General Relativity Conflict and Rivalries : Einstein's Polemics with Physicists [1 ed.] 9781443887809, 9781443883627

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General Relativity Conflict and Rivalries

General Relativity Conflict and Rivalries: Einstein's Polemics with Physicists By

Galina Weinstein

General Relativity Conflict and Rivalries: Einstein's Polemics with Physicists By Galina Weinstein This book first published 2015 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2015 by Galina Weinstein All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-4438-8362-X ISBN (13): 978-1-4438-8362-7

“When I accompanied him [Einstein] home the first day we met, he told me something that I heard from him many times later: ‘In Princeton they regard me as an old fool.’ (‘Hier in Princeton betrachten sie mich als einen alten Trottel’.) […] Before he was thirty-five, Einstein had made the four great discoveries of his life. In order of increasing importance they are: the theory of Brownian motion; the theory of the photoelectric effect; the special theory of relativity; the general theory of relativity. Very few people in the history of science have done half as much. […] For years he looked for a theory which would embrace gravitational, electromagnetic, and quantum phenomena. […] Einstein pursued it relentlessly through ideas which he changed repeatedly and down avenues that led nowhere. The very distinguished professors in Princeton did not understand that Einstein’s mistakes were more important than their correct results. Einstein, during my stay in Princeton, was regarded by most of the professors there more like a historic relic than as an active scientist”. Leopold Infeld, “As I see It”, Bulletin of the Atomic Scientists, 1965, 9.



CONTENTS

Preface ........................................................................................................ ix Chapter One ................................................................................................. 2 From Zurich to Berlin 1. Einstein and Heinrich Zangger .......................................................... 2 2. From Zurich to Prague ....................................................................... 3 3. Back to Zurich ................................................................................... 9 4. From Zurich to Berlin ...................................................................... 12 Chapter Two .............................................................................................. 28 General Relativity between 1912 and 1916 1. The Equivalence Principle.............................................................. 28 2. Einstein's 1912 Polemic with Max Abraham: Static Gravitational Field.......................................................................... 37 3. Einstein's 1912 Polemic with Gunnar Nordström: Static Gravitational Field.......................................................................... 47 4. Einstein's 1912-1913 Collaboration with Marcel Grossmann: Zurich Notebook to Entwurf Theory .............................................. 50 5. Einstein's 1913-1914 Polemic with Nordström: Scalar Theory versus Tensor Theory ..................................................................... 84 6. Einstein's Polemic with Gustav Mie: Matter and Gravitation ...... 101 7. 1914 Collaboration with Grossmann and Final Entwurf Theory ............................................................................ 105 8. Einstein's Polemic with Tullio Levi-Civita on the Entwurf Theory ............................................................................ 117 9. Einstein's 1915 Competition with David Hilbert and General Relativity ...................................................................................... 131 10. Einstein Answers Paul Ehrenfest's Queries: 1916 General Relativity ............................................................... 172 11. The Third Prediction of General Relativity: Gravitational RedShift.................................................................. 182 12. Erich Kretschmann's Critiques of Einstein's Point Coincidence Argument ................................................................. 192 13. Einstein and Mach's Ideas ............................................................ 206 14. Einstein's Reaction to Karl Schwarzschild's Solution .................. 211

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15. The Fourth Classic Test of General Relativity: Light Delay ........ 239 Chapter Three .......................................................................................... 242 General Relativity after 1916 1. Einstein's 1916 Polemic with Willem de Sitter, Levi-Civita and Nordström on Gravitational Waves ....................................... 242 2. Einstein's Polemic with de Sitter: Matter World and Empty World ......................................................................... 267 3. Bending of Light and Gravitational Lens: Einstein and Arthur Stanley Eddington ...................................................... 285 4. Einstein's Interaction with Hermann Weyl and the Cosmological Constant ................................................................ 306 5. Einstein's 1920 Matter World, Mach's Ether and the Dark Matter...................................................................... 322 6. Einstein's 1920 Polemic with Eddington on de Sitter's World ..... 327 7. Einstein's Reaction to the Aleksandr Friedmann Solution ........... 331 8. Einstein's Reaction to the Georges Lemaître Solution ................. 334 9. Edwin Hubble's Experimental Results ......................................... 337 10. The Lemaître-Eddington Model ................................................... 341 11. Einstein and the Matter World: the Steady State Solution ........... 350 12. Einstein's Collaboration with de Sitter ......................................... 354 13. Einstein's Reaction to Lemaître's Big Bang Model ...................... 359 14. Einstein's Interaction with George Gamow: Cosmological Constant is the Biggest Blunder ............................ 366 15. Einstein, Gödel and Backward Time Travel ................................ 373 References ............................................................................................... 382 Index ........................................................................................................ 405



PREFACE

We can pose many thought-provoking questions in regards to Einstein's achievements: Was the theory of general relativity the invention of Albert Einstein, who would close himself off in an office with his violin, pipe and a pile of papers? Or, was it a culmination of Einstein's multilateral interactions with other scientists? And, to what extent did Einstein's loyal friend from school, Marcel Grossmann, contribute to the mathematics of the general theory of relativity? Other questions focus on the general topic of how scientists debate and, in the process, modify their ideas. Was the theory of general relativity a physical-conceptual theory containing innovations in a synthesis developed from his interactions with friends and colleagues? Was the theory of general relativity a product of debates and conflicts between Einstein and other scientists? In this book I present an approach that focuses on the work of an individual scientist, Albert Einstein, and his interaction with and response to many eminent and non-eminent scientists. According to this approach, the ongoing discussions between Einstein and other scientists have all contributed to the edifice of general relativity and relativistic cosmology. Mara Beller argues that, the scientists, with whom Einstein implicitly or explicitly interacted, form a complicated web of collaboration (Beller 1999). I have analysed the works of those scientists who were, in any way, connected to the development of the general theory of relativity and relativistic cosmology, focusing on their implicit and explicit responses to Einstein's work. This analysis has uncovered latent undercurrents, which could not have been exposed by means of tracking the intellectual pathway of Einstein to his general theory of relativity. The new interconnections and meanings disclosed have revealed the central figures who influenced Einstein during his development of the general theory of relativity and in the construction of the edifice of relativistic cosmology; this includes the themes, outlooks, suppositions and contributions of these scientists as opposed to Einstein's Weltanschauung (worldview).

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Furthermore, current history presupposes that all the efforts invested by physicists like Max Abraham, Gunnar Nordström, Gustav Mie and David Hilbert, which presented differing outlooks and discussions revolving around the theory of gravitation, were relegated to the background. Thus, those works that did not embrace Einstein's overall conceptual concerns (these primarily included the heuristic equivalence principle and "Mach's ideas") were rejected, and authors focused on Einstein's prodigious scientific achievements. I emphasise the limits of the simplistic hero worship narrative, and the associated problems tracing Einstein's intellectual pathway to the general theory of relativity. I demonstrate in this book that one should not discard Einstein's response to the works of Max Abraham, Gunnar Nordström, Gustav Mie, Tullio Levi-Civita, David Hilbert and others. On the contrary, Einstein's responses to these works constitute a dynamic (explicit or implicit) interaction that assisted him in his development of the general theory of relativity. The issue of a mutual interaction and inspiration profoundly touches central historiographic questions and topics, such as: What is the character and nature of innovations? Indeed, a penetrating look reveals a picture that attests to the fact that the general theory of relativity was the product of a much more complex process. Therefore, by combining the creative processes and discoveries of numerous scientists with that of Einstein, we can better understand their influence on Einstein's path to the general theory of relativity. In other words, the construction of the historical account should reflect Einstein's interactions with other scientists. By incorporating this perspective, the historical account of the development of the theory of relatively diverges from its focus on the creative process of the humble genius who never wore socks, and who would close himself off in an office, claiming "I will a little tink" (Hoffmann 1968; Regis 1987, 20). Rather, this account focuses on Einstein's active interactions with other scientists in Europe and later the United States and their influence on his work. This leads to the important conclusion that the general theory of relativity was not developed as a single, coherent construction by an isolated, brooding individual; instead, it illuminates the reality that general relativity was developed through Einstein's conflicts and interactions with other scientists, and was consolidated by his creative processes during these exchanges.

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After 1917, scientists Willem de Sitter, Sir Arthur Stanley Eddington, Aleksandr Friedmann, Georges Lemaître and several others found in Einstein's general theory of relativity an initial theory from which they developed cosmological models. They elaborated the theory of general relativity until it evolved into a new version that contained cosmological solutions and models not found in Einstein's 1915 and 1916 papers. By addressing Einstein's response, interaction and competition with these scientists, and the post-1916 responses of other scientists to his work, I wish to shed light on Einstein's way of thinking and organisation of ideas, as well as the impact he had on scientists with whom he interacted, all of which contributed to the construction of the general theory of relativity and relativistic cosmology. Two examples have been provided to support this claim: 1) In 1918 Felix Klein demonstrated to Einstein that the singularity in the de Sitter solution to the general relativity field equations was an artefact of the way in which the time coordinate was introduced. Einstein failed to appreciate that Klein's analysis of the de Sitter solution showed that the singularity could be transformed away. In his response to Klein, Einstein simply reiterated the argument of his critical note on the de Sitter solution. Einstein, however, was usually trusted as the authority on scientific matters. In 1917-1918 the physicist-mathematician Hermann Weyl's position corresponded exactly to Einstein's when he criticised de Sitter's solution; Weyl's criticism revealed the influence of Einstein's authority in physics even on first-rate mathematicians (like Weyl): In 1922, Erich Trefftz constructed an exact static spherically symmetric solution for Einstein's vacuum field equations with the cosmological term. The Trefftz metric represents a model for a spherical closed (finite) universe, a de Sitter static universe devoid of matter whose material mass is concentrated in just two spherical bodies on opposite sides of the world. Einstein identified a problem with this line element. In 1922 he demonstrated that Trefftz's solution contained a true singularity in the empty space between the two bodies. Consequently, time stands still in the de Sitter empty space between the two masses. This signifies there are other masses distributed between the two masses. Einstein said that Weyl had already shown that many masses existed somewhere in-between the two bodies. Indeed, to keep the two bodies apart at a constant distance (in a static closed world), Weyl had to introduce a true singularity at the mass horizon (somewhere in the empty space between the two bodies). He introduced the true singularity and concluded that a zone of matter exists

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between the two bodies. Weyl was misled by the apparent de Sitter singularity into believing that the mass in de Sitter's world is distributed on a mass horizon, and this induced him to introduce a true singularity. Weyl therefore omitted part of the space-time around the horizon and replaced it by the Schwarzschild interior solution of Einstein's field equations with the cosmological constant. Instead of removing the apparent de Sitter singularity, Weyl introduced a true singularity by joining two solutions: the de Sitter and Schwarzschild interior solutions (Goenner 2001, 111112). In March 1918, before publishing the book Space-Time-Matter, Weyl instructed his publisher to send Einstein the proofs of his book. In the same month, Weyl also instructed his publisher to send David Hilbert the proofs of his book. Hilbert looked carefully at the proofs of Weyl's book but noticed that the latter did not even mention his first Göttingen paper from November 20, 1915, "Foundations of Physics". Though Weyl mentioned profusely Einstein's works on general relativity, no mention was made of Hilbert's paper. Einstein received the proofs page-by-page from the publisher and read them with much delight and was very impressed. However, Einstein, an initial admirer of the beauty of Weyl's theory, now raised serious objections against Weyl's field theory. Einstein's objection to Weyl's field theory was Weyl's attempt to unify gravitation and electromagnetism by giving up the invariance of the line element of general relativity. Weyl persistently held to his view for several years and only later finally dropped it. 2) Einstein also seemed to influence Sir Arthur Stanley Eddington when he objected to what later became known as "black holes". In 1922, during discussion sessions at the Collège de France in Paris, Jacques Hadamard questioned Einstein about the Schwarzschild solution to his field equations and its practical relevance for astronomy. In Schwarzschild's solution a singularity exists at r = 0 (a quantity that becomes infinite). Hadamard was questioning what would actually happen in reality if, mathematically, the singularity could really become infinite in our world? Could this practically and physically occur? While it may not happen in our solar system, it may certainly be possible elsewhere in the universe. This question reportedly embarrassed Einstein. He said that if the radius term could really become zero or infinite (be singular) somewhere in the universe, then it would be an unimaginable disaster for his general theory of relativity. Einstein considered this a catastrophe, and jokingly called it, the "Hadamard catastrophe". Hence, according to Einstein, the

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Schwarzschild singularity, r = 0, characterised a catastrophic region. He did not think the "Hadamard catastrophe" was possible, and he did not want to think about the physical effects of this case. In 1939, Einstein repeated his previous claims, speaking clearly against the Schwarzschild singularity: He stated the impossibility of the Schwarzschild singularity, it did not exist in physical reality. Eddington, however, seemed to have been influenced by Einstein's viewpoint. In his controversy during the Royal Astronomical Society meeting of 1935 with Subrahmanyan Chandrasekhar, Eddington argued that various accidents may intervene to save a star from contracting into a diameter of a few kilometres. This possibility, according to Eddington, was a reductio ad absurdum of the relativistic degeneracy formula. Chandrasekhar later said that gravitational collapse leading to black holes is discernible even to the most casual observer. He, therefore, found it hard to understand why Eddington, who was one of the earliest and staunchest supporters of the general theory of relativity, should have found the conclusion that black holes may form during the natural course of the evolution of stars, so unacceptable. However, it is very reasonable that Eddington, who was one of the earliest and staunchest supporters of Einstein's classical general relativity, found the conclusion that "black holes" were so unacceptable, because he was probably influenced by Einstein's objection to the Schwarzschild singularity. Einstein's 1929 new unified field theory was based ondistant parallelism. In May 1929, Einstein received two letters from Élie Joseph Cartan pointing out that Einstein's basic mathematical idea of distant parallelism had been previously worked out by him (Cartan) in great detail in several publications, and that he had even explained the idea to Einstein when they met briefly in Paris in the spring of 1922 at Jacques Hadamard's home. Cartan explained to Einstein that in his 1922 articles devoted to the new theory of general relativity he had introduced the notion "teleparallelism". It was a special case of a more general notion of the Euclidean connection that he (Cartan) had already advanced and published when Einstein gave his 1922 lectures at the Collège de France. Einstein confessed that he had understood nothing of all the explanations that Cartan had given him in Paris in 1922, still less was it clear to him how they could be made of any use to physical theory. Einstein nevertheless told Cartan that the manifolds he himself had used in his unified field theory were a special case of those studied by Cartan.

