. I had already discovered earlier that Mercury’s perihelion motion had come out too small. In addition, I found that the equations were not covariant for substitutions corresponding to a uniform rotation of the (new) reference system. Finally, I found that the consideration I made last year on the determination of Lagrange’s H function for the gravitational field was thoroughly illusory, in that it could easily be modified such that no restricting conditions had to be attached to H , thus making it possible to choose it completely freely” (Schulmann et al., 1998a, Doc. 177). 221 See also Sect. 1.1.2.

1.3 Einstein’s Education and His Way to General Relativity with a Special. . .

45

Nachlass (Janssen, 2007, Figs. 1 and 2), or unpublished calculations in Einstein’s hand on the recto of a letter to Naumann (Schulmann et al., 1998c, p. 180) were used in order to recapitulate the period when Einstein noticed, considered, and solved these problems, especially the problem on Mercury’s perihelion and the rotation problem, see, for instance, Janssen (1999), Earman and Janssen (1993), Janssen (2007) and Klein et al. (1995, pp. 344–359). Although he had worries in midOctober, in November 1915, Einstein published four papers culminating in the final field equations, which we will briefly discuss in the following.222 In his first November paper on November 4 (Einstein, 1915e, p. 782), Einstein addressed the problem concerning the covariance under rotations. He introduced what is nowadays called the November tensor223 { Rim = −

∂

.

im l

} +

∂xl

Σ ⎧ il ⎫ { ρm } ρ

ρ

l

(1.16)

,

where it is { μν } .

τ

=

Σ1 σ

2

⎛ gσ τ

∂gμσ ∂gμν ∂gνσ + − ∂xν ∂xμ ∂xσ

⎞ (1.17)

the Christoffel symbol according to Einstein (1914a, p. 1046).224 Einstein then derived the field equations to be Rμν = −κTμν ,

.

(1.18)

which are covariant under “Transformationen von der Determinante 1”225 (Einstein, 1915e, p. 783). As we already mentioned above, the analysis of the Zurich notebook revealed that Einstein considered the November tensor .Rμν already in 1912–1913 as it appears in the Zurich notebook (Renn & Sauer, 2007, pp. 192–193).226 In an addendum to his first November paper, on November 11, he used the Ricci tensor227 in order to publish modified field equations that are generally covariant by the restriction that the trace of the energy momentum tensor vanishes according to electromagnetic fields (Einstein, 1915f). From a letter by Einstein to Hilbert on

222 For

a detailed analysis, see Norton (1984). here use Einstein’s notation and we wrote down the right hand side as in Einstein (1915e, p. 782). We note that .Rim should not be confused with today’s notation of the Ricci tensor. 224 Nowadays, we denote the Christoffel symbol by .┌ τ and use superscripts for the coordinates, νμ see Fliesßach (2016, p. 57). 225 “[T]ransformation of a determinant equal to 1” (Kox et al., 1997, p. 103). 226 For the Zurich notebook, see Einstein (2007). The November tensor appears for the first time on page 22R, see Janssen et al. (2007, pp. 606,645). See also footnote 186. 227 He denoted the Ricci tensor by .G . im 223 We

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1 Introduction

November 18, 1915, we learn that the difficulty for Einstein was not to find generally covariant equations, but rather to find them such that they reduce to the classical equations in the Newtonian limit:228 Die Schwierigkeit bestand nicht darin allgemein kovariante Gleichungen für die .gμν zu finden; denn dies gelingt leicht mit Hilfe des Riemann’schen Tensors. Sondern schwer war es, zu erkennen, dass diese Gleichungen eine Verallgemeinerung, und zwar eine einfache und natürliche Verallgemeinerung des Newton’schen Gesetzes bilden. (Schulmann et al., 1998b, Doc. 148)

We saw in the Entwurf theory that he tried to find an expression for .┌μν in Eq. (1.10) such that both requirements are satisfied. On the same day when he wrote this letter, Einstein used the preceding two papers to predict the correct orbit of Mercury (Einstein, 1915c). By using two approximations, Einstein concluded:229 Die Rechnung liefert für den Planeten Merkur ein Vorschreiten des Perihels um .43'' in hundert Jahren, während die Astronomen .45'' ± 5'' als unerklärten Rest zwischen Beobachtungen und NEWTONSCHER Theorie angeben. Dies bedeutet volle Übereinstimmung. (Einstein, 1915c, p. 839)

It was this paper, where Einstein also predicted that the light deflection is twice as strong as in Einstein (1911a).230 While Mercury’s perihelion advance had already been observed at that time (Verrier, 1859), the correct light deflection was only observed in 1919 (Dyson et al., 1920). Both events proved Einstein’s theory, and especially due to the observation of the light deflection, Einstein became an international celebrity, see Earman and Glymour (1980) for a historical discussion.231

228 “The

difficulty was not in finding generally covariant equations for the .gμν ’s; for this is easily achieved with the aid of Riemann’s tensor. Rather, it was hard to recognize that these equations are a generalization, that is, a simple and natural generalization of Newton’s law” (Schulmann et al., 1998a, Doc. 148). 229 Capitalization suppressed. “The calculation yields, for the planet Mercury, a perihelion advance of .43'' per century, while the astronomers assign .45'' ±5'' per century as the unexplained difference between observations and the Newtonian theory. This theory therefore agrees completely with the observations” (Kox et al., 1997, Doc. 24). 230 See footnote 133. For a historical discussion of this paper, see Earman and Janssen (1993). 231 Nowadays, we speak of four confirmations of predictions of general relativity. In addition to the exact measurement of Mercury’s perihelion advance and light deflection, the gravitational redshift was observed in 1960 (Pound & Rebka, 1960). For a detailed historical discussion of the experiments on the redshift at that time, see Hentschel (1996). Recently in 2015, the Laser Interferometer Gravitational Wave Observatory (LIGO) observed gravitational waves predicted by Einstein’s general relativity (Abbott et al., 2016). For this observation, Weiss, Thorne, and Barish were awarded the Nobel Prize in 2017 (The Nobel Committee for Physics, 2017b,a). Previous to this direct observation, gravitational waves were observed indirectly in 1978 (Hulse & Taylor, 1975; Taylor et al., 1979). For this detection, Hulse and Taylor were awarded the Nobel Prize in 1993 (Nobel Media, 2021b). Einstein published on gravitational waves several times (Einstein, 1916b, 1918; Einstein et al., 1938; Einstein & Rosen, 1937; Einstein, 1915b, 1916a). However, he also changed his mind on the existence of gravitational waves several times. For a detailed

1.3 Einstein’s Education and His Way to General Relativity with a Special. . .

47

Although Einstein made the remarkable calculation on Mercury’s perihelion advance, he saw himself as “schlechter Rechner”232 who preferred to work “vorstellungsmäßig”233 as written down in a diary of the student Rudolf Jakob Humm who visited Einstein in Berlin in 1917 (Seelig, 1954, p. 188), see also Rowe (2020). In accordance with this statement, Einstein had several assistants over the years who apparently made calculations for him. For instance, his assistant Walther Mayer234 who followed Einstein to Princeton in 1933 frequently performed mathematical computations for Einstein and even was nicknamed “Einstein’s calculator” (Topper, 2013, p. 137). Similar remarks can be found in Einstein’s publications as in Einstein (1923b) or Einstein (1925b), where Jakob Grommer235 made a “direkte Ausrechnung”236 (Einstein, 1923b, p. 364) or supported Einstein “bei allen rechnerischen Untersuchungen auf dem Gebiete der allgemeinen Relativitätstheorie in den letzten Jahren”237 (Einstein, 1925b, p. 419). Mayer and Grommer were not the only mathematical assistants of Einstein. Other assistants who worked for him and who had an especially strong mathematical background were Ludwig Hopf,238 Emil Nohel,239 Hermann Muentz,240 Banesh Hoffmann,241 Valentine Bargmann,242

historical discussion, see generally Buchwald (2020) and Renn (2020) therein as well as Kennefick (2007), Kennefick (2014), Cattani and De Maria (1993), and Franklin (2010). 232 “Bad calculator.” Translation by the author. 233 “Imagination-like.” Translation by the author. 234 Mayer worked with Einstein in 1930–1934 (Calaprice et al., 2015, p. 145). Before Einstein moved to Princeton, he insisted on an appointment for Mayer (Pais, 1982, pp. 492–493). Einstein and Mayer published together several papers (Einstein & Mayer, 1930, 1931b,b, 1932a,b, 1933b,c,a). 235 Grommer studied mathematics in Göttingen and obtained his Ph.D. with David Hilbert in 1913 (Hilbert, 1935, p. 433). He was considered as an outstanding mathematician (Pais, 1982, pp. 487– 488). Grommer worked together with Einstein from 1917 to 1928 (Calaprice et al., 2015, p. 145). They published two papers together (Einstein & Grommer, 1923, 1927). 236 “[D]irect calculation” (Kormos Buchwald et al., 2015a, p. 168). 237 “[W]ith all calculations in the area of general relativity in the last few years” (Kormos Buchwald et al., 2018b, p. 44). 238 Hopf obtained his Ph.D. with Sommerfeld in 1909 and worked in 1910 with Einstein publishing two papers (Einstein & Hopf, 1910b,a), see Pais (1982, p. 485) and Calaprice et al. (2015, p. 145). 239 Nohel studied mathematics and obtained his Ph.D. in 1912 or 1913 and worked in 1911–1912 with Einstein, see Pais (1982, pp. 485–486) and Calaprice et al. (2015, p. 145). 240 Muentz worked with Einstein on distant parallelism in 1928–1929 and later became professor of mathematics (Pais, 1982, pp. 491–492). Even though they did not publish together, Einstein mentioned Muentz in Einstein (1929e,a). 241 Hoffmann worked on projective relativity with Veblen from 1929 and obtained his Ph.D. in 1932 before becoming an assistant of Einstein from 1935–1937 (Pais, 1982, pp. 495–496). They published one paper together with Infeld (Einstein et al., 1938). 242 Bargmann became professor for mathematical physics and worked in 1937–1944 with Einstein (Pais, 1982, p. 496). They published two papers together (Einstein et al., 1941; Einstein & Bargmann, 1944).

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1 Introduction

Ernst Straus,243 and John Kemeny.244 By comparing the full list of Einstein’s assistants in Calaprice et al. (2015, pp. 145–146), we see that Einstein, in his Princeton years, had assistants mainly of mathematical education. According to Michelmore (1962), Einstein is reported to have complained on mathematics in Princeton over and over again, saying: If only I knew more mathematics[.] [. . . ] I’m like a man struggling to climb a mountain without being able to reach its peak.

It is safe to assume that many of Einstein’s mathematical assistants had to make calculations for Einstein as it was the case with Mayer. In fact, we will encounter in Sect. 4.2.2 an example where his assistant Peter Bergmann was supposed to make a calculation that took several months. We will also learn that he had to doublecheck his result with other assistants, in our case with Leopold Infeld.245 Nevertheless, in contrast to his own statements, the paper on Mercury’s perihelion advance shows that Einstein in fact was a good calculator. This was also acknowledged by Hilbert in a short letter to Einstein on November 19, 1915:246 [H]erzlichste Gratulation zu der Ueberwältigung der Perihelbewegung. Wenn ich so rasch rechnen könnte, wie Sie, müsste bei meinen Gleich[un]g[en] entsprechend das Elektron kapituliren und zugleich das Wasserstoffatom sein Entschuldigungszettel aufzeigen, warum es nicht strahlt. (Schulmann et al., 1998b, Doc. 149)

By the analysis of research notes, we will support our statement that Einstein was a good calculator who frequently used clever and sophisticated tricks.247 Let us now come back to Einstein’s fourth and final November paper (Einstein, 1915b) from November 25, 1915. In this paper, Einstein dropped the restriction of the vanishing of the trace of the energy momentum tensor and derived the final field equations Gim

.

243 Straus

⎛ ⎞ 1 = −κ Tim − gim T , 2

(1.19)

worked in 1944–1948 with Einstein (Calaprice et al., 2015, p. 146), and they published two papers (Einstein & Straus, 1945, 1946). In 1948, Straus obtained his Ph.D. and later became professor of mathematics. 244 Kemeny was a mathematician and worked with Einstein in 1948–1949 although he had recommended Einstein not to choose him as the subject was “as far from my specialty in mathematics as you can get” (Pais, 1982, p. 497). 245 Infeld was Einstein’s assistant in 1936–1941 (Calaprice et al., 2015, pp. 145). For a short biography, see Calaprice et al. (2015, pp. 137–138). 246 “[C]ongratulations on conquering perihelion motion. If I could calculate as rapidly as you, in my equations the electron would correspondingly have to capitulate, and simultaneously the hydrogen atom would have to produce its note of apology about why it does not radiate” (Schulmann et al., 1998a, Doc. 149). 247 See, for instance, Sect. 4.3.7.

1.3 Einstein’s Education and His Way to General Relativity with a Special. . .

49

where it is Gim = −

Σ∂

.

l

{

im l

∂xl

} +

Σ ⎧ il ⎫ { mρ } lρ

ρ

l

+

Σ∂ l

{ } il l

∂xm

−

Σ ⎧ im ⎫ ⎧ ρl ⎫ lρ

ρ

l

.

(1.20) Nowadays, we call .Gim the Ricci tensor which is usually denoted by .Rim and which is the contracted Riemann tensor .R lilm .248 Einstein did not use this notation, instead, he used the notation .Rim for the November tensor from Eq. (1.16).249 Using Einstein’s notation, it is ⎛ ⎞ 1 im i im Tim − gim T = κT , (1.21) .g Gim = Gi =: G = −κg 2 where G is nowadays called the Ricci scalar R. It follows that 1 Gim − gim G = −κTim , 2

.

(1.22)

which is the form of Einstein’s field equations that is commonly known. The left hand side of Eq. 1.22 is nowadays called the Einstein tensor and usually denoted by .Gim . At the same time, Hilbert was working on the field equations as well. On November 20, 1915, he submitted a paper that was published on March 31, 1916 (Hilbert, 1915). The published version contains the same field equations. However, it turned out that these equations were not the original equations from November 20 but had been revised, see Corry et al. (1997), Sauer (1999), and Sauer (2005b); Rowe (2001). Thus, Einstein was the first one who published the field equations on the theory of general relativity. In 1916, he published a comprehensive article on the new theory (Einstein, 1916a).250 We already quoted from this paper at the end of Sect. 1.3.4.3 when showing that Einstein’s view on mathematics and on Minkowski’s work changed over the years of developing his theory of general relativity. In fact, Einstein acknowledged not only Minkowski’s work but also the works and contributions by Gauss, Riemann, Christoffel, Ricci, Levi-Civita, and his friend Grossmann, showing

248 See

Fliesßach (2016, pp. 96–97) and Sect. 3.1. also footnotes 223 and 227. 250 Even before Einstein submitted his review paper on March 20, 1916, Karl Schwarzschild found the first exact solution of the field equations, first for the case of a mass point (Schwarzschild, 1916b) and then for a spherically symmetric mass distribution (Schwarzschild, 1916a). 249 See

50

1 Introduction

the significance of mathematical concepts for Einstein in 1916. The quote continues as follows:251 Die für die allgemeine Relativitätstheorie nötigen mathematischen Hilfsmittel lagen fertig bereit in dem „absoluten Differentialkalkül“, welcher auf den Forschungen von Gauss, Riemann und Christoffel über nichteuklidische Mannigfaltigkeiten ruht und von Ricci und Levi-Civita in ein System gebracht [. . . ] wurde. [. . . ] Endlich sei an dieser Stelle dankbar meines Freundes, des Mathematikers Grossmann, gedacht, der mir durch seine Hilfe nicht nur das Studium der einschlägigen mathematischen Literatur ersparte, sondern mich auch beim Suchen nach den Feldgleichungen der Gravitation untersützte. (Einstein, 1916a, p. 769)

For a long time, this preface, and with it Einstein’s praise of mathematics, was not received in the English speaking world, since the first page of Einstein’s review paper was not translated in the most common English source for Einstein’s article, see Dickenstein (2009), Lorentz et al. (1952, pp. 109–164), and Kox et al. (1997, Doc. 30). A discussion of Einstein’s review paper can be found in Sauer (2005a). He emphasized the significance of Einstein’s theory over the past decades: In essence, Einstein’s general theory of relativity of 1916 remains today the accepted theory of the gravitational field. (Sauer, 2005a, p. 821)

We finally finish this section with two quotes by Einstein from later years where he spoke about physical theories, in particular about his general theory of relativity, and the role of mathematics playing in the developing of those theories. In June 1933 when Einstein gave the Herbert Spencer Lecture at Oxford “on the method of theoretical physics” (Holton, 1968, p. 650),252 Einstein spoke with respect to the interaction between experience, physics and mathematics:253 251 “The mathematical tools that are necessary for general relativity were readily available in the “absolute differential calculus,” which is based upon the research on non-Euclidean manifolds by Gauss, Riemann, and Christoffel, and which has been systematized by Ricci and Levi-Civita [. . . ]. [. . . ] Finally, I want to acknowledge gratefully my friend, the mathematician Grossmann, whose help not only saved me the effort of studying the pertinent mathematical literature, but who also helped me in my search for the field equations of gravitation” (Schulmann et al., 1998c, p. 146). 252 For the first time, Einstein spoke in English, reading a translation from his German version (Fölsing, 1994, p. 758). 253 The German text was “Nach unserer bisherigen Erfahrung sind wir nämlich zum Vertrauen berechtigt, daß die Natur die Realisierung des mathematisch denkbar Einfachsten ist. Durch rein mathematische Konstruktion vermögen wir nach meiner Überzeugung diejenigen Begriffe und dijenige gesetzliche Verknüpfungen zwischen ihnen zu finden, die den Schlüssel für das Verstehen der Naturerscheinungen liefern. Die brauchbaren mathematischen Begriffe können durch Erfahrung wohl nahe gelegt, aber keinesfalls aus ihr abgeleitet werden. Erfahrung bleibt natürlich das einzige Kriterium der Brauchbarkeit einer mathematischen Konstruktion für die Physik. Das eigentlich schöpferische Prinzip liegt aber in der Mathematik. In einem gewissen Sinn halte ich es also für wahr, daß dem reinen Denken das Erfassen des Wirklichen möglich sei, wie es die Alten geträumt haben. Um dieses Vertrauen zu rechtfertigen, muß ich mich notwendig mathematischer Begriffe bedienen. Die physikalische Welt wird dargestellt durch ein vierdimensionales Kontinuum. Nehme ich in diesem eine Riemannsche Metrik an und frage nach den einfachsten Gesetzen, denen eine

1.4 Einstein’s Unified Field Theory Program

51

Our experience hitherto justifies us in believing that nature is the realisation of the simplest conceivable mathematical ideas. I am convinced that we can discover by means of purely mathematical constructions the concepts and the laws connecting them with each other, which furnish the key to the understanding of natural phenomena. Experience may suggest the appropriate mathematical concepts, but they most certainly cannot be deduced from it. Experience remains, of course, the sole criterion of the physical utility of a mathematical construction. But the creative principle resides in mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed. In order to justify this confidence, I am compelled to make use of a mathematical conception. The physical world is represented as a four-dimensional continuum. If I assume a Riemannian metric in it and ask what are the simplest laws which such a metric system can satisfy, I arrive at the relativist[ic] theory of gravitation in empty space. If in that space I assume a vector-field or an anti-symmetrical tensor-field which can be inferred from it, and ask what are the simplest laws which such a field can satisfy, I arrive at [. . . ] Maxwell’s equations for empty space. (Einstein, 1934, pp. 36–37)

Einstein’s opinion that nature needs to be described by the simplest possible mathematical laws did not change for the rest of his life. In his autobiographical notes, he wrote in 1949 about the equations of general relativity:254 Eine noch so umfangreiche Sammlung empirischer Fakten kann nicht zur Aufstellung so verwickelter Gleichungen führen. Eine Theorie kann an der Erfahrung geprüft werden, aber es gibt keinen Weg von der Erfahrung zur Aufstellung einer Theorie. Gleichungen von solcher Kompliziertheit wie die Gleichungen des Gravitationsfeldes können nur dadurch gefunden werden, dass eine logische einfache mathematische Bedingung gefunden wird, welche die Gleichungen völlig oder nahezu determiniert. (Einstein, 1949b, p. 88)

1.4 Einstein’s Unified Field Theory Program After presenting his general theory of relativity in 1915 and 1916 (Einstein, 1915a, 1916a), Einstein tried to find a unified field theory in which the theory of gravitation and the theory of electromagnetism should be unified. In this section, we will briefly outline Einstein’s efforts on this unification with a special focus on his fivedimensional approaches.255 Based on our analysis of the subsequent chapters, we

solche Metrik genügen kann, so gelange ich zu der relativistischen Gravitationstheorie des leeren Raumes. Nehme ich in diesem Raume ein Vektorfeld an, beziehungsweise das aus demselben abzuleitende antisymmetrische Tensorfeld und frage nach den einfachsten Gesetzen, denen ein solches Feld genügen kann, so komme ich auf die Maxwellschen Gleichungen des leeren Raumes” (Einstein, 1953b, pp. 153–154). 254 “No ever so inclusive collection of empirical facts can ever lead to the setting up of such complicated equations. A theory can be tested by experience, but there is no way from experience to the setting up of a theory. Equations of such complexity as are the equations of the gravitational field can be found only through the discovery of a logically simple mathematical condition which determines the equations completely or [at least] almost completely” (Einstein, 1949a, p. 89). 255 For further literature on Einstein’s unified field theory program, we refer to Sauer (2014), Sauer (2006), Van Dongen (2010), Goenner (2004), Goenner (2014), Vizgin (1994), and Goldstein and Ritter (2003).

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1 Introduction

will discuss Einstein’s 1938 program on the five-dimensional generalized Kaluza theory more comprehensively in Sect. 1.5. A first approach to the unified field theory was made by Hermann Weyl in 1918 (Weyl, 1918) who addressed the “inconsistency” that in Riemannian geometry the direction of two vectors at points, which are not infinitesimally close to each other, is not comparable, in contrast to the length of the vectors (Weyl, 1918, p. 466). As shown in Sauer (2014, p. 290,291), Einstein’s reaction to Weyl’s suggestions was very ambivalent, but he picked up Weyl’s idea in 1921, even though with reserved statements about the physical meaning (Einstein, 1921). In 1919, Theodor Kaluza, at the time Privatdozent in Königsberg,256 sent a manuscript to Einstein with a new approach, introducing a fifth dimension to the four dimensions of general relativity in order to include electromagnetism as an effect of the fifth dimension (Wünsch, 2009, p. 328). Initially, Einstein had again ambivalent opinions on Kaluza’s idea and did not help him publish these ideas (Kormos Buchwald et al., 2004, Doc. 40, Doc. 48). However, in 1921, Einstein changed his opinion, mentioning that he prefers Kaluza’s approach to Weyl’s (Kormos Buchwald et al., 2009, Doc. 270). Eventually, Einstein did submit Kaluza’s paper to the Prussian Academy for publication (Kormos Buchwald et al., 2009, Doc. 318). In fact, after Kaluza’s publication (Kaluza, 1921), Einstein used the idea of the five-dimensional space in many of his own publications (Wünsch, 2009; Sauer, 2014). At this point, we recall that the idea of a five-dimensional theory was not entirely new as we already saw in Sect. 1.3.4.6. The Finnish physicist Gunnar Nordström postulated already in 1914 a five-dimensional theory to unify the electromagnetic and gravitational fields (Nordström, 1914c,d, 1915). At this time, Einstein was well aware of at least some of Nordström’s contributions to a gravitation theory.257 In 1923, Einstein published on Kaluza’s theory together with Grommer showing that258 die KALUZA’sche Theorie keine von den .gμν allein abhängige zentralsymmetrische Lösung besitzt, welche als (singularitätsfreies) Elektron gedeutet werden könnte. (Einstein & Grommer, 1923, p. 5)

We will see in the following that such a singularity-free particle solution played an important role in Einstein’s search for a unified field theory. Until 1927 when he came back to the five-dimensional approach, Einstein considered further approaches to unified field theory. One approach was initiated by Eddington trying to generalize Weyl’s approach by putting the concept of affine connection into the focus as the fundamental mathematical quantity instead of the

256 For

a comprehensive biography of Kaluza, see Wünsch (2000). a more comprehensive account of this time period and these contributions, see Norton (1992) and Isaksson (1985). See also footnote 217 for the discussion whether Einstein knew Nordström’s five-dimensional approach. 258 “Thus it is proven that Kaluza’s theory possesses no centrally symmetric solution dependent on the .gμν ’s alone that could be interpreted as a (singularity-free) electron” (Kormos Buchwald et al., 2012b, p. 33). 257 For

1.4 Einstein’s Unified Field Theory Program

53

metric, see Eddington (1921) and Eddington (1923, pp. 213–240) as well as Sauer (2014, p. 293). Einstein published on this approach in Einstein (1923f), Einstein (1923a), Einstein (1923e), and Einstein (1923d) deriving new field equations.259 However, he concluded:260 Ein singularitätsfreies Elektron liefern diese Gleichungen nicht. (Einstein, 1923e, p. 140)

Einstein also pursued his own approaches as in Einstein (1923b), where he considered quantum phenomena, or in Einstein (1925b), where he used both the affine connection and the metric as fundamental mathematical quantities (Sauer, 2014, pp. 293–294).261 In the latter, he concluded with posing the question:262 Die nächste Frage ist nun die, ob die hier entwickelte Theorie die Existenz singularitätsfreier zentralsymmetrischer elektrischer Massen begreiflich erscheinen läßt. (Einstein, 1925b, p. 419)

While he then in 1927 came back to an approach from 1919 where he worked on a geometric interpretation (Einstein, 1919, 1927a),263 he also came back to Kaluza’s approach. In 1926, the Swedish physicist Oscar Klein, coming from a quantum theoretical point of view, improved Kaluza’s five-dimensional theory (Klein, 1926a). Only 1 year later, Einstein published his own results in two brief communications (Einstein, 1927b,c) generalizing Kaluza’s theory. His results, however, were very similar to Klein’s (Sauer, 2014, p. 295).264 In the following years 1928–1931, Einstein then considered the so-called Fernparallelismus265 approach where a tetrad field is used to define both a curvature-free connection and a metric tensor field. For a detailed discussion of this approach and the role of extensive collaborations with colleagues as Lanczos, Muentz, Grommer, and Weitzenböck, see Sauer (2006).266 Einstein published several times on this approach both alone as in Einstein (1928b), Einstein (1928a), Einstein (1929d), Einstein (1929e), Einstein (1929a), Einstein (1930a), and Einstein (1930b) and

259 Hilbert pointed out in two lectures from 1923 that Einstein ended up with equations that Hilbert already formulated in 1915/1916 (Hilbert, 1924, p. 2). For a detailed discussion, see Majer and Sauer (2005). 260 “These equations do not yield a singularity-free electron” (Kormos Buchwald et al., 2015b, p. 51). 261 It is this paper Einstein (1925b) where Einstein for the first time used the term “unified field theory” in a title, see Sauer (2014, p. 294). 262 “Now, the next question is whether the theory developed here makes the existence of singularity-free, centrally symmetric electric masses understandable” (Kormos Buchwald et al., 2018b, p. 44). 263 See Sauer (2014, p. 295). 264 See also Einstein’s own comment in the supplement of his second note (Einstein, 1927c). Peruzzi and Rocci recently presented a review of these two communications and illustrated the differences between Klein’s and Einstein’s results (Peruzzi & Rocci, 2018). 265 Distant parallelism. 266 Einstein frequently mentioned his assistants in his publications, see Sauer (2006).

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together with his assistant Mayer in Einstein and Mayer (1930) and Einstein and Mayer (1931b). Only a few months later than the last publication on distant parallelism, Einstein again came back to the five-dimensional approach, apparently inspired by his distant parallelism approach (Sauer, 2006, p. 434). In Einstein and Mayer (1931a) and Einstein and Mayer (1932a), Einstein and Mayer assumed a four-dimensional spacetime and attached a five-dimensional space to each point of the four-dimensional space-time. One of the main differences to Kaluza’s theory was that the new formulation is based on a four-dimensional space-time such that it fits to the physical experience that “die Welt unserer Erfahrung dem Anscheine nach vierdimensional ist”267 (Einstein & Mayer, 1931a, p. 542). By this assumption, they succeeded to derive field equations compatible with general relativity (Einstein & Mayer, 1931a, pp. 553–554) and generalized them in Einstein and Mayer (1932a). During the politically difficult times in 1933 and Einstein’s emigration to the United States,268 they then turned to semivectors and spinors (Einstein & Mayer, 1932b, 1933b,c,a) trying to incorporate quantum theory and the Dirac equation (Van Dongen, 2004, p. 229).269 The next publication on unified field theory is then the already mentioned article “on the generalization of Kaluza’s theory of electricity” from 1938 co-authored by his assistant Peter Bergmann270 (Einstein & Bergmann, 1938).271 Very similar to his motivation from 1931, the goal of this generalization was to “ascribe physical reality to the fifth dimension” (Einstein & Bergmann, 1938, p. 683). In a short remark after presenting Kaluza’s original theory, Einstein and Bergmann pointed out one of the main difficulties to Kaluza’s theory:272 Many fruitless efforts to find a field representation of matter free from singularities based on this [Kaluza’s] theory have convinced us, however that such a solution does not exist. We tried to find a rigorous solution of the gravitational equations, free from singularities, by taking into account the electromagnetic field. We thought that a solution of a rotationalsymmetrical character could, perhaps, represent an elementary particle. Our investigation was based on the theory of “bridges” (Einstein and Rosen, Phys. Rev., 48: 73 (1935)[)]. We convinced ourselves, however, that no solution of this character exists. (Einstein & Bergmann, 1938, p. 688)

267 “[B]y all appearance, the world is four-dimensional according to our experience.” Translation by the author. 268 Einstein became citizen of the USA in 1940, see Calaprice et al. (2015). As already mentioned in footnote 234, Mayer followed Einstein to Princeton upon Einstein’s request in 1933. However, the collaboration ended shortly afterward in 1934, see Van Dongen (2004, pp. 227–228). 269 The politically difficult times are also reflected in Einstein and Mayer’s publication history, see Sauer (2014, pp. 298–299) and Van Dongen (2004, pp. 224–229). 270 We will give a short biography of Peter Bergmann in Sect. 1.6. 271 For a detailed discussion, see Chap. 3. 272 The first sentence of this quote can be found embedded in the text on page 688, while the following sentences are quoted from a footnote. The reference they referred to is Einstein and Rosen (1935).

1.4 Einstein’s Unified Field Theory Program

55

We conclude that Einstein investigated Kaluza’s theory by imposing a rotational symmetry in order to find particle solutions free from singularity. As stated in Van Dongen (2002, pp. 192–197), it is safe to assume that Einstein also tried to find such solutions for his new field equations derived in Einstein and Bergmann (1938, p. 695).273 Simultaneously to this publication, Einstein composed the Washington Manuscript. This manuscript basically contains the same theory and field equations, and however, it has a different internal structure that Einstein preferred over the structure of the publication, see Sect. 3.4. Although Einstein was initially very euphoric about his new theory,274 he again encountered difficulties as he concluded in his follow-up publication with his assistants Bergmann and Valentine Bargmann275 from 1941: It seems impossible to describe particles by non-singular solutions of the field equations. (Einstein et al., 1941, pp. 224–225)

We see that this statement is very similar to the statement above on Kaluza’s theory. However, Einstein did not explain how he came to this conclusion. Instead, he again chose a slightly uncertain formulation (“It seems impossible”)276 (Einstein et al., 1941, p. 224). They also concluded that the new field equations derived in Einstein et al. (1941) could not explain the differences in strength between electromagnetic and gravitational fields. Finally in 1943, Einstein together with Wolfgang Pauli proved:277 that the field equations of the relativistic gravitational theory and of its five-dimensional generalization do not admit any non-singular stationary solution which represents a field of non-vanishing total mass or charge. (Einstein & Pauli, 1943, p. 131)

This was a generalization to higher dimensions of Einstein’s proof in Einstein (1941).278 For a detailed discussion of Einstein’s five-dimensional approach during the time period 1938–1943 and on his failure of finding particle solutions, we here refer to Van Dongen (2002) and Van Dongen (2010). In Chap. 4, we will come back to this subject and look at the manuscript pages where Einstein further investigated this theory. As a result of the proofs from 1941 to 1943, Einstein tried a new approach by coming back to a four-dimensional approach (Einstein & Bargmann, 1944; Einstein, 1944). There, they still held on to the concept of covariance of the field equations,

273 We

will come back to this in detail in Chap. 4. Sect. 3.3. 275 We will give a short biography of Valentine Bargmann and Peter Bergmann in Sect. 1.6. 276 Our italics. Above, Einstein chose the formulation “Many fruitless efforts [. . . ] have convinced us [. . . ]” (Einstein & Bergmann, 1938, p. 688). 277 Italics suppressed. 278 As pointed out in Van Dongen (2002, pp. 197–206), in 1983, regular, static, and stable solutions of the field equations corresponding to particles (so-called solitons) have been found in the framework of Kaluza’s theory (Sorkin, 1983; Gross & Perry, 1983). For further discussion of an additional restriction used in Einstein and Pauli’s proof, see also Giulini (2009, sec. 3.6.2). 274 See

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1 Introduction

but they dropped the assumption of the existence of a Riemann metric (Einstein & Bargmann, 1944, p. 1) by introducing so-called bivectors. Finally, in his last 10 years from 1945 to 1955, Einstein returned to the Riemann metric and assumed an asymmetric metric that led to a substantially more complex and less known mathematical framework (Sauer, 2014, p. 301). He published several papers on this approach (Einstein, 1948, 1950a,b; Einstein & Straus, 1946; Einstein & Einstein, 1954; Einstein & Kaufman, 1956); and (Einstein, 2014, Appendix II), while his last results were presented by his assistant Bruria Kaufman alone in July 1955 shortly after Einstein’s death (Kaufman, 1956).

1.5 Outline of Einstein’s 1938 Program Based on Subsequent Analysis In Sect. 1.3.4, we saw that Einstein worked on his theory of general relativity for several years and finally published the respective field equations in a short paper in 1915, while he wrote a comprehensive review article on the new theory some months later in 1916. As shown in Sect. 1.4, he then unsuccessfully worked on finding a unified field theory for the rest of his life. While he frequently published on his new approaches and on the respective field equations, he never fully accomplished the theory and hence never published a comprehensive account on it in contrast to 1916. In fact, it is a challenging task to expose the ideas lying behind Einstein’s theories. Questions remain unsolved of how Einstein tested the new field equations, how he included matter, or what further assumptions and requirements he made. One promising approach in order to shed light into this is the analysis of unpublished documents as correspondence and research notes. There, Einstein did not explicitly comment on his ideas, but we have calculations, equations, or formulas that resulted from certain assumptions. Based on our reconstruction and analysis of the subsequent chapters, we can make some preliminary statements on Einstein’s 1938 program that we will briefly outline in the present section.279 A detailed discussion on the following steps will be presented in the subsequent chapters. Let us start with Einstein and Bergmann’s publication from 1938.280

279 We

here note that (Van Dongen, 2010, pp. 139–156) also looked at Einstein and Bergmann’s publication (Einstein & Bergmann, 1938) as well as at some letters from their correspondence with a special focus on quantization. We here will specify and expand these ideas. 280 We will discuss Einstein and Bergmann’s publication (Einstein & Bergmann, 1938) comprehensively in Chap. 3 and we will also derive the respective field equations there.

1.5 Outline of Einstein’s 1938 Program Based on Subsequent Analysis

57

They generalized Kaluza’s five-dimensional theory by giving the fifth dimension reality and by assuming periodicity with respect to the fifth dimension .x 0 . Accordingly, for the five-dimensional metric components .γμν ,281 they assumed γμν (x 1 , x 2 , x 3 , x 4 , x 0 + nλ) = γμν (x 1 , x 2 , x 3 , x 4 , x 0 ),

.

(1.23)

where .λ does not depend on .x μ and .n ∈ Z.282 Einstein and Bergmann then chose a special coordinate system such that new metric components .gαβ with .g0α = 0 exist. Furthermore, they considered two types of coordinate transformations and defined tensors: With respect to so-called fourtransformations, tensors transform similarly to the four-dimensional theory, while they are invariant with respect to so-called cut-transformations. In particular, the new metric components .gab are tensors.283 Introducing .ϕm as the analogue of the electromagnetic potential in the fourdimensional case, Einstein and Bergmann derived two sets of field equations by a variational principle. The variation with respect to the metric components .g ik yields the ten equations ⎛

⎞ ⎛ ⎞ 1 1 1 1 m mn Rkl + Rlk − Rgkl + α2 2ϕkm ϕl − ϕmn ϕ gkl .α1 2 2 2 2 ⎞ ⎛ 1 mn rs rs + α3 − ∂0 g ∂0 gmn gkl + 2g ∂0 gkr ∂0 gsl − 2∂0 ∂0 gkl − ∂0 gkl g ∂0 grs 2 ⎛ ⎞ )2 1 ( mn + α4 gkl g ∂0 gmn + 2g mn ∂0 ∂0 gmn + 2∂0 gmn ∂0 g mn = 0, (1.24) 2 which we identify with the gravitational field equations. Similarly, the variation with respect to the electromagnetic potential .ϕm yields the four equations ⎛ ⎛ .

⎛ ⎞ ⎞√ s r − 4α2 ∇t ϕ st − g ks ∂0 ┌kr −g dx 0 = 0, α1 g km ∂0 ┌km

(1.25)

which we identify with the electromagnetic field equations. They also derived five identities for these field equations on the basis of Noether’s theorem. Here, .ϕmn = ∂n ϕm − ∂m ϕn is an antisymmetric tensor that corresponds to the electromagnetic field tensor in the four-dimensional theory. As .ϕm does not depend on .x 0 , the

281 Here,

Greek indices run from 1 to 4 and 0. Latin indices only run from 1 to 4. interpret .λ as a coordinate distance between two corresponding space points. 283 We recall that the components with index zero vanish, which is why we can use Latin indices here. 282 We

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integral taken over one period of .x 0 in the second set of the field equations does not vanish. Furthermore, the quantities .α1 to .α4 are constants, and .Rkl and R are the five-dimensional analogues to the Ricci tensor and Ricci scalar. The analysis of extensive archival documents that we will mainly discuss in Chap. 4 allows us to specify Einstein’s 1938 program beyond their publication. In contrast to Einstein’s theory of general relativity,284 the field equations in 1938 do not contain a term for the energy–momentum tensor. Instead, Einstein tried to consider matter as a solution of the field equations.285 In particular, he was looking for particle-like solutions. Usually, he considered time-like stationarity and spherical spatial symmetry resulting in equations that depend only on a radial coordinate .r = √ 1 2 (x ) + (x 2 )2 + (x 3 )2 . However, he also derived two kinds of equations belonging to an anisotropic and an isotropic case indicating that he considered further symmetries as well. Assuming spherical symmetry, he introduced specific metric components .χik with χik = αδik + β

.

xk xixk , χk4 = −iγ · , and χ44 = δ, 2 r r

(1.26)

where .α, .β, .δ, and .η are functions of r.286 The fourth coordinate .x 4 can be associated with the time coordinate. In addition, he also used the slightly modified metric components 1 ψik = χik − ηik χ , 2

.

(1.27)

where .ηik = diag(1, 1, 1, −1) is the Minkowski metric and .χ = χkk = 3α + β − δ is the trace. As it seems, Einstein not only tried to unify gravitation and electromagnetism but also considered quantization. This becomes noticeable by the wave-like components of the metric. These parts contain the additional factor ⎛ ⎞ 0 2π i νx 4 + xλ

e

.

(1.28)

describing a wave that is traveling along the fifth dimension, where .ν is its frequency. We see that the wavelength .λ is denoted accordingly to the coordinate distance .λ of the periodic fifth dimension.

284 See

Eq. 1.22. also Lehmkuhl (2019) and Lehmkuhl (2017). 286 It is .η = iγ . 285 See

1.5 Outline of Einstein’s 1938 Program Based on Subsequent Analysis

59

Einstein’s procedure in 1938 was to plug the metric components into the field equations or the identities in order to get algebraic equations. In fact, he derived287 .

−

4π f α (3α + 2β) λ ┌ ⎞┐ ⎛ 2 2 2π ' 2 η (−α + β + δ) + η 3α ' + β ' + δ ' − α − β + δ + λ r r r − 2α2 ϕ 4τ ;τ = 0

(1.29)

and called this equation Stromgleichung. As we will argue in the following chapters, this equation can be associated with the second set of field equations or with the fifth identity. We see that the functions .α, .β, .δ, and .η as well as their derivatives appear according to the metric components. Denoting the first line in Eq. (1.29) by .I4 as it apparently was done by Einstein yields I4 = κϕ 4τ ;τ

.

(1.30)

for a constant .κ. We see that it can directly be associated with Maxwell’s equation j 0 = cρ =

.

c 0k F ,k , 4π

(1.31)

where .Fik = ∂i Ak − ∂k Ai is the electromagnetic field tensor, .Ai the four-potential, and .j 0 = cρ the zeroth component of the four-current with the charge density .ρ.288 In fact, Einstein frequently also considered I4 ≈ −

.

4π f α(3α + 2β) ≈ −2α2 ϕ4τ,τ ≈ −Δϕ λ

(1.32)

as an approximation of the Stromgleichung where we associate .ϕ with the electrostatic potential according to Poisson’s equation in the four-dimensional theory. Plugging the metric components into the first set of field equations or into the first four identities, Einstein and his assistants apparently derived further algebraic equations ⎞ ┌ ┐ ⎛ ⎞ 1 ⎛ α3 3 α4 ⎞ 1 ' α3 2 − α4 − f rα + − + r β− rf + 2f η .−α +δ + 4 2 r 4 2 2 ⎛ α α4 ⎞ 3 rδ = 0, (1.33) + − + 4 2 '

287 It

'

⎛

seems as if Bergmann derived this equation. In fact, it took him several months. here note that the time coordinate corresponds to .x 0 in the four-dimensional case and to .x 4 in the five-dimensional case. 288 We

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1 Introduction

⎛ ⎞ 2 1 ' 1 ' 1 f α− f + f β + α3 η − f ' δ = 0, .2f α + (1.34) 2 r 2 2 ⎛ ⎞ ⎞ ⎛ 5 ' 6 1 1 2 f − f ' δ' f − f α ' − f ' β ' + α3 η' + . r 2 r 2 2 ┐ ┌ ⎛ ⎞ 1 + 2α3 − 6α4 − 2f 2 f + f '' α 2 ⎞ ⎛ ⎞ ⎛ 6 4 2 1 '' 2 ' 1 '' + 2 f − f − f − 2α4 f β − f η + 2α4 f − f δ = 0, r 2 r 2 r (1.35) '

and ⎞ ┐ α3 1 ' ⎛ α3 α4 ⎞ ' 1 ' ' 3 α − + rβ − rf η − α4 − f 2 r − 2 r 4 2 2 4 ┐ ┌ ⎛ ⎞ 1 ' α4 α3 δ r+ + − + 2 r 4 ⎞ ⎛ ⎞ ⎛ 1 3 α3 5 α4 2 ' 2 β − α4 − f − 2rff α + 2 + f − α3 − + 4 2 4 2 r ⎛ ⎞ 1 '' 5 ' 2 − rf + f + f η 2 r 2 ⎞ ⎛ α α4 3 δ=0 (1.36) + − + 4 2

┌ ⎛ .

calling them wave equations. These four equations were considered by Einstein on his manuscript pages. In particular, by the analysis of these research notes, we see that the particle solutions should be localized, a fact that he apparently tried to describe by using .δdistributions.289 Assuming that the particle is localized at .r = 0, the equations from above should be regular at the origin. In order to avoid singularities at the location of the particles, Einstein introduced a new variable .ρ 2 = a 2 + r 2 and rewrote all equations in terms of .ρ.290 At .r = ∞, the solutions should have a certain falloff. In particular, the potential .ϕ should behave like .r −1 in accordance with the four-dimensional case. Einstein imposed this condition by looking at power series expansions of the functions .α, .β, .δ, .η, and .ϕ in terms of .ρ −1 and demanding that certain coefficients in the power series expansion of .ϕ vanish, such that .ϕ ≈ ϕ1 ρ −1

289 See

Chap. 5. the notation .ρ stands for the variable replacing r that should not be confused with the charge density in the four-dimensional theory.

290 Here,

1.6 Einstein’s Princeton Assistants

61

is a good approximation. We will argue that it is this situation, where Einstein also considered projective geometry.291 Einstein and Bergmann discussed not only the power series expansion of .ϕ but also the respective starting powers in the power series expansions of the further functions such that the Stromgleichung and their approximations hold. In doing so, one important further requirement that needed to be satisfied was that the integral ⎛ .

⎛∞ I4 dV = 4π

I4 r 2 dr

(1.37)

0

converges for large r.292 By our analogy to the five-dimensional case, this implies a finite charge for the particles. In summary, by our analysis, we learn about the metric components that Einstein considered in order to transfer the differential equations from the publication into five algebraic equations.293 We also see how he further investigated these new equations by assuming space-like spherical symmetry, time-like stationarity, localized particles with finite charge, no singularities at the origin, a certain fall-off at infinity, and a connection between quantization and the periodic fifth dimension. However, the documents that we know do not allow a full and consistent interpretation. For instance, there is a contradiction on the nature of the function .η. We will learn about several documents that suggest that .η is a real-valued function, while it seems to be a pure imaginary quantity in other documents. Contradictions like this and further unknown assumptions exacerbate the full derivation of the algebraic equations starting from the field equations or the identities.

1.6 Einstein’s Princeton Assistants We already saw in Sects. 1.4 and 1.3.4.7 that Einstein’s assistant Walther Mayer accompanied Einstein to Princeton where they worked together until 1934 and published several papers.294 Einstein’s subsequent assistant was Nathan Rosen with whom he published four papers (Pais, 1982, pp. 494–495).295 When this 291 See

Chap. 2. a consequence, the powers appearing in .I4 needed to be chosen accordingly, where the approximation for .ϕ should also hold with respect to the Stromgleichung. 293 In fact, Einstein derived one further equation. However, this is a combination of two of the five equations. 294 In Sect. 1.3.4.7, we also mentioned several of Einstein’s assistants with mathematical background. 295 We already know Einstein et al. (1935) and Einstein and Rosen (1935). In addition, they published on gravitational waves (Einstein & Rosen, 1937) and on the two-body problem (Einstein & Rosen, 1936). 292 As

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1 Introduction

collaboration ended, Banesh Hoffmann joined Einstein in 1935, who obtained his Ph.D. in the field of projective relativity under the supervision of Veblen (Pais, 1982, pp. 495–496).296 He stayed with Einstein until 1937. In 1936, Leopold Infeld joined them. Together, they published on the problem of motion (Einstein et al., 1938). Einstein and Infeld then also published two further articles (Einstein & Infeld, 1940, 1949) on the same problem and one popular scientific book (Einstein & Infeld, 1938a) together. Infeld worked with Einstein in Princeton until 1941. In the same year when he became Einstein’s assistant (1936), Peter Gabriel Bergmann joined them as well. He stayed there until 1941 and published two articles on the five-dimensional approach, see Sect. 1.4. The first paper was only published by Einstein and Bergmann (Einstein & Bergmann, 1938), while the second paper was also co-authored by Valentine Bargmann (Einstein et al., 1941) who joined them in 1937. V. Bargmann297 stayed until 1944 and published one further article (Einstein & Bargmann, 1944) together with Einstein.298 After V. Bargmann, Einstein then worked with Ernst Straus (1944–1948), John Kemeny (1948–1949), Robert Kraichnan (1949–1950), and Bruria Kaufmann (1950–1955), see Sects. 1.3.4.7 and 1.4.299 In this section, we will give a short biography of P. Bergmann and V. Bargmann, the two assistants, that we will encounter the most in the following chapters.

1.6.1 Peter Gabriel Bergmann For a more comprehensive biography, we refer to Schmutzer (2003) and Salisbury (2012), which are also our two sources for this section. Peter Gabriel Bergmann started studying chemistry and physics in Dresden in 1931. After two semesters he moved to Freiburg specializing on theoretical physics and attending courses by Gustav Mie. After two further semesters, he moved to Berlin. Due to the difficult political situation, Bergmann fled from Germany soon after his move to Berlin and finally became Philipp Frank’s student in Prague, who was Einstein’s successor. There, he obtained his Ph.D. as a 21-year-old student in 1936 (Schmutzer, 2003, p. 411). In fact, by an extant correspondence, we learn that Bergmann’s mother initially asked Einstein whether he might become Bergmann’s supervisor (AEA 6-220), who then recommended Pauli in Zurich (AEA 6-221).300 As mentioned above, Bergmann came to Princeton becoming Einstein’s assistant

296 We

will come back to Hoffmann and his role in projective relativity in Sect. 2.1.2. See also footnote 241. 297 For a better distinction between the names Bergmann and Bargmann, we frequently write down the initials of their first names as well. 298 See also Sect. 1.4. 299 For a full list of Einstein’s assistants and collaborators, see Calaprice et al. (2015, pp. 145–146). 300 See also Sect. 2.1.3.

1.7 Structure of the Book

63

after obtaining his Ph.D. in 1936 where he stayed until 1941. He then got brief appointments at Black Mountain College, Lehigh University, Columbia University, and the Woods Hole Oceanographic Institute before joining Syracuse University in 1947 where he stayed until his retirement in 1982 (Salisbury, 2012). In 1942, he wrote the well-known book “Introduction to the Theory of Relativity” with a foreword by Einstein (Bergmann, 1942). We will see in Sect. 3.4.2 that after Einstein’s death in 1955, Bergmann helped to organize his estate by corresponding with Einstein’s secretary Helen Dukas.

1.6.2 Valentine Bargmann For two biographies of Valentine Bargmann, we here refer to Klauder (1999) and Lieb et al. (1976) which are also the two sources for this section. Bargmann studied in Berlin from 1925 to 1933. He then moved to Zurich where he obtained his Ph.D. in physics under the supervision of Gregor Wentzel. In 1937, he then emigrated to the United States. By the help of John von Neumann, he was soon accepted in Princeton and became Einstein’s assistant in the same year. Einstein and Bargmann worked together until 1944 and published two papers (Einstein et al., 1941; Einstein & Bargmann, 1944).301 Even after his collaboration with Einstein, he stayed in Princeton becoming officially a lecturer in physics in 1946. Except for a short stay at the University of Pittsburgh, he stayed there ever since. In addition to his contributions to both physics and mathematics, he was well known for his lectures.

1.7 Structure of the Book We will start with the question on Einstein’s mathematical knowledge. In both the Prague notebook and on the Princeton manuscripts, we found sketches on projective geometry that will be discussed in Chap. 2. The appearance of such sketches alone is intriguing as Einstein rarely drew any sophisticated sketch at all. The fact that the field was well known among mathematicians but not necessarily among physicists makes it particularly interesting as Einstein as theoretical physicist was evidently proficient in this subject. His mathematical skills in projective geometry will be assessed with respect to his own education that we already discussed in Sect. 1.3.1. Following this, we will discuss further sources where Einstein in one way or the other encountered projective geometry in his later years in Sect. 2.1. In doing so, we will also present the double page of the Prague notebook and the five most important manuscript pages with respect to projective geometry. We will then give a brief

301 See

also footnote 242.

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overview of the underlying theory in Sect. 2.2. The analysis starts in Sect. 2.3, where we comprehensively discuss several manuscript pages. We summarize our results briefly in Sect. 2.4. We will then analyze the double page of the Prague notebook and connect the respective sketches with the sketches on the manuscript pages in Sect. 2.5. We finish the chapter with a conclusion and a short conjecture on the purpose of Einstein’s ideas about projective geometry in Sect. 2.6. We were able to connect some parts of these manuscript pages with Einstein and Bergmann’s publication from 1938 (Einstein & Bergmann, 1938). In Chap. 3, we will analyze several documents such as an unpublished manuscript, further research notes, and the publication itself in order to investigate how Einstein explored this new theory. After briefly outlining the theory of general relativity and deriving the respective field equations in Sect. 3.1, we will discuss Einstein and Bergmann’s theory in Sect. 3.2. In Sect. 3.3, we look at Einstein’s own and positive opinion on the generalization of Kaluza’s theory that he expressed in several letters from around that time. By analyzing a correspondence between Einstein and Bergmann, we found the unpublished Washington manuscript that Einstein composed even before the publication was printed. The manuscript has essentially the same theoretical content as the publication and will be discussed in Sect. 3.4. There, we will see that these two works differ in their internal structure. This is an intriguing fact for the discussion of Einstein’s way of theorizing, especially as he frankly communicated his preference to the manuscript. We finish our considerations with respect to the Washington manuscript with some concluding remarks in Sect. 3.5. Thereafter, we will then be able to analyze several manuscript pages and connect them to either the publication from 1938 or the Washington manuscript in Sect. 3.6. We finish the chapter with a brief conclusion in Sect. 3.7. We will see that many of the mentioned manuscript pages contain not only considerations that are related to the publication or the Washington manuscript but also additional and idiosyncratic calculations. By a comprehensive study of all available manuscript pages, we identified a batch of sheets that contain such calculations. On these pages, which also include the pages with projective geometry, Einstein examined and further developed the generalized Kaluza theory. Extant correspondence between his assistants Peter Bergmann and Valentine Bargmann from 1938 to 1939 will help us to contextualize the manuscript pages. In Sect. 4.1, we will re-date the respective letters from 1938. By using the results of our analysis of both letters and manuscript pages, we will discuss Einstein’s calculations subsequent to the publication in Sect. 4.2 and look at Einstein’s further investigation of the new field equations from the publication. In Sect. 4.3, we will then date, contextualize, reconstruct, and comment on several further manuscript pages. In particular, we will finally date the manuscript pages with projective geometry accurately in Sect. 4.3.1. We will also discuss further calculations that allow us to draw conclusions on Einstein’s mathematical skills. We will finish this chapter with a short conclusion in Sect. 4.4. Another example of such skills is Einstein’s interest in the at that time relatively new field of .δ-functions. We found manuscript pages on this subject that are connected to Einstein’s correspondence in 1939. We will discuss the respective

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notes in Chap. 5, and we will refer to further working sheets on the same subject. In Sect. 5.1, we briefly introduce .δ-functions before analyzing three manuscript pages as well as corresponding letters from 1939 in Sects. 5.2.1 to 5.2.5. We will also briefly come back to an expression that we already encountered on some documents from 1938 in Sect. 5.2.6, and we will discuss further related sheets in Sect. 5.2.7. Again, we finish this chapter with a short conclusion in Sect. 5.3. We end in Chap. 6 with a short conclusion.

Chapter 2

Einstein and Projective Geometry

In this chapter, we will discuss research notes by Albert Einstein that can be identified with considerations on projective geometry and more precisely with involutions.1 These notes are interesting for several reasons. First, we found these notes in Einstein’s Prague notebook, on his Princeton manuscripts, and also on a separate manuscript page. The respective notes were written at different time periods. While a double page of the Prague notebook was written between 1912 and 1915, the notes on the Princeton manuscript pages can be dated to 1938. The third source might have been written between 1929 and 1932. The double page of the Prague notebook can be dated by the preceding and succeeding pages of the same notebook. On these pages, Einstein made several calculations on gravitational lensing. We already mentioned these pages in Sect. 1.1.2 and showed how the discovery of these notes rewrote the history of gravitational lensing. They show that Einstein considered this phenomenon twice between 1912 and 1915 and not only in 1936 when he published on it Einstein (1936), see also Sauer (2008); Renn et al. (1997); Renn and Sauer (2003a).2 The Princeton manuscripts with research notes on projective geometry can be dated by further calculations that we find on these pages. We were able to bring these calculations in connection with Einstein and Bergmann’s publication (Einstein and Bergmann 1938) and with an extant correspondence. This is the reason why we do not only discuss these pages in the present chapter, but also in Sects. 3.6 and 4.3.1, where we then finally can date the pages accurately.3 The third source was found by browsing through all unidentified pages containing research notes in Einstein’s

1 Some

previous results of this chapter have been published in Sauer and Schütz (2021). the Princeton manuscripts, there are also further pages from 1936 that contain calculations on gravitational lensing (Schütz 2017). By analyzing Einstein’s research notes from both 1912/15 and 1936, we showed how Einstein explored gravitational lensing along different pathways that we called the “space of implications” (Sauer and Schütz 2019). 3 For an accurate dating, we will also use the correspondence that we will re-date in Sect. 4.1. 2 On

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Schütz, Einstein at Work on Unified Field Theory, Einstein Studies 17, https://doi.org/10.1007/978-3-031-52127-0_2

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hand that were available in the Albert Einstein Archives and the database of the Einstein Papers Project. We can only date this page by a remark in the database. We summarize that we have another example where Einstein considered a subject at least twice in a time period of more than 20 years apart. Second, not only did Einstein consider projective geometry in these two different time periods, the sketches that we find on the respective pages are also very similar to each other. In fact, we find two sketches in the Prague notebook that are almost identical to two sketches on the manuscript page AEA 62-787r. We would only need to rotate one of these sketches by .90◦ in order to receive the other sketch that was drawn more than 20 years later. A similar situation appears between two further sketches in the Prague notebook and two sketches on the page AEA 62-789. One of these sketches can also be found on our third source AEA 124-446 where Einstein even used an equivalent notation. Although we have found no evidence that Einstein in Princeton recalled or even used his notes from 1912/15, the striking similarity between these sketches is fascinating. As we will see throughout the chapter, these similarities also helped us to reconstruct and understand Einstein’s research notes. For instance, two sketches in the Prague notebook stand for a transition from a projectivity to a hyperbolic involution. However, Einstein skipped one important intermediate step, namely the situation of an elliptic involution. This missing step made the interpretation of the sketch very difficult. However, on the manuscript page AEA 62-787r, Einstein drew the intermediate step that facilitated the interpretation of the transition on AEA 62-787r and also helped us to understand the sketches in the Prague notebook. To illustrate the transitions between the sketches, not only for the example above, we created computer-based animations by using the dynamic geometry software GeoGebra. We will refer to these animation when discussing the respective sketches. Third, the research notes on projective geometry and the way Einstein considered it are connected to a purely mathematical subject. While the underlying mathematical theory might seem to be elementary from a modern mathematical point of view, it is an intriguing fact that Einstein as a theoretical physicist of the first half of the twentieth century considered this mathematical subject. It is all the more interesting as he already came in contact with it during his student years at ETH as we learned in Sect. 1.3.2. By the analysis of his research notes, we will be able to assess Einstein’s knowledge in projective geometry. It is also an interesting coincidence that we find sketches very similar to Einstein’s sketches in Grossmann’s research notes from the time when he himself taught projective geometry at ETH. It directly reminds us of Grossmann showing Einstein the mathematical tools for his theory of general relativity, even though we were not able to find any strong evidence that Grossmann and Einstein communicated on projective geometry after their student years. Fourth, it is interesting to investigate whether Einstein used projective geometry for his own studies. By this thought, one is tempted to connect Einstein’s considerations on projective geometry with the theory of projective relativity. We will look at this; however, we did not find any evidence for a connection between these two subjects, although Einstein considered projective relativity already in 1931/32. Instead, we will conjecture that he considered projective geometry while elaborating

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his generalization of Kaluza’s theory in the context of power series expansions at infinity. We will come back to this after having discussed both the theory in Chap. 3 and the further development based on correspondence and research notes in Chap. 4. Even if our conjecture is wrong, it is interesting to see that Einstein either had not forgotten very specific statements in projective geometry or that he had relearned it in 1938. Fifth, Einstein very rarely used any sketches, graphical representations, or geometric constructions in his research notes. We frequently find the common sketch of a coordinate system or frame of reference on some of his sheets; however, we usually do not find any sophisticated or elaborated sketch on a physical or mathematical subject. Instead, most of Einstein’s research notes consist of algebraic calculations.4 Hence, the pages with sketches on projective geometry are a rarity in the sense that they contain geometric considerations and sketches in Einstein’s hand.5 We will start our analysis with briefly discussing sources where Einstein came in contact with projective geometry after his studies at ETH.6 While we already mentioned the Princeton manuscripts and the Prague notebook generally in Sect. 1.1, we will present the respective pages on projective geometry in Sect. 2.1.1. As another source, we will discuss an entry of one of his diary in Sect. 2.1.2 showing that he considered Veblen and Hoffmann’s works on projective relativity. Finally, we briefly discuss Einstein and Bergmann’s correspondence in Sect. 2.1.3, where Bergmann wrote about the point at infinity, a notion that is very well known in projective geometry. As we will analyze research notes on projective geometry, we will first briefly discuss the theory in Sect. 2.2. We will then analyze the research notes in Sect. 2.3. We will start with the manuscript page AEA 62-785r (Sect. 2.3.1) and then discuss the pages AEA 62-787r (Sect. 2.3.2) and AEA 62-789 (Sect. 2.3.3) as well as the back page AEA 62-789r (Sect. 2.3.4).7 In Sect. 2.3.5, we will analyze the manuscript page AEA 124-446 before pointing to some further research notes in Sect. 2.3.6. In Sect. 2.4, we summarize our results and briefly point out similarities between the individual manuscript pages. We will then finally discuss the double page of the Prague notebook in Sect. 2.5 and connect Einstein’s considerations with his sketches appearing on the Princeton manuscripts. We will finish our analysis with a short conclusion in Sect. 2.6 and a conjecture on the purpose behind Einstein’s sketches in Sect. 2.6.1.

4 We

will encounter this fact in Sect. 4.3 when analyzing further manuscript pages in the context of Einstein’s elaboration of his new theory. 5 The sketches in Einstein’s notes are also interesting in light of Einstein’s reply on the question about his “internal or mental images” (Hadamard 1945, 140), where he answered that “the psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be ‘voluntarily’ reproduced and combined” (Einstein 1945, 142). 6 In Sect. 1.3, we discussed his studies at school and university with a special focus on geometry. 7 We will come back to these pages and analyze further entries in Sects. 3.6 and 4.3.

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2.1 Sources In this section, we briefly present our main sources containing the sketches on projective geometry. We will also briefly mention a diary where Einstein might have come in contact with projective geometry in the context of projective relativity. We note that we already discussed Einstein’s mathematical education with focus on geometry comprehensively in Sect. 1.3. In particular, he attended several courses on projective geometry at ETH as we showed in Sect. 1.3.2.

2.1.1 The Prague Notebook and the Princeton Manuscripts The double page of the Prague notebook is shown in Fig. 2.1. As already explained in Sect. 1.1.2, the pages before and after the double page date from 1912 and 1915,

Fig. 2.1 The double page of Einstein’s Prague notebook containing sketches on projective geometry. Reproduced from AEA 3-013, pages 49–50] and Klein et al. (1993, 588). This figure has been published in Sauer and Schütz (2021). © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama

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respectively, and contain calculations on gravitational lensing8 such that we can assume that the double page was written between 1912 and 1915. It starts on the left page with a sketch and the text “eau glyceriné” both written in pencil and not in Einstein’s hand. We also find four letters below this expression that probably belong to the first sketch. They overlap with a sketch in Einstein’s hand that was written in ink. This is also the first sketch on projective geometry. It is followed by two further sketches and two short text passages. Einstein’s writing continues on the right hand side with four x’s that we will interpret as a sketch as well. The considerations on projective geometry then end with a fifth sketch that was canceled out and a calculation consisting of five lines next to it. We find two further sketches, one carrying the letters H and M and probably drawn in Einstein’s hand as well as the expression .sin(ωt) · sin(ωt + α).9 As shown in Sauer and Schütz (2021), the entries not related to projective geometry might be related to Brownian motion and to magnetism, respectively. It is possible that they were written in context of Einstein’s discussions with Jean Perrin and Paul Langevin who met in Paris in March 1913, see Klein et al. (1993, Doc. 437). If so, the sketches on projective geometry possibly were written at the same time. In the following, we will only be concerned about the notes on projective geometry. The four pages from the Princeton manuscript are shown in Fig. 2.2. The top left page is AEA 62-785r, which contains a calculation consisting of two lines that are not directly related to projective geometry. We were able to link these calculations to the context of Einstein and Bergmann’s publication and to an extant correspondence. We will discuss these lines in Sect. 4.3.1.1. It is followed by a second calculation, two sketches, and one sketch that is crossed out. These notes are related to projective geometry and will be discussed in the following. The front of the page AEA 62-785 will be discussed in Sect. 3.6.4. We will see that it contains calculations on tensor densities and can be connected to a letter from Einstein to Bergmann from July 23, 1938. The top right page in Fig. 2.2 is AEA 62-787r that starts with a more comprehensive calculation that is not directly related to projective geometry. This calculation, however, was also written in the context of Einstein’s further considerations on his new theory as we will show in Sect. 4.3.1.2. The same holds for the front of the page AEA 62-787. At the second half of the manuscript page, we find four sketches in the context of projective geometry. The two pages at the bottom of Fig. 2.2 are the pages AEA 62-789 (bottom right) and its back page AEA 62-789r (bottom left). We will claim that AEA 62-789 is the continuation of AEA 62-789r. There, Einstein started with calculations that are connected to the further development of his new theory that we will analyze in Sect. 4.3.1.3. He then drew five sketches on projective geometry. At the top of AEA 62-789, we find three further sketches on projective geometry, where two of

8 See

Sauer (2008); Renn et al. (1997); Renn and Sauer (2003a). multiplication dot could also be a plus sign or an equal sign. It reminds us of considerations as in Klein et al. (1993, 359). 9 The

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Fig. 2.2 Manuscript pages AEA 62-785r, 62-787r, 62-789, and 62-789r. This figure has been published in Sauer and Schütz (2021). © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama

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Fig. 2.3 Manuscript page AEA 124-446. Parts of this page have been published in Sauer and Schütz (2021). © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama

them were crossed out. The rest of the page contains two passages of texts and calculations that can be connected to the so-called Washington manuscript that was written by Einstein in the process of publishing their article on the generalization of Kaluza’s theory. We will discuss it in Sect. 3.6.1. By a detailed analysis of these pages, we will be able in Sect. 4.3.1.4 to state that the manuscript pages were written between mid-June and mid-July 1938. Our third source is shown in Fig. 2.3. It contains three sketches and a short calculation consisting of two lines. We will not discuss the sketch at the top of the page as it is very generic. The two further sketches as well as the short calculation, however, contain considerations on projective geometry. The manuscript page is undated. It seems as if the back of the page shows an undated letter. According to the database of the Albert Einstein Archives, the manuscript page comes from the Stern family. It was dated tentatively between April 1929 and September 1932 according to Einstein’s stay in his summer house in Caputh. By our analysis, we will also refer to the back of the letter AEA 6-250, which is shown in Fig. 2.4. As we will argue in Sect. 4.1, Einstein wrote this letter between June 22 and June 29, 1938. It contains an equation that is, up to a sign mistake, equivalent to an equation appearing on the manuscript page AEA 62-789r as we will see in Sect. 4.3.1.3. It furthermore contains two sketches that might be connected to projective geometry.