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Einstein was a rebel and cynic. Characteristically, he was personally unconcerned about the priority question and immediately sprang into action, insisting that this issue had to be rectified. He puzzled over what to do and what to write that would satisfy all just claims. Finally, on Einstein's invitation, Cartan wrote an historical note explaining distant parallelism and the matter was settled. Indeed in 1929, Einstein wrote an exposition of his unified field theory for the journal Mathematische Annalen. It was published together with Cartan's essay about the history of distant parallelism. Einstein, however, was more than delighted to find a very able mathematician such as Cartan interested in his unified field theory, because almost all physicists believed that a fundamental description of physical reality was not possible on the basis of his unified field theory. He employed Cartan's mathematical abilities to answer a series of mathematical queries regarding distant parallelism and Einstein's proposed field equations. Cartan was very pleased that Einstein introduced him to his new research on his unified field theory and thanked him for the trust he put in his abilities as a mathematician. In this book I discuss Einstein's work on unified field theory that deals with the synthesis of gravitation and electromagnetism. I do not discuss the quantum aspects of the theory. In 1917 and 1918 the mathematical tools of classical general relativity were elaborated by Levi-Civita and Weyl. They introduced the concept of parallel transport in a Riemannian space as a means of giving an invariant interpretation to the curvature of space. One thinks of particles as moving along geodesic lines in curved, four-dimensional space-time. An affine connection defines the amount of curvature of geodesic lines. According to Einstein, the general theory of relativity assumes its simplest form when expressed in a generally covariant form. However, in 1923 Élie Cartan formulated Newtonian theory of gravity as a geometric-dynamical theory and provided a generally covariant formulation of Newtonian gravity in space-time – in terms of an affine connection. It is thus possible to redo Newtonian gravity as a theory of curved space-time. Furthermore, the Newton-Cartan theory is written in a generally covariant form as in Einstein's theory of general relativity. Suppose we possess a generally covariant formulation of both theories – Einstein's general relativity and Newtonian non-relativistic theory.

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Einstein explained that Newtonian physics ascribes independent and real existence to space and time, i.e. Newtonian physics assumes a fixed, nondynamical background space-time structure. General relativity, unlike Newtonian physics and even special relativity, is a backgroundindependent theory. In general relativity, "there is no such thing as an empty space, i.e. a space without field". Furthermore, space-time does not claim existence on its own, but only as a structural quality of the field (Einstein 1952, 155, 176). Space-time is dynamical, and ceases to exist when a singularity is reached. Hence Einstein's general theory of relativity is a dynamical background-independent theory. John Stachel calls our attention to the fact that in general relativity, the behaviour of measuring rods and clocks (chrono-geometry, i.e. in general the metric) is determined by the inertio-gravitational field. Both the chrono-geometrical and the interio-gravitational structures are dynamical fields. Wherever there is a chrono-geometric structure there is always also an affine inertio-gravitational structure. Chrono-geometrical and inertiogravitational structures obey field equations coupling them to each other and to all other physical processes. Thus physical processes do not take place in space-time. Space-time is just an aspect of the totality of physical processes. In Newtonian physics, however, the measurement of time and space is unaffected by the presence of an inertio-gravitational field. We thus define compatibility of chronometry and geometry with the inertiogravitational field (Stachel 2002, 2007b, 429). In conclusion, I would like to present a more balanced historical account of the development of the general theory of relativity and relativistic cosmology; an account that will reflect the complicated interactions leading to the solidifying of Einstein's general theory of relativity and cosmology. The main conclusion of my essay is that if we take Einstein's interactions with his colleagues and friends into account, we may arrive at a revised historical explanation of Einstein's general relativity and of relativistic cosmology. Carefully examining the historical progression that led to the theory of relativity reveals pluralistic debates and interactions, based on different aesthetic, philosophic and scientific presuppositions that were dominant in influencing Einstein during his development of the general theory of relativity. In this book I examine Einstein's interactions and debates with several scientists between 1912 and 1948.

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It should be noted that the number of physicists actively engaged in research in general relativity remained small between 1925 and 1955. Referring to those years, Peter Bergmann once noted: "You only had to know what your six best friends were doing and you would know what was happening in general relativity" (Pais 1983, 268). Jean Eisenstaedt has characterized the period that extended from 1925 to 1955 as the "low water mark" of general relativity (Eisenstaedt 1989). Until 1955 the relativity community had a strange mix: There was a small group of specialized physicists, Einstein's friends and assistants and a small group of specialized astronomers/cosmologists and mathematicians (who became friends and colleagues of Einstein's). This state of aơairs lasted until the 1955 Bern conference marking the Jubilee (50th anniversary) of Einstein's theory of special relativity (Kennefick 2007, 174-175). Einstein had an aversion to pomp and ceremony and did not feel the need to celebrate his achievements. He did not live long enough to participate in this conference. Indeed, the conference was held in July 1955 (shortly after Einstein died in April 1955). Although it was decided to have the conference in Bern where Einstein had published his special relativity, the papers presented were largely devoted to general relativity: cosmology, unified field theory and methods of solutions of the field equations. The conference was attended by physicists interested in working in relativity theory and it would later come to be known as GR0, the zeroth conference in a series which continues to this day. This book is loosely based on my PhD thesis that was written between 1995 and 1998 at the Hebrew University of Jerusalem. Although little of it remains, I would like to thank the late Prof. Mara Beller, who was inspirational as my PhD supervisor. The book has benefited greatly from my endless conversations with her at the Van Leer Jerusalem Institute during my PhD studies. I would also like to thank Prof. Asa Kasher of the Tel-Aviv University for believing in me. Last, but not least, many thanks to CSP for supporting this project, and especially to Sophie Edminson and Amanda Millar for their patience and kind help in completing the present book and my first book, Einstein’s Pathway to the Special Theory of Relativity, CSP, 2015. Galina Weinstein, November 2015

CHAPTER ONE FROM ZURICH TO BERLIN

1. Einstein and Heinrich Zangger Working alongside Einstein at the patent office was his close friend Michele Besso. During his employment there, Einstein enjoyed considerable freedom in what he called the worldly cloister, where he spent considerable time ruminating and pondering his best ideas; brooding upon his theories, and inventing his most beautiful concepts. During this time at the patent office, he contemplated the problem of gravitation, and in doing so, invented a new thought experiment: A man falling freely from a roof under the influence of gravity. He published his first paper on the topic on December 4, 1907, "On the Relativity Principle and the Conclusions Drawn from It" (Einstein 1907). When no complicated mathematics entered into the theory, the extension of the 1905 special relativity principle apparently turned out to be quite natural and simple. Guided by Galileo's principle of free fall, Einstein postulated the principle of equivalence – he assumed the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system. Using this new principle, with physical reference systems and measuring rods and clocks, he arrived at new results: bending of light rays in a gravitational field, and gravitational redshift (Einstein 1907, 411-462). Einstein understood the importance of Galileo's law of free fall: All bodies experience the same acceleration in a gravitational field, which can be formulated as the law of the equality of inertial and gravitational mass. He then connected between the Newtonian result of the equality of the inertial mass and the gravitational mass (which was quite accidental from the point of view of classical mechanics) and his principle of equivalence. Equality of gravitational and inertial mass was, therefore, essential for Einstein and played a crucial role in constructing his theory on the basis of the equivalence principle.

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Michele Besso introduced Einstein to Heinrich Zangger, a doctor of forensic medicine, the originator of shock treatment, and director of the Forensic Medicine Institute of the University of Zurich. Zangger was devising a new, experimentally simple method for determining Avogadro's number as an outgrowth of his research on milk as a colloidal system. The method consisted of the microscopic observation of the irregular path, due to Brownian motion, of small mercury droplets falling through a liquid. For advice on the theoretical analysis of this experiment, Zangger turned to a mechanical engineering professor at the Swiss Polytechnic, Aurel Stodola, a specialist in the field of steam turbines and thermodynamics. Like Zangger, he had wideranging interests that included physics. A letter from Michele Besso to Zangger, written more than two decades later, suggests that Stodola may have learned of Einstein from Besso, who was one of his students. Aware of Einstein's work in Brownian motion, Stodola directed Zangger to him. Zangger consulted Einstein at the patent office in Bern, where the physicist was then working.Einstein and Zangger became close friends. In time, Einstein, his mother, and the whole Einstein family would be among Zangger's patients (Medicus 1994, 459; Seelig 1954, 129; 1956a, 109). Einstein would write to Zangger about his personal and professional difficulties. He would also tell him his opinions concerning the controversies and polemics he had with physicists like Max Abraham and David Hilbert (see Chapter 2, Sections 2 and 9).

2. From Zurich to Prague From 1909 Einstein was finally accepted into the academic world, as associate professor at the University of Zurich. This appointment brought him, for the first time, a position with the conventional view of a certain public prestige. Between 1907 and 1911, Einstein was occupied with research on radiation and the quantum of light, and did not publish papers on gravitation. Nonetheless, he was pondering the problem of gravitation from 1907, during the four years he was working at the University of Zurich, as he himself explained it during the 1933 Glasgow talk, "The Origins of the General Theory of Relativity". Einstein explained that between 1908 and 1911 he considered his 1907 principle of equivalence, and checked whether this was an indication for an extension of the special principle of relativity to coordinate systems in

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non-uniform motion with respect to each other, once one wanted to reach a causal theory of the gravitational fields (Einstein 1933; 1934, 252; 1954, 287). However, in spring 1911 Einstein left his comfortable position at the University of Zurich and moved to the German University of Prague, KarlFerdinand University. Philipp Frank explains why Einstein took this step (Frank, 1949, 130-131, 135-137; 1947, 75-79). From the financial point of view, the position of an associate professor in the University of Zurich was unattractive; his income was no larger than it had been at the patent office. Also, he had to pay for things which gave him no pleasure for life, but which were required by this position. Although Einstein loved Zurich, he was too busy teaching to undertake decent research at the University of Zurich. Beyond regular teaching, he was occupied with training students, administrative duties, and financial problems. Thus, progress on his research on gravitation slowed, and he concerned himself with the quantum problem and heavy administrative and teaching duties. He would sometimes joke that in the relativity theory he could well put a clock at every point in space (in this way synchronising distant clocks in one reference frame); however, in reality he found it difficult to set up only one clock at one point in space, because he had no time for any research. Prague boasted two universities, one Czech University and the other a German University. In the fall of 1910, a vacancy arose in the teaching chair of theoretical physics in the German University. The German University was certainly not the center of research in physics, and while Einstein and his first wife Mileva Mariü were not thrilled about moving countries, he looked forward to this position and was eager to accept it: This position offered Einstein, for the first time in his life, a full professorship with adequate salary and significant research time. Indeed, the position was first offered to Einstein. The decision was formally made by the Kaiser of Austria. He deferred, however, the final decision to the Ministry of Education. Physicist Anton Lampa, from the Ministry of Education, was in charge of the selection. Lampa's philosophical Weltanschauung (view-point) was for the most part

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influenced by his teacher, Ernst Mach's, positivistic views. Mach was the first rector of the German University in Prague. Philipp Frankreports that Lampa was ambitious, and he tried to appear as a man who cared about ethical and modern educational ideas. He wanted to advance freedom of teaching; but there was a big gap between his high ideals and his real scientific abilities. It had always been his dream to climb to the realms of the extraordinary and the genius. Lampa knew he was not a genius. He was thus willing to accept the presence of more important people who could follow Mach. When he thought of candidates for the position to follow Mach's tradition, he had two physicists in mind: Einstein (who never considered himself a genius) and Austrian physicist, Gustav Jaumann (who considered an unrecognised or neglected genius). Mach denied the reality of atoms, but Einstein (in his work on the Brownian motion) objected to Mach's stand. Einstein would later create his theory of gravitation in Prague, and would be an ardent advocate of Mach's ideas. However, this was not anticipated by Lampa and others, because as a candidate for the position, his work on gravitation was still in its infancy. At first, the candidates were classified on the basis of their achievements. Einstein had greater achievements to his credit than Jaumann, and so Einstein was the preferred candidate. The Ministry of Education, however, first offered the position to the "neglected genius", Jaumann, since the government preferred to appoint Austrians rather than foreigners. Jaumann, offended by Einstein's preferred status, rejected the offer. He told the minister, that if Einstein was the preferred candidate with greater achievements, then he would have nothing to do with a university that ignores the true merit. The government thus overcame its aversion to foreigners and offered the position to Einstein. In Einstein's case, therefore, the background to this appointment was typical in many ways, because the personal sympathies and antipathies of the deciding people played a certain role in the decision.1

  1

A similar incidence had already occurred to Einstein in 1908 with the position at the University of Zurich. Alfred Kleiner had persuaded Friedrich Adler to accept the position. The faculty at the University of Zurich was not eager to accept Einstein the Israelite, who had no understanding of how to get on with important people. So they accepted Adler first; however Adler had found a more interesting

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Lampa received a letter of recommendation from Max Planck who wrote on Einstein's work on the theory of relativity that it "probably exceeds in audacity everything that has been achieved so far in speculative science and even in epistemology; non-Euclidean geometry is child's play by comparison". Recommending Einstein for the position in Prague, Planck went on to compare Einstein to Copernicus (Pais 1982, 192). Einstein was appointed to the German University of Prague. However, as a Jew not everything went smoothly, and he was forced to confront his religious status. In 1896 he renounced his legal affiliation to the Jewish religious community and thus become konfessionslos (without religion). Imperial Austrian authorities would not accept claims to be konfessionslos, which he had signified on his Swiss citizenship a decade earlier (in 1901). To avoid this difficulty, Einstein stated he was of the Jewish religion, and in the questionnaire that he had to fill out he simply wrote his religion was "of Mosaic faith", as Jewish was then called in Austria (Stern 1999, 102103). Einstein moved to Prague as a full professor at end of March 1911. Just before that in January 1911 he received an invitation from Hendrik Antoon Lorentz to a lecture in Leiden. In 1892 Lorentz advanced a version of the theory of the electron, based on Maxwell's electromagnetic theory, in order to explain electromagnetic and optical phenomena in bodies at rest and in motion. Lorentz demarcated ponderable matter from the imponderable luminiferous stationary immobile ether. Lorentz published his electron theory in several papers and in his seminal work, Attempt at a Theory of Electrical and Optical Phenomena in Moving Bodies (Lorentz 1895), which Einstein read before creating the special theory of relativity. Although Einstein gave up the immobile ether, he always mentioned Lorentz and his influence on his thought. Einstein corresponded with Lorentz in 1909 on his research on the quantum of light and radiation, and he appreciated Lorentz as a profound thinker (Einstein to Jakob Laub, May 17, 1909, CPAE 5, Doc. 160). He was extremely excited to meet Lorentz for the first time, and he travelled with Mileva to Leiden. Einstein engaged in long conversations with Lorentz after his lecture. Bernard Cohn interviewed the elderly Einstein on Sunday morning in April 1955, two weeks before he died. According to Cohn, Einstein met Lorentz in Leiden through Paul Ehrenfest. He

  job at the German Museum in Munich. Upon Adler's suggestion, the position was awarded to Einstein instead.

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remarked that he had admired and loved Lorentz perhaps more than anyone else he had ever known, and not only as a scientist (Cohn 1955). Frank reports that when Einstein arrived in Prague he was certainly unlike the average professor at the German University. He was konfessionslos but became Mosiac (Jewish); he was married to a Slav wife, and was suddenly plunged into a milieu where nationality, race, and religion were burning issues; and he was certainly a little extraordinary among the average professors at the German University in Prague. Everyone was curious to meet Einstein, whose reputation as an extraordinary genius preceded his arrival. In Prague it was customary for a newly arrived faculty member to visit all his colleagues. Initially, Einstein accepted the advice of Lampa and the mathematician Georg Pick to follow this tradition, and he began his more than forty visits and, at the same time, toured the old city of Prague. However, Einstein was a rebel and cynic; he also had an aversion to pomp and ceremony. He spurned the rules of bourgeois life and opted for bohemian alternatives. He soon began to feel the banality of the boring conversations about trivial matters that took place during these visits, and so refused to continue doing them. Consequently, the professors whom he had not visited were offended; for the notorious professor had already visited several others. Frank, who later replaced Einstein, explained that the professors whom Einstein did not visit thought he was capricious. Frank said that the true explanation was that these colleagues lived in urban areas of the city that did not interest Einstein, or their names were too far back in the faculty directory. It was pure coincidence. Instead of wasting time with formalities, in Prague, Einstein came back to work on the problem of gravitation. This was typical of Einstein (Frank, 1949, 139-140; 1947, 79). Generally, Einstein enjoyed his time in Prague, even though life was not as pleasant as it had been in Zurich. He felt alien in Prague, and all drinking water had to be boiled first. The population for the most part spoke no German and was strongly anti-German. The students at the university, too, were less intelligent and industrious as in Zurich, but Einstein said he had a fine Institute with a magnificent library. In 1911, in Prague, when Einstein returned to intensive work on gravitation, he understood that one could test the theory by experiment. He published his first paper on the topic in June 1911 in the Annalen der

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Physik, "On the Influence of Gravitation on the Propagation of Light". He began his paper by saying that he was returning to the theme of gravitation, because his previous presentation of the subject did not satisfy him. However, the main reason for returning to the subject was his realisation that the consequence from his 1907 gravitation theory – rays of light, passing close to the Sun, are deflected by its gravitational field – was observable through experimental examination. From September 1911, Erwin Freundlich took upon himself to test the bending of light (Einstein to Erwin Freundlich, September 21, 1911, January 8, 1912, CPAE 5, Doc. 287, 336). In the body of the 1911 paper Einstein returned to Galileo's experimental principle of free fall. He was guided by this experimental principle towards formulating the more mature equivalence principle of 1911. He still did not leave the comfortable framework of the physical frame of references, the system of measuring rods and clocks that gave physical meaning for points in space-time. His new gravitation theory was thus a coordinate-dependent theory (Einstein 1911, 898). During the months he worked in Prague, he also published two papers discussing the theory of the static gravitational field, the first in February and the second in March 1912: "The Speed of Light and the Statics of the Gravitational Fields" and "On the Theory of the Static Gravitational Fields". In his theory on the static gravitational field Einstein was guided by Galileo's principle of free fall, the equality of inertial and gravitational mass, and the equivalence principle; he also used the same method of research he had used with special relativity: the coordinate-dependent methodology. His 1911 paper concluded that the velocity of light in a gravitational field is a function of the place. Accordingly, in February 1912, within static gravitational theory, Einstein replaced the gravitational potential by the variable speed of light. Thus, he offered a theory of static fields which violated his own light postulate from the special theory of relativity. In March 1912, he discovered that the equivalence principle is not universally valid: The principle of equivalence is valid only locally. Einstein's first paper on static gravitational fields also dealt with the uniformly rotating disc. Until 1912, Einstein discussed uniformly accelerated systems; the February 1912 paper was the first time he tried to extend the relativity principle to uniformly rotating systems.

From Zurich to Berlin

9

Einstein could not yet formulate his 1912 theory of static gravitational field in terms of Minkowski's four-dimensional space-time formalism. It seems that the reason for not using Minkowski's four-dimensional formalism was Einstein's inability to incorporate the equivalence principle and his basic idea of gravitational potential, which was replaced by the variable speed of light, into Minkowski's space-time formalism. According to Minkowski, the speed of light is constant. However, any gravitation theory that provides the speed of light in an empty space with an unchangeable value, that is to say, assumes the constancy of the velocity of light, cannot explain the deflection of light near the Sun. After 1912, Einstein was able to use Minkowski's formalism in his theory of gravitation. He would show that light rays that moved in straight lines signified an affiliation with Euclidean geometry, and deflected light rays signified an affiliation with non-Euclidean geometry.

3. Back to Zurich Hardly six months after Einstein's departure from Zurich, and his friend and eternal lifesaver, Marcel Grossmann, whose father helped Einstein obtain the position in the patent office in Bern, asked Einstein whether he would be interested in a post at the Eidgenössische Polytechnische Schule (Swiss Federal Polytechnic School). Einstein had been a student at the school, which he used to call the Zürcher Polytechnikum (the Zurich Polytechnic School or Zurich Polytechnic). The institute was later named the Eidgenössische Technische Hochschule [ETH] (Swiss Federal Institute of Technology). While a student at the Polytechnic, Einstein often skipped classes, and failed to attend all required lectures, and used Marcel Grossmann's notes for studying before sitting for an examination. Grossmann had become a professor at the Zurich Polytechnic in 1907 and was appointed dean of mathematics in 1911. Einstein was happy to exchange Prague for Zurich. He was most certainly inclined, in principle, to accept a teaching position in theoretical physics at the Polytechnic. Pierre Weiss from the Polytechnic approached two very important scientists for recommendations, scientists whom Einstein met in the first Solvay congress in Brussels in 1911 – Marie Curie and Henri Poincaré (Einstein to Grossmann, November, 18, 1911, CPAE 5, Doc. 307; Seelig 1954, 154, 162-163; 1956a, 129, 133).

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Zangger also fought for Einstein's appointment to the Polytechnic. Zangger learned, probably from Grossmann, that Einstein was unhappy in Prague and wanted to return to Zurich. Although he was a professor at the Medical School of the Cantonal University, Zangger immediately employed his good connections with the federal government on his friend's behalf. Not all of the professors at the Polytechnic were enthusiastic about the prospect of Einstein's appointment, because it did not fit with their own research priorities and personal ambitions. In particular, the school president, Robert Gnehm, a former chemistry professor, was not in favour of this appointment; although he admitted that Einstein was an eminent scientist, Gnehm felt that he would be an unnecessary luxury. He argued that theoretical physics was already well covered, observing that in the preceding year, for example, there had been courses on the theory of absolute magnetic and electric measurements, the theory of alternating currents, electrical oscillations, electro-mechanics, experimental radioactivity, and ionization and radioactivity. He furthermore noted that Einstein was not a good teacher and that, as a physicist, he would require laboratory space. This was not quite true, since Einstein was occupied with basic principles, that is to say, theoretical physics. In response, Zangger directly contacted one of the federal councilors, Ludwig Forrer, whom he knew personally. Zangger, who had attended several hours of Einstein's lectures agreed he was not a smooth orator, and therefore not a good teacher for students who were too lazy to think and who only wanted to memorise their work. However, he felt that high level students would find in Einstein a firstclass teacher (Medicus 1994, 460-461). Thus Einstein was finally elected to the office of Professor of Theoretical Physics in the Zurich Polytechnic. Einstein was a celebrity by the time he left Prague. Newspapers speculated about reasons for his short sixteen-month stay in Prague. The political landscape in Austria-Hungary was compared to the situation in Germany. However, Einstein was returning to Zurich not Germany. There were also suggestions that because Einstein was a Jew he had been treated badly by the education authorities in Vienna and therefore did not wish to stay in Austria. Einstein sent a letter to the ministry of Vienna saying he had no cause for dissatisfaction in Prague. His decision to leave Prague was due solely to the fact that when he left Zurich he promised that he would be

From Zurich to Berlin

11

pleased to return there under acceptable conditions. In August 1912, Einstein returned to Zurich as a full professor with a top salary grade to the Physical Institute of the Federal Polytechnic, an institute that he cherished. After arriving back in Zurich in 1912, Einstein stopped using a variable speed of light to describe the gravitational field, as he had attempted to do in 1912 in Prague. Instead, he turned to the metric tensor field, a mathematical object of ten independent components that characterize the geometry of space and time. Einstein needed Marcel Grossmann's help to proceed in his search for a gravitational theory at the basis of which was the fundamental metric tensor. Einstein asked Grossmann to help him solve this gravitational problem by providing him with mathematical tools. Grossmann was happy to collaborate on the problem of gravitation with Einstein (Einstein 1955, 15-16). Grossmann searched the literature, and brought the works of Bernhard Riemann, Gregorio Curbastro-Ricci, Tullio Levi-Civita and Elwin Bruno Christoffel to Einstein's attention. With Grossmann's help Einstein searched for gravitational field equations for the metric tensor, which stayed covariant under non-linear coordinate transformations. The law of motion of material points in the gravitational field was provided by the geodesic line equation. Einstein detailed his struggles with the new mathematical tools Grossmann brought him, in a small blue notebook – named by scholars the Zurich Notebook. Einstein became fascinated with Riemann's calculus, and he filled 43 pages of this notebook with calculations. He continued to receive new mathematical tools from Grossmann, whose name he jotted in the notebook to indicate which tensors were received from him. Grossmann's name was written on top of one of the pages, where Einstein considered candidate field equations with a gravitational tensor constructed from the Ricci tensor; an equation Einstein would return to in his November 4, 1915, paper on general relativity. However, as indicated in the Zurich Notebook, he finally chose non-covariant field equations. Einstein's collaboration with Grossmann led to two joint papers on these limited-covariant field equations: The first of these was published before the end of June 1913. The second, published almost a year later. The first paper was titled "Outline of a Generalised Theory of Relativity and of a Theory of Gravitation", and referred to by scholars as the Entwurf (Outline) paper (Einstein and Grossmann 1913). A remark to the paper, written by Einstein, contained the well-known Hole Argument.

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Einstein left Zurich in March-April 1914; this ended his collaboration with Grossmann.

4. From Zurich to Berlin Einstein did not stay long in Zurich. In January 1913, Fritz Haber thought of bringing Einstein to the Kaiser Wilhelm Institute of Physical Chemistry and Electrochemistry in Berlin. The Kaiser Wilhelm Society was an organisation created in 1911 by Kaiser Wilhelm II. Haber wrote to Hugo Krüß, a worker at the Prussian Ministry of Education, that Einstein's closest colleagues were Max Planck, Emile Warburg, and Heinrich Rubens, who also worked in disciplines related to his specialty (Fritz Haber to Hugo Krüß, January 4, 1913, CPAE 5, Doc. 428). Einstein was still unaware of this state of affairs in Berlin (the thought of bringing him to Berlin). At the time, he was at the Zurich Polytechnic working on his new gravitation theory and troubled by various calculations. He was attempting to extract expressions of broad covariance from the Ricci tensor while imposing coordinate conditions. His disappointment led him to attempt establishing field equations while starting from the requirement of the conservation of momentum and energy. Still, Einstein did not fully relinquish. He was trying to find a way to recover the new field equations from the gravitational tensor, the Ricci tensor he had extracted from the Riemann tensor. But, he found it impossible; the next pages in his notebook indicate he was led to noncovariant field equations, which he also established through energymomentum considerations. Einstein and Grossmann very likely had already prepared a paper for submission relating these non-covariant field equations (the Entwurf theory). The time now was late spring 1913. At the same time, in Berlin, between January and May 1913, the emphasis seemed to have shifted from Haber's original proposal. By late spring, Max Planck and Walther Nernst had modified Haber's proposal, combining the idea of Einstein's membership in the Academy of Prussian Sciences with the prospect of his directorship of the Kaiser Wilhelm Institute of Physics. Einstein would remain in Haber's institute until 1917, and a special membership in the Academy of Prussian Sciences was conferred to him. To ensure Einstein's election to the Prussian Academy, the four most important German physicists and members of the Academy – Planck,