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Fig. 2.4 The back of the letter AEA 6-250, which has the archival number AEA 6-250.1. © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama

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2.1.2 Projective Relativity and the Pasadena Diary In Sect. 1.4, we briefly discussed Einstein’s considerations on unified field theory. We showed that in 1927, he published on the five-dimensional approach in two brief communications (Einstein 1927b,c). Approximately at this time, Oswald Veblen started thinking about a new approach, namely using projective geometry to describe the four dimensions with five coordinates (Veblen 1933, 3–5). As professor for mathematics in Princeton, he would later be one of the persons being responsible that Einstein joined the Institute for Advanced Study in 1933 (Feffer 1998, 493,494).10 In 1930, Veblen published his ideas together with Banesh Hoffmann, who earned his Ph.D. under the supervision of Veblen in 1932 and was from 1935 to 1937 Einstein’s colleague at the Institute for Advanced Study (Pais 1982, 495). Based on Klein’s work, they tried to solve the unification problem using projective geometry (Veblen and Hoffmann 1930). Veblen published the basic ideas behind this approach in a more comprehensive matter in Veblen (1933). In 1932, Veblen sent his manuscript to Einstein asking him for criticism (AEA 23-159). When Einstein published his papers in 1931 and 1932 together with his assistant Mayer (Einstein and Mayer 1931a, 1932a),11 the results mainly yielded the same results as the approaches by Veblen and Hoffmann in 1930 (Veblen 1933, 5), which is why Veblen planned to send their previous publication to Einstein as a reprint together with “other related papers” in October 1931 (AEA 23-156). Einstein responded that he is “sehr neugierig auf Eure Lösung” (AEA 23-157).12 In 1931, Schouten and van Dantzig showed that Einstein and Mayer’s theory can be replaced by their own approach using projective geometry, very similar to Veblen and Hoffmann’s approach (Schouten and Dantzig 1931, 1932). They published more papers regarding this subject13 as well as Wolfgang Pauli did (Pauli 1933b,a). In 1942, Peter Bergmann, who worked from 1936 to 1941 with Einstein (Pais 1982, 496), finally showed that the theories using projective geometry and Kaluza’s theory are equivalent, see Bergmann (1942, 268–270) and Pauli (1963, 277). From December 1931 to February 1932, Einstein was in the U.S.A. from which a diary is extant (AEA 5-258 and 29-136 to 29-141). The first entry is from December 3, when Einstein was already aboard the ship. In this entry, pictured in Fig. 2.5, he wrote:14 10 In

fact, in 1927, Veblen already offered Einstein a position in Princeton via telegram (AEA 23149). Einstein declined this offer arguing that he already has been getting old in Berlin and is going to stay there. He ends the letter comparing himself to an old flowering plant: “You should not move an old flowering plant since it then withers prematurely” (“einen alten Blumenstock soll man nicht mehr versetzen, weil er sonst vorzeitig eingeht.”) (AEA 23-150). Translation by the author. See also (Kormos Buchwald et al. 2021, Docs. 52, 55) for the two documents. 11 See Sect. 1.4. 12 “Very excited about your solution.” Translation by the author. 13 For an overview, see Veblen (1933, 5) or Goenner (2004, 71,72). 14 “Yesterday and today, I put together the proof that all Euclidean constructions can be done with the compass alone. Tania Ehrenfest didn’t want to believe me. With her, I read in Leiden Veblen

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Fig. 2.5 First entry in Einstein’s Pasadena diary from 1931/32 (AEA 29-136, p. 3). © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama

Gestern und heute konstatierte ich mir den Beweis dafür zusammen, dass alle euklidischen Konstruktionen mit dem Zirkel allein ausgeführt werden können. Tania Ehrenfest hatte es nicht glauben wollen. Mit ihr habe ich in Leiden Veblen und Hofmanns Theorie der Elektrizität (fünfdimensional projektive Theorie) gelesen. Kam mir recht künstlich vor. Was die wohl zu unserer Theorie sagen werden? (AEA 29-136, p. 2–3)

This entry refers directly to Veblen and Hoffmann’s publication in 1930 that was sent to Einstein by Veblen in 1931. He also mentioned our theory that probably refers to Einstein and Mayer’s first publication on unified field theory in 1931. Although it is tempting to connect any appearance of projective geometry in Einstein’s notes with projective relativity, we did not find any evidence for Einstein considering projective relativity while drawing the sketches on projective geometry. We will come back to this in Sect. 2.6.1.

2.1.3 Bergmann Correspondence and the Point at Infinity After his work with Mayer, Einstein’s next publication on unified field theory was in 1938, together with Bergmann and again based on Kaluza’s idea (Einstein and Bergmann 1938), see Sects. 1.4 and 1.5. From this time period, a correspondence is extant between Einstein and Bergmann. This correspondence more or less began in 1933 when Bergmann’s mother asked Einstein whether her son could become a Ph.D. student under Einstein’s supervision (AEA 6-220). Einstein recommended Pauli in Zurich as a supervisor (AEA 6-221), but eventually, Bergmann went to Prague and obtained his Ph.D. with Einstein’s successor Philipp Frank in 1936, see Sect. 1.6. In the same year, the conversation between Einstein and Bergmann really took off, when Bergmann applied for a collaboration with Einstein (AEA 6-222). Even after this collaboration ended in 1941 (Pais 1982, 496), the correspondence

and Hoffmann’s theory of electricity (five-dimensional projective theory). Seemed quite artificial to me. What they will say about our theory?” Translation by the author.

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continued—although less frequently. After Einstein’s death, Bergmann continued corresponding with Einstein’s former secretary Helen Dukas.15 In their correspondence, Bergmann wrote to Einstein on July 17, 1938 about a problem that was pointed out by Einstein in AEA 6-242.16 In this context, he wrote:17 Sie sehen also, daß man mit einer Entwicklung um den unendlich fernen Punkt starten kann. (AEA 6-264).

The notion unendlich ferner Punkt is well known in the context of projective geometry and was also introduced in the first lecture of projective geometry at ETH in summer 1897 (Grossmann 1897, 1). As we found manuscript pages with sketches on projective geometry from the same time period, it seems as if Bergmann discussed projective geometry with Einstein in this correspondence. However, the notion of unendlich ferner Punkt in Bergmann’s letter was probably used in the context of power series expansions at infinity rather than in the direct context of projective geometry. It is possible that Einstein and Bergmann tried to solve their problem related to power series expansions by using projective geometry. This would explain why Bergmann then used the term unendlich ferner Punkt in the context of power series expansions. We will come back to the relation between projective geometry and power series expansions in Sect. 2.6.1.

2.2 Theoretical Background In the following, we will first give an overview of important literature on projective geometry. We will then introduce the notations that are used in the subsequent analysis. At the end of the section, we will briefly discuss Pascal’s theorem and involutions on a conic in order to facilitate the reconstruction of Einstein’s sketches.

2.2.1 Literature In this section, we present some literature dealing with the theory of projective geometry, which is necessary for the subsequent analysis of Einstein’s notes. Furthermore, many of the books mentioned in this section were written during Einstein’s lifetime. This makes them of special interest to us as Einstein might have known some of the books.

15 We

will look at some letters of this correspondence in Sect. 3.4. will re-date the letters in Sect. 4.1. 17 “Hence, you see that one can start with an expansion around the point at infinity.” Translation by the author. 16 We

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A good overview of the theory of projective geometry is given in The Real Projective Plane from 1949 and on Projective Geometry from 1964 by Coxeter, while we will frequently refer to the respective second editions (Coxeter 1961) and (Coxeter 1987). Another book that is important for our analysis is (Enriques 1915) that especially deals with statements about the radical axis and involutions on a line (Enriques 1915, 124–128), see also Coxeter and Greitzer (1967, 27–36) and Johnson (2007, 32). For a first introduction to projective geometry, we refer to Duschek (1963, 111–116) as well as Blaschke (1954, 11–75), which is the third edition, published in 1954. It discusses, among others, the variations of Pascal’s theorem (p. 70) and involutions on a line (p. 75). Blaschke has also published many books in “Grundlehren der mathematischen Wissenschaften” about Einstein’s theory of relativity during Einstein’s lifetime. Pascal’s theorem and its different versions on hexagons, quadrangles, and triangles are discussed in Grassmann (1909, 95–101) from 1909.18 The visualizations of the different versions look very similar to Einstein’s sketches on AEA 62-787r and in his notebook (Klein et al. 1993, Appendix A), page 49. However, they are not identical. In addition, we will show that Einstein drew the sketches in the context of projectivities and involutions. Nevertheless, the book is a good example of how to study the different versions of Pascal’s theorem in 1909 (Grassmann 1909, 95–101). In Samuel (1988, 42–43), real and imaginary circles are discussed. The figure on page 43 has similarities to Einstein’s sketch on AEA 62-785r. Imaginary elements are also discussed in Locher-Ernst (1940, 164–194). In Kowol (2009) on projective geometry and Cayley–Klein geometry, the imaginary theory is discussed from a modern point of view. Interestingly, the figure in Kowol (2009, 214) is almost the same one that Einstein drew on AEA 62-785r. Pottmann and Wallner refer to Einstein’s construction as right angle involution and call the points where the circles intersect Laguerre Points in Pottmann and Wallner (2010, 25,26). The subject is also discussed in Whicher (1985, 154), in which a figure on page 154 looks a bit similar to Einstein’s imaginary case on AEA 62-785r. We found the same discussion with sketches similar to Einstein’s sketches also in older books from the beginning of the twentieth century in, for example, (Emch 1905, 13–14), (Hatton 1913, 101), (Askwith 1917, 82), or (Dowling 1917, 131). In Seidenberg (1962), two different approaches for studying projective geometry are explained in a comprehensive way: as a continuation of Euclidean geometry and as an independent discipline (Seidenberg 1962, 1). In Kaplansky’s “Linear Algebra and Geometry” (Kaplansky 1969), we find an introduction of projective geometry (Kaplansky 1969, 97) followed by an analytic discussion of cross ratios and harmonic conjugates (Kaplansky 1969, 105–113). At the end of the

18 The author of Grassmann (1909) is not Hermann Günther Graßmann (1809–1877) who is known

for his Ausdehnungslehre, but his son Hermann Ernst Graßmann (1857–1922), see also Engel (1922) and Gutzmer1 (1922) for the full name.

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chapter, he discussed conics and higher-dimensional spaces (Kaplansky 1969, 113– 129). For another analytic discussion of projective geometry, which is helpful for understanding involutions on a line and Einstein’s first equation on AEA 62-785r, see Faulkner (1960, 35). In general, there are two methods to connect the projective plane with the affine case. The first method is to interpret the affine plane as part of the projective plane minus one line. This approach is illustrated in Coxeter (1987, chapter 12.8) and Coxeter (1961, chapter 8). The second method is to take the affine or Euclidean space as given and to identify the lines and planes that pass through a given point O as the points and lines of the projective plane, respectively. This approach is comprehensively illustrated in Brannan et al. (2012, 137–151). In a special book that treats “the most famous problems of mathematics,” we found many theorems whose visualizations look like some of Einstein’s sketches (Dörrie 1958, 261–284). Although the book is from 1958, the first edition was published in 1932. This book is of interest to us because of Einstein’s predilection for amusing problems in mathematics as addressed in “Mathematische Mußestunden” by Hermann Schubert (1898) as already mentioned in Sect. 1.1.2, see also Rowe (2011). In addition to the mentioned literature possibly read by Einstein that shows similarities to Einstein’s sketches, further literature that Einstein might have read are (Juel 1934) and the well-known (Veblen and Young 1910, 1918), which presented the field from an axiomatic point of view. As mentioned before, Einstein clearly could also have read Fiedler’s textbook on descriptive geometry (Fiedler 1871) or (Fiedler 1883, 1885, 1888) and his lectures as well as Grossmann’s lectures as Grossmann (1907). We finally note that relations from special and general relativity can be visualized by using projective geometry. Such representations can be found frequently in the literature as in Liebscher (1999).19 Although Einstein’s sketches sometimes look similar to those visualizations, we did not find any evidence that Einstein considered such representations. We looked through the above-mentioned texts with a special interest in sources that might have been used by Einstein himself. However, we have not been able to identify any specific work that might have affected him or that might have served as a source for Einstein. We found many books that partly treat the same subjects as on Einstein’s research notes. However, at one point or the other, they differ to his considerations.

19 See

especially (Liebscher 1999, chapter 8).

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2.2.2 Notations We use capital italic letters to describe points and lower case italic letters for lines. If a line l passes through the points P and Q, we write .l = P Q. If two distinct lines o and p pass through a point S, we write .S = o · p. Given three real numbers .x1 , x2 , and .x3 , not all zero, the point .(x) is an ordered set of three numbers .(x) = (x1 : x2 : x3 ), where for a nonzero real number .λ, it is .(λx1 : λx2 : λx3 ) = (x1 : x2 : x3 ). The three numbers .x1 , x2 , and .x3 are called the homogeneous coordinates of the point .(x). A line .[X] = [X1 : X2 : X3 ] is defined analogously. We write .X ^ Y for a projectivity between two points X and Y , see also Coxeter (1987, 9–10). An involution is a projectivity of period two, which means that the projectivity interchanges pairs of points (Coxeter 1987, 41,45). As an involution is determined by any two of its pairs (Coxeter 1987, 45), we denote the involution ' ' ' ' ' .AA B ^ A AB by .(AA )(BB ) for any two pairs of the involution. Given four collinear points .A = (a), .B = (b), .C = (a + b), and .D = (a + μb), the parameter .μ is called cross ratio, in formula .(AB, DC) = μ (Coxeter 1961, 194). Let D be the harmonic conjugate of C with respect to A and B; we write .H (AB, CD), see also Coxeter (1987, 22).

2.2.3 Pascal Line and Involution on a Conic In this section, we formulate Pascal’s theorem that allows us to define projectivities on conics. This concept will lead us immediately to involutions on conics. Given a hexagon ABCDEF inscribed in a conic, Pascal’s theorem says that the three pairs of opposite sides meet in three collinear points .P = AB · DE, .Q = BC · EF , and .R = AF · CD (Coxeter 1961, 103). The line joining the three collinear points is called Pascal line. Let us now investigate how to construct the corresponding point of a given projectivity on a conic. For that purpose, we define the axis of the projectivity ' ' ' ' ' ' .ABC ^ A B C as the Pascal line of the hexagon .AB CA BC inscribed in a conic. By Pascal’s theorem, this line is defined by two of the three points .P = AB ' · A' B, ' ' ' ' .Q = B C · BC , and .R = AC · CA (Coxeter 1961, 105). Given seven points .A, B, C, X, A' , B ' , C ' on a conic and a projectivity .ABC ^ ' A B ' C ' with its axis o, then we locate the point .X' with .X ^ X' as the second intersection of the line AT with the conic, where .T = A' X · o (Coxeter 1961, 106). Since the only possibility for X being invariant is that X lies on o, it follows that the projectivity is elliptic (has no invariant points) if o is an exterior line, parabolic (has one invariant point) if o is a tangent, or hyperbolic (has two invariant points) if o is an interior line (Coxeter 1961, 105,106). In particular, the intersections between o and the conic determine the invariant points.

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Fig. 2.6 Involution on a conic. Inspired by Coxeter (1961, 109)

Finally, we can consider involutions on a conic. As an involution is determined by any two of its pairs (Coxeter 1987, 45), we can regard the involution .(AA' )(BB ' ) as the special case of the projectivity .ABC ^ A' B ' C ' for .C = B ' and .C ' = B (Coxeter 1961, 108). Thus, the hexagon .AB ' CA' BC ' becomes the quadrangle .AB ' A' B, and the axis o of the involution becomes .(AB ' · A' B)(AB · A' B ' ), which is one of its diagonal sides. By Coxeter (1987, 75), its pole is the opposite vertex .O = AA' ·BB ' . This point is called the center of the involution (Coxeter 1961, 108). Let us visualize this situation by an example. In Fig. 2.6a, the quadrangle ' ' .AB A B is inscribed in a conic without a sense around the conic. The axis o, colored in red, is defined by the points .P = AB ' · A' B and .Q = AB · A' B ' , which are two diagonal points of the quadrangle (green lines). By Coxeter (1987, 75), the third diagonal point O (blue lines) is the pole of the axis o. The same situation is visualized in (b), but now the quadrangle is inscribed anti-clockwisely, which is why we get an elliptic involution, while in (a) it is a hyperbolic involution. In both situations, moving the point B to the point .A' and .B ' to A gives us the result that the tangents in the points A and .A' (orange lines) meet each other on the axis (Coxeter 1961, 108,109). Clearly, the same holds for the tangents in B and .B ' . With the above considerations, we get the theorem that given an involution on a conic, the lines joining two corresponding points are concurrent and meet in O. Conversely, given a conic, then any pencil through a given point O determines an involution on the conic (Coxeter 1961, 108, 109). It follows immediately that, if existing, the points of contacts of the tangents of the conic through O are the invariant points of the involution (Coxeter 1961, 109). Let us now consider the situation apart from projectivities and involutions, similar to Grassmann (1909, 95–101). In Fig. 2.7a, a hexagon .AA' BC ' CB ' is given. Note that this hexagon is different to the situation from above. The opposite sides ' ' ' ' ' ' .AA to .CC , .A B to .CB , and .AB to .BC are in same color, respectively. By

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Fig. 2.7 From the inscribed hexagon to the quadrangle and to the triangle

Pascal’s theorem, their three intersections are collinear (red line). We now look what happens if we move .C ' to B and C to .B ' . Then, as visualized in (b), we get the quadrangle .AA' BB ' with opposite sides .AA' to .BB ' , .AB ' to .A' B, and AB to .A' B ' . If b and .b' denote the tangents in B and .B ' , respectively, we see that the points ' ' ' ' ' ' .AA · BB , .AB · b, and .A B · b are collinear. Going one step further, we let .A ' ' move to A and get the triangle .ABB , where A is the opposite vertex of .BB , .B ' of AB, and B of .AB ' , see Fig. 2.7c. We get the result that the three sides and the tangents in the vertices opposite to the side meet each other in three collinear points (Grassmann 1909, 100,101). What do these results imply in terms of involutions? The situation in Fig. 2.7b is the same as in Fig. 2.6a. This means that in Fig. 2.6, the center .O = AA' · BB ' of the involution is collinear with .AB ' · b and .A' B · b' (which are not be drawn in Fig. 2.6). Going one step further to the triangle in Fig. 2.7c, the point A represents, in terms of involutions, an invariant point since it is a tangent through the center O.

2.3 Analysis of Princeton Manuscripts and Further Working Sheets As we already mentioned in Sect. 2.1.1, the double page of the Prague notebook, the four Princeton manuscript pages AEA 62-785r, 62-787r, 62-789, and 62-789r as well as another working sheet AEA 124-446 contain very similar sketches on projective geometry.20 In this section, we will discuss them. As we will see, most of the sketches can be interpreted in the context of involutions. This especially holds for the sketches appearing in the Prague notebook that we also find in a very similar form on the Princeton manuscripts. This fact is of special interest to us as the double page in the Prague notebook was written by Einstein at the time when he finished his general theory of relativity.

20 These

pages are shown in Figs. 2.1, 2.2, and 2.3.

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Parts of the following results have been published in Sauer and Schütz (2021), which focuses on the individual sketches of the Prague notebook and links them with sketches from other sources. In contrast to this, we will here first discuss the manuscript pages individually as entity and discuss the Prague notebook afterward. We will also briefly discuss further considerations on the manuscript pages and refer to subsequent sections where we will again look at the manuscript pages from another point of view. For a best possible impartial interpretation, we will discuss the manuscript pages in the order they appear in reel 62. We will start in Sect. 2.3.1 with the page AEA 62-785r containing both calculations and sketches on involutions. Einstein started with the general equation of an involution and determined the center of the involution to be the corresponding point to the point at infinity. He then derived the constant of the involution and considered the cases of two double points that can be real or imaginary corresponding to the cases of a hyperbolic and elliptic involution, respectively. In the sketches, Einstein constructed pairs of an involution by using the constant of the involution derived in his short calculation as well as the radical axis. We can identify the two cases of a hyperbolic and elliptic involution in his sketches as well as the case of the parabolic involution that only has one real invariant point.21 In addition to the considerations on involutions, we find three equations at the top of the page that can be connected to Einstein and Bergmann’s correspondence from 1938. On the page AEA 62-787r, we find four sketches all showing a conic that will be discussed in Sect. 2.3.2. The sketch on the right depicts Pascal’s theorem in two different versions. In the first version, all opposite sides are not parallel, while in the second case, one pair of opposite sides is parallel, which causes the Pascal line to be parallel to these lines as well. The fact that Einstein drew three additional lines in the center of the conic shows that he drew this sketch in order to consider a projectivity. He then made the transition to a quadrangle inscribed in a conic that stands for a elliptic involution. The transition was made by letting to points falling together. By moving one point such that the sense of direction changes, Einstein created the case of a hyperbolic and parabolic involution, respectively. This page contains calculations related to Einstein and Bergmann’s correspondence from 1938 as well. The page AEA 62-789 will be discussed in Sect. 2.3.3. It contains three sketches on projective geometry. The first sketch is very generic and has many different interpretations. Together with the second sketch, however, it seems as if Einstein constructed pairs of an involution on a line by using the complete quadrangle. In the first case, he constructed the two double points of a hyperbolic involution as well as one further pair. In the second case, he constructed two pairs of an involution. The third sketch is very generic, and we will give several possible interpretations. The rest of the manuscript page contains calculations related to Einstein and Bergmann’s

21 We

already note here that a parabolic involution is no real involution in a strict sense.

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publication from 1938. More precisely, it is connected to Einstein’s Washington manuscript rather than to the publication.22 The back page AEA 62-789r contains five more sketches on projective geometry and will be discussed in Sect. 2.3.4. One of these sketches shows most of the lines of a complete quadrangle. Two further sketches probably are derivations of this sketch in the case that two lines become parallel. For the remaining two sketches, we cannot provide an explicit interpretation. The top of the page contains calculations that again can be connected to Einstein and Bergmann’s correspondence. As the page AEA 62-789r contains the sketches at the bottom of the page, while AEA 62789 contains the sketches at the top, we argue that AEA 62-789r was written before AEA 62-789. In Sect. 2.4, we will give a short summary and point out the similarities between the four manuscript pages. We will then look at the working sheet AEA 124-446 in Sect. 2.3.5. There, one sketch is about the complete quadrangle interpreted as two perspectivities resulting in an involution. The other sketch and attendant calculations are about cross ratios. In Sect. 2.5, we finally discuss the double page of the Prague notebook. The first page shows a complete quadrangle. Due to the introduced notation, it becomes clear that Einstein considered two perspectivities resulting in a hyperbolic involution with two invariant points where one pair has been interchanged. Such a sketch with equivalent notation appears also on the manuscript page AEA 124-446. Next to this sketch in the Prague notebook, we find another sketch that has been crossed out. However, we can interpret it as the construction of pairs of an involution on a line. Thus, the two sketches appearing next to each other in the Prague notebook are very similar to the two sketches appearing next to each other on the page AEA 62-789. At the bottom of the left page, we find two sketches with conics. The first sketch shows Pascal’s theorem, while the second sketch is a derivation of Pascal’s theorem showing a hyperbolic involution. These two sketches also appear on the manuscript page AEA 62-787r. There, we find two more sketches showing intermediate or further steps of the transition. At the top of the right page, we find a very generic sketch. Introducing a specific notation, the sketch fits to Einstein’s calculation beneath the sketch. There, he considered the harmonic relation and derived that two double points of the involution separate each pair of the involution by the harmonic relation. The calculation can be connected to Einstein’s calculation on the manuscript page AEA 62-785r as in both situations, Einstein investigated the general equation of an involution. As Einstein also constructed pairs of an involution in the Prague notebook, he constructed them on AEA 62-785r as well. In fact, we will argue that Einstein considered hyperbolic involutions in all sketches of the Prague notebook and that this is also the reason why he crossed out the second sketch on the right page. As already mentioned in Sect. 2.1.1, the double page also contains considerations on different subjects probably from 1913 that will not be discussed here.

22 We

will come back to this in Sect. 3.6.1.

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For a better understanding, we will show many figures with own transcriptions of Einstein’s sketches if necessary. The purpose of these figures is not to make an accurate replica of the sketches but rather to support the understanding and interpretation. By our analysis, we will see that sketches appear on different manuscript pages that are very similar to each other. Furthermore, many sketches can be transformed into each other. For a better visualization and understanding of these transitions, we made several video sequences using the interactive geometry application GeoGebra. We will refer to them when appropriate.23

2.3.1 AEA 62-785r We will start our analysis with the manuscript page AEA 62-785r that is shown in Fig. 2.8. The front of the page AEA 62-785 will be discussed in Sect. 3.6.4. The back page is written in black ink and in Einstein’s hand. It starts with two short and one long equations at the top of the page. After a short break, it continues with the word Involution followed by equations and text. The paragraph ends with a short horizontal line. On the second part of the page, there are further calculations and three sketches. The first two equations are .η = rη and .β = r 2 β. Beneath them, we find the equation .

┌

┐ ┌ ⎛ α α3 3 α4 ⎞ 3 ┐ 3 − α4 − (ν + ϕ)2 rα + r − + r β 4 2 4 2 ⎛ ⎞ ⎛ α 1 2 ' α4 ⎞ 3 − r ϕ + 2rν + 2rϕ η + − + rδ = 0. 2 4 2

− α' + δ' +

(2.1)

The first two equations introduce the substitutions .η and .β frequently used on many manuscript pages as well as in a letter from Einstein and Bergmann’s correspondence.24 Equation (2.1) belongs to an equation from another sheet AEA 6259 belonging to the correspondence. We will discuss these equations and their meaning comprehensively in Chap. 4.25 We will then come back to the present manuscript page in Sect. 4.3.1.1 in order to discuss these equations and their relation to projective geometry in more detail.

23 The animations are accessible at https://doi.org/10.1007/s00407-020-00270-z as supplementary

material to Sauer and Schütz (2021) (visited on 06/10/2021). Sauer and Schütz (2022) discuss the usage of such animations for historical studies as well as for didactic representations. 24 See the letter AEA 6-245. 25 For Eq. (2.1), see Sect. 4.2.2.

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Fig. 2.8 Manuscript page AEA 62-785r. This page has been published in Sauer and Schütz (2021). © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama

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The next passage begins with the word Involution that indicates the subject of projective geometry. The next line reads axy + b (x + y) + c = 0.

.

(2.2)

This equation can be associated with an involution on a line. We argue that the context is as follows: We have given a line in the projective plane that can be defined by any two distinct points .P = (p) and .Q = (q). On this line, we can define any involution by three real numbers a, b, and c with the condition .b2 /= ac. We can express any point X on this line by .(xp + q) for a .x ∈ R ∪ {∞}. It follows that in this case, the corresponding point of X is .Y = (yp + q) with the condition in Eq. (2.2) for a .y ∈ R ∪ {∞}. Einstein obviously considered such an involution and a pair .x, y. In the next line, Einstein considered .x = ∞ and obtained ay + b = 0.

.

(2.3)

Indeed, by .x = ∞, it follows that .X = P = (p) and Eq. (2.3) follows. It does not seem as if Einstein considered homogeneous coordinates here. Instead, he probably thought of x as a nonzero real number, divided Eq. (2.2) by x and then considered the limit .x → ∞. We conclude that if x is the point at infinity, the corresponding point y needs to satisfy the condition in Eq. (2.3). Two cases emerge: Either y is also the point at infinity that makes this point becoming an invariant point or y is real-valued. In the latter case, y is the center of the involution as it is the corresponding point of the point at infinity. Einstein considered the second case as he wrote down: “Reeller Punkt. Als Anfang genommen. .b = 0.”26 Even though the German word Ursprung as origin instead of “Anfang” would fit better in this context, we argue that he chose y to be the origin, namely .y = 0. In this case, it immediately follows that .b = 0 by Eq. (2.3), written down by Einstein. Einstein’s next line reads xy + c = 0.

(2.4)

xy + ~ c=0

(2.5)

.

By .b = 0, Eq. (2.2) implies .

with ~ .c = c/a for .a /= 0. As c and ~ .c are constants, Einstein simply wrote down c instead of ~ .c implicitly changing the meaning of the constant c. Let us summarize Einstein’s procedure so far. He considered .x = ∞ and derived the condition in Eq. (2.3) for the corresponding point y by using the general equation

26 “Real

point. Taken as the beginning. .b = 0.”

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of an involution. In the case that the point at infinity is not an invariant point, y is real-valued (“Reeller Punkt”), and we get y=−

.

b a

(2.6)

for .a /= 0. Einstein then took this point as the origin (.y = 0) and obtained .b = 0. Speaking in homogeneous coordinates, it is then .X = P and .Y = Q using the notation from above. We can identify the points P as the point at infinity and Q as the origin of the line. These points are corresponding points of the involution described in Eq. (2.4), while we recall that Einstein changed the meaning of c. Any other pair .X = (xp + q) and .Y = (yp + q) of the involution on the line needs to satisfy the condition from Eq. (2.4) as well. This result can be associated directly with the constant of an involution with the corresponding point of the point at infinity being the center of the involution. Einstein then finished his considerations with the comment: “Für .x = y zwei Wurzeln. Können reel oder imaginär sein.”27 Einstein misspelled the German word “reell.” Assuming .x = y, we get the quadratic equation ax 2 + 2bx + c = 0

.

(2.7)

by Eq. (2.2). When considering Eq. (2.4) instead of Eq. (2.2), we derive x 2 + c = 0.

(2.8)

.

In both cases, the two solutions of the equations are the invariant points of the involution. If the discriminant .Δ1 = 4b2 − 4ac belonging to Eq. (2.7) or .Δ2 = −c belonging to Eq. (2.8) is positive, the solutions are real and the involution has two real invariant points (hyperbolic involution). In the case that the discriminant is negative, the solutions are imaginary and no real invariant point exists (elliptic involution). The case .Δ = 0 is excluded because of the condition .b2 /= ac. As Einstein did not write this condition down, we can associate the case .Δ = 0 with the parabolic involution where only one real invariant point exists.28 Einstein finished his first paragraph with drawing a short horizontal line. In the second paragraph, Einstein then started with writing down | |x + y1 a : b : c = || 1 x2 + y2

.

| 1|| 1|

| | |1 x y | : || 1 1 || 1 x2 y2

| | |x y x + y1 | |. : || 1 1 1 x2 y2 x2 + y2 |

(2.9)

= y two roots. Can be real or imaginary.” note at this point that in a strict sense, a parabolic involution is not an involution as it is not a one-to-one correspondence anymore. 27 “For .x 28 We

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Although the colon in Eq. (2.9) indicates projective coordinates, we argue that it works only as a symbol for separation of corresponding terms in this case. We state that Einstein considered | | | | | | | | | | |x1 + y1 1| | , b = |1 x1 y1 | , and c = |x1 y1 x1 + y1 | , .a = | (2.10) |x2 y2 x2 + y2 | |1 x2 y2 | |x2 + y2 1| where .xi , yi ∈ R ∪ {∞} are corresponding points for .i = {1, 2}. As an involution is determined by any two of its pairs, let .(xi p + q) be the corresponding points of .(yi p + q), respectively. Thus, the involution is determined, and we get the linear equation system axi yi + b (xi + yi ) + c = 0

(2.11)

a = (x1 + y1 ) − (x2 + y2 )

(2.12)

b = x2 y2 − x1 y1

(2.13)

c = x1 x2 y1 + x1 y1 y2 − x1 x2 y2 − x2 y1 y2 .

(2.14)

.

by Eq. (2.2). Choosing .

implies .

and .

Indeed, these are the values of the determinants considered by Einstein in Eq. (2.9). In fact, denoting the involution in terms of determinants is not unusual, see for example (Blaschke 1954, 74). This interpretation is consistent with Einstein’s second line where he considered the case “für .x = ∞” and derived29 | | |1 x1 y1 | | | |1 x2 y2 | x2 y2 − x1 y1 . (2.15) .y = − | | =+ |x1 + y1 1| x2 + y2 − x1 + y1 | | |x2 + y2 1| As on the first part of the manuscript page, he again considered the case .x = ∞ and used Eq. (2.3) in order to get y=−

.

b a

(2.16)

29 The last plus sign in the denominator of the expression on the right hand side probably should be a minus sign. It is possible that Einstein corrected this in his calculations in the process when he corrected the minus sign to a plus sign that appears after the second equal sign.

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Fig. 2.9 Reconstruction of Einstein’s first sketch on AEA 62-785r. Einstein did not draw the gray circles. The notations are changed. Einstein denoted the points by .xi , O, and .yi instead of using capital letters. He did not name the two intersections points .S1 , .S2 , and the line e

for .a /= 0. He then inserted the relations from Eq. (2.10). As above, Eq. (2.15) expresses the constant of the involution. Defining the center of the involution .Y = (yp + q) as the origin (.y = 0), we get x1 y1 = x2 y2 ,

.

(2.17)

the constant of the involution. With Eq. (2.15), Einstein’s calculations on this manuscript page end. On the right side of the page, next to the calculation of the second paragraph, the first sketch begins. The reconstruction of the sketch with all important lines is shown in Fig. 2.9. Einstein did not draw the gray circles. We argue that Einstein denoted the corresponding points .(xi p + q) and .(yi p + q) with .xi and .yi , respectively. To avoid multiple definition and stressing that in our usual notation .xi and .yi do not describe points, we denote the points with capital letters .Xi and .Yi . In his sketch and in Fig. 2.9, these points are the intersections of the black circles with the x-axis. The vertical line is the radical axis of the two circles since it passes through the intersection points .S1 and .S2 of the circles. In this case, it is OX1 · OY1 = OX2 · OY2 ,

.

(2.18)

where .OX1 denotes the distance between the points O and .X1 . We associate this sketch directly with the involution on a line. Taking the point O as the center of the involution, the product .OXi · OYi is the constant of the involution. This implies that the points .X1 , .Y1 and .X2 , .Y2 are indeed corresponding points of the involution on the line e. Moreover, this sketch deals with a hyperbolic involution meaning that two real invariant points exist. We can find these points by looking for circles that pass through .S1 and .S2 and touch the x-axis, see the gray circles in Fig. 2.9. Einstein did not name the invariant points, but he marked them. Due to the constant of the involution, the distances between the invariant points and

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Fig. 2.10 Reconstruction of Einstein’s second sketch on AEA 62-785r. Einstein did not draw the gray circle and line. The notations are changed: Einstein only used the notations .x1 , O, and .y1

the center, respectively, are equal. Einstein indicated this by drawing a half-circle through the two invariant points with center O. Let us now summarize the results and connect Einstein’s calculation with his sketch. By Eq. (2.2), the general equation of an involution, we find the corresponding point to the point at infinity (.x = ∞). This led us to Eq. (2.3). In the case that this point is not an invariant point, we can set .b = 0 and get Eq. (2.4). This is what Einstein possibly had in mind when he called this point “Anfang” and denoted it with O in the sketch. By setting .x = y, we find the invariant points of the involution that can be real or imaginary according to his comment on the first part. In his sketch, he indicated these points by the semicircle and marked the intersection with the xaxis. By knowing two pairs of the involution, the involution is uniquely defined, and we can determine a, b, and c, as in Eq. (2.9). Analogously, we could also construct the corresponding point of any other point by drawing a circle through each of the two known pairs such that the radical axis is the joining line of the intersections .S1 and .S2 of the circles. Any other circle through .S1 and .S2 that intersects or touches the x-axis determines a pair of corresponding points of the involution. The second sketch was drawn at the bottom left. Above it, Einstein wrote “Im imaginären Falle.”30 A reconstruction of the sketch is depicted in Fig. 2.10a and b. Einstein’s sketch permits two different interpretations that will be discussed in the following. Let us first consider Fig. 2.10a. We can identify the line e with the line of the involution since it carries the corresponding points .X1 , .Y1 , and O in analogy to Fig. 2.9. Once again, we use capital letters for the points contrary to Einstein. We notice that, in contrast to the first sketch, the corresponding points are now on two different sides of O. In Fig. 2.9, both .X1 and .Y1 are on the left side of O, where now .X1 is on the left and .Y1 is on the right. This is only possible when the line of the involution is placed such that it separates the two fixed points .S1 and .S2 where the circles intersect. In order to see this, we move the line e in the situation of Fig. 2.9

30 “In

the imaginary case.”

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up such that it comes to lie between .S1 and .S2 . This means that no circle exists that passes through both .S1 and .S2 and also touches the line e of the involution, since all circles passing through .S1 and .S2 need to cross the line e twice. Thus, the involution is elliptic and has only imaginary and no real invariant points.31 We argue that Einstein indicated this by titling the sketch with “Im imaginären Falle,” which is clearly connected to his calculations where he commented that the solutions of the quadratic equation (2.7) or (2.8) could also be imaginary. Although Einstein did not draw the second circle in Fig. 2.10a, he marked the point .S1 with a dot.32 The second intersection point .S2 can lie anywhere else on the lower semicircle. Its exact position is not important for the following considerations. Nevertheless, we argue that the vertical line through O and .S1 is the radical axis (in comparison to the first sketch) which is why .S2 would be the second intersection of this line and the circle. In this case, both lines .p1 and .p2 stand for a line of an involution that passes through one of the intersection points .S1 or .S2 . If .p1 is the line of the involution, we can find only one circle that passes through .S1 and .S2 and also touches the line .p1 . This is the black circle Einstein drew. It determines the only invariant point .S1 . Any other circle intersects .p1 twice as, for example, the gray circle. In accordance with the categorization of projectivities in Sect. 2.2.3, we call this involution a parabolic involution. Indeed, this is not unusual as the term appears in the literature.33 However, the parabolic involution is not an ordinary involution as it is not a one-to-one correspondence anymore: The point .S1 is the corresponding point to any other point on .p1 .34 Any circle that determines the corresponding points of the involution passes through .S1 , as, for example, the gray circle. Thus, .C1 is not only the only invariant point of the involution, but also the center of the involution as it is the corresponding point to the point at infinity. Instead of drawing further circles in order to determine further pairs of the involution, we can also alter line .p1 . See, for example, the line .p2 that passes through .S1 and P . It is the line of another parabolic involution. The dark circle determines the corresponding pair .S1 and P . Einstein might have drawn a second line .p2 instead of a second circle in order to investigate parabolic involutions. All in all, we argue that Einstein considered different cases in this sketch. This could also be the reason why he scratched out his third sketch, which only consists of a circle and a line indicating the center of the involution. It seems as if Einstein wanted to discuss a third case but instead of drawing a new figure, he added the case to the second sketch.

31 Einstein

did not draw the point .S2 . However, as mentioned above, by his notations .x1 , O, and it becomes clear that the pair .x1 , .y1 is separated by the center O corresponding to an involution where the line e passes between .S1 and .S2 . 32 Note that for the following interpretation, it is not necessary to draw the second gray circle; however, it facilitates conceiving the interpretation. 33 See, for example, (Emch 1905, 13–14). 34 This is the reason why the case .b2 = ac usually is excluded as mentioned above, see Footnote 28.

.y1 ,

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By the discussion above, the line h in Fig. 2.10a has no interpretation. We could see it as a hyperbolic involution where the black circle does not intersect the line of the involution. However, the line also permits another interpretation of the entire sketch. We note that the interpretation above seems more likely to us, but for the sake of completeness, we will present alternative interpretations that serve as a caveat. For this purpose, we consider Fig. 2.10b. We can define a radical axis even if the circles do not intersect. Let us assume this in Fig. 2.10b where the second circle would lie above the given one. We can then assume r to be the radical axis of the two circles. In this situation, the line .g1 could work as the line of the involution with center .O ' where P and .P ' are corresponding points of the involution, similar to Fig. 2.9. As of yet, we have assumed that the point at infinity is not an invariant point. Let us drop this assumption and ask what happens if the point at infinity is an invariant point. We can get to this situation by rotating the line .g1 to .g2 around P . The point .O ' then moves to the left until it becomes the point at infinity and the radical axis r and the line .g1 become parallel. They meet in the point at infinity. We saw above that in this case, the corresponding point of .x = ∞ is .y = ∞ that is not a real point anymore. Given this situation, the circle drawn in Fig. 2.10b determines the second invariant point of the involution on .g2 , which is P . It is clear that this is the only additional invariant point: Any circle that determines corresponding points has its center on the vertical line through O, where O is the center of the given circle. Let us now look at the line .g3 . It either represents the elliptic involution as above fitting to Einstein’s title, or it represents a second case where the point at infinity is an invariant point. If so, the black circle determines one pair .X1 , .Y1 of the involution on the line .g3 . With the same argumentation as above, the second invariant point has to be O. Furthermore, the second invariant point is the harmonic conjugate of the point at infinity with respect to .X1 and .Y1 . In fact, O is the midpoint of .[X1 Y1 ]. Einstein may have indicated this by naming the center O.35 The second interpretation gives a meaning to all lines in Einstein’s sketch except for the two short vertical lines at the margin. However, the scenario described is in contradiction to Einstein’s calculation where he had taken a real point as the center of the involution implying that the point at infinity is not an invariant point. It seems more likely that he investigated all three cases of involutions (hyperbolic, elliptic, and parabolic) in the case that the point at infinity is not an invariant point fitting to his calculations. In summary, Einstein considered involutions both algebraically and geometrically. He made calculations on hyperbolic, elliptic, and parabolic involutions where he considered the point at infinity in particular. We associated the first sketch to 35 Such

a treatment of involutions is also discussed by Enriques in paragraph 40 (Enriques 1915, 124–128) in 1915. Its first edition was printed in 1903. Enriques, too, initially considered the case that the point at infinity is not an invariant point and dropped this condition later. However, the calculations and sketches are fundamentally different which is why we do not think that Enriques’ book had served as a reference for Einstein.

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Fig. 2.11 Grossmann’s transcription of the first lecture on imaginary elements in 1897 (Grossmann 1897, 71). In this, the term “Doppelpunkt” was used. We will come back to this in Footnote 112 when discussing a calculation in Einstein’s notebook. Original located at ETHBibliothek Zürich, Hs 421:12

the discussion of a hyperbolic involution fitting to Einstein’s calculations. For the second sketch, two different interpretations exist, while we prefer the first one: It explains the text “Im imaginären Falle” and gives a coherent interpretation of most of the lines. Furthermore, it fits to Einstein’s calculations as well. The second interpretation provides a good explanation of the sketch as an entity but cannot interpret the text and does not fit to the calculations. All in all, it seems as if Einstein knew this particular subject of projective geometry very well. As mentioned in Sect. 1.3.2, Einstein attended two courses on projective geometry in 1897/98. The first one dealt, among others, with imaginary elements where the introduction of this section was transcribed by Grossmann (1897). This passage is shown in Fig. 2.11 and deals with the difference between elliptic and hyperbolic involutions and their imaginary and real invariant elements in particular. It says:36

36 “[...] but those can be either real, then they are double points of the considered hyperbolic involution, or they are imaginary, such that the involution is elliptic.” Translation by the author.

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[. . . ] diese können aber entw[eder] reell sein, dann sind es die Doppelpunkte der betr[achteten] hyperbolischen Involution, od[er] sie sind imaginär, so ist jene Involution elliptisch. (Grossmann 1897, 71)

Obviously, Einstein knew these relations even in 1938. He either had not forgotten the subject or looked it up again, while we did not find literature that might have served as a source for him. In addition, one of Grossmann’s own lectures from 1907 in Basel can be associated with the present manuscript page. From this lecture, a manuscript is extant which was written by Grossmann himself (Grossmann 1907). On pages 17.2 and 18.1, Grossmann discussed the difference between hyperbolic and elliptic involutions with corresponding sketches as well as the center of an involution. However, the sketches are not the same as in Einstein’s sketches, and we have no indication that Einstein used or even knew this manuscript. We will come back to this manuscript in the Sects. 2.3.2 and 2.3.3. For a better understanding of the relations between the individual sketches on the present manuscript page, we created animations visualizing and representing both the sketches individually as well as the transitions between them.37 The relations between the sketches on the present manuscript page can be seen in the video sequence “62785r.mp4.”38

2.3.2 AEA 62-787r In the following, we will reconstruct and interpret four sketches on the manuscript page AEA 62-787r, which is shown in Fig. 2.12. The first part of the page contains calculations and text written in pencil, which can be connected to calculations on the front of the page AEA 62-787. We will discuss them in Sect. 4.3.1.2 in more detail. The calculations are connected to Einstein and Bergmann’s correspondence from 1938. The second part of the present manuscript page contains four sketches, two written in black ink and two written in pencil. We will argue that these sketches are related to projective geometry and involutions. We will also state that the four sketches are connected to each other. Starting from the sketch on the right, we can make a transition in order to get to the sketch at the bottom left. By a further transition, we also get to the situation shown in the two other sketches. We will first discuss the sketches separately and then connect them with each other. We will start with the top left sketch. It is drawn in black ink and is shown in Fig. 2.13a and b.39 The sketch shows a circle with four points on it. With the notation

37 We

used the interactive geometry application GeoGebra. animation is accessible at https://doi.org/10.1007/s00407-020-00270-z as supplementary material to Sauer and Schütz (2021) (visited on 06/10/2021). 39 The following discussion does not distinguish between clockwise and counter-clockwise senses which is why we call both as ordered sense. 38 The

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Fig. 2.12 Manuscript page AEA 62-787r. This page has been published in Sauer and Schütz (2021). © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama

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Fig. 2.13 Reconstruction of Einstein’s top left sketch on AEA 62-787r. Einstein did not use notations in this sketch

in Fig. 2.13a, the lines .A' P , .A' B ' , AB, and SP are dashed, while the lines AP , AS, and .SA' are solid. The last two lines are tangents of the conic through the point S. The sketch can clearly be identified with Fig. 2.6a. By the theory presented in Sect. 2.2.3, given the quadrangle .AB ' A' B inscribed in the conic, the opposite sides ' ' ' ' ' ' .AB , .A B, and AB and .A B as well as the tangents in A, .A and B, .B meet each ' other in collinear points. Einstein did not draw the tangents in B and .B , but we can find the intersections .P = AB ' · A' B, .Q = AB · A' B ' , and .S = a · a ' , where a and ' ' 40 Note that Einstein did not mark .a denote the tangents in A and .A , respectively. any points. As shown in Sect. 2.2.3, we can regard the sketch as an involution on a conic where A, .A' and B, .B ' are corresponding points, respectively.41 In this case, the line .(AB ' · A' B)(AB · A' B ' ) = P Q is the axis of the involution .AA' B ^ A' AB ' and cuts the circle twice. Therefore, it is an interior line, and the involution is a hyperbolic involution, where the intersections of P Q with the conic are the invariant points. The center of the involution named as O in Fig. 2.6a had not been drawn by Einstein.42 We argue that indeed Einstein had the quadrangle with the sense shown in Fig. 2.13a in mind: Assume the quadrangle .AB ' A' B as shown in Fig. 2.13b, and P would still be the diagonal point where the opposite sides .AB ' and .BA' meet. The diagonal point .A' B ' · AB would be missing, but instead, we could identify the point .O = AA' · BB ' as the center of the involution. In this case, the elliptic 40 In

the following, we denote the tangent in a point X on the conic by x. will see below that Einstein indeed considered involutions here as he drew three lines in the sketch on the right that do not have a meaning otherwise. 42 It would lie outside the conic. 41 We

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Fig. 2.14 Reconstruction of Einstein’s bottom left sketch on AEA 62-787r. Einstein did not use notations in this sketch and did not draw the gray lines. However, he drew the circle around the point S. The left bottom corner is torn off which is why the point in this corner is not visible on Einstein’s page

involution can be determined by lines through O. The intersections of the secants through O and the conic are then corresponding points. However, in this scenario, the intersection of the tangents in A and B would be meaningless.43 This situation changes in the bottom left sketch, and we will argue that there, Einstein had an ordered sense of the quadrangle in mind. A small part of the sketch is missing since the left corner of the paper on which Einstein wrote has been torn off. Einstein drew the sketch in black ink. In Fig. 2.14a and b, the sketch is fully presented according to our suggestion. This time, only the lines .B ' P and SQ, in terms of Figure (a), are dashed. Let us now consider the situation in Fig. 2.14a. The quadrangle .AB ' A' B is inscribed in the conic with an ordered sense. As in the situation above, .P = AB ' · A' B and .Q = AB · A' B ' are two of the three diagonal points which is why the line P Q is the axis of the involution .AA' B ^ A' AB ' . It passes also through the intersection S of the tangents in A and .A' . The point S is the only point marked by Einstein in this sketch as he drew a small circle around it.44 The intersection Q, which is not visible in Einstein’s sketch as the part of the paper has been torn off, can be determined by the intersection of the Pascal line SP and the side AB. The two opposite lines AB and .A' B ' of the quadrangle meet each other on the Pascal line. Einstein, however, did not draw the line .A' B ' . Moreover, in Einstein’s sketch, the lines AB and .A' B ' would not even meet in Q due to the inaccuracy of his drawing. This might be the reason why he did not draw .A' B ' .

43 In

situation (a), the tangent points are corresponding points if speaking in terms of involutions. will see that Einstein also drew a circle around one point in the sketch on the right. We will connect these two points with each other below. 44 We

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Fig. 2.15 Reconstruction of Einstein’s middle sketch on AEA 62-787r. Einstein did not use notations in this sketch and did not draw the gray lines

Since the axis P Q of the involution is an exterior line, the involution is elliptic. The center of the involution .O = AA' ·BB ' lies inside but was not drawn by Einstein similar to before. With the same argumentation as above, we argue that Einstein considered an ordered quadrangle as in Fig. 2.14a in contrast to the situation in Fig. 2.14b. Assuming the situation in (b), O would be the center of the involution and P still one diagonal point. However, the tangents in A and B meeting in S would be meaningless.45 Thus, if interpreting the two left sketches in Einstein’s notes as involutions, we should introduce the notations shown in Figs. 2.13a and 2.14a. Let us now look at the sketch in the middle that is shown in Fig. 2.15. All drawn lines are solid and in pencil. In this sketch, there are only three, again not labeled, points on the conic. In the situation of Fig. 2.15a, we denote the points with A, .B ' , and B in analogy to before. While the first two sketches showed a quadrangle inscribed in a conic, we have here an inscribed triangle. We already saw when discussing Fig. 2.7c that indeed the intersections between each side of the inscribed triangle and the tangent through the opposite vertex are collinear. Thus, the points .O = BB ' · a, .U = AB ' · b, and ' ' .V = AB · b are collinear. Note that Einstein did not draw the gray lines .b and AB 46 as well as the intersection V . In terms of involutions, the point .O = a · BB ' is the center of the involution .ABB ' ^ AB ' B. Hence, A becomes an invariant point. Any secant through O determines a pair of the involution, as, for example, B and ' .B . By this interpretation, Einstein would have drawn the center of the involution

45 Note

that the situation in Fig. 2.14a is equivalent to the situation in Fig. 2.6b except for the counter-clockwise sense of the quadrangle in Fig. 2.6b. 46 This situation, not in terms of involutions, is also depicted in Grassmann (1909, 100,101) similar to Einstein’s sketches. However, we argue that Einstein considered involutions here.

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explicitly as a distinct point in contrast to the situations before.47 In addition, the point S would again be meaningless as intersection of the tangents a and b. This is the reason why we suggest another notation, which is shown in Fig. 2.15b. Instead of moving the point .A' to A as we did in Fig. 2.7b,48 we can move the point ' .A to B. We then get a situation equivalent to Fig. 2.15b. In this case, the points ' ' .A and B as the corresponding points to A and .B fall together with the center of the involution O, implying that O lies on the conic. Since all lines except for the tangent through O cut the conic in a second point, only one invariant point exists. It also implies that the point O is the corresponding point to any other point lying on the conic. Similar to the discussion of Fig. 2.10a, we call this involution a parabolic involution and denote it by .AOB ' ^ OAO. We again have to note that this is not a real involution since O is the corresponding point of any other point on the conic. In this situation, the Pascal line and the axis of the involution is the line P O. In analogy to before, the point P still is the intersection of .AB ' with .A' B, where the latter becomes the tangent o. The point S is the intersection of a and .a ' = o. We see that by this interpretation, it is necessary to draw the center O of the involution as it falls together with the points .A' and B. It is possible that Einstein did not draw the center O explicitly, but rather the corresponding points to A and ' .B . By this interpretation, it makes sense that Einstein did not mark the point O in the situations before, but in Fig. 2.15b. The point S as intersection of a and o gets a meaning as well as it is also the intersection of the tangents a and .a ' . We note that in this situation, the point U and the line U P become meaningless. It is possible that Einstein drew this line in order to complete the sketch in the sense of Fig. 2.7c but without a direct implication in terms of involutions. At this point, we summarize that Einstein considered a hyperbolic involution in the above left sketch, an elliptic involution in the bottom left sketch, and a parabolic involution in the middle sketch. In all cases, he investigated involutions on a conic. Einstein considered these cases also on the page AEA 62-785r as we have seen in Sect. 2.3.1. There, he treated involutions on a line together with a short calculation. Let us now look at the fourth and last sketch on the right side that is drawn in pencil. This sketch is the only one with labeled vertices. For the sake of simplicity, we did not accurately transcribe it in Fig. 2.16, but rather used colors and altered both the lengths of the line segments and the angles. In Einstein’s sketch, we find six marked points numbered by 1 to 6. He also drew a seventh point and denoted it by .6' . We will argue in the following that he combined two different situations into his sketch: He first considered the hexagon with points 1 to 6 inscribed in a conic and then the hexagon with points 1 to 5 with the additional point .6' . He apparently moved the point 6 from the first hexagon in a certain way to receive a second hexagon. In Einstein’s sketch, all lines belonging to the first situation are drawn solid. The lines that change by the transition .6 → 6' are drawn dashed. For

47 We

argued that Einstein had the notation suggested in Figs. 2.13a and 2.14a in mind, where the center of the involution O does not appear. 48 The situation in Fig. 2.7b is equivalent to Figs. 2.14b or 2.13a.

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Fig. 2.16 Reconstruction of Einstein’s right sketch on AEA 62-787r. Einstein did not use colors and numbered the points A to .F ' by 1 to .6' . He did not name the other points but drew the two green segments thick and also drew a circle around .S1

better comprehensibility and in analogy to our convention, we use the letters A to F or .F ' for the points 1 to 6 and .6' , respectively. We note that Einstein drew the two opposite sides AB and ED thick and also drew a circle around the point .S1 . We will come back to this later when connecting all sketches on the manuscript page with each other. Let us start with the first case. Assuming the hexagon ABCDEF inscribed in the conic, Pascal’s theorem states that its opposite sides meet each other in pairs in three collinear points, meaning that the points .S1 = AB · DE, .S2 = BC · EF , and .S3 = AF · CD are collinear. The joining line is the Pascal line drawn in red. Since this line is an exterior line, a projectivity or an involution possibly considered by Einstein is elliptic. In Fig. 2.16, the opposite sides of the hexagon ABCDEF are in the same color, respectively. The Pascal theorem explains all solid lines appearing in Einstein’s sketch except for the line segments AD, BE, and CF lying inside the conic.49 In his sketch, Einstein drew the line BE twice, where one of the lines meets the lines AD and CF in one common point. However, as we can see in our reconstruction in Fig. 2.16, in general this is not the case.50 These three lines get a meaning only if interpreting the sketch as a projectivity. As we saw in Sect. 2.2.3, we can interpret Pascal’s theorem as a projectivity on a conic, where the Pascal line of the hexagon ABCDEF is the axis of the projectivity .AEC ^ DBF . In this case, the points A and D, E and B

mentioned above, the dashed lines belong to the second situation with point .F ' . note at this point that Einstein did not only draw BE twice but also doubled other lines as CD or .S1 S3 . Thus, it probably is a coincidence that one of his lines BE meets AD and CF in one common point.

49 As

50 We

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as well as C and F are corresponding points of the projectivity. These points were connected by Einstein by drawing the lines AD, BE, and CF . The fact that Einstein drew these lines shows that he did not only consider Pascal’s theorem but interpreted it in terms of projective geometry. Taking Einstein’s sketch shown in Fig. 2.16 as starting point and interpreting as described above, it becomes evident that all further sketches on the manuscript page indeed were drawn in terms of projective geometry and especially in terms of involutions as well. We will show this in the following after discussing the second hexagon in Fig. 2.16 with the point .F ' . Considering the hexagon .ABCDEF ' , the point .F ' was chosen by Einstein such that the side .AF ' becomes parallel to its opposite side CD. By the transition .F → F ' , the lines F A and F E become meaningless and are replaced by the dashed lines ' ' 51 By the transition .F E → F ' E, the point .S moves and becomes .F A and .F E. 2 ' ' .S causing that the Pascal line moves as well, becoming .S1 S drawn dashed by 2 2 ' Einstein. Furthermore, by the transition .F A → F A , the lines .F A' and CD do not meet each other anymore in the affine plane causing that they only meet in the point at infinity. As a direct consequence, the Pascal line .S1 S2' becomes parallel to these two sides and meets them in the point at infinity as well. Thus, the three parallel lines meet each other in the point at infinity. Thus, in projective geometry, the Pascal line still passes through .S1 , .S2' , and .S3' = AF ' · CD. The transition ' .F → F has been visualized in an animation as well. It can be found in the video sequence “62787rPoint6.mp4.”52 We have now discussed all three sketches individually in terms of projective geometry. While we have a strong argument for the last sketch to be drawn in this context, the context of involutions was slightly imposed in the other cases. This changes when connecting the four sketches with each other. The basis of the following transition is the fact that Einstein marked the point S in Fig. 2.14a in the same way as the point .S1 in Fig. 2.16. While Einstein numbered and marked the points in Fig. 2.16, the point S in Fig. 2.14a is the only point marked by Einstein in the three remaining sketches. In Fig. 2.14a, we see that the marked point S is the intersection point of the tangents a and .a ' on the conic. In Fig. 2.16, the marked point .S1 is the intersection point of the lines AB and DE, which are the only two bolded lines. Einstein’s markings suggest that the two lines AB and DE from Fig. 2.16 meeting in .S2 become the tangents a and .a ' in Fig. 2.14 meeting in S. The situation is shown in Fig. 2.17. Picture (a) shows the inscribed hexagon as in Fig. 2.16, but without the lines belonging to the additional point .F ' . Picture (c) shows the inscribed quadrangle belonging to the elliptic involution as in Fig. 2.14, but with a changed notation. Picture (b) is the intermediate step of the transition from picture (a) to (c). Starting at (a), we let the points A and E move to the points

51 The

line F C becomes meaningless as well; however, Einstein did not draw a replacement for it. animation is accessible at https://doi.org/10.1007/s00407-020-00270-z as supplementary material to Sauer and Schütz (2021) (visited on 06/10/2021).

52 The

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Fig. 2.17 The connection between Einstein’s hexagon and the bottom left quadrangle on AEA 62787r. Einstein did not draw the lightened lines. Picture (a) shows Einstein’s inscribed hexagon without the objects belonging to the additional point .F ' . Picture (c) shows Einstein’s sketch on elliptic involution, while picture (b) is an intermediate step of the transition from (a) to (c)

B and D such that they finally fall together, respectively. The two lines AB and CD drawn as thick lines by Einstein in (a) then become tangents of the conic as shown in (b). The intersection point .S1 in both cases was marked by Einstein. Furthermore, we moved the point F to the left, but without changing the orientation. The transition from (b) to (c) is simply made by altering the eccentricity of the ellipse and slightly repositioning the points. We then get Einstein’s sketch from Fig. 2.14 with the notation corresponding to the previous hexagon. In doing so, the intersection point .S2 = DF · BC moves from the right to the left. In pictures (b) and (c), the line DF is drawn lightened as Einstein did not draw it in his sketch. In fact, Einstein drew his sketch (c) inaccurately such that the line DF would meet the line BC as well as the Pascal line .S1 S2 rather on the right side of .S1 instead of the left side as it is the case in our picture (b).53 The procedure described above is very similar to the discussion at the end of Sect. 2.2.3. Let us interpret the transition from the hexagon to the quadrangle as the transition from a projectivity to an involution as we did in Sect. 2.2.3. We then interpret the hexagon ABCDEF in (a) as a projectivity .AEC ^ DBF on a conic with the axis through the points .S1 = AB · DE, .S2 = BC · EF , and .S3 = AF · CD. Einstein also drew the three lines connecting the pairs of the projectivity. By the transition .A → B and .E → D, the projectivity .AEC ^ DBF becomes an involution .BDC ^ DBF with the axis of the involution being the line .S1 S3 . The interior line AD becomes BE implying that .S4 is the center of the involution as it

53 We

note that between these two sketches, one position for F exists where DF and .S1 S3 become parallel. Einstein considered such a situation in his inscribed hexagon by introducing the point .F ' .

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Fig. 2.18 The transition from the elliptic (a) to the hyperbolic (b) and parabolic (c) involutions

is the intersection of the two lines BD and CF that connect corresponding points. However, Einstein did not draw this line as we see in (c). We conclude that Einstein in (a) considered a projectivity as he drew the interior lines AD, BE, and F C. As he drew the lines AB and CE bolded and marked the points .S1 in (a) and (c) by a circle, it becomes clear that he considered the transition from a hexagon to a quadrangle as shown in Fig. 2.17. The resulting quadrangle then stands for an elliptic involution on a conic. In the same way, we can connect Fig. 2.17c with the two remaining sketches from Figs. 2.13 and 2.15. Figure 2.18a shows again Einstein’s sketch at the bottom left as in Fig. 2.17c. Figure 2.18b shows Einstein’s sketch at the top left, while (c) is his middle sketch. We can go from (a) to (b) by moving the point C up such that it comes to lie between the points F and D. This situation is shown in Fig. 2.18b, where we also needed to slightly rearrange the points. By moving C, we change the orientation, which does not change the points related to each other. The involution in (b) still connects the same points as in (a) since no two points fell together and which is why we still describe the involution by .BDC ^ DBF . However, the axis of the involution .S1 S3 becomes an interior line by the change of orientation. Hence, the situation in (b) becomes a hyperbolic involution where the two intersection points between the axis and the conic are the invariant points. Accordingly, the center of the involution .BD · CF becomes an exterior point in (b) as it was an interior point in (a).54 By the transition (a) to (b), the point C needs to pass the point D. Thus, there is one intermediate step where it is .C = D. This situation is shown in (c). By the transition .C → D, the point .S2 also falls together with .C = D. Hence, the orange line CD falls together with the green line d as well as with the red line .S1 S3 .55 Thus, the axis of the involution (red line) only touches the conic in one point D. This point

54 Einstein 55 We

did not draw the center in (a) nor (b). named the points .D = C = S2 only by D and colored the respective lines red.