From Zurich to Berlin

13

Nernst, Rubens and Warburg submitted a request to the Prussian Ministry of Education (Seelig 1954, 172-173; 1956a, 143-144). They first announced in the physical-mathematical class of the Prussian Academy of Sciences that they would submit a proposal for membership at the next session. The identity of the candidate was not given. Two weeks later they proposed Einstein for election as a regular member of the Academy. Max Planck, the presiding secretary of the class, read aloud a text of proposals to the physical-mathematical class on June 12, 1913. The text stated (Proposal for Einstein's Membership in the Prussian Academy of Sciences, CPAE 5, Doc. 445; Doc. 1 in Kirsten and Treder 1979, 96-97): Einstein was born in March 1879 in Ulm, raised in Munich, was a citizen of Zurich from 1901, and was employed as a technical expert in the patent office from 1902 to 1909. In 1905 he was awarded his doctoral degree from the University of Zurich, habilitated in 1908 in Bern, accepted an appointment as extraordinary professor of theoretical physics at the University of Zurich in 1909, and an ordinary professor at the German University in Prague the following year. From there he returned to Zurich, to the Polytechnic in 1912. Planck noted that thanks to Einstein's papers in the field of theoretical physics, published for the most part in the Annalen der Physik, Einstein achieved at a young age, within the circle of his specialty, a worldwide reputation. His name became widely known in his famous 1905 treatise on the electrodynamics of moving bodies, which established the principle of relativity. However, Planck informed the attendants that as fundamental as this idea of Einstein's has proved to be for the development of the principles of physics, its applications were, for the present, still at the very limit of the measurable. Planck mentioned Einstein's tackling of other central questions that proved to be much more meaningful for applied physics. He was the first to demonstrate the significance of the quantum hypothesis for the energy of atomic and molecular motions; he derived from this hypothesis a formula for the specific heats of solid bodies. Planck summarised his proposal to the physical-mathematical class by saying that, among the major problems with which modern physics was so

14

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rich, there was no one in which Einstein did not take a position in a remarkable manner. However, Planck added that Einstein in his speculations, occasionally, overshot the target, as for example in his light quantum hypothesis. This should not be counted against him he continued; because without taking a risk, even in the most exact science, one could not be driven to real innovation. In 1905, Planck was co-editor of the Annalen der Physik, and he accepted Einstein's paper on light quanta for publication, even though he disliked the idea of light quanta. In regard to Einstein's theory of gravitation at that time, Planck commented that Einstein was working intensively on his new theory of gravitation; with what success, only the future would tell. Planck ended his proposal by saying that he and the three undersigned members (Nernst, Rubens, and Warburg) were aware that their proposal to accept so young a scholar as a full member of the Prussian Academy was unusual; but they believed not only that the unusual circumstances adequately justify the proposal, but also that the interests of the academy really required that the opportunity that now manifested itself to obtain such an extraordinary power be taken in its full possibility. Planck thought that even if they could not guarantee the future, they could be convinced that the recommended previous accomplishments of the nominee fully justified Einstein's appointment to the most distinguished state scientific institute. Additional evidence indicated that the entire world of physics would consider Einstein's entrance to the Berlin Prussian Academy of Sciences as a special gain for the academy. Louis Kollros, a former classmate of Einstein, wrote that, Planck and Nernst speculated about Einstein as an award-winning chicken-hen, but Einstein was worried he might not be able to lay any more eggs (Kollros 1955, 29-30); because he was probably worried about not being able to find generally covariant gravitational equations. Planck, however, had the right intuition. Already in 1905, he accepted Einstein's relativity paper for publication in the Annalen. He was also the first known scientist to respond to it in 1906. Planck had the feeling that the principle of relativity was the right direction. Now Planck believed that Einstein was the right person. He would not always believe in Einstein's new theory of gravitation, but his first intuition was to bring him to Berlin.

From Zurich to Berlin

15

Walter Nernst took the occasion of the physical-mathematical class meeting on June 12, 1913, to make a confidential announcement that Einstein's salary might be raised to 12,000 marks, thanks to a personal commitment by Leopold Koppel (a financer from the board of directors of Haber's institute) to donate 6,000 marks a year for a 12-year period to the Academy; the latter expressed his willingness to Haber to help defray the costs of an appointment for Einstein (Haber to Krüß, January 4, 1913, and Proposal for Einstein's Membership in the Prussian Academy of Sciences, June 12, 1913, CPAE 5, Doc. 428, note 5, Doc. 445, note 2; Kirsten and Treder, 1979, 97). Meanwhile in Zurich, Einstein was fully immersed in his new Entwurf gravitation theory. His joint paper with Grossmann – the Entwurf paper – was published before the end of June 1913 (Einstein and Grossmann 1913). In June 1913, Michele Besso came from Gorizia, near Trieste to visit Einstein in Zurich, and they both tried to solve the new Entwurf field equations to attain the perihelion advance of Mercury in the field of the static Sun in the Einstein-Besso manuscript. During the same month, Einstein was visited By Paul and Tatyana Ehrenfest from Leiden. They stayed in a pension in Zurich and spent a great deal of time with Einstein and his family. Ehrenfest met Grossmann and Besso during this time (Klein 1970, 294). Einstein was thus surrounded by two of his best friends – Besso and Ehrenfest – while he pursued solutions to the Einstein-Grossmann Entwurf field equations, and Mercury's perihelion precession problem. On July 3, 1913, the physical-mathematical class had voted twenty-one to one in Einstein's favour, and a week later the plenary session debated the issue. On July 11, 1913, Planck and Nernst caught a train from Berlin to Zurich, at Koppel's suggestion. They went to Zurich to personally influence Einstein in favour of their plan. Einstein, who at aged thirty-four was younger than them and had only garnered eight years' academic teaching experience, was already being offered positions from universities throughout the world (Frank, 1949, 178-179; 1947, 107-108). Planck questioned Einstein on his current research, to which Einstein described general relativity as it was then (the Entwurf theory). Many years later Einstein told Abraham Pais that Planck had advised him, as an older friend, against this theory, for he would not succeed; and even if he did, no one would believe him (Pais 1982, 239).

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Planck and Nernst came to Einstein with the following plan: A research institute for physics that did not yet exist, and there was no chance that such institute would be founded in the near future. Einstein, however would be (in 1917) the first director of this institute in preparation. In the meantime he would lead the research in other institutes, where physics has been carried under his instruction. Einstein was also to become a member of the Royal Prussian Academy of Science in Berlin. It was considered a great honor to be a member of this corporate body, and many significant professors from the University of Berlin never achieved membership. For most, membership was honorary, and did not provide a viable stipend. However, a few positions existed that were supported by foundations (Koppel's) with large salaries, so that they could be undertaken full-time. Such a position was offered to Einstein. Moreover, Einstein was the youngest member of this prestigious academy. Both in the Prussian Academy and in the Kaiser Wilhelm Institute, Einstein's main professional occupation should be the organisation of research. He was also to have the title of professor at the University of Berlin, but there were no obligations, except the right of lecturing as much or as little as he liked. He should have nothing to do with the administrative work of the university, with examinations, the appointment of new professors, etc. His only obligation was to be active in organizing research in the Prussian Academy and the Kaiser Wilhelm Institute. He could teach and engage in research as much as he liked. Einstein would earn a regular salary of 900 deutsche Marks per year. In addition to this, the Prussian Academy of Sciences later approved (on November 22, 1913) the special personal salary of 12,000 deutsche Marks per year (the personal commitment by Leopold Koppel). Both salaries would be paid to him within the month in which he moved to Berlin (The Prussian Academy of Sciences to Einstein, November 22, 1913, CPAE 5, Doc. 485). Max Planck and Walter Nernst made great efforts to bring Einstein to Berlin. They traveled to Zurich to personally offer him the opportunity to undertake physical research at the Kaiser Wilhelm Institute, become a member of the Prussian Academy of Sciences, teach at the university if he wished, and have opportunity to use his time for his own research. There were great advantages offered by this invitation to Berlin. Besides the academic honors that the Prussian Academy bestowed on Einstein, he was about to move to Berlin with a much larger salary than he had in Zurich.

From Zurich to Berlin

17

The democrat Einstein, who loved Switzerland, at first hesitated to reply to the coveted agreement. During their visit to Zurich, Planck and Nernst made a Sunday excursion on July 13, 1913, while Einstein thought their offer over. Despite Einstein's unusual talents he could still expect to be stimulated by new ideas, since it was always fruitful to receive the criticism of so many scientists capable of independent thinking working in many different fields. He would thus enjoy endless opportunities to meet with leading physicists, chemists, and mathematicians in Berlin. This was a great advantage, because Berlin was the center of theoretical physics at that time. On the other hand, it was difficult for Einstein to return to the Germany from which he had fled as a pupil. He had renounced his German citizenship two decades earlier in 1896, preferring being stateless, and then Swiss, than to undertaking military service in Germany. It seemed to him even a kind of betrayal of his convictions to become a member of a group with which he did not harmonise in so many respects, simply because it was connected with a pleasant position for himself. It was for him a struggle between his personality as a scientific investigator who could benefit by moving to Berlin, and his aversion to memberships within certain social groups (Frank, 1949, 176-182; 1947, 107-109; Kollros 1955, 29-30). Additional personal factors also influenced his decision. Einstein had an uncle in Berlin, Rudolf Einstein, a fairly successful businessman. In 1898 when Einstein was young, his father Hermann founded an electrical factory in Milan. Rudolf Einstein was persuaded to finance the enterprise, even though he had lost money in such earlier ventures. For a time around 1900 Hermann Einstein seems to have been free of financial worries. Rudolf had a daughter Elsa, who was now a widow. Einstein remembered that his cousin Elsa, as a young girl, had often been in Munich and had impressed him as a friendly, happy person, and he remembered his uncle's generosity. The prospects of being able to enjoy the pleasant company of this cousin in Berlin enabled him to think of the Prussian capital somewhat more favourably. Einstein was, therefore, looking forward to the position, first of all because of Elsa, but also because of Haber and Planck. Both men had made a really touching impression, but most important to him was that he had Elsa,

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someone with whom he could connect more deeply. Hence, it was first the company of Elsa and then that of Planck and Haber that eventually drew Einstein to accept the offer. Einstein met Planck and Nernst at the train station in Zurich and indicated to his visitors, by a prearranged signal – the waving of a white cloth – that he accepted the offer. Planck and Nernst left Zurich for Berlin on the night train of Monday and arrived home on the morning of Tuesday, July 15, 1913 (Einstein to Elsa Löwenthal, 14? July, 1913, CPAE 5, Doc. 451, note 2). The final vote on Einstein's membership in the Prussian Academy took place a few days later. On Thursday, July 24, 1913, the plenary session of the Prussian Academy of Sciences elected Einstein as a member with a final vote of forty-four to two. The result was reported to the Prussian Ministry of Education four days later. It was already almost August 1913. Elsa's assistance with Einstein's appointment was probably quite effective, because Haber knew how to estimate the influence of Einstein's friendly cousin Elsa. Haber saw the result of Elsa's assistance with Einstein's appointment without watching her arranging anything, but by the way it was all done he could easily recognise who did it. Einstein therefore said that Haber "knows his Pappenheimer" (Einstein to Elsa Löwenthal, July 14?, after 19-before 24 July, August 11, December 2, 1913, CPAE 5, Doc. 451, 454, 466, 488). Einstein was to come to Berlin in the spring of 1914 and be Jacobus Vant Hoff's successor, sit in the Prussian academy with a sufficient salary and without any further obligations, so that he could devote himself to scientific work. Vant Hoff was granted the first research professorship in the Prussian Academy that included an annual salary of 10,000 marks and an honorary professorship at the University of Berlin, free of teaching obligations. He died in 1911, and his chair had been vacant since then. Indeed Einstein was to receive 12,000, and, like Vant Hoff, to have no teaching obligations (Einstein to Elsa Löwenthal, July 14?, 1913, CPAE 5, Doc. 451, note 3). Besso visited Einstein in Zurich again in August 1913, shortly after Planck and Nernst left Zurich. Together, they continued to work on the Entwurf field equations, to find the advance of the perihelion of Mercury. Einstein, however, had become dissatisfied with his Entwurf field equations (Einstein to Lorentz, August 14, 1913, CPAE 5, Doc. 467).