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is the only invariant point of the parabolic involution. Furthermore, the center of the involution .CF · BD falls together with D.56 Thus, the point D is a corresponding point to each point on the conic as the intersection points between the conic and any line through the center D are corresponding points. We also see this as the pair C, F becomes D, F by the transition .C → D, where B, D is a pair of the involution as well. Using our notation, the involution .BDC ^ DBF in (a) and (b) becomes the involution .BDD ^ DBF or .BDF ^ DBD in (c).57 Two possible scenarios emerge: Either Einstein first considered the transition (a) to (b) and then (b) to (c), or he first considered the transition from (a) to (c) and then went further to (b).58 The positions of Einstein’s sketches do not favor any of these options. The fact that he drew the sketches in (a) and (b) in ink, while he drew (c) in pencil, slightly favors the option (a) to (b) to (c). However, Einstein obviously changed his pen once as he also drew the hexagon in pencil.59 From the transition’s point of view, we can also find arguments for both options: By the transition (a) to (b) to (c), he first changed the positions of the quadrangle and then went to a triangle. In the other case, he would have changed a quadrangle to a triangle and then back to a quadrangle. However, by moving the point C for the transition (a) to (b), the sketch (c) is an intermediate step supporting option (a) to (c) to (b). We conclude that Einstein started with the sketch on the right that stands for a projectivity on a conic. In this sketch, Einstein considered an additional case where two opposite sides become parallel resulting in the fact that these two sides meet the axis of the projectivity (Pascal line) in the point at infinity. He then let two sides of the conic become tangents and drew the sketch at the bottom left depicting an elliptic involution. Starting from this situation, he moved one point such that he received the situations in his top left sketch and in his middle sketch. The former stands for a hyperbolic involution, while the latter stands for a parabolic involution. In each case, he drew the axis of the involution indicating the type of the projectivity or involution as it directly shows whether invariant points exist or not. Only in the case of the parabolic involution, Einstein drew the center of the involution. However, this was necessary as it falls together with another point on the conic. The transitions between the individual sketches described in this section are visualized in the video sequence “MS62787r.mp4.”60

56 This is the only sketch where we can explicitly identify the center of the involution in one of Einstein’s sketches. However, we argue that Einstein did not draw the point D in order to point to the center. He rather drew the center as it falls together with points that he also drew in the previous sketches. 57 Once more, we remark that a parabolic involution is not an involution in a strict sense. 58 Clearly, we receive the situation in (c) not only by moving point C in (a), but also by moving point C in (b). 59 As we argued above, the first step was to go from the hexagon (projectivity) to the quadrangle (elliptic involution). 60 The animation is accessible at https://doi.org/10.1007/s00407-020-00270-z as supplementary material to Sauer and Schütz (2021) (visited on 06/10/2021).

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We finally mention some lectures Einstein might have known in which we found similar sketches. In Sect. 2.3.1, we noted that Grossmann discussed involutions on a line in a lecture on projective geometry on pages 17.2 and 18.1 from 1907 (Grossmann 1907). Astonishingly, just a few pages further on page 20.1, Grossmann wrote down the theorem that the diagonal triangle is self-polar if the quadrangle is inscribed in a conic. Beneath, there is a sketch that is almost the same as in Fig. 2.15. Only one line in Einstein’s sketch is missing. Furthermore, a sketch equivalent to Fig. 2.15 appears in Grossmann’s lecture notes on “Geometrische Theorie der Invarianten I” held by Carl Friedrich Geiser (Grossmann 1898a, 49). Einstein and Grossmann attended this course in 1898 (Stachel 1987a, 366) of which a transcript is extant, written by Grossmann. On page 49, they discussed the situation from Fig. 2.15, even with the same notation as we suggested. We will find a further connection to this lecture in Sect. 2.3.3.

2.3.3 AEA 62-789 The manuscript page AEA 62-789 is written in black ink. It is shown in Fig. 2.19. While Einstein drew three sketches on the first third of the page, the rest of the page contains calculations and text. In Sect. 3.6.1, we will associate these considerations with Einstein and Bergmann’s publication from 1938 (Einstein and Bergmann 1938) and to the Washington manuscript written by Einstein.61 In fact, we will come to the conclusion that the paragraph on the present manuscript page is connected to Einstein’s Washington manuscript rather than to the publication. The manuscript page has a back page AEA 62-789r whose entries precede the entries on AEA 62789 as the sketches on AEA 62-789r appear at the bottom, while the sketches on AEA 62-789 appear at the top of the page. We will argue that the three sketches on the present manuscript page all deal with projective geometry and probably also with involutions. Einstein crossed out the two sketches at the top, while the sketch beneath on the left is not crossed out. The problem occurring is that the two left sketches are very generic and appear in many different contexts. Let us, therefore, start with the right sketch that is more specific. Einstein’s sketch is shown in Fig. 2.20. Einstein did not draw the gray line segments. He denoted the points on the line g as shown in the second row in Fig. 2.20. In the following, we will use our notation shown in the first row. He marked the points H and G by a cross and drew a dashed line through E and F but did not name these objects. He also marked the points A, .B ' , B, and .A' . Furthermore, he drew one further line that goes below the points F , E and passes the line g between B and .A' . We interpret this line as a cancelation since the line is crooked and it seems very arbitrarily chosen. Einstein obviously crossed out his sketch.

61 We

will discuss the publication in Chap. 3.

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Fig. 2.19 Manuscript page AEA 62-789. This page has been published in Sauer and Schütz (2021). © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama

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Fig. 2.20 Reconstruction of the right sketch on AEA 62-789. Einstein did not draw the gray line segments. Einstein denoted the points as shown in the second row

A reasonable context for this sketch is the construction of a pair of an involution. Given a complete quadrangle with the vertices F , E, H , and G and a line g not passing through any of its vertices, then the opposite sides F G and H E, GE and H F as well as F E and GH meet the line g in three pairs of an involution. This implies that given Einstein’s sketch, the points A, .A' and B, .B ' are two pairs of the involution. By drawing the quadrangle, we find another pair C, .C ' of the involution. Indeed, the way Einstein drew the lines indicates this context. Given the points A, .B ' , B, .A' as well as the line g, we first need to construct another point C. This can be made by drawing four line segments through A, .B ' , B, and .A' , respectively, such that the lines through A and .B ' meet each other in F and the lines through B and .A' in E. These are the solid line segments in Einstein’s sketch. The line through E and F determines the point C by crossing the line g. Einstein did not determine the point C. The reason for this might have been that the line EF crosses the line g only near the right margin of Einstein’s physical sheet. Hence, he only indicated it by drawing a segment (dashed line). The corresponding point .C ' can then be constructed by extending the line segments through A, .B ' , B, and .A' such that the lines through ' ' .B and .A meet each other in G and the lines through A and B in H . Einstein drew these extensions as dashed segments. The points G and H are the points that were marked by a cross in Einstein’s sketch. The line connecting them meets g in the point .C ' . Einstein did not draw the line GH , perhaps because it intersects the line g far away on the left side. In fact, the intersection point .C ' lies onto another sketch on the left. This eventually might have been the reason why Einstein did not draw GH and crossed out the sketch instead. Einstein’s chosen notations A, B, C, and D does neither support nor object this interpretation. Let us now investigate the two sketches on the left of the page. Figure 2.21a shows Einstein’s sketch at the top, and Figure (b) shows the sketch beneath. Einstein crossed out Fig. 2.21a. The crossing line neither passes through nor begins or starts at any specific point, unlike all other lines in the sketch.

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Fig. 2.21 Reconstruction of two of Einstein’s sketches on AEA 62-789. Figure (a) was crossed out by Einstein

Both sketches in Fig. 2.21 are very generic and known in many contexts of projective geometry. For example, we encounter Figure (a) as an example of a complete quadrangle, in its most famous context of the fourth harmonic point and its construction,62 and, at least in parts, also when discussing self-polar triangles and involutions on a conic. A fourth context will be provided in Sect. 2.3.5 as well as in Sect. 2.5 where we will interpret two sketches that are very similar to Fig. 2.21a. These sketches appear on the working sheet AEA 124-446 as well as in Einstein’s Prague notebook. By the notation Einstein used there, it is clear that he considered two perspectivities carried out consecutively resulting in an involution. This involution interchanges two points on the horizontal line, while the two remaining points are invariant under the involution. This context is very specific, and we do not have any evidence that Einstein might have considered it on the present manuscript page as he did not use any notation. However, this interpretation provides a possible context for the sketch in Fig. 2.21a as well.63 The situation for the sketch in Fig. 2.21b is similar. For example, we encounter it in the very basic context of perspectivities. We will see in Sect. 2.3.5 that Einstein drew a very similar sketch on the manuscript page AEA 124-446 as well. As he there again used a more specific notation, we will argue that he considered perspectivities and cross ratios of sets of points and lines as well as the preservation of the cross ratio under perspectivities. We also encounter the sketch from Fig. 2.21b when showing that a polarity forms an involution on a not self-conjugate line.64 Furthermore, the sketch can be found in different books that were published during Einstein’s lifetime. Enriques, for example, used it to introduce cross ratios (Enriques 1915, 106) whereas Dowling used it to construct imaginary elements (Dowling 1917, 139).

62 We

will come back to this context in Sect. 2.3.4 when discussing Einstein’s bottom left sketch on the back page AEA 62-789r, see Fig. 2.25. 63 Astonishingly, we will see in Sect. 2.5 that on the double page of the Prague notebook, we also find a sketch similar to Fig. 2.20. We will discuss the relation between the two sketches in Einstein’s first line of the present manuscript page from Figs. 2.20 and 2.21a. 64 For example, see Coxeter (1987, 60,61).

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Fig. 2.22 Interpretation of Einstein’s top left sketch on AEA 62-789 that connects the first two of his sketches

We see that many contexts are possible and it is very difficult to specify them. One possibility to find a more specific context is to combine the two sketches. We looked through literature in order to find contexts where both sketches from Fig. 2.21 appear together. Indeed, for the proof that a line cannot contain more than two self-conjugate points, both sketches can be used, see Coxeter (1987, 61). Enriques (1915, 163–165) gives the proof as well; however, he used only Fig. 2.21a. Another possibility to find a more specific context is to use Einstein’s sketch at the top right. This sketch is not as generic as the two remaining sketches as we saw above. Indeed, by considering Einstein’s sketch in Fig. 2.20, a reasonable and very likely interpretation emerges for Einstein’s sketch in Fig. 2.21a. In fact, we will argue that this is the context of the present manuscript page. For this, we introduce the notation shown in Fig. 2.22. As Fig. 2.20 shows the construction of a pair of an involution when two pairs A, .A' and B, .B ' of distinct points are given, Fig. 2.22 provides a construction of the corresponding point C of .C ' if two invariant points .A = A' and .B = B ' of an hyperbolic involution on the line g are given. The lines a, b, and .c' through A, B, and .C ' (dashed lines) meet each other in .E = a · b, .G = a · c' , and .H = b · c' . By naming the intersection of BG and AH by F , the line F E meets the line of the involution in the desired point C (solid lines) as the opposite sides of the quadrangle with vertices F , H , E, and G meet the line g in corresponding points. The way of how Einstein drew the lines allows many different contexts. However, it does not contradict our interpretation. We also do not know the reason why Einstein crossed out this sketch. We conclude that both sketches at the top of the present manuscript page were probably drawn in the context of involutions. More precisely, they both show the visualization of the construction of a corresponding point of an involution in two different cases that are shown in Fig. 2.23. Einstein’s first sketch is shown in (a) as in Fig. 2.21a. His second sketch is shown in (b) as in Fig. 2.20. Einstein’s left sketch (a) shows a quadrangle with vertices F , H , E, and G. The line of the involution g passes through two diagonal points A and B such that these points are invariant points of the involution (.A = A' and .B = B ' ). By choosing the point .C ' and constructing the complete quadrangle, we find the point C on the line g as the corresponding point to

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Fig. 2.23 Transition from Einstein’s left (a) to the right (b) sketch on AEA 62-789

C ' . By moving the point H up, we get the quadrangle from Einstein’s second sketch (b). In this case, the line g does not pass through any diagonal point. Hence, we get three pairs of distinct points, namely A and .A' , B and .B ' as well as C and .C ' .65 Finally, we note that Grossmann, in his manuscript from 1907, discussed the construction of an involution and the respective theorem on pages 18.2 and 19.1 (Grossmann 1907). We saw in Sect. 2.3.1 that sketches similar to those on the page AEA 62-785r appear on pages 17.2 and 18.1. After these pages, the two mentioned pages 18.2 and 19.1 follow. The next page 19.2 deals with involution on a conic and a sketch that we do not find in Einstein’s manuscript. However, on the following page 20.1, a sketch appears that reminds us of the situation from Fig. 2.15 on AEA 62-787r. We do not claim that Einstein drew his sketches while reading Grossmann’s manuscript because the pages were written in different time periods. However, the option that Einstein discussed these topics with Grossmann during their joint collaboration from 1912 to 191466 cannot be excluded. This becomes especially clear since, as we will see in Sect. 2.5, sketches on projective geometry also appear in the Notebook that was written between 1912 and 1915.

.

2.3.4 AEA 62-789r Let us now look at the back page AEA 62-789r that is shown in Fig. 2.24. The page is written in black ink and contains some calculations at the top and five sketches that deal with projective geometry. The calculations at the top can be identified with considerations related to the further development of Einstein and Bergmann’s 1938 publication. We will discuss these calculations in Sect. 4.3.1.3 in more detail. Directly beneath the calculations, we find five sketches. We can provide a context for the first sketch at the top and the two sketches at the bottom. We will argue

65 The transition described here is visualized in the video sequence “Notebook3.mp4.” The animation is accessible at https://doi.org/10.1007/s00407-020-00270-z as supplementary material to Sauer and Schütz (2021) (visited on 06/10/2021). 66 See Sect. 1.3.4 and Sauer (2015a, 242–251).

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Fig. 2.24 Manuscript page AEA 62-789r. This page has been published in Sauer and Schütz (2021). © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama

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Fig. 2.25 Reconstruction of Einstein’s sketch at the bottom left on AEA 62-789r. Einstein did not use any notation and did not draw the gray line. For the discussion of point D, see Footnote 70

that these constructions all belong to the context of the harmonic relation and, in particular, consider the case that one of the points is the point at infinity. We cannot provide an explicit context for the two sketches in the middle. For the right one, we found a sketch appearing in Grossmann’s lecture notes from 1897 that looks similar but is not equivalent. The remaining sketch in the middle is too generic. In fact, we argue that Einstein had not finished this sketch. Let us first briefly discuss the calculations at the top of the page. We find power series expansions of the functions f , .α, .η, and .β as well as the derivatives .f ' and .α ' . In the middle, Einstein wrote down an expression for .ρ that serves as a substitution for r. In Chap. 4, we will look at these relations in more detail. We also note that we find the long equation ⎛

⎞ ⎛ ⎛ ⎞ ⎞ ⎞ ⎛ 1 r r 1 1 .2 f0 + f1 −α1 2 + −f1 2 α1 ρ 2 ρ ρ ρ ⎞ ┐ ┌ ⎛ 3 1 1 r 1 1 r 1 r − f1 3 β3 3 + α3 η2 2 + f1 3 δ1 = 0 − 2r f0 + f1 ρ 2 ρ 2 ρ ρ ρ ρ

(2.19)

on the present manuscript page. As it was the case with Eq. (2.1) on AEA 62-785, Eq. (2.19) can be connected to an equation that appears on an extra sheet AEA 6-259 from Einstein and Bergmann’s correspondence, see Sect. 4.2.2. It is also connected to an equation on the back of the letter AEA 6-250.67 In addition, Einstein implicitly used the substitutions .η and .β that also appear on AEA 62-785r, see Sect. 2.3.1. Let us now continue with the sketch at the bottom left.68 It is shown in Fig. 2.25; however, Einstein did not use any notation. We use a notation that is based on the subsequent situations. This is also the reason why we exceptionally name points as 69 .α or .β instead of using capital letters.

67 The

back of the letter has the archival number AEA 6-250.1. Its connection to Eq. (2.19) will be discussed in Sect. 4.3.1.3. 68 Einstein probably did not start with this sketch. 69 We find these notations in the sketches at the bottom right and at the top of the present manuscript page.

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Fig. 2.26 Reconstruction of Einstein’s sketch at the top of AEA 62-789r. Einstein only named the points A, B, .α, and .β

Einstein’s sketch shows a triangle with vertices A, B, and .β and three lines inside that meet in .α. Einstein did not draw the line QD; however, he marked a point by a dot which position fits to the position of D.70 Thus, the sketch reminds us of the construction of the fourth harmonic point and is very similar to Einstein’s top left sketch on the back page AEA 62-789r, see Fig. 2.21a. There, we listed some possible contexts that fit to Einstein’s sketch on the present page as well. In the following, we will argue that Einstein considered the construction of the fourth harmonic point71 as this fits to the subsequent sketches.72 Clearly, we can find the fourth harmonic point D of C if A, B, and C are given. Assuming the points A, B, and C lying on one line as in Fig. 2.25, we construct the fourth harmonic conjugate as follows: Choose an arbitrary point .β, not lying on AB, and draw .Aβ and .Bβ. Then, choose a point P on .Bβ arbitrarily and draw AP . The line joining B and .α = AP · Cβ intersects .Aβ in Q. The desired point D then is the intersection of AB with QP . Einstein did not draw the final step. Einstein’s sketch at the top of the page shows the situation from Fig. 2.25 for the case that the point C is the midpoint of the line segment .[AB]. The sketch is shown in Fig. 2.26.

70 In

fact, the dot does not belong to the sketch at the bottom left from Fig. 2.25 but rather to the sketch at the bottom right from Fig. 2.27. It clearly lies on a dashed horizontal line that belongs to the sketch on the right. However, the position fits to the intersection D in Fig. 2.25. A possible scenario is that Einstein first drew the sketch on the right and thereafter the sketch on the left. As he did not have enough space for the line QP , he might have marked the position D by using the line belonging to the other sketch. We need to take into account that one line is missing in the sketch on the right as well. There, the left line connecting .A' with .β has not been drawn, see Fig. 2.27. The dot could also stand for the intersection .Q' between the missing line and the horizontal dashed line. However, the position of the dot does not fit to the position of the missing intersection .Q' . 71 This corresponds to the context given at the second bullet point in Sect. 2.3.3. 72 As we will see below, there, Einstein considered the case that one point of the harmonic relation lies in the middle of two other points.

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Fig. 2.27 Reconstruction of Einstein’s bottom right sketch on AEA 62-789r. By our interpretation, Einstein mistakenly drew two more lines and one further point. He named neither the points Q, ' ' .Q , P , .P nor the lines f and g. He also did not draw the gray lines. For the discussion of the point ' .Q , see Footnote 70

Einstein did not draw the points C and D in Fig. 2.26. Starting from the complete quadrangle as in Fig. 2.25, we can ask what happens if the point D becomes the point at infinity in affine plane. In this case, the lines .g = QP and .f = AB become parallel. Thus, the two lines g and f meet in the point at infinity. The line .αβ then meets f in the midpoint of .[AB]. We can construct point .α similar to Fig. 2.25 as follows: Choosing .β arbitrarily not lying on AB allows us to choose a point P on .Bβ. The point Q needs to be chosen such that QP is parallel to AB as the intersection of these two lines is the point D at infinity. Then, the intersection between the lines AP and BQ is .α. Clearly, by choosing a different position for the point P on .Bβ, the position of .α changes. However, the final line .αβ still meets f in the midpoint of .[AB]. The points .A' and .B ' depict the same situation with different positions of the two points on f . Einstein obviously considered two different situations as all lines belonging to .A' and .B ' are dashed, while all lines belonging to A and B are solid (except for f and g).73 The fact that .A' P ' and .B ' Q' meet in the same point .α as AP and BQ implies that the distance between the points A and .A' is equal to the distance between B and .B ' . We note that similar to the situation above in Fig. 2.25, the final line (this time .αβ) has not been drawn in both situations. The bottom right sketch shown in Fig. 2.27 is very similar to the previous sketch from Fig. 2.26 except for the fact that now the line g lies beneath f . Einstein drew the line f solid and the line g dashed in both sketches implying that he indeed interchanged these two lines. The construction still works as described above such that the line .αβ meets f in the midpoint of .[AB]. The difference to the situation before is that we now choose

73 We introduced the notations .A'

and .B ' . Einstein only used the notations A, B, .α, and .β. However, Einstein used the notations .A' and .B ' in his sketch at the bottom right, see Fig. 2.27.

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the point P on .Bβ not between the points B and .β but beneath B. As before, he did not draw the final line .αβ that meets f in the fourth harmonic point. Einstein’s initial drawing was incorrect which is why he crossed it out. He first drew two lines through A and B meeting in another position for .α on line g. He then crossed out the point .α, which is why we ignored this point as well as the related lines in Fig. 2.27. Einstein again considered a second situation by replacing A and B by .A' and ' ' ' .B . In doing so, he explicitly used the notations .A and .B . As before, all lines 74 belonging to the second situation are dashed lines. Einstein did not draw .βQ' and .B ' Q' . A reason for this might be that the intersection between .βQ' and g lies onto another sketch on the left, which is Fig. 2.25. As we already discussed in Footnote 70, Einstein marked a point on the line g by a dot that could stand for ' ' .Q = βA · g. However, the position of Einstein’s dot does not fit to the position of ' .Q , such that the dot could also stand for the point D from Fig. 2.25. We also note that Einstein’s sketch was drawn inaccurately. In his sketch, the distance .A' A is not equal to the distance .BB ' . This means that the intersection point ' ' ' ' .Q B · A P would not be .α if drawn accurately. We also note that in Einstein’s sketch, the lines f and g are not parallel. Instead, they would meet each other on the left side.75 We conclude that all three sketches deal with the harmonic relation. In Fig. 2.25, Einstein apparently constructed the fourth harmonic point in the case that none of the three given points is the point at infinity. This changes in Figs. 2.26 and 2.27 where one of the points is considered to be the point at infinity in affine plane. In this case, the fourth harmonic point corresponding to the point at infinity is the midpoint of the line segment between the two remaining points. He made this construction for four different cases: In Fig. 2.26, the line g is above f , while in Fig. 2.27, the lines are interchanged. In both cases, he then considered two further situations by replacing the points A and B by .A' and .B ' , respectively. It might be the case that Einstein considered another question as for the construction of the fourth point as the final lines are missing in all cases. Nevertheless, it is evident that the context of Einstein’s considerations at the present page is related to the harmonic relation as well as to the construction of the fourth harmonic point. The transitions between these three sketches in Figs. 2.25, 2.26, and 2.27 described above are visualized in the video sequence “Notebook1.mp4.”76 Finally, we will briefly discuss the remaining two sketches. The sketch on the right side of the manuscript page is shown in Fig. 2.28a. It shows a horizontal line with a right triangle where two of its points lie on the horizontal line. The 74 In

contrast to before, the line BQ belonging to the first situation was drawn dashed as well. the lines are not parallel at all, Einstein might have made this on purpose. 76 This video sequence also shows transitions belonging to the Prague notebook as well as to the manuscript pages AEA 124-446 and 62-789. We will discuss these in Sects. 2.3.5 and 2.5. The transition of the three sketches discussed here begins from 02:14 min on. The animation is accessible at https://doi.org/10.1007/s00407-020-00270-z as supplementary material to Sauer and Schütz (2021) (visited on 06/10/2021). 75 As

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Fig. 2.28 Einstein’s sketch on the right of AEA 62-789r and the comparison with a sketch in Grossmann’s lecture notes from 1897. (a) Image has been published in Sauer and Schütz (2021). © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama. (b) Original located at ETH-Bibliothek Zürich, Hs 421:12. (a) Einstein’s sketch on AEA 62-789r. (b) Sketch appearing in Grossmann’s lecture notes on projective geometry held by Fiedler in 1897 (Grossmann 1897, 41)

point at the right angle is the center of a circle that does not touch or cross the horizontal line. Einstein then drew a tangent on this circle parallel to one side of the triangle and indicated that these two lines meet in the point at infinity. Although we cannot provide a specific context, it seems as if this sketch was also written in the context of projective geometry. In particular, it discusses the point at infinity. A sketch very similar to Einstein’s appears on page 41 of Grossmann’s transcript of Fiedler’s lecture from 1897 (Grossmann 1897), see Fig. 2.28b. The most characteristic resemblances are the two parallel lines, the brace and the symbol indicating that the two lines meet in the point at infinity. Grossmann’s sketch appears in the chapter “Von den Kegelschnitten.”77 It shows two concentric circles and a hexagon .AB ' CA' BC ' inscribed in the larger circle according to a short comment next to the sketch. The lines .AA' , .BB ' , and .CC ' connecting the pairs of the involution touch the inner circle. The two opposite sides .AB ' and .A' B of the hexagon are parallel and meet in the point at infinity, which was explicitly indicated in the sketch.78 Clearly, this case reminds us of Fig. 2.16 where Einstein drew a hexagon with point .6' such that two opposite sides become parallel.79

77 “About

conics”. though Grossmann did not draw these lines, it seems as if the opposite sides .B ' C, .BC ' and .AC ' , .CA' are parallel as well. Thus, the opposite sides of the hexagon .AB ' CA' BC ' meet each other in the point at infinity such that the Pascal line becomes the line at infinity. This is supported by a comment written in pencil next to the sketch. 79 In our discussion, we used the notation .F ' instead of .6' . 78 Even

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Fig. 2.29 Manuscript page AEA 124-446. Parts of this page have been published in Sauer and Schütz (2021). © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama

The fifth sketch in the middle of the manuscript page contains four lines and three points. We have not found any specific context for this sketch. It also seems as if the sketch is not complete. It also might be a previous and wrong version of Einstein’s sketch at the bottom right.

2.3.5 AEA 124-446 Before discussing the double page in the Prague notebook, we will briefly discuss another manuscript page that does not belong to reel 62. In the database of the Einstein Papers Project and the Albert Einstein Archives, we found another working sheet AEA 124-446 with unidentified sketches and calculations that belong to the context of projective geometry.80 The manuscript page is shown in Fig. 2.29. The back of the manuscript page is an undated letter. According to the database of the Albert Einstein Archives, the manuscript page comes from the Stern family and was dated between April 1929 and September 1932 according to Einstein’s stay in

80 We

searched through all entries of the database containing unidentified calculations.

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Fig. 2.30 Reconstruction of Einstein’s sketch on the left of AEA 124-446. Einstein named all objects except of c, b, d, g, f , h, and S

Caputh. It contains three sketches and a short calculation. The sketch at the top is very generic depicting a triangle that will not be discussed here. The sketch on the left of the page is reproduced in Fig. 2.30. Einstein drew four lines a, b, c, and d where he only named a. All lines pass through S and meet a horizontal line g in A, B, C, and D, respectively.81 He then marked the angles .γ1 , .γ2 , .δ1 , and .δ2 as shown in Fig. 2.30. He also drew two crossing lines from the bottom left to the top right of the sketch. They can be seen as cancelations; however, we will provide a context for them in the following using a short calculation that can be found beneath the sketch. Einstein started this calculation with writing down .

81 He

BC BD AC AD = : . : AC AD BC BD

named all points except for S.

(2.20)

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The left hand side is the cross ratio .(AB, CD) of the points A, B, C, and D in the affine plane,82 while the right hand side is its inverse. Einstein then calculated further by canceling the right hand side of Eq. (2.20) getting .

AC 2 AD 2 1 1 : = : = 1. 1 1 BC 2 BD 2

(2.21)

In Einstein’s second line, he wrote down .

ac sin γ1 ad sin δ1 : . bc sin γ2 bd sin δ2

(2.22)

This is the cross ratio .(ab, cd) defined for four lines a, b, c, and d meeting each other in one common point S in the affine plane.83 He then canceled the line segments and got the expression .

sin γ1 sin δ1 . : sin γ2 sin δ2

(2.23)

In fact, it is .(ab, cd) = (AB, CD). We can see this by looking at the four triangles CBS, ADS, BDS, and ACS 84 and their respective areas .ACBS , .AADS , .ABDS , and .AACS . All these four triangles have the same altitude h with base g. Thus, it is AC AD : = BC BD

1 2 1 2

· AC · h

1 2 1 2

· AD · h

AACS AADS : . ABCS ABDS

(2.24)

2AACS 2AADS ac sin γ1 ad sin δ1 sin γ1 sin δ1 : = : = : 2ABCS 2ABDS bc sin γ2 bd sin δ2 sin γ2 sin δ2

(2.25)

.

· BC · h

:

· BD · h

=

Furthermore, with the law of sines, it is .

showing the assertion. The fact that Einstein wrote down the intermediate step from Eq. (2.22) indicates that he indeed considered the idea of such a proof.85 Moreover, it immediately follows that the cross ratio is invariant under perspectivities. This is indicated by

see this, we impose the coordinates .A = (a, 0), .B = (b, 0), .C = (c, 0), and .D = (d, 0). By Kaplansky (1969, 107), it is then .(AB, CD) = (d − b)/(d − a) · (c − a)/(c − b) equivalent to the left hand side of Eq. (2.20). In doing so, we regarded the notation XY as the distance between the two points X and Y , where here X and Y stand for the points A, B and C, D, respectively. 83 See, for example, (Wagner 2017, 12). 84 The notation XY S stands for the triangle with vertices X, Y , and S. 85 This proof can be found in most of the textbooks on projective geometry. See, for example, (Enriques 1915, 105–110). 82 To

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Fig. 2.31 Three sketches very similar to each other. By the notations in (a) and (c) and by related calculations, the sketches visualize cross ratios of points and lines as well as its preservation under perspectivities. (a) and (b) © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama. (c) Original located at ETH-Bibliothek Zürich, Hs 421:10.1. (a) Einstein’s sketch on the manuscript page AEA 124-446. (b) Einstein’s sketch on the manuscript page AEA 62-789. (c) Sketch in Grossmann’s lecture notes from 1896/97 on “Darstellende Geometrie I” held by Fiedler, see (Grossmann 1896–97, 38)

Einstein as well as he drew the two crossing lines f and h. By the intersections between f and h with a, b, c, and d, we get two new sets of four points with the same cross ratio. We argue that this is the reason why Einstein drew these two lines. We encountered a very similar sketch already on the manuscript page AEA 62789, see Fig. 2.21b. There, however, we did not find expressive notations nor related calculations. Figure 2.31b shows it again,86 while (a) shows the sketch on the present manuscript page. We found another sketch fitting to this context in Grossmann’s lecture notes from 1896/97 on “Darstellende Geometrie I,” see picture (c) from Grossmann (1896–97, 38). The lecture was held by Fiedler and attended by Einstein Stachel (1987a, 363). On the respective page 38 of Grossmann’s notes, we find the equations .

sin ac sin ax AC AX : = : BC BX sin bc sin bx

(2.26)

as well as .

ΔAT C ΔAT X : = ΔBT C ΔBT X

1 2T A · T C 1 2T B · T C

· sin ac · sin bc

:

1 2T A · T X 1 2T B · T X

· sin ax · sin bx

=

sin ac sin ax : sin bc sin bx (2.27)

next to the sketch. Grossmann named the objects S, D, and d from Fig. 2.30 by T , X, and x, respectively, and used the same notations for the objects A, B, C, a, b, and

86 See

also our reconstruction in Fig. 2.21b.

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Fig. 2.32 Reconstruction of Einstein’s sketch in the middle of AEA 124-446. He used the same notations for the points except for P and Q. Instead, he marked them with thick dots. He did not use the notations f and g

c. He furthermore denoted the area of the triangle AT X by .ΔAT X and the angle at a·x by ax. Hence, we can identify the calculations in Grossmann’s lecture notes with the calculations on the present manuscript page by Einstein. Moreover, as we argued that Einstein also investigated the invariance of the cross ratio under perspectivities by drawing the two additional lines, Grossmann’s notes continue with this subject as well. Let us now discuss Einstein’s next sketch that is shown in Fig. 2.32. A similar sketch appears on the manuscript page AEA 62-789 as well, see Fig. 2.21a. There, we stated that this generic sketch could have been written in different contexts. However, due to a second sketch next to it, we finally stated that Einstein considered the construction of pairs of an involution on a line. On the present manuscript page, Einstein introduced a specific notation shown in Fig. 2.32. He did not name the points P and Q but marked them by thick dots. We also introduced the notation of the lines for a better understanding. Einstein used three different notations: first, the notations A, B, C, and D without primes; second, the same notations with a prime; and third, with two primes.87 We thus argue that he considered three different situations. Starting from the notation without primes, we find four points A, B, C, and D lying on g. As Einstein marked the points P and Q by thick dots, we argue that he considered perspectivities with centers P and Q. The prime notation of the points suggests that he first considered the perspectivity on Q such that A goes to .A' , B goes to .B ' , C goes to .C ' , and D remains invariant and becomes .D ' . The points with primes all lie on the line f . The second perspectivity then has P as its center

.

87 The

notation D is barely readable in Einstein’s sketch.

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such that .A' goes to B which is also denoted by .A'' , .B ' goes to A which is also denoted by .B '' , .C ' goes to C which is also denoted by .C '' , and .D ' again remains invariant and becomes .D '' . We see that the points with two primes then again lie on the line g. Thus, the product of the two perspectivities can be described as a hyperbolic involution .ABCD ^ BACD, where the points C and D are the two invariant points and the points A and B are interchanged. We will see in Sect. 2.5 that Einstein considered such an involution also in his Prague notebook. There, he did not use such a comprehensive notation; however, a short comment tells us that he apparently had the same procedure in mind.88

2.3.6 Further Notes For the sake of completeness, we briefly mention further sheets containing research notes that might be connected to projective geometry. We found them in the process of browsing through all extant research notes with geometric sketches in the database of the Albert Einstein Archives and the Einstein Papers Project. The first page is the manuscript page AEA 31-224 that contains two sketches that might have been drawn in the context of projective geometry. However, we were not able to connect it with the research notes discussed above.89 The second document is the back of the letter AEA 6-250 with the archival number AEA 6-250.1. In Sect. 4.1, we will date the letter to the end of June 1938. On the back of the letter, we find an equation that is directly related to Eq. (2.19) on AEA 62-789r. We also find two sketches that might have been drawn in the context of projective geometry. Hence, the back of the letter could provide a direct link between Einstein’s calculations on the generalization of Kaluza’s theory and the sketches on projective geometry. We will come back to this in Sect. 4.3.1.3 when discussing the calculations on AEA 62-789r as well as the equation on the back of the letter AEA 6-250.

2.4 Summary of the Analysis Let us briefly summarize the results from the previous sections and connect the individual pages with each other. We discussed the manuscript pages AEA 62-785r, 62-787r, 62-789, 62-789r, and 124-446 and came to the conclusion that they were all

88 The transition from the sketch in the Prague notebook to the sketch on the present manuscript page is visualized at the beginning of the video sequence “Notebook1.mp4.” The animation is accessible at https://doi.org/10.1007/s00407-020-00270-z as supplementary material to Sauer and Schütz (2021) (visited on 06/10/2021). 89 See also sketches in AEA 80-914 and 90-914.

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written in the context of projective geometry. More precisely, Einstein considered different types of involutions with a special focus on the point at infinity. On AEA 62-787r, Einstein started with a sketch showing Pascal’s theorem in two situations: In the first situation, the opposite sides of the inscribed hexagon meet each other at real points in the affine plane, respectively. In the second situation, two opposite sides became parallel such that they meet each other in the point at infinity. Three additional lines that do not belong to Pascal’s theorem show that Einstein looked at this sketch in terms of a projectivity. Starting from this, he derived three further sketches by letting certain points fall together or by changing the orientation. These sketches then stand for different types of involutions. At the bottom left, Einstein considered an elliptic involution with no real invariant points. The sketch at the top left shows a hyperbolic involution with two invariant points, while the sketch in the middle can be interpreted as a so-called parabolic involution with only one invariant point.90 These cases were considered by Einstein on AEA 62-785r as well. As he considered projectivities and involutions on conics on AEA 62-787r, he here considered involutions on lines. At the top of the manuscript page, we found a short calculation starting from the general equation of involutions. He then chose the origin of the line to be the corresponding point of the point at infinity. In doing so, the general equation of the involution became a more specific equation that could directly be connected with the constant of an involution. He then looked for invariant points and considered two cases, namely that they are real or imaginary. In the case of two real solutions, the involution is a hyperbolic one. Einstein drew a sketch belonging to this case. He constructed two pairs of the hyperbolic involution as well as the two invariant points by using the radical axis of two intersecting circles. For the case of no real solutions, Einstein drew another sketch corresponding to an elliptic involution with no real invariant points. In this sketch, he also considered the case that only one invariant point exists corresponding to the vanishing of the discriminant. In terms of involutions, this is again a parabolic one. The construction of pairs of an involution on a line was also considered by Einstein on the manuscript page AEA 62-789. Instead of using the radical axis as it was the case on AEA 62-785r, he constructed the pairs by using a complete quadrangle. This is possible as the opposite sides of the quadrangle meet a separate line in pairs of an involution. He apparently considered two cases: First, he constructed a pair of an involution by starting from two invariant points corresponding to a hyperbolic involution. In the second case, he started from two pairs consisting of distinct points, respectively. He then constructed a third pair of this involution. Two of the three sketches on AEA 62-789 are very similar to the sketches that we find on the working sheet AEA 124-446. Due to detailed notations and short calculations, however, we argue that Einstein here first considered cross ratios of

90 Again, we note that a parabolic involution is not an involution in the strict sense as it is not a one-to-one correspondence anymore.

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points and lines and their invariance under perspectivities. In his second sketch, he then carried out two perspectivities that result in a hyperbolic involution that interchanges two pairs, and let two other points invariant. The complete quadrangle from AEA 62-789 appears in a similar form also on its back page AEA 62-789r. Here, Einstein apparently investigated the harmonic relation in three different cases. For the first case, he considered three real points lying on a line and the fourth real harmonic point. In the further two cases, he set one of the three given points to be the point at infinity, similar as he considered the point at infinity on AEA 62-787r. In this case, the fourth harmonic point divides the line segment of the two remaining points into two equal parts. The point at infinity was obviously also considered in another sketch on this manuscript page that might be connected to a sketch appearing in Grossmann’s lecture notes from 1897. We also pointed out similarities between Einstein’s sketches on his working sheets and Grossmann’s own lecture (Grossmann 1907). In addition to the sketches on projective geometry, the pages from reel 62 all contain calculations that are not directly connected to projective geometry. As we already mentioned in the respective discussions, we will look at these calculations in more detail in Chap. 4. For now, we conclude that they all belong to the context of Einstein and Bergmann’s publication on the generalization of Kaluza’s theory of electricity (Einstein and Bergmann 1938) that we will discuss in Chap. 3. We can date them roughly to summer 1938. In Sect. 4.3.1.4, we will argue that they were written between mid-June and July 6, 1938. Finally, we note that we identified further documents that might contain considerations on projective geometry as, for instance, the document AEA 6-250.1 that is the back of the letter AEA 6-250.

2.5 Prague Notebook and Its Connection with Princeton Manuscripts In this section, we will reconstruct and interpret five sketches that were drawn by Einstein in his Prague Notebook on pages 49 and 50 (Klein et al. 1993, Appendix A). The double page is shown in Fig. 2.33.91 As already described in Sect. 1.1.2, the pages before and after the pages 49 and 50 deal with gravitational lensing and were written in 1912 and 1915, respectively (Sauer 2008, 6). Therefore, it seems natural that the pages intermediately were written between 1912 and 1915, which makes the double page especially interesting as it was the time period when Einstein was finishing his general relativity. The double page begins with a sketch in pencil, probably not in Einstein’s hand. We also find four letters beneath this sketch that overlap with the subsequent sketch. As mentioned in Sect. 2.1.1, the first sketch and the four letters probably were written in 1913 and do not belong to projective 91 We

already know this figure from Fig. 2.1.

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Fig. 2.33 Double Page from the Prague Notebook (pp. 49–50), see AEA 3-013 and Klein et al. (1993, 588). This figure has been published in Sauer and Schütz (2021). © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama

geometry. The same holds for the last two sketches and the short calculation on magnetism. In between, we find five sketches, two short comments, and a short calculation that were all written in the same black ink and in Einstein’s hand. In the following, we will only consider these five sketches, and we will refer to them as the first, second, third, fourth, and fifth sketches.92 Let us now start with the first sketch, which is shown as reconstruction in Fig. 2.34. Clearly, the sketch shows a strong resemblance to Fig. 2.22 on AEA 62789 and to Fig. 2.32 on AEA 124-446. This is why any context described in Sect. 2.3.3 would also provide a context for this sketch. However, on the present manuscript page, Einstein introduced notations that imply a specific interpretation. Einstein considered four points A, B, .A' , and .B ' all lying on the line g. He then drew a complete quadrangle and named three intersections by .β, .α ' , and .β that all lie on the line h that also passes through A. Furthermore, he marked the two remaining points by a thick dot, respectively. Einstein’s notation of the first four points suggests that he considered a projectivity or involution with pairs A, 92 In doing so, we count Einstein’s four crosses including the brace at the second page as one sketch.

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Fig. 2.34 Reconstruction of Einstein’s first sketch on the double page. Einstein did not name the points R and S but marked them by drawing two thick dots. We adapted Einstein’s notation even though we usually use capital letters for the notation of points. For a better understanding, we also introduced the notations g and h

A' and B, .B ' . In the case of an involution, it would be an elliptic involution, since the corresponding pairs separate each other.93 By the construction of the complete quadrangle, it is .H (AA' , BB ' ). Indeed, given an elliptic involution as it is the case in Fig. 2.34 and a pair of corresponding points A, .A' , one (and only one) other pair B, .B ' exists with .H (AA' , BB ' ), see Grassmann (1909, 166). This interpretation, however, does not explain the further points Einstein named and drew. It is conspicuous that Einstein named the points on g by Latin letters, while the points on h by the respective Greek letters. Only the point A does not have an analogue, probably because it is the intersection between h and g. The fact that Einstein also drew the remaining two points S and R as thick dots suggests that he considered two perspectivities, where by the first perspectivity, points on g go to h, and by the second perspectivity, the points go back to g. Indeed, considering the perspectivity with center R, the point A remains invariant, while the point B goes to .β, .A' goes to .α ' , and .B ' goes to .β ' . Thus, all points with Latin letters go to the points carrying the corresponding Greek letters.94 When carrying out another perspectivity with center S, the point A again remains invariant, while .β goes to .B ' , ' ' ' .α goes back to .A , and .β goes to B. Thus, by carrying out the two perspectivities consecutively, we get an hyperbolic involution .AA' BB ' ^ AA' B ' B with the two invariant points A and .A' , where B and .B ' are being interchanged. In fact, we find a short comment beneath this sketch saying: “B u .B ' vert.” that probably stands for “B und .B ' vertauscht”95 supporting our interpretation. We see that the notation on g indicates an elliptic involution, while the notation of the remaining points, the thick dots, and the comment beneath the sketch suggests that Einstein considered two perspectivities carried out consecutively resulting in a hyperbolic involution. We encountered a very similar sketch on the manuscript page AEA 124-446, see Fig. 2.32. There, Einstein considered the reverse order .

93 This

becomes clear by either the discussion of Figs. 2.9 and 2.10a in Sect. 2.3.1 or by Coxeter (1961, 53). 94 The point A apparently does not have an analogue as it remains invariant. 95 “B and .B ' interchanged.”

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as he first carried out the perspectivity on S and thereafter the perspectivity on R, speaking in the notation of Fig. 2.34.96 Starting from Fig. 2.34, we get to the situation from Fig. 2.32 on AEA 124-446 by moving point .β up such that point A vanishes at infinity and reappears to meet line g on the right hand side of the complete quadrangle. This transition is visualized at the beginning of the video sequence “Notebook1.mp4.”97 Changing the notations .B → A, .A' → C, .B ' → B, ' ' ' ' ' .A → D, .β → B , .α → C , and .β → A finally leads us to Fig. 2.32. We recall that in Fig. 2.22 when discussing the first sketch on the manuscript page AEA 62-789 that is very similar to the sketch in the Prague notebook, we suggested another interpretation. There, Einstein probably constructed pairs of a hyperbolic involution on a line by using the complete quadrangle as opposite sides of this quadrangle meet a line in corresponding points. In the case that the line passes through two diagonal points of this quadrangle, these two points are the invariant points of a hyperbolic involution. We came to this interpretation by the fact that another and more specific sketch on AEA 62-789 appears next to the mentioned sketch that shows the construction of pairs of an involution in the case that the line does not pass through any diagonal point, see Einstein’s second sketch on AEA 62789 in Fig. 2.20. Einstein’s notation in the Prague notebook in his first sketch does not suggest that he initially considered such a construction. However, we will see in Fig. 2.41 that the fifth sketch in the Prague notebook on the second page is very similar to the second sketch on AEA 62-789. Even though this sketch is on the right page of the double page, it lies next to Einstein’s first sketch on the left page. Hence, it is possible that Einstein considered this context in Fig. 2.34 as well. We will come back to this subject after having discussed the fifth sketch in Fig. 2.41. We will then look at the five mentioned sketches (the two sketches on AEA 62-789 and their analogues in the Prague notebook as well as the sketch on AEA 124-446) in Fig. 2.42 at the end of this section.98 Let us now consider the next sketch in the Prague notebook that is shown in Fig. 2.35. Einstein only used the notation 1 for the point C. By our notation, he drew a hexagon ABCDEF inside a conic, while we colored its opposite sides AB and DE, BC and EF as well as CD and AF in same color. Einstein marked the intersection point .S1 = ED · AB by a thick dot. Furthermore, he marked the sides ED and AB as well as the extensions of the sides CB and CD by a little bar. We drew them thick instead. All sides of the hexagon were drawn as solid lines, while their extensions are dashed.

96 By

the notation used in Fig. 2.32, he first carried out the perspectivity with center Q and then the perspectivity with center P . 97 The animation is accessible at https://doi.org/10.1007/s00407-020-00270-z as supplementary material to Sauer and Schütz (2021) (visited on 06/10/2021). 98 See also the video sequences “Notebook3.mp4” and “Notebook1.mp4.” The animations are accessible at https://doi.org/10.1007/s00407-020-00270-z as supplementary material to Sauer and Schütz (2021) (visited on 06/10/2021).

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Fig. 2.35 Reconstruction of Einstein’s second sketch on the double page. Einstein marked the two sides ED and AB. We drew them thick. He also marked the two extensions of the sides CD and CB that we drew thick as well. He also drew a thick dot for the point .S1 as depicted. He did not use any notation except for the point C that he numbered by 1. While we drew opposite sides of the inscribed hexagon in same colors, Einstein did not use any color. As depicted, he drew all sides of the hexagon solid, while the extensions are dashed lines

Clearly, we can identify this sketch with Pascal’s theorem implying that the intersections points .S1 , .S2 = EF · BC, and .S3 = CD · AF meet in three collinear points. The joining line (red dashed line) is the Pascal line. We saw in Fig. 2.16 when discussing the manuscript page AEA 62-787r that Einstein there considered Pascal’s theorem as well. We argued that he furthermore considered the case that two opposite sides become parallel99 and that Einstein interpreted Pascal’s theorem in terms of projectivities as he drew three lines AD, BE, and F C connecting corresponding points of the projectivity .AEC ^ DBF . We then showed that he considered the transition .A → B and .E → D resulting in an inscribed quadrangle, see Fig. 2.17. This quadrangle then stood for an elliptic involution .BDC ^ DBF . By moving the point C to another position and changing the orientation, he then derived another quadrangle standing for a hyperbolic involution, see Fig. 2.18b. He then also let the points C and D fall together resulting in an inscribed triangle standing for a parabolic involution, see Fig. 2.18c.100 By Einstein’s notation and markings on the Prague notebook in Fig. 2.35, we see that he marked the important points for the transition to the quadrangles described above. Indeed, by the transition .A → B and .E → D, the sides ED and AB become tangents meeting in .S1 . Einstein marked the sides ED and AB as well as the point .S1 .101 The second transition of moving C is also indicated in the sketch as

99 This

caused that the Pascal line became parallel as well. again note that it is not clear whether Einstein first considered the hyperbolic or the parabolic involution. 101 We note that Einstein marked these objects on the manuscript page AEA 62-787r as well, see Fig. 2.16. 100 We

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Fig. 2.36 Reconstruction of Einstein’s third sketch on the double page. Einstein only used a notation for the point C by calling it 1. The tangents were drawn as thick solid lines, while all further lines are dashed

he named the point C by using the notation 1 and marked both lines CD and CB that pass through it. In fact, we find a sketch directly beneath the hexagon showing an inscribed quadrangle, see Fig. 2.36. In this sketch, we find the notation 1 again. Einstein also drew two tangents as thick lines. All other lines are dashed. The fact that he marked the lines ED and AB as well as the point .S1 in Fig. 2.35 and drew two thick tangents in Fig. 2.36 shows that he indeed let the points A and E falling together with the points B and D, respectively. Furthermore, the marked point 1 that we find in both sketches was moved as well. While it lay between B and D in Fig. 2.35, it now lies between D and F in Fig. 2.36. Einstein thus changed the orientation. Due to the above considerations and especially due to the similarities between the sketches with those on AEA 62-787r, we argue that Einstein considered the projectivity .AEC ^ DBF in Fig. 2.35 even though he did not draw the three joining lines of corresponding points. He then made the two transitions .A → B and .E → D as well as moving the point C in one step leading him to Fig. 2.36. This sketch then shows the hyperbolic involution .BDC ^ DBF with two invariant points that are determined by the Pascal line .S1 S3 crossing the conic twice.102 This interpretation is in accordance with the considerations made in Fig. 2.34, where we argued that he considered a hyperbolic involution as well. Figure 2.37 visualizes the two transitions from Einstein’s hexagon (a) to his quadrangle (c). We also drew a sketch with the intermediate step (b) where the point C has not been moved. Einstein skipped this step on the Prague notebook. However, we find it on the manuscript page AEA 62-787r where we interpreted

102 By

the theory presented in Sect. 2.2.3, given the quadrangle DF BC inscribed in a conic, the opposite sides DF , BC and DC, F B as well as the tangents in D, B and C, F meet each other in four collinear points. In Fig. 2.36, we did not draw the tangents in C and F . The joining line is the Pascal line.

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Fig. 2.37 The transition from Einstein’s hexagon (a) to the quadrangle (c) on the double page of the Prague notebook. For a better visualization, we drew corresponding lines in same color and made an intermediate step (b). For the transition (a) to (b), we let the points E and A fall together with D and B without moving the point C. We also reposition points B, D, and F . For the transition from (b) to (c), we then move the point C up such that it comes to lie between D and F . (a) Einstein inscribed hexagon from Fig. 2.35 describing a projectivity. (b) Intermediate step showing an elliptic involution that had not been drawn by Einstein in the Prague notebook. This situation only appears on AEA 62-787r. (c) Einstein’s inscribed quadrangle from Fig. 2.36 describing a hyperbolic involution

it as an elliptic involution with no invariant points.103 For a better understanding, we drew corresponding lines in same color. This transition is also visualized in the video sequence “Notebook2.mp4.”104 Beneath Einstein’s quadrangle from Fig. 2.36, he wrote the comment: “Zugleich Konstruktion des Zentrums.”105 We argue that Einstein referred to the center of the involution that we introduced in Sect. 2.2.3. By drawing the lines CF and BD joining the corresponding points of the involution .BDC ^ DBF , we get the center of the involution as their joint intersection, see Fig. 2.6a. As this had not been drawn by Einstein, he made his comment: Even without explicitly drawing the center, one can imagine the position by the shown construction. As it lies outside the conic, two lines exist that pass through the center and touch the conic. These two touching points then determine the invariant points of the hyperbolic involution.106 For the sake of completeness, we note that we also took different interpretations of this 103 The

Pascal line does not cross the conic. animation is accessible at https://doi.org/10.1007/s00407-020-00270-z as supplementary material to Sauer and Schütz (2021) (visited on 06/10/2021). 105 “At the same time construction of the center.” 106 We recall that any secant through the center of the involution determines a pair of the involution as the two intersections with the conic as shown in Sect. 2.2.3. We also recall that the touching points between the two tangents through the center of the involution are the intersections between Pascal line and the conic. 104 The

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Fig. 2.38 Similarities between the two sketches on AEA 62-787r (a) and in the Prague notebook (b) showing the inscribed hexagon and the projectivity on a conic. Images have been published in Sauer and Schütz (2021). © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama. (a) Einstein’s right sketch on AEA 62-787r, see Fig. 2.16. (b) Einstein’s sketch from Fig. 2.35 in the Prague notebook

sketch into account where Einstein might have drawn the center of the involution. However, in this case, certain lines and points as the intersection of the tangents become meaningless as it was already the case in Fig. 2.13. As a further similarity between the Prague notebook and the manuscript page AEA 62-787r, we recall that Einstein there neither drew the center of the involution except for the case of the parabolic involution in Fig. 2.15b as it fell together with a point of the involution.107 Similarly astonishing are the markings Einstein used on both AEA 62-787r and the Prague notebook. As we saw in Fig. 2.16, Einstein there marked the two sides AB and DE as well as their intersection analogously to the situation from Fig. 2.35 in the Prague notebook. These sides then became tangents after the transition. In order to emphasize the similarities between the sketches on the manuscript page and those on the Prague notebook, we placed Einstein’s original sketches next to each other in Figs. 2.38 and 2.39. Just alone looking at them bear the strong similarities between them. In order to go from the inscribed hexagon from AEA 62-787r in Fig. 2.38a to the hexagon from the Prague notebook shown in (b), we only need to move some points along the conic without the need of changing the orientation. For the quadrangle in Fig. 2.39, we only need to rotate (b) clockwise by .90◦ in order to get (a). The transitions of how we can derive the sketches from AEA 62-787r starting from those in the notebook are visualized in the video sequence “NotebookToMS.mp4.”108

107 We

argued that Einstein did not draw the center explicitly, but only the point of the involution. animation is accessible at https://doi.org/10.1007/s00407-020-00270-z as supplementary material to Sauer and Schütz (2021) (visited on 06/10/2021).

108 The

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133

Fig. 2.39 Similarities between the two sketches on AEA 62-787r (a) and in the Prague notebook (b) showing the inscribed quadrangle and the hyperbolic involution on a conic. Images have been published in Sauer and Schütz (2021). © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama. (a) Einstein’s top left sketch on AEA 62-787r, see Fig. 2.13. (b) Einstein’s sketch from Fig. 2.36 in the Prague notebook

Fig. 2.40 Einstein’s first sketch on page 50 of his Prague notebook. He did not use a notation. Furthermore, he drew crosses instead of circles

We close the discussion of the two sketches from Figs. 2.35 and 2.36 by the remark that they share one common line in Einstein’s drawing. It is the line CB in Fig. 2.36. As it has no specific meaning in Fig. 2.35 (we did not draw it there), we argue that Einstein simply extended this line such that it overlaps with the other sketch. Let us now consider the next page of the notebook (page 50). It begins with a small sketch that only shows four crosses where the three left crosses are combined by a brace, see Fig. 2.40. It is a very generic sketch and allows many different interpretations. For instance, it could stand for the harmonic relation of four points. In fact, it possibly is not even a complete sketch. Beneath this sketch, however, we find a short calculation consisting of five lines. Imposing a specific notation as in Fig. 2.40 enables us to connect the sketch directly with the calculation. In this case, we can interpret both the sketch and the calculation in terms of hyperbolic involutions, where the brace gets a meaning by pointing to the two invariant points. Hence, let us call the points O, .X1 , A, and .X2 and impose the coordinates .O = (0 : 0 : 1), .X1 = (x1 : 0 : 1), .A = (a : 0 : 1), and .X2 = (x2 : 0 : 1) with .a /= 0 in the projective plane, all lying on the line .x2 = 0. We choose .x3 = 0 to be the line at infinity. In the affine plane, we then express the coordinates as .O = (0, 0), .X1 = (x1 , 0), .A = (a, 0), and .X2 = (x2 , 0) such that the points .X1 , A, and .X2 have

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the distances .x1 , a, and .x2 from O. Equivalently to the first equation on AEA 62785r in Eq. (2.2), we can express the involution generally by rxy + s (x + y) + t = 0,

.

(2.28)

where x, y is a pair of the involution and r, s, and t are real numbers determining the involution. Imposing the assumption that both O and A are invariant points, we get t =0

(2.29)

ra 2 + 2sa = a (ra + 2s) = 0

(2.30)

.

from O being invariant and .

from A being invariant. By setting .r = 2, we get .s = −a. In the case that the two points .X1 = (x1 , 0) and .X2 = (x2 , 0) are corresponding points of the involution, it is 2x1 x2 − a (x1 + x2 ) = 0.

(2.31)

x1 x2 : = −1. a − x1 a − x2

(2.32)

.

This can be recast to .

We recall that the cross ratio .(OA, X1 X2 ) of the four points is (OA, X1 X2 ) =

.

a − x2 a − x1 (x2 − a) x1 = : = −1, x2 x1 x2 (x1 − a)

(2.33)

where we used Eq. (2.32) in the last equation. This implies that the four points O, A, .X1 , and .X2 form a harmonic set. So far, this result is no surprise since the two invariant points of a hyperbolic involution are harmonic conjugates to any other pair of the involution. In the Prague notebook, Einstein worked the calculation the other way around starting from the harmonic relation in Eq. (2.32). He then rearranged it and derived x1 (a − x2 ) = −x2 (a − x1 )

(2.34)

2x1 x2 − a(x1 + x2 ) = 0,

(2.35)

.

as well as .

2.5 Prague Notebook and Its Connection with Princeton Manuscripts

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corresponding to Eq. (2.31). As we have seen above, this equation describes a hyperbolic involution. He then implicitly set .x1 = x2 = x and derived x 2 − ax = 0.

.

(2.36)

By setting .x1 = x2 = x, Einstein determined the invariant points of the involution. The quadratic equation in Eq. (2.36) thus gives us the coordinates of the two invariant points. Einstein concluded109 x = 0 and x = a

.

(2.37)

implying that .O = (0, 0) and .A = (a, 0) are the invariant points of the hyperbolic involution. Next to Eq. (2.37), Einstein wrote down the word “Doppelv.” or “Doppelp.” Both expressions can be interpreted in the context of Einstein’s calculation. The former stands for “Doppelverhältnis,”110 which was Einstein’s starting point in Eq. (2.32) and probably was considered in Fig. 2.40. The latter stands for “Doppelpunkt,”111 a term frequently used for invariant points.112 These double points were determined by Einstein by setting .x1 = x2 = x and satisfy Eq. (2.37). As Einstein’s comment is placed next to this equation, we argue that it stands for “Doppelp.” even though it looks more like “Doppelv.” We summarize that by imposing the notation from Fig. 2.40, the sketch and the short calculation become directly connected to each other and can be interpreted as an investigation of hyperbolic involutions. We note that the imposed notation was not arbitrarily chosen as Einstein used the notations .x1 , .x2 , and a in his calculation. Finally, we note that Einstein’s calculation and especially his final solution in Eq. (2.37) are wrong in a strict sense, since it is .x = x1 /= a as well as .x = x2 /= 0 in Einstein’s first Eq. (2.32). Einstein’s calculation and considerations bear a strong resemblance to the manuscript page AEA 62-785r. There, he also considered the equation of an involution, see Eq. (2.2). Furthermore, next to the calculation, he then made a construction of pairs of an involution on a line for the hyperbolic, elliptic, and parabolic cases, see Figs. 2.9 and 2.10. We saw in Fig. 2.34 that Einstein considered a hyperbolic involution on a line in the Prague notebook as well as he interchanged the points B and .B ' , while the points A and .A' remained invariant. As we saw above, the calculation in the Prague notebook determines pairs of a hyperbolic

109 Instead

of writing “and,” Einstein used the German abbreviation “u” for “und.” ratio”. 111 “Double point”. 112 For instance, see Grassmann (1909, 111), Enriques (1915, 68), or Coxeter (1961, 32). As we know from Grosmann’s lecture notes, Fiedler used this term in his lecture attended by Einstein as well, see Grossmann (1897, 71). We discussed this passage already in Fig. 2.11 when analyzing the manuscript page AEA 62-785r. 110 “Cross

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Fig. 2.41 Reconstruction of Einstein’s fifth sketch on the double page. Einstein did not draw the gray line and did not name the objects B and .B ∗ . Moreover, he corrected the notation for the points ∗ ∗ .A and .C . For a better visualization, we also introduced the notation for the points E, F , G, and H as well as for the line g

involution. As we will see in the following, he even drew another sketch right next to this calculation where he constructed pairs of an involution in full analogy to the manuscript page AEA 62-785r. The only difference is that he used the radical axis for the construction on the manuscript page AEA 62-785r, while he used a complete quadrangle in the Prague notebook. The respective sketch is shown in Fig. 2.41. It is the fifth and last sketch on projective geometry. Einstein crossed it out. He used the notation of the points A and C as in Fig. 2.41 and corrected the notations for the points .C ∗ and .A∗ afterward as he interchanged them: .A∗ became .C ∗ and vice versa. Imposing the initial notation as depicted in Fig. 2.41 allows the following interpretation. Given the quadrangle with vertices E, F , G, and H , its opposite sides meet a line g not passing through any of its vertices in pairs of an involution. The opposite sides of the quadrangle are GF and H E, GE and H F as well as H G and EF .113 Thus, the points A and .A∗ as well as C and .C ∗ are corresponding points of the involution, fitting to Einstein’s initial notation. The third pair B, .B ∗ was not drawn by Einstein. He only drew the line EF dashed but did not name B and also did not draw the line H G.114 It seems as if the goal of this construction was to construct these two points. However, after drawing the dashed line, Einstein crossed out the sketch and renamed the points .C ∗ and .A∗ . The reason for this becomes clear when considering all sketches on the Prague notebook together. We saw that he considered a hyperbolic involution in his first sketch by carrying out two perspectivities that interchange the two points B and ' ' .B and let the two points A and .A invariant. In his second and third sketches, he considered the transition from a projectivity on a conic to a hyperbolic involution.

113 We 114 We

recall that opposite sides do not meet each other in a vertex. note that Einstein used the notations A and C, but not B.

2.5 Prague Notebook and Its Connection with Princeton Manuscripts

137

In doing so, he skipped the intermediate step of an elliptic involution. In his fourth sketch as well as in the related calculation, Einstein considered a hyperbolic involution as well by setting .x1 = x2 = x corresponding to the two invariant points. In his fifth sketch from Fig. 2.41, however, he drew the complete quadrangle such that the corresponding points A, .A∗ and C, .C ∗ separate each other. This means that the involution is an elliptic one. We can see this, for instance, by looking at Einstein’s construction of pairs of an hyperbolic involution using the radical axis on the manuscript page AEA 62-785r, see Fig. 2.9. There, any two pairs .X1 , .Y1 and .X2 and .Y2 constructed by circles through .S1 and .S2 separate each other if and only if the line e lies between .S1 and .S2 corresponding to an elliptic involution.115 When Einstein noticed that the involution in his fifth sketch is an elliptic involution, he apparently renamed the points such that the involution became a hyperbolic one. However, in this case, the construction does not work anymore as the opposite sides of the complete quadrangle do not meet each other in corresponding points. As a consequence, Einstein crossed out this sketch. Hence, we argue that Einstein considered hyperbolic involutions in all of his sketches on projective geometry. Einstein’s fifth sketch appears on the second page of the Prague notebook. Thus, it lies next to the first sketch from Fig. 2.34. This allows an alternative interpretation of his first sketch. Considering the complete quadrangle with vertices .β, S, .β ' , and R, the opposite sides are .Rβ and .β ' S meeting at B, RS and .ββ ' meeting at .α ' as well as .Rβ ' and .βS meeting at .B ' . Hence, the line g passes through the two diagonal points B and .B ' of the complete quadrangle. We can interpret Einstein’s first sketch as a hyperbolic involution on the line g with the two invariant points B and .B ' , where A and .A' is another pair of the involution. Although Einstein’s notation as well as his additional comment indicates that he considered two perspectivities as shown above, it is possible that Einstein considered it this way in the process of drawing his fifth sketch as both sketches lie next to each other. Starting from Einstein’s first sketch from Fig. 2.34, we get the situation from his fifth sketch in Fig. 2.41 by moving point S slightly to the upper right such that it comes to lie near .β ' . The situation is visualized at the beginning of the video sequence “Notebook3.mp4.”116 We recall that such a situation already appeared on the manuscript page AEA 62789, see Sect. 2.3.3. There, Einstein also drew two sketches next to each other. The first one showed the involution where the separate line passed through two diagonal points of the complete quadrangle, while in the second sketch, the line did not pass through any diagonal point. We come from the fifth sketch in the Prague notebook shown in Fig. 2.41 to the second sketch on AEA 62-789 shown in Fig. 2.20 by moving the point G to the upper right and renaming the points.117 Both sketches show the construction of pairs of an involution by using a complete quadrangle, where the line of the involution does not pass through any diagonal

115 See

also Coxeter (1961, 53) and Footnote 93. animation is accessible at https://doi.org/10.1007/s00407-020-00270-z as supplementary material to Sauer and Schütz (2021) (visited on 06/10/2021). 117 We also need to slightly move the remaining points. 116 The

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Fig. 2.42 Similarities between the three sketches on AEA 62-789 (a), 124-446 (b), and in the Prague notebook (c) showing either the construction of points of an involution starting from two invariant points or two perspectivities carried out consecutively resulting in a hyperbolic involution. Images have been published in Sauer and Schütz (2021). © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama. (a) Einstein’s top left sketch on AEA 62789, see Fig. 2.22. (b) Einstein’s middle sketch on AEA 124-446, see Fig. 2.32. (c) Einstein’s sketch from Fig. 2.34 in the Prague notebook

Fig. 2.43 Similarities between the two sketches on AEA 62-789 (a) and in the Prague notebook (b) showing the construction of points of an involution starting from two pairs of distinct points. Images have been published in Sauer and Schütz (2021). © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama. (a) Einstein’s top right sketch on AEA 62789, see Fig. 2.20. (b) Einstein’s sketch from Fig. 2.41 in the Prague notebook

point. Next to both sketches, we find the sketches where the line passes through two diagonal points. In the Prague notebook, this sketch is the first one shown in Fig. 2.34, while it is also the first sketch on AEA 62-789 shown in Fig. 2.22. Again, we note that Einstein’s introduced notation in the Prague notebook suggests that he, there, initially had in mind the context of perspectivities resulting in a hyperbolic involution. This was also the case in Fig. 2.32 from AEA 124-446 with an even more explicit notation. All these sketches (the two sketches from the Prague notebook, the two sketches from AEA 62-789 as well as the sketch from AEA 124-446) are shown in Figs. 2.42 and 2.43. The transition between the sketches in the Prague notebook and those on AEA 62-789 in the context of the construction of pairs of an involution is also

2.6 Concluding Remarks

139

visualized in the video sequence “Notebook3.mp4.”118 The connection between Einstein’s first sketch in the Prague notebook and the sketch on AEA 124-446 in the context of perspectivities is visualized at the beginning of the video sequence “Notebook1.mp4.”119 This video sequence also shows the connection between the sketches from Fig. 2.42 and the first, fourth, and fifth sketches on AEA 62-789r that all show the complete quadrangle and its derivations as well, see Figs. 2.25, 2.26, and 2.27.120 We conclude this section by summarizing that in his Prague notebook, Einstein drew five sketches and made one calculation that are all related to projective geometry. More precisely, we argue that all considerations are about hyperbolic involutions with two invariant points. We saw that the first sketch is connected to the fifth sketch showing the construction of pairs of an involution on a line by using a complete quadrangle. By the notation introduced in Einstein’s first sketch, he initially considered the construction of a hyperbolic involution by carrying out two perspectivities consecutively. In his fifth sketch, he constructed an elliptic involution. As he intended to construct a hyperbolic involution, he renamed the points and crossed out the construction. The second and third sketches belong together and show the transition from a projectivity on a conic to a hyperbolic involution on a conic by using Pascal’s theorem. The fourth sketch is linked to the calculation starting from the harmonic relation and computing the two invariant points of the corresponding hyperbolic involution. Furthermore, all of these considerations have an analogue in the Princeton manuscripts. The first and fifth sketches appear on AEA 62-789, the second and third sketches appear on AEA 62-787r, and the calculations as well as the fourth sketch are connected to the calculation on AEA 62-785r. Furthermore, the first sketch with a very specific notation appears on the working sheet AEA 124-446.

2.6 Concluding Remarks In this chapter, we analyzed and interpreted several sketches and calculations appearing in Einstein’ research notes from different time periods. The double page of the Prague notebook was written between 1912 and 1915, probably in 1913. The manuscript page AEA 124-446 was dated to the time period between 1929 and 1932

118 The

animation is accessible at https://doi.org/10.1007/s00407-020-00270-z as supplementary material to Sauer and Schütz (2021) (visited on 06/10/2021). 119 The animation is accessible at https://doi.org/10.1007/s00407-020-00270-z as supplementary material to Sauer and Schütz (2021) (visited on 06/10/2021). 120 There, Einstein considered one point being the point at infinity.

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according to the database of the Albert Einstein Archives, and the manuscript pages AEA 62-785r, 62-787r, 62-789, and 62-789r can be dated to summer 1938.121 All considerations on the research notes can directly be linked to the subject of projective geometry. More precisely, Einstein considered involutions with a special focus on the point at infinity. Moreover, we argue that in the Prague notebook, Einstein drew sketches that are all connected to hyperbolic involutions. The appearance of these sketches alone is an intriguing fact as it is evidence that Einstein as a theoretical physicist of the twentieth century knew the subject of projective geometry very well even though this field of mathematics did not play a prominent role in his physical theories. The presented sketches and calculations are an example of evidence-based statements about what Einstein did know about projective geometry. It is all the more astonishing that the sketches in the Prague notebook are strikingly similar to those on the manuscript pages. In fact, it is another instance of the appearance of equivalent subjects in Einstein’s research notes that were written some quarter century apart that might entertain the same underlying trains of thought.122 We visualized the strong connections between the different sources in several video sequences by using the interactive geometry application GeoGebra. We saw that Einstein attended certain courses on projective geometry during his student years that discussed phenomena that also appear on his research notes. This shows that he either had not forgotten this subject until 1938, or he relearned it back then. Both options show that Einstein had a certain interest in and a specific knowledge of projective geometry and that he maintained an active command of basic notions of projective geometry throughout his life. We saw in Sect. 1.3 that Einstein remembered Geiser’s lectures on infinitesimal geometry during his development of the theory of general relativity. By our analysis of his research notes, we have now encountered another example of Einstein using mathematical tools that he learned at ETH. In addition, we found strong connections between Einstein’s sketches and Grossmann’s own lecture on projective geometry from 1907.123 In the following, we will conjecture on the purpose and motivation behind Einstein’s ideas on projective geometry.

121 We already gave a short summary of the interpretation of the sketches appearing on the manuscripts in Sect. 2.4. For a short summary of the main results related to the sketches from the Prague notebook, see the last paragraph of the preceding Sect. 2.5. 122 We recall that Einstein’s research notes on gravitational lensing appeared in the Prague notebook as well as in a publication and on research notes from 1936, see Renn et al. (1997); Einstein (1936); Sauer and Schütz (2019); Schütz (2017). 123 See our discussion in the last paragraph of Sect. 2.3.3.

2.6 Concluding Remarks

141

2.6.1 Conjecture on the Purpose of Einstein’s Ideas About Projective Geometry Looking at each sketch individually, it would have been likely that Einstein was just doodling. This alone would be interesting as it shows Einstein’s knowledge in projective geometry. However, the fact that Einstein investigated a very specific subject in projective geometry, namely involutions and particularly hyperbolic involutions, both in the Prague notebook and on the Princeton manuscripts and the fact that these sketches are directly connected to each other suggest that he drew these sketches and made the related calculations with a certain purpose in mind. In the following, we will conjecture on this purpose. It substantially has been formulated in Sauer and Schütz (2021). As we saw in Sect. 2.1.2, Einstein knew Veblen and Hoffmann’s five-dimensional projective theory from 1930. As the sketches on projective geometry can be dated to 1938124 when Einstein was working on a five-dimensional approach in order to generalize Kaluza’s theory,125 it might be tempting to associate any explicit consideration of projective geometry on Einstein’s part with projective relativity of some form or other. Perhaps Einstein might have had in mind to interpret the four-dimensional space-time as some projected version of the five-dimensional space-time. In fact, we will see in Chap. 4 that in his calculations on the fivedimensional approach, he imposed centrally symmetry such that the number of dimensions could be reduced. However, we have not been able to establish any convincing link along these lines. Instead, we suggest to draw our attention to the further calculations on the respective pages of the Princeton manuscripts and to Bergmann’s letter AEA 6264 that we have already mentioned in Sect. 2.1.3. Some of the text passages and calculations on these pages directly belong to the publication (Einstein and Bergmann 1938) or the Washington manuscript that we will discuss in Chap. 3. The other parts of the pages can be connected to the correspondence and to the further development of the new theory. We will look at this in Chap. 4 and already gave a summary of Einstein’s further considerations in Sect. 1.5. Einstein and his assistants were looking for particle-like solutions in the framework of their new theory.126 The new theory was based on Kaluza’s fivedimensional approach assuming a five-dimensional metric that is periodic with respect to the fifth dimension. By elaborating their theory, Einstein and his assistants assumed further specifications as stationarity with respect to the fourth, time-like coordinate and spatial, three-dimensional spherical symmetry. By the latter, they

124 See

also Sect. 4.3.1. will discuss Einstein and Bergmann’s publication on this subject in Chap. 3. 126 This was also stated in their follow-up paper (Einstein et al. 1941). See also Van Dongen (2002, 2010). 125 We

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derived certain equations that only depend on a radial coordinate r.127 In order to find particle-like solutions of the field equations that can be interpreted physically, Einstein required regularity at the origin .r = 0 and a certain fall-off at .r = ∞.128 Furthermore, they considered several power series expansions of new functions .α(r), .β(r), .δ(r), and .η(r) that also appear in the metric components. It is in this context, where Bergmann wrote about the point at infinity. After writing down his deliberations on the power series expansions, he concluded that “man mit einer Entwicklung um den unendlich fernen Punkt starten kann” (AEA 6-264).129 This is the reason why we claim neither that Einstein’s considerations on projective geometry are related to projective relativity nor that Bergmann’s expression was used in the context of projective geometry. We rather conjecture that Einstein considered projective geometry and particularly involutions in order to get a better understanding of power series expansions at infinity. As we will discuss in Chap. 4, a power series expansion at infinity usually is done by considering the transformation .r → 1/r and then expanding the power series expansion at the origin. Clearly, the general equation of an involution that Einstein considered in both his Prague notebook and on the Princeton manuscripts in Eqs. (2.28) and (2.2) is a general case of this transformation. Interpreting it in terms of involutions, the point at infinity goes to the origin, while the origin goes to the point at infinity. We thus conjecture that Einstein used projective geometry to investigate the properties of such power series expansions at infinity, for example, by looking at a more general and sophisticated case or by trying to get a geometric understanding of it. Our conjecture needs to be further elaborated and tested. If it is right, it would allow us to draw implications on Einstein’s heuristics during the time period of formulating the theory of general relativity as we find these sketches already in the Prague notebook from 1912/15. We recall that these sketches are embedded in research notes on gravitational lensing. In fact, Einstein there considered the magnification factor of gravitational lenses and derived a diverging expression on the page preceding the double page.130 However, in contrast to Einstein’s considerations on projective geometry, the point at infinity does not play a decisive role in his considerations on gravitational lensing. We will come back to our conjecture in Sect. 4.3.1.5 after having discussed Einstein and Bergmann’s publication, their correspondence, and the further calculations on the manuscript pages.

127 We will connect them with the new field equations. We also identified the metric components that they probably considered, see Sect. 4.2. 128 For these purposes, Einstein introduced a new variable .ρ, see also Chap. 4. 129 “One can start with an expansion around the point at infinity.” Translation by the author. We already looked at this quote in Sect. 2.1.3. 130 See Klein et al. (1993, 587) and Sauer (2008, 17).

Chapter 3

Different Pathways to the Generalization of Kaluza’s Theory

We already saw that the manuscript pages discussed in Chap. 2 contain text passages and calculations that cannot directly be associated with projective geometry. In particular, the second part of the manuscript page AEA 62-7891 contains considerations that are related to Einstein and Bergmann’s publication “on a generalization of Kaluza’s theory of electricity” (Einstein and Bergmann, 1938). We can see this, for instance, by the appearance of terms as Schnitt-Transf[ormation]2 and Vierertransf[ormation]3 that are introduced in the publication (Einstein and Bergmann, 1938, p. 686). In this chapter, we will look at this publication in more detail such that we can analyze the rest of the manuscript page. In this process, we will learn about further documents that were written in the context of the generalization of Kaluza’s theory. For instance, there is a preliminary, German version of the publication extant (AEA 1-133). We will also learn about a third document, the so-called Washington manuscript entitled “Einheitliche Feldtheorie”4 that has never been published. The title alone is a noteworthy fact as it is the only document published or unpublished that carries this title without any further specification. It is written and composed around the same time, however, without Bergmann being co-author.5 The extant versions of this manuscript are dated to July 6, 1938. At that time, Einstein and Bergmann still

1 We

discussed the sketches on this page in Sect. 2.3.3. Our italics. 3 Four-transformation. Our italics. 4 “Unified field theory.” 5 Some results of the discussion of the Washington manuscript have been published in Sauer and Schütz (2020). 2 Cut-transformation.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Schütz, Einstein at Work on Unified Field Theory, Einstein Studies 17, https://doi.org/10.1007/978-3-031-52127-0_3

143

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3 Different Pathways to the Generalization of Kaluza’s Theory

discussed the proofs of their publication as we will learn from their correspondence. From the Washington manuscript, two apparently identical typescripts are extant with handwritten equations and comments (AEA 2-121 and 5-008) as well as a holographic version (AEA 97-487). Further documents as AEA 2-119 are also extant that contain previous text passages of the Washington manuscript. It is interesting to see that Einstein chose a different structure in order to discuss substantially the same theory in his Washington manuscript compared with the publication. As we learn from the correspondence between Einstein and Bergmann, Einstein preferred his Washington manuscript over the publication. We will discuss and compare these two different pathways of exploring the new theory which sheds light on Einstein’s modus operandi. In fact, it reminds us of how he explored the implications of gravitational lensing in several documents along different pathways by starting from various assumptions, see the “space of implications” in Sauer and Schütz (2019). We will see that Einstein built his new theory in the Washington manuscript upon three axioms. We will connect this axiomatic formulation with earlier statements of Einstein’s on the role of axioms in theoretical physics. In doing so, we will argue that he changed his opinion over the years and that the Washington manuscript is an example for Einstein’s goal to find the “fewest possible mutually independent hypotheses” (Einstein, 1922a, p. 1).6 In addition, we will not only look at the content of the manuscript but also address the question of why Einstein composed it. We will also investigate the circumstances that finally led to a donation to the Library of Congress and discuss the role of Einstein’s good friend Elias Avery Lowe who was professor for paleography at the Institute for Advanced Study. It will also allow us to briefly shed light on the public role of Einstein’s later works. As mentioned above, we will also discuss further manuscript pages. In fact, we found several pages that can be associated with the publication and with the Washington manuscript. While some of them will only be mentioned (AEA 62-373r, 62-800, 62-807, 63-026, 63-026r, and 63-325), we will discuss the manuscript pages AEA 62-789, 62-798, 62-802, 62-794, and 62-785 in more detail. Two of them (AEA 62-789 and 62-785) are connected to the manuscript pages with projective geometry. The page AEA 62-798 is interesting as it contains also calculations on the further elaboration of Kaluza’s theory that we will address in Chap. 4. The two remaining manuscript pages AEA 62-802 and 62-794 are particularly interesting as Einstein investigated the Lagrange density in order to derive the new field equations by neglecting derivatives of the connections. As we will see, this is a method that he already used in his review paper from 1916. We will start in Sect. 3.1 with a derivation of Einstein’s field equations of the general theory of relativity from a modern point of view. We will also present another Lagrange density that is not invariant anymore, but which is nevertheless suitable for a derivation of the field equations. This will help us later to analyze the manuscript pages. In Sect. 3.2, we will then look at Einstein and Bergmann’s

6 Translation

by Kormos Buchwald et al. (2012b, p. 249).

3 Different Pathways to the Generalization of Kaluza’s Theory

145

theory from 1938 in great detail by deriving the respective field equations. We will base our derivation on Einstein and Bergmann’s publication and provide additional information and calculations. For instance, Einstein and Bergmann discussed the derivation of the field equations of the classical theory of general relativity but only briefly outlined the derivation of the new field equations. We will carry out the respective derivations in more detail. The underlying theory will also be the basis for the subsequent Chap. 4. We will first discuss the space structure in Sect. 3.2.1 and then derive the field equations for the theory of general relativity for a second time in Sect. 3.2.2. This time, however, we look at Einstein and Bergmann’s derivation using tensor densities. This will be especially helpful as the derivation of the field equations of the generalized Kaluza theory in Sect. 3.2.3 is then based on the same ideas. By analyzing correspondence from that time period, we will learn in Sect. 3.3 about Einstein’s very optimistic view on his new theory. Further correspondence7 leads us to the unpublished Washington manuscript that Einstein composed simultaneously to the publication. We will discuss it in Sect. 3.4 and start with a short characterization of the manuscript. We will then look at the correspondence between Einstein and Bergmann in Sect. 3.4.1. There, we will not only learn more about the manuscript itself but also about the fact that Einstein preferred his Washington manuscript over the publication. In Sect. 3.4.2, we will show that different versions of the manuscript exist and shed light on the question of why Einstein composed this manuscript. We will also look at its donation to the Library of Congress. While we discuss the different versions of this manuscript and their differences in Sect. 3.4.3, we will point out the differences between the manuscript and the publication in Sect. 3.4.4. In this chapter, we will also discuss the internal structure of the Washington manuscript and look at its axiomatic formulation. In doing so, we will draw a connection to some statements Einstein made between 1915 and 1925. We will then briefly discuss a correction of a text passage in the publication in Sect. 3.4.5 and finish this section with some concluding remarks in Sect. 3.5. In Sect. 3.6, we will contextualize several manuscript pages. In particular, the previous sections will enable us to connect some considerations on the manuscript page AEA 62-789 in Sect. 3.6.1 with the Washington manuscript. This is especially interesting as it is one of the manuscript pages with sketches on projective geometry, see Sect. 2.3.3. We will come back to this manuscript page and its back page in Sect. 4.3.1 where we then also date this manuscript page accurately. In Sect. 3.6.2, we then briefly look at a passage on the manuscript page AEA 62-798 and connect it to the publication. In Sect. 3.6.3, we will then investigate the two pages AEA 62802 and 62-794. On these two pages,8 Einstein considered the derivation of the Lagrange density that he used in order to derive the field equations of his new theory. We will see that Einstein used certain relations and tricks that he did not use in neither the publication nor the Washington manuscript. In particular, we will

7 We 8 The

will discuss this correspondence in Chap. 4. page AEA 62-794 is the continuation of AEA 62-802.

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3 Different Pathways to the Generalization of Kaluza’s Theory

show that he neglected the terms containing derivatives of the Christoffel symbols. We then connect this procedure to other publications Einstein’s as his review paper from 1916 (Einstein, 1916a). In Sect. 3.6.4, we will discuss the manuscript page AEA 62-785 and show that Einstein there derived the covariant derivative of tensor densities. We will also connect it to the Washington manuscript and we will see that Einstein communicated his results with Bergmann in the letter AEA 6-266. This is especially interesting as the back of the page (AEA 62-785r) is one of the manuscript pages containing sketches on projective geometry.9 Finally, we briefly conclude this chapter in Sect. 3.7.

3.1 Einstein’s Field Equations of General Relativity We first derive the field equations using the variational principle as it is done in modern textbooks.10 We consider Hamilton’s principle .δWG = 0, where ⎛ WG =

LG d n x

.

(3.1)

is the action with the Lagrange density .LG . The Lagrange density is a tensor √ density11 which means that it can be written as .LG = −gR, where R is the Ricci scalar.12 It is the contraction of the Ricci tensor λ λ σ σ Rμν = R λμλν = ∂λ ┌μν − ∂ν ┌μλ + ┌σλ λ ┌μν − ┌σλ ν ┌μλ

.

(3.2)

which in turn is the contraction of the Riemann tensor R

.

ρ

ρ

μλν

ρ

ρ σ σ = ∂λ ┌μν − ∂ν ┌μλ + ┌σ λ ┌μν − ┌σρ ν ┌μλ .

(3.3)

We derive the field equations by varying the action with respect to the covariant metric components .gμν . Thus, we consider the expression ⎛ δWG =

.

(√ √ ) √ −gg μν δRμν + −gRμν δg μν + g μν Rμν δ −g dn x.

(3.4)

Using λ λ δRμν = ∇λ (δ┌μν ) − ∇ν (δ┌μλ )

.

9 See

Sect. 2.3.1. Carroll (1997), Landau et al. (1989), Goenner (1996), Fliesßach (2016). 11 See also Sect. 3.2.2. 12 We will come back to this in Sect. 3.6.3.1. 10 See

(3.5)

3.1 Einstein’s Field Equations of General Relativity

and Gauss’s theorem, it is ⎛ ⎛ ⎞ ⎛ √ μν σ λ n √ , dn−1 xnσ −g g μν δ┌μν − g μσ δ┌μλ . d x −gg δRμν =

147

(3.6)

∂V

V

with the boundary .∂V of V and the normal vector .nσ . This integral vanishes as it is ρ δ┌μν = 0 on .∂V . By

.

√ 1√ δ −g = −gg μν δgμν , 2

.

(3.7)

we get ⎛ ⎞ √ 1 dn x −g −g μσ g νλ Rμν δgσ λ + g μν Rδgμν 2 ⎛ ⎞ ⎛ 1 μν σ μ λν n √ = − d x −g g g Rσ λ − g R δgμν . 2 ⎛

δWG =

.

(3.8)

By the fundamental lemma of calculus of variations, we get Einstein’s contravariant field equations 1 R μν − g μν R = 0 2

.

(3.9)

and the covariant field equations 1 Rαβ − gαβ R = 0 2

.

(3.10)

in vacuum. In order to obtain the field equations for the non-vacuum case, we consider the action ⎞ ⎛ ⎛ 4 c LG + LM dn x, (3.11) .W = 8π G with the normalized Lagrange density for gravitation and .LM the Lagrange density for matter. We then get Einstein’s field equations 1 8π G Rαβ − gαβ R = 4 Tαβ . 2 c

.

(3.12)

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3 Different Pathways to the Generalization of Kaluza’s Theory

Before going further to Einstein and Bergmann’s publication, we note that it is also possible to consider only ~G = L

.

√

⎛ ⎞ σ σ −gg μν ┌σλ λ ┌μν − ┌σλ ν ┌μλ

(3.13)

instead of the Lagrange density .LG . We see that the derivatives of the Christoffel symbols do not occur in the new Lagrange density.13 Clearly, this Lagrange density is not an invariant scalar anymore; however, it can nevertheless be used to derive the field equations, see Blau (2019, pp. 385–387).

3.2 Einstein and Bergmann’s Generalized Kaluza Theory In the following, we look at the theory of Einstein and Bergmann’s 1938 publication “On a Generalization of Kaluza’s Theory of Electricity” (Einstein and Bergmann, 1938). There is also a German draft version extant in AEA 1-133. This manuscript is in Einstein’s hand and contains almost the same content as their publication. Both the publication and the German draft first introduce Kaluza’s five-dimensional theory and then the generalized theory. This is supposed to “make the reading easier” (Einstein and Bergmann, 1938, p. 683). In addition to these two documents, the Washington manuscript14 exists that we will discuss comprehensively in Sect. 3.4. It is also written in German and dated July 6, 1938. In this manuscript, Einstein introduced the theory independently of the previous Kaluza theory in order to emphasize the logical structure of the new theory.15 In the following, we will introduce Einstein and Bergmann’s generalized theory based on their publication, the German draft, and the Washington manuscript. We will use our own notation.16 For further literature on this theory, we refer to Bergmann (1942, pp. 272–279) and Van Dongen (2010, pp. 139–143). For a discussion of Einstein and Bergmann’s paper, see also Witten (2014), who devoted his article to Einstein and Bergmann’s attempt of interpreting the fifth dimension physically.

13 In

Sect. 3.6.3.1, we will see that Einstein considered such a Lagrange density on one of his manuscript pages. 14 Einstein referred to this manuscript as Washington manuscript. For instance, see the letters AEA 6-256 or 6-271. 15 This is one of the reasons why Einstein preferred his manuscript version over the publication. Both a holographic and typed version are extant. See the documents with the archival numbers AEA 97-487, 2-121, and 5-008. Some passages and related calculations can also be found on several manuscript pages as on AEA 2-119 or 62-789, see also Sect. 3.6. 16 For instance, in Einstein and Bergmann’s publication, they did not use the notation .∂ A or a b .∇a Ab , but .Ab,a or .Ab;a instead.

3.2 Einstein and Bergmann’s Generalized Kaluza Theory

149

3.2.1 Space Structure and Properties Einstein and Bergmann considered a five-dimensional space with the metric .

dσ 2 = γμν dx μ dx ν ,

(3.14)

where Greek indices stand as a representation of .1, 2, 3, 4, 0. Latin indices run only from 1 to 4. The main difference to Kaluza’s theory in Kaluza (1921) is that Einstein and Bergmann now gave a specific physical reality to the fifth dimension by interpreting it as “approximately” four-dimensional (Einstein and Bergmann, 1938, pp. 687–688). While Kaluza assumed a space which is invariant along the 0 17 Einstein and Bergmann considered a space that is .x dimension like a cylinder, 0 periodic with respect to the .x dimension. Hence, we can find coordinates such that γμν (x 1 , x 2 , x 3 , x 4 , x 0 + nλ) = γμν (x 1 , x 2 , x 3 , x 4 , x 0 ),

.

(3.15)

where .λ does not depend on the coordinates .x μ and .n ∈ N. This means that a point P of the physical space is represented by an infinite number of points .Pn . Similar to Kaluza’s theory, they then demanded that given two points P and .P1 , which represent the same physical point, one and only one geodesic line that they called A-line exists which connects the two points P and .P1 . Hence, all points .Pn that represent the same physical point lie on this A-line. Einstein and Bergmann then introduced a so-called special coordinate system. Given all A-lines, we consider a four-dimensional hypersurface at .x 0 = 0 that cuts each A-line once and only once. A point Q on the .x 0 = 0 surface is then defined by the four coordinates .x a . The coordinates also define an A-line through this point along which .x a is constant. Any point P that is lying on this A-line is then characterized by the four coordinates .x a of Q and the distance between P and Q measured along the A-line. Einstein and Bergmann then introduced ⎛P1 b=

dσ

.

(3.16)

P

for two corresponding, consecutive points P , .P1 lying on one A-line, where b depends only on .x a . For the .x 0 coordinate of the point P , they set 1 .x = b

⎛P

0

dσ, Q

17 Kaluza

called this “Zylinderbedingung” (Kaluza, 1921, p. 967).

(3.17)

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3 Different Pathways to the Generalization of Kaluza’s Theory

where we recall that Q is the intersecting point between the A-line and the .x 0 = 0 surface.18 By this convention, two corresponding, consecutive points on one A-line differ in their .x 0 coordinate by . Δ x 0 = 1.19 Accordingly, it is .

dσ 2 = b2 d(x 0 )2

(3.18)

for any line element along an A-line. Clearly, on the A-lines, it is .

dx 0 1 d2 x 0 dx a = 0 and = as well as = 0. dσ dσ b dσ 2

(3.19)

We have now introduced the special coordinate system in which the five coordinates x 0 to .x 4 correspond to any space point. In the following, we will derive specific relations for the metric components .γμν . On geodesic lines, it generally is20

.

.

ν μ d2 x α α dx dx = 0. + ┌ μν dσ dσ dσ 2

(3.20)

By Eqs. (3.19) it follows that ⎛ α ┌00

.

dx 0 dσ

⎞2 = 0,

(3.21)

and therefore α ┌00 =

.

1 ασ γ (2∂0 γ0σ − ∂σ γ00 ) = 0 2

(3.22)

implying21 .2∂0 γμ0 − ∂μ γ00 = 0. Hence, it is ∂0 γ00 = 0

(3.23)

2∂0 γa0 − ∂a γ00 = 0.

(3.24)

.

and .

we assumed that P is lying on the positive .x 0 side. a strict sense, it is .b = P P 1 /λ, where .P P 1 stands for the integral in Eq. (3.16) and .λ was chosen as .λ = 1. In his Washington manuscript, Einstein more precisely set .x 0 · P P 1 /λ = σ . We will come back to this in Sect. 3.4.4.4. 20 For instance, see (Fliesßach, 2016, p. 72). 21 It is .det(γ μν ) /= 0. 18 Here, 19 In

3.2 Einstein and Bergmann’s Generalized Kaluza Theory

151

We note that Eq. (3.23) is no surprise as Eq. (3.18) holds along an A-line. In particular, it is .γ00 = b2 . Hence, taking the integral of Eq. (3.24) along the A-line over one period leads us to ⎛P1 .

⎛ (2∂0 γa0 − ∂a γ00 ) dx 0 = −∂a ⎝γ00

P

⎛P1

⎞ dx 0 ⎠ = −∂a b2 = 0,

(3.25)

P

where we used the periodicity of .γ0a and Eq. (3.23) in the first equality. Hence, both γ00 and b are not only independent of .x 0 but also of .x a and we can set .b = 1 in the whole space implying

.

γ00 = 1 and ∂0 γa0 = 0

.

(3.26)

by Eq. (3.24).22 Einstein and Bergmann then introduced the contravariant unit vector .Aμ = (0, 0, 0, 0, 1) in the special coordinate system such that it is Aα = Aβ γβα = γ0α .

(3.27)

.

They also introduced the new metric tensor gαβ = γαβ − Aα Aβ .

(3.28)

.

It is clear that by Eq. (3.26), all components of .gαβ where .α = 0 and/or .β = 0 vanish. Thus, we basically consider a four-dimensional metric tensor .gab . The difference to Kaluza’s theory is that the components .gab depend on .x 0 as .γab and hence .gab are periodic functions of .x 0 .23 Einstein and Bergmann then chose .gab and .Am as the field variables. We will see that .Am then are regarded as the electromagnetic potential. The next step is to look at possible coordinate transformations. Given an A-line, the four coordinates .x a have been chosen arbitrarily such that we can introduce new coordinates .x ' a with the transformation rules x ' = x ' (x 1 , · · · , x 4 ),

.

a

a

x' = x0 0

(3.29)

that still correspond to a special coordinate system. Einstein and Bergmann called transformations of this kind Four-transformations. Furthermore, the four-

= 1, one cannot let the coordinate distance be . Δ x 0 = 1 anymore but need to let it variable by setting . Δ x 0 = λ. See also Footnote 19. In Einstein’s Washington manuscript, he directly chose this approach, see Sect. 3.4.4.4. 23 We note that .A = γ does not depend on .x 0 because of Eq. (3.26). m a0 22 At this point, Einstein and Bergmann remarked that if setting .b

152

3 Different Pathways to the Generalization of Kaluza’s Theory

dimensional hypersurface at .x 0 = 0 was chosen arbitrarily. It is also possible to choose another surface according to the A-line if the .x a do not change. Thus, transformations of the kind x' = xa,

.

a

x ' = x 0 + f (x 1 , · · · , x 4 ), 0

(3.30)

preserve the special coordinate system as well and are called Cut-transformations. It follows that ∂x a .

∂x

= 0,

'0

∂x 0 ∂x

'0

= 1, and

∂x 0 = 0 for four-transformations, ∂x ' a

(3.31)

while it is .

∂x a = 1, ∂x ' a

∂x a ∂x ' b

= 0 (a /= b),

∂x a ∂x ' 0

= 0,

∂x 0 ∂f ∂f = − 'a = − a ∂x ' a ∂x ∂x

∂x 0 ∂x ' 0

= 1, and (3.32)

for cut-transformations. In the last equality, we used .

∂f ∂x α ∂f ∂f = a a = ' ∂x ∂x ' a ∂x α ∂x

(3.33)

as f does not depend on .x 0 . Einstein and Bergmann called the quantity .a s a contravariant vector,24 if it transforms with respect to four-transformations as a' =

.

s

∂x ' s t a ∂x t

(3.34)

and if it is invariant (.a ' s = a s ) with respect to cut-transformations.25 The definitions of a covariant vector and further tensors are analogous.26

24 We

here note that s runs from 1 to 4, but .a s depends on .x 0 to .x 4 . see that the four-transformation is defined according to the transformation of tensors in the four-dimensional theory of general relativity. We will come back to this fact when looking for covariant derivatives. 26 See Eqs. (12a) and (12b) in Einstein’s Washington manuscript AEA 5-008, 97-487, and 2-121. For a covariant vector .as , it is .a ' s = ∂x t /∂x ' s · at with respect to four-transformations and .a ' s = as for cut-transformations. 25 We

3.2 Einstein and Bergmann’s Generalized Kaluza Theory

153

Since the metric components .γμν are tensors in the five-dimensional space,27 they generally transform as γ ' μν =

.

∂x α ∂x β γαβ . ∂x ' μ ∂x ' ν

(3.35)

With respect to four-transformations, they transform as γ ' mn =

.

∂x α ∂x β ∂x a ∂x b ∂x 0 ∂x b ∂x a ∂x 0 γa0 m n γαβ = m n γab + m n γ0b + ' ' ' ' ' ' ∂x ∂x ∂x ∂x ∂x ∂x ∂x ' m ∂x ' n +

∂x 0 ∂x 0 ∂x a ∂x b γ = γab , 00 ∂x ' m ∂x ' n ∂x ' m ∂x ' n

γ ' 0m =

∂x α ∂x β ∂x β ∂x a γαβ = ' m γ0β = ' m γ0a , m ' 0 ∂x ∂x ∂x ' ∂x

γ ' 00 =

∂x α ∂x β ∂x ' 0 ∂x ' 0

and

γαβ = γ00 = 1,

(3.36)

while they transform with respect to cut-transformation as γ ' mn =

.

∂x a ∂x b ∂x 0 ∂x b ∂x a ∂x 0 ∂x α ∂x β γ = γ + γ + γa0 αβ ab 0b ∂x ' m ∂x ' n ∂x ' m ∂x ' n ∂x ' m ∂x ' n ∂x ' m ∂x ' n +

∂x 0 ∂x 0 ∂f ∂f ∂f ∂f γ0n − n γm0 + m n , m n γ00 = γmn − ' ' m ∂x ∂x ∂x ∂x ∂x ∂x

γ ' 0m =

∂x α ∂x β ∂x β ∂f ∂f γ = γ = γ0m − m γ00 = γ0m − m , αβ m ' m 0β 0 ∂x ' ' ∂x ∂x ∂x ∂x

γ ' 00 =

∂x α ∂x β ∂x ' 0 ∂x ' 0

γαβ = γ00 = 1.

and (3.37)

We see that the five-dimensional metric components .γμν are not invariant with respect to cut-transformations such that they are no tensors under these transformations.28

27 By

the above definition, we will see that the components .γmn are no tensors with respect to cut-transformations. 28 The considerations from Eqs. (3.36) and (3.37) cannot be found in the publication nor the Washington manuscript in such a detail. However, we find comprehensive calculations related to those transformations on the manuscript page AEA 62-789, see Sect. 3.6.1. Furthermore, the paragraph where Einstein concluded that the metric components .γμν are no tensors were discussed by Einstein several times. For instance, we find the respective paragraphs in many letters and on manuscript pages. It is possible that these considerations might have motivated Einstein to compose the Washington manuscript. We will discuss them in Sects. 3.4.5, 3.6.1, and 3.6.2.

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3 Different Pathways to the Generalization of Kaluza’s Theory

This is also the reason why Einstein and Bergmann introduced the fourdimensional metric components .gmn and .g mn . Here, the contravariant components are defined as29 l gmn g ln = δm .

.

(3.38)

In fact, these are tensors under cut- and four-transformations. We will show it for gmn . They transform with respect to four-transformation as

.

g ' mn = γ ' mn − γ ' 0m γ ' 0n =

.

=

∂x b ∂x a ∂x b ∂x a γ − γ γ0b ab 0a n m m ∂x ' ∂x ' n ∂x ' ∂x '

∂x a ∂x b ∂x a ∂x b − γ γ gab = (γ ) ab 0a 0b ∂x ' m ∂x ' n ∂x ' m ∂x ' n

(3.39)

and with respect to cut-transformations, they are invariant because of g ' mn = γ ' mn − γ ' 0m γ ' 0n

.

⎛ ⎞⎛ ⎞ ∂f ∂f ∂f ∂f ∂f ∂f = γmn − m γ0n − n γm0 + m n − γ0m − m γ0n − n ∂x ∂x ∂x ∂x ∂x ∂x = γmn − γ0m γ0n = gmn .

(3.40)

Furthermore, the components defined by30 Amn := ∂n Am − ∂m An

.

(3.41)

are components of a tensor as well, because of ⎛ Amn = ∂n

.

=

⎞ ⎛ ⎞ ∂x ' b ' ∂x ' a ' Ab A a − ∂m ∂x n ∂x m

∂x ' a ∂x ' b ∂x ' b ∂x ' a ' ' ' ∂ A + A ∂ − A ∂ − ∂m A' b a a b n n m ∂x m ∂x m ∂x n ∂x n

∂x ' a ∂x ' α ∂ ∂x ' b ∂x ' α ∂ ' A − A' b a ∂x n ∂x m ∂x ' α ∂x m ∂x n ∂x ' α ⎛ ⎞ ∂ ∂x ' a ∂x ' b ' ∂ ∂x ' a ∂x ' b ' ' = = m A − A A ab , a b ∂x ' a ∂x m ∂x n ∂x ∂x n ∂x ' b =

(3.42)

29 See also page 9 in Einstein’s Washington manuscript AEA 2-121 or 5-008. In AEA 97-487, it is on page 7. 30 We introduced .A in Eq. (3.27). m

3.2 Einstein and Bergmann’s Generalized Kaluza Theory

155

with respect to four-transformation, where we used Eq. (3.31) in the fourth equality. With respect to cut-transformation, it is ⎛ ⎞ ⎛ ⎞ ∂f ∂f Amn = ∂n A' m − ' m − ∂m A' n − ' n ∂x ∂x

.

=

∂A' m ∂ ∂f ∂ ∂f ∂A' n − n 'm + m 'n − m n ∂x ∂x ∂x ∂x ∂x ∂x

∂A' m ∂A' n ∂ ∂f ∂ ∂f ∂x ' α ∂A' m ∂x ' α ∂A' n − − + = − ∂x n ∂x m ∂x n ∂x m ∂x m ∂x n ∂x n ∂x ' α ∂x m ∂x ' α ∂f ∂A' m ∂f ∂A' n ∂A' m ∂A' n ∂A' n ∂A' m − − = − = A' mn , − = m n m n n ∂x ' ∂x ' ∂x ' 0 ∂x ' ∂x ' 0 ∂x ' ∂x ' m ∂x ' (3.43)

=

where we used Eq. (3.33) in the third equality, Eq. (3.32) in the fifth, and Eq. (3.26) in the sixth equality. We see that in Einstein and Bergmann’s generalized fivedimensional theory, .Am itself is not a tensor according to Eq. (3.37), but the antisymmetrical derivative .Amn is a tensor. Let us briefly recall the classic electromagnetic case: Consider the electromag~k − ∂k A ~i , where .A ~i denotes the classic electromagnetic netic tensor .Fik = ∂i A potential. The potential is defined up to the gradient of a time-independent function. In analogy to the five-dimensional theory, we therefore can regard .Am as the electromagnetic potential, where only .Amn = ∂n Am − ∂m An is a tensor. Before we address the variational principle regarding the generalized Kaluza theory, we will discuss the formation of new tensors according to Einstein and Bergmann (1938, pp. 692–693), where we will find expressions for both the connection and the curvature tensor. Given a tensor, we can always form a new tensor by differentiation with respect to .x 0 , since it is ∂ .

∂x ' 0

=

∂ ∂x α ∂ = 0 0 ∂x α ' ∂x ∂x

(3.44)

with respect to both four-transformations and cut-transformations according to Eqs. (3.31) and (3.32). Hence, differentiating a scalar .ρ with respect to .x 0 remains a scalar. This also holds for vectors and tensors. Differentiating them with respect to .x 0 does not change their tensor character. Given a contravariant vector .a r , it is ∂a ' r .

∂x ' 0

=

∂ ∂x ' 0

⎛

∂x ' r s a ∂x s

⎞ = as

∂ ∂x ' r ∂x ' r ∂ s ∂x ' r ∂a s + a = ∂x s ∂x ' 0 ∂x s ∂x 0 ∂x ' 0 ∂x s

(3.45)

with respect to four-transformations and .

∂a r ∂a ' r ∂a ' r = = ∂x 0 ∂x 0 ∂x ' 0

(3.46)

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3 Different Pathways to the Generalization of Kaluza’s Theory

with respect to cut-transformations. Analogously, this holds for tensors of other kinds as well. We will now look at the differentiation with respect to .x a and want to find the covariant differentiation. Given the scalar .ψ, it is for cut-transformations .

∂ψ ' ∂ψ ∂f ∂ψ ∂x α ∂ψ = a − a 0, a a = ' α ' ∂x ∂x ∂x ∂x ∂x ∂x

(3.47)

where we used Eq. (3.32) in the second equality. We see that the factor of the additional term also appears in .A' a from Eq. (3.37). Hence, the operator ⎛ .

∂ ∂ − Aa 0 ∂x a ∂x

⎞ (3.48)

acting on a tensor does not affect the invariance with respect to cut-transformations. We will explicitly show this for the scalar .ψ and for the contravariant vector .a r . It is ⎛ .

∂ ∂ ' a −Aa ' ∂x ∂x ' 0

⎞

ψ' =

⎞ ⎛ ∂ψ ∂ψ ∂f ∂ψ ∂f − − A − a a a a 0 ∂x ∂x ∂x ∂x ∂x 0 ⎛ ⎞ ∂ ∂ = − Aa 0 ψ ∂x a ∂x

(3.49)

where we used Eqs. (3.37), (3.44), and (3.47) in the first equality. For the contravariant vector .a r it follows that ⎞ ⎞ r ⎛ ⎛ r ∂ ∂a ∂ ∂a r ∂a r ∂f 'r ' ∂a ' a = ' a − A a 0 = ' a − Aa − a . a −Aa ' 0 ' ∂x ∂x ∂x ∂x ∂x ∂x 0 ∂x ⎞ ⎛ ∂a r ∂x α ∂a r ∂f = ' a α − Aa − a ∂x ∂x ∂x ∂x 0 ⎞ ⎛ ⎞ r ⎛ r ∂a ∂a ∂f ∂a r ∂f − A = − − a ∂x a ∂x a ∂x 0 ∂x a ∂x 0 ⎞ ⎛ ∂ ∂ ar, = − A (3.50) a ∂x a ∂x 0 where we used the invariance of .a r under cut-transformations and Eq. (3.46) in the first, Eq. (3.37) in the second, and Eq. (3.32) in the fourth equality. Let us now turn to four-transformations. By Eq. (3.34), we know that tensors transform with respect to four-transformation as usual tensors in the four-

3.2 Einstein and Bergmann’s Generalized Kaluza Theory

157

dimensional tensor algebra.31 Thus, we know that the expression32 1 ∂a Bs − g lm (∂a gsm + ∂s gam − ∂m gsa ) Bl 2

.

(3.51)

behaves like a tensor with respect to four-transformations, if .Bs is a covariant vector. We now need to modify Eq. (3.51) to make it invariant with respect to cuttransformation without changing its behavior with respect to four-transformations. If we substitute .∂a in expression (3.51) by the operator (3.48), we get 1 ∂a Bs − Aa ∂0 Bs − g lm [(∂a gsm − Aa ∂0 gsm ) + (∂s gam − As ∂0 gam ) 2

.

− (∂m gsa − Am ∂0 gsa )] Bl .

(3.52)

Comparing with expression (3.51), the new expression (3.52) contains the additional terms Aa ∂0 Bs and

.

1 lm g Bl (Aa ∂0 gsm + As ∂0 gam + −Am ∂0 gsa ) . 2

(3.53)

However, with respect to four-transformations, .Aa = γ0a has tensorial character because of Eq. (3.36).33 In addition, the quantities .Bl and .g lm are also tensors, wherefore .∂0 Bs and .∂0 gab have tensorial character as well because of the discussion around Eq. (3.44). We conclude that expression (3.52) still behaves like a tensor with respect to four-transformations. We also know that the operator .∂a − Aa ∂0 acting on a tensor leaves the components of the tensor invariant with respect to cut-transformations. Thus, expression (3.52) is also invariant with respect to cuttransformation and therefore a tensor with respect to four- and cut-transformations. Hence, we define the covariant derivative as l ∇a Bs = ∂a Bs − Aa ∂0 Bs − ┌sa Bl ,

.

(3.54)

where l ┌sa =

.

31 See

1 lm g [(∂a gsm − Aa ∂0 gsm ) + (∂s gam − As ∂0 gam ) − (∂m gsa − Am ∂0 gsa )] . 2 (3.55)

also Footnote 25. expression corresponds to the covariant derivative in the four-dimensional theory of relativity. 33 We recall that .γ mn are no tensors as they are not invariant with respect to cut-transformations, see Eq. (3.37). 32 This

158

3 Different Pathways to the Generalization of Kaluza’s Theory

l = ┌ l and thus, the connection is torsion free. We set for a It obviously is .┌sa as contravariant vector .B l s l ∇a B s = ∂a B s − Aa ∂0 B s + ┌la B.

.

(3.56)

The rule for more general tensors is analogous. We conclude that the connection defined in Eq. (3.55) is metric compatible because of ∇a gmn = ∂a gmn − Aa ∂0 gmn

.

1 − δnr [(∂a gmr − Aa ∂0 gmr ) + (∂m gar − Am ∂0 gar ) − (∂r gam − Ar ∂0 gam )] 2 1 r − δm [(∂a gnr − Aa ∂0 gnr ) + (∂n gar − An ∂0 gar ) − (∂r gan − Ar ∂0 gan )] 2 = 0.

(3.57)

It follows analogously that .∇a g mn = 0. We now want to find an expression for the Riemann tensor. Similarly as in the four-dimensional theory, we look at the difference .∇b ∇a Bs −∇a ∇b Bs for a covariant vector with the components .Bs . It is t t ∇b ∇a Bs = ∂b ∂a Bs − Ab ∂0 ∂a Bs − ┌ab ∂t Bs − ┌bs ∂a Bt − Aa ∂b ∂0 Bs − ∂0 Bs ∂b Aa

.

t t t + Ab Aa ∂0 ∂0 Bs + ∂0 Bs Ab ∂0 Aa + ┌ab At ∂0 Bs + ┌bs Aa ∂0 Bt − Bt ∂b ┌as t t t r t r t − ┌as ∂b Bt + Ab Bt ∂0 ┌as + Ab ┌as ∂0 Bt + ┌ab ┌rs Bt + ┌bs ┌ar Bt r t t r + ┌bt ┌as Br − ┌br ┌as Bt .

(3.58)

Since the 1., .(2. + 5.), 3., .(4. + 12.), 7., 9., .(10. + 14.), 15., and .(16. + 17.) terms on the right side are symmetric in a and b, respectively, they cancel each other by taking the difference .∇b ∇a Bs − ∇a ∇b Bs . In addition, the 8. term vanishes because of Eq. (3.26). It follows that ∇b ∇a Bs − ∇a ∇b Bs = − (∂0 Bs ∂b Aa − ∂0 Bs ∂a Ab ) ( t t t t − Bt ∂b ┌as − ∂a ┌bs − Ab ∂0 ┌as + Aa ∂0 ┌bs ) t r t r + ┌br ┌as − ┌ar ┌bs = −Bt R tsab − ∂0 Bs Aab ,

.

(3.59)

where we used again the notation .Aab = ∂b Aa − ∂a Ab and introduced ) ( ) ( t t t t t r t r − ∂a ┌sb − ┌ar R tsab = ∂b ┌sa − Ab ∂0 ┌sa − Aa ∂0 ┌sb ┌sb + ┌br ┌sa .

.

(3.60)

We will call it Riemann tensor in analogy to the four-dimensional theory. The last term on the right hand side of Eq. (3.59) is a tensor because of the discussion around Eq. (3.45) and because of Eqs. (3.42) and (3.43). The left hand side and

3.2 Einstein and Bergmann’s Generalized Kaluza Theory

159

Bt are obviously tensors as well. Thus, the Riemann tensor must be a tensor. It is antisymmetric in its last two indices.34 s is a tensor as well as it is Furthermore, the derivative .∂0 ┌am

.

t ∇a (∂0 Bm ) − ∂0 (∇a Bm ) = ∂a ∂0 Bm − Aa ∂0 ∂0 Bm − ┌am ∂0 B t ( ) t − ∂0 ∂a Bm − Aa ∂0 Bm − ┌am Bt

.

t = Bt ∂0 ┌am ,

(3.61)

because of .∂0 Aa = 0. Since the partial derivative of a tensor with respect to .x 0 is again a tensor according to our discussion around Eq. (3.44), the left side of t is a tensor. Eq. (3.61) is a tensor, which implies that .∂0 ┌am On page 694, Einstein and Bergmann finally listed the invariants used to set up the action:35 H1 = R,

.

H2 = Amn Amn ,

H3 = ∂0 g mn ∂0 gmn , and H4 = g mn ∂0 gmn g rs ∂0 grs . (3.62)

In order to find the field equations, we will use Hamilton’s principle and solve ⎛ δW = δ

.

√ −g (α1 H1 + α2 H2 + α3 H3 + α4 H4 ) d5 x = 0

(3.63)

by varying .g mn and .Am , where .α1 to .α4 are constants. The integral will be taken over a region of the coordinates .x 1 to .x 4 and over one period along the .x 0 direction. In accordance with Einstein and Bergmann’s publication, we also introduce the notation Hi =

.

√

−gαi Hi .

(3.64)

In Einstein and Bergmann’s publication, they then directly derived the new fourteen field equations with a short reference to their appendix. There, they first introduced tensor densities and then derived the field equations by the variation of the Riemann tensor in the four-dimensional theory. In the second part of the appendix, they then transferred the procedure to the five-dimensional theory and briefly outlined the derivation of the first part of the field equations. Compared to our derivation of the four-dimensional field equations in Sect. 3.1, the main difference is that Einstein and Bergmann used tensor densities. As Einstein considered them also on one of his manuscript pages, we will look at Einstein and Bergmann’s appendix in more detail in the following Sects. 3.2.2 and 3.2.3. We also note that Einstein chose

analogy to the four-dimensional theory, we call .Rik = R lilk Ricci tensor and .R = g ik Rik Ricci scalar. See also Eq. (3.2). 35 See also Eq. (25) in Einstein’s Washington manuscript. 34 In

160

3 Different Pathways to the Generalization of Kaluza’s Theory

a different approach in his Washington manuscript by abandoning the appendix and including the mathematical part into the main text. We will come back to this in Sect. 3.4.

3.2.2 Field Equations of General Relativity Einstein and Bergmann used the common Christoffel symbols s ┌mn =

.

1 sl g (∂n gml + ∂m gnl − ∂l gmn ) , 2

(3.65)

∂s g 2g

(3.66)

where it is n ┌ns =

.

with the determinant .g = |gik |. Einstein and Bergmann then referred to Hlavaty (1928) and introduced tensor densities accordingly. A tensor density is a tensor times a factor such that it transforms as a scalar density. A scalar density . ρ of weight n transforms as (n)

| a |n | ∂x | ρ ' = || b || ρ , ∂x ' (n) (n)

(3.67)

.

where the additional factor is the Jacobi determinant. Hence, ordinary tensors can be seen as tensor densities of weight 0. Multiplying two tensor densities of weights m and n yields a tensor density of weight .(m + n). In particular, multiplication of a scalar density of weight 1 and a tensor gives a tensor density of weight 1. Given a tensor density .Bba , the covariant derivative gets an additional term as well such that (n)

it is36 l r ∇s Bba = ∂s Bba − ┌sb Bla + ┌lsa Bbl − nBba ┌sr .

.

(n)

(n)

(n)

(n)

(3.68)

(n)

This holds for tensors of higher rank analogously. By this definition, the differentiation rule of products holds.

36 See

also Sect. 3.6.4.

3.2 Einstein and Bergmann’s Generalized Kaluza Theory

161

We will now show that the determinant g of the covariant metric components is √ a scalar density of weight 2 and that . −g is a scalar density of weight 1. Given the transformation law of the metric components ' gab =

.

∂x m ∂x n gmn , ∂x ' a ∂x ' b

(3.69)

it follows that | a |2 | ∂x | | .g = | | ∂x ' b | g. '

(3.70)

Multiplying with .(−1) and taking the square roots lead us to | a| √ | ∂x | √ −g ' = || b || −g. ∂x '

.

(3.71)

The last two equations show the assertion. It follows that the covariant derivative of the determinant g is r ∇s g = ∂s g − 2g┌sr = 0,

.

(3.72)

where we used Eq. (3.66) in the second equality. Analogously, it is √ √ ∂s g √ √ √ r ∇s −g = ∂s −g − −g┌sr = ∂s −g − −g 2g √ √ √ √ −2 −g∂s −g = ∂s −g − −g = 0. 2g

.

(3.73)

Furthermore, it is l l ∇s gmn = ∂s gmn − ┌ms gln − ┌sn gml

.

1 1 r = ∂s gmn − δnr (∂s grm + ∂m grs − ∂r gms ) − δm (∂n grs + ∂s grn − ∂r gsn ) = 0, 2 2 (3.74) √ and thus .∇s (gmn −g) = 0 because of the product rule. Similarly, it is √ mn −g) = 0. With these preliminary considerations, we can now compute .∇s (g ⎛ δ

.

H1 dτ = 0

(3.75)

162

3 Different Pathways to the Generalization of Kaluza’s Theory

√ √ with .H1 = −gR = −gg km δil R iklm . By our definition of the Riemann tensor from Eq. (3.3), it is i i r r i r r i δR iklm = ∂l δ┌km − ∂m δ┌kl + ┌rli δ┌km + ┌km δ┌rli − ┌rm δ┌kl − ┌kl δ┌rm

.

i r r i r i i i r = ∂l δ┌km + ┌rli δ┌km − ┌kl δ┌rm − ┌lm δ┌kr − ∂m δ┌kl − ┌mr δ┌kl r r i + ┌km δ┌rli + ┌lm δ┌kr i i = ∇l δ┌km − ∇m δ┌kl .

(3.76)

This is equivalent to Eq. (3.5). Einstein and Bergmann, however, derived i i δR iklm = ∇m δ┌kl − ∇l δ┌km .

.

(3.77)

This sign difference happened due to a different definition of the Riemann tensor. Considering the five-dimensional Riemann tensor from Eq. (3.60), we see that its four-dimensional analogue is the negative version of our definition in Eq. (3.3).37 i As the integral of the variation of .Rklm does not depend on this sign, both versions 38 yield the same results. With δgab = −gbr gas δg sr

.

(3.78)

it is √ √ √ √ 1√ −gg km g ab δgab δ( −gg km ) = −gδg km + g km δ −g = −gδg km + 2 ⎞ ⎞ ⎛ ⎛ √ √ 1 1 = −g δg km − g km δra gas δg sr = −g δg km − g km gsr δg sr , 2 2 (3.79)

.

37 Einstein also considered different conventions regarding the contraction of the Riemann tensor. On July 12, Einstein wrote to Bergmann that they contracted the Riemann tensor .R iklm with respect to the first and fourth index in order to get the Ricci tensor .Rik (AEA 6-256). This is different to the usual convention as in Eq. (3.2). In one of his subsequent letters AEA 6-271 on July 15, he corrected himself writing that this mistake only appeared in “his manuscript.” Bergmann came on July 16 to the same result that they used the correct convention, see his letters AEA 6-258 and 6-263. Indeed, contracting the Riemann tensor with respect to the first and fourth index yields the negative version of the Ricci tensor. In Einstein’s Washington manuscript, he did not consider the four-dimensional case at all. However, his five-dimensional Eq. (27) corresponds to the convention from Eq. (3.77). 38 We will use Eq. (3.76) instead of Eq. (3.77).

3.2 Einstein and Bergmann’s Generalized Kaluza Theory

163

where we used Eq. (3.7) for the second equality. We can now solve Eq. (3.75) by ⎛ δ

.

⎛ H1 dτ = δ

√ dτ −gg km δil R iklm

⎛ =

dτ ⎛

=

dτ

┌ √

┐ √ −gg km δil δR iklm + δil R iklm δ( −gg km )

┌ √

⎛ ⎞ i i −gg km δil ∇l δ┌km − ∇m δ┌kl

⎞ ┐ ⎛ √ 1 +Rkm −g δg km − g km gsr δg sr 2 ⎛ ⎞⎞ ┌ ⎛√ ⎛ l m = dτ ∂l −g g km δ┌km − g kl δ┌km +

⎞ ┐ ⎛ √ 1 −gδg km , Rkm − Rgkm 2

(3.80)

where we used .δ(δil ) = 0 in the second equality and Eqs. (3.79) and (3.76) in the √ third equality. In the fourth equality, we used .∇l (g km −g) = 0 and therefore39 √

.

⎛ ⎞ √ ⎞ ⎛ i i l l = −g g km ∇l δ┌km −gg km δil ∇l δ┌km − ∇m δ┌kl − g km ∇m δ┌kl ⎞ ⎛ √ l m − g kl ∇l δ┌km = −g g km ∇l δ┌km ⎛√ ⎞⎞ ⎛ l m −g g km δ┌km − g kl δ┌km = ∇l ⎛√ ⎞⎞ ⎛ l m . −g g km δ┌km − g kl δ┌km = ∂l

(3.81)

The last equality is satisfied, because the expression inside the brackets is a vector density .K l of weight 1.40 Thus it is ∇l K l = ∂l K l + ┌lrl K r − K l ┌lrr = ∂l K l ,

.

(3.82)

where we used Eq. (3.68). The considerations above using tensor densities correspond to the considerations in Sect. 3.1, where we did not use tensor densities, see especially the transition from the covariant to the partial derivatives in Eq. (3.6). In particular, we see that the first part of the integral in the last line of Eq. (3.80) is

39 In

Einstein and Bergmann (1938, p. 699), Einstein and Bergmann considered the respective negative version of this expression due to the different definition of the Riemann tensor, see the discussion around Eqs. (3.76) and (3.77). 40 Note that .δ┌ i is a tensor according to Eq. (3.76), while .√−g is a scalar density of weight 1 by km Eq. (3.71).

164

3 Different Pathways to the Generalization of Kaluza’s Theory

equivalent to the second line of Eq. (3.6). Using the same arguments and Gauss’s k vanish on the boundary theorem, this part does not contribute as the variations of .┌lm of the integration region. Thus, we get the field equations from Eq. (3.80) as 1 Rkm − Rgkm = 0. 2

(3.83)

.

3.2.3 Field Equations in Generalized Kaluza Theory In Eq. (3.83), we already have derived the four-dimensional field equations by the variation of the Ricci scalar. We now perform the variational principle in the generalized Kaluza theory. We saw that tensors in the five-dimensional theory are defined with respect to their behavior under four- and cut-transformations. They transform under fourtransformations as in the four-dimensional theory and are invariant with respect to cut-transformations, see our discussion around Eq. (3.34). In analogy, Einstein and Bergmann defined a scalar density of weight n if it is | a |n | ∂x | | .ρ = | | ∂x ' b | ρ (n) (n) '

(3.84)

with respect to four-transformations41 and if they are invariant .

ρ' = ρ

(n)

(3.85)

(n)

with respect to cut-transformations. The respective rules also hold for tensordensities.42 Comparing the covariant derivatives of tensors in the five-dimensional theory from Eqs. (3.54) and (3.56) with the respective derivatives of the four-dimensional theory yields for the covariant derivatives of tensor densities .Bba of weight n in the (n)

five-dimensional theory in analogy to Eq. (3.68)43 l r ∇s Bba = ∂s Bba − As ∂0 Bba − ┌sb Bla + ┌lsa Bbl − nBba ┌sr .

.

(n)

41 This

(n)

(n)

(n)

(n)

(3.86)

(n)

is similar to Eq. (3.67). We here note that Einstein and Bergmann used Greek symbols in their definition for both scalar densities in the four- and five-dimensional case, see their Eqs. (A 1) and (A 11). In his Washington manuscript, he used Latin indices, see his Eq. (13). 42 Similarly as before, ordinary tensors are tensor densities of weight 0 and the product of two tensor densities of weights m and n is a tensor density of weight .(n + m). 43 Again, the product rule holds. See also Sect. 3.6.4.

3.2 Einstein and Bergmann’s Generalized Kaluza Theory

165

We now show that .g = |gmn | is a scalar density of weight 2 with .∇s g = 0. √ √ Similarly, . −g is a scalar density of weight 1 with .∇s −g = 0. It is | a |2 | ∂x | | .g = | | ∂x ' b | g '

(3.87)

with respect to four-transformation similarly to Eq. (3.70).44 With respect to cuttransformations, it is g' = g

.

(3.88)

by Eq. (3.40). Indeed, the determinant g is a scalar density of weight 2 according to the definition in Eqs. (3.84) and (3.85). It follows similarly to the considerations in √ Eq. (3.71) that . −g is a scalar density of weight 1. Let us now show that the respective covariant derivatives vanish. By Eq. (3.55), it is ┌sll =

.

1 1 lr g (∂s glr − As ∂0 glr ) = ~ ┌sll − As g lr ∂0 glr , 2 2

(3.89)

a is the common Christoffel connection from Eq. (3.65) in the four┌bc where .~ dimensional theory. It follows that

┌sll =

.

1 1 1 ∂s g − As g lr ∂0 glr = (∂s g − As ∂0 g) . 2g 2 2g

(3.90)

Hence, it is ∇s g = ∂s g − As ∂0 g − 2┌sll g = 0

.

(3.91)

(2)

and √ √ √ √ ∇s −g = ∂s −g − As ∂0 −g − ┌sll −g

.

(1)

√ √ −g −g 1 1 ∂s g + As ∂0 g = 0, = − √ ∂s g + √ As ∂0 g − 2 −g 2 −g 2g 2g (3.92)

where we used Eq. (3.90) in the second equality.

44 See

Eq. (3.39) for the transformation of .gmn .

166

3 Different Pathways to the Generalization of Kaluza’s Theory

Finally, we note that given a vector density .B s , the divergence of the vector (1)

density is ∇s B s = ∂s B s − As ∂0 B s = ∂s B s − ∂0 (As B s ),

.

(1)

(3.93)

where we used Eq. (3.86) in the first equality and the fact that .As does not depend on .x 0 in the second equality.45 Similarly as in Eq. (3.82), we conclude that on the right hand side, only partial derivatives appear, while the left hand side is a covariant derivative.46 In the following, we will derive the field equations of the generalized Kaluza theory. For this, we consider Eq. (3.63) and split the integral into four separate integrals that we denote by .W1 to .W4 , respectively.47 We again note that they need to be taken over an arbitrary region of the coordinates .x 1 to .x 4 and over one period of .x 0 , where .δg km depends on .x α and .δAm only on .x a .

3.2.3.1

First Action

We first compute48 ⎛ δW1 = δ

.

√ −gα1 H1 d5 x = δ

⎛ =

α1

⎛

√ −gα1 R d5 x = δ

⎛

√ α1 −gg km δil R iklm d5 x

⎛√ ⎞ √ −gg km δil δR iklm + δil R iklm δ( −gg km ) d5 x.

(3.94)

For the variation of the Riemann tensor, it is i i i i i i δR iklm = ∂m δ┌kl − ∂0 ┌kl δAm − Am ∂0 δ┌kl − ∂l δ┌km + ∂0 ┌km δAl + Al ∂0 δ┌km

.

r r r i i r − ┌km δ┌lri − ┌lri δ┌km + ┌kl δ┌mr + ┌mr δ┌kl i i i i = ∇m δ┌kl − ∇l δ┌km − δAm ∂0 ┌kl + δAl ∂0 ┌km ,

(3.95)

similarly to Eq. (3.76).49 We see that compared to the four-dimensional case, we get two additional terms that will cause that this variation now contributes to .δW1 .

45 See

Eq. (3.26). the four-dimensional theory, we used this fact in Eq. (3.80). 47 Einstein and Bergmann outlined the derivation briefly for the first integral. 48 We note that we used .R = g km δ l R i i klm in the third equality but could also have used .R = g mk δil R imlk . 49 Here, we used the five-dimensional Riemann tensor from Eq. (3.60) that is the negative version of its four-dimensional analogue from Eq. (3.3). See also the discussion around Eq. (3.77). 46 In

3.2 Einstein and Bergmann’s Generalized Kaluza Theory

167

√ For the variation of . −gg km , we use Eq. (3.7). It follows that ⎛

√ α1 δil R iklm δ( −gg km ) d5 x

.

⎛ =

α1 δil R iklm

⎛√

√ ⎞ −gδg km + g km δ −g d5 x

⎞ ⎛ √ 1 α1 −gδil R iklm δg km + g km g ab δgab d5 x 2 ⎞ ⎛ ⎛ √ 1 l i = α1 −g δi R klm − Rgkm δg km d5 x. 2 (3.96) ⎛

=

We here note that in the five-dimensional case, it is .Rkm /= Rmk ,50 but .g km δil R iklm = √ R = g mk δil R imlk . Hence, the integral resulting from the variation of . −gg km contributes51 ⎞ ⎛ ⎛ √ 1 1 1 Rkm + Rmk − Rgkm δg km d5 x. (3.97) . α1 −g 2 2 2 For the first variation of .R iklm , it follows that ⎛ .

√ α1 −gg km δil δR iklm d5 x ⎛

⎛ ⎞ √ i i i i α1 −gg km δil ∇m δ┌kl d5 x − ∇l δ┌km − δAm ∂0 ┌kl + δAl ∂0 ┌km

⎛

⎛ ⎞ √ i i d5 x α1 −gg km δil −δAm ∂0 ┌kl + δAl ∂0 ┌km

⎛

⎞ ⎛ √ l r α1 −g g km ∂0 ┌km δAl d5 x − g kl ∂0 ┌kr

= = =

(3.98)

by using Eq. (3.95). We will show in the following why the two terms with the derivatives do not contribute and can be neglected in the second equality. First, we note that √ √ √ ∇m ( −gg km ) = −g∇m g km + g km ∇m −g = 0,

.

(3.99)

l = ∂ ┌l . does not hold for the four-dimensional case as it is .∂k ┌ml m kl look into Einstein’s handwritten German version of the publication in AEA 1-133 reveals that he initially derived this equation with the term .Rkm instead of .1/2 · Rkm + 1/2 · Rmk and corrected it afterward, see pages (7) and (8) in AEA 1-133. 50 This 51 A

168

3 Different Pathways to the Generalization of Kaluza’s Theory

because of the discussion around Eq. (3.57) and because of Eq. (3.92). Introducing √ i ,52 it follows that the vector density . B m = −gg km δil δ┌kl (1)

√ √ i i −gg km δil ∇m δ┌kl = ∇m ( −gg km δil δ┌kl ) = ∇m B m = ∂m B m − ∂0 (Am B m ), (3.100)

.

where we used Eq. (3.93) in the last equality. The first term on the right hand side does not contribute as the variations vanish at the boundary of the .x a ,53 while the second term can be neglected as we take the integral over one period along .x 0 and as .B m is periodic with respect to .x 0 . Hence, the integral ⎛ .

√ i 5 d x α1 −gg km δil ∇m δ┌kl

(3.101)

does not contribute. The same argumentation holds for the second term √ √ i i ~l , ) = ∇l B −gg km δil ∇l δ┌km = ∇l ( −gg km δil δ┌km

.