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19

Einstein was busy working on his theory of gravitation. During September 1913, he spoke at two conferences: A lecture before the annual meeting of the Natural Science Society in Frauenfeld on September 9, 1913, and at the eighty-fifth congress of the German Natural Scientists and Physicists in Vienna (September 23, 1913) (Einstein 1913a, 1913b). Both talks were dedicated to his new gravitation theory. He prepared the latter talk in August. However, his colleagues – especially those from the Göttingen School – attacked his new theory because his gravitational equations were not generally covariant. In November 1913, Einstein found an elegant way out. He realised he could easily prove that a theory with generally covariant field equations could not exist (Einstein to Hopf, November 2, 1913, CPAE 5, 1913, Doc. 480). Indeed, Einstein already possessed the Hole Argument and he also had the new offer in Berlin. On October 7, 1913, Planck sent a letter to a possible influencing professor who requested material in order to scientifically assess Einstein's candidacy for the position in Berlin. The recipient of the letter could not be located. Planck sent him the material and the following supporting letter on behalf of Einstein. Planck took exactly the first sections from the supporting letter that he had read earlier aloud to the physicalmathematical class on June 12, 1913, and reproduced them in the letter to the professor (Planck to unknown recipient, October 7, 1913, Doc. 6 in Kirsten and Treder 1979, 100). The June 1913 letter was also signed by Nernst, Rubens and Warburg. However, in light of the similarity to Planck's October 1913 letter to the unknown professor, it is possible that Planck had also written the June 1913 letter read to the physicalmathematical class. On November 30, 1913, Einstein sent his resignation letter to Robert Gnehm, the president of the Federal School Council of the Polytechnic. Einstein reported to him that he had received an offer from the Prussian Academy of Sciences, and he requested to have him released, effective April 1, 1914. On December 6, 1913 a meeting of the Swiss Federal Council gathered in order to approve Einstein's request. Robert Gnehm reported to the meeting that he tried to persuade Einstein not to resign his position, but he understood that the position in Berlin was important to Einstein, given the fact that it was free of any teaching obligation. Two days after the meeting, on December 8, 1913, Gnehm addressed a request for approval of

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Einstein's resignation after detailing the inducements offered to Einstein to stay – an increase in salary, lifetime tenure, elimination of any teaching obligation, and occasional grants from a research fund of the Polytechnic. A day before, on December 7, 1913, Einstein had already written to the Prussian Academy of Sciences that he declared herewith that he accepted their election. On December 10, 1913, the Prussian Academy notified the Ministry of Education of Einstein's acceptance of the election, and the plenary session of the Prussian Academy was notified on December 18, 1913. Einstein informed the Academy that he would like to take up his new duties during the first days of April 1914 (Einstein to the Prussian Academy of Sciences, December 7, 1913, CPAE 5, Doc. 493). On December 11, 1913 the Swiss Federal Council approved, upon Einstein's request, his release from the professorship at the Zurich Polytechnic. Einstein also quitted the winter semester 1913-14 in Zurich, which ended on March 21, 1914 (Einstein to Robert Gnehm, November 30, 1913; Robert Gnehm to Einstein, December 15, 1913; Einstein to the Prussian Academy of Sciences, December 7, 1913, CPAE 5, Doc. 487, note 1, 493, 494, note 4). At the end of December 1913 Einstein's friends and colleagues from Zurich organised a farewell dinner at "Crown Hall" in Zurich. They all felt sorry he was leaving; but Einstein told them he was delighted to be able to devote all his time to his research (Kollros 1955, 29-30). Until January 1914, there was still no Kaiser Wilhelm Institute for Physics in Berlin. But, in early January a new initiative was developed during a meeting held at the Prussian Ministry of Education. Max Planck suggested delaying the creation of an institute until Einstein had fully settled in Berlin. In early January 1914, Walter Nernst proposed, as one alternative, the creation of a scientific committee to supervise research, composed of Planck, Einstein, Haber, Warburg, Rubens, Max von Laue and Ernst Beckmann. Einstein hoped that the question of a physics institute would be resolved upon his arrival in Berlin. However, by February still nothing had become of the institute. Already a year before, in late January 1913, Friedrich Schmidt of the Prussian Ministry of Education believed that a theoretical physical institute was unnecessary – because Einstein was not an experimentalist. This could explain why they postponed establishing an institute for Einstein.

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In early February 1914, Nernst, Haber, Planck, Rubens, and Warburg circulated a proposal to the Prussian government, the Kaiser Wilhelm Society, and to the Koppel Foundation for the creation of the Institute of Physics Research of the Kaiser Wilhelm Society. Based on Nernst's proposal of early January 1914, and on the need to minimise overhead costs, it was suggested that the institute occupy modest quarters, that experimental work would be conducted in already existing laboratories, and that the administrative burden would be placed on a scientific committee, of which Einstein was proposed as permanent honorary secretary. In late March 1914, the Kaiser Wilhelm society discussed an offer by the Koppel foundation to support the opening costs of the institute, and architects were consulted on its construction. During the same month, Einstein told his friend Zangger that he would probably get his own institute and an assistant. However, Einstein's membership in the Prussian Academy did not confer the right of having an assistant. Einstein first needed to receive the directorship of an institute, and only then could he get an assistant. Already in August 1913, Einstein was worried about not being able to have an assistant. He did not know then what to do, because he was offered neither an institute nor an assistant in Berlin. He was not allowed to take an assistant, but since he loved to work in collaboration with others, he wanted to request an assistant. He thought of Adriaan Fokker, Lorentz's student who once worked with Einstein in Zurich (during the winter semester of 1913-1914). With Fokker Einstein, would discover the covariant formalism of Gunnar Nordström's theory early in 1914.2 Without an institute and without the ability to employ an assistant, Einstein preferred the company of his close friends, Michele Besso and Paul Ehrenfest, and loyal colleagues, Lorentz, Erwin Freundlich, and others. Einstein was thirty-five years old when he left Zurich on March 20 or 21, 1914. Before arriving in Berlin he traveled to Holland, where he spent a week between March 23 and 29 in Leiden and Haarlem, visiting his

  2

Einstein to Lorentz, August 14, 1913, to Ehrenfest November, 1913, to Elsa Löwenthal, February 1914, and to Zangger, March 10, 1914; Haber to Krüss, January 4, 1913, CPAE 5, Doc. 428, notes 5, 9, 467, 484, note 3, 509, note 5, 513, 549, 598; Pais 1982, 487.

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friends Lorentz and Ehrenfest. He used the opportunity to talk with them about gravitation. This helped advance the Entwurf theory, but Einstein was still troubled with the restricted covariance of the field equations. Einstein traveled to Leiden with Lorentz's former student, Fokker. He arrived in Berlin on April 6, 1914 (Klein 1970, 296). Einstein left Zurich alone. His wife, Mileva Mariü, and children, spent the beginning of the year staying at Fritz and Clara Haber in Berlin, to search for an apartment there. During this trip, Mileva found an apartment very close to Haber's institute in Berlin-Dahlem and within proximity to Erwin Freundlich. However, Mileva and Einstein's two sons returned to Zurich at the end of March, while Einstein arrived alone in Berlin in early April (Einstein to Elsa Löwenthal, after December 2, 1913, CPAE 5, Doc. 489, note 3). Einstein's arrival in Berlin marked the end of his effective collaboration with Marcel Grossmann and the end of his marriage to Mileva. In Berlin, Einstein sat in the Prussian Academy of Sciences as a young man among older, proud and authoritative men, with some great achievements (Frank 1949, 182). Einstein's abode was in Dahlem, and his office was at Haber's institute, the Kaiser Wilhelm Institute of Physical Chemistry and Electrochemistry. He gave an inaugural address to the Leibniz-session of the Academy on July 2, 1914. The talk was general and dealt with the principles of theoretical physics. Towards the end of his speech he mentioned briefly that one arrives at a definite extension of the relativity theory, if one essentially specifies a principle of relativity in an extended sense for nonuniform motions. One is quickly led thereby to a general theory of gravitation, which includes dynamics (Einstein 1934, 175; 1954, 223). Planck gave a reply speech to this. On July 7, 1914, Einstein wrote a private response to Planck (Einstein to Planck, July 7, 1914, CPAE 8, Doc. 18; Kirsten and Treder 1979, 14-105). Recall that Planck was already a sceptic about Einstein's new gravitation theory and had advised Einstein against pursuing it. Planck thought that Einstein would not succeed; and even if he did, no one would believe him (Pais 1982, 239). When Planck heard Einstein's inaugural speech about extending the principle of relativity, thereby arriving at a definite theory of gravitation, he felt that Einstein was too quickly willing to renounce his most basic principle from his special theory of relativity, namely, the principle of the

From Zurich to Berlin

23

constancy of the velocity of light. Planck, who spoke in June and October 1913 about Einstein's special theory of relativity with unflagging enthusiasm, could not accept Einstein renouncing the principle so quickly. Einstein tried to explain to Planck that he did not actually renounce the principle of the constancy of the velocity of light. On the contrary, in his new gravitation theory, he imposed two conditions on the metric tensor: 1) Correspondence with Newton's theory: obtaining the analogue expression to Poisson's equation: ‫׏‬ଶ ĭ ൌ ͶɎ ɏǡ where U is the mass density, G the gravitational constant and ) is the scalar potential. 2) Correspondence with special relativity as a limiting case, and this enabled the introduction of the constancy of speed of light. Einstein's bending of light could not be accommodated with a constant light velocity. However, Einstein felt he was doing the right thing by going to Berlin. Erwin Freundlich was especially important for demonstrating the bending of light, the theoretical assumption of his theory. Freundlich – who had a post in the Berlin observatory – was in Berlin waiting since autumn 1911 to demonstrate Einstein's prediction regarding the bending of light rays in the gravitational field of the Sun. Einstein very likely thought that his personal relationships could prove fruitful to Freundlich, and that being in Berlin would assist in expediting the mission. Hence, it was not only Elsa and Planck who drew Einstein to Berlin. It was also Freundlich in Berlin. He wished to oversee the light rays' demonstration, which would prove his new theory of gravitation by experiment. On August 1, 1914, Germany declared war on Russia, and World War I broke out a day later. On August 21, 1914, there was a total eclipse of the Sun whose path passed through Feodosiya in southern Russia. During a total eclipse of the Sun, it is possible to take pictures of the field of stars surrounding the darkened location of the Sun, because during its occultation, the light emanating from the Sun does not interfere with

24

Chapter One

visibility of fainter objects. The pictures taken during the solar eclipse are compared with pictures of the same region of the heavens taken at night. According to Einstein's prediction, those stars closest to the limb of the Sun during the eclipse are found to be displaced slightly, by amounts that are inversely proportional to the distance of the stellar image from the Sun (Bergmann 1968, 66). Erwin Freundlich had planned to mount an expedition to test Einstein's prediction of the deflection of light in the gravitational field of the Sun already in 1911, and he had discussed the plans in correspondence with Einstein. However, the Berlin observatory and its director Hermann Struve refused to provide the funds. Einstein thus had to provide them from his private resources. The funds eventuated from the chemist Emil Fisher and from the Krupp Foundation. On July 19, 1914, Freundlich left Berlin with his colleague Walther Zurheilen accompanied by a technician and other members of the Royal Prussian Observatory. After a week of travel they arrived at Feodosiya in the Crimea on July 25, where they met the well-equipped expedition of the Argentine observatory of Córdoba. Freundlich had borrowed parts of a telescope from the Argentine observatory and had assembled four cameras altogether. However, Freundlich and his German team were converted from visiting astronomers to enemy aliens. On August 4, Freundlich was interned as a prisoner of war in Odessa, and his instruments impounded. Einstein, in Berlin, was worried and reported to his friend Ehrenfest about his good astronomer Freundlich that was going to experience captivity instead of the solar eclipse in Russia. However, a few weeks later (on August 29) Freundlich was exchanged along with other Germans for Russian officers in similar circumstances, but was forced to leave his instruments behind (equipment that consisted of a telescope [valued at 20,000 Marks], a tent [975 marks], 2 chronometers [700 Marks], meteorological instruments [261 Marks], and miscellaneous items [150 marks]). Freundlich returned to Berlin by the end of September 1914, just in time to read Einstein's new October 1914 Entwurf review article. Einstein's first big project on Gravitation in Berlin was to be complete by October 1914 a

From Zurich to Berlin

25

summarizing long review article on his Entwurf theory. The paper was published in November of this same year.3 The outbreak of World War I in August 1914 complicated the solareclipse expedition and further delayed the establishment of the Kaiser Wilhelm Physics Institute. However, the institute was finally set up in 1917 long after Einstein had developed his generally covariant field equations. The verification of the deflection of light in the gravitational field shared a similar fate (Einstein to Elsa Löwenthal, February, 1914, CPAE 5, Doc. 509, note. 5).

  3 Einstein to Freundlich, mid-August, August 26, December 7, January 20, 1913, CPAE 5, Doc. 468, 472, 492, 506; George Hale to Einstein November 8, 1913, CPAE 5, Doc. 483; Einstein to Ehrenfest, August 19, 1914, CPAE 8, Doc. 34, note 4; Einstein to Zannger, March 10, 1914, CPAE 5, Doc. 513; Earman and Glymour 1980a, 61-62.

CHAPTER TWO GENERAL RELATIVITY BETWEEN 1912 AND 1916

1. The Equivalence Principle Between 1905 and 1907, Einstein tried to extend the special theory of relativity so that it would explain gravitational phenomena. He reasoned that the most natural and simplest path to be taken was to retain the scalar gravitational potential ) and to complete the Newtonian gravitational field equation, i.e. Poisson's equation for gravity: ‫׏‬ଶ ĭ ൌ ͶɎ ɏǡ (where U is the mass density and G the gravitational constant) by adding a term differentiated with respect to time in such a way, so that the special theory of relativity is satisfied. Einstein also tried to adapt the law of motion of the mass point in a gravitational field to the special theory of relativity. However, he was unsure what path should be taken, because special relativity required the inertia of energy and this complicated the problem. He found that the vertical acceleration of the mass point in the vertical gravitational field, that is to say, the acceleration of a falling mass point, was not independent of its horizontal velocity, or the internal energy of the system. This contradicted Galileo's experimental law, according to which the vertical acceleration of a body in the vertical gravitational field is independent of the horizontal component of its velocity. Einstein was thus not satisfied with his route, because his result did not fit in with Galileo's law of free fall, which states that all bodies are accelerated in the gravitational field in the same way (as long as air resistance is neglected). The violation of Galileo's law of free fall raised Einstein's strong suspicions that this law was a matter of the utmost importance, and he

General Relativity between 1912 and 1916

29

realised that in it must lie the key to a deeper understanding of inertia and gravitation. Indeed this law, which may also be formulated as the law of the equality of inertial and gravitational mass, was illuminating for Einstein. Inertial mass is resistance to acceleration and gravitational mass is a measure of the force on an object in a gravitational field (Einstein 1933; 1934, 250-251; 1954, 286-287). Then "the breakthrough came suddenly one day. I was sitting on a chair in my patent office in Bern. Suddenly a thought struck me: If a man falls freely, he would not feel his weight". Einstein developed a thought experiment. This was the happiest thought of his life. He imagined an observer freely falling from the roof of a house; for the observer there is during the fall – at least in his immediate vicinity – no gravitational field. If the observer lets go of any bodies, they remain relative to him, in a state of rest or uniform motion, regardless of their particular chemical and physical nature. The observer is therefore justified in interpreting his state as being (locally) at rest (Einstein 1922a, 47; 1920a, CPAE 7, Doc. 31, 265). Isaac Newton had already realised that Galileo's law of free fall was connected with the equality of the inertial and gravitational mass. In the opening paragraph of the Principia, Newton defines "the quantity of matter", and says it is proportional to the mass of the body. Newton wrote that "the quantity of matter" is proportional to the weight, as he had found by experiments on pendulums (Newton 1726, Book I, 9). In approximately 1685, therefore, Newton realised that there was an (empirical) equality between inertial and gravitational mass. For Newton, however, this connection was accidental. Einstein, on the other hand, said that Galileo's law of free fall could be viewed as Newton's equality between inertial and gravitational mass, but for him the connection was not accidental. Einstein's 1907 breakthrough was to consider Galileo's law of free fall as a powerful argument in favour of expanding the principle of relativity to systems moving non-uniformly relative to each other. Einstein realised that he might be able to generalise the principle of relativity when guided by Galileo's law of free fall; for if one body fell differently from all others in the gravitational field, then with the help of this body an observer in free fall (with all other bodies) could find out that he was falling in a gravitational field (Einstein 1921, CPAE 7, Doc. 31, 268). In September 1907, the editor of the Yearbook for Radioactivity and Electronics, Johannes Stark, asked Einstein to write a review article on the