(1)

(3.102)

~l . for a certain vector density .B Hence, by using expression (3.97) and Eq. (3.98), the final result of the variation is ⎞ ┌ ⎛ ⎛ ⎛ √ √ 1 1 1 5 Rkl + Rlk − Rgkl δg kl .δW1 = δ −gα1 H1 d x = α1 −g 2 2 2 ⎞ ┐ ⎛ s r δAs d5 x. + g km ∂0 ┌km − g ks ∂0 ┌kr (3.103) Einstein and Bergmann’s additional explanations in the appendix of their joint publication end here. They only give the final results of the three further variations 54 .δW2 , .δW3 , and .δW4 .

i are tensors, while .√−g is a tensor density of weight 1. objects .g km and .δ┌kl 53 See our discussion at the end of Sect. 3.2.2 and around Eq. (3.6). 54 We do not find further explanations in Einstein’s Washington manuscript, either. 52 The

3.2 Einstein and Bergmann’s Generalized Kaluza Theory

3.2.3.2

169

Second Action

We will now compute the second action in Eq. (3.63). We will look at ⎛ δW2 = δ

.

⎛ =δ ⎛ =

√ −gα2 H2 d5 x √ −gα2 Amn Amn d5 x = δ

⎛

√ α2 Amn Ast −gg ms g nt d5 x

( √ √ α2 d5 x Amn Ast −gg ms δg nt + Amn Ast −gg nt δg ms

(√ √ ) +Amn Ast g ms g nt δ −g α2 d5 x −gg ms g nt Amn δAst ⎛ ) √ ms nt + −gg g Ast δAmn = α2 d5 x (L1 + L2 ) ,

(3.104)

where we denote the term in brackets in the second line by .L1 and the term in brackets in the third line by .L2 . We first note that .Amn = ∂n Am − ∂m An is antisymmetric and, therefore, .An l = g nm Aml , .An l = g lm Anm , and .Amn = g ms g nt Ast are antisymmetric as well. Concerning the variations with respect to .g kl on the right hand side of Eq. (3.104), it is ) ) √ ( ( √ −g Amn Amt δg nt + Amn As n δg ms = −g Amk Aml + Akn Al n δg kl ) ( √ = −g Akm Al m + Akn Al n δg kl ( ) √ (3.105) = −g 2Akm Al m δg kl .

.

√ The variation with respect to . −g becomes √ √ 1√ 1 Amn Amn δ −g = Amn Amn −gg ab δgab = − Amn Amn −ggab δg ab , 2 2 (3.106)

.

by using Eq. (3.7) in the first equality. Thus, the first part of the integral in √ Eq. (3.104) containing variations with respect to . −g and .g kl is L1 =

.

√

⎞ ⎛ 1 m mn −g 2Akm Al − Amn A gkl δg kl . 2

(3.107)

170

3 Different Pathways to the Generalization of Kaluza’s Theory

Let us now find an expression containing variations with respect to .As in .L2 . It is ) ( √ √ −g Ast δAst + Amn δAmn = 2 −gAst δAst = 2 −gAst (∂t δAs − ∂s δAt ) ) ( √ = 2 −g ∂t (Ast δAs ) − δAs ∂t Ast − ∂s (Ast δAt ) + δAt ∂s Ast ) ( √ = 4 −g ∂t (Ast δAs ) − δAs ∂t Ast ( ) ┐ ┌ √ = 4 −g ∂t (Ast δAs ) − δAs ∇t Ast + At ∂0 Ast − ┌trs Art − ┌trt Asr , (3.108)

L2 =

.

√

where we used the fact that .Ast is an antisymmetric tensor. We note that .∂0 At = 0 by Eq. (3.26) and therefore .∂0 δAt = 0. Thus, we get ) ( √ √ √ 4 −g ∂t (Ast δAs ) − δAs At ∂0 Ast = ∂t (4 −gAst δAs ) − 4Ast δAs ∂t −g √ √ − ∂0 (4 −gδAs At Ast ) + 4δAs At Ast ∂0 −g. (3.109)

.

√ We now introduce the tensor density . B t = 4 −gAst δAs of weight 1 and get (1)

) ( √ 4 −g ∂t (Ast δAs ) − δAs At ∂0 Ast = ∂t B t ( √ √ ) − ∂0 (At B t ) − 4Ast δAs ∂t −g − At ∂0 −g .

.

(3.110)

Similarly as before, the variations vanish at the boundary of .x a and .B t is periodic with respect to .x 0 . Hence, the integral ⎛ .

[∂t B t − ∂0 (At B t )] d5 x

(3.111)

~2 of .L2 that contributes to the vanishes. By Eqs. (3.108) and (3.110), the part .L integral in Eq. (3.104) is ( ) ( √ √ √ ) ~2 = −4 −gδAs ∇t Ast − ┌trs Art − ┌trt Asr − 4Ast δAs ∂t −g − At ∂0 −g . L (3.112)

.

By Eq. (3.92), it is √ √ √ √ ∇t −g = ∂t −g − At ∂0 −g − ┌tll −g = 0.

.

(3.113)

Thus, the second part on the right hand side of Eq. (3.112) becomes .

( √ √ √ ) − 4Ast δAs ∂t −g − At ∂0 −g = −4Ast δAs ┌tll −g

(3.114)

3.2 Einstein and Bergmann’s Generalized Kaluza Theory

171

~2 can be simplified to and .L ( ) √ √ ~2 = −4 −gδAs ∇t Ast − ┌trs Art − ┌trt Asr − 4Ast δAs ┌tll −g L ⎛ ⎞ √ = −4 −gδAs ∇t Ast − ┌trs Art − ┌trt Asr + ┌ltl Ast √ = −4 −gδAs ∇t Ast , (3.115)

.

where we used that .┌trs = ┌rts is symmetric in t and r, while .Art = −Atr is antisymmetric in t and r which is why .┌trs Art = 0. We conclude that the second action can be recast as ⎞ ┐ ┌ ⎛ ⎛ √ 1 .δW2 = d5 xα2 −g 2Akm Al m − Amn Amn gkl δg kl − 4∇t Ast δAs 2 (3.116) by Eqs. (3.107) and (3.115).

3.2.3.3

Third Action

The third part of the action is ⎛ δW3 = δ

.

⎛ =

⎛

√

d x −gα3 H3 = δ 5

√ d5 xα3 −g∂0 g mn ∂0 gmn

( ) √ √ √ d5 xα3 ∂0 g mn ∂0 gmn δ −g + −g∂0 g mn ∂0 δgmn + −g∂0 gmn ∂0 δg mn . (3.117)

We first note that it is ∂0 g st = −g as g bt ∂0 gab

(3.118)

δgab = −gas gbt δg st .

(3.119)

.

and .

√ As usual, for the variation with respect to . −g in Eq. (3.117), it follows by using Eq. (3.7) that √ 1√ 1√ −g∂0 g mn ∂0 gmn gkl δg kl . −g∂0 g mn ∂0 gmn g kl δgkl = − ∂0 g mn ∂0 gmn δ −g = 2 2 (3.120)

.

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3 Different Pathways to the Generalization of Kaluza’s Theory

Concerning the variation with respect to .gmn , we recast the second term inside the brackets of Eq. (3.117) in order to get ∂0 g mn ∂0 δgmn = −g am g bn ∂0 gab ∂0 (−gmr gns δg rs ) ( ) = −g am g bn ∂0 gab −gmr gns ∂0 δg rs − gmr δg rs ∂0 gns − gns δg rs ∂0 gmr ⎛ ⎞ = ∂0 grs ∂0 δg rs + δg rs g bn ∂0 grb ∂0 gns + g am ∂0 gas ∂0 gmr ( ) = ∂0 gmn ∂0 δg mn + δg kl g rs ∂0 gkr ∂0 gsl + g sr ∂0 gsl ∂0 grk

.

= ∂0 gmn ∂0 δg mn + 2δg kl g rs ∂0 gkr ∂0 gsl ,

(3.121)

where we used Eqs. (3.118) and (3.119) in the first equality and only changed dummy indices in the second to last equality. The last term on the right hand side of Eq. (3.121) already contains the variation with respect to .g kl . Multiplying the first √ term by . −g yields the last term in the brackets of Eq. (3.117). Thus, we get ⎞ ┌ ⎛ 1 − ∂0 g mn ∂0 gmn gkl + 2g rs ∂0 gkr ∂0 gsl δg kl 2 ┐ +2∂0 gmn ∂0 δg mn . (3.122) ⎛

δW3 =

.

√ d5 xα3 −g

The last term in the brackets of Eq. (3.122) can be recast as √ √ √ 2 −g∂0 gmn ∂0 δg mn = 2 −g∂0 (∂0 gmn δg mn ) − 2 −gδg mn ∂0 ∂0 gmn ,

.

(3.123)

where the second term contains the variation .δg mn . For the first term, it is √ √ √ 2 −g∂0 (∂0 gmn δg mn ) = ∂0 (2 −g∂0 gmn δg mn ) − 2∂0 gmn δg mn ∂0 −g √ √ = ∂0 (2 −g∂0 gmn δg mn ) − −g∂0 gmn δg mn g ab ∂0 gab . (3.124)

.

Thus, by Eqs. (3.123) and (3.124), we get ⎛ δW3 =

.

⎛ √ 1 d xα3 −g − ∂0 g mn ∂0 gmn gkl + 2g rs ∂0 gkr ∂0 gsl − 2∂0 ∂0 gkl 2 ) √ −∂0 gkl g rs ∂0 grs δg kl + d5 xα3 ∂0 (2 −g∂0 gmn δg mn ). (3.125) 5

3.2 Einstein and Bergmann’s Generalized Kaluza Theory

173

Since .gmn is a tensor and the partial derivative with respect to .x 0 yields a tensor as √ well,55 the expression . B = 2 −g∂0 gmn δg mn is a scalar density of weight 1. Since (1)

.

B is periodic with respect to .x 0 , the term .∂0 B does not contribute to the integral.

(1)

(1)

Hence, we finally get ⎛ δW3 =

.

3.2.3.4

⎛ √ 1 d xα3 −g − ∂0 g mn ∂0 gmn gkl + 2g rs ∂0 gkr ∂0 gsl − 2∂0 ∂0 gkl 2 ) −∂0 gkl g rs ∂0 grs δg kl . (3.126) 5

Fourth Action

For the fourth and last action, it is ⎛ ⎛ √ √ −gα4 H4 d5 x = d5 xα4 δ( −gH4 ) .δW4 = δ

(3.127)

with √ √ √ δ( −gH4 ) = g mn ∂0 gmn g rs ∂0 grs δ −g + −g∂0 gmn g rs ∂0 grs δg mn √ + −gg mn g rs ∂0 grs ∂0 δgmn √ √ + −gg mn ∂0 gmn ∂0 grs δg rs + −gg mn ∂0 gmn g rs ∂0 δgrs

.

= L1 + L2 + L3 + L4 + L5 ,

(3.128)

where we denoted the terms on the right hand side by .L1 to .L5 , respectively. For L1 , we again use Eq. (3.7) and get

.

L1 = −

.

( )2 1√ 1√ −ggkl δg kl g mn ∂0 gmn g rs ∂0 grs = − −ggkl δg kl g mn ∂0 gmn . 2 2 (3.129)

The terms .L2 and .L4 containing the variations with respect to .g kl are identical by interchanging the dummy indices. Hence, it is √ √ √ −g∂0 gmn g rs ∂0 grs δg mn + −g∂0 grs g mn ∂0 gmn δg rs = 2 −gδg kl g mn ∂0 gkl ∂0 gmn . (3.130)

.

55 See

Eqs. (3.44) and (3.46).

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3 Different Pathways to the Generalization of Kaluza’s Theory

The terms .L3 and .L5 are also identical by interchanging the dummy indices. For these two terms, it is .

√ √ 1 (L3 + L5 ) = −gg mn g rs ∂0 gmn ∂0 δgrs = − −gg mn g rs ∂0 gmn ∂0 (gar gbs δg ab ) 2 ⎛ ⎞ √ = − −gg mn g rs ∂0 gmn gar gbs ∂0 δg ab + gar δg ab ∂0 gbs + gbs δg ab ∂0 gar ⎛ ⎞ √ = − −gg mn ∂0 gmn gbs ∂0 δg sb + δg sb ∂0 gbs + δg ar ∂0 gar ⎛ ⎞ √ (3.131) = − −gg mn ∂0 gmn gbs ∂0 δg sb + 2δg kl ∂0 gkl ,

where we used Eq. (3.119) in the second equality. We have to take into account that the following equations have to be multiplied by the factor 2 at the end. The first term on the right hand side of Eq. (3.131) can be recast as .

√ √ −gg mn gbs ∂0 gmn ∂0 δg sb = − −gg mn gbs ∂0 (∂0 gmn δg sb ) √ + −gg mn gbs δg sb ∂0 ∂0 gmn .

−

(3.132)

We again take the first term of Eq. (3.132) and get .

−

√ −gg mn gbs ∂0 (∂0 gmn δg sb ) ⎛ √ √ ⎞ = −g mn gbs ∂0 ( −g∂0 gmn δg sb ) − ∂0 gmn δg sb ∂0 −g ⎞ ⎛ √ 1√ sb ef sb mn −gg ∂0 gmn δg ∂0 gef = −g gbs ∂0 ( −g∂0 gmn δg ) − 2 ( )2 √ 1√ −g g mn ∂0 gmn gbs δg sb . = −g mn gbs ∂0 ( −g∂0 gmn δg sb ) + 2

(3.133)

We see that the second term on the right hand side is the negative of the term on the right hand side of Eq. (3.129). The first term of Eq. (3.133), however, can be recast as .

√ √ − g mn gbs ∂0 ( −g∂0 gmn δg sb ) = −∂0 ( −gg mn gbs ∂0 gmn δg sb ) √ + −ggbs ∂0 gmn δg sb ∂0 g mn √ + −gg mn ∂0 gmn δg sb ∂0 gbs .

(3.134)

3.2 Einstein and Bergmann’s Generalized Kaluza Theory

175

At this point, we note that the term in the second line of Eq. (3.134) together with the second term on the right hand side of Eq. (3.131) (both multiplied by 2) and the term on the right hand side of Eq. (3.130) cancel each other. All together, we get 5 Σ .

Li =

i=1

( )2 √ 1√ −ggkl δg kl g mn ∂0 gmn + 2 −gg mn gbs δg sb ∂0 ∂0 gmn 2 √ √ + 2 −ggbs ∂0 gmn δg sb ∂0 g mn − 2∂0 ( −gg mn gbs ∂0 gmn δg sb ), (3.135)

where the term in the second line does not contribute to the integral due to its periodicity with respect to .x 0 . We finally get for the fourth action ⎛ δW4 =

.

3.2.3.5

√

d xα4 −ggkl 5

⎛

⎞ )2 1 ( mn mn mn g ∂0 gmn + 2g ∂0 ∂0 gmn + 2∂0 gmn ∂0 g δg kl . 2 (3.136)

The Field Equations

In Eqs. (3.103), (3.116), (3.126), and (3.136), the variations .δg kl are arbitrary functions of .x α , which is why the sum of the coefficients of .δg kl vanishes by the fundamental lemma of calculus of variations. Hence, we get the first set of ten field equations56 ⎛

⎞ ⎛ ⎞ 1 1 1 1 m mn Rkl + Rlk − Rgkl + α2 2Akm Al − Amn A gkl .α1 2 2 2 2 ⎛ ⎞ 1 mn rs rs + α3 − ∂0 g ∂0 gmn gkl + 2g ∂0 gkr ∂0 gsl − 2∂0 ∂0 gkl − ∂0 gkl g ∂0 grs 2 ⎛ ⎞ )2 1 ( mn g ∂0 gmn + 2g mn ∂0 ∂0 gmn + 2∂0 gmn ∂0 g mn = 0, + α4 gkl (3.137) 2 √ where we canceled the factor . −g. By Eq. (3.26), the variation .δAs does not depend on .x 0 , but only on .x a . Thus, the sum of the coefficients of .δAs only vanishes for the integral taken over .x a , but the integral over one period of .x 0 remains. Thus, we get the second set of four field equations ⎛ ⎛ .

56 The

⎛ ⎞ ⎞√ s r − 4α2 ∇t Ast −g dx 0 = 0. − g ks ∂0 ┌kr α1 g km ∂0 ┌km

equations are symmetric in k and l.

(3.138)

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3 Different Pathways to the Generalization of Kaluza’s Theory

We find Eqs. (3.137) and (3.138) in Einstein and Bergmann’s publication as Eqs. (35) and (36).

3.2.3.6

The Identities

Finally, by expressing the invariance of the Hamilton integral with respect to infinitesimal coordinate transformations ⎛ ⎞ 1 4 α 'α .x = x + ξ α x' , · · · , x' , (3.139) Einstein and Bergmann derived the five identities ⎛ 2

.

⎛ ∇s Gk s dx 0 + Ask

I s dx 0 = 0

(3.140)

and ⎛⎛

⎛ .

Grs ∂0 g rs dx 0 − ∇s

⎞ I s dx 0

=0

(3.141)

for the field equations, where the integrals again have to be taken over one period of √ x 0 . The notation .Gkl / −g stands for the left hand side of Eq. (3.137) and .I s is the argument of the integral in Eq. (3.138).

.

3.3 Einstein’s Opinion on the New Theory Einstein was very enthusiastic about his new theory. Even though this is not untypical for Einstein, we want to look at some of his statements. He mentioned it several times during the decisive period of completing his publication. Since Einstein’s letters frequently are undated, we first have to date the letters in question. For this purpose, we have to look closer at the family situation around 1938. Einstein’s son Hans Albert Einstein started working for the Greenville Sediment Load Laboratory in South Carolina in 1938 (Ettema and Mutel, 2006). He, his wife, and their two children moved there in 1938 upon Einstein’s request (Isaacson, 2007, pp. 443–444). Einstein then saw their grandchildren for the first time as he wrote in a letter to his sister Maria57 (AEA 29-425): “Wir sind schon ein paar Tage hier an einer Meeresbucht zusammen, und ich habe endlich meine Enkelchen in natura gesehen und piepen hören.” This letter is undated, however, by

57 Maria Winteler-Einstein was frequently called Maja or Maya. For a short biography, see Stachel (1987a, p. 389).

3.3 Einstein’s Opinion on the New Theory

177

writing about the bay, he referred to his summer residence in Nassau Point, where he stayed for sailing from June 15, 1938.58 In his letter to Maria, he also wrote: “Am ersten Juli beginnt Albert mit seiner wässrigen Praxis” (AEA 29-425), which refers directly to Hans Albert’s new position, where he was put in charge for studies on the sediment transport in streams and rivers (Ettema and Mutel, 2006). This implies that Einstein wrote his letter to Maria between June 15 and July 1 in 1938. In a letter to Mileva (AEA 75-618), Einstein wrote also about Hans Albert’s new position. He started the letter with: “Wir sind noch ein paar Tage vergnügt miteinander, bevor der Adn in seine ledernen Hosen schlupfen muss” (AEA 75618). Adn is a nickname for Einstein’s son Hans Albert that Einstein frequently used, see for instance (Schulmann et al., 1998b, Doc. 22).59 Since Hans Albert started his new position on July 1 according to Einstein’s letter to Maria (AEA 29425), the letter AEA 75-618 was written before. Einstein also mentioned his sailboat in this letter, which implies that he already stayed in Nassau point at the time, together with his son’s family. This indicates that the letter to Mileva (AEA 75618) was written between June 15 (when Einstein went to Nassau Point) and July 1 (when Hans Albert started working in Greenville). In addition, he also mentioned his grandchildren as well as “die Sache mit dem Haus” and suggested that Mileva should move (AEA 75-618). These both issues were discussed by Mileva, Maria, and Einstein in several letters around that period, see for instance AEA 97-242, 144-451, 75-949, and 75-661. Moreover, we know from a second letter to Maria (AEA 97-242) that his son’s family did not stay until June 30. Thus, we can date the letter to Mileva (AEA 75-618) between June 15 and June 30, 1938. Let us now have a look at some of Einstein’s statements about his theory. According to him, he worked “ganz fest wissenschaftlich, sogar erfolgreicher als in vielen früheren Jahren” (AEA 75-618 to Mileva, end of June 1938). Hence, he still worked a lot on his scientific problems and explicitly compared his work at the time with his works in earlier years. He then came to the conclusion that his current work is more successful. A very similar and enthusiastic statement can be found in a letter to Maria from June 30: Wissenschaftlich habe ich eine wunderbare Zeit; erstaunlich dass manchmal noch etwas spriessen kann in einem alten Gehirn. Es gelingt eben noch etwas, was durch die Arbeit von Jahren vorbereitet und zur Reife gekommen ist. (AEA 97-242)

In this passage, he referred to his previous works of the past 20 years as kind of preparatory work, which then culminated in his “final” work from 1938. In a second letter to Maria, which was written some days before (AEA 29425, end of June 1938), it becomes clear what he was looking for: “[I]ch hab das Gefühl, dem Rätsel der Elektrizität auf der Spur zu sein.” He did not mention the

58 This becomes clear by the letter AEA 54-240 written on June 7, where he wrote down his summer address. 59 See also Kox et al. (2009, p. 613).

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3 Different Pathways to the Generalization of Kaluza’s Theory

gravitational part of the unified field theory, but only the electricity. This is also expressed by the title that Einstein and Bergmann chose for their publication. In a letter to Maurice Solovine from June 27, he wrote: Ich arbeite mit meinen jungen Leuten an einer überaus interessanten Theorie, mit der ich die gegenwärtige Wahrscheinlichkeits-Mystik und Abkehr vom Realitäts-Begriff auf dem physikalischen Gebiete zu überwinden hoffe. (AEA 21-236)

Clearly, the two mentioned collaborators are P. Bergmann and V. Bargmann.60 According to this quote, the goal of his work was to replace the emerged quantum theory by going back to his field theory. At the same time, he was uncertain whether he will be successful or not: “Sprechen Sie aber nicht darüber, weil ich noch nicht weiss, ob ich damit zu Ende komme” (AEA 21-236). It is remarkable that Einstein wrote such a statement just a few days before his publication. This directly shows that his theory presented in the Annals of Mathematics was not a completed theory. Some weeks later, he sounded much more optimistic regarding his success and the implications for the future. In a letter to Besso from August 8, he wrote: Wissenschaftlich habe ich eine sehr interessante Zeit. Du weisst ja, dass ich nie an im Wesentlichen statistische Grundlagen der Physik geglaubt habe trotz der Erfolge der Quantentheorie. Nun habe ich in diesem Jahre nach zwanzig Jahren vergeblichen Suchens eine aussichtsreiche Feld-Theorie gefunden, die eine ganz natürliche Fortsetzung der relativistischen Gravitationstheorie ist. Sie liegt auf der Linie der Kaluzaschen Idee vom Wesen des elektrischen Feldes. (AEA 70-368)

We again find the comparison with his past 20 years which seemed to have been used only for the search for his final theory, which he then found in 1938. This new theory replaced the quantum theory as a continuation of his general relativity. But he used the word aussichtsreich, which again indicates that the theory he had established was not a complete one. The same optimism is expressed in a letter to Mileva some weeks before on July 20: Ich habe dieses Jahr etwas Wunderbares in meiner Wissenschaft gefunden. Dies wird sich später wahrscheinlich als entscheidender Fortschritt herausstellen. (AEA 75-949)

In this statement, Einstein again used a word that expresses some kind of uncertainty: wahrscheinlich. But he is very much convinced that his theory will be a success, which only is not visible yet, but will be in some years. Einstein commented the new theory similarly optimistic in several letters to Bergmann in 1938.61 For instance, after having discovered a serious problem on July 15,62 he wrote to Bergmann on the same day: “[M]ein Vertrauen in die Theorie ist sehr groß” (AEA 6-271). Similarly, around July/August, he wrote: “Sie werden

60 P. Bergmann worked with Einstein from 1936 to 1941. V. Bargmann worked with him from 1937 to 1944. See also Sect. 1.6. 61 We will date and discuss the correspondence between Einstein and Bergmann in Chap. 4. 62 In a previous letter, Einstein wrote: “Unser Problem ist viel bösartiger als ich gedacht hatte” (AEA 6-242).

3.4 Einstein’s Washington Manuscript

179

sehen, dass wir das ganze Problem in Princeton in kurzer Zeit gelöst haben werden. Und ich habe alles Vertrauen in die Theorie” (AEA 6-269). He stayed optimistic for the next months, which is evident from a letter to his son from October 1938: Wissenschaftlich geht es trotz allem sehr schön vorwärts. Ich habe Hoffnung, dass wir das Rätsel der elektrischen Atomistik wirklich lösen werden. Dies wird sich in den nächsten Monaten entscheiden. (AEA 75-946)

On December 23, 1938, he again wrote to Solovine expressing his optimism: In der wissenschaftlichen Arbeit bin ich auf eine wunderbare Arbeit gestossen, an der ich mit zwei jungen Kollegen mit grossem Eifer arbeite. Es besteht Hoffnung, auf diese Weise die mir unerträgliche statistische Grundlage der Physik zu überwinden. Es ist eine Erweiterung der allgemeinen Relativitätstheorie von grosser logischer Einfachheit. (Einstein, 1960, p. 76)

However, between this time period and 1941, he changed his mind. In his publication from 1941, together with Bargmann and Bergmann, they stated: [T]his fact [. . . ] causes serious difficulties for the physical interpretation of the theory. It seems impossible to describe particles by non-singular solutions of the field equations. (Einstein et al., 1941, pp. 224–225)

He expressed his problems also in a letter to his son, which is dated between 1939 and 1944 by the Albert Einstein Archives: Ich habe hart gearbeitet, aber nicht viel Erfolg damit gehabt. Ich sehe zwar, wo die Sache hinaus muss und glaube, dass die Idee richtig war, die ich damals bei Euch gehabt habe. Aber man stösst auf ganz neue mathematische Aufgaben, die zu dick sind für mein altes Gehirn. So habe ich wenig Hoffnung, die Sache noch zu einem überzeugenden Ende zu bringen. (AEA 75-905.)

3.4 Einstein’s Washington Manuscript In July 1938, Einstein and Bergmann published their article “On a Generalization of Kaluza’s Theory of Electricity” (Einstein and Bergmann, 1938). As we showed above in Sect. 3.2, one of the aims of the article was to interpret Kaluza’s fifth dimension of the space-time realistically.63 Einstein himself was very much convinced of their theory which is evident from several letters, see Sect. 3.3. It is astonishing that Einstein, while working with Bergmann on the theory, apparently wrote another unpublished manuscript on his own about the same subject dated to July 6, 1938. This manuscript is entitled “Einheitliche Feldtheorie”64 and is, to the best of our knowledge, the only published or unpublished document that carries

63 The

majority of the results presented in the following subsections and in Sect. 3.5 have been published in Sauer and Schütz (2020). 64 “Unified Field Theory.”

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3 Different Pathways to the Generalization of Kaluza’s Theory

this title without any further specification.65 The role of such titles and its public attention is demonstrated in example in Sauer (2006, pp. 416–416): The paper (Einstein, 1929e) entitled “Zur einheitlichen Feldtheorie”66 coming closest to the title of the Washington manuscript received an enormous public attention ending up in Einstein himself publishing two popular accounts on his latest research results in the New York Times and in the London Times, see Einstein (1929b,c).67 With respect to the Washington manuscript, Einstein himself frequently referred to it as “my Washington manuscript” pointing to the fact that he composed it without Bergmann’s collaboration.68 As we will see in Sect. 3.4.2, he donated this manuscript to the Library of Congress located in Washington, which explains its name. In the introduction of his manuscript, Einstein mentioned his collaboration with Bergmann; however, he did not write about their upcoming joint publication:69 In den letzten Monaten habe ich zusammen mit meinem Assistenten P. Bergmann eine einheitliche Feldtheorie entwickelt, welche durch Verallgemeinerung von Kaluza’s Theorie des elektrischen Feldes entstanden ist. Im Folgenden soll diese Theorie unabhängig von ihren historischen Wurzeln dargestellt werden, damit ihre logische Struktur möglichst deutlich hervortrete. (AEA 97-487)

We see that Einstein pointed out the idiosyncrasy of his manuscript, namely that the theory is expounded without including Kaluza’s original theory. In the following, we will first look at Einstein and Bergmann’s correspondence in Sect. 3.4.1, where we will see that Einstein preferred the Washington manuscript over the publication. In Sect. 3.4.2, we will learn about different versions of the Washington manuscript and discuss the donation to the Library of Congress. There, we will also suggest some possible reasons why Einstein composed the manuscript. We will look at the different versions in more detail in Sect. 3.4.3. In Sect. 3.4.4, we will then comprehensively discuss some differences between the Washington manuscript and the publication with a special focus on the axiomatic structure of the manuscript. In particular, there is one paragraph that Einstein considered and changed several times, see Sect. 3.4.5. Finally, we will finish with some concluding remarks in Sect. 3.5.

65 Further titles in German, English, or French containing similar terms as “unified field theory” are Einstein and Mayer (1930), Einstein and Mayer (1931a), Einstein and Mayer (1932a), Einstein (1925b), Einstein (1928a), Einstein (1929a), Einstein (1929e), Einstein and de Donder (1929), Einstein (1930c), Einstein (1930a), Einstein (1930b), Einstein (1953a), see also Sect. 1.4. 66 ”On the unified field theory.” 67 See also Pais (1982, p. 346). For Einstein’s general public image, see Friedman and Donley (1989). 68 See the discussion of their correspondence in Sect. 3.4.1 and, for instance, the letter AEA 6-271. 69 “In the last months, I have developed, together with my assistant P. Bergmann, a unified field theory, which emerged as a generalization of Kaluza’s theory of the electric field. In the following this theory shall be presented independently from its historical roots, in order that its logical structure may come to the fore as clearly as possible” (Sauer and Schütz, 2020, p. 7).

3.4 Einstein’s Washington Manuscript

181

3.4.1 Einstein and Bergmann Correspondence Einstein and Bergmann’s article was received on April 8, 1938 by the Annals of Mathematics and issued in July 1938 (Einstein and Bergmann, 1938). As indicated in the letter to Besso from which we already quoted in Sect. 3.3, the article had not been printed until the beginning of August. On August 8, 1938, Einstein wrote to his friend about their publication in AEA 70-368:70 “Wenn die Arbeit gedruckt ist sende ich sie Dir.” Around this time, a correspondence between Einstein and Bergmann took place while Einstein stayed in Nassau Point, New York and Bergmann in Robinhood, Maine.71 In July 1938, Einstein and Bergmann had two main topics to discuss: The continuation and elaboration of their new theory and the corrections concerning their paper in press. Because of time pressure for the publication process, Einstein and Bergmann temporarily sent each other several letters per day. For instance, between Friday, July 15 and Monday, July 18, they sent in total six different letters.72 We learn from several letters that they discussed the proofs of their paper until late July as, for instance, in AEA 6-266, where Einstein on July 22 or 23, 1938 asked Bergmann for help concerning an English term: Ich sende Ihnen die Korrekturen trotz des Zeitverlustes noch einmal zu, weil ich den korrekten englischen Ausdruck einer nötigen Korrektur nicht finden kann. (AEA 6-266)

In the midst of this correspondence, Einstein mentioned a so-called Washington Manuskript. The first time this expression appears is in AEA 6-256 that we dated to July 12: Ich habe die ganze Theorie neu bearbeitet für das Washington-Manuskript, indem ich nicht von dem Kaluza’schen Spezialfall ausgegangen bin, sondern gleich die neue Theorie systematisch dargestellt und den Anhang in den Text hineingearbeitet habe. Die ganze Sache nimmt so eine überaus schöne Form an, und ich habe wirklich Freude daran. (AEA 6256)

This quote directly indicates that Einstein’s manuscript covers the same theory as the publication, albeit presented differently. Einstein frequently compared the publication with the Washington manuscript and always came to the result that he prefers the exposition given in his manuscript. For instance, in the letter AEA 6-256 on July 12, Einstein pointed out a mistake to Bergmann. A few days later on July 15, Einstein then noticed that the mistake appeared only in his manuscript and not in their paper: “Der Fehler scheint nur

70 “As

soon as the work is printed, I will send it to you.” Translation by the author.

71 Einstein was at Nassau Point from June 15, 1938, see AEA 54-240 or the discussion in Sect. 3.3. 72 The

majority of the letters is not dated accurately. We will re-date them in Sect. 4.1.

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3 Different Pathways to the Generalization of Kaluza’s Theory

in meinem Manuskript zu sein” (AEA 6-271). In the same letter, he then again expressed preference for the treatment in his Washington manuscript when he wrote: Es ist schade, dass wir in unserer Abhandlung von Kaluza ausgegangen sind und die Tensordichten apart behandelt haben. In meinem Washington-Manuskript ist es viel schöner und übersichtlicher [. . . ] (AEA 6-271).

In the following sections, we will encounter further examples where Einstein compared the publication with the Washington Manuscript and then came to the result that he prefers the version of the manuscript.73 We already saw that Einstein composed the Washington manuscript as sole author. In fact, Bergmann apparently did not know the exact content of it. On July 15, for instance, he asked for the manuscript so that he could read it: Wenn Ihnen nicht daran liegt, das Manuskript möglichst rasch nach Washington zu schicken, würde ich mich freuen, wenn Sie es mir vorher nochmal schicken würden. Ich könnte es in 1-2 Tagen lesen und evtl. direkt nach Washington schicken, falls Sie mir die Adresse schreiben. Natürlich nur, falls die Sache nicht eilt. Aber es interessiert mich natürlich, wie die Sache so aussieht. (AEA 6-258)

We see that Einstein apparently planned to send the manuscript to Washington74 and that Bergmann expressed great interest in Einstein’s manuscript. This is understandable as it apparently contained substantially the same theory as their publication, but with a better internal structure. The letter also shows once again that Bergmann was not involved in the process of composing the manuscript. The next reaction to Bergmann’s request known to us is Einstein’s letter AEA 6-269 that he wrote between July 29 and August 3, 1938. At that time, Einstein already sent the manuscript to Washington and downplayed the significance it would have for Bergmann: [. . . ] Ich warte aber damit, bis wir wieder in Princeton sind, weil bei dieser schwerfälligen Korrespondiererei doch nicht viel herauskommt. Dann gebe ich Ihnen auch das nach Washington gesandte Manuskript, das für Sie doch nur formales Interesse hat. (AEA 6-269)

3.4.2 The Donation to the Library of Congress In the database of the Einstein Papers Project, we found a 15-page typescript with the title “Einheitliche Feldtheorie” and the archival numbers AEA 2-121 and AEA 5-008. The typescripts are two identical Xerox copies. This typescript is

73 For

instance, see the letter AEA 6-265. We will come back to this in Sect. 3.4.5, where we will look at several versions of a paragraph that appears in the publication, in their draft version of the publication, in the Washington manuscript and on further manuscript pages. We will see that he apparently was unsatisfied with the respective paragraph in their publication and preferred the version of the Washington manuscript. 74 We will address this issue further in Sect. 3.4.2.

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written in German and dated July 6, 1938 with a typed signature by Einstein. There, Einstein introduced the generalized Kaluza theory that he had “developed together with P. Bergmann in the last months” (page 1, AEA 2-121). In addition to these two typescripts, we found a manuscript page that contains a text passage from the Washington manuscript: In AEA 2-119, the last paragraph of the Washington manuscript can be found almost verbatim handwritten by Einstein. It seems as if AEA 2-119 is a previous version of the last paragraph of the manuscript. In Sect. 3.6.1, we will also see that the manuscript page AEA 62-789 also contains a draft version of a paragraph of the Washington manuscript as well as pertinent calculations related to it. In the following, we will learn about another version of the manuscript that was donated to the Library of Congress. After Einstein’s death, Bergmann, who then was at Syracuse University,75 helped Dukas in the organization of Einstein’s writings. On April 27, 1964, Dukas thanked Bergmann for sending her copies of all documents in Bergmann’s possession related to Einstein (AEA 6-321). In this letter, she also mentioned a copy of a typescript from July 1938 entitled “Einheitliche Feldtheorie,” which Bergmann sent to her. However, she could not find a holographic manuscript in her files as she wrote on April 27, 1964: Da komme ich an die “Einheitliche Feldtheorie”, deren getipptes Manuskript Sie mir sandten, vom July 1938. Ich kann kein handschriftliches Ms.finden, von dem es doch offensichtlich abgeschrieben ist. Die Gleichungen sind,meiner Ansicht nach, von Prof. E. eingesetzt worden. Auch eine Kopie ist nicht vorhanden, die ich doch sicher gemacht haette,wenn ich die Sache getippt haette. (AEA 6-321)

Nor did she find a copy of the typescript. This fact made her skeptical such that she even conjectured that she had not made the typescript by herself. Hence, she asked Bergmann whether his wife did the transcription (AEA 6-321). Dukas herself remembered that she went away for holidays in July or August 1938: Es fiel mir naemlich ein, dass ich im Juli oder August 38 fuer einige Wochen auf Ferien ging– wir waren damals auf Long Island (Nassau Point, Peconic). (AEA 6-321)

As mentioned in a letter by Einstein from July 13 (AEA 54-583), Dukas came back on August 1, 1938. Some months after her first letter to Bergmann, Dukas convinced herself that she made the typed transcription since she recognized the types of her typewriter. However, she was not able to recall any circumstances that led Einstein to compose the manuscript. On January 5, 1965, she wrote to Bergmann: Es scheint von mir getippt zu sein–jedenfalls sieht es wie meine alte Remington portable aus und die Gleichungen sind definitiv in Prof. E’s Handschrift. Ich kann mich aber absolut nicht erinnern, wofuer diese Arbeit bestimmt war, wo sie erschienen ist etc. Sie ist in keiner Bibliographie vermerkt. (AEA 6-322)

75 See

also Sect. 1.6.

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Since she was away for “einige Wochen” (AEA 6-321), it seems likely to us that she made the transcription in a hurry and forgot to make a copy as she usually would have done. She also mentioned in the second letter to Bergmann (AEA 6-322) that the manuscript could have been a gift for the Library of Congress. This is the reason why Bergmann wrote to the Library of Congress on January 14, 1965. According to Bergmann, he had “no indication or recollection of the circumstances leading to the presentation of the holographic manuscript to the Library of Congress” (AEA 6-324). This statement supports the conjecture that Bergmann, indeed, was not involved in the composition of the manuscript. He then asked the Library of Congress whether they could provide any information about Einstein’s contribution. J. C. Broderick, at the time specialist of the Manuscript Division of the Library of Congress, answered Bergmann on February 5, 1965 (AEA 5-007 and AEA 6-325): “This manuscript, a twelve-page autograph document, came to the Library in July 1938, the gift of Dr. Einstein.” Let us briefly recollect that the document with the archival numbers AEA 2-121 and 5-008 is a 15-page typed manuscript rather than a twelve-page autograph manuscript. In the catalog of the Manuscript Division of the Library of Congress, however, a document “Einheitliche Feldtheorie” is listed, signed by Einstein and presented to the Library in 1938 (Manuscript Division, Library of Congress, 2010). In collaboration with the Albert Einstein Papers, we asked for this document and any relevant correspondence. We gratefully acknowledge that the Library of Congress provided a copy of these documents. Indeed, the document in question is a twelvepage autograph document, signed by Einstein on July 6, 1938 (AEA 97-487).76 By an analysis of the several versions of the manuscript, we see that the copies AEA 2121 and 5-008 are more or less typed versions of the holograph, see Sect. 3.4.3. Let us now try to reconstruct the circumstances that led to Einstein’s contribution. When Einstein sent the manuscript to the Library of Congress, he enclosed a cover letter (AEA 97-494) dated July 13, 1938 and addressed to Dr. Herbert Putnam, at the time Director of the Library of Congress. We know by the memorandum of receipt (AEA 97-493) that the Library of Congress received the manuscript on July 19, 1938. Einstein finished the manuscript on July 6 mentioned it in his correspondence with Bergmann for the first time on July 12 (AEA 6-256) and sent it to the Library one day later on July 13. In the meantime, Dukas had to prepare a transcription of the manuscript. This transcription then apparently was given to Bergmann as Einstein indicated in the letter to Bergmann AEA 6-269.77 As mentioned above, Dukas did not make a copy of this transcription nor of the manuscript itself, probably since she made the transcription in a hurry. This is the reason why Bergmann was in possession of the original version of the typescript, while no copy remained in Einstein’s or Dukas’ hands. After Einstein’s death, Bergmann then made a copy and sent it to Dukas as indicated in AEA 6-321. Dukas categorized the typescript and made a second Xerox copy of it. This explains the two archival numbers AEA 5-008

76 In the archives of the Library of Congress, the document carries the archival number MSS19596. 77 We

quoted the respective passage at the end of Sect. 3.4.1.

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and 2-121.78 The Albert Einstein Archives only had Xerox copies of the typescript version (AEA 2-121, 5-008), while we have not been able to locate the original typescript, which probably remained in Bergmann’s possession. The above discussion implies that Einstein already sent the manuscript to the Library when Bergmann asked for it on July 15 (AEA 6-258). In his cover letter to Putnam, Einstein wrote:79 [M]ein lieber Freund, Professor E. Lowe, teilte mir mit, dass Sie gerne ein Manuskript von mir für die Library of Congress hätten. Ich sende Ihnen deshalb beiliegend eine für diesen Zweck angefertigte Ausarbeitung meiner neuesten Theorie, auf die ich ganz besonderen Wert lege. (AEA 97-494)

Putnam apparently asked for a contribution by Einstein explicitly. However, he did not ask Einstein himself, but Einstein’s friend and colleague Prof. Elias Avery Lowe. The fact that Lowe was the person who initiated Einstein’s contribution is also confirmed by a preliminary press release by the Library of Congress (AEA 97-492). Dated on July 20, 1938, it says: It was announced at the office of the Librarian of Congress today that through the good offices of Professor E. Lowe it had received directly from Professor Albert Einstein a holograph manuscript of twelve pages entitled “Einheitliche Feldtheorie”. (AEA 97-492)

It continued with briefly discussing the importance for the scientific world: Men of science working in the same general field will naturally be much interested; and the manuscript is now on exhibition with other scientific material. (AEA 97-492)

As mentioned above, Putnam apparently did not ask Einstein for this contribution. Since Putnam was not present at the library at the time of reception, he asked his Chief Assistant Librarian Martin A. Roberts (1875–1940)80 to send letters of appreciation to Einstein (AEA 97-491) and Lowe (AEA 97-489 and 97-490). In his letter to Lowe, Roberts wrote on July 22, 1938: We are in receipt from Professor Albert Einstein of an original manuscript of his “Einheitliche Feldtheorie” which he informs us was sent at your kind suggestion. (AEA 97-489 and 97-490)

This sentence almost reads as if the library did not expect Einstein’s contribution. However, Putnam did not send this letter personally and we do not know to which extent Roberts was included in Putnam’s agreement with Lowe. Let us now address the question why Einstein donated the manuscript at all and why Putnam apparently asked Lowe instead of Einstein for the contribution.

78 It looks like as if AEA 5-008 is the Xerox copy Bergmann sent to Dukas and AEA 2-121 then is the Xerox copy of the Xerox copy. 79 “My good friend, Professor E. Lowe, informs that you would like to have one of my manuscripts for the Library of Congress. I am sending you herewith a specially prepared copy of my newest theory which I consider particularly worthy” (AEA 97-488). This is a translation of Einstein’s cover letter. 80 For a tribute to Martin Roberts, see Davis (1940). A short characterization as “devoted and selfless public servant” (MacLeish, 1944, p. 11) can also be found in MacLeish (1944, pp. 1–11).

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For this, let us briefly look at the person Prof. E. Lowe in more detail.81 From 1913 to 1948, he was lecturer and reader in paleography at Oxford University (John, 1994). The letter of appreciation, which was sent to Einstein on July 19, reveals that Lowe worked also as the Consultant in Paleography to the Library of Congress (AEA 97-491).82 In 1936, he became professor at the Institute for Advanced Study in the faculty Historical Studies where he retired in 1945 (John, 1994). He kept his appointment in Oxford and became professor emeritus in Princeton. From 1929 on until his death in 1969, Lowe worked on his greatest work Codices Latini Antiquiores, where he supplied facsimiles to all extant Latin literary manuscripts which had been copied before the ninth century (John, 1969).83 This means that Einstein and Lowe were Princeton colleagues from 1936 on. Moreover, as Einstein wrote in his letter to Putnam (AEA 97-494), Lowe was a “good friend” of Einstein. In 1911, Lowe married Helen Tracy Lowe-Porter who later became Einstein’s translator (John, 1994).84 Einstein not only had a good relation to Lowe but also to his wife. He expressed his satisfaction by her translations several times, for example, in a letter to her husband in 1940: Ich lasse Ihrer Frau herzlich danken, dass sie mir schon wieder mit ihrem prächtigen Englisch beisteht; ich fühle mich ähnlich wie ein Stammler, sobald ich etwas englisch sagen soll. (AEA 55-635)

The good relation between Einstein and “the Lowes” is also shown in a letter from Lowe in celebration of Einstein’s birthday in 1945 (AEA 30-815). As Lowe’s daughter wrote in the memoir of her parents, not only Einstein himself but the Einstein family were “family friends” (Lowe, 2006, p. 55). In another letter of appreciation in 1939,85 Einstein mentioned that Lowe had trouble with several institute members:86 Hoffentlich geniessen Sie mit der kleinen Familie den Sommer und hat auch Ihr lieber Mann etwas Freiheit, um den vielen Aerger zu vergessen, den er durch die abscheulichen Instituts-Menschen hat. Wenn ich nur wüsste, was da zu machen ist. (AEA 53-892)

81 Lowe’s

literary estate is located in the Morgan Library (Mayo and Sharma, 1990). also John (1994, p. 4). 83 For further information about Lowe, see also two obituaries of Lowe by two of his former colleagues (John, 1970; Bischoff, 1970). 84 She was also translator of Thomas Mann (Romero, 1980). 85 The letter (AEA 53-892) is undated. However, Einstein directly referred to Porter-Lowe’s translation of his comment to Gandhi’s birthday. In a letter from January 12, 1939, Einstein was asked to send a paper regarding Gandhi’s birthday “not later than 1st March, 1939” (AEA 28459.1). Einstein’s letter to Lowe Porter (AEA 53-892) was sent, in all probability, during the same time period. This is supported by the fact that, according to his letter, Einstein wrote his letter in summer. 86 We know from several letters (AEA 38-093, 38-094, 52-503 and 53-783) about discrepancies in the institute at the time. The professors of the faculty demanded a say regarding the succession of Flexner who was director at the institute from 1930 to 1939 (Bonner, 1998). However, it is not clear if these are the troubles Einstein insinuated. 82 See

3.4 Einstein’s Washington Manuscript

187

This letter shows that Einstein tried to support Lowe concerning the institute matters. It is remarkable that Einstein also advocated Lowe in a very personal manner when Lowe was about to retire in 1944. Einstein apparently endorsed Lowe’s wish for better pension arrangements as he expressed in letters to Leidesdorf (AEA 55-632) and to Fulton (AEA 55-633) acknowledging Lowe’s work on October 16, 1944: “I am well informed as to the high esteem which this scholar [Lowe] enjoys in the scientific world” (AEA 55-632 and 55-633). The above examples show that, indeed, Einstein had an amicable relation to Lowe. Considering this good relation between Einstein and Lowe, it seems logical that Lowe, who was as paleographer naturally interested in handwritten documents, suggested a holographical contribution by Einstein to the Library of Congress. In fact, we have found no evidence that Einstein at any time intended to publish the manuscript. In the following, we will suggest three possible not mutually exclusive reasons that led Einstein to the composition of the Washington manuscript. First, we should take into consideration the good and personal relation between Lowe and Einstein and the fact that Lowe worked as a consultant for paleography for the Library of Congress. We saw that Einstein wrote the manuscript and donated it to the Library at Lowe’s suggestions. It is possible that Lowe was working on a collection of autographs and in this process asked Einstein whether he could contribute to it. Einstein then might have composed the manuscript as a friendly turn for Lowe. Einstein himself was also interested in such projects: The magazine Modern Review, for instance, covered Einstein’s visit to a graphologist who analyzed Einstein’s handwriting in 1930 (Anonymous, 1930, p. 191).87 Furthermore, such contributions from Einstein are not unusual. In 1924, for example, Einstein recorded his voice for the Lautabteilung an der PreuSSischen Staatsbibliothek, which collected voices of famous people for historical interest (Mahrenholz, 2003).88 Another example is Einstein’s handwritten copy (AEA 5-025) of his famous paper “Zur Elektrodynamik bewegter Körper”89 (Einstein, 1905e) in 1943. The manuscript was auctioned in order to promote the sale of war bonds. Afterward, it was presented to the Library of Congress (Manuscript Division, Library of Congress, 2010; Brasch, 1945). At the same time in late 1944 or early 1945, the Library of Congress acquired a second manuscript of Einstein’s entitled “Das BiVektor Feld,”90 which also was auctioned in the interest of war bonds (Manuscript Division, Library of Congress, 2010; Brasch, 1945).

87 See

also AEA 121-159. record, which is lost, contains the archival number Aut 56 and the identification number 16189 with the title “Bericht seiner Forschungsresultate” in the Lautarchiv, see https:// www.lautarchiv.hu-berlin.de/objekte/lautarchiv/16189/ (visited on 06/10/2021). See also (Kormos Buchwald et al., 2015b, Doc. 208). For a voice recording already from 1921, see Janssen et al. (2002, Doc. 50a) in Kormos Buchwald et al. (2012a). 89 “On the electrodynamics of moving bodies” (Stachel, 1989b, Doc. 23). 90 “The bi-vector field.” 88 Einstein’s

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Einstein also donated manuscripts even before 1938. For instance, this was the case with a manuscript of his publication “Die Grundlage der allgemeinen Relativitätstheorie”91 (Einstein, 1916a), see Kormos Buchwald et al. (2009, Doc. 330). This, however, was preceded by a dispute between the astronomer Erwin Freundlich and Einstein. Freundlich apparently thought that Einstein had given the manuscript to him as a present as it becomes clear from a letter by Berliner, who tried to mediate between Einstein and Freundlich, see Kormos Buchwald et al. (2009, Doc. 337). Einstein, however, stated that he never had donated Freundlich the manuscript. He only gave him the manuscript with the request to send it back to Einstein, see Kormos Buchwald et al. (2009, Doc. 330) and Kormos Buchwald et al. (2009, Doc. 339). The disagreement between them severely strained their relationship, such that Einstein “wanted nothing to do” with him anymore (Kormos Buchwald et al., 2009, Doc. 340). In the end, Einstein regained the manuscript and intended to send it to Oppenheim in 1922, who was also a mediator between them and who was to sell it in order to support the Hebrew University in Jerusalem, see Kormos Buchwald et al. (2012a, Doc. 146). However, it turned out differently and Einstein donated it to Heinrich Loewe in order to support several institutions, among them the university library in Jerusalem. This was confirmed by Heinrich Loewe in 1923 (AEA 36-860). It was not until 1925 that the manuscript, indeed, was received by the university library as confirmed by Leo Kohn, the secretary of the “Vorbereitenden Komités für die Universität Jerusalem,” see AEA 36-863. The manuscript is still in possession of the Jewish National and University Library in Jerusalem as part of the Schwadron Collection, see Kormos Buchwald et al. (2009, p. 386). For a more detailed discussion of the donation, see Gutfreund and Renn (2015, pp. 1–5). As a second conjecture regarding the occasion of composing this manuscript, we observe that Abraham J. Karp stated that Einstein presented the manuscript to the library in gratitude for being able to live in the USA (Karp, 1991, p. 207) after Einstein applied to obtain U.S. citizenship in May 1935.92 Einstein was also very active helping friends and colleagues to emigrate to the USA as we learn, for instance, from many letters. Indeed, it is possible that he, for reasons of gratefulness, composed and donated the manuscript. However, Karp did not provide any further evidence for this conjecture. In addition, in this case, the question arises why the composition and donation happened due to Lowe’s suggestions. A third reason for composing the manuscript emerges when looking at the differences between the publication and the manuscript.93 As we already saw in Sect. 3.4.1, Einstein preferred the presentation of the theory given in his manuscript over the publication. It is possible that he reflected the theory after submitting it to

91 “The

foundation of the general theory of relativity” (Kox et al., 1996, Doc. 30).

92 See the declaration of intention AEA 123-350. After a required five-year waiting period, Einstein

became a U.S. citizen in 1940 (Calaprice et al., 2015, p. 24). 93 We will do this in Sect. 3.4.4.

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189

the Annals of Mathematics and that during this process, the idea of composing a new manuscript occurred to him that would present the theory more clearly.94

3.4.3 Differences Between the Two Versions of the Manuscript As already explained above, two versions of the manuscript exist: one 15-page version with the archival numbers AEA 5-008 and 2-12195 and the twelve-page version contributed to the Library of Congress (AEA 97-487). Both manuscripts are written in German, bear the title “Einheitliche Feldtheorie,” and are dated July 6, 1938. The version AEA 2-121 is a typed written document with a typed signature Einstein’s while the version AEA 97-487 is a holographic manuscript with a signature. The equations in the typed version (AEA 2-121) are in Einstein’s hand. The typed version of the manuscript (AEA 2-121) is a transcription of the holographic version. The original text of the typed manuscript is in some cases different from the holographic version. These deviations, however, were corrected manually afterward. On page 1 of the typed version (AEA 2-121), for example, the word “sie” that does not appear in the holographic version (AEA 97-487), was manually crossed out. In the places where words or sentences were missing in the typed version, they were added manually afterward. This, for example, happened on page 9 of the typed version (AEA 2-121) with a heading. Furthermore, the holographic version (AEA 97-487) contains many words or even paragraphs that are crossed out which do not appear in the typed version (AEA 2-121). For example, Einstein started at the end of page 2 of the holographic version a new paragraph and crossed it out afterward. This paragraph does not appear in AEA 2121, which implies that AEA 2-121, indeed, is a transcript of AEA 97-487. The same phenomenon appears at several passages of the manuscript. There are further minor differences as in Eq. (17). While we find the factor .

∂f ∂ρ ∂xa ∂x0

(3.142)

in the typed version (AEA 2-121), it is .

∂ρ ∂f ∂x0 ∂xa

(3.143)

in the holographic version (AEA 97-487). A particular interesting difference appears in Eqs. (25) and (26) concerning the subscripts. In the holographic document, he mainly wrote the subscripts underneath 94 See 95 We

121.

also Sect. 3.4.5. will refer to this manuscript in the following only by quoting the archival number AEA 2-

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the letters, except for two exceptions: He corrected .H3 to .H in Eq. (25) and wrote 3

down .α2 in Eq. (26), which probably is a mistake. In his field Eqs. (29a) and (30a), he then used the usual subscripts .α2 , .α3 , and .α4 . In the typed version, where Einstein added the equations manually, he first wrote down all subscripts in Eqs. (25) and (26) next to the letters as in .H1 and then corrected all of these subscripts such that they are underneath the letters as in .H . In the first set of field Eqs. (29a), however, he did 1

not correct the subscripts and let them as .α2 , .α3 , and .α4 . In the second set of field Eqs. (30a), he then corrected it again and changed .α2 to .α . This confusion appears 2

also in many letters between Einstein and Bergmann from 1938. For example, he used both .ϕ2 and .ϕ in his letter AEA 6-250 at the end of June. In AEA 6-252 2

on July 1, he corrected the other way around: He crossed out the subscript of .ϕ and 1

wrote down .ϕ1 . We find the same phenomenon also on some of his research notes.96

3.4.4 Axiomatic Structure of the Manuscript and its Differences to the Publication In this section, we will discuss the differences between Einstein and Bergmann’s publication (Einstein and Bergmann, 1938) and Einstein’s Washington manuscript. In both the publication and the Washington manuscript, Kaluza’s original five-dimensional theory is extended by giving the fifth dimension a physical interpretation and replacing the cylinder condition by a periodicity condition. On this basis, Einstein and Bergmann derived the new field equations ⎛

⎞ ⎛ ⎞ 1 1 1 1 m mn Rkl + Rlk − Rgkl + α2 2Akm Al − Amn A gkl .α1 2 2 2 2 ⎞ ⎛ 1 + α3 − ∂0 g mn ∂0 gmn gkl + 2g rs ∂0 gkr ∂0 gsl − 2∂0 ∂0 gkl − ∂0 gkl g rs ∂0 grs 2 ⎞ ⎛ )2 1 ( mn + α4 gkl g ∂0 gmn + 2g mn ∂0 ∂0 gmn + 2∂0 gmn ∂0 g mn = 0 (3.144) 2 and .

96 For

⎛ ⎛ ⎛ ⎞ ⎞√ s r − 4α2 ∇t Ast − g ks ∂0 ┌kr −g dx 0 = 0 α1 g km ∂0 ┌km

(3.145)

instance, see AEA 62-787 and 62-788. We will come back to these pages in Sect. 4.3.

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191

as we have derived them already in Eqs. (3.137) and (3.138), see also Einstein and Bergmann (1938, p. 695).97 As explained comprehensively in Sect. 3.2, Einstein and Bergmann considered the five-dimensional metric components .gαβ , which effectively reduce to four-dimensional metric components .gab as their zero components vanish. In general, the metric components are periodic functions of 0 .x , which was pointed out as the substantial difference to Kaluza’s theory in Einstein and Bergmann (1938, p. 691).98 The quantities .Amn are the antisymmetric derivatives of .Am , which represent, in Einstein and Bergmann’s interpretation, the electric potential. In contrast to the metric components, .Am does not depend on .x 0 , see Eqs. (3.26) and (3.27). Analogously to the notation used in the fourdimensional general theory of relativity, .Rik is the five-dimensional Ricci tensor and i .┌ kl are the five-dimensional analogues to the Christoffel symbols, see Eqs. (3.55) and (3.60). The quantities .αi are constants that were discussed by Einstein in a later unpublished manuscript entitled “Ein Gesichtspunkt für eine spezielle Wahl der in der verallgemeinerten Kaluza-Theorie auftretenden Konstanten” (AEA 1-136)99 as well as in the follow-up publication (Einstein et al., 1941). The substance and the theory itself presented in the Washington manuscript is the same as in their publication. However, there are significant differences that were partly pointed out by Einstein himself. We will discuss them in the following.

3.4.4.1

The Title and Authorship

The first difference appears in the title. While Einstein and Bergmann chose the title On a Generalization of Kaluza’s Theory of Electricity (Einstein and Bergmann, 1938), Einstein simply used the title Einheitliche Feldtheorie in his manuscript. Clearly, a further difference is the authorship: The publication is a joint work by Einstein and Bergmann, while the Washington manuscript was composed by Einstein alone. With respect to the fact that the Washington manuscript does not contribute any new substance of the five-dimensional theory, one might have expected that Bergmann would be co-author of the manuscript as well. On the contrary, it seems as if Bergmann did not even know about Einstein composing the manuscript and did not know the text of the Washington manuscript until both of them met again in Princeton. Clearly, both changes in title and authorship are compatible with our suggestion that Lowe asked Einstein as consultant of the Library of Congress to compose the manuscript in order to obtain an autograph manuscript. This also fits to the fact that Einstein apparently never intended to publish it.

97 As already mentioned above and as we will see in the following, Einstein interchanged the notations .αi and .α . i

98 See

also Einstein et al. (1941, p. 217). 99 The manuscript has been dated to 1941 by the Albert Einstein Archives. See also Van Dongen (2010, p. 143).

192

3.4.4.2

3 Different Pathways to the Generalization of Kaluza’s Theory

The Axiomatic Structure

As already quoted in Sect. 3.4, the introduction of Einstein’s Washington manuscript reads as follows:100 In den letzten Monaten habe ich zusammen mit meinem Assistenten P. Bergmann eine einheitliche Feldtheorie entwickelt, welche durch Verallgemeinerung von Kaluza’s Theorie des elektrischen Feldes entstanden ist. Im Folgenden soll diese Theorie unabhängig von ihren historischen Wurzeln dargestellt werden, damit ihre logische Struktur möglichst deutlich hervortrete. (AEA 97-487)

Presenting the theory independently of Kaluza’s theory and building it directly on axioms is one of the main differences to the publication, where Einstein and Bergmann first gave an overview of Kaluza’s theory (Einstein and Bergmann, 1938, pp. 683–688). They then altered Kaluza’s theory by ascribing a “physical reality to the fifth dimension” (Einstein and Bergmann, 1938, p. 683). Instead of developing their theory as an separate theory, as it was done by Einstein in his manuscript, they rather extended Kaluza’s theory in their joint publication. In the Washington manuscript, Einstein defined at the very beginning three axioms that characterize “die Raumstruktur, welche der Theorie zugrundeliegt, vollständig”101 (p.2, AEA 2121). In the first axiom, Einstein introduced the five-dimensional space with the Riemann metric .

dσ 2 = γμν dxμ dxν ,

(3.146)

which can be locally transformed into a diagonal metric .

dσ 2 = dx12 + dx22 + dx32 − dx42 + dx02

(3.147)

by a suitable choice of a local coordinate system. At this point we note that contrary to the publication, Einstein used subscript for the coordinates (.xi instead of .x i ). Einstein’s second axiom then postulated that the space is closed along the .x0 coordinate such that the metric components are periodic functions of .x0 , that is γμν (x1 , x2 , x3 , x4 , x0 + nλ) = γμν (x1 , x2 , x3 , x4 , x0 )

.

(3.148)

for a constant .λ and a integer n. Hence, a point P can then be associated with an infinite number of further points corresponding to the values of n which have been called homologe Punkte102 (p.1, AEA 2-121).

100 Translation

in Footnote 69. the space structure, which underlies the theory.” Translation by the author. 102 Homolog points. Translation by the author. 101 “completely

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193

Finally, the third axiom required the existence of a unique, singularity free, and closed space-like geodesic through each point:103 Durch jeden Punkt unseres Raumes soll es eine und nur eine in sich singularitätsfrei geschlossene “raumartige” geodätische Linie geben. (AEA 2-121, p. 2)

This implies that between two homolog points, there exists one and only one geodesic that also passes through any further homolog point. The fact that Einstein characterized the theory explicitly by only three axioms is interesting with respect to his own statements on axioms in theoretical physics. In fact, the axiomatic formulation of the new theory in three independent postulates in his Washington manuscript clearly is an example of his own understanding of the goal of theoretical physics. In his article (Einstein, 1922a) “über die gegenwärtige Krise der theoretischen Physik,”104 he claimed the following at the very beginning:105 Es ist Ziel der theoretischen Physik, ein auf möglichst wenigen von einander unabhängigen Hypothesen ruhendes logisches Gedankensystem zu schaffen, das den ganzen Komplex der physikalischen Prozesse kausal zu erfassen gestattet. (Einstein, 1922a, p. 1)

This point of view might explain why Einstein preferred his Washington manuscript so much compared with the publication, which does not emphasize the axiomatic foundation of the new theory by generalizing Kaluza’s theory rather than building up an isolated theory. However, we note that Einstein apparently had some ambiguous opinions about such an axiomatic point of view over the years. For example, when competing with Hilbert in finding the field equations of general relativity,106 Einstein contrary to Hilbert did not aim to build his theory on a minimum number of axioms as he explicitly expressed in his review paper from 1916:107 Es kommt mir in dieser Abhandlung nicht darauf an, die allgemeine Relativitätstheorie als ein möglichst einfaches logisches System mit einem Minimum von Axiomen darzustellen. Sondern es ist mein Hauptziel, diese Theorie so zu entwickeln, daß der Leser die psychologische Natürlichkeit des eingeschlagenen Weges empfindet und daß die zugrunde gelegten Voraussetzungen durch die Erfahrung möglichst gesichert erscheinen. (Einstein, 1916a, p. 777)

103 “Through each point of our space there shall exist one and only one geodesic line that is closed without singularities and ‘space-like”’ (Sauer and Schütz, 2020, p. 8). 104 “On the present crisis of theoretical physics” (Kormos Buchwald et al., 2012b, Doc. 318). 105 “The goal of theoretical physics is to create a logical system of concepts based on the fewest possible mutually independent hypotheses, allowing a causal understanding of the entire complex of physical processes” (Kormos Buchwald et al., 2012b, p. 249). 106 See Sect. 1.3.4. 107 “It is not my purpose in this discussion to represent the general theory of relativity as a system that is as simple and logical as possible, and with the minimum number of axioms; but my main object is to develop this theory in such a way that the reader will feel that the path we have entered upon is psychologically the natural one, and that the underlying assumptions will seem to have the highest possible degree of security” (Kox et al., 1997, Doc. 30).

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We see that in this quote, Einstein put emphasis on a physical understanding rather than on a mathematical formulation. A similar statement on the axiomatic method can be found in contemporary correspondence. On November 23, 1916, Einstein wrote to Weyl and criticized the axiomatic approach:108 Der Hilbertsche Ansatz für die Materie erscheint mir kindlich, im Sinne des Kindes, das keine Tücken der Aussenwelt kennt. Ich suche vergeblich nach einem physikalischen Anhaltspunkte dafür, dass die Hamilton’sche Funktion für die Materie sich aus den .ϕν , und zwar ohne Differentiation, bilden lasse. Jedenfalls ist es nicht zu billigen, wenn die soliden Überlegungen, die aus dem Relativitätspostulat stammen, mit so gewagten, unbegründeten Hypothesen über den Bau des Elektrons bezw. der Materie verquickt werden. Gerne gestehe ich, dass das Aufsuchen der geeigneten Hypothese bezw. Hamilton’schen Funktion für die Konstruktion des Elektrons eine der wichtigsten heutigen Aufgaben der Theorie bildet. Aber die “axiomatische Methode” kann dabei wenig nützen. (Schulmann et al., 1998b, Doc. 278, p. 366)

Einstein here refers to Hilbert’s axiomatic method of the theory of general relativity used in Hilbert (1915). There, Hilbert postulated two axioms, one of them saying that the Hamilton function is an invariant with respect to an arbitrary transformation (Hilbert, 1915, p. 396).109 Einstein himself referred to the variational principle as an axiom in 1925 when writing an appendix to Eddington’s unified field theory that was based on the affine connection,110 see Einstein (1925a, p. 367). We see that Einstein was very skeptical about an axiomatic foundation of a theory in theoretical physics in 1915/16. However, it seems as if his opinion changed over the years as we saw in our examples from 1922 to 1925. The Washington manuscript and its axiomatic approach substantiate this change of view, especially in the light of Einstein’s positive view about it. The transition from Einstein and Bergmann’s publication as an extension of Kaluza’s theory toward an axiomatic structure in the Washington manuscript, however, also reflects Einstein’s tensions about the role of axiomatics. 3.4.4.3

The Mathematical Appendix

As quoted above, Einstein pointed out a second main difference between their publication and the manuscript in the letter AEA 6-256 to Bergmann on July 12, namely that he incorporated the mathematical appendix of their publication into 108 “Hilbert’s assumption about matter appears childish to me, in the sense of a child who does not know any of the tricks of the world outside. I am searching in vain for a physical indication that the Hamilton function for matter can be formed from the .ϕν ’s, without differentiation. At all events, mixing the solid considerations originating from the relativity postulate with such bold, unfounded hypotheses about the structure of the electron or matter cannot be sanctioned. I gladly admit that the search for a suitable hypothesis, or for the Hamilton function for the structural makeup of the electron, is one of the most important tasks of theory today. The “axiomatic method” can be of little use here, though.” (Schulmann et al., 1998a, p. 266). 109 For a more detailed discussion on Hilbert’s axiomatic method, see, for instance, Majer and Sauer (2005, 2006), Sauer (2002). 110 See Sect. 1.4.

3.4 Einstein’s Washington Manuscript

195

the main text.111 In the publication, Einstein and Bergmann treated tensor densities and the mathematical part of the derivation of the field equations in the appendix (Einstein and Bergmann, 1938, pp. 696–701). This is needed to derive the field Eqs. (35) and (36) on page 695. In Einstein’s manuscript, we find the tensor densities as well as the derivation of the field equations embedded in the text. In fact, the manuscript does not have an appendix. For instance, we can associate the Eqs. (13), (20), and (27) from the Washington manuscript in AEA 2-121 with the Eqs. (A 11), (A 12), and (A 16b) from the appendix of the publication. We also note that Einstein and Bergmann first presented the derivation of the field equations in the classic theory of relativity by using tensor densities (Einstein and Bergmann, 1938, pp. 696–699) before assigning the procedure to the fivedimensional generalized Kaluza theory. The derivation of the field equations in the four-dimensional case is completely missing in Einstein’s manuscript. This makes sense as he also did not consider Kaluza’s theory. Instead of generalizing existing theories, he rather developed it from scratch by formulating the three axioms.

3.4.4.4

Further Differences

Apart from these main differences that were partly mentioned by Einstein in the correspondence, there exist more minor differences. A rather confusing difference between the two works is the notation. In the publication, they used .Aa = γ0a for the components of the metric with index zero (Einstein and Bergmann, 1938, p. 691), while Einstein used the notation .ϕm = γ0m in his manuscript (page 7, AEA 2-121). This obviously also affected the notation of the antisymmetric derivative. In the publication, they used .Aab = Aa,b − Ab,a (Einstein and Bergmann, 1938, p. 963), while Einstein used .ϕmn = ϕm,n − ϕn,m in his manuscript. Einstein himself apparently was confused by this. For example, in the holographic version of the Washington manuscript he first erroneously wrote .Amn and corrected it to .ϕmn (AEA 97-487, page 9). In other cases, he first wrote .ϕa and then .Aa , as in a letter to Bergmann (AEA 6-266). A similar phenomenon, where Einstein might have confused the content of the Washington manuscript with the content of the publication happened when he wanted to correct the convention of the contraction of the Riemann tensor in a letter to Bergmann on July 12 (AEA 6-256) but noticed himself some days later on July 15 that this mistake was only in his manuscript (AEA 6-271), probably referring to the Washington manuscript.112

111 “Ich habe die ganze Theorie neu bearbeitet für das Washington-Manuskript, indem ich nicht von dem Kaluza’schen Spezialfall ausgegangen bin, sondern gleich die neue Theorie systematisch dargestellt und den Anhang in den Text hineingearbeitet habe” (AEA 6-256). 112 Bergmann also replied to Einstein’s first letter on July 16 that he could not find this mistake in his manuscript (AEA 6-263). The dating of the sequence will be carried out in Sect. 4.1.

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3 Different Pathways to the Generalization of Kaluza’s Theory

Concerning the subscripts discussed in Sect. 3.4.3, Einstein and Bergmann used, in the relevant cases, always subscripts directly underneath the letters in contrast to the Washington manuscript, see, for instance, (Einstein and Bergmann, 1938, p. 694) and our discussion in Sect. 3.4.3. As already briefly mentioned in Sect. 3.2.1, the derivation of the special coordinate system differs between the publication and the Washington manuscript. While our presentation in Sect. 3.2.1 is mostly in analogy to Einstein and Bergmann’s publication, Einstein chose a slightly different explanation in his Washington manuscript. For instance, instead of choosing the distance b as in Eq. (3.18), Einstein set113 .

PP1 σ = 0 λ x

(3.149)

corresponding to Eqs. (3.146) and (3.148). We saw that Einstein and Bergmann indirectly chose .λ = 1 by setting the coordinate distance between two corresponding consecutive points P and .P 1 equal to 1. As already mentioned in Footnote 22, they then needed to undo this choice in order to set .b = 1. In his Washington manuscript, Einstein was more consistent and let .λ variable until he set it equal to .P P 1 .114 We also note that Einstein chose .α1 = α = 1 in his Washington manuscript in contrast 1

to the publication.115 We also briefly want to point out one interesting mistake in Einstein and Bergmann’s publication. They denoted two consecutive points on one A-line, which represent the same point in the physical space, by P and .P ' . Furthermore, they ◦

denoted the intersection point of an A-line and the surface .x 0 = 0 by .P . Since they set the coordinate distance between P and .P ' as . Δ x 0 = λ = 1, it then was for a point P on the same A-line116 x0 =

.

1 b

⎛

P ◦

dσ,

(3.150)

P

where ⎛ b=

P'

dσ.

.

(3.151)

P

Einstein and Bergmann numbered these Eqs. (17) and (17a). However, they omitted the lower limit P of the integral in their Eq. (17a) (Einstein and Bergmann, 1938,

113 See

his Eq. (6). his discussion of Eq. (7b). 115 For instance, see his Eq. (26). 116 See our Eqs. (3.16) and (3.17). 114 See

3.4 Einstein’s Washington Manuscript

197

p. 690). On pages (10) and (11) of the German version of the publication (AEA 1133), Einstein numbered the respective equations as (12) and (12a) and used the correct limits, but a different notation. He denoted two corresponding, consecutive points for the second integral by .P ' and .P '' and the point on the surface .x 0 = 0 by ◦

P0 instead of .P . Interestingly, this topic was also discussed by Einstein and Bergmann when they sent each other the final corrections. On July 22/23,117 Einstein wrote in a letter to Bergmann:118

.

⎛ P '' ⎛ P '' Bei (17a) habe ich unzweckmässig korrigiert. Es stand . P [.] Es sollte heissen . P ' , ◦ '' ⎛P während ich unnötigerweise . ◦ ' gesetzt habe. Man könnte auch die P’s ganz weglassen und P

im Text sagen: erstreckt über eine Periode. Machen Sie dies nach Gutdünken. Um meine falsche Korrektur unschändlich zu machen, können Sie am Rand ein Papierchen darüber kleben. (AEA 6-266)

We do not know how Bergmann decided and which corrections they finally sent, but neither the additional sentence nor the correct limits appear in the final publication.

3.4.5 A Correction to the Proofs By means of the correspondence between Einstein and Bergmann that took place in summer 1938, we can reconstruct the process of changing a text passage in their publication that might have triggered Einstein to compose the Washington manuscript or to choose a different structure for it.119 For this, we look at a paragraph of the publication that was found to be in need of correction by the time when Einstein was working on his Washington manuscript. The paragraph in question can be found in a preliminary version of the publication. This version has the archival number AEA 1-133 and is written in German. The corresponding paragraph is on page 6 and reads in its first and crossed out version as follows: Hieraus ersieht man, dass die .γab nicht ohne Weiteres als Tensor[größe] vierdimensionaler Metrik interpretierbar sind. Die Existenz einer solchen ergibt sich aber aus folgender Überlegung: sind P und .P ' [. . . ] (AEA 1-133)

The three dots indicate the subsequent geometric considerations. Einstein wanted to correct this paragraph around July 1, just a few days before he finished his Washington manuscript on July 6. In a letter to Bergmann on July 1 (AEA 6-252),

117 We

will date the letters in Sect. 4.1. see in AEA 1-133 that Einstein introduced new numbers for the equations, see page (11). Equations (12) and (12a) then became (17) and (17a) accordingly. 119 In the following discussion, we use the sequence of the letters that we will suggest in Sect. 4.1. 118 We

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3 Different Pathways to the Generalization of Kaluza’s Theory

Einstein suggested a new formulation of the paragraph in question, namely: Hieraus ersieht man, dass die .γab nicht invariant sind bezüglich Schnitttransformationen. Es empfiehlt sich deshalb, statt dieser Grössen solche einzuführen, welche diese Eigenschaft besitzen, und zwar auf Grund folgender überlegung: Sind P und .P ' . . . (AEA 6-252)

We find almost the identical formulation written above the canceled lines in AEA 1133, page 6. There, it says in its corrected version: Hieraus ersieht man, dass die .γab nicht invariant sind bezüglich Schnitt-Transformationen. Es empfiehlt sich deshalb, statt der Grössen .γmn neue Grössen einzuführen, die diese Eigenschaft besitzen und zwar auf Grund folgender Überlegung: sind P und .P ' [. . . ] (AEA 1-133)

Thus, Einstein changed two things: First, he replaced the part “tensor quantities of a four dimensional metric” by “invariant with respect to cut transformations” and second, he added that it is useful to find new quantities that have this invariance under cut-transformations. Einstein discussed the tensor analysis with respect to the generalized Kaluza theory and the special coordinate system coherently from page 5 on in his Washington manuscript.120 In the publication, however, they discussed first the Kaluza theory with the respective paragraph on page 686 before transferring the theory to its generalization. The definition of a tensor and especially its invariance under cut-transformations in the generalized version only appears on page 692. This might have been the reason why he wanted to replace the word tensor in the quoted paragraph. However, this is also the problem for Bergmann in his reply on July 5 (AEA 6254), where he pointed out that it would be unclear to the reader why invariance under cut-transformations is desired: Sie schlagen eine Änderung vor, über deren Wünschenswertigkeit ich mir nicht ganz klar bin. Es ist zu beachten, daß wir bisher noch keine allgemeine Definition des Tensors im „angepassten System“ eingeführt haben und das auch in diesem Abschnitt nicht tun. (AEA 6-254)

He also added that die Formulierung “tensor of a four dimensional metric” nicht überflüssig [ist]. (AEA 6-254)

Bergmann only had the English version of the publication and quoted the paragraph that he did not want to change: We see, therefore, that .γmn can not be regarded as a tensor of a four dimensional metric. The existence of such a tensor can be deduced, however, by the following consideration: . . . (AEA 6-254)

120 We here use the typed version AEA 5-008 or 2-121. In the holographic version AEA 97-487 it starts on page 4.

3.4 Einstein’s Washington Manuscript

199

According to Einstein’s objection, Bergmann then suggested a corrected version: We see, therefore, that the .γab are not invariant with respect to cut-transformations. It seems more appropriate to introduce, instead of the .γab , such quantities [as] possess that property. This may be done as follows: . . . (AEA 6-254)

Einstein agreed with Bergmann’s suggestion on July 12 (AEA 6-256). Some paragraphs before, he mentioned the Washington manuscript and that he preferred it over the publication.121 On July 21, he again came back to the respective paragraph when discussing the corrections: Sie haben einiges vergessen, was Sie mir früher bestätigt haben, z.B. bei der Einführung der die Bemerkung, dass .γmn „kein Tensor sei“ ist an dieser Stelle sinnlos. (AEA 6-265)

.gmn

We note that this does not correspond to Bergmann’s comment. As before, Einstein then directly commented that the structure of the Washington manuscript is much better: Das Schlechteste an unserer Darstellung ist durch die Einführung des Anhangs verschuldet, statt dass man Tensoren und Tensordichten gleich zusammen behandelt. (AEA 6-265)

This is exactly what Einstein optimized in the Washington manuscript. The story continues as Einstein did not find an appropriate English phrase. On July 22/23 (AEA 6-266), he suggested again a new version: Statt der Grössen .γmn , welche gemäss (14) bezüglich Schnitt-Transformation nicht invariant sind, kann man mit Vorteil neue Grössen .gmn auf Grund folgender Überlegung einführen: [. . . ] AEA 6-266

Bergmann was supposed to translate this sentence. We see that in Einstein’s suggestion, he deleted the phrase “Es empfiehlt sich” in comparison to his letter AEA 6-252 and added “mit Vorteil.” In the final version, we do not find this phrase anymore: Instead of the .γmn which are not invariant with respect to cut-transformations (according to (14)) we introduce new functions .gmn by the following consideration: [. . . ] (Einstein and Bergmann, 1938, p. 686)

Bergmann apparently did not translate it. We see that Einstein spent a lot of time in order to find the correct phrase for this short paragraph. This is also supported by the fact that Einstein considered this paragraph on some manuscript pages as well. In Sects. 3.6.1 and 3.6.2, we will discuss the manuscript pages AEA 62-789 and 62-798, where further versions of this paragraph appear. We also saw that Einstein frequently mentioned his manuscript and expressed his preference for it several times when writing about this paragraph. Hence, it is possible that the making of the Washington manuscript and his initial objection concerning this paragraph on July 1 in AEA 6-252 are connected with each other. We note that in this letter, Einstein did not mention the

121 We

already quoted this paragraph in Sect. 3.4.1.

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3 Different Pathways to the Generalization of Kaluza’s Theory

Washington manuscript. Einstein mentioned the Washington manuscript only in the letter AEA 6-256 on July 12.

3.5 Concluding Remarks on the Washington Manuscript Einstein composed a manuscript that he frequently called “my Washington manuscript” which contains essentially the same theory as Einstein and Bergmann’s publication. However, Bergmann did not figure as co-author anymore. Instead, Einstein did not let Bergmann participate in the process of its composition and finally donated it to the Library of Congress upon Lowe’s request. Einstein apparently never intended to publish it. We conjectured that Lowe asked Einstein for it in order to get an autograph manuscript for the Library of Congress for paleographic reasons. It seems as if Einstein welcomed this idea and that he used this opportunity in order to restructure their new theory. In fact, the manuscript significantly differs from the publication in many aspects which led to the fact that Einstein preferred his manuscript version over the publication. In addition to the obvious differences in title and authorship, the Washington manuscript treats the derivation of the field equations in a direct mathematical way without relegating mathematical details into an appendix. Furthermore, Einstein chose a different structure for the manuscript. In fact, he based the theory exclusively on axioms rather than extending and generalizing Kaluza’a theory as it was done in the publication. In doing so, he made the historical origin of the theory disappear. In contrast to the publication, we also do not find any treatment of the regular theory of general relativity in the Washington manuscript. The differences in structure led to an analysis of Einstein’s view on axiomatics in physical theories. We saw that Einstein’s attitude toward the role of axioms was somewhat ambivalent and that the transition from the publication to the Washington manuscript serves as an example for Einstein’s change of view from being critical about the role of axiomatics in physical theories toward formulating the goal of theoretical physics as finding as few independent hypotheses as possible. Let us finally consider the last paragraph of the Washington manuscript.122 It says:123 Die im Vorstehenden entwickelte Theorie gibt eine formal völlig befriedigende einheitliche Auffassung von der Struktur des physikalischen Raumes. Weitere Untersuchungen müssen zeigen, ob sie eine (von statisischen Elementen freie) Theorie der Elementar-Teilchen sowie der Quanten-Phänomene enthält. (AEA 2-121, p. 5)

122 A

previous version of this paragraph is extant in AEA 2-119. suppressed an additional blank space inside of the parenthesis. In English, it says: “The theory developed here provides a unified conception of the structure of physical space which is completely satisfactory from a formal point of view. Further investigations will have to show whether it contains a theory (free of statistical elements) of elementary particles and of the quantum phenomena” (Sauer and Schütz, 2020, p. 10). 123 We

3.6 Manuscript Pages Dealing with the 1938 Theory

201

We do not find such a concluding paragraph in Einstein and Bergmann’s publication. The last paragraph of the Washington manuscript supports the claim that we already made in Sects. 1.4 and 1.5, namely that Einstein was searching for particle solutions. As we showed there, Einstein did not find such solutions and finally gave up the fivedimensional approach, see Einstein et al. (1941), Einstein and Pauli (1943).124 We note that in their follow-up paper in 1941, Einstein and his assistants gave a brief description of their generalization of Kaluza’s theory by going back in spirit to Einstein’s Washington manuscript rather than to their earlier publication (Einstein and Bergmann, 1938) by starting with recapitulating the three axioms (Einstein et al., 1941, p. 213). We finally finish our considerations on the Washington manuscript with noting that despite its highly technical character, the Library of Congress made it accessible for a broader audience showing the public’s fascination on Einstein’s unified field theory.

3.6 Manuscript Pages Dealing with the 1938 Theory This section is about several manuscript pages that contain calculations related to Einstein and Bergmann’s 1938 publication (Einstein and Bergmann, 1938) as well as to the Washington manuscript. The Einstein Papers Project and the Albert Einstein Archives have gathered a vast amount of documents related to Einstein’s life. In this database, they collected approximately 2000 pages containing unidentified calculations, of which approximately 1750 pages belong to reel 62 and 63 (Sauer, 2019, p. 373). By looking through all these pages, several manuscript pages could be identified with content directly related to Einstein and Bergmann’s 1938 publication or to the Washington manuscript. We will reconstruct and interpret some of them in this section.125 We will start with the manuscript page AEA 62-789 in Sect. 3.6.1 that contains calculations related to the coordinate transformations and to the introduction of the tensor .gmn . Thereafter, we will briefly look at the manuscript page AEA 62798 in Sect. 3.6.2 which contains similar considerations. We will argue that the next two pages AEA 62-802 and 62-794 belong together and we will discuss them comprehensively in Sect. 3.6.3. There, Einstein derived the Lagrange density by using several relations that are not known to us from the publication. We will also show how Einstein neglected in his derivation the derivatives of the Christoffel symbols and link this procedure to earlier works as to his review paper from 1916 (Einstein, 1916a). After this, we will look at page AEA 62-785 in Sect. 3.6.4 that contains calculations on how to derive the derivative of a scalar density. We will

124 See

also Van Dongen (2002, 2010). In Chap. 4, we will use further manuscript pages in order to investigate the question of how Einstein proceeded in order to find such particle solutions. 125 For further analyses, see Chaps. 2, 4, 5.

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3 Different Pathways to the Generalization of Kaluza’s Theory

connect these considerations with a letter from Einstein to Bergmann from July 1938. In addition, the manuscript pages AEA 62-373r, 62-800, 62-807, 63-026, 63026r, and 63-325 seem also to be connected to either Einstein and Bergmann’s publication or to Kaluza’s theory. We will come back to these pages briefly in Sect. 4.3.8. We will here stick to our notation even if Einstein used a slightly different notation on the manuscript pages. For example, we frequently denote the partial derivative of the tensor .As with respect to .x a by .∂a As . Einstein mostly used the notation .As,a . As another example, he frequently denoted the coordinates .x a with a subscript .xa .

3.6.1 AEA 62-789 We discussed the page AEA 62-789 also in Sect. 2.3.3, where we looked at the sketches at the top of the page. Here, we are interested in the rest of the page which is related to the Washington manuscript and to Einstein and Bergmann’s publication (Einstein and Bergmann, 1938).126 The calculations begin with γ ' μν =

.

∂x α ∂x β γαβ . ∂x ' μ ∂x ' ν

(3.152)

This is the general transformation law for a tensor .γμν and reminds us of the metric components as in Eq. (3.35). We do not find this equation in Einstein and Bergmann (1938) nor in the handwritten German version of the publication in AEA 1-133. However, it appears as Eq. (14) in the Washington manuscript.127 On the manuscript page, Einstein then continued with splitting the transformation for .γ ' μν into .γ ' mn , .γ ' m0 , and .γ ' 00 and derived γ ' mn =

.

∂x a ∂x b ∂x a ∂x 0 ∂x 0 ∂x b ∂x 0 ∂x 0 m n γab + m n γa0 + m n γ0b + ' ' ' ' ' ' ∂x ∂x ∂x ∂x ∂x ∂x ∂x ' m ∂x ' n

(3.153)

and analogously γ ' m0 =

.

126 In

∂x a ∂x b ∂x a ∂x 0 ∂x 0 ∂x b ∂x 0 ∂x 0 γ + γ + γ + ab a0 0b m m m ∂x ' ∂x ' 0 ∂x ' ∂x ' 0 ∂x ' ∂x ' 0 ∂x ' m ∂x ' 0

(3.154)

the following, Einstein used subscripts for the coordinates, while we use superscripts. AEA 5-008, 2-121, and 97-487.

127 See

3.6 Manuscript Pages Dealing with the 1938 Theory

203

as well as γ ' 00 =

.

∂x a ∂x b

∂x a ∂x 0

∂x 0 ∂x b

∂x 0 ∂x 0

∂x ' 0 ∂x '

∂x ' 0 ∂x '

∂x ' 0 ∂x '

∂x ' 0 ∂x ' 0

γ + 0 ab

γ + 0 a0

γ + 0 0b

.

(3.155)

This is equivalent to our procedure in Eqs. (3.36) and (3.37). He also noted that it is γ00 = 1 as we derived in Eq. (3.26). Accordingly, Einstein then wrote

.

γ ' mn =

.

∂x a ∂x b γab ∂x ' m ∂x ' n

(3.156)

∂x a γ0a ∂x ' m

(3.157)

and γ ' 0m =

.

for four-transformations and γ ' mn = γmn −

.

∂f ∂f ∂f ∂f γ0m − m γ0n + m n ∂x n ∂x ∂x ∂x

(3.158)

for cut-transformation. He first wrote down primes in all denominators and crossed them out afterward. This is in accordance with Eq. (3.33).128 He furthermore concluded γ ' 0m = γ0m −

.

∂f ∂x m

(3.159)

for four-transformations. The last equations are all equivalent to our considerations from Eqs. (3.36) and (3.37). In fact, he already indicated .γ ' 00 = 1 in Eq. (3.155). We also find the transformation law .x 0 = x ' 0 − f (x ' 1 · · · x ' 4 ) that holds for cuttransformations according to Eq. (3.30). At the end of the page, Einstein first argued that .γ0m is not a tensor with respect to cut-transformations and compared .γ0m with the electric potential by introducing the notation .ϕm : Wegen des Verhaltens bei Schnitt-Transformationen ist .γ0m kein Vektor. Denn .γ0m ändert sich bei dieser um einen willkürlichen Gradienten einer Funktion .f (x1 ·· x4 ). Dies ist genau das Verhalten des elektrischen Potenzials[.] Um dies äusserlich erscheinen zu lassen, wollen wir im Folgenden .γ0m durch .ϕm ersetzen. (AEA 62-789)

128 As

we will see below, f does not depend on .x 0 .

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3 Different Pathways to the Generalization of Kaluza’s Theory

He then introduced the tensor components .gmn : Es folgt ferner, dass gegenüber Schnitt-Transformationen .gmn ≡ γmn − γ0m γ0n eine Invariante ist. Da sich diese Grüssen bezüglich Vierertranformationen wie die .γmn verhalten, so ist .gmn ein Tensor in dem von uns definierten Sinne. (AEA 62-789)

These considerations correspond to the definition of a tensor around Eqs. (3.34) as well as to Eqs. (3.39) and (3.40). Hence, one is tempted to connect this passage to the publication (Einstein and Bergmann, 1938). However, we will argue that this paragraph is a preliminary version of a passage in the Washington manuscript.129 Let us first note that Einstein used the notation .ϕm (not .Am ), which he also used in his Washington manuscript but not in the publication. Since Einstein, as we explained above, got confused by this notation several times, this argument is not sufficient enough. For a second argument, we will look at the respective paragraph in his Washington manuscript.130 Einstein first considered Eqs. (3.156) to (3.159) in his Eqs. (14a) and (14b) on page 5. He then followed on page 6: Wegen der zweiten Gleichung (15b) [this should be (14b)] ist .γ0m kein Vektor; denn .γ0m ändert sich bei Schnitt-Transformation um den Gradienten einer willkürlichen Funktion .f (x1 · · x4 )[.] Dies ist genau das Verhalten des elektrischen Potentials. Wir wollen deshalb statt .γ0m in der Folge .ϕm schreiben. Es folgt ferner aus (14b), dass .gmn ≡ γmn − γ0m γ0n · · · · (15) Schnitt-Transformationen gegenüber invariant ist. Da sich die .gmn gemäss (14a) gegenüber Vierer-Transformationen wie die .γmn verhalten, so ist .gmn ein Tensor in dem von uns definierten Sinne. (AEA 97-487)

This is also the version from the typed manuscript AEA 2-121 and 5-008. In the holographic version AEA 97-487, Einstein first wrote “[. . . ]; denn .γ0m ändert sich bei dieser um einen Gradienten einer willkürlichen Funktion [. . . ],” but corrected it afterward. We see that the first version is almost identical to the version on AEA 62789. We conclude that the paragraphs appearing in the Washington manuscripts and on AEA 62-789 are very similar and almost identical. In contrast, Einstein and Bergmann chose a slightly different approach in their publication. On page 686, they also considered the transformations of .γ0m and .γmn according to Eqs. (3.158) and (3.159). However, they already had introduced the notation .Am such that they considered the transformation of .Am rather than of .γ0m . They then concluded that .γmn is not a vector and introduced .gmn after some geometric considerations. Such geometric considerations appear neither in Einstein’s Washington manuscript nor on AEA 62-789. The comparison between .Am and the electric potential then only appears one paragraph later on page 687 of the publication. We also note that Einstein and Bergmann derived these relations only for Kaluza’s theory while they referred to this when discussing the generalized version on page 692.

129 We 130 We

already looked at further versions of this paragraph in Sect. 3.4.5. will here use the version AEA 97-487.

3.6 Manuscript Pages Dealing with the 1938 Theory

205

We conclude that Einstein chose two different approaches to introduce the metric components .gmn and that he compared the functions .ϕm with the electric potential in two different situations. Since the approach on AEA 62-789 is similar and almost identical to the approach in his Washington manuscript, we argue that AEA 62-789 is a preliminary version of the Washington manuscript.131 Finally, we note that the manuscript page AEA 62-798 contains another version of the same passage. As we will argue in Sect. 3.6.2, this paragraph probably is directly related to Einstein and Bergmann’s publication.

3.6.2 AEA 62-798 This manuscript page is divided into two parts. The first part contains unidentified calculations that have a certain similarity to the calculations on the manuscript pages AEA 62-785r, 62-787r, and 62-789r, see Sects. 2.3.1, 2.3.2, and 2.3.4. We will investigate these calculations in Chap. 4. Here, we will briefly discuss the second part as it contains a text passage that is related to Einstein and Bergmann’s publication. In fact, it is another version of the paragraphs that we discussed in Sects. 3.6.1 and 3.4.5. The paragraph starts with an English sentence that is crossed out. It reads: [T]he .γmn are not invariant wit[h] regard to a cut-transformation[.] (AEA 62-798)

The fact that it was written in English shows that the sentence is connected to the publication rather than to the Washington manuscript. Einstein then continued with writing down: Wir sehen, dass die .γmn Schnitt-Transformationen gegenüber nicht invariant sind. Zu Grössen, welche schnitt-invariant sind und gleichzeitig gegenüber Vierer-Transformationen Tensor-Charakter besitzen kann man wie folgt gelangen. (AEA 62-798)

He then wrote down another version of this paragraph: Statt der Grössen .γmn , welche gemäss (14) bezüglich Schnitt-Transformationen nicht invariant sind, kann man auf Grund folgender Überlegung gewisse Grössen .gmn einführen. (AEA 62-798)

The first part of the second version reminds us of the suggested formulation from the letter AEA 6-266 that we quoted in Sect. 3.4.5. However, the second part then differs.132 The fact that Einstein referred to the Eq. (14) shows that the paragraph indeed is related to the publication. In his Washington manuscript, the respective equation is (14b) as we already saw in Sect. 3.6.1. This is also supported by the fact that

131 We also note that the present manuscript page is not related to the draft version of the publication in AEA 1-133. We find the respective paragraph on pages 5–6 of AEA 1-133. 132 In particular, the phrase “mit Vorteil” is missing here.

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3 Different Pathways to the Generalization of Kaluza’s Theory

the paragraphs on the present manuscript page refer to subsequent considerations that probably are the geometric considerations that do not appear in the Washington manuscript. Finally, we note that both versions on the present manuscript page differ from the two German versions in the German draft AEA 1-133, such that it is possible that this version here was even written before the draft.

3.6.3 AEA 62-802 and AEA 62-794 In this section, we reconstruct and interpret the manuscript pages AEA 62-802 and AEA 62-794.133 We argue that page AEA 62-794 was written after and in the context of AEA 62-802. Einstein first derived specific expressions for parts of the connection on the manuscript page AEA 62-802 and then used them on page AEA 62-794 in order to derive the action of the Einstein-Bergmann theory as given in Eq. (3.62). The page AEA 62-802 starts with some expressions. Among them, we find .gab + ϕa ϕb that corresponds to .γab according to Eq. (3.28)134 as well as .1 + ϕs ϕ s that we will again encounter below.135 After these expressions, Einstein apparently looked at the difference .∇b ∇c Aa − ∇c ∇b Aa , where .Aa is an arbitrary tensor.136 Einstein first calculated s ∇b Aa = ∂b Aa − ϕb ∂0 Aa − ┌ab As .

.

(3.160)

This follows immediately by Eq. (3.54). He further correctly computed s s ∇c ∇b Aa = ∂c (∇b Aa ) − ϕc ∂0 (∇b Aa ) − ┌ac ∇b As − ┌bc ∇s Aa .

.

(3.161)

Hence, it is s s ∇c ∇b Aa = ∂c ∂b Aa − ∂c ϕb ∂0 Aa − ϕb ∂c ∂0 Aa − ∂c ┌ab As − ┌ab ∂c As

.

s s − ϕc ∂0 ∂b Aa + ϕc ∂0 ϕb ∂0 Aa + ϕc ϕb ∂0 ∂0 Aa + ϕc ∂0 ┌ab As + ϕc ┌ab ∂0 As s s s l s − ┌ac ∂b As + ┌ac ϕb ∂0 As + ┌ac ┌sb Al − ┌bc ∇s Aa .

(3.162)

We see that the 1., .(3. + 6.), .(5. + 11.), 8., .(10. + 12.), and 14. terms are symmetric in b and c. They cancel each other by taking the difference .∇b ∇c Aa − ∇c ∇b Aa .

we will not use Einstein’s notation .Aa,b or .Aa;b , but .∂b Aa and .∇b Aa instead. recall that Einstein used the notation .ϕb = γ0b in the Washington manuscript instead of .Ab as it was the case in the publication (Einstein and Bergmann, 1938). 135 See Eq. (3.176). 136 In particular, it is .A /= γ in contrast to Sect. 3.2. a 0a 133 Again, 134 We

3.6 Manuscript Pages Dealing with the 1938 Theory

207

Furthermore, the seventh term vanishes because of Eq. (3.26). Thus, we get ( ) s s ∇b ∇c Aa − ∇c ∇b Aa = ∂0 Aa (∂c ϕb − ∂b ϕc ) + As ∂c ┌ab − ∂b ┌ac ⎛ ⎞ ( ) s s s l s l − Al ┌ac − ϕb ∂0 ┌ac ┌sb − ┌ab ┌sc − As ϕc ∂0 ┌ab ( s s s = ∂0 Aa (∂c ϕb − ∂b ϕc ) + As ∂c ┌ab − ∂b ┌ac − ϕc ∂0 ┌ab ⎞ s l s l s . +ϕb ∂0 ┌ac − ┌ac ┌lb + ┌ab ┌lc (3.163)

.

After Eq. (3.161), we find three lines of expressions separated off by two horizontal lines in Einstein’s notes that correspond to the right hand side of Eq. (3.162). However, he neglected the first and last term. Obviously, these two terms are symmetric in b and c and vanish when taking the difference from Eq. (3.163). In fact, he commented this for the last term on the right hand side of Eq. (3.161) by writing down “symm[etrisch] in b u[nd] c” beneath the term. Einstein also underlined five terms in his three separated lines. One of the terms is symmetric in b and c and the four others cancel each other by pairs when taking the difference. Beneath the second horizontal line, Einstein wrote down further six terms. Among them, we find the 2., 4., 9., and 13. terms from the right hand side of Eq. (3.162). These are exactly the terms that do not vanish when taking the difference. He also wrote down the 10. and 12. term and underlined them. It seems as if he first did not see the symmetry for these two terms, wrote them down, and underlined them indicating that they will vanish as well. He then concluded that the expression s s s l s l ∂c ┌ab − ∂b ┌ac − ┌lb ┌ac + ┌lc ┌ab

.

(3.164)

is a tensor as .(∂c ϕb − ∂b ϕc ) ∂0 Aa is a tensor. Indeed, .(∂c ϕb − ∂b ϕc ) ∂0 Aa is a tensor if .Aa is a tensor.137 Since the difference of the covariant derivatives is also a tensor, s − ϕ ∂ ┌s the rest of Eq. (3.163) is a tensor as well.138 However, the terms .ϕc ∂0 ┌ab b 0 ac are missing in Einstein’s expression (3.164). Einstein then made some further calculations that we cannot interpret. We will skip them and continue with the last part of the manuscript page. Because of the following calculations, we argue that page AEA 62-794 is the continuation of the present manuscript page. Einstein starts with the first line of .

137 See 138 In

┌ s ┐ ab

1 [∂b (gas + ϕa ϕs ) + ∂a (gbs + ϕb ϕs ) − ∂s (gab + ϕa ϕb )] 2 1 (3.165) = [∂b γas + ∂a γbs − ∂s γab ] , 2

=

Eqs. (3.42) to (3.44). fact, this corresponds to the five-dimensional Riemann tensor, see Eq. (3.60).

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3 Different Pathways to the Generalization of Kaluza’s Theory

where we used .gab = γab − ϕa ϕb for the second line.139 We here note that in the four-dimensional theory, it is l ┌ab = γ ls

.

┌ s ┐

(3.166)

ab

for the Christoffel symbols if .γab are the four-dimensional metric components. In the generalized Kaluza theory, Einstein and Bergmann denoted the five-dimensional metric components by .γμν and defined the connection as in Eq. (3.55). By .ϕn = γ0n , it follows that .

┌ 0 ┐ ab

=

1 1 [∂b γa0 + ∂a γb0 − ∂0 γab ] = [∂b ϕa + ∂a ϕb − ∂0 gab ] . 2 2

(3.167)

Einstein did not write down the second expression. For the second equality, we used ∂0 ϕa = 0.140 Einstein also computed

.

.

┌ a ┐

=

b0

1 [∂b ϕa − ∂a ϕb + ∂0 gab ] , 2

(3.168)

which follows analogously. After a short horizontal line, the next equation reads t ┌ab = g ts

.

┌ s ┐ ab

− ϕt

┌ 0 ┐ ab

.

(3.169)

Assuming that Einstein used the connection from Eq. (3.55), he would have neglected some terms in Eq. (3.169) as it is 1 ts g [(∂a gbs − ϕa ∂0 gbs ) + (∂b gas − ϕb ∂0 gas ) − (∂s gab − ϕs ∂0 gab )] 2 1 1 = g ts (∂a γbs + ∂b γas − ∂s γab ) − g ts (ϕb ∂a ϕs + ϕs ∂a ϕb + ϕa ∂0 gbs ) 2 2 1 1 − g ts (ϕa ∂b ϕs + ϕs ∂b ϕa + ϕb ∂0 gas ) + g ts (ϕa ∂s ϕb + ϕb ∂s ϕa + ϕs ∂0 gab ) 2 2 ┌ ┐ 1 s − g ts ϕs (∂a ϕb + ∂b ϕa − ∂0 gab ) = g ts ab 2 1 − g ts [ϕb (∂a ϕs − ∂s ϕa + ∂0 gas ) + ϕa (∂b ϕs − ∂s ϕb + ∂0 gbs )] 2 ┌ s ┐ ┌ 0 ┐ 1 ts − ϕ t ab − g [ϕb (ϕsa + ∂0 gas ) + ϕa (ϕsb + ∂0 gbs )] , (3.170) = g ts ab 2

t ┌ab =

.

139 See 140 See

Eq. (3.28). Eq. (3.26).

3.6 Manuscript Pages Dealing with the 1938 Theory

209

where we used the notation .ϕmn = ∂n ϕm − ∂m ϕn in the last equality. We see that our derivation differs by the last term on the right hand side compared to Einstein’s Eq. (3.169). As already indicated above, let us therefore assume Eq. (3.166) as well as γ ts = γ at γ bs γab = g ts + ϕ t ϕ s

(3.171)

.

according to Eq. (3.28). Then, it is t ┌ab = γ ts

.

┌ s ┐

= g ts

ab

┌ s ┐

+ ϕtϕs

ab

┌ s ┐ ab

= g ts

┌ s ┐ ab

+ ϕt

┌ 0 ┐ ab

(3.172)

as .ϕ s vanishes for all s except for .s = 0.141 Hence, Latin indices here apparently include .s = 0. Einstein then further calculated a ┌0b = g as

.

┌ s ┐ 0b

,

(3.173)

which follows from .

┌ 0 ┐ 0b

=

1 (∂b ϕ0 + ∂0 ϕb − ∂0 g0b ) = 0 2

(3.174)

where we used Eq. (3.26) and .g0b = 0. He also derived 0 ┌0b = −ϕ s

.

┌ s ┐ 0b

.

(3.175)

By our assumptions, this holds if .g 0s = −ϕ s . Einstein also derived 0 ┌ab = −ϕ s

.

┌ s ┐ ab

⎞ ┌ ┐ ⎛ 0 . + 1 + ϕl ϕ l ab

(3.176)

By Eq. (3.172) and .g 0s = −ϕ s as before, this holds if .ϕ 0 = 1 + ϕl ϕ l . We are not aware of such relations. As already mentioned in Footnote 141, it is also possible that Einstein used other relations than .ϕ 0 = 1 and .ϕ s = 0 for .s /= 0 here as otherwise, Eqs. (3.175) and (3.176) could be simplified. In fact, we will see that Einstein did this on the manuscript page AEA 62-794. We note that Einstein considered these terms also at the top of the present manuscript page. Let us now continue with the manuscript page AEA 62-794. It starts with the sentence “Alles[,] was mit undifferenziertem .ϕ multipliziert ist, muss sich wegheben

141 See

the discussion around Eq. (3.27). It is also possible that Einstein here used another relation as it seems as if he did not use the vanishing of .ϕ s for .s /= 0 in the following, see Eqs. (3.175) and (3.176).

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3 Different Pathways to the Generalization of Kaluza’s Theory

(bezw. gehen in bekannter Weise ein [)].”142 We saw above that Einstein split the t in terms with .g ts and with .ϕ t on the manuscript page terms of the connection .┌ab AEA 62-802, see Eq. (3.169). If we neglect the terms multiplied with .ϕ t or .ϕt in the respective connections according to Einstein’s comment, it is t ┌ab ≈ g ts

.

┌ s ┐ ab

≈

1 ts g [∂b gas + ∂a gbs − ∂s gab ] 2

(3.177)

by Eqs. (3.165) and (3.169). Einstein simply wrote down t t ┌ab ∼ ┌ab ,

.

(3.178)

(g)

where the subscript probably refers to the gravitational part analogously to the fourdimensional theory. Furthermore, it is 0 ┌ab ≈

.

┌ 0 ┐ ab

=

1 (∂b ϕa + ∂a ϕb − ∂0 gab ) 2

(3.179)

by Eqs. (3.176) and (3.167), which corresponds to Einstein’s next approximation. Analogously, it is a ┌0b = g as

.

┌ s ┐ 0b

=

1 as g (∂b ϕs − ∂s ϕb + ∂0 gsb ) 2

(3.180)

by Eqs. (3.165) and (3.173), which corresponds to Einstein’s third approximation. By Eq. (3.175), it furthermore is 0 ┌0b ≈0

.

(3.181)

as well as b ┌0b = g bs

.

┌ s ┐ 0b

=

1 bs 1 g (∂b ϕs − ∂s ϕb + ∂0 gbs ) = g as ∂0 gas 2 2

(3.182)

by Eqs. (3.165) and (3.173) corresponding to Einstein’s last two approximations. We conclude that Einstein here on page AEA 62-794 used the relations that he wrote down at the end of the manuscript page AEA 62-802. Einstein then considered the function ⎛ ⎞ st a b a 0 b b b a ┌sb + ┌0a .H = g ┌ta + 2┌s0 ┌ta − ┌sta ┌ab − ┌st0 ┌0b ┌0b . (3.183)

142 “All

terms multiplied by non-derivatives of .ϕ must be canceled out or act as usual.” Translation by the author.

3.6 Manuscript Pages Dealing with the 1938 Theory

211

We will see that .H corresponds to the invariants from Eq. (3.62). By neglecting the multiplications of .ϕt and .ϕ t , it follows from Eqs. (3.178) to (3.180) and (3.182) that ⎛ H=g

.

st

a b ┌ta ┌sb g g

1 b + g al (∂s ϕl − ∂l ϕs + ∂0 gls ) (∂t ϕa + ∂a ϕt − ∂0 gat ) − ┌sta ┌ab 2 g g

1 − g st (∂t ϕs + ∂s ϕt − ∂0 gst ) g lm ∂0 glm 4 1 + g bm (∂a ϕm − ∂m ϕa + ∂0 gma ) g al (∂b ϕl − ∂l ϕb + ∂0 glb ) . 4

⎞

(3.184)

Einstein did not write down Eq. (3.184) but computed 1 H = H + g st g al (∂s ϕl − ∂l ϕs + ∂0 gls ) (∂a ϕt + ∂t ϕa − ∂0 gat ) g 2

.

1 + g st g lm ∂0 glm (∂t ϕs + ∂s ϕt − ∂0 gst ) 4 1 + g al (∂b ϕl − ∂l ϕb + ∂0 glb ) g bm (∂a ϕm − ∂m ϕa − ∂0 gma ) . 4 Hence, it seems as if Einstein used the notation ⎛ H=g

.

g

st

a b ┌ta ┌sb g g

(3.185)

⎞

b − ┌sta ┌ab g g

(3.186)

and used wrong signs twice in the second and third row of Eq. (3.185). The latter, he corrected in his next equation. As we explained above, we argue that the index “g” indicates the gravitational case similar to Eq. (3.178). The corresponding part √ of the action is .H1 = −gR with g

g

⎛ ⎞ b a b a R = g st R lslt = g st ∂t ┌sll − ∂l ┌stl − ┌ba ┌st + ┌ta ┌sb ,

.

g

(3.187)

g

where we used the Riemann tensor analogously to Eq. (3.60) by neglecting all terms multiplied by .ϕ t or .ϕt .143 Hence, we see that .R from Eq. (3.187) and .H g

g

from Eq. (3.186) only differ by the partial derivatives of the connection. We saw in Sect. 3.1 that these terms can be neglected by the variational principle. This probably is the reason why Einstein implicitly neglected these terms on the present manuscript page. We will come back to this in Sect. 3.6.3.1. 143 We could also have used Eq. (3.3) according to the four-dimensional theory. We here note again

that we then would have got the negative version due to a different definition of the Riemann tensor. See also our discussion around Eq. (3.76).

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3 Different Pathways to the Generalization of Kaluza’s Theory

In the following, we will again use the notation .ϕab = ∂b ϕa − ∂a ϕb , where .ϕab is an antisymmetric tensor as we showed in Eqs. (3.42) and (3.43). By Eqs. (3.184) and (3.186), it is 1 H = H + g st g al (ϕls + ∂0 gls ) (∂t ϕa + ∂a ϕt − ∂0 gat ) g 2

.

1 − g st g lm ∂0 glm (∂t ϕs + ∂s ϕt − ∂0 gst ) 4 1 + g al g bm (ϕlb ϕma + ϕma ∂0 glb + ϕlb ∂0 gma + ∂0 gma ∂0 glb ) . 4

(3.188)

We note that .

1 st al 1 1 g g ϕls (∂t ϕa + ∂a ϕt − ∂0 gat ) = ϕ at (∂t ϕa + ∂a ϕt ) − ϕ at ∂0 gat 2 2 2

(3.189)

as well as .

1 al bm 1 g g ϕma ∂0 glb = ϕ bl ∂0 glb . 4 4

(3.190)

Thus, it is .

1 st al 1 g g ϕls (∂t ϕa + ∂a ϕt − ∂0 gat ) + g al g bm (ϕma ∂0 glb + ϕlb ∂0 gma ) 2 4 1 = ϕ at (∂t ϕa + ∂a ϕt ) . (3.191) 2

Using this, we get from Eq. (3.188) 1 1 H = H + g st g al ∂0 gls (∂t ϕa + ∂a ϕt − ∂0 gat ) − g st g lm ∂0 glm (∂t ϕs + ∂s ϕt − ∂0 gst ) g 2 4

.

1 1 + g al g bm (ϕlb ϕma + ∂0 gma ∂0 glb ) + ϕ at (∂t ϕa + ∂a ϕt ) . 2 4

(3.192)

This is almost equivalent to Einstein’s next equation. One of the two sign mistakes from Eq. (3.185) still appeared in his equation. In addition, he did not write down the last term in Eq. (3.192). However, he commented that the terms with the symmetric √ derivatives of .ϕ (“symmetr. Abl. der .ϕ”) vanish if we take the integral of .H g. By symmetric derivatives, he probably referred to .∂t ϕa + ∂a ϕt .144 We note that he first claimed that his equation cannot be true since it still contains terms with .∂b ϕa +∂a ϕb but crossed out this sentence afterward.

note that Einstein called the quantities .ϕab = ∂b ϕa − ∂a ϕb “antisymmetrical derivatives,” see for example Einstein and Bergmann (1938, p. 686).

144 We

3.6 Manuscript Pages Dealing with the 1938 Theory

213

Thus, by neglecting these symmetric derivatives, we formulate the Hamilton function as 1 1 1 1 H = H + ∂0 g at ∂0 gat + g lm ∂0 glm g st ∂0 gst − ϕ am ϕam − ∂0 g bl ∂0 glb g 2 4 4 4

.

1 1 1 = H + ∂0 g ls ∂0 gls + g lm ∂0 glm g st ∂0 gst − ϕ ls ϕls , g 4 4 4

(3.193)

where we used Eq. (3.118) and .ϕma = −ϕam . This is, except for the signs, equivalent to Einstein’s last equation on the first part of AEA 62-794. We see that the terms correspond to the parts .H1 to .H4 from Eq. (3.62). We will now briefly discuss the difference between .H from Eq. (3.186) and .R g

g

from Eq. (3.187) with respect to the Lagrange density and connect them to Einstein’s review article (Einstein, 1916a).

3.6.3.1

Neglecting the Derivatives of the Christoffel Symbols

As we saw above, Einstein implicitly used a Lagrange density without the partial derivatives of the Christoffel symbols as it was ⎛ ⎞ R − H = g st ∂t ┌sll − ∂l ┌stl .

(3.194)

.

As in Sect. 3.1, these derivatives do not contribute to the variational principle such that both R and .H can be used for deriving the field equations. This probably was the reason why Einstein on the manuscript page AEA 62-794 considered .H instead of R. In his review paper from 1916, Einstein himself emphasized this fact (Einstein, 1916a, pp. 803–805). Here, we will briefly outline his argumentation.145 It is l l a l a Rsr = ∂l ┌sr − ┌ar ┌sl − ∂r ┌sll + ┌al ┌sr √ √ a l l = ∂l ┌sr − ┌sla ┌ar − ∂r ∂s ln −g + ┌sr ∂a ln −g,

.

see Einstein (1916a, pp. 800–801). Einstein then chose Lagrange density reduces to

√ −g = 1 such that the

.

⎛ ⎞ l . R = g st Rst = g st ∂l ┌stl − ┌sla ┌at

.

145 We

(3.195)

(3.196)

note that Einstein contracted the Riemann tensor in Einstein (1916a, pp. 800–801) with respect to the first and fourth index. We already encountered such a situation in the correspondence between Einstein and Bergmann, see Footnote 37. Here, we will use the contraction with respect l instead of .R l and he initially used to the first and third index. Einstein also used the notation .Bsrt srt another notation for the Christoffel symbol, see Einstein (1916a, p. 802).

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3 Different Pathways to the Generalization of Kaluza’s Theory

He then argued that one does not need to consider the derivatives of the Christoffel symbol and that l H = g st ┌sla ┌at

.

(3.197)

suffices (Einstein, 1916a, pp. 803–805). This is equivalent to the discussion from √ above with the difference that here, Einstein chose . −g = 1 such that the first term on the right hand side of Eq. (3.194) vanishes anyhow. For showing the equivalence of the two Lagrange densities, Einstein considered both R and .H as functions of .g ab and .∂t g ab . It then is l l l δH = δg st ┌sla ┌ta + g st δ┌sla ┌ta + g st ┌sla δ┌ta .

(3.198)

l l a l a g st δ┌sla ┌ta = g ts δ┌ta ┌sl = g st δ┌ta ┌sl ,

(3.199)

l l δH = δg st ┌sla ┌ta + 2g st ┌sla δ┌ta .

(3.200)

.

By .

it is .

We now use that it is both ⎞ ⎛ l l l = δg st ┌sla ┌ta ┌sla δ g st ┌ta + g st ┌sla δ┌ta

.

(3.201)

and ⎞ 1 ⎞ ┌ ⎛ ⎛ ┐ 1 l = ┌sla δ g st g lr (∂t gar + ∂a gtr − ∂r gta ) = ┌sla δ g st g lr ∂a gtr . ┌sla δ g st ┌ta 2 2 (3.202)

.

The last equality holds as it is ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ ┌sla δ g st g lr ∂t gar = ┌lsa δ g sr g lt ∂r gat = ┌sla δ g lr g st ∂r gta .

.

(3.203)

Using Eq. (3.201) in the first and Eq. (3.202) in the second equality, Eq. (3.200) yields ⎞ ⎞ ⎛ ⎛ l l l − δg st ┌sla ┌ta δH = 2┌sla δ g st ┌ta = ┌sla δ g st g lr ∂a gtr − δg st ┌sla ┌ta .

.

(3.204)

Finally, we get l δH = −┌sla δ∂a g sl − δg st ┌sla ┌ta

.

(3.205)

3.6 Manuscript Pages Dealing with the 1938 Theory

215

by using .

− ∂a g sl = g st g lr ∂a gtr .

(3.206)

Einstein then concluded that it is .

∂H ∂H l = −┌sta = −┌sla ┌ta and ∂ (∂a g st ) ∂g st

(3.207)

by replacing the variations in Eq. (3.205) by derivations with respect to .g st and .∂a g st . We can now put these expressions into the Euler-Lagrange equation146 .

∂H − ∂a ∂g st

⎛

∂H ∂ (∂a g st )

⎞ =0

(3.208)

and get .

l − ┌sla ┌ta + ∂a ┌sta = 0,

(3.209)

which fits to R from Eq. (3.196) showing the assertion.

3.6.4 AEA 62-785 The manuscript page AEA 62-785 has a back page AEA 62-785r that contains geometric sketches, see Sect. 2.3.1. On the front of the page, Einstein derived the covariant derivative of tensor densities. It begins with the equation147 x ' = x 0 + f,

.

0

(3.210)

which points to the context of cut-transformations as in expression (3.30), where f depends on .x 1 to .x 4 . Einstein’s next equation can be associated with the partial derivative of a scalar .ρ that transforms with respect to cut-transformations as .

146 For

∂ρ ∂ρ ∂x s ∂ρ ∂f ∂ρ ∂x α , a = a − a = α ' s ' ' ∂x ∂x ∂x ∂x ∂x ∂x 0 ∂x a

(3.211)

instance, see Misner et al. (1973, p. 323), N˘astase (2019, pp. 49–50), or Franklin (2017, p. 87) for a detailed derivation of the Euler-Lagrange equation in field theories. 147 Einstein used subscripts for the coordinates.

216

3 Different Pathways to the Generalization of Kaluza’s Theory

according to Eq. (3.32). Einstein did not write down the second expression. His third equation reads148 ϕa' = ϕa −

.

∂f , ∂x a

(3.212)

which is the transformation of .ϕa with respect to cut-transformations according to Eq. (3.37). We recall that Einstein and Bergmann used the notation .Aa := γ0a in their publication, while Einstein used the notation .ϕa := γ0a in his Washington manuscript. We argue that he also used this notation on the present manuscript page. As we derived in Sect. 3.2.1, the operator ⎛ .

∂ ∂ − ϕa 0 a ∂x ∂x

⎞ (3.213)

leaves, acting on a tensor, the components of the tensor invariant with respect to cut-transformations. With respect to four-transformations, it is ⎛ .

∂ ' ∂ a − ϕa ' ∂x ∂x ' 0

⎞

∂x α ∂ρ ∂ρ ∂ρ ∂x α ' ∂ρ − ϕ = − ϕ α a ∂x ' a ∂x α ∂x ' a ∂x ' a ∂x 0 ∂x 0 ⎞ ⎛ ∂ρ ∂x s ∂ρ = 'a − ϕs 0 , (3.214) ∂x s ∂x ∂x

ρ' =

where we used Eqs. (3.44) and (3.31). This equation is equivalent to Einstein’s fourth equation on the present manuscript page. We conclude that the operator .∂a − ϕa ∂0 acting on a scalar behaves like a vector. We do not find this discussion explicitly in Einstein and Bergmann (1938), but in Einstein’s Washington manuscript.149 In the following, Einstein then considered the scalar density . ρ of weight n (n)

without using the index indicating the weight of the tensor density.150 It transforms as | α |n | ∂x | ' | .ρ = | (3.215) | ∂x ' β | ρ, which was written down by Einstein. We recall that this holds in the fourdimensional theory as well as in the five-dimensional theory with respect to four-transformations, see Eqs. (3.67) and (3.84).151

148 Einstein

used the notation .f,a for the derivative .∂a f . the discussion of his Eq. (17). 150 We note that Einstein used the same notation for a scalar on the first part of the manuscript page. 151 Scalar densities are invariant under cut-transformations, see Eq. (3.85). 149 See

3.6 Manuscript Pages Dealing with the 1938 Theory

217

Let us denote the Jacobi determinant by D, we get from Eq. (3.215) lg ρ ' = n lg D + lg ρ,

.

(3.216)

which we find in Einstein’s notes next to Eq. (3.215). Einstein then investigated the transformation of the quantity .ρ −1 ∂a ρ. By using Eq. (3.215) in the third equality, we get .

1 ∂ρ ' ∂ lg ρ ' ∂ρ ' ∂ lg ρ ' ∂ lg D ∂ lg ρ ∂ lg D ∂D ∂ lg ρ ∂ρ a = a = a =n a + a =n a + ' ' ' ' ' ' ' ' ρ ∂x ∂ρ ∂x ∂x ∂x ∂x ∂D ∂x ∂ρ ∂x ' a ⎞ ⎛ ∂s ρ ∂x s n ∂D ∂x α 1 ∂ρ ∂x α ∂s D + = + = n . (3.217) D ∂x α ∂x ' a ρ ∂x α ∂x ' a D ρ ∂x ' a

This holds with respect to four-transformations in the generalized Kaluza theory152 as well as for transformations in the four-dimensional theory. Einstein only wrote down the first and last expression. On the left side of the page, he then continued with g ' = D 2 g,

.

(3.218)

which means that the determinant g is a scalar density of weight 2, see Eqs. (3.70) and (3.87).153 Similarly to Eqs. (3.216) and (3.217), it follows that .

⎞ ⎛ ∂s g ∂x s 1 ∂g ' ∂ lg g ' ∂ lg D ∂ lg g ∂s D + = = 2 + = 2 , ∂x ' a ∂x ' a ∂x ' a D g ∂x ' a g ' ∂x ' a

(3.219)

where we used Eq. (3.218) in the second equality. On the right side of the page,154 Einstein started a slightly separated calculation. He considered .∂s D and crossed out his first equation. Given the Jacobi matrix ⎛ Jαβ =

.

∂x α ∂x ' β

⎞ (3.220) α=0,...,4;β=0,...,4

and its determinant D, it is ∂s D = D tr (J −1 ∂s J )

.

152 See

(3.221)

Eq. (3.31). note that in the five-dimensional theory, it is a scalar density as it is also invariant under cut-transformations, see Eq. (3.88). 154 We note that the considerations on the right side of the page were not needed in the subsequent calculations. 153 We

218

3 Different Pathways to the Generalization of Kaluza’s Theory

by Jacobi’s formula, where the inverse of the Jacobi matrix is Jγ−1 δ =

⎛

∂x ' γ ∂x δ

.

⎞ (3.222)

. γ =0,...,4;δ=0,...,4

It follows that ⎛ ∂s D = D tr

4 Σ

.

⎛

⎞ Jγ−1 α ∂s Jαβ

= D⎝

α=0

4 4 Σ Σ

⎞ −1 Jβα ∂s Jαβ ⎠ = D

β=0 α=0

∂x ' β ∂x α . ∂s ∂x α ∂x ' β (3.223)

Einstein only wrote down the first and last expression of Eq. (3.223). He then concluded ∂s D = −D

.

∂x α ∂ 2 x ' β , ∂x ' β ∂x α ∂x s

(3.224)

which follows by ⎛ 0 = ∂s

.

∂x ' β ∂x α ∂x α ∂x ' β

⎞

∂x α ∂x ' β ∂x ' β ∂x α ∂ + ∂s . s ∂x α ∂x ' β ∂x ' β ∂x α

=

(3.225)

At the end of the present manuscript page, Einstein then looked at vector densities. He first noted that the expression ρ (s) .

(3.226)

s

g2

is a scalar. This holds since . ρ is a scalar density of weight s and .g s/2 is, by similar (s)

considerations as in Eq. (3.71) or at the beginning of Sect. 3.2.3, a scalar density of weight s as well. Hence, expression (3.226) is a tensor density of weight 0 and thus a tensor.155 Einstein continued and concluded that the two expressions ∂a ρ (s) .

ρ

−

s ∂a g 2 g

(s)

155 See

also our discussion at the beginning of Sect. 3.2.2 or in Footnote 42.

(3.227)

3.6 Manuscript Pages Dealing with the 1938 Theory

219

and ∂a ρ (s) .

ρ

l − s┌al

(3.228)

(s)

are vectors. For the first expression (3.227), it is156 .

⎞ ⎛ ⎞ ⎛ ∂l ρ ∂x l ∂l D ∂l g ∂x l 1 ∂ρ ' s 1 ∂g ' ∂l D s + 2 + − = s − ρ ' ∂x ' a 2 g ' ∂x ' a D ρ ∂x ' a 2 D g ∂x ' a ⎛ ⎞ s ∂l g ∂x l ∂l ρ − = 'a (3.229) ∂x ρ 2 g

with respect to four-transformations, where we used Eqs. (3.217) and (3.219). However, it is not invariant with respect to cut-transformations as it is .

1 ∂ρ ' s 1 ∂g ' 1 ∂ρ s 1 ∂g 1 ∂ρ ∂x α s ∂g ∂x α − = − = − 2 g ' ∂x ' a ρ ∂x ' a 2 g ∂x ' a ρ ∂x α ∂x ' a 2g ∂x α ∂x ' a ρ ' ∂x ' a ⎞ ⎞ ⎛ ⎛ ∂g ∂ρ ∂f ∂g ∂f 1 ∂ρ s − 0 a − − 0 a , = 2g ∂x a ρ ∂x a ∂x ∂x ∂x ∂x (3.230)

where we used the invariance of tensor densities with respect to cut-transformations in the first and Eqs. (3.32) and (3.211) in the third equality. We conclude that expression (3.227) is not invariant with respect to cut-transformation and thus it is not a tensor. Hence, we argue that Einstein here considered the four-dimensional case rather than the five-dimensional generalization. Clearly, by Eq. (3.66), expression (3.228) is then also a tensor. As last expression, Einstein wrote down l ∂a ρ − s ρ ┌al

(3.231)

.

(s)

(s)

and concluded that this is a vector density, which follows immediately by the multiplication of . ρ with expression (3.228). (s)

Let us now connect these calculations with Einstein and Bergmann’s 1938 theory. As in Eq. (3.68), the covariant derivative of a scalar density . ρ of weight s is (s) r ∇a ρ = ∂a ρ − s ρ ┌ar .

.

(s)

156 We

(s)

(s)

ignore the indices that indicate the weight of the tensor density.

(3.232)

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3 Different Pathways to the Generalization of Kaluza’s Theory

In fact, the right hand side is Einstein’s expression (3.231). We can, therefore, interpret the end of the page as an introduction of the covariant derivative of a scalar density, which itself is a scalar density. We can support our interpretation by means of a letter (AEA 6-266) from Einstein addressed to Bergmann that was written on July 22/23, 1938. It mainly discusses corrections concerning their publication and directly addresses passages on the pages 686 and 690 of their publication.157 At the end of the page we find an attached remark:158 P.S. (Hat nichts mit dem Manuskript zu thun) Die einfachste Einführung der Differentiation der skalaren Dichte ist so[:] ρ

.

(k) k g2

ist ein Skalar, also die logarithm. Ableitung davon

ein Vektor. (im gewöhnlichen Riemann-Fall). Im Kaluza-Fall ⎛ ⎞ tritt die Ableitung . ∂x∂ a − Aa ∂x∂ 0 dafür ein. (AEA 6-266)

Hence, he directly referred to expression (3.226). This is a scalar and thus its logarithmic derivative159 ⎛ρ ⎞ ∂ ρ a ∂ ⎝ (s)s ⎠ = (s) − s ∂a g . lg ∂x a ρ 2 g g2

(3.233)

(s)

is a vector, which is Einstein’s expression (3.227). As we saw above, replacing the second term in the last expression by using Eq. (3.66) and multiplication by . ρ yields (s)

Einstein’s expression (3.231) which is a scalar density of weight s. This procedure is equivalent to Einstein’s discussion subsequent to Eq. (18a) in his Washington manuscript.160 Indeed, replacing the partial derivative by the operator (3.213) according to the quote from his letter AEA 6-266 yields161 ⎛ .

∂ ∂ − ϕa 0 a ∂x ∂x

⎞

⎛ρ ⎞ ⎛ ⎞ 1 (s) l lg ⎝ s ⎠ = ∂a ρ − ϕa ∂0 ρ − s ρ ┌al ρ g2 (s) (s) (s)

(3.234)

(s)

157 See

also our discussions in Sect. 3.4.4.4 and 3.4.5, where we already quoted from this letter. first wrote .ϕa instead of .Aa in the last line. In their joint publication, they used the notation .Aa , while Einstein used .ϕa in his Washington manuscript as well as on the present manuscript page. 159 This is similar to Einstein’s considerations in Eqs. (3.217) and (3.219). 160 See pages 7–8 in AEA 97-487 and pages 9–10 in AEA 2-121 and 5-008. 161 This is in accordance with our discussion around Eq. (3.48), where we introduced the operator (3.213). 158 Einstein

3.7 Concluding Remarks

221

where we used Eq. (3.90). Multiplication by . ρ then again yields a scalar density of (s)

weight s.162 In fact, we see that the expression in brackets on the right hand side of Eq. (3.234) is in accordance with our definition of the covariant derivative of scalar densities in the five-dimensional case from Eq. (3.86). We also note that the derivation given above is equivalent to Einstein’s discussion around Eq. (19) of his Washington manuscript. We conclude that Einstein derived the covariant derivative of scalar densities in the four-dimensional case on the present manuscript page by using the logarithmic derivative. He also considered the analogue for the five-dimensional theory. Furthermore, the manuscript page is connected to the letter AEA 6-266 from July 22/23 as Einstein there wrote to Bergmann how to derive the respective covariant derivatives. It is also connected to Einstein’s Washington manuscript and especially to the discussion of Eq. (19).163 As Einstein finished his Washington manuscript on July 6, we argue that the present manuscript page was written around the same time.