30

Chapter Two

theory of relativity. Einstein replied that he would be happy to write the paper. On November 1, 1907, Einstein told Stark that he had finished the first part of the work and was working on the second part. The first part dealt with special relativity. Einstein estimated that the whole paper would be forty printed pages, and hoped to send it by the end of the month (Einstein to Stark, November 1, 1907, CPAE 5, Doc. 63). The paper was published on December 4, 1907, and the second part included gravitation. While writing this paper, Einstein suddenly arrived at a breakthrough, which boosted his research towards the general theory of relativity. Einstein very likely arrived at this breakthrough sometime during November 1907. In his 1907 paper, "On the Relativity Principle and the Conclusions Drawn from It" Einstein invoked a new principle, the equivalence principle. Einstein assumed the complete physical equivalence of a homogeneous gravitational field and a corresponding (uniform) acceleration of the reference system. Acceleration in a space free of homogeneous gravitational fields is equivalent to being at rest in a homogeneous gravitational field (Einstein 1907, 454). Einstein considered a uniformly accelerated reference system, accelerated relative to a non-accelerated system in the direction of its x-axis, with J the magnitude of the acceleration (temporarily constant). The accelerated system is instantaneously at rest relative to the nonaccelerated system when the clocks of the latter read the time t = 0. The clocks of the former are then synchronised to read the time W. What is the rate of the clock in the next time element in the uniformly accelerated reference system (the coordinates of a point event are [K] ? Einstein found that the time is: ߬ሺͳ ൅ ߛߦ Τܿ ଶ ሻ. The clock, therefore, runs faster in the next time element than an identical clock reading the time W. According to the 1907 equivalence principle, time measurements in a uniformly accelerated reference system are also valid for a coordinate system in which a homogeneous gravitational field is acting. Consequently, for an observer located somewhere in space, the clock in a

General Relativity between 1912 and 1916

31

(higher) gravitational potential runs faster than an identical clock located at the coordinate origin (lower gravitational potential). Einstein sets: Ȱ ൌ ߛߦǡ where ) is the gravitational potential, so that he obtains: ߬ሺͳ ൅ ȰΤܿ ଶ ሻǤ He concludes that clocks, and more generally any physical process, proceed faster the higher the gravitational potential at the position of the process taking place. Likewise, clocks run more slowly the lower the gravitational potential at the position of the process taking place. Therefore, if a clock indicating a certain time is located in a certain gravitational potential then it runs ሺͳ ൅ ȰΤܿ ଶ ሻtimes faster than an identical clock located at the coordinate origin, near a massive object (at a stronger gravitational potential) (Einstein 1907, 455-458). This is gravitational time dilation: The closer clocks are to a material source, the more slowly they run. Einstein considered clocks located at different gravitational potentials and whose rates could be controlled precisely; these produced spectral lines. From the above he arrived at a gravitational redshift: the wavelength of light coming from the Sun's surface is larger by about one part in two millionths than that of light produced by the same substance on earth (Einstein 1907, 458-459). Einstein obtains equations of similar form for the effect of gravitation on electromagnetic phenomena (the velocity of light). He again examines the two systems: In a uniformly accelerated reference system, accelerated relative to a non-accelerated system in the direction of its x-axis, J is the magnitude of the acceleration (temporarily constant). He again obtains the term ሺͳ ൅ ߛߦ Τܿ ଶ ሻand according to his equivalence principle also the term ሺͳ ൅ ȰΤܿ ଶ ሻǤ He recognised that, according to his principle of equivalence, the velocity of light in a gravitational field is a function of the place: ܿሺͳ ൅ ߛߦ Τܿ ଶ ሻ ՜ ܿሺͳ ൅ ȰΤܿ ଶ ሻ.

32

Chapter Two

He reasoned that it follows from this result that the light rays do not move along the [-axis, but are bent by the gravitational field (Einstein 1907, 461). In 1907, Einstein presented a new principle, the equivalence principle. He did not yet present a new theory, but rather an extension of the method that he had used in the special theory of relativity. He was guided by Galileo's principle of free fall and the equality of inertial and gravitational masses. He postulated the principle of equivalence, and with physical reference systems and measuring rods and clocks, he arrived at three new results: 1. the bending of light rays in a gravitational field; 2. that clocks run at different rates when placed in different gravitational potentials; and 3.the gravitational redshift. In 1907, Einstein thus chose the natural route, following his special theory of relativity coordinate-dependent theory. When no complicated mathematics enters into the theory, the extension of the special principle of relativity into uniformly accelerating systems becomes quite natural and simple. In June 1911, Einstein published a whole paper dedicated to gravitation titled, "On the Influence of Gravitation on the Propagation of Light". In the body of the paper, Einstein returned to Galileo's experimental principle of free fall. He was guided by this experimental principle towards formulating the more mature equivalence principle of 1911. Einstein considered two systems: a system K at rest in a homogeneous gravitational field, and a uniformly accelerated system K' placed in a space free of gravitational fields and moving with uniform acceleration. If we suppose that the systems K and K' are physically, exactly equivalent, then we may just as well define the system K as being found in space free from homogenous gravitational fields, and consider K as uniformly accelerated. Einstein obtained by theoretical consideration of the processes which take place relatively to a uniformly accelerating reference system K', information as to the course of processes in a homogeneous gravitational field (Einstein 1911, 899-900). Recall that between 1905 and 1907, Einstein tried to adapt the law of motion of the mass point in a gravitational field to the special theory of relativity. However, special relativity required the inertia of energy and this complicated the problem. According to special relativity the inertial mass of a body increases with the energy it contains; if the increase of energy amounts to E, then the increase in inertial mass is equal to E/c2, where c denotes the velocity of light. Einstein found that the vertical acceleration of the mass point in the vertical gravitational field, i.e., the

General Relativity between 1912 and 1916

33

acceleration of a falling mass point, was not independent of its horizontal velocity, or the internal energy of the system. This contradicted Galileo's experimental law of free fall, which states that all bodies are accelerated in the gravitational field in the same way. The violation of Galileo's law of free fall raised Einstein's strong suspicions that this law was a matter of the utmost importance. This law, which may also be formulated as the law of the equality of inertial and gravitational mass, was illuminating for Einstein (Einstein 1933; 1934, 250-251; 1954, 286-287). In his 1911 paper, in Section §2, Einstein solved the above problem, which had bothered him from 1905 to 1907, in terms of the equivalence principle. According to special relativity, the inertial mass of a body increases with the energy it contains. Einstein wondered whether this increase of inertial mass corresponded to an increased gravitational mass. If no such correspondence existed, then Galileo's principle of free fall, according to which there is equal falling of all bodies in a gravitational field, would be violated. For a body would fall in the same gravitational field with varying acceleration according to its energy content. In order to extend the above result from special relativity to gravitation, Einstein used the equivalence principle: a system at rest in a homogenous gravitational field K and a uniformly accelerating system in a space free of gravitational fields K' are equivalent with respect to all physical processes; thus the laws of nature with respect to K will be in agreement with those with respect to K' (Einstein 1911, 900-903). Einstein considered two systems S1 and S2, and provided each one with measuring instruments. The two systems are situated on the z-axis of the system K (S1 is near the origin of the axes). They are separated by a distance h from each other. The gravitational potential in S2 is greater than that in S1 by J·h (the higher up an object is the greater its gravitational potential). S2 emits a definite quantity of energy towards S1.

Chapter Two

34

Einstein uses the principle of equivalence of K and K', and in place of the system K in a homogeneous gravitational field he considers a gravitational-free system K', which moves with uniform acceleration in the direction of positive z, and with the z-axis of which the systems S1 and S2 are connected. In system K', S2 radiates energy to S1, a process we observe from another system K0, which is non-accelerating. By special relativity to first-order approximation we get: ‫ܧ‬ଵ ൌ ‫ܧ‬ଶ ൬ͳ ൅

ߛ݄ ൰ǡ ܿଶ

E2 – the radiation energy emitted from S2 towards S1. E1 – the radiation arriving at S1. By Einstein's principle of equivalence the same energy relationship should hold if the same process takes place in the system K. In this case the above equation reads: ‫ܧ‬ଵ ൌ ‫ܧ‬ଶ ൬ͳ ൅

Ȱ ൰Ǥ ܿଶ

)is the potential of the gravitation vector in S2. Einstein concluded that the energy arriving at S1 is greater than the energy (measured by the same means), emitted at S2, with that energy being the potential energy of the mass E/c2 in the gravitational field. He demonstrated that the increase in gravitational mass is equal to E2/c2, which is therefore equal to the increase in inertial mass. This result emerged from the equivalence principle, the equivalence of K and K', according to which the gravitational mass with respect to K is exactly equal to the inertial mass with respect to K'. Max von Laue wrote to Einstein that he had carefully studied his 1911 paper, but, because he could not accept Einstein's equivalence principle, he did not believe in the accuracy of his theory. He could not accept the full equivalence of the systems K (a system at rest in a homogenous gravitational field) and K' (an accelerated system with no gravitational

General Relativity between 1912 and 1916

35

field). He believed that a mass could cause the gravitational field in system K, but could not do so in the accelerated system K'. Therefore, the presence, or else the absence of such a mass, will decide whether there is a gravitational field or whether it is an accelerating system being dealt with (Laue to Einstein, December 27, 1911, CPAE 5, Doc. 333). Einstein replied to the above objection against the equivalence principle, which was raised again in 1918, within the context of the clock paradox: Consider a Galilean coordinate system K, and U1 and U2 – two identical clocks (Uhr) that are free from outside influences. These run at the same pace when in close proximity and also at any distance from each other, if they are both at rest relative to K. Consider the clock paradox thought experiment: Let A and B be two distant points of the system K. A is the origin of K, and B is a point on the positive x-axis. The two clocks are initially at rest at point A. They run at the same rate, and the positions of the hands are the same. We now impart a constant velocity v in the positive direction of the x-axis to clock U2, so that it moves towards B. At B we imagine the velocity reversed, so that clock U2 returns to A. As it arrives at A, the clock is decelerated so that it is once again at rest relative to U1; clock U2 lags behind U1 (Einstein 1918, 698-700). When seen from system K, the process is interpreted in the following manner: U1 remains at rest in K. At point A, Clock U2 is accelerated by an external force along the positive x-axis until it reaches the velocity v. U2 moves from point A until it reaches point B. At point B, Clock U2 is accelerated by an external force acting along the negative x-axis until it reaches the velocity v in the negative direction. U2 moves back with constant velocity v in the direction of the negative x-axis. An external force brings U2 to rest at point A. When U1 and U2 meet after the to and fro motion, U2 lags behind U1. Consider a uniformly accelerating coordinate system K', that is co-moving with clock U2. Relative to K', it is clock U1 that is moving, with clock U2 remaining at rest. Einstein described the viewpoint of an observer on the accelerating system K'. This observer explains the clock paradox in the following manner: 1) According to the principle of equivalence, due to an external force acting along the positive x-axis upon the observer and the clock U2, the observer can consider himself as being at rest in a homogeneous

36

Chapter Two

gravitational field that acts in the direction of the negative x-axis. At point A', the clock U1 is accelerated in this field until it reaches the velocity v, whereupon the field vanishes. 2) U1 moves with constant velocity v from point A' until it reaches point B' on the negative x-axis. It seems that clock U1, moving with velocity v, has a slower rate than clock U2 which remains at rest in the accelerating system K'. 3) U1 is now at point B'. A homogeneous gravitational field in the direction of the positive x-axis causes clock U1 to accelerate in the direction of the positive x-axis until it reaches the velocity v, whereupon the gravitational field vanishes. An external force along the negative x-axis acts upon clock U2 and prevents it from moving in this gravitational field. U1 is thus accelerated towards U2 by the gravitational field and, U1 is therefore at a location of higher gravitational potential than U2. U1 has a faster rate than U2 because clocks at a higher gravitational potential run faster. 4) U1 moves with constant velocity v in the direction of the positive x-axis towards U2. Again U1, moving with velocity v, has a slower rate than clock U2 which remains at rest in the accelerating system K'. 5) Finally, a gravitational field in the negative x-axis brings clock U1 to rest, whereupon the gravitational field vanishes. During all this time, clock U2 is kept at rest in the accelerating system K' by an external force. The running ahead of U1 during the turning point 3) at B' amounts to precisely twice as much as the lag behind during 2) and 4). Hence, again from the viewpoint of the system K', U2 lags behind U1 after the to and fro motion. It was later claimed against Einstein that he solved the clock paradox by taking into account the influence the gravitational field has on clocks with respect to the system K' (an accelerated system with no gravitational field): This gravitational field is not real, but fictitious. Its existence is only invoked by the choice of coordinates and it is dependent on an accelerating system K'. Real gravitational fields, like in Newtonian gravitational theory, are generated by masses, and cannot be made to vanish by the suitable choice of a coordinate system. Therefore, it was claimed against Einstein that a fictitious field (non-inertial field) cannot have an influence on the rate of a clock. Then, indeed, relative to K' it is clock U1 that is moving,

General Relativity between 1912 and 1916

37

with clock U2 remaining at rest; and it follows that U1 should finally run behind U2, in contradiction with Einstein's conclusion. Einstein replies to this claim that the distinction real – unreal is not helpful, because, with respect to K' the gravitational field "exists" in the same way as any other physical entity that can only be defined with reference to a coordinate system, even though it does not "exist" with respect to the system K. Einstein began his 1911 paper by stating that rays of light experience deflection when passing close to the Sun's gravitational field. Accordingly, Einstein explained that rays of light, passing close to the Sun, experience by its gravitational field a deflection which follows from his theory (that is to say, follows from an analysis based on the equivalence principle). Einstein concluded that a ray of light passing near the Sun would undergo a deflection amounting to 0.83 arc seconds (Einstein 1911, 898). Max von Laue was also sceptical about the astronomical test Einstein proposed: the deflection of light. He thought that if the deflection that Einstein claimed in his paper was to be observed, then it could always be blamed on the variation of the refractive index in the Sun's atmosphere (Laue to Einstein, December 27, 1911, CPAE 5, Doc. 333).

2. Einstein's 1912 Polemic with Max Abraham: Static Gravitational Field An important conclusion of Einstein's 1911 paper was that the velocity of light in a gravitational field is a function of the place (Einstein had already mentioned this point in his 1907 study): if the velocity of light is c0 at the origin of the coordinates, then at a place with certain gravitational potential, ) it should be (Einstein 1911, 906): ܿ ൌ ܿ଴ ሺͳ ൅ ȰΤܿ ଶ ሻǡ which is dependent on that gravitational potential, and is thus different from c0. Consider a source that emits radiation and thus vibrates with a certain frequency. This is also a definition of a clock. Based on his 1907 analysis, Einstein concluded that if we measure time in a location close to the origin with clock U, we must measure the time in a location near the source far

38

Chapter Two

from the origin with a clock that goes ͳ ൅ ȰΤܿ ଶ slower than the clock U if we compare it with the clock U in the same place. If time slightly runs slowly, then light would appear to be delayed or slowed down. The delay (the Shapiro delay) depends on how low the gravitational potential is (See Chapter 2, Section 15). Accordingly, Einstein offered a theory of static gravitational fields which violated his own light postulate from the special theory of relativity, and as a consequence this result excludes the general validity of the Lorentz transformation. Einstein's 1911 paper drew the attention of other scientists to develop their own gravitation theory. In December 1911, a short time after the publication of Einstein's 1911 paper, Max Abraham from Göttingen submitted a paper titled, "On the Theory of Gravitation". Abraham's theory was formulated in terms of Hermann Minkowski's four-dimensional space-time formalism of special relativity, and in terms of Einstein's 1911 relation between the variable velocity of light and the gravitational potential (Abraham 1912a, 1-4). Since Minkowski took the constancy of the speed of light to be one of his fundamental principles, Abraham tried the impossible: Abraham began his paper by saying that in a recently published paper, Einstein (Einstein 1911) proposed the hypothesis that the speed of light c depends on the gravitational potential ). He offered to develop a theory of the gravitational potential which satisfied the principle of relativity and derive from it a relation between c and ), which in firstorder approximation was equivalent to Einstein's (Abraham 1912a, 1). Abraham derived the following relation: ܿ ଶ Τʹ െ ܿ଴ଶ Τʹ ൌ Ȱǡ and claimed it was equivalent to Einstein's 1911 relation: ܿ ൌ ܿ଴ ሺͳ ൅ ȰΤܿ ଶ ሻǡ but better serves to represent the independence from the arbitrary chosen origin of coordinates (Abraham 1912a, 2). In February 1912, Abraham published a correction to his paper (Abraham 1912a, 176). An oversight, highlighted by a friendly note from Einstein, had to be corrected.

General Relativity between 1912 and 1916

39

Following Minkowski's 1908 theory, Abraham wrote that we consider: ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ ݑ†ƒݖ‬ൌ ݈݅ ൌ ݅ܿ‫ݐ‬ǡ as the coordinates of the four-dimensional space (Abraham 1912a, 1). Einstein informed Abraham that this should rather be: We consider: ݀‫ݔ‬ǡ ݀‫ݕ‬ǡ ݀‫ ݑ݀†ƒݖ‬ൌ ݈݅݀ ൌ ݅ܿ݀‫ݐ‬ǡ as components of a displacement ds in four dimensional space (Abraham 1912a, 176). In 1908 Minkowski showed that the Lorentz transformations are invariant under the quadratic form (Minkowski 1908, 106): ‫ ݔ‬ଶ ൅ ‫ ݕ‬ଶ ൅ ‫ ݖ‬ଶ െ ܿ‫ ݐ‬ଶ Ǥ Minkowski also arrived at: ܿ ଶ ݀‫ ݐ‬ଶ െ ݀‫ ݔ‬ଶ െ ݀‫ ݕ‬ଶ െ ݀‫ ݖ‬ଶ Ǥ Abraham then came up with the following idea from Einstein's correction: ݀‫ ݏ‬ଶ ൌ ݀‫ ݔ‬ଶ ൅ ݀‫ ݕ‬ଶ ൅ ݀‫ ݖ‬ଶ െ ܿ ଶ ݀‫ ݐ‬ଶ is the square of the four-dimensional line element, where the speed of light c is determined by (Abraham 1912a, 176): ܿ ଶ Τʹ െ ܿ଴ଶ Τʹ ൌ ȰǤ Abraham, therefore, wished to incorporate Einstein's idea of a nonconstant light velocity into the special theory of relativity (Pais 1982, 230): He tried to incorporate the above relation between c and ) into the fourdimensional line element – and eat one's cake and have it too. Abraham had effectively introduced above the general four-dimensional line element. However, for the time being Abraham's expression remained an isolated mathematical formula without context and physical meaning

40

Chapter Two

that, at this point, was indeed neither provided by Abraham's nor by Einstein's physical understanding of gravitation (Renn 2007, 311-312).4 From Abraham's correction we can infer that Einstein read Abraham's paper of 1912 on the theory of gravitation, and then Einstein corrected it and responded to it by a theory of his own. We can also infer that Einstein was already deeply involved in Minkowski's four-dimensional formalism even though he did not yet implement it in his theory of gravitation. Shortly afterward, in February 1912, Einstein sent a letter to the editor of the Annalen der Physik, Wilhelm Wien, in which he submitted to him the paper "The Speed of Light and the Statics of the Gravitational Fields" for publication in the Annalen (Einstein 1912a). Einstein considered Abraham's theory of gravitation as completely unacceptable (Einstein to Wien, February 24, 1912, CPAE 5, Doc. 365). In his paper, Einstein considered a relationship between the velocity of light and the gravitational potential in the static gravitational field, so that the gravitational field was determined by c. Einstein wrote a linear field equation that corresponds to Poisson's equation in Newtonian theory: ‫ ܿ׏‬ൌ ߢܿߩǡ where N denotes the universal gravitational constant, U the matter density, ‫ ׏‬the Laplacian operator, and c is the velocity of light. He then obtained the same form of linear field equation in a static gravitational field: ‫׏‬ĭ ൌ ߢܿ ଶ ߩǡ where, ‫ ܭ‬ൌ ߢܿ ଶ denotes a gravitational constant. Einstein considered a system in a state of uniform acceleration K', accelerated in the direction of its x-axis, and Jis the magnitude of the acceleration (temporarily constant), with respect to a non-accelerated

  4

In 1914, Abraham also suggested in his paper "The New Mechanics" that "all forces propagate in the theory of relativity with the speed of light, even gravity" (Abraham 1914a, 17). However this could not be considered as gravitational waves and this actually also remained as an isolated result, until later when proposed as gravitational waves (equivalent to the electromagnetic waves) in 1916 by Einstein as a solution to his linearized field equations.

General Relativity between 1912 and 1916

41

system. He found the law of motion of a material point that holds in the accelerating system K'. Guided by the equivalence principle, the same equation at which he had arrived for K' should hold well for a motion of a material point in a static gravitational field. Einstein thus obtained the same form of equation for a motion of a material point in a static gravitational field (Einstein 1912a, 359, 361-362). Einstein then reformulated his findings from his 1911 Annalen paper considering the time: If we measure time with a clock in a higher gravitational potential, we must measure the time in a lower gravitational potential with a clock that goes ͳ ൅ ȰΤܿ ଶ more slowly than that clock if compared with the clock at the higher gravitational potential. We can summarise this insight as gravity bends time (Einstein 1912a, 365-366). This is gravitational time dilation: The closer clocks are to a material source, the more slowly they run. In the same paper, Einstein considered a system K with coordinates x, y, z in a state of uniform rotation (disc) in the direction of its x-coordinate and referred to it from a non-accelerated system. The origin of K possesses no velocity. Hence, Einstein considered a rigid body already in a state of uniform rotation observed from an inertial system. The system K is equivalent to a system at rest K' in which there exists a certain kind of mass-free static gravitational field. In 1912 Einstein extended the 1911 equivalence principle to uniformly rotating systems. The systems K and K' are physically equivalent. A uniformly rotating system (disc) K is equivalent to a system at rest K' in a static gravitational field. A centripetal force causing a centripetal acceleration on an observer on a uniformly rotating disc acts towards the centre of the disc. The observer feels this as if there is an inertial force, a centrifugal force, acting on the uniformly rotating disc K. According to the equivalence principle, the situation on the disc is exactly equivalent (feels like) to one in the system at rest K' in a static gravitational field. Einstein explained the rotating disc problem very succinctly. Euclidean geometry could not be applied to the rotating disc. We take a great number of small measuring rods (all equal to each other) and place them end-toend across the diameter 2R and circumference ʹߨܴ of the uniformly rotating disc. From the point of view of a system at rest all the measuring rods on the circumference are subject to the Lorentz contraction. An

42

Chapter Two

observer in the system at rest concludes that in the uniformly rotating disc K the ratio of the circumference to the diameter is different from S (Einstein 1912a, 356). Einstein did not explicitly speak about the possibility that Euclidean geometry is unacceptable on a uniformly rotating disc; but this impression was implicitly contained in the above description. In March 1912, Einstein wrote Wien again, and asked him to return the paper and not publish the manuscript because he had discovered that not everything in the paper was correct (Einstein to Wien, March 11, 1912, CPAE 5, Doc. 371). Recall that in 1911 Einstein found that, since mass and energy are equivalent (different forms of the same thing), the energy arriving at S1 is greater than the energy, which has been emitted at S2. The increase in gravitational mass is equal to E/c2, and therefore is equal to the increase in inertial mass resulting from the theory of relativity (Einstein 1911, 902903). Related to this finding is another discovery, a finding that Einstein arrived at in his February paper of 1912: Einstein derived equations that included the energy of the material point in a static gravitational field. According to special relativity, this is related to the mass of the material point. The gravitational mass of the material point depends on the kinetic energy in the same manner as the material point's inertial mass. The kinetic energy cannot be distinguished from the gravitational energy, and it depends on the mass, the velocity of the material point, and on the velocity of light. Thus, it depends on the gravitational potential. The energy of a material point at rest in the gravitational field is mc (Einstein 1912a, 363). Sometime between February and March 1912, Einstein realised that since the energy would exert gravitational influence just as mass would, energy contained within the gravitational field itself would also exert gravitational influence. Any change in the strength of the gravitational field would produce an extra variation as the change in the energy within the field fed back into the system as a whole. In other words, Einstein's linear equation from February 1912 was not consistent with the principle of action and reaction and the principles of energy and momentum. Einstein very likely already realised that his linear field equation should be replaced by a non-linear field equation; however, he finally decided to

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allow the publication of this paper, and immediately submitted another correcting paper to the Annalen, "On the Theory of the Static Gravitational Fields". Einstein arrived at a non-linear equation for the static gravitational field (Einstein 1912b, 450-453, 456): ͳ …‫ ܿ׏‬െ ሺ݃‫ܿ݀ܽݎ‬ሻଶ ൌ ߢܿ ଶ ıǤ ʹ The second term on the left hand side of the equation is the energy density of the gravitational field multiplied by c, and ı denotes the mass density and the energy density. Einstein included the basic electromagnetic equations into his theory of the static gravitational field. In this case the source of the gravitational field is the mass density augmented with locally measured energy density (Pais 1982, 205). Einstein was guided by electrostatic theory (the electrostatic field and the electrostatic interaction between charged particles) to treat the static gravitational field for material particles. His notebook from 1912 (Einstein's Scratch Notebook) is filled with doodles, diagrams, wiring diagrams, notes and equations. On page 39, he wrote the above non-linear equation for the static gravitational field and, on page 40, he drew wiring diagrams. The wiring diagrams in his notebook are an indication that electrostatic theory was a heuristic guide for Einstein in his search for a static theory of gravitation ("Einstein's Scratch Notebook", CPAE 3, 3740). Einstein also discovered that the principle of equivalence is valid only locally. He found this important but difficult step, because he was departing from the foundations of the unconditional principle of equivalence. He realised that the latter holds only for infinitely small fields, and the derivation of the equations of motion of the material point apply only to infinitely small space (Einstein 1912b, 455-456). Consider a non-accelerated observer in a static gravitational field, as contrasted with a uniformly accelerated observer in a region of free space where the gravitational field can be neglected. In this case the principle of equivalence makes the assertion that the results obtained by the two observers in performing any given physical experiment will be precisely identical, provided that the observer in free space is given an acceleration, relative to a non-accelerated observer (of the special theory of relativity),

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which is equal and opposite to the gravitational acceleration found by the other observer. In 1912, Einstein found an alternative expression to the above principle of equivalence. Having asserted the complete equivalence between the two observers, the equivalence will have to persist when we make analogous changes in their states of motion. Thus if the observer in the static gravitational field is himself allowed to fall freely with the natural acceleration due to gravity g, and the forced acceleration given to the observer in free space is reduced to zero, they must still obtain identical results in any given experiment that they may perform. Hence, for a freefalling observer in a static gravitational field the effects of gravitation would be canceled. It is thus always possible in the case of a static gravitational field to transform (locally) to space-time coordinates such that the effects of gravity will not appear. We may maintain for a sufficiently small region that the effects of gravitation could be removed by the use of free-falling frames of reference (Tolman 1934, 174-175). Hence according to the new principle of equivalence, acceleration (freefall) in a static gravitational field is locally equivalent to being at rest in a space free of static gravitational fields. This embodies the main idea of Einstein's 1907 thought experiment: An observer falls freely from the roof of a house. During the fall, there is no gravitational field for the observer, in his immediate vicinity. According to the new principle of equivalence, one cannot distinguish Newtonian inertial systems (material particles in free fall following geodesic lines in curved non-flat space-time) from local inertial systems (material particles in free fall following geodesic lines in flat space). We cannot, therefore, locally separate gravity from inertia. Einstein would later say that gravity and inertia are described by a single inertiogravitational field. Einstein did not forget Abraham, but mentioned that the matter was not as easy as Abraham thought. Between March and June 1912, he wrote to his friends and colleagues about the amusing polemic in which he was engaged with Abraham. He told his best friend, Besso, that Abraham's theory was created off the top of his head, from mere mathematical beauty considerations, torn off and completely untenable. That was almost Abraham's opinion of Einstein's theory, except for the mathematical beauty. He noted to Wien that Abraham told him he had converted to his (Einstein's) theory; and he reported to Zangger that he was engaged in an

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amusing polemic with Abraham; the latter had accepted his (Einstein's) main new results concerning gravitation (Einstein to Besso, March 26, 1912, to Wien, May 17, 1912, and to Zangger, May 20 and June 5 1912, CPAE 5, Doc. 377, 395, 398, 406). Actually, Abraham understood quite the opposite: He thought that it was Einstein who had converted to his theory. According to Abraham's understanding, Einstein corrected his February static gravitational theory because he borrowed some equations from him (Abraham 1912b, 1058). In his June 1912 reply to Einstein, Abraham said that it would be careless to reject Einstein's results, some of which (that is, his expression for the energy density) were precisely similar to those found in Abraham's theory; results that Einstein independently formulated by the principle of equivalence (Abraham 1912b, 1058). Remember that in February 1912 Abraham published a correction to his paper after he had received a friendly note from Einstein. Einstein, therefore, read Abraham's 1912 paper on the theory of gravitation, corrected it and responded to it by a theory of his own – his February 1912 "The Speed of Light and the Statics of the Gravitational Fields". Abraham then blamed Einstein for presenting results similar to his own and formulating these results by the principle of equivalence. According to Einstein, Abraham accepted Einstein's main new results concerning gravitation from 1911, and then published his 1912 theory of gravitation. Einstein explained that after he had published his 1911 paper, Abraham created a theory of gravitation that rendered information and drew conclusions from his 1911 paper. However, Abraham's system of equations could not be reconciled with the principle of equivalence. Einstein thought that Abraham's theory was completely wrong and anticipated differences between him and Abraham. Einstein attacked Abraham's theory because of what he considered the incompatibility between Abraham's simultaneous implementation of both a variable speed of light and Minkowski's formalism, and because Abraham's theory could not be reconciled with a theory based on the equivalence principle (Einstein to Hopf, February 20, 1912, CPAE 5 Doc. 364). Abraham attacked Einstein stating that in his earlier 1911 theory, Einstein had given up the essential postulate of the constancy of the speed of light by accepting the effect of the gravitational potential on the speed of light. Abraham claimed that in Einstein's February 1912 work (Einstein 1912a),

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the requirement of the invariance of the equations of motion under Lorentz's transformations also fell, and this dealt the death blow to the special theory of relativity. Abraham remarked that even the author himself of the special theory of relativity was now convinced by its inconsistency (Abraham 1912b, 1056). In July 1912, Einstein responded to Abraham that the principle of the constancy of the velocity of light could be maintained only insofar as one restricted oneself to local space-time regions of constant gravitational potential. On the other hand, this was not the limit of validity of the principle of relativity, but was that of the constancy of the velocity of light, and thus of the special theory of relativity.Hence, this situation by no means implies the failure of the principle of relativity. Einstein intended to generalise the principle of relativity to accelerated systems and thus indeed to demonstrate the limited validity of the special theory of relativity to inertial systems of reference. He explained to Abraham that special relativity would always retain its significance as the simplest theory for the important limiting case of space-time events in a constant gravitational potential (Einstein 1912d, 1062-1063). Abraham's final criticism was of Einstein's March paper (Einstein 1912b, 456). He did not like Einstein's use of the new local principle of equivalence. It appeared to Abraham as a fluctuating basis. He insisted that Einstein's new equivalence principle, applicable only to local regions, was actually giving up the equivalence principle (Abraham 1912b, 1058). Einstein replied by describing his equivalence principle of 1912: It could only apply consistently to infinitely small regions of space. Einstein said that he knew that it did not supply a satisfactory basis, but therein he did not see any reason for rejecting the equivalence principle because it applied to the infinitely small (Einstein 1912d, 1063). Einstein was confident in this principle, which was a natural extrapolation of the most general experimental propositions of physics (i.e. Galileo's free-fall law and the experimental equivalence of inertial and gravitational masses). He still lacked the proper mathematical tools that could provide a solid foundation for his theory. He was, therefore, unable to supply a sound response to Abraham. Finally, Einstein answered Abraham's plagiarism blames. Einstein said bluntly that his result contradicted the fundamental equations of Abraham's theory. However, Abraham did not accept Einstein's reply. He

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showed that his expression for the energy density in a static field, which followed from his 1912 theory of gravitation, exactly coincided with Einstein's expression for the energy density in the field (Einstein 1912e, 1064). On August 16, 1912, Einstein wrote to Ludwig Hopf that Abraham slaughtered him along with the theory of relativity in two massive attacks, under the "nostrification" (borrowing, yearning to possess) of his results. Einstein described Abraham's theory as a stately steed that lacks three legs. Einstein said that Abraham noted that the knowledge of the mass of energy came from Robert Mayer. Therefore, Abraham also blamed Einstein for plagiarising the mass-energy equivalence (Einstein to Hopf, August 16, 1912, CPAE 5, Doc. 416).

3. Einstein's 1912 Polemic with Gunnar Nordström: Static Gravitational Field Max Abraham criticised Einstein's 1912 theory of static gravitational fields, claiming he failed to implement Minkowski's reformulation of special relativity in terms of a four-dimensional space-time. In 1912, the Finnish physicist Gunnar Nordström from Göttingen, who had been Minkowski's student between 1906 and 1907, proposed a theory of gravitation that followed Minkowski's space-time formalism. He claimed Einstein's 1911 hypothesis – that the speed of light c depends upon the gravitational potential – led to considerable difficulties for the principle of relativity, as the discussion between Einstein and Abraham showed (Abraham 1912b; Nordström 1912, 1126). Nordström said that Abraham, however, followed Einstein's hypothesis of the variable speed of light. He thus had to abandon special relativity and change Minkowski's theory (this is not quite true. Abraham tried the impossible: He wished to incorporate Einstein's idea of a non-constant light velocity into Minkowski's fourdimensional space-time formalism of special relativity). Nordström questioned whether it was possible to replace Einstein's hypothesis of a variable speed of light with a different one (which left the speed of light c constant), and still adapt the theory of gravitation to the special principle of relativity in such a way that gravitational and inertial mass were equal. Instead of the velocity of light, Nordström considered the rest mass m of the mass point to vary with the gravitational potential ), and he supplied an expression for the dependence of the mass on the gravitational potential (Nordström 1912, 1126).

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݉

݀Ȱ ݀݉ ൌ ܿଶ Ǥ ݀‫ݐ‬ ݀‫ݐ‬

Integration yields: మ

݉ ൌ ݉଴ ݁ ஍Τ௖ Ǥ The rest mass m when ) = 0 is m0. Nordström's theory, however, had a little problem. Nordström wrote at the end of his paper that according to his theory, mass points could not really exist, because within mass points the gravitational potential ) = – f, and hence the mass would be zero (Nordström 1912, 1129). Einstein could not accept Nordström's above theory. Before submitting his paper, Nordström had probably sent Einstein a copy, because he raised Einstein's criticisms in his paper, even though Nordström did not accept them. Nordström wanted to replace Einstein's hypothesis of a variable speed of light with a variable mass hypothesis, while leaving the speed of light c constant and still adapting the theory of gravitation to the special principle of relativity in such a way that gravitational and inertial mass would be equal. Einstein explained in his 1911 paper that, according to the inertia of energy from special relativity, the inertial mass of a body increases with the energy it contains. If an increase of inertial mass corresponds to an increase of gravitational mass, then Galileo's principle of free fall is not violated, and a body would not fall in a gravitational field with varying acceleration according to its energy content (Einstein 1911, 900-901). However, in Nordström's theory, which contains varying mass, a body falls in the same gravitational field with varying acceleration according to its energy content. Therefore, Galileo's principle of free fall is violated. An increase of inertial mass cannot correspond to an increase of gravitational mass, and of course the principle of equivalence is violated. Nordström admitted that his theory was not compatible with Einstein's principle of equivalence. However, he did not see a sufficient reason to reject the theory. For even though Einstein's principle of equivalence was extraordinarily ingenious, Nordström thought it still provided great difficulties, especially when one reconciled it with Minkowski's spacetime formalism and special relativity – because then one had to assume that the speed of light c depended upon the gravitational potential.

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Nordström, therefore, reasoned that other attempts at treating gravitation were desirable and he wished to provide a contribution to such an attempt (Nordström 1912, 1129). Remember that, quite immediately after formulating special relativity, Einstein rejected the idea of taking the Poisson equation and adapting it to special relativity – rendering it Lorentz covariant. He attempted to treat the gravitational law within the framework of the special theory of relativity. Sometime between 1905 and 1907, Einstein had already tried to extend the special theory of relativity in such a way so as to explain gravitational phenomena. Actually, this was the most natural and simplest path to be taken. Einstein tried to establish a field-law for gravitation by retaining the scalar gravitational potential, and completing the Poisson equation in an obvious way by a term differentiated with respect to time in such a way, so that the special theory of relativity was satisfied without changing the postulate of the constancy of the velocity of light. Finally, Einstein rejected this idea (Einstein 1934, 250-251, 1954, 286-287). However, scientists at that time, and even afterwards in 1912, attempted this path. Nordström, in his 1912 theory, did just that. Abraham and Nordström – both from Göttingen (Minkowski's great mathematical centre) – criticised Einstein's 1912 theory of static gravitational fields for its failure to implement Minkowski's reformulation of special relativity in terms of a four-dimensional space-time. Einstein, however, could not yet formulate his 1912 theory of static gravitational field in terms of Minkowski's four-dimensional space-time formalism. In fact, Einstein was already busily involved with Minkowski's formalism in 1912, and he had started to study Minkowski's four-dimensional reformulation of the special theory of relativity in earnest around 1910 – before his polemic with Abraham had erupted (Einstein 1955, 9-17, 1214). However, he did not use this formalism in his theory of static gravitational fields of 1912 because he recognised that Minkowski's basic idea of a constant light speed contradicted his theory, at the basis of which stood the idea of a variable speed of light. However, Einstein understood that Minkowski's formalism was crucial for the further development of his theory of gravitation, and that a successful application of Minkowski's formalism to the problem of gravitation called for a mathematical generalisation of this formalism and a reformulation of his 1912 theory of gravitation.

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4. Einstein's 1912-1913 Collaboration with Marcel Grossmann: Zurich Notebook to Entwurf Theory Sometime between February and March 1912, Einstein started to write a manuscript on the special theory of relativity at the request of the Leipzig physicist, Erich Marx, who hoped to have a contribution on relativity theory from Einstein for his Handbuch der Radiologie (Handbook of Radiology). Although the precise circumstances remain obscure, it appears that Einstein worked on this article off and on in Prague and Zurich from 1912 to 1914, producing a 72-page text (Rowe 2008, 55). It was probably in Zurich where he wrote a lengthy exposition on Minkowski's fourdimensional space-time formalism. By 1912, Einstein had come to understand the great importance of Minkowski's concepts, and incorporated his mathematical structure into his own way of thinking. The 1912 manuscript on the special theory of relativity is the first evidence of Einstein's use of Minkowski's formalism (Einstein 1912f, 43-54, introduction, 34). In May 1912, Einstein found the appropriate starting point for a generalisation of Minkowski's formalism. The principles of Euclidean geometry determine the relative positions of rigid bodies. Einstein realised that the most basic concept in geometry is the distance among bodies. The concept of the straight line may be reduced to that of distance. Distance is denoted by a rigid body on which two material points have been specified. We express a distance in terms of the space-interval between these two points. Einstein applied Pythagoras' theorem to infinitely near points: ݀‫ ݏ‬ଶ ൌ ݀‫ ݔ‬ଶ ൅ ݀‫ ݕ‬ଶ ൅ ݀‫ ݖ‬ଶ ǡ where ds2 denotes the interval between them. In space-time every event is determined by the space-coordinates x, y, z, and the time-coordinate t. Thus, the physical description is fourdimensional. Two infinitely near points can be connected by means of a light-signal if the relation: ݀‫ ݏ‬ଶ ൌ ݀‫ ݔ‬ଶ ൅ ݀‫ ݕ‬ଶ ൅ ݀‫ ݖ‬ଶ െ ܿ ଶ ݀‫ ݐ‬ଶ ൌ Ͳǡ

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holds for them. ds is invariant and has a value that, for infinitely close space-time points, is independent of the particular inertial system selected. ds may be measured by means of measuring-rods and clocks. In 1908, Minkowski built up, on the basis of the invariant ds, a fourdimensional geometry, which is somewhat analogous to the threedimensional Euclidean geometry. The four-dimensional Minkowski flat space-time is distinguished from that of Euclidean geometry in that ds2 may be greater or less than zero. Corresponding to this we differentiate between timelike and spacelike line elements. The boundary between them is marked out by the light-cone ds2 = 0 which starts out from every point. If we consider only elements which belong to the same time-value (distances at rest): ܿ ଶ ݀‫ ݐ‬ଶ ൌ Ͳǡ and as before, Euclidean geometry holds for these elements (Einstein 1929). Consider a flat four-dimensional space-time, three space time coordinates and one time coordinate: ‫ ݔ‬ൌ ‫ݔ‬ଵ ǡ ‫ ݕ‬ൌ ‫ݔ‬ଶ ǡ ‫ ݖ‬