3.7 Concluding Remarks In this chapter, we first looked at the theory of general relativity from a modern point of view and then comprehensively presented Einstein and Bergmann’s generalization of Kaluza’s five-dimensional approach in order to find a unified field theory. The motivation of their new theory was to give the additional fifth dimension reality. In doing so, they replaced Kaluza’s cylinder condition by a periodicity condition which results in the fact that the metric components are now periodic functions of the fifth coordinate. By the analysis of correspondence from around that time, we know that Einstein had high hopes in this new theory. In 1941, however, Einstein had to drop it as they had not been able to find particle-like solutions. Einstein considered this new theory along different pathways. In addition to the English publication from 1938, there is also a German draft extant that was discussed in Einstein and Bergmann’s correspondence. Einstein also composed the so-called Washington manuscript that essentially has the same physical content as the publication but has never been published. Instead, he donated it to the Library of Congress. It is entitled “Unified Field Theory” and is the only writing, published or unpublished, that carries this title without any further specification. It is interesting to see that Bergmann no longer figured as co-author as Einstein composed his manuscript alone. In addition, it substantially differs in its internal structure as Einstein presented the theory independently of its historical roots and thus detached it from Kaluza’s theory. Instead, he focused on its axiomatic structure. Furthermore,

162 We

saw in our discussion around Eq. (3.48) that the operator leaves tensors invariant with respect to cut-transformations. 163 See pages 7–8 in AEA 97-487 and pages 9–10 in AEA 2-121 and 5-008.

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3 Different Pathways to the Generalization of Kaluza’s Theory

Einstein incorporated the mathematical appendix of the publication into the main text. We also saw that different versions of this manuscript are extant, namely a holographic, a typed, and a copied version. The differences between manuscript and publication are especially interesting as Einstein preferred the structure of his manuscript over the presentation in the publication.164 Finally, Einstein wrote down both text passages and calculations on several documents that are directly connected to their publication or to the Washington manuscript. For instance, the second part of the manuscript page AEA 62-789 that also contains sketches on projective geometry as well as the page AEA 62785165 can be associated with the Washington manuscript, while the manuscript page AEA 62-798 was rather written in the context of the publication. We also looked at the two manuscript pages AEA 62-802 and 62-794 which are connected to each other, where Einstein derived the action of the Einstein-Bergmann theory by using a modified Lagrange density. In the following two chapters, we will look at these and further working sheets in more detail. We will see that Einstein there tested and further developed the new theory.

164 We

already gave a short conclusion on our analysis of the Washington manuscript in Sect. 3.5, where we also addressed the questions of why Einstein composed it and why he donated it to the Library of Congress. 165 Its back page AEA 62-785r also contains sketches on projective geometry.

Chapter 4

Einstein’s Further Considerations on the Generalized Kaluza Theory

We already saw that certain Princeton manuscript pages are connected to Einstein and Bergmann’s theory from 1938.1 We identified these pages, since they contain preliminary passages, equations, and formulas or phrases like SchnittTransformation that could be connected to the publication (Einstein & Bergmann, 1938) or to the Washington manuscript. Among these pages belongs AEA 62-789 on which we identified sketches at the top of the page with projective geometry and considerations on the second part of the page with Einstein and Bergmann’s theory.2 However, the rest of the manuscript pages AEA 62-785r, 62-787r, and 62-789r that contain sketches on projective geometry do not contain such notes that can be directly connected to the publication or to the Washington manuscript.3 These pages rather contain unidentified calculations. Another manuscript page with such calculations is AEA 62-798 which we briefly discussed in Sect. 3.6.2. We saw that the first part of this page contains calculations that are somehow similar to those on the pages with projective geometry, while its second part could be associated with Einstein and Bergmann’s publication. A similar connection appears on the manuscript page AEA 62-789 and its back AEA 62-789r. We already argued that AEA 62-789 is the continuation of AEA 62-789r, see Sect. 2.3.3. Einstein first considered the unidentified calculations at the top of AEA 62-789r before he drew the sketches on projective geometry that continue at the top of AEA 62-789. He then finally wrote down some calculations and text passages in the context of the Washington manuscript. Hence, it seems as if the unidentified calculations are connected to both the sketches on projective geometry and to the 1938 theory.

1 See

Sects. 3.4 and 3.6. Sects. 2.3.3 and 3.6.1. 3 See Sects. 2.3.1, 2.3.2, and 2.3.4. 2 See

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Schütz, Einstein at Work on Unified Field Theory, Einstein Studies 17, https://doi.org/10.1007/978-3-031-52127-0_4

223

224

4 Einstein’s Further Considerations on the Generalized Kaluza Theory

One characteristic of these calculations is long equations that can be found on the three manuscript pages AEA 62-785r, 62-789r, and 62-798. For instance, on AEA 62-789r, Einstein wrote down the equation ⎞ ⎞ ⎛ ⎞ ⎛ ⎛ ⎞ ⎛ 1 r r 1 1 α1 −α1 2 + −f1 2 2 f0 + f1 ρ 2 ρ ρ ρ ┐ ⎞ ┌ ⎛ 1 r3 1 1 − f1 3 β3 3 − 2r f0 + f1 ρ 2 ρ ρ

.

+ α3 η2

r 1 1 r + f1 3 δ1 = 0, 2 2 ρ ρ ρ

(4.1)

where he apparently set 1 f = f0 + f1 , ρ

.

1 α = α1 , ρ

η = η2

r , and ρ2

β = β3

r2 . ρ3

(4.2)

The equations on AEA 62-785r bear certain resemblances. Here, Einstein wrote down ┐ ┌ ┌ ⎛ α α4 ⎞ 3 ┐ 3 α3 3 ' ' 2 r β + − α4 − (ν + ϕ) rα + r − .−α +δ + 2 4 2 4 ⎞ ⎛ ⎛ α α4 ⎞ 1 2 ' 3 rδ = 0, r ϕ + 2rν + 2rϕ η + − + (4.3) − 2 2 4 where we have η = rη

.

and

β = r 2 β.

(4.4)

Although there are certain differences as, for instance, the fact that the quantity .ρ does not appear on AEA 62-785r, but on AEA 62-789r, the equations seem to be related to each other. For example, on both pages, Einstein extracted the factors r and .r 2 in the quantities .η and .β, respectively. We also see that on both pages, Einstein used the notations .α, .β, .η, and .δ. In addition, the quantities .α3 and .α4 appear which immediately reminds us of the constants in the new field equations as in Eq. (3.137).4 This intuition, however, is wrong as we will see in the following sections. By browsing through the extant working sheets with unidentified research notes in the database, we found many further pages that contain such kinds of calculations. This holds especially for many manuscript pages on reel 62 with archival numbers between AEA 62-783 and 62-805.

4 In

Eq. (3.137), Einstein and Bergmann used the subscript directly beneath the letters. In his Washington manuscript, however, he used the notation .αi , see also our discussion in Sect. 3.4.3.

4 Einstein’s Further Considerations on the Generalized Kaluza Theory

225

We summarize that we have three kinds of notes: First, the sketches on projective geometry that we discussed in Chap. 2, second, the notes related to the publication and to the Washington manuscript that we be discussed in Chap. 3, and third, the unidentified calculations, which we will address in the present chapter.5 It is natural to ask whether these three kinds of notes are related to one another. In fact, it turned out that the unidentified calculations are directly connected to Einstein and Bergmann’s new theory. This is somehow surprising as the calculations at first glance have nothing in common with the equations of the new theory. In order to draw the connection between these two kinds of calculations, it is necessary to look at Einstein and Bergmann’s correspondence in more detail. There, we will find a connection between the characteristics of the unidentified calculations mentioned above and the field equations. By analyzing further correspondence between Einstein and V. Bargmann, we will even be able to suggest an expression for the five-dimensional metric components in dependence of the above-mentioned quantities .α, .β, .δ, and .η. This will then enable us to better understand the unique material that the manuscript pages offer. In fact, it allows us to analyze and reconstruct how Einstein and his assistants elaborated and further investigated the new theory. Such information can only be found on unpublished documents since Einstein never published on it.6 The further analysis will also substantiate our conjecture on the purpose of Einstein’s considerations on projective geometry from Sect. 2.6.1 by connecting them with the unidentified calculations. Due to the extent of our analysis of both the letters and manuscript pages, we will only present the main results in this chapter. For a full and detailed analysis of the individual letters and manuscript pages, we refer to Schütz (2021). As already discussed in Sects. 1.4 and 1.5 and as also claimed in Van Dongen (2010, pp. 130–156) and Van Dongen (2002, pp. 192–197), the context of Einstein’s further investigations was to find particle-like solutions of the new field equations.7 We recall that Einstein had this goal throughout his attempts of formulating a unified field theory and that it was this problem that motivated him to generalize Kaluza’s theory, see Sect. 1.4 and Einstein and Bergmann (1938, p. 688). Similarly, he finished his Washington manuscript with formulating the task of looking for particle solutions in the framework of the new theory, see Sect. 3.5 and page 15 in AEA 2-121. While Einstein had very high hopes for his new theory,8 we learn from his follow-up paper co-authored by both of his assistants V. Bargmann and P. Bergmann that he had not succeeded in finding such solutions (Einstein et al., 1941, pp. 224–225) and gave up on this approach for good at the latest with his proof 5 In

Chap. 5, we will learn about the fourth kind: calculations on an integral form that stands for the potential and certain functions appearing in the metric components. 6 For instance, one step in Einstein’s program was to consider power series expansions at the origin and at infinity. We find the first terms of such power series expansions already in Eq. (4.2) on AEA 62-789r. For a brief summary, see also Sect. 1.5. 7 Van Dongen also discussed parts of Einstein’s correspondence with P. Bergmann, V. Bargmann, and Pauli (Van Dongen, 2002, pp. 192–197). 8 See Sect. 3.3.

226

4 Einstein’s Further Considerations on the Generalized Kaluza Theory

“on the non-existence of regular stationary solutions of relativistic field equations” written together with Pauli in 1943 (Einstein and Pauli, 1943).9 In the process of analyzing the research notes, we will also discuss Einstein’s repertoire of mathematical tools on the basis of selected manuscript pages. In Chap. 2, we already looked at Einstein’s knowledge in projective geometry. In the present chapter, we will see that Einstein was very much experienced in making long and sophisticated calculations by using ingenious mathematical tools in order to facilitate the respective subsequent calculations. First, we will re-date the letters from Einstein and Bergmann’s correspondence in Sect. 4.1. It took place in summer 1938 and was preliminary categorized and dated by Helen Dukas. However, we will need to slightly correct her dating. The main results of our analysis of the correspondence and the manuscript pages will already be used in Sect. 4.2 in order to outline the context of Einstein’s considerations subsequent to their publication. In particular, we will address the question of how Einstein further developed the new theory and especially the field equations. There, we will start in Sect. 4.2.1 with recalling the most important relations of the theory. Thereafter, we will look at specific documents from Einstein and P. Bergmann’s correspondence from 1938 in Sect. 4.2.2 in order to learn about the so-called Stromgleichung as well as certain wave equations. These equations are the connection between Einstein and Bergmann’s publication and the calculations on the working sheets. In Sect. 4.2.3, we briefly point to a difficulty emerging by Einstein’s notation. We will then in Sect. 4.2.4 analyze further documents as, for instance, letters from the correspondence between Einstein and V. Bargmann in order to propose an expression for the five-dimensional metric in terms of the functions .α(r), .β(r), .δ(r), and .η(r). We will argue that Einstein used these metric components and derived the Stromgleichung and the wave equation starting from the field equations or from the identities of Einstein and Bergmann’s theory. In doing so, we will look at the Stromgleichung in more detail and connect it with the fourdimensional case. We will then propose an ansatz that Einstein apparently used in order to derive the respective equations. In Sect. 4.3, we will then finally look at several manuscript pages that were written in this context. On these documents, Einstein frequently considered the above-mentioned equations. We will start with a categorization of the manuscript pages.10 Instead of discussing each manuscript page individually, we rather give short summaries in most of the cases. In Sect. 4.3.1, we will discuss the calculations on the manuscript pages that also contain sketches on projective geometry. This will enable us to date them accurately and to specify our conjecture on the purpose of considering projective geometry. In Sect. 4.3.2, we will mention several manuscript pages where Einstein considered power series expansions of the functions .β and .η at the origin and at infinity. In Sect. 4.3.3, we give an overview of manuscript pages that contain investigations of the above-mentioned wave equations as well as on the Laplace operator. As we already saw in Eq. (4.3), we will encounter some equations

9 We

already quoted the respective passages in Sect. 1.4. will see that the different categories partly overlap.

10 We

4.2 Context of Einstein’s Further Calculations

227

written in terms of r and some in terms of .ρ. In particular, the wave equations are written in terms of r. Einstein then rewrote these equations such that they depend on .ρ. Manuscript pages containing similar considerations will be presented in Sect. 4.3.4. Further manuscript pages that are presented in Sect. 4.3.5 contain some idiosyncratic relations. They probably were written before the main correspondence started. Thereafter, we will come back to the substitution .ρ in Sect. 4.3.6 and present manuscript pages that contain further expressions for .ρ(r). In Sect. 4.3.7, we will then present an interesting manuscript page where Einstein used mathematical tools in order to find a certain expression for a power series expression. In fact, we will learn that he first tried to derive it by computing the derivatives, a process that led him to several mistakes and that can be reconstructed by a careful analysis. Finally, in Sect. 4.3.8, we point to further research notes that were probably written in the same context. We end this chapter with a brief conclusion in Sect. 4.4.

4.1 Einstein and Bergmann’s Correspondence: Dating the Letters The letters from summer 1938 were already given a preliminary date by Helen Dukas. However, we need to re-date some of the letters and we propose a new sequence of the letters, which is shown in Table 4.1.11

4.2 Context of Einstein’s Further Calculations We will start this section with recalling the most important relations of Einstein and Bergmann’s generalized Kaluza theory. We will then show that the general equations from the publication are connected with the more specific equations considered by Einstein on his manuscript pages and in the correspondence. In order to do so, we will use some important results of our analysis of both Einstein’s research notes and correspondence.12 We will also present a suggestion of how Einstein might have derived the specific equations from the general field equations. In particular, we will present the metric components that were considered by Einstein and his assistants in Sect. 4.2.4.

11 In the table, PB stands for Peter Bergmann and AE for Albert Einstein. P stands for Princeton, NP for Nassau Point, and R for Robinhood. In most of the cases, we took the locations from Dukas’ notes. In AEA 54-240, Einstein wrote to Razovsky that his summer address will be in Nassau Point from June 15. The abbreviation “D” stands for Dukas and refers to the date that was suggested by Dukas if it differs from our suggestion. For a comprehensive argumentation of our suggested dating, see Schütz (2021). 12 We note that during the correspondence, Einstein and Bergmann frequently used terms like “De Broglie Frequenzen.” For a detailed discussion, see Van Dongen (2010, 2004, 2002).

228

4 Einstein’s Further Considerations on the Generalized Kaluza Theory

Table 4.1 Sequence of letters of the Einstein–Bergmann correspondence from 1938. For the abbreviations used here, see Footnote 11 AEA 6-240 6-262 6-245 6-246 6-247 6-249 6-250 6-251 6-252 6-253 6-254 6-256 6-242 6-271 6-258 6-263 6-264 6-248 6-265 6-266 6-261 6-267 6-268 6-269 6-270 6-259 6-260 6-272 6-243 6-244 6-255

From/To PB, P to AE, P AE, NP to PB, R AE, NP to PB, R PB, R to AE, NP PB, R to AE, NP AE, NP to PB, R AE, NP to PB, R PB, R to AE, NP AE, NP to PB, R AE, NP to PB, R PB, R to AE, NP AE, NP to PB, R AE, P/NP to PB, R AE, NP to PB, R PB, R to AE, NP PB, R to AE, NP PB, R to AE, NP AE, NP to PB, R AE, NP to PB, R AE, NP to PB, R AE, NP to PB, R AE, NP to PB, R PB, R to AE, NP AE, NP to PB, R PB, R to AE, NP PB AE, NP to PB, R PB, R to AE, NP AE to PB

Date on letter 5-5-38 Monday 6-17 6-18-38 6-20-38 Tu, 6-21

Th, 8-4-38

Our dating Th, 5-5-38 Mo, 6-13-38 Fr, 6-17-38 Sa, 6-18-38 Mo, 6-20-38 Tu, 6-21-38 6-22 – 6-29-38 Th, 6-30-38 Fr, 7-1-38 Mo, 7-4-38 Sa, 7-9-38 Tu, 7-12-38 Fr, 7-15-38 Fr, 7-15-38 Fr, 7-15-38 Sa, 7-16-38 Su, 7-17-38 Mo, 7-18-38 Th, 7-21-38 7-22/23-38 Su, 7-24-38 Mo, 7-25-38 Th, 7-28-38 7-29 – 8-3-38 Th, 8-4-38

8-15-38

8-5 – 8-14-38 Mo, 8-15-38

6-30-38 7-1-38 7-4-38 7-5-38 Tuesday Friday Friday 7-15-38 7-16-38 7-16-38 Monday Thursday 7-22/23 Sunday 7-25-38 7-28-38

Notes Calcs from 1937 D: 7-14/15

D: 6-25 – 6-28

D: 7-5-38 D: 7-6/7-38 D: Early June 1938 D: 8-5-38 3rd page is 6-270p3

D: 6-20-38 D: 7-18 – 7-20-38 D: 7-14/15-38

D: 7-30/31-38 3rd p. is 6-258p3 Calcs D: 7-16/17-38 Different context Related to AEA 6-243 Copies of publication

4.2.1 Most Important Relations of the Generalized Kaluza Theory In their joint publication on the generalization of Kaluza’s theory, Einstein and Bergmann considered the five-dimensional metric components .γμν , where γ00 = 1

.

(4.5)

4.2 Context of Einstein’s Further Calculations

229

according to Eq. (3.26). It is furthermore13 .

∂ γa0 = 0, ∂x 0

(4.6)

which means that the components .γa0 of the metric do not depend on .x 0 . In Einstein and Bergmann (1938, p. 691 and eq. (8)), they then introduced the contravariant unit vector ⎧ 0, μ = {1, 2, 3, 4} μ .A = (4.7) 1, μ = 0 with the covariant components A = Aβ γβα = γ0α ,

.

(4.8)

see Eq. (3.27). We note that by Eq. (4.6), it is ∂ Aa = 0 ∂x 0

(4.9)

gαβ = γαβ − Aα Aβ ,

(4.10)

.

in particular. In the next step, they introduced .

see Eq. (3.28). By Eqs. (4.5) and (4.8), it is g0β = 0.

.

(4.11)

This is the reason why Einstein and Bergmann considered the components .gab as the components of a four-dimensional metric, which are periodic functions of .x 0 . Furthermore, as defined in Einstein’s Washington manuscript, for the fourdimensional metric, g ac gcb = δba

.

(4.12)

holds, see Eq. (3.38).

13 As usual, if not stated otherwise, Latin indices run from 1 to 4, while Greek indices run from 0 to 4.

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4 Einstein’s Further Considerations on the Generalized Kaluza Theory

For vectors .Bs and .B s , the covariant derivative is defined as14 .

l Bs;a = Bs,a − Aa Bs,0 − Bl ┌sa

(4.13)

s B s;a = B s,a − Aa B s,0 + B l ┌la ,

(4.14)

and .

where l ┌sa =

.

) ( ) ( )┐ 1 lm ┌ ( g gms,a − Aa gms,0 + gma,s − As gma,0 − gas,m − Am gas,0 , 2 (4.15)

see Eqs. (3.54) and (3.55). The covariant derivatives of tensors of general rank are being built analogously. In particular, it follows that gmn;a = g mn;a = 0,

(4.16)

.

see Eq. (3.57). Einstein and Bergmann also introduced the antisymmetric derivatives Aab = Aa,b − Ab,a

(4.17)

.

as in Eq. (3.41). They then derived the field equations in a general form from the variational principle with the Lagrange density √ .H = −g (α1 H1 + α2 H2 + α3 H3 + α4 H4 ) , (4.18) where H1 = R, and H2 = Amn Amn ,

.

(4.19)

see Eqs. (3.62) to (3.64).15 The first set of field equations became ⎛

⎞ ⎛ ⎞ 1 1 1 1 Rkl + Rlk − Rgkl + α2 2Akm Al m − Amn Amn gkl 2 2 2 2 ⎞ ⎛ 1 mn rs rs + α3 − ∂0 g ∂0 gmn gkl + 2g ∂0 gkr ∂0 gsl − 2∂0 ∂0 gkl − ∂0 gkl g ∂0 grs 2 ⎞ ⎛ )2 1 ( mn + α4 gkl g ∂0 gmn + 2g mn ∂0 ∂0 gmn + 2∂0 gmn ∂0 g mn = 0. (4.20) 2

α1

.

14 In the following, we will adopt Einstein’s notation of the covariant and partial derivatives, where it is .∂a Bs = Bs,a and .∇a Bs = Bs;a . 15 As usual, we use the notation .α instead of .α and .H instead of .H . i i i

i

4.2 Context of Einstein’s Further Calculations

The second set was ⎛ ) )√ ( ( mn s n − 4α2 Ast ;t −g dx 0 = 0, . α1 g ┌mn,0 − g ms ┌mn,0

231

(4.21)

where the integral is over one period of .x 0 , see Eq. (3.138). This is equivalent to equation (30) in Einstein’s Washington Manuscript, where he set .α1 = 1.16 The integral in Eq. (4.21) remains, since .δAm does not depend on .x 0 .

4.2.2 Wave Equations and the Stromgleichung In this section, we give a brief summary about the technical subject discussed in the correspondence, especially about the most important equations.17 During their entire correspondence of that period, Einstein and Bergmann referred to certain equations, which also appear on the manuscript pages, sometimes in a slightly modified form. We argue that these equations are the four equations .F1 to .F4 on AEA 6-259 as well as the so-called Stromgleichung on AEA 6-258.18 The first equation .F1 on AEA 6-259 is19 .

┐ ⎛ ⎞ ┌ ⎞ 3 α4 ⎞ 1 ' 1 ⎛ α3 α3 − α4 − f 2 rα + − + r β− rf + 2f η 4 2 r 4 2 2 ⎞ ⎛ α α4 3 rδ = 0. (4.22) + − + 4 2

− α' + δ' +

⎛

The second equation .F2 is 1 2f α ' + f ' α − 2

.

16 In

⎛

⎞ 1 2 1 f + f ' β + α3 η − f ' δ = 0, r 2 2

(4.23)

the Washington Manuscript, Einstein used the notation .ϕm instead of .Am . also discussed corrections of their publication Einstein and Bergmann (1938) as well as the Washington manuscript. 18 The third page of this letter containing the Stromgleichung was numbered AEA 6-270 but belongs to the letter AEA 6-258. We will refer to this page as AEA 6-258. 19 We note that on AEA 6-259, the equation might be read to contain the quantity .α instead of .α 2 3 in the last bracket on the left hand side. However, by the analysis of many manuscript pages, we argue that it is actually .α3 , see also Sect. 4.3. 17 They

232

4 Einstein’s Further Considerations on the Generalized Kaluza Theory

while the third equation .F3 is ⎞ ⎛ ⎞ ⎛ 5 ' 6 1 1 2 f − f ' δ' f − f α ' − f ' β ' + α3 η' + . r 2 r 2 2 ┐ ┌ ⎛ ⎞ 1 + 2α3 − 6α4 − 2f 2 f + f '' α 2 ⎞ ⎛ ⎞ ⎛ 6 4 2 1 '' 2 ' 1 '' + 2 f − f − f − 2α4 f β − f η + 2α4 f − f δ = 0. r 2 r 2 r (4.24) The fourth equation .F4 is ⎞ ┌ ⎛ ┐ α3 1 ' ⎛ α3 3 α4 ⎞ ' 1 ' ' − α4 − f 2 r − α − + rβ − rf η . 4 2 r 4 2 2 ┐ ┌ ⎛ ⎞ α4 1 ' α3 r+ δ + − + 4 2 r ⎞ ⎛ ⎞ ⎛ 1 3 5 α4 α3 2 ' 2 − α4 − f − 2rff α + 2 + f − α3 − β + 4 2 4 2 r ⎞ ⎛ 1 '' 5 ' 2 rf + f + f η − 2 r 2 ⎞ ⎛ α α4 3 δ = 0. + − + (4.25) 4 2 Einstein computed also .F4 − F1 /r on AEA 6-259, which is ⎞ ⎛ ⎛ α α3 3 α4 ⎞ ' 1 ' ' ⎛ α3 α4 ⎞ ' 3 − α4 − f 2 rα ' − + rβ − f rη + − + rδ − 2rff ' α . 4 2 4 2 2 4 2 ⎞ ⎛ ⎞ ⎛ 1 '' ' 2 rf + 2f η = 0. (4.26) + f − α3 β − 2 Einstein and Bergmann called equations (4.22) to (4.25) wave equations as well as field equations throughout their correspondence.20 The so-called Stromgleichung is21 .

4π f α(3α + 2β) λ ┌ ⎛ ⎞┐ 2 2 2π ' 2 η −α + β + δ) + η(3α ' + β ' + δ ' − α − β + δ + λ r r r

−

= 2α2 ϕ 4τ ;τ .

20 For

(4.27)

instance, see AEA 6-251 or 6-248. wrote down this equation in AEA 6-258 with some ambiguity concerning the sign of the term .(2δ)/r. However, we learn from the letter AEA 6-268 that the sign is positive. 21 Bergmann

4.2 Context of Einstein’s Further Calculations

233

It seems as if Bergmann sent these equations to Einstein at the beginning of their correspondence on May 5, 1938 in AEA 6-240. In this letter, Bergmann used the terms “Wellengleichungen” and “Stromgleichung.” However, he wrote that he was not sure about the correctness of the Stromgleichung as he had only calculated it once completely and a second time without the final operations. He also mentioned that the Stromgleichung had not yet been checked by another person (Infeld).22 Two months later, on July 15, he sent the Stromgleichung to Einstein again upon his request in AEA 6-258. Still, he was not sure about the correctness as we learn from the letter AEA 6-268 written on July 28, where he was finally convinced about its correctness. Hence, Bergmann needed almost 3 months to check the correctness of the Stromgleichung. We have no indication that Einstein as well would have computed the Stromgleichung. We note that Eqs. (4.22) and (4.23) are differential equations of first order while Eqs. (4.24), (4.25), and (4.27) are differential equations of second order, since the term .ϕ 4τ ;τ contains second derivatives. Einstein and Bergmann considered all Eqs. (4.22) to (4.25) as first-order differential equations as they only considered derivatives of the functions .α, .β, .δ, and .η.23 The equations are written in terms of the variable r. As indicated in AEA 6240, all terms appear to be real-valued. However, by our analysis, it is under consideration whether .η is a real or a pure imaginary quantity.24 The prime notation stands for derivatives with respect to r. It is .f = ω−2π ϕ/λ. Einstein and Bergmann frequently used the approximation .f ≈ ω throughout the correspondence, where .f0 = ω = 2π ν is the angular frequency and .ν is the frequency. The wavelength is denoted by .λ.25 They also called the quantity f frequently “Frequenz.”26 By assuming .f ≈ f0 = ω, Eqs. (4.24) and (4.25) become differential equations of first order. Concerning the coordinates, the .x 4 -dimension is the time dimension t. The coordinate .x 0 describes the dimension in which the space is periodic according to the publication Einstein and Bergmann (1938). The coordinates .x 1 to .x 3 are spacelike coordinates. The index .τ in Eq. (4.27) probably runs from 1 to 4.

22 “Nun noch zur Rechensicherheit: Die Wellengleichungen sind zweimal gerechnet, von Anfang bis Ende, die Stromgleichung zweimal bis auf die letzten Operationen (Zusammenführung der Glieder und Berücksichtigung des Realitätscharakters). Infeld habe ich diese Woche noch nicht gesehen, sodaß alle Rechnungen nicht mit den Resultaten von jemand anderem verglichen sind” (AEA 6-240). 23 For instance, see AEA 6-251 or 6-268. 24 See the letters AEA 6-240, 6-268, and 6-270. 25 As we saw in Sect. 3.2.1, in their publication, and in the Washington manuscript, .λ could be interpreted as the coordinate distance between two corresponding points. See also Van Dongen (2002, p. 195) and Van Dongen (2010, p. 150). 26 “Frequency”.

234

4 Einstein’s Further Considerations on the Generalized Kaluza Theory

4.2.3 The Usage of Different Notations Einstein and Bergmann used several notations which can easily be confused. This section aims to clarify these notations. First, Einstein used both the notations .αi and .α interchangeably. In their i

publication, they used the notation .α to describe constants appearing in the Lagrange i

density from Eq. (4.18), see Einstein and Bergmann (1938, p. 694). In his typescript of the Washington manuscript with the archival numbers AEA 2-121 and 5-008, he first used the notation .αi and later corrected it to .α . In the holographic version of the i

manuscript with the archival number AEA 97-487, he used both notations .α and .αi , i

see equation (26). For the sake of simplicity, we will use only the notation .αi . As we will see, the same constants appear also in many equations both in the correspondence and on the manuscript pages, where Bergmann and Einstein mostly used the notation .αi as in equations .F1 to .F4 on AEA 6-259. As indicated in AEA 6270, the constants .αi appearing on AEA 6-259 differ from the same constants in their publication by the factor .4π 2 /λ2 . Second, the notations .αi and .α not only stand for the constants appearing in i

the Hamilton function but also for the coefficients in the power series expansion of .α as in Eq. (4.2). This also holds for the functions .β, .η, .δ, and .ϕ, respectively. For instance, in the letter AEA 6-262, Einstein used the notation .ϕ for the second 2

coefficient in the power series expansion of .ϕ, while he used both notations .ϕ and .ϕ2 2

in AEA 6-250. In AEA 6-252, he only used the notation .ϕ2 . The same phenomenon appears on many manuscript pages: On AEA 62-787, he used the notation .α for the 1

first coefficient in the power series expansion of .α while using the notation .α1 on AEA 62-789r. As in Eq. (4.1) on AEA 62-789r, it happens that the notations .α1 and .α3 appear in one equation, where .α1 stands for the first coefficient in the power series expansion of .α, but .α3 for the constant factor appearing in the Hamilton function. We will come back to this equation in Sect. 4.3.1.3 in more detail. In another context, .α3 can also stand for a coefficient in the power series expansion of .α as in AEA 6-253. Third, the constants .α3 and .α4 appearing in the Hamilton function are connected to constants .C1 and .C2 appearing in a correspondence between Einstein and V. Bargmann in 1939. These constants play a decisive role in the determination of the frequency as we can see on the manuscript page AEA 62-805. In accordance with the letter AEA 6-264, the frequency .f02 = ω2 = α3 belongs to the so-called anisotropic case, while the frequency f02 = ω2 = α3

.

α3 − 4α4 α3 + 2α4

(4.28)

belongs to the isotropic case.27 27 On

the manuscript page AEA 62-805, Einstein derived a wrong equation due to a sign mistake.

4.2 Context of Einstein’s Further Calculations

235

We find very similar equations in the correspondence between Einstein and V. Bargmann in 1939. On AEA 6-208, for example, Bargmann wrote about the expression C1 ·

.

C1 + 4C2 . C1 − 2C2

(4.29)

It is likely that the quantities .C1 and .C2 are connected to the constants .α3 and .α4 , respectively. This is also supported by the letter AEA 6-208, where Bargmann wrote down the Lagrange density .H3 containing the factor .C1 . In Einstein and Bergmann’s publication, .H3 was multiplied by the factor .α3 , see Einstein and Bergmann (1938, p. 694). By Eqs. (4.28) and (4.29), it seems as if it is .C1 = −α3 and .C2 = α4 or .C1 = α3 and .C2 = −α4 . In the letter AEA 6-208, Einstein also used the relation .C1 = 2C2 . Einstein used similar relations for .α3 and .α4 on some manuscript pages.28 Fourth, the notation .ϕ4 does not only stand for the fourth coefficient in the power series expansion of .ϕ as in AEA 6-265 but also for the fourth component of the potential .ϕa as at the end of the letter AEA 6-261. It is, indeed, confusing that Einstein used the same notation .ϕ4 earlier in the very same letter also as a coefficient. In fact, the fourth component .ϕ4 of the potential .ϕa is frequently abbreviated by .ϕ. When Einstein then looked at the power series expansion of .ϕ, he used the notation .ϕ4 for the coefficient in its power series expansion. We finally remark that Einstein and Bergmann used the notation .Aa for the potential in their publication instead of .ϕa .29

4.2.4 Einstein’s Ansatz The aim of this section is to discuss an ansatz Einstein might have considered in order to investigate the general field equations (4.20) and (4.21). We address the question of how Einstein and Bergmann derived the Stromgleichung (4.27) starting from the field equations. Let us recall the second set of the field equations ⎛ .

28 For

) )√ ( ( mn s n − 4α2 Ast ;t −g dx 0 = 0, α1 g ┌mn,0 − g ms ┌mn,0

(4.30)

instance, see AEA 36-778, 62-798, or 62-803.

29 Einstein himself confused this when he wrote the letter AEA 6-266 to Bergmann and erroneously

wrote .ϕa instead of .Aa .

236

4 Einstein’s Further Considerations on the Generalized Kaluza Theory

as well as the so-called Stromgleichung .

−

4π f α (3α + 2β) λ ┌ ⎞┐ ⎛ 2 2 2 2π ' η (−α + β + δ) + η 3α ' + β ' + δ ' − α − β + δ + λ r r r − 2α2 ϕ 4τ ;τ = 0,

(4.31)

which was written down by Bergmann on AEA 6-258.30 The prime notation stands for derivatives with respect to r. Einstein introduced the notation .ϕm for the electric potential in his Washington manuscript31 as well as the antisymmetric derivatives ϕmn = ϕm,n − ϕn,m ,

.

(4.32)

equivalently to Eq. (4.17) in his publication.32 In this way, we can identify the term 4τ in Eq. (4.31) with the term .α Ast in Eq. (4.30) for .s = 4. .α2 ϕ 2 ;τ ;t We recall that the integral in Eq. (4.30) remains because .ϕm does not depend on 0 .x . In the following, we will assume that the argument vanishes as it is the case for the remaining field equations. In this case, it is .

) α1 ( mn s n g ┌mn,0 − g ms ┌mn,0 − 2α2 Ast ;t = 0, 2

(4.33)

and the second set of the field equations becomes structurally very similar to the Stromgleichung in the case for .s = 4. This becomes especially clear when considering .α1 = 2, which was indicated by Bergmann in the letter AEA 6-258p3.33 He wrote down H = 2R + α2 M + · · · ,

.

(4.34)

which probably refers to the variational principle from Eq. (4.18). There, .H1 = R can be identified with the gravitational part, while .H2 = Amn Amn can be associated with the electromagnetic part of the Hamilton function. Identifying .H1 with R and 34 it follows that Einstein and Bergmann chose .α = 2.35 .H2 with M, 1

30 The

respective page is numbered AEA 6-270p3, although it belongs to the letter AEA 6-258. page 7 of AEA 2-121. 32 See page 12 of AEA 2-121. 33 This letter was wrongly numbered as AEA 6-270. 34 The notation M possibly refers to Maxwell. 35 We note at this point that Einstein probably chose .α = 1 in his Washington manuscript, 1 see equation (26) on AEA 2-121 or 97-487. Furthermore, Bergmann and Einstein concluded in 31 See

4.2 Context of Einstein’s Further Calculations

237

Considering .α1 = 2, we get the relation 4 n g mn ┌mn,0 − g m4 ┌mn,0

.

4π f α (3α + 2β) λ ┌ ⎛ ⎞┐ 2π ' 2 2 2 η (−α + β + δ) + η 3α ' + β ' + δ ' − α − β + δ + λ r r r =−

(4.35)

by comparing Eq. (4.31) with Eq. (4.33) for .s = 4. We can now ask what assumptions satisfy this equation. Einstein apparently chose a special ansatz in order to further investigate the field equations of the generalized Kaluza theory. Such an ansatz needs to express the metric components .gmn or .γμν in terms of .α, .β, .δ, and .η. These quantities are then functions of r. As we a . This might be the reason saw in Eq. (4.15), the potentials .Am and .ϕm appear in .┌bc why we get terms with the wavelength .λ and the frequency f on the right hand side of Eq. (4.35), namely in the case that the potential depends on .λ and f . In the following, we will first look for an interpretation of the Stromgleichung using the considerations of the classical four-dimensional general relativity. By using these considerations, we will find an expression for the potential .A4 = ϕ4 . Thereafter, we will discuss an ansatz that was considered by Einstein at least in 1939, while it is likely that he also considered it in 1938, maybe in a slightly varied form in order to derive the Stromgleichung. 4.2.4.1

Interpretation of the Stromgleichung

In the letter AEA 6-262, Einstein mentioned a so-called Integralbedingung für die Stromdichte36 ⎛ . (4.36) I4 dV = 4π r 2 , which holds for large r. Thus, Einstein obviously called the quantity .I4 the current density. An expression equivalent to Eq. (4.36) is very well known in the classical case of the four-dimensional theory. Let us briefly recall some relations.37

their publication that the “constant .α /α corresponds to the gravitational constant” (Einstein and 2 1

Bergmann, 1938, p. 696), while two constants “cannot be eliminated from the theory” (Einstein and Bergmann, 1938, p. 696). These two constants could be .α3 and .α4 , which might be the reason why they appear in Eqs. (4.22) to (4.25). As we already saw in Sect. 4.2.3, Einstein used the notations .α i

and .αi interchangeably. We finally note that Einstein composed a manuscript about those constants titled “Ein Gesichtspunkt für eine spezielle Wahl der in der verallgemeinerten Kaluza-Theorie auftretenden Konstanten” (AEA 1-136). 36 “Integral condition for the current density”. 37 For instance, see also Fliesßach (2016, chapter 6).

238

4 Einstein’s Further Considerations on the Generalized Kaluza Theory

We assume the metric components .gmn = ηmn of Minkowski space,38 the - where .Ai is the four-potential and .ϕ the scalar potential. potential .Ai = (ϕ, A), Furthermore, let Fik =

.

∂Ai ∂Ak − k ∂x i ∂x

(4.37)

be the electromagnetic tensor. Considering the charge density .ρ(-r , t), the fourcurrent becomes dx i = (cρ, ρ v-) = (cρ, j-) dt

ji = ρ

.

(4.38)

with the velocity of the charge .v-.39 In particular, the zeroth component of the fourcurrent is .j 0 = cρ. For the charge Q in a certain volume V , it is 1 .Q = c

⎛ j 0 dV .

(4.39)

V

With these notations, the second group of Maxwell’s equations is 4π i j. c

(4.40)

4π 0 j = 4πρ. c

(4.41)

F,kik =

.

In particular, we get F,k0k =

.

We can now compare Eq. (4.41) with Einstein and Bergmann’s Stromgleichung (4.27) on AEA 6-258 and see that Eq. (4.41) might be the analogue to the Stromgleichung in classical electrodynamics, while Bergmann and Einstein used the index four instead of zero.40 Moreover, in the classical theory, it follows that j 0 = cρ =

.

c 0k F . 4π ,k

38 Here, .η mn stands for the Minkowski metric. 39 At this point we need to be aware of the different

(4.42)

meanings of .ρ. Here, .ρ describes the charge density, while on the manuscript pages and in the correspondence, Einstein used the variable .ρ as a replacement of r in order to avoid singularities. 40 Both indices correspond to the time dimension.

4.2 Context of Einstein’s Further Calculations

239

It seems very plausible that Einstein, indeed, denoted the quantity .I4 as “current density” as it is the analogue to the zeroth component of the four-current .j 0 = cρ with the charge density .ρ in the four-dimensional theory. This implies that Einstein and Bergmann chose their notation in the generalized Kaluza theory such that I4 = k · ϕ 4τ ;τ

.

(4.43)

with k constant. By identifying k with .2α2 , it follows that 4π f α(3α + 2β) λ ┌ ⎛ ⎞┐ 2π ' 2 2 2 ' ' ' η −α + β + δ) + η(3α + β + δ − α − β + δ + r r r λ

I4 = −

.

(4.44)

from Eq. (4.27). The above considerations also imply that Einstein refers to the fourth component .ϕ4 of the potential when writing about “the potential .ϕ” in the correspondence, analogously as the scalar potential .ϕ is the zeroth component of the four-potential i .A in the four-dimensional theory. This interpretation is also indicated by Einstein in his letter AEA 6-261. Let us come back to the integral condition from Eq. (4.36). The term ⎛ . I4 dV (4.45) V

is then the analogue to the charge from Eq. (4.39). Rewriting in spherical coordinates, the integral expression in Eq. (4.45) becomes ⎛ .

⎛ ∞ ⎛ 2π ⎛ π I4 dV =

I4 r 2 sin θ dr dθ dϕ, 0

0

(4.46)

0

which simplifies in case of centrally symmetry to ⎛ .

⎛ ∞ I4 dV = 4π

I4 r 2 dr.

(4.47)

0

This integral was discussed by Bergmann in AEA 6-251. Finally, we want to discuss an approximation of the Stromgleichung. As in AEA 6-251 or 6-261, Einstein and Bergmann mostly approximated the term .ϕ 4τ ;τ with .Δϕ which reduces to 2 Δϕ = ϕ '' + ϕ ' r

.

(4.48)

240

4 Einstein’s Further Considerations on the Generalized Kaluza Theory

in the case of centrally symmetry. The Stromgleichung can then be approximated generally by41 I4 ≈ −

.

4π f α(3α + 2β) ≈ 2α2 ϕ 4τ ;τ ≈ −2α2 ϕ4τ,τ ≈ −Δϕ. λ

(4.49)

As in Eqs. (4.42) and (4.43), we associate .I4 with .j 0 and write Poisson’s equation42 generally as Δϕ = −I4

.

(4.50)

in order to avoid confusion between the two different meanings of .ρ. We will refer to .I4 as Strom or current in analogy to Einstein and Bergmann.43

4.2.4.2

The Potential

In this section, we will consider the question about the potential .Am = ϕm appearing in the generalized Christoffel symbols in Eq. (4.35) before we look for expressions of the metric components. In the correspondence between Einstein and Bergmann, we frequently find the expression ⎛ ϕ⎞ f = 2π ν − λ

.

(4.51)

as in AEA 6-247. We learn from the correspondence that both f and .ϕ can be written as power series expansions of r.44 One of the most important requirements for Einstein is that .ϕ becomes approximately proportional to .r −1 for large r.45 As we saw in Sect. 4.2.4.1, we can identify the potential .ϕ with the fourth component .ϕ4 of .ϕa . This implies ⎞ ⎛ λ f = .A4 = ϕ4 = λ ν − (ω − f ) . 2π 2π

(4.52)

We conclude that Eq. (4.52) explains the appearance of .λ and f on the right hand side of Eq. (4.35). However, we also get an additional variable .ν or .ω that we do not find in Eq. (4.35).

41 See

the letters AEA 6-251, 6-258, and 6-261. also AEA 6-250 and 6-258. 43 For instance, see AEA 6-251. 44 For instance, see AEA 6-247 and 6-253. 45 For instance, see AEA 6-261. The behavior of .ϕ for large r is also discussed in Van Dongen (2010, p. 151) and Van Dongen (2002, pp. 195–196). 42 See

4.2 Context of Einstein’s Further Calculations

4.2.4.3

241

Ansatz in Bargmann Correspondence, 1939

In the following sections, we will look for an ansatz for the metric components .gab and .γμν that Einstein used in order to investigate the general field equations (4.20) and (4.21). As we already explained, such an ansatz should depend on the functions .α(r), .β(r), .δ(r), and .η(r). In order to find such an ansatz, we looked through all sheets of papers that are declared as CALC in the Einstein Archives, including reels 62 and 63 as well as through all letters written in summer 1938 between May and July, including the correspondence between Einstein and Bergmann. We also looked at a second correspondence that took place in summer 1939 between Einstein, V. Bargmann, and P. Bergmann encompassing the five letters AEA 6-273, 6-206, 6-274, 6-275, and 6-208. In the present section, we will look at this correspondence in more detail. We also refer to Chap. 5, where we will discuss an integral condition that was investigated by Einstein throughout the same correspondence. The subject of the correspondence from 1939 still is the continuation of the generalized Kaluza theory. Indeed, it is in the second letter AEA 6-206 from July 6, 1938 where we find an expression of the metric components depending on the four functions. V. Bargmann wrote46 “Bisher haben wir die GröSSen .χμν benutzt” (AEA 6-206, page 7), indicating that they used these relations already before. He then gave expressions for these quantities, namely47 χik = αδik + β

.

xi xk xk , χk4 = −iγ · , and χ44 = δ. r r2

(4.53)

These quantities give us expressions for the metric components in dependence on the quantities .α, .β, .δ, and .η.48 In the same letter, Bargmann also used the trace χ = 3α + β − δ,

.

(4.54)

see (AEA 6-206, page 6). Equation (4.54) gives us information about how to interpret the quantities in Eq. (4.53). Let us assume that the Latin indices in Eq. (4.53) run from 1 to 3.49 We also interpret .δik as the Kronecker symbol ⎧ 1 i=k .δik = (4.55) 0 i /= k.

46 “Until

now, we have used the quantities .χμν .” here adopt Bargmann’s notation and use subscripts for coordinates. 48 The function .η is connected to the quantity .γ from Eq. (4.53) by .η = iγ , see AEA 6-270 and 6-240. We note that .γ should not be misread as the determinant or trace of the metric components .γμν . 49 Since Bargmann considered the indices .k4 and 44 separately, it stands to reason that the Latin indices run either from 0 to 3 or from 1 to 3. As we will see, it is likely that the Latin indices run from 1 to 3. Greek indices will, if not stated otherwise, run from 1 to 4 in the following. 47 We

242

4 Einstein’s Further Considerations on the Generalized Kaluza Theory

The quantities .χik obviously are symmetric. Since Bargmann did not write down expressions for .χ4k , we assume that .χk4 = χ4k such that all .χμν become symmetric. Furthermore, let .xμ be the coordinates, while it is r=

.

/

x12 + x22 + x32 ,

(4.56)

which can be interpreted as a spatial distance. Using the signature .ημν = diag(1, 1, 1, −1), we then get χ = χ ρρ = ηρν χνρ = 3α + β

.

x12 + x22 + x32 − δ = 3α + β − δ, r2

(4.57)

which is equivalent to Eq. (4.54).50 Clearly, assuming .ημν = diag(1, 1, 1, 1) and .χ44 = −δ instead of Eq. (4.53) yields the correct trace as well. In this case, Bargmann merely would have made a sign mistake; however, the subsequent equations would then become incorrect. Considering the failure of any other interpretation, we argue that the above assumptions are, indeed, the underlying assumptions in Bargmann’s letter. Bargmann continued his letter with writing: Mit den ursprünglich eingeführten Größen .ψμν , den in .x 0 periodischen Anteilen der .gμν , hängen sie durch die Gleichung zusammen: .ψμν = χμν − 12 ημν χρρ . (AEA 6-206, p. 8)

This sentence is of special interest to us, since V. Bargmann directly refers to the metric components .gμν . It is not clear to which metric components of the generalized Kaluza theory he refers: the five-dimensional or four-dimensional metric components. We recall that in their publication, Einstein and Bergmann denoted the five-dimensional metric components by .γμν and the four-dimensional ones by .gab .51 However, it is possible that they changed their notation. In either case, the .ψμν are the parts of .gμν , which are periodic in .x 0 . We also emphasize that Bargmann again indicated that they introduced these components initially (“ursprünglich eingeführt”), possibly when deriving the Stromgleichung 1 year earlier. Bargmann continued with setting ψik = αδik + β

.

xi xk xk , ψk4 = −iγ , and ψ44 = δ. 2 r r

(4.58)

50 We also tried several different assumptions; however, they either yielded complicated expressions for the variable r without any physical meaning or led to incorrect equations in the following. As an example, we also calculated with different signatures as .(−1, −1, −1, 1), .(−1, 1, 1, 1), or .(1, −1, −1, −1) or assumed that the indices run from 0 to 3. 51 There, Greek indices run from 0 to 4 and Latin indices from 1 to 4.

4.2 Context of Einstein’s Further Calculations

243

He stated that by52 1 ψμν = χμν − ημν χρρ , 2

.

(4.59)

it is then 1 1 α = − (α + β − δ) , β = β, γ = γ , δ = (3α + β + δ) , and χ = −χ . 2 2 (4.60)

.

Let us first note that Eq. (4.58) is structurally equivalent to Eq. (4.53). Bargmann replaced the quantities .α, .β, .γ , and .δ by the overline versions .α, .β, .γ , and .δ. Instead of denoting the resulting quantities by the overline notation .χ μν , too, he called them .ψμν . Furthermore, he explicitly used the Minkowski metric in Eq. (4.59), which we introduced in Eq. (4.57). Interpreting the quantity .χρρ in Eq. (4.59) as the trace .χ , we can calculate the components .ψμν as 1 xi xk 1 ψik = χik − ηik χ = αδik + β 2 − ηik (3α + β − δ) 2 2 r ⎞ ⎛ 1 1 xi xk 1 xi xk 3 = α − α − β + δ δik + β 2 = − (α + β − δ) δik + β 2 , 2 2 2 2 r r (4.61)

.

xk 1 ψk4 = χk4 − ηk4 χ = −iγ , r 2

.

52 A

(4.62)

form as in Eq. (4.59) was frequently used by Einstein when deriving the linearized field equations by assuming weak gravitational fields as it makes possible to reduce them to wave equations. For instance, Einstein used it when investigating gravitational waves as in Einstein (1916b, eq. (8a)), Einstein (1918, eq. (3)), or Einstein and Rosen (1937, p. 43). He also used it in the publication on the problem of motion in 1938, see Einstein et al. (1938, eq. (1,18)), or when considering the Newtonian limit in Einstein (1922c, eq. (99)). See also Einstein’s Zurich notebook that we already discussed in Sect. 1.3.4.5 and especially Janssen et al. (2007, pp. 632–637). By this form, the linearized field equations in the classical theory of relativity can be derived | | in a simpler form. The weak fields are being expressed by introducing small perturbations .|hμν | 0, it is .

∂ 2 g(a0 ) = σ (σ − 1) a0σ −2 e−τ a0 −τ σ a0σ −1 e−τ a0 −σ τ a0σ −1 e−τ a0 +a0σ τ 2 e−τ a0 ∂a02 ⎛ ⎞ = e−τ a0 a0σ −2 σ (σ − 1) − 2a0 τ σ + a02 τ 2 = − e−σ a0σ −2 σ < 0, (5.11)

which is why the local extremum of .g(x) at .a0 indeed is a local maximum. In the next line, Einstein wrote “.σ u[nd] .τ gross”24 fitting to our considerations from above that the peak of the .δ-function becomes higher and narrower for large .σ and .τ , see Fig. 5.2. Einstein also introduced the substitution x = a + ξ,

(5.12)

.

where it seems as if Einstein did not differentiate between the notation a and .a0 here and in the following.25 It is thus a = a0 = σ/τ

(5.13)

.

such that the function .g(x) becomes f (ξ ) := g(a + ξ ) =

.

⎛σ τ

+ξ

⎞σ

σ

e−τ ( τ +ξ ) .

(5.14)

While the local maximum of .g(x) was at .x = a = σ/τ , the local maximum of .f (ξ ) is at .ξ = 0. By Eq. (5.8), it follows that ⎞⎞ ⎛ ⎛ ξ . lg f (ξ ) = lg g(a + ξ ) = σ lg (a + ξ ) − τ (a + ξ ) = σ lg a + lg 1 + a − τ (a + ξ ) .

(5.15)

Einstein only wrote down the last expression, which he recast to .

24 “.σ

⎞ ⎛ 1 ξ2 ξ − τa − τξ lg f (ξ ) = lg g(a + ξ ) ≈ σ lg a + − 2 a2 a ⎞ ⎛σ 1 ξ2 − τ ξ − σ 2. = (σ lg a − τ a) + 2 a a

and .τ large”. becomes especially clear by Eq. (5.34).

25 This

(5.16)

5.2 Analysis of Related Manuscript Pages and Correspondence

287

Here, Einstein explicitly used the first terms of the Taylor expansion (Mercator series) .

lg (1 + ε) = ε −

ε2 + ··· . 2

(5.17)

Indeed, for .ε > −1, it is .

lg (1 + ε) =

∞ Σ

(−1)n+1

n=1

εn ε2 ≈ε− , 2 n

(5.18)

which holds for small .ε. As Einstein used this approximation for .ε = ξ/a in Eq. (5.17), it is apparently .a >> ξ and thus .σ >> ξ τ . As Einstein considered large .σ and .τ , he obviously looked at small .ξ corresponding to the neighborhood of the position of the peak of .f (ξ ). Einstein then concluded “Funkt[ion] selbst appr[oximiert] durch .Ba σ e−τ a · 2

e

− 21 σ ξ 2 a

.”26 By “Funkt[ion],” Einstein apparently referred to a function 2

− 12 σ ξ 2

h(ξ ) := B · a σ e−τ a e

.

a

≈ B · f (ξ ) = B · g(a + ξ ),

(5.19)

where the approximation from Eq. (5.16) was used.27 Thus, he still considered the original function .g(x) but moved the peak to the origin by the substitution .x = a +ξ and now added a normalization factor B, which he investigated in the following. Einstein did not differentiate between these functions and did not name them. In effect, Einstein approximated the peak of the function .g(x) = x σ e−τ x by a 2 2 Gauss function .h(ξ ) ∼ e−(σ ξ )/(2a ) . We note that this approximation is only valid for small .ξ . In particular, if .σ, τ > 0, it is ⎧ f (ξ ) →

.

∞ σ even −∞ σ odd

(5.20)

for .ξ → −∞, while the approximation .h(ξ ) from Eq. (5.19) tends to zero. Both functions are visualized in Fig. 5.3 for the values .B = 1, .σ = 41, and .τ = 14 as in Fig. 5.2b. The role of the factor B becomes clear by Einstein’s subsequent considerations. In the next line, he wrote down the expression σ −τ a

Ba e

.

⎛+∞ 1 ξ 2 − σ e 2 a2 dξ, −∞

−1σ

ξ2

itself approximated by .Ba σ e−τ a · e 2 a2 ”. 27 The middle term on the right hand side of Eq. (5.16) vanishes due to Eq. (5.13). 26 “Function

(5.21)

288

5 Considerations on Delta Functions

Fig. 5.3 Plots of .g(x) = x σ e−τ x (a) and .h(ξ ) = B · a σ e−τ a e .τ = 14. It is .a = σ/τ according to Eq. (5.13)

− 12 σ

ξ2 a2

(b) for .B = 1, .σ = 41, and

which we interpret as being equal to the integral of .h(ξ ) over the entire real line, ⎛∞

⎛∞ h(ξ ) dξ =

.

−∞

2

1 ξ σ −τ a − 2 σ a 2

B ·a e

dξ = Ba e σ

e

−τ a

−∞

⎛+∞ 1 ξ 2 − σ e 2 a2 dξ.

(5.22)

−∞

In particular, in this case, it is .B /= B(ξ ). In order to solve this integral, Einstein made a short auxiliary calculation. We find the two lines ⎛∞ .

e−x dx = I = 2

√

(5.23)

π

−∞

and ⎛∞ I =

.

2

e−r 2π r dr = π. 2

(5.24)

0

Einstein obviously used polar coordinates, a standard trick to solve the Gaussian integral, in order to solve the integral from Eq. (5.22) as we will show in the following.28 Let us introduce the substitution .A = −σ/(2a 2 ). It is ⎞2 ⎛ ∞ ⎛ ⎛∞ ⎛∞ ⎛∞ ⎛∞ ( 2 2) 2 −Aξ −Ax 2 −Ay 2 ⎠ ⎝ . e dξ = e dx e dy = e−A x +y dx dy. −∞

−∞

−∞

−∞ −∞

(5.25) 28 We

only find the two-line equations (5.23) and (5.24) in Einstein’s notes.

5.2 Analysis of Related Manuscript Pages and Correspondence

289

By introducing polar coordinates ⎛ ⎞ ⎛ ⎞ x r cos ϕ = y r sin ϕ

Ф

.

(5.26)

and by computing the corresponding Jacobi determinant ⎛ .

det DФ = det

⎞

∂x ∂ϕ ∂y ∂ϕ

∂x ∂r ∂y ∂r

⎛ = det

cos ϕ −r sin ϕ sin ϕ r cos ϕ

⎞ = r,

(5.27)

it is ⎛∞ ⎛∞ .

) ( −A x 2 +y 2

e

⎛2π⎛∞ ⎛2π⎛∞ ( ) 2 −A r 2 cos2 ϕ+r 2 sin2 ϕ dx dy = e r dr dϕ = e−Ar r dr dϕ

−∞ −∞

0 0

⎛∞ =

0 0

e−Ar 2π r dr. 2

(5.28)

0

We get Einstein’s integral from Eq. (5.24) by .A = 1. By the substitution .u = Ar 2 , it follows that ⎛ .

⎝

⎛∞

⎞2 e−Aξ dξ ⎠ = 2

−∞

π . A

(5.29)

Again, for .A = 1 we get Einstein’s equation (5.23), where he denoted the integral by I . Reading Einstein’s two lines from Eqs. (5.23) and (5.24), it seems as if he went the other way around starting with the result in Eq. (5.23). However, we argue that Einstein first wrote down the equation ⎛∞ .

e−x dx = I, 2

(5.30)

−∞

then calculated ⎛∞ I =

.

2

e−r 2π r dr = π, 2

(5.31)

0

√ and only thereafter added the result . π to the first line. Otherwise, the calculation does not make any sense as he was interested in I instead of .I 2 .

290

5 Considerations on Delta Functions

Einstein used this result in order to get ⎛∞ e

.

/

2

− 12 σ ξ 2 a

−∞

dξ =

2 √ a π. σ

(5.32)

We do not find this equation in Einstein’s notes; however, he wrote down the right hand side directly beneath the integral from expression (5.21). In the next and last step of the first part of the manuscript page, Einstein normalized the integral (5.21) such that .h(ξ ) satisfies the condition ⎛∞ h(ξ ) dξ = 1

.

(5.33)

−∞

according to the .δ-function, see Eq. (5.22). We get / 1 = Ba e

.

σ

−τ a

·a

√ 1 2π = 2π a0σ +1 e−σ σ − 2 B σ

(5.34)

and find this equation in a very similar form in Einstein’s notes. In the second equality, he replaced a by .a0 = σ/τ according to our assumption from Eq. (5.13). He also made a mistake in the second equality as he apparently forgot to observe the minus sign of the power of .σ . Equation (5.34) is now a condition for the normalization factor B introduced in Eq. (5.19) such that the function .h(ξ ) as approximation for .g(x) = x σ e−τ x can serve as .δ-function. Einstein commented this by “(Gleichung für B) bei gewähltem 29 and ended his calculation by a horizontal line. .σ ” We see that the first part of the manuscript page is well structured. The content is self-contained and we can reconstruct the idea behind Einstein’s calculations. This changes for the second part of the page. In the first line, he wrote down δ(x − a0 ) = Bx σ e−τ x

.

(5.35)

starting from the result of the first part. As we saw, Einstein approximated the right hand side of Eq. (5.35) that we called .B · g(x) by the Gauss function that serves as a .δ-function and that we denoted by .h(ξ ), see also Eq. (5.19). Here, Einstein now takes the function .Bg(x) as .δ-function, where B satisfies equation (5.34). At this point it is important to remark once more that .Bg(x) does not converge for .x → −∞. This is probably the reason why Einstein in the following only considered integrals from zero to infinity.

29 “(Equation

for B) for a chosen .σ ”.

5.2 Analysis of Related Manuscript Pages and Correspondence

291

We first note that Einstein wrote down the expression ⎛ Bx

.

(τ x)3 (τ x)2 ··· − 1 − τx + r! 2!

σ

⎞ ,

(5.36)

which is .Bg(x), where .e−τ x is written as a power series expansion. Einstein did not further consider this expression. The next line reads ⎛ Ax n dx = An

.

(5.37)

and proceeded with ⎛

⎛∞

A(α0 ) =

Aδ(x − α0 ) dx =

.

ABx σ e−τ x dx.

(5.38)

0

Clearly, Einstein used Eq. (5.35) for the second equality. Explicitly writing down the .δ-function suggests that it is .A = A(x) and .A(α0 ) = A(x = α0 ). Einstein stopped his calculations here and wrote down a horizontal line and30 ⎛∞ .

A(α) e−α

2r2

dα = ϕ(r),

(5.39)

0

where he first wrote down . dx instead of . dα.31 He commented this line by “.ϕ(r) für grosse r[.] Nur kleine .α tragen bei.”32 In the following, we will interpret these equations as well as Einstein’s comment. In Fig. 5.4, we plotted the Gauss function lr (α) = e−α

.

2r2

(5.40)

for the values .r = 0.5 (red), .r = 2 (blue), and .r = 10 (green). Clearly, if r is large, .lr (α) is large for small .α and small for large .α.33 Thus, by integrating from .α = 0 to .α = ∞ for large r, .lr (α) contributes only for small .α. Otherwise, for small r, the peak becomes wider such that .lr (α) also contributes for large .α. If .lr (α) is now multiplied by a factor .A(α) as in Eq. (5.39), the values of .A(α) for small .α

30 We here note that the left hand side is closely related to what is often called Weierstrass transform. 31 Einstein frequently conflated the two different variables .α and x as we will also see in Eq. (5.2.4). 32 “.ϕ(r) for large r[.] Only small .α contribute.” 33 See, for instance, the green graph in Fig. 5.4.

292

5 Considerations on Delta Functions

Fig. 5.4 Graph of the Gauss 2 2 function .lr (α) = e−α r for the values .r = 0.5 (red), .r = 2 (blue), and .r = 10 (green)

are important for large r, while its values for large .α are only important for small r. Einstein apparently was interested in large r such that the values of .A(α) for small .α contribute. This is probably the idea behind Einstein’s comment. Again, we note that Einstein did not draw any sketch here. Let us now come to Eq. (5.39). The notation .ϕ(r) reminds us of the potential that should behave like .r −1 for large r, see Chap. 4. We get this directly if .A(α) is constant. Assuming .A(α) = c constant, it is ⎛∞ ϕ(r) =

.

A(α) e−α

2r2

dα =

√ π 1 c ∼ r −1 , 2 r

(5.41)

0

see Eq. (5.29). In this case, the function considered here by Einstein fits to the potential .ϕ(r) for large r according to his comment. Although we do not find any evidence for .A(α) being constant on this manuscript page, we will see in the following that this interpretation is very likely.

5.2.2 Related Correspondence In fact, during the correspondence between Einstein and V. Bargmann that took place in summer 1939, Einstein wrote about this discussion supporting our interpretation. Let us first look at some passages from the correspondence. In AEA 6-273 on June 30, 1939, Einstein wrote34

34 “I

think I have now found the calculation method adapted to our problem and will present the main thing here under restriction to the centrally symmetrical problem. An r-symmetrical function can be represented in the form [.. . .]. The particular form is determined by A. The behavior of f for large r is determined by the progression of A for small .α, the behavior of f for small r by the progression of A for large .α.” Translation by the author.

5.2 Analysis of Related Manuscript Pages and Correspondence

293

Eine r-symmetrische Funktion lässt sich darstellen in der Form ⎛∞ 2 2 .f (r) = A(α) e−α r · · · · ·(1)[.] 0

Die besondere Form wird durch A bestimmt. Das Verhalten von f für grosse r wird durch den Verlauf von A für kleine .α bestimmt, das Verhalten von f für kleine r durch den Verlauf von A für grosse .α. (AEA 6-273)

Hence, Einstein here considered a function .f (r) depending on the radial coordinate r as part of a spherically symmetric function .Ф(r, θ, ϕ) = f (r) equivalently to Eq. (5.39) and discussed the behavior of .f (r) and the influence of the factor .A(α) as we showed above around Fig. 5.4. He continued: Soll .f (r) für grosse r rasch absinken, so muss A klein sein für kleine .α. Soll .f (r) = 1r werden, so hat man .A = konst. zu wählen. Eine bei .r = 0 abgerundete Funktion, die

sich für grosse r wie . 1r verhält, erhält man durch die Wahl [Picture from AEA 6-273]35 wobei g den Abrundungs-Radius bestimmt. Das Willkürliche dieser Wahl liegt in dem plötzlichen Abfall von A bei .r = g. (In Wahrheit ist ein rascher Abfall statt eines plötzlichen zu erwarten). Dies wäre die Darstellung für .ϕ.

We see that Einstein indeed considered .f (r) → ϕ(r) ≈ 1/r which supports our interpretation that .ϕ(r) stands for the potential. In this case, one has to set .A(α) as constant while Einstein here introduced a .g ∈ R with ⎧ c, α < g .A(α) = (5.42) 0, α > g for a .c > 0.36 Why did not Einstein simply choose a constant .A(α) ? Considering .f (r), only small .α contribute for large r as we explained above. Thus, the function −1 for large r in both cases when choosing .A .f (r) behaves like .r (α) constant or as shown in Eq. (5.42). The difference lies in the behavior for small r. For a constant −1 for small r as well, and thus it is not “abgerundet”37 .A(α) , .f (r) behaves like .r (AEA 6-273) at .r = 0. We can make this clear by looking again at .lr (α) from Eq. (5.40). For .r = 0, this is a constant and the integral in Eq. (5.41) becomes infinite. By choosing a factor .A(α) that vanishes for large .α, the integral becomes finite, while the behavior for large r does not change. The next paragraph reads as follows: Die Glockenfunktionen für die .ψ werden durch solche A dargestellt, die für grosse .α ebenfalls stark abfallen, für kleine .α ebenfalls. Sie lassen sich durch die Dirac’sche .δFunktion für A approximieren, was auf die Gauss’sche Funktion für r hinauskommt. In Wahrheit fällt A hier für kleine .α rasch, aber nicht unendlich rasch ab. (AEA 6-273)

35 © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama. 36 Einstein

did not use the notation c. In the letter AEA 6-246 from 1938, Bergmann wrote about a similar function in order to describe the potential .ϕ. 37 “Rounded” or “smooth”.

294

5 Considerations on Delta Functions

We recall that Einstein, Bergmann, and Bargmann frequently wrote about the .ψ quantities and that they contain expressions of the functions .α(r), .β(r), .δ(r), and .η(r), see our discussion from Sect. 4.2. More precisely, by using .ψ, Einstein here referred directly to the functions .α(r), .β(r), .δ(r), and .η(r) as it becomes clear by the subsequent letters AEA 6-274 and 6-275 written on July 9, 1939 and at the end of July 1939, respectively.38 In AEA 6-274, he wrote: Eine erste brauchbare Näherung dürfte man erhalten, wenn man für .α,

Dirak’sche .δ-Funktionen und für .ϕ die A-Funktion 274]39 ansetzt. (AEA 6-274)

β

γ

. 2, . r r

, .δ [.. . .]

[Picture from AEA 6-

In AEA 6-275, he again wrote: Was mir zunächst auffiel, war, dass man eine Funktion vom allgemeinen Charakter von .ϕ

[Picture from AEA 6-275]40 während für die

erhält, wenn man A so wählt

[Picture from AEA 6-275]41 den zu erwartenden

die .δ-Funktion Charakter darstellt. (AEA 6-275)

.ψ-Grössen

We first note that all passages from the three letters fit together with respect to the potential .ϕ as it should be expressed by ⎛∞ f (r) =

.

A(α) e−α

2r2

dα,

(5.43)

0

which transitions into .ϕ(r) for .A(α) chosen as in Eq. (5.42). Although we have not quoted it here, Eq. (5.43) can be found on all three letters AEA 6-273, 6-274, and 6-275. So far, this supports our interpretation of the very last part of the manuscript page AEA 62-783.

38 We

briefly point out the problem that Einstein used .α not only for a variable as in .A(α) but also for the function .α(r). We will here and in the following always write about .α(r) instead of .α when referring to the function unless we quote Einstein. 39 © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama. 40 © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama. 41 © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama.

5.2 Analysis of Related Manuscript Pages and Correspondence

295

5.2.3 An Expression for the Functions α, β, δ, and η We saw that in AEA 6-273 and 6-275, Einstein wrote about the .ψ quantities, while he replaced them in AEA 6-274 by the functions β γ , , and δ. r2 r

α,

.

(5.44)

We encountered these functions in Chap. 4 frequently and showed in Sect. 4.2 that these functions appear in the spherically symmetrical metric components. According to the three quoted letters, these functions should now be represented by the function ⎛∞ f (r) =

.

A(α) e−α

2r2

dα

(5.45)

0

similarly to the potential .ϕ(r), see Eq. (5.43). While for .ϕ(r), .A(α) needs to be constant or chosen as in Eq. (5.42), Einstein apparently wanted to express the .ψ quantities by expressing .A(α) as .δ-function as, for instance, A(α) ≈ δ(α − α0 ).

(5.46)

.

On the manuscript page AEA 62-783, Einstein investigated such .δ-functions and it stands to reason that it is then A(α) = Bα σ e−τ α ,

(5.47)

.

see Eq. (5.35). The idea behind Einstein’s calculations on AEA 62-783 would then be to approximate the .ψ quantities as ⎛∞ f (r) =

A(α) e

.

−α 2 r 2

0

⎛∞ dα =

σ

Bα e 0

−α02 r 2

∼e

,

−τ α −α 2 r 2

e

⎛∞ dα ≈

δ (α − α0 ) e−α

2r2

0

(5.48)

which corresponds to Einstein’s comment in AEA 6-273. This in turn would give us an explicit expression for the .ψ quantities. We note that in this case, Einstein would have used a different notation on AEA 62-783 as we then interpreted the function A from Eq. (5.38) as a representa2 2 tion for .e−α r .

296

5 Considerations on Delta Functions

Interpreting .A(α0 ) from Eq. (5.38) as the quantity .A(α) from Eq. (5.45) would yield ⎛ ⎞ ⎛∞ ⎛∞ ⎝ A(x)Bx σ e−τ x dx ⎠ e−α 2 r 2 dα .f (r) = 0

0

0

0

⎛ ⎞ ⎛∞ ⎛∞ 2 2 ≈ ⎝ A(x)δ (x − α0 ) dx ⎠ e−α r dα ⎛∞ ∼ A(α0 )

−α 2 r 2

e

√ dα =

π A(α0 ) 1 , r 2

(5.49)

0

which seems unlikely to us as he could then also have chosen .A(α) in Eq. (5.45) as being constant.

5.2.3.1

Further Procedure

Before analyzing the manuscript page AEA 62-795, we will briefly look at the letters AEA 6-273 and 6-274 in more detail. It becomes clear by the quote above from the letter AEA 6-274 that the procedure of using the .δ-function for .A(α) only is an approximation: “Eine erste brauchbare Näherung dürfte man erhalten [.. . .]” (AEA 6-274). We will here investigate for what purpose Einstein considered this approximation. Starting from Eq. (5.43), Einstein considered the power series expansion of −α 2 r 2 around .α = 0 that we already found on the manuscript page AEA 62-783, .e see Eq. (5.36). In both letters, Einstein recast .f (r) to ⎛∞ f (r) =

.

⎛ ⎞ 1 4 4 2 2 A(α) 1 − α r + α r ± · · · dα 2

0

⎛∞

⎛∞ A dα − r

= 0

2

r4 α A dα + 2

⎛∞ α 4 A dα ± · · ·

2

0

0

(5.50)

5.2 Analysis of Related Manuscript Pages and Correspondence

297

and called these new integrals “Momenten-Integrale” (AEA 6-273). In AEA 6-273 he described the following procedure: Wenn A angenommen wird, (z. B. für .α und . rδ2 in unserer Bezeichnungsweise die Dirak’sche .δ Funktion)42 so bestimmen sich die höheren Momenten-Integrale aus den beiden ersten. Denkt man sich alle Variablen im Nullpunkt in Potenzreihen nach r entwickelt, so liefern die Differenzialgleichungen Beziehungen zwischen den Entwicklungs-Koeffizienten, d.h. auch zwischen den Momenten-Integralen. Führt man diese durch die Annahmen über die A auf die beiden ersten zurück, so erhält man die Gleichungen, um die eingeführten Konstanten zu bestimmen, welche die Funktionen A bestimmen. (AEA 6-273)

The procedure described here reminds us of the letter AEA 6-265, where Einstein tried to determine the constants .ε appearing in .ϕ(r), a appearing in the substitution .ρ, and A appearing in the functions .α(r), .β(r), .δ(r), and .η(r). There, Einstein tried to plug the power series expansions of the mentioned functions into the Stromgleichung (which is a differential equation) in order to determine the constants. By this procedure, the coefficients .αi , .βi , .δi , and .ηi in the power series expansions appear as new unknown constants. In the present letter, the constants .ε and a do not appear. In fact, it seems as if they did not use the substitution of .ρ in 1939 at all. However, the quantity A resulting from the functions is being discussed here as well as the coefficients of the functions. Einstein also wrote about differential equations that might be the Stromgleichung and the equations .F1 to .F5 .43 The procedure described by Einstein in the present letter is to plug the functions .α(r), .β(r), .δ(r), .η(r), and .ϕ(r) into the differential equations yielding relations between the coefficients of the power series expansions and relations between the integrals from Eq. (5.50). Einstein wrote that the higher integrals can be determined by the first two integrals. This fact then should enable Einstein to determine the coefficients and constants which in turn determine the factor A for the functions .α(r), .β(r), .δ(r), .η(r), and .ϕ(r). Concerning the accuracy of his approximation method, Einstein concluded: Natürlich wird man die Anzahl der Konstanten auch hier zunächst stark reduzieren müssen, um für den Anfang überhaupt durchzukommen. Aber man kann dies nachträglich immer mehr verbessern. Die Kontrolle liegt natürlich darin, dass die höheren Momentenintegrale nicht zu falsch herauskommen dürfen, wenn man die niederen (und höheren) durch die Differentialgleichungen bestimmt hat und aus diesen vermöge der bestimmten Zahlenkonstanten die höheren Momentenintegrale berechnet. (AEA 6-273)

Let us briefly discuss the question of how the higher integrals in Eq. (5.50) might be determined by the first two integrals. As in AEA 6-274, let ⎛∞ I2n =

α 2n A dα

.

(5.51)

0

42 It

seems as if Einstein erroneously wrote .δ/r 2 instead of .β/r 2 . This ambiguity occurs several times in the correspondence. 43 See Eqs. (4.22) to (4.27). It could also be another differential equation resulting from Einstein and Bergmann’s field equations.

298

5 Considerations on Delta Functions

be the integrals from Eq. (5.50). Let us further assume .A = δ(α − α) for a specific α. It is then

.

.

I2n I2(n−1)

= α2.

(5.52)

This, however, is regarded by Einstein only as an approximation. In AEA 6-274, Einstein wrote: Da nämlich die A in der Gegend eines bestimmten .α rasch abfallen sollen [the ideal I2n case would be .δ-functions], muss für grosse n die Größe . I2(n−1) einen gewissen Werte .α 2 annähernd zustreben. (AEA 6-274)

It seems as if Einstein wanted to use .I2 /I0 = α 2 in order to approximate the higher integrals. We will come back to this in Sect. 5.2.4.

5.2.4 AEA 62-795 On the manuscript page AEA 62-795, we find many calculations written in ink as well as a sketch in pencil. It is depicted in Fig. 5.5. The calculations seem to continue on its back AEA 62-795r that we will discuss in Sect. 5.2.5. The manuscript page starts with ⎛g

−α 2 x 2

e

.

0

⎞ ⎛g ⎛ g3 g5 α4x 4 2 2 1−α x + · · · dα = g − x 2 + x 4 · · · . dα = ϕ = 3 10 2 0

(5.53) In the first expression, Einstein again first wrote down . dx instead of . dα.44 The second equality tells us that he investigated the expression for the potential. This is probably the reason why the factor A does not appear here. As we saw in the previous sections, Einstein defined A as ⎧ A(α) =

c,

α g,

.

(5.54)

see Eq. (5.42). In doing so, he wanted the function .f (r) from Eq. (5.43) to describe the potential .ϕ(r) (Einstein here used the variable x instead of r).45 This is also the

44 Einstein

frequently interchanged these variables, see also Eq. (5.39). saw in Sect. 5.2.2 that in this case, .f (r) = ϕ(r) behaves like .r −1 for large r but is also “abgerundet” (AEA 6-273) at .r = 0.

45 We

5.2 Analysis of Related Manuscript Pages and Correspondence

299

Fig. 5.5 The manuscript page AEA 62-795. © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama

300

5 Considerations on Delta Functions

reason why the integral runs from zero to g, which is the so-called AbrundungsRadius (AEA 6-273). The second equations reads46 ⎛∞ Ae

.

−α 2 x 2

⎛∞ ⎛ ⎛∞ ⎛∞ ⎞ 2 2 2 dα = A 1 − α x + · · · dα = A dα − x α 2 A dα · · · .

0

0

0

0

(5.55) In this equation, the factor A appears. While Einstein investigated the potential .ϕ(r) in the first equation, it seems likely that in the second equation, he looked at the .ψ quantities, which are the functions .α(r), .β(r), .δ(r), and .η(r). These functions appear in the spherically symmetrical metric components and were discussed in Chap. 4 in great detail. Again, we emphasize that Einstein used the same notation for different objects: .α(r) is a function dependent on r, while .α in Eqs. (5.53) and (5.55) is a variable. Similarly, .δ(r) is also one of these four functions, while .δ(α − α0 ) is the .δ-function.47 We recall that the .ψ quantities can be approximated by setting A equal to the .δ-function, see Sects. 5.2.2 and 5.2.3. Beneath these two lines, we find the comment: “Wird mit höheren Potenzen allmählich unsicherer wegen Willkür des Ansatzes.”48 In AEA 6-273, we already encountered the notion “Willkür.” There, he wrote: “Das Willkürliche dieser Wahl liegt in dem plötzlichen Abfall von A bei .r = g. (In Wahrheit ist ein rascher Abfall statt eines plötzlichen zu erwarten).” We see that Einstein referred to the approximation of cutting the argument of the integral at some point. For .ϕ(r), this cut at .r = g is caused by the definition in equation (5.54). Similarly, the .δ-function for A describing the .ψ quantities makes a similar cut at the position .α = α. In AEA 6-274, Einstein commented this similarly: “Da nämlich die A in der Gegend eines bestimmten .α rasch abfallen sollen [.. . .].” In order to interpret Einstein’s comment on the present manuscript page, let us look at the integrals on the right hand side of equation (5.55). As we already discussed in equations (5.51) and (5.52), it is ⎛∞

⎛∞ A dα − x

.

0

⎛∞ α A dα + x

2

2

0

α 4 A dα ± · · · = 1 − x 2 α 2 + x 4 α 4 − x 6 α 6 ± · · ·

4 0

(5.56)

46 Einstein

omitted the limits in the last integral. saw in Sect. 5.2.1 that Einstein investigated the .δ-function .δ(α − α0 ) on the manuscript page AEA 62-783. On the present page AEA 62-795, Einstein rather used the notation .α instead of .α0 . 48 “Becomes more uncertain with higher powers due to the arbitrariness [Willkür] of the approach [Ansatz].” 47 We

5.2 Analysis of Related Manuscript Pages and Correspondence

301

Fig. 5.6 Sketch drawn by Einstein on the manuscript page AEA 62-795. © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama

when choosing .A = δ(α − α). However, this is only an approximation. If the factor A is not the .δ-function, but only “rasch abfallend” at .α = α, the difference 49 becomes larger for large between the right hand side of Eq. (5.56)┌ and “in reality” ┐ powers of .α: In reality, the values .α ∈ α − ε, α + ε for small .ε contribute to the integrals as well. For higher powers of .α, this contribution becomes larger. This might be the idea behind Einstein’s comment. The same argumentation holds for the approximation of cutting A at .r = g in order to describe .ϕ(r). It would also explain why Einstein wanted to use the first two integrals (good approximation) in his approximation method described in Sect. 5.2.3.1. On the present manuscript page, Einstein proceeded with a sketch that is depicted in Fig. 5.6. It seems as if Einstein drew two coordinate systems and connected the origins with each other. Furthermore, the two lines passing through the origin of the first coordinate system remind us of a light cone. Einstein apparently used the labels A and x for the axes. However, we did not find a reasonable explanation for his sketch. We note that by Eq. (5.55), the factor A would rather depend on .α instead of x in accordance with Eqs. (5.45) and (5.50). The label of the vertical axis could also be t.50 49 This means in the case that is not an approximation. Einstein referred to this case as “In Wahrheit” (AEA 6-273). 50 Although we have not found any specific connection between these two manuscript pages, we note that on AEA 31-438, Einstein drew a coordinate system with the same label of the vertical axis and also with a cross or x on the right edge of the horizontal axis. There, we also find the notations r and .α.

302

5 Considerations on Delta Functions

Let us now look at the calculations beneath the sketch. Einstein investigated the equation51 ⎛∞ .

A e−α

2x2

dα = e−x

(5.57)

0

and apparently considered its convolution with .x 2σ ⎛∞ ⎛∞ Ae

.

0

−α 2 x 2

⎛∞ x

2σ

dx dα =

0

e−x x 2σ dx.

(5.58)

0

Einstein only wrote down the first equation, while we will show in the following that he mainly considered the second equation. The convolution with the factor .x 2σ was indicated by Einstein next to Eq. (5.57), but the integral with respect to the variable x does not appear in his calculation on the left hand side of the manuscript page. However, on the right hand side, he computed the two integrals ⎛∞

−x

e

.

⎛∞ x

2σ

dx

and

0

e−α

2x2

x 2σ dx.

(5.59)

0

In the following, we will show that he found expressions for these two integrals and then plugged them into Eq. (5.58). Einstein treated the factor A as if it does not depend on x. Let us start with the first integral, which he evaluated as52 ⎛∞

−x

e

.

⎛∞ x

2σ

dx =

0

2σ x 2σ −1 e−x dx = (2σ )!,

(5.60)

0

which follows by partial integration.53 For the second integral, he found ⎛∞ e

.

0

51 Einstein

−α 2 x 2

x

2σ

dx = α

−(2σ +1)

⎛∞

e−x x 2σ dx, 2

(5.61)

0

first wrote .A2 instead of A and did not use any limits for the integral. did not write down the limits of the integral on the left hand side. 53 Although Einstein did not use this notion on the present manuscript pages, the left hand side of Eq. (5.60) is the Gamma function .┌ (2σ + 1). 52 He

5.2 Analysis of Related Manuscript Pages and Correspondence

303

which follows from substituting .αx → x, introduced the notation ⎛∞ Iσ =

.

e−x x 2σ dx 2

(5.62)

0

and proceeded with ⎛0 Iσ = −

.

x 2σ −1 ⎛ −x 2 ⎞ 2σ − 1 d e = 2 2

1

⎛∞

x 2σ −2 e−x dx. 2

(5.63)

0

We first note that the notation .Iσ should not be confused with the notation .I2n from Eq. (5.51). Furthermore, Einstein did not write down the limits of the integrals on the right hand sides of Eqs. (5.61) and (5.63) and used different limits for the integral in the second expression of Eq. (5.63). In his notes, Eq. (5.63) is rather the continuation of Eq. (5.61), and it thus reads as if Einstein did not compute .Iσ but the integral from the left hand side of Eq. (5.61). In addition, he apparently did not use any auxiliary 2 2 calculation. For the first equality in Eq. (5.63), he used . d(e−x ) = −2x e−x dx. 2 2 The last equality follows by partial integration with . d(e−x ) = ( d(e−x )/ dx) dx. By Eqs. (5.62) and (5.63), it follows iteratively that ⎞ ⎞⎛ ⎞ ⎛ ⎛ 3 1 1 Iσ −1 = σ − σ− Iσ −2 Iσ = σ − 2 2 2 ⎞⎛ ⎞ ⎛ ⎞ ⎛ 3 1 1 σ− · . . . · σ − (σ − 1) − I0 = σ− 2 2 2 ⎞ ⎞⎛ ⎛ 1 3 1 · . . . · I0 . σ− = σ− 2 2 2

.

(5.64)

Equivalent considerations can be found in Einstein’s notes.54 He then proceeded with calculating the value of .I0 by looking at ⎛ . (2I0 ) = ⎝2

⎞2

⎛∞

2

e 0

−x 2

⎛

dx ⎠ = ⎝

⎞2

⎛∞

−∞

e

−x 2

dx ⎠ =

⎛∞

e−r 2π r dr = π 2

(5.65)

0

according to Eqs. (5.30) and (5.31). Einstein did not write down the second and third expressions but directly computed the integral using polar coordinates as it

54 While

we looked at .Iσ −(σ −1) in order to find the factor before .I0 , Einstein rather considered .σ steps of odd numbers.

304

5 Considerations on Delta Functions

was already the case on manuscript page AEA 62-783, see Sect. 5.2.1.55 He finally wrote down I = α −(2σ +1) Z,

(5.66)

√ ⎞ ⎛ 1 π 1 · ... · . .Z = σ− 2 2 2

(5.67)

.

where, consequently, it is

Hence, it is .Z = Iσ by Eqs. (5.64) and (5.65) as well as ⎛∞ I=

.

e−α

2x2

x 2σ dx

(5.68)

0

according to Eq. (5.61). We note that on the manuscript page AEA 62-783, Einstein used the notation I differently, namely for the integral .I0 , see Eq. (5.30). Einstein finished his auxiliary calculations here by a horizontal line and apparently came back to Eq. (5.57) on the left side of the manuscript page. He convoluted56 ⎛∞ .

A e−α

2x2

dα = e−x

(5.69)

0

with .x 2σ in order to get ⎛∞ Aα

.

−(2σ +1)

√ ⎛ ⎞ ⎞⎛ 1 1 π 3 dα = (2σ )! σ− · ... · σ− 2 2 2 2

(5.70)

0

by apparently having Eq. (5.58) in mind and using Eqs. 5.60 and (5.66) to (5.68) as well as .A /= A(x). Beneath equation (5.70), Einstein wrote on AEA 62-795 “Erfüllen die Gleichung für zwei aufeinanderfolgende grosse .σ und lokalisiertes A.”57 Einstein then proceeded with four equations. The first two equations belong to the second part (“lokalisiertes A”) of his comment, while the last two equations belong to the first part. He did not indicate which quantities satisfy (“erfüllen”) the equation.

55 By a correction of the integral limits in Einstein’s notes, we see that he apparently had in mind the third expression. He first wrote down the lower limit as .−∞ and corrected it to zero afterward. 56 Equation (5.69) is Eq. (5.57). 57 “Satisfy the condition for two consecutive large .σ and localized A.”

5.2 Analysis of Related Manuscript Pages and Correspondence

He started with the equations ⎛ . A dα = A and

α = α.

305

(5.71)

It is not clear how he derived these conditions. It might be the case that these conditions hold because the quantities not known to us satisfy equation (5.70) for two consecutive .σ if A is localized at .α = α. It is possible to interpret the function .A = A(α) as having properties similar to the .δ-function: While the .δ-function increases and decreases “infinitely steeply” ⎛ around a certain value .α = α with . δ(α) dα = 1,58 the function .A(α) does not. Its maximum as well as the width of its peak has finite values. It seems as if Einstein had in mind the approximation ⎛ (5.72) . A(α) dα ≈ A(α) · ΔA =: A, where the width of the peak is denoted by .ΔA and .A(α) vanishes for .α /∈ ┐ ┌ α − ε, α + ε for a small .ε. Einstein apparently approximated ⎛∞ .

A(α)α −(2α+1) ≈ A · α −2(σ +1)

(5.73)

0

as we will see in the following. The idea behind this might be similar to the idea of the .δ-function in Eq. (5.4): The function .A(α) increases and decreases fast around ┐ ┌ α − ε, α + ε such that only the values in the .α = α and vanishes for .α /∈ neighborhood of .α = α contribute. For A being the .δ-function, it would be .A = 1. Einstein used Eq. (5.73) as he derived Aα

.

−(2σ +1)

√ ⎛ ⎞⎛ ⎞ 1 3 1 π σ− σ− · ... · = (2σ )!, 2 2 2 2

(5.74)

which follows from Eq. (5.70). As we saw in Einstein’s comment above, this holds for “two consecutive large .σ ,” which is why Einstein substituted .σ → σ + 1 and derived √ ⎛ ⎞⎛ ⎞ 1 1 1 π −(2σ +3) σ+ σ− · ... · = (2σ + 2)(2σ + 1)(2σ )! (5.75) .Aα 2 2 2 2 as well. By Eqs. (5.74) and (5.75), it follows that α

.

58 See

−2

⎛ ⎞ ⎛ ⎞ 1 1 σ+ = (2σ + 2)(2σ + 1) = 4(σ + 1) σ + 2 2

Eq. (5.3).

(5.76)

306

5 Considerations on Delta Functions

corresponding to Einstein’s next line. He then concluded 1 α= √ . 2 σ +1

(5.77)

.

We note that for large .σ , .α is small. Using Eq. (5.77), he apparently approximated A by

.

A∼ ⎛

.

4 1 ⎞ √ ( √ )2σ +1 . π 2 σ +1 σ− ! (2σ )! 1 2

(5.78)

This follows from Eq. (5.74), where Einstein apparently used ⎛ .

σ−

⎞⎛ ⎞ ⎞ ⎛ 1 3 1 1 σ− · ... · . != σ− 2 2 2 2

(5.79)

√ √ If so, he made a mistake as the factor .4/ π should rather be .2/ π in Eq. (5.78).59 This, however, did not affect Einstein’s further considerations where he approximated .lg A for large .σ and neglected all constant factors. For this approximation, Einstein made a further auxiliary calculation on the right side beneath the horizontal line deriving Stirling’s formula. The second term in Einstein’s second expression is barely readable. However, it seems as if he considered the relations

.

lg (σ !) = lg σ + lg (σ − 1)! =

σ Σ i=1

⎛σ lg i ≈

lg x dx = σ lg σ − σ + 1 ≈ σ lg σ − σ. 1

(5.80) Einstein wrote down the first, second, fourth, and sixth expressions. Einstein’s approximation is only correct since the integral yields a term .σ lg σ , which is why the additional term .lg(σ )/2 not written down by Einstein can be neglected. He then used Eq. (5.80) in order to approximate .lg A. It is ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎛ ⎞ 1 4 1 1 + σ− + lg √ lg σ − . lg A ∼ 2σ lg 2σ − 2σ − σ − 2 2 2 π ⎞ ⎛ √ − (2σ + 1) lg 2 σ + 1 (5.81)

59 In

Eq. (5.74), the factor .1/2 appears directly before the term Einstein’s mistake.

√ π /2 which probably caused

.

5.2 Analysis of Related Manuscript Pages and Correspondence

307

√ by Eq. (5.78). As Einstein considered large .σ , the constant term .lg (4/ π ) can be neglected. He also neglected a factor .1/2 in the last term and derived .

⎞ ⎛ ⎞ ⎞ ⎛ ⎛ 1 1 1 + σ− − (2σ + 1) lg 2 lg σ − lg A ∼ 2σ lg 2σ − 2σ − σ − 2 2 2 − (2σ + 1) lg (σ + 1)

(5.82)

instead of Eq. (5.81). For a next approximation, he then apparently used .

⎞ ⎛ 1 ≈ lg σ ≈ lg (σ + 1) lg σ − 2

(5.83)

and only considered terms that contain a term of the order of magnitude of .σ lg σ , such that it is .

lg A ∼ 2σ lg 2σ − 3σ lg σ.

(5.84)

Einstein concluded “A nimmt sehr schnell ab.”60 Indeed, it is ⎛ ⎞σ 4 .A ≈ σ

(5.85)

for large r and .A decreases as .σ −σ . We summarize that Einstein considered the potential .ϕ(r) and probably the .ψ quantities at the top of the page. After an unidentified sketch, he then considered ⎛ .

A e−α

2x2

dα = e−x .

(5.86)

The correspondence suggests that the left hand side stands for the .ψ quantities.61 Hence, by Eq. (5.86), these quantities would behave like .e−r , which is similar to the factor discussed by Einstein in the letter AEA 6-271. In the following, Einstein then convoluted with .x 2σ and considered the integral over x, see Eq. (5.58). He then made a comprehensive auxiliary calculation and derived equation (5.70). Einstein commented on this equation by “Erfüllen die Gleichung,” while he did not indicate which quantities satisfy the equation. It seems plausible that Einstein wrote about the .ψ quantities here. If so, the behavior of the .ψ quantities might imply the conditions from Eq. (5.71) that Einstein imposed on .A(α). In the following he then investigated whether A decreases fast for large .σ and came to the conclusion that it does. This might have convinced him that the function .A(α) can be approximated by

60 “A

decreases very fast.” Sects. 5.2.2, 5.2.3, and 5.2.3.1.

61 See

308

5 Considerations on Delta Functions

the .δ-function according to the correspondence. Such an interpretation is supported by the back page AEA 62-795r, where Einstein wrote down “Also .δ(α − α) gute Approximation.”62 We will discuss this in Sect. 5.2.5.

5.2.5 AEA 62-795r In this section, we will discuss the manuscript page AEA 62-795r, which is the back of the page AEA 62-795. It contains calculations mainly written in ink on the first part that are related to the manuscript pages AEA 62-795 and 62-783. On the middle part, Einstein made considerations that are connected to Brownian motion, also written in ink, while the last part is written in pencil and is again related to the first part. The manuscript page is shown in Fig. 5.7. At the top, we find the expression .f (x) of the integral that we already encountered in the correspondence and on both manuscript pages AEA 62-795 and 62-783. We saw that it stands for the potential .ϕ for a constant A and for the .ψ quantities63 for an A that can be approximated by the .δ-function .δ(α − α). In fact, at the end of the first part, Einstein wrote down such a .δ-function, which indicates that the context of the present manuscript is about the .ψ quantities.64 The first line reads ⎛ ⎛ ⎛ ⎞ 2 2 2 2 . (5.87) A e−α x dα = A e−α x −2α 2 + 4α 4 x 2 dα. Denoting the left hand side by .f (x) and assuming .A = A(α) according to the correspondence, it is ∂f (x) = . ∂x ∂ 2 f (x) = ∂x 2

⎛ ⎛

⎛ ⎞ 2 2 A −2α 2 x e−α x dα

and

⎞ ⎛ 2 2 A −2α 2 + 4α 4 x 2 e−α x dα.

(5.88)

Einstein wrote down parts of these two derivatives in the top right corner of the manuscript page. For the first derivative, he apparently first wrote down the term 2 2 .2α x instead of .2αx indicating that he considered the derivative with respect to .α instead of x, but he corrected it afterward. We see that Eq. (5.87) can be considered as .f (x) = f '' (x), where the prime notation stands for derivatives with respect to

− α) good approximation”. quantities are probably the functions .α(r), .β(r), .δ(r), and .η(r). 64 Except for the last equations. 62 “Hence, .δ(α 63 The .ψ

5.2 Analysis of Related Manuscript Pages and Correspondence

309

Fig. 5.7 The manuscript page AEA 62-795r. © The Hebrew University of Jerusalem, Israel. Digital image photographed by Ardon Bar Hama

310

5 Considerations on Delta Functions

x.65 We recall that on the front of the page AEA 62-795, Einstein set f (x) = e−x ,

(5.89)

.

see Eq. (5.57), which solves the differential equation.66 In the next two lines, we find the three expressions ⎛ 2 2 . A e−α x 4α 4 x 2 dα ⎛

A2α 3 d e−α

− ⎛

e−α

2

2x2

2x2

,

and

⎛ ⎞ Aα 3 dα.

(5.90)

Clearly, the first expression is the second term on the right hand side of Eq. (5.87), in which there is the only term containing the variable x except for the exponential function. The first and second expressions in (5.90) are obviously equivalent as it is ∂ e−α ∂α

2x2

.

= −2αx 2 .

(5.91)

In the subsequent calculation, Einstein used the prime notation for derivatives with respect to .α, see Eq. (5.96). Assuming this, the argument of the third expression in (5.90) is not equal to the argument of the first two expressions unless we use further assumptions as A(α) =

.

c α2 x 2 e a3

(5.92)

for a constant c, where .A(α) would also depend on x and, in particular, diverge for large .α. This seems unlikely to us. Instead, we argue that Einstein used the product rule and calculated ⎛∞ .

−

⎛∞ ⎛ ⎞ ┌ ┐∞ ⎞ ⎛ 2 2 −α 2 x 2 3 −α 2 x 2 = −2 A(α)α e A(α)2α d e + 2 d A(α)α 3 e−α x 3

0

0

0

⎛∞ =2

e−α

2x2

⎛ ⎞' A(α)α 3 dα

(5.93)

0

65 Below,

we will use the prime notation for derivatives with respect to .α. here note that any linear combination of .ex and .e−x solves the differential equation, in particular the hyperbolic sine and cosine functions .sinh x and .cosh x. These functions have been considered by Einstein on the manuscript page AEA 62-223 that might be connected to the context of the present manuscript page, see also Sect. 5.2.7.

66 We

5.2 Analysis of Related Manuscript Pages and Correspondence

311

such that the variable x only appears as argument of the exponential function. Under the assumption that .A(α) vanishes for large .α,67 the first term in the second expression vanishes for .x /= 0.68 For .x = 0, the convergence of the term depends on .A(α). In fact, Einstein derived an expression for .A(α) such that it vanishes for large .α, see Eqs. (5.100) and (5.101). Einstein only wrote down the first and last expressions beneath each other by using the notation A instead of .A(α). He also did not write down limits, but obviously the calculation holds for the lower limit being 69 .−∞ as well. We find some further expressions partly crossed out written down beneath the first expression in (5.90), where Einstein apparently considered the argument of the exponential function as well as its derivatives. Finally, we note that Einstein corrected the power of .α in the first and last expressions of Eq. (5.93) from 2 to 3. This mistake might have resulted from considering the derivative with respect to x instead of .α as it was already the case in his auxiliary calculation in the top right corner. The mistake appeared also in Einstein’s next equation and was then corrected. Einstein then considered the condition ⎞ ⎛ ⎞' ⎛ 2 − 2 Aα 3 dα = 0, .A 1 + 2α (5.94) where the appearance of the differential . dα probably is a mistake. Clearly, he set the arguments of the integrals in Eq. (5.87) to be equal by using Eqs. (5.91) and (5.93). In the next lines, he neglected the differential and derived ⎞ ⎛ A 1 − 4α 2 − 2A' α 3 = 0,

.

(5.95)

where we can identify the auxiliary calculation .

⎞' ⎛ − 2 Aα 3 = −6Aα 2 − 2A' α 3 ,

(5.96)

which clearly supports our interpretation that the prime notation stands for derivatives with respect to .α. Recasting equation (5.95) led him to 1 A' = . 2 A

67 This

⎛

1 2 − 3 α α

⎞ dα.

(5.97)

seems likely to us as Einstein wanted to approximate it by the .δ-function. variable x probably stands for the radial component r, see Sect. 5.2.2. 69 We chose the limits according to the front of the page AEA 62-795 as well as to the correspondence, see Sect. 5.2.4 or AEA 6-273. 68 The

312

5 Considerations on Delta Functions

First, we note that Einstein used the notation a instead of .α inside the brackets. This notation was then also used by him in the following two equations. We argue that he interchanged the notations .α and a. Second, Einstein apparently made a mistake by recasting equation (5.95) as he omitted a factor 2 in the second term inside the brackets of Eq. (5.97). This led him to a wrong expression for .A(α) as we will see in the following. Third, the differential . dα appears here again. It is possible that Einstein wrote this down because of his next step where he integrated and derived70 .

lg A = −

1 1 − lg α + konst., 4 α2

(5.98)

which follows by71 ⎛ .

1 ∂A dα = lg A. A ∂α

(5.99)

1 − 41 12 e α α

(5.100)

Finally, he derived A=

.

and drew a box around this equation. We note that the correct factor in Eq. (5.97) would yield A=

.

1 − 14 12 e α . α2

(5.101)

The graphs of these two functions are shown in Fig. 5.8.72 Einstein commented his equation with “In Wahrheit bei oberen a abgeschnitten. Also .δ(α − α) gute Approximation”73 indicating again that he investigated whether the .δ-function for .A(α) is a good approximation. We see that Einstein here used the notation .α again. Furthermore, he used the notation .α which we only know from the front of the page AEA 62-795. There, the function .A(α) had its peak at .α = α, see Sect. 5.2.4. Einstein stopped his calculations here. However, it is interesting to see that it is74 ⎛∞ f (x) =

.

√ 1 − 14 12 −α 2 x 2 e α e dα = π e−x 2 α

(5.102)

0

70 Einstein

here used the notation a instead of .α. also Eq. (5.9). 72 We note that a graph as in Fig. 5.8b was discussed by Einstein on the manuscript page AEA 62322r, see Sect. 5.2.7. There, we find a graph that is similar to .A(α) around .α = 0 with the comment that .A(α) should vanish at .−∞ and .∞. 73 “In truth, cut of at higher a. Hence, .δ(α − α) good approximation.” 74 Again, Einstein did not indicate whether the lower limit is 0 according to the correspondence or .−∞. However, this does not make any difference for our considerations here. 71 See

5.2 Analysis of Related Manuscript Pages and Correspondence

313

Fig. 5.8 Plots of .A(α) in Einstein’s version as in Eq. (5.100) (a) and of .A(α) as in Eq. (5.101) (b) when correcting his mistake of the missing factor 2

if we take the correct form for .A(α) as in Eq. (5.101). Equation (5.102) holds as it is ⎛∞ .

1 − 12 −α 2 x 2 e 4α dα = e−x α2

0

⎛∞

⎛

1 − e α2

1 2α −αx

⎞2

dα = e

−x

1 −x e 2

⎛ ⎞2 1 − αx− 2α

2x e

dα

0

0

=

⎛∞

⎛∞

⎛

⎞ −⎛αx− 1 ⎞2 √ 2α α −2 + 2x e = π e−x ,

−∞

(5.103) where we used the substitutions .u = (2αx)−1 in the second and .u = αx − (2α)−1 as well as Eq. (5.29) in the fourth equality.75 Clearly, the condition from Eq. (5.87) is satisfied. We see that .f (x) takes on the form demanded by Einstein on the front of the page AEA 62-795 in Eq. (5.57). There, Einstein started from .f (x) = e−x and looked how .A(α) behaves around its peak without deriving an explicit expression for .A(α). On the present manuscript page, Einstein started from .f (x) = f '' (x), where the prime notation stands for derivatives with respect to x, and derived an explicit expression for .A(α). While Einstein only implicitly indicated that he tried to approximate .A(α) by the .δ-function on the front of the page AEA 62-795,76 he explicitly commented that the .δ-function is a good approximation for .A(α) on the present manuscript page. As we know from the correspondence, Einstein tried to approximate the .ψ quantities, which are related to .ξ(r) ∈ {α(r), β(r), δ(r), η(r)} by approximating

75 In

the third equality, we added the second and third expressions. for example, his comments discussed in Sect. 5.2.4 or Eq. 5.71.

76 See,

314

5 Considerations on Delta Functions

A(α) by the .δ-function (see Sects. 5.2.3 and 5.2.2).77 The present manuscript page suggests that at least one of these quantities behaves like .f (r) = e−r . At the end of the first part of the manuscript page, we find the equation

.

⎛g

−α 2 x 2

e

.

1 dα = x

0

⎛xg

e−y dy 2

(5.104)

0

and the substitution .αx = y written in pencil. The equality is obviously true. Einstein here investigated the potential .ϕ(r) which results from78 ⎛∞ f (r) =

.

A(α) e−α

2r2

dα

(5.105)

0

for an A chosen as in Eq. (5.42), while he apparently set .c = 1 on the present manuscript page.79 As we already saw, Einstein wanted to get .ϕ(r) ∼ r −1 for large r, which is in accordance with Eq. (5.104).80 It is possible that Einstein checked this property by his substitution. Beneath equation (5.104), Einstein drew a horizontal line and started new calculations and little sketches written in ink. These considerations are obviously connected to the theory of Brownian motion. For instance, the two sketches can be found in a similar way in Einstein (1908) as well as many of its equations; see, for instance, the numbered equations (0), (1), (4), and (6a) as well as figures 93 and 94 in Einstein (1908). As we have not found any promising connection between the calculations on Brownian motion and those on the manuscript pages and in the correspondence discussed in this chapter, we will skip them. At the bottom of the manuscript page, we find the equation ⎛ 2 2 .f = A e−α x dα (5.106) with the comment:81 In der Umg[ebung] von .x = 0 soll Betr[ag] v[on] .f '' oberhalb einer gew[issen] Grenze liegen. (AEA 62-795r)

77 We again note that .δ(r) should not be confused with the .δ-function and .α(r) not with the variable .α.

78 Einstein used the variable r in the correspondence and on AEA 62-783, while he used the variable x on the manuscript pages AEA 62-795 and 62-795r. 79 Einstein also looked at this on the front of the page AEA 62-795, on the manuscript page AEA 62-783, as well as in the correspondence with Bargmann. For the function .f (r), see also Eq. (5.39), (5.43), and (5.53). 80 The integral on the right hand side of Eq. (5.104) is the error function, which tends to .√π /2 for large x and positive g. 81 “In the neighborhood of .x = 0, absolute value of .f '' should lie above a certain limit.”

5.2 Analysis of Related Manuscript Pages and Correspondence

315

In fact, it is the first time that this equation appears on the present manuscript page, even though Einstein used the expression in Eq. (5.87).82 He then considered the power series expansion of the exponential expression such that it is83 ⎛ f =

⎛

α4x 4 ± ··· A 1−α x + 2 2 2

.

⎞ dα

(5.107)

⎛ ⎞ A −4α 2 + 2 · 3α 4 x 4 ± · · · dα

(5.108)

and computed the second derivative with respect to x as f '' =

⎛

.

according to his comment. We note that Einstein missed the factor 4 of the first term in .f '' and only wrote down the right hand sides of Eqs. (5.107), and (5.108). The calculation then stops. We note at this point that Einstein made similar 2 considerations on the manuscript page AEA 31-219. There, he considered .e−ax 2 2 instead of .e−α x .

5.2.6 An Expression Similar to δ-Function from 1938 In this section, we briefly discuss the appearance of functions similar to the .δfunction that we found in the research notes from 1938. We note that he considered the arctangent function as r 1 arctan , r a 1 g(x) = x arctan , ax 1 ρ(r) = arctan r r

f (r) =

.

or (5.109)

on the documents AEA 6-258,84 62-791, 62-799, and 6-248. The second function obviously transforms to the first function for .x = r −1 . In Fig. 5.9, the graph of the first function is shown for .a = 1, .a = 0.5, and .a = 0.2. Trying to use this function as .δ-function, the problem is that it does not vanish fast enough for .a → 0 for

82 We already encountered it several times as on the manuscript pages AEA 62-795, 62-783 as well as in the letters AEA 6-273, 6-274, and 6-275. 83 See also Eq. (5.55). Einstein only wrote down the first two terms in the brackets. 84 This is a letter from Bergmann to Einstein.

316

5 Considerations on Delta Functions

Fig. 5.9 The graph of .f (r) = r −1 arctan(r/a) for .a = 1, .a = 0.5, and .a = 0.2

r > 0 as it is .f (r) → π/(2r). Slightly modifying the function by taking the inverse argument yields

.

a 1 f~(r) = arctan , r r

.

(5.110)

which would solve this problem as it is .f~(r) → ∞ for .r → 0 and .a > 0 as well as f~(r) → 0 for .a → 0 and .r > 0. We also note that by

.

π⎞ x 1⎛ , arctan + ε→0 π a 2

Θ(x) = lim

.

(5.111)

where .Θ(x) is the Heaviside step function,85 the function h(r) =

.

can also be used as a .δ-function.

85 See

Bartelmann et al. (2018a, 4–7).

a2 1 1 arctan 2 2π a r

(5.112)

5.2 Analysis of Related Manuscript Pages and Correspondence

317

As Einstein did not consider these modified functions on the respective sheets and as we did not find any evidence that he effectively used the arctangent function in order to describe a .δ-function, we will not discuss this further here.

5.2.7 Further Manuscript Pages We here present further manuscript pages that might be connected to the contexts presented in the previous sections and chapters. It might be worthwhile to look at these manuscript pages in future projects. We will present these pages briefly and point out similarities to the discussed subjects. As already mentioned, on the manuscript page AEA 31-219, Einstein considered the function ⎛∞ y=

.

z(a) e−ax da 2

(5.113)

0

and derived the first and second derivatives with respect to x similarly to the procedure on AEA 62-795r, see Sect. 5.2.5. On the manuscript page AEA 62-223, Einstein considered the function ϕ=

.

A sin αr , r cos αr

(5.114)

where the notation reminds us of the potential .ϕ(r). We also find the equations 1 , cos αr 1 β = Cr , cos αr 1 γ = Dr 2 , cos αr 1 δ=E . cos αr α=B

.

and (5.115)

It seems as if the quantity .α on the right hand side of the equations is not equal to the quantity .α on the left hand side of the first equation. The left hand side reminds us of the functions discussed in the correspondence. In fact, the equations are very similar to equations used by V. Bargmann in his letter AEA 6-206 (page 6). We also find the expressions .

eαr − e−αr eαr + e−αr

and

1 − e−2αr , 1 + e−2αr

(5.116)

318

5 Considerations on Delta Functions

which both are equivalent to .tanh(αr) = sinh αr/ cosh αr. As we have seen in Sect. 5.2.5, Einstein considered on the manuscript page AEA 62-795r a differential equation that is solved by .sinh(x) or .cosh(x). Similar and equivalent expressions appear also on the manuscript page AEA 62300. There, we also find the expression 1 − e−αr , r 2

.

(5.117)

which can be found on many manuscript pages as on AEA 62-317. In fact, it seems as if the manuscript pages AEA 62-317 to AEA 62-322r in reel 62 belong together and might be connected to the contexts discussed in the previous sections and chapters.86 For instance, on AEA 62-317r, we find the equations ⎛ .

r n e−α

2r2

dr =

1

zn , α n+1

(5.118)

which is for .r = x, .n = 2σ , and .zn = Iσ equivalent to an equation on AEA 62795, see Eqs. (5.61) and (5.62). The subsequent manuscript page AEA 62-318 might belong to Einstein and Bergmann’s publication as well as to the Washington manuscript, while the manuscript page AEA 62-319 then contains again expressions similar to the trigonometric functions from above. The last manuscript page of this sequence is AEA 62-322r. Einstein here drew a little sketch with the comment “Soll beiderseits im .∞ verschwinden.” The graph drawn by him and the additional condition fit to the function considered by Einstein on AEA 62-795r. There, he looked at ⎛ 2 2 . (5.119) A e−α x dα and almost derived the condition A=

.

1 − 14 12 e a . α2

(5.120)

As we saw in Sect. 5.2.5, Einstein made a mistake (factor 2) and derived another expression for A. However, the graph of the correct function .A(α) fits to Einstein’s sketch and comment on the manuscript page AEA 62-322r, see also Fig. 5.8b. In addition, we find the expressions Σ .

An e−nx

and

ΣΣ

An

xm m n (−1)m m!

(5.121)

86 It is possible that the manuscript pages AEA 62-320, 62-320r, 62-321, and 62-321r do not belong to this sequence.

5.2 Analysis of Related Manuscript Pages and Correspondence

319

which show similarities to Eq. 5.55 on AEA 62-795 by the transformations .n → α 2 and .x → x 2 . On AEA 62-795, Einstein wrote down the first line of ⎛∞ ⎛ ⎛∞ ⎛ ⎞ 2 2 .f (x) = A 1 − α x + · · · dα = A dα − x 2 α 2 A dα ± · · · 0

=

Σ

0

⎛∞

n=0 0

A

x 2n 2n α (−1)n dα, n!

(5.122)

87 see √ also the letter AEA 6-275. Similar calculations as well as the expression a 2 + r 2 can be found on the manuscript page AEA 52-574. This expression appears in Einstein’s letter AEA 6-275 as well. Let us briefly come back to the manuscript page AEA 62-317, where we find expression (5.117). There, Einstein also wrote down the expression

.

.

1 + A e−αr +B e−βr , r

(5.123)

which we can associate with expressions on the manuscript pages AEA 26-417 and 28-458.88 On these two pages, Einstein also wrote about “horizontale Tangenten im Nullpunkt”89 and considered functions as .(1 + Ar) e−αr which reminds us of the letter AEA 6-271. In this context, the manuscript page AEA 36-765 might be interesting as well. It also contains equations similar to those appearing on the manuscript page AEA 36-778. Although the manuscript page seems not to be connected to the discussed context, we note that the function ⎛∞ .

f (α) e−α

2r2

dα

(5.124)

0

which we encountered several times in the previous sections also appears on the manuscript page AEA 62-575. Similarly, on AEA 38-526, Einstein considered the function .e−r r n that reminds us of Eq. (5.5) from AEA 62-783.

87 Einstein

wrote down the second line in a different form in his letter AEA 6-275. also the manuscript page AEA 28-459. 89 “Horizontal tangents at the origin”. 88 See

320

5 Considerations on Delta Functions

5.2.8 AEA 63-058r Although the manuscript page AEA 63-058r probably does not contain considerations on neither .δ-functions nor Einstein and Bergmann’s publication, we will briefly discuss it here as we find interesting calculations on the differential equation y ' = −2xy.

(5.125)

.

Interpreting the prime notation as derivative with respect to x, the function f (x) = e−x

2

(5.126)

.

obviously solves the differential equation. As this solution can easily be derived by separation of variables, Einstein surely had known it. Indeed, we find the expression from the right hand side of Eq. (5.126) also next to Eq. (5.125). In the following, Einstein then considered the sequence ⎛x yi+1 (x) = 1 −

2~ x yi (~ x ) d~ x

.

(5.127)

0

and showed that .yi (x) → e−x for .i → ∞ and .x < 1 when starting with 2

⎧ y=

.

0

1, x ≤ 1

(5.128)

0, x > 1,

which he indicated by a little sketch.90 He then calculated91 ⎛x y (x) = 1 −

2~ x y (~ x ) d~ x=

.

1

90 Einstein

0

0

⎧ ⎛x ⎪ ⎪ ⎪ 1 − 2~ x d~ x = 1 − x2, ⎨

x1

0 ⎛1

(5.129) 2~ x d~ x = 1 − 1 = 0,

0

did not use the notation .~ x but simply x. He did not indicate that the function .y changes

its value at .x = 1, and however, this becomes clear by the subsequent calculation. 91 Einstein skipped the second equality, respectively.

0

5.2 Analysis of Related Manuscript Pages and Correspondence

321

for the first step and proceeded with92 ⎛x y (x) = 1 −

2~ x y (~ x ) d~ x=

.

2

0

1

⎧ ⎛x ( ) ⎪ ⎪ ⎪ 1 − 2~ x 1 −~ x 2 d~ x = 1 − x 2 + 12 x 4 , ⎨ ⎪ ⎪ ⎪ ⎩1 −

0 ⎛1 0

x 1. (5.130)

Finally, it is y (x) =

.

3

⎧ ⎞ ⎛x ⎛ ⎪ ⎪ ⎪ x = 1 − x 2 + 12 x 4 − 16 x 6 x 1 −~ x 2 + 12 ~ x 4 d~ ⎨1 − 2~ ⎪ ⎪ ⎪ ⎩1 −

0 ⎛1 0

⎞ ⎛ ⎛x x − 2~ x 4 d~ x· 2~ x 1 −~ x 2 + 12 ~ 1

1 2

d~ x=

1 3

−

⎛

x 1, (5.131)

where Einstein made a mistake in the last equality as he derived the term .1/6 instead of .5/6. After this third step, Einstein stopped as he apparently recognized that in the n-th step, the .(n + 1)-th term in .y (x) for .x < 1 is n

(−1)n

.

x 2n 2n x 2n = (−1)n , 2 · 4 · . . . · 2n n!

(5.132)

which are the coefficients of the power series expansion of .e−x . He then apparently made the substitution .ρ = x 2 such that it is 2

1 1 3 ρ , y = 1 − ρ + ρ2 − 2·3 2 3

.

(5.133)

which we find in Einstein’s notes. Furthermore, he substituted .

ρ 1+ρ

(5.134)

ρ=

1 −1 1−σ

(5.135)

σ =

and recast this to .

92 Einstein

skipped the first equality.

322

5 Considerations on Delta Functions

by using 1−σ =

.

1 . 1+ρ

(5.136)

He then used the power series expansion of .(1 − σ )−1 at .σ = 0 in order to derive ρ = σ + σ2 + σ3 + σ4 + ···

.

(5.137)

and considered93 ρ = σ + σ2 + σ3 + ··· ,

.

ρ 2 = σ 2 + 2σ 3 + · · · ,

and

ρ3 = σ 3 + · · ·

(5.138)

such that it is94 1 1 y = 1 − σ − σ2 − σ3 − ··· . 2 6 3

.

(5.139)

We summarize that on the present manuscript page, Einstein solved the differential equation (5.125) using an interesting way as he considered the sequence of functions from Eq. (5.127). By the two substitutions .ρ and .σ , he then expressed the power series expansion of .e−x in three different forms as in Eqs. (5.131), (5.133), and (5.139). Finally, we remark that Einstein considered the differential equation (5.125) or similar approximations as .y on the manuscript pages AEA 63-050, 62-373, and i

63-042r. On the manuscript page AEA 63-042, we find considerations equivalent to Eqs. (5.134) to (5.137). There, Einstein commented “Entw[icklung] [der] .ρ in Entw[icklung] [der] .σ verwandelt” (AEA 63-042).

5.3 Concluding Remarks In this chapter, we analyzed three further manuscript pages AEA 62-783, 62-795, and 62-795r and connected them to Einstein and V. Bargmann’s correspondence that took place in summer 1939. These documents are of special interest to us as they

93 Einstein only wrote down the dots in the first equality and neglected it in the second and the third. 94 Again, Einstein did not write down the dots on the right hand side.

5.3 Concluding Remarks

323

all contain considerations on .δ-functions that Einstein might have used in order to search for particle solutions of the 1938 field equations. On the mentioned documents, Einstein considered a function .f (r) that should describe both the potential .ϕ(r) and the so-called .ψ quantities that we could identify with the functions .α(r), .β(r), .δ(r), and .η(r) that appear in the metric components of the five-dimensional theory.95 In particular, the function .f (r) looks like ⎛ f (r) =

.

A(α) e−α

2r2

dα,

(5.140)

where .f (r) becomes the potential .ϕ(r) if .A(α) is constant or chosen as in Eq. (5.42).96 In this case, it behaves like .r −1 for large r, which was one of the most important requirements for Einstein. In the case that .f (r) should describe the .ψ quantities, Einstein approximated .A(α) with the .δ-function. In doing so, he considered several functions that could serve as .δ-functions and approximated them by the Gauss function. Finally, we identified further manuscript pages that might have been written in the same context. In particular, we found some manuscript pages on which Einstein used an interesting way in order to solve a differential equation.

95 We 96 In

here note that the function .δ(r) should not be confused with the .δ-function. Eq. (5.140), .α should not be confused with the function .α(r).

Chapter 6

Conclusion

In the present book, we analyzed more than 100 documents mostly written by Albert Einstein and most of them until now unidentified. The documents are related to the five-dimensional Einstein–Bergmann approach from 1938 and consist of research notes, correspondence, scratch notebooks, unpublished manuscripts, and publications. They were taken from the database and the holdings of the Albert Einstein Archives at The Hebrew University of Jerusalem and the Einstein Papers Project at the California Institute of Technology (Caltech). Using these documents, we investigated Einstein’s practice of theorizing, his way of calculating, and the heuristics of his modus operandi, all in the context of his work on a unified field theory of gravitation and electromagnetism. Based on our analysis, we also looked at Einstein’s mathematical skills in the light of his education as a theoretical physicist, where we especially focused on his knowledge of and interest in projective geometry. After establishing the four-dimensional theory of general relativity in 1915, Einstein’s ambitious goal was to unify gravitation and electromagnetism in a field theory that allows for particle-like solutions and quantum effects. Based on Kaluza’s theory, Einstein and his assistant Peter Bergmann published an article in 1938, where they introduced an additional fifth dimension that was to be interpreted physically. One of the main differences to Kaluza’s approach was that the new metric components were periodic with respect to the fifth dimension. Letters from that time period show that Einstein was very much convinced of the viability of this new approach. Exploring the logical structure of the theory, Einstein composed another unpublished manuscript even before the published article was printed. We discussed both this so-called Washington manuscript and the circumstances that led Einstein to its composition. It contains thematically the same content as the publication but substantially differs in its internal structure. In addition, we dated, contextualized, reconstructed, and commented on about 50 further unpublished research documents that belong to the same context. Furthermore, we re-dated letters from extensive correspondence between Einstein and his assistants Peter © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Schütz, Einstein at Work on Unified Field Theory, Einstein Studies 17, https://doi.org/10.1007/978-3-031-52127-0_6

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6 Conclusion

Bergmann and Valentine Bargmann from the years 1938 and 1939, where we found calculations and considerations on the further development of the new theory. These letters, for the first time, were contextualized, reconstructed, commented on, and interpreted. The mentioned documents show in detail how Einstein tried to find particle solutions for the field equations based on the new theory by considering certain assumptions and requirements for the five-dimensional metric components. The analysis of the research notes also showed that Einstein considered projective geometry and especially involutions, although this mathematical subject did not play a prominent role in his publications. Some sketches and calculations show significant similarities with each other, although they were written more than a quarter century apart. The existence of such sketches shows that Einstein also in his later years actively used mathematical tools which he had learned during his studies at the ETH in Zurich. Our detailed analysis of the respective manuscript pages as well as of surrounding calculations and correspondence suggests that Einstein tried to gain a geometric understanding of power series expansions around the point at infinity. He frequently used such power series expansions in his considerations on a unified field theory.

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Index of Research Notes, Letters, and Further Documents

A AEA 1-133, 9, 143, 148, 167, 197, 198, 202, 205 AEA 1-136, 9, 191, 237 AEA 121-159, 187 AEA 123-350, 188 AEA 124-446, 7, 10, 68, 69, 73, 82, 84, 109, 116, 118–124, 126–128, 138, 139 AEA 144-451, 177 AEA 2-119, 9, 144, 148, 183, 200 AEA 2-121, 10, 144, 148, 152, 154, 179–202, 204, 220, 221, 225, 234, 236 AEA 21-236, 178 AEA 22-289, 276 AEA 22-290, 276 AEA 22-290.1, 10, 276 AEA 23-149, 75 AEA 23-150, 75 AEA 23-156, 75 AEA 23-157, 75 AEA 23-159, 75 AEA 26-417, 319 AEA 28-434, 10, 275, 276 AEA 28-435, 10, 275 AEA 28-436, 275 AEA 28-458, 10, 319 AEA 28-459, 10, 319 AEA 28-459.1, 186 AEA 28-512, 10, 275 AEA 29-136, 10, 75, 76 AEA 29-141, 10, 75 AEA 29-234, 19 AEA 29-425, 176, 177 AEA 3-013, 10, 70, 126 AEA 30-815, 186

AEA 31-219, 10, 315, 317 AEA 31-224, 10, 123 AEA 31-438, 301 AEA 36-765, 10, 319 AEA 36-778, 9, 235, 267, 268 AEA 36-860, 188 AEA 36-863, 188 AEA 38-093, 186 AEA 38-094, 186 AEA 38-526, 319 AEA 47-569, 12 AEA 5-007, 184 AEA 5-008, 10, 144, 148, 152, 154, 179–202, 204, 220, 221, 234 AEA 5-025, 187 AEA 5-242.1, 9, 267, 276 AEA 5-258, 9, 75 AEA 52-503, 186 AEA 52-574, 9, 319 AEA 53-783, 186 AEA 53-892, 186 AEA 54-240, 177, 181, 227 AEA 54-583, 183 AEA 55-632, 187 AEA 55-633, 187 AEA 55-635, 186 AEA 6-206, 10, 241, 242, 244, 246, 247, 249, 279, 317 AEA 6-208, 10, 235, 241, 244, 247, 249, 254, 279 AEA 6-220, 62, 76 AEA 6-221, 62, 76 AEA 6-222, 76 AEA 6-240, 10, 228, 233, 241, 246, 251

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Schütz, Einstein at Work on Unified Field Theory, Einstein Studies 17, https://doi.org/10.1007/978-3-031-52127-0

351

352

Index of Research Notes, Letters, and Further Documents

AEA 6-242, 10, 77, 178, 228, 256, 260, 264, 265, 267, 280 AEA 6-243, 228 AEA 6-244, 228 AEA 6-245, 10, 85, 228, 251, 255, 257, 259–261, 263, 267, 275 AEA 6-246, 10, 228, 252, 257, 261, 263, 267, 268, 293 AEA 6-247, 10, 228, 240, 261, 263, 264 AEA 6-248, 10, 228, 232, 267–269, 315 AEA 6-249, 10, 228, 263, 275 AEA 6-250, 9, 10, 73, 74, 113, 123, 125, 190, 228, 234, 240, 255, 263, 264, 275 AEA 6-250.1, 10, 74, 113, 123, 125, 263, 264 AEA 6-251, 10, 228, 232, 233, 239, 240, 247, 266, 267, 279 AEA 6-252, 10, 190, 197, 198, 228, 234, 247, 261, 275 AEA 6-253, 10, 228, 234, 240, 247, 252, 255, 261, 262, 264, 266, 267 AEA 6-254, 10, 198, 199, 228 AEA 6-255, 228 AEA 6-256, 10, 148, 162, 181, 184, 194, 195, 228 AEA 6-258, 10, 162, 182, 185, 228, 231–233, 236, 238, 240, 247, 269, 274, 315 AEA 6-259, 9, 85, 113, 228, 231, 232, 234, 247, 248, 256, 262 AEA 6-260, 10, 228 AEA 6-261, 10, 228, 235, 239, 240, 247, 251, 259, 261 AEA 6-262, 10, 228, 234, 237, 247, 251, 255, 261, 263, 266, 267, 275 AEA 6-263, 10, 162, 195, 228 AEA 6-264, 10, 77, 141, 142, 228, 234, 264, 265, 267, 268, 280 AEA 6-265, 10, 199, 228, 235, 264, 275, 279, 280, 297 AEA 6-266, 10, 146, 181, 195, 197, 199, 220, 228, 235 AEA 6-267, 10, 228 AEA 6-268, 10, 228, 232, 233, 248, 251, 253, 256 AEA 6-269, 10, 182, 184, 228 AEA 6-270, 10, 228, 231, 233, 234, 236, 241, 246, 247, 249–251 AEA 6-271, 10, 148, 162, 178, 180, 182, 195, 228, 247, 281, 307, 319 AEA 6-272, 10, 228 AEA 6-273, 10, 241, 279, 292–298, 300, 301, 311, 315 AEA 6-274, 10, 241, 279, 294–298, 300, 315 AEA 6-275, 10, 241, 279, 294, 295, 315, 319 AEA 6-321, 183, 184

AEA 6-322, 183, 184 AEA 6-324, 184 AEA 6-325, 184 AEA 62-065, 9, 275 AEA 62-223, 9, 310, 317 AEA 62-300, 10, 318 AEA 62-317, 10, 318, 319 AEA 62-317r, 10, 318 AEA 62-318, 10, 318 AEA 62-318r, 10, 318 AEA 62-319, 10, 318 AEA 62-319r, 10, 318 AEA 62-320, 318 AEA 62-320r, 318 AEA 62-321, 318 AEA 62-321r, 318 AEA 62-322, 10, 318 AEA 62-322r, 10, 312, 318 AEA 62-373, 10, 322 AEA 62-373r, 9, 144, 202, 275 AEA 62-403, 275 AEA 62-403r, 275 AEA 62-495r, 10, 275 AEA 62-575, 319 AEA 62-575r, 4 AEA 62-783, 9, 224, 280, 281, 283–292, 294–296, 300, 304, 308, 314, 315, 319, 322 AEA 62-785, 9, 71, 85, 113, 144, 146, 201, 215–222, 257 AEA 62-785r, 9, 69, 71, 72, 78, 79, 82–95, 100, 111, 113, 123, 124, 134–137, 139, 140, 146, 205, 215, 222–224, 251, 255–257, 259, 264–266, 268 AEA 62-786, 9, 255, 257, 266 AEA 62-786r, 9 AEA 62-787, 9, 71, 95, 190, 234, 255, 257–260, 264, 266, 267 AEA 62-787r, 9, 68, 69, 71, 72, 78, 82–84, 95–106, 111, 123–125, 129–133, 139, 140, 205, 223, 255, 257–261, 264–266 AEA 62-788, 9, 190, 247, 251, 255, 262, 263, 267, 268 AEA 62-789, 9, 68, 69, 71, 72, 82–84, 106– 111, 116, 121–126, 128, 137–140, 143–145, 148, 183, 199, 201–205, 222, 223, 264–266 AEA 62-789r, 9, 69, 71–73, 82, 84, 106, 109, 111–118, 123, 125, 139, 140, 205, 223–225, 234, 255, 261–266 AEA 62-790, 9, 267 AEA 62-791, 9, 269–274, 315 AEA 62-792, 9, 267, 268

Index of Research Notes, Letters, and Further Documents AEA 62-792r, 9, 266, 267 AEA 62-794, 9, 144, 145, 201, 206–215, 222 AEA 62-795, 9, 280, 281, 296, 298–308, 310–315, 318, 319, 322 AEA 62-795r, 9, 280, 281, 298, 308–315, 317, 318, 322 AEA 62-796, 9, 255, 260, 266 AEA 62-797, 9, 267 AEA 62-798, 9, 144, 145, 199, 201, 205–206, 222–224, 235, 267, 268 AEA 62-799, 10, 268, 269, 315 AEA 62-800, 10, 144, 202, 275 AEA 62-802, 10, 144, 145, 201, 206–215, 222 AEA 62-803, 235 AEA 62-803r, 10, 267, 268 AEA 62-805, 10, 224, 234, 266, 267 AEA 62-807, 10, 144, 202, 275 AEA 63-026, 10, 144, 202, 275 AEA 63-026r, 10, 144, 202, 275 AEA 63-030, 10, 276 AEA 63-030r, 10, 276 AEA 63-042, 10, 322 AEA 63-042r, 10, 322 AEA 63-050, 10, 322 AEA 63-058r, 10, 281, 320–322 AEA 63-325, 10, 144, 202, 275 AEA 70-368, 178, 181 AEA 70-755, 18 AEA 71-539, 17, 18 AEA 74-593, 16 AEA 75-618, 177 AEA 75-661, 177 AEA 75-905, 179 AEA 75-946, 179 AEA 75-949, 177, 178

353

AEA 80-914, 10, 123 AEA 90-914, 123 AEA 97-242, 177 AEA 97-487, 10, 144, 148, 152, 154, 179–202, 204, 220, 221, 234, 236 AEA 97-488, 185 AEA 97-489, 185 AEA 97-490, 185 AEA 97-491, 185, 186 AEA 97-492, 185 AEA 97-493, 184 AEA 97-494, 184–186

E Einstein Besso manuscript, 3, 44

P Prague notebook, 4, 5, 7–9, 27, 29, 44, 63, 67–73, 78, 82–84, 109, 116, 118, 123, 125–142, 266

W Washington manuscript, 5, 9, 10, 55, 64, 73, 84, 106, 141, 143–146, 148, 150–154, 159, 162, 164, 168, 179–201, 204–206, 216, 220, 221, 223–225, 229, 231, 233, 234, 236, 264, 265, 281, 318

Z Zurich notebook, 3, 7, 33–41, 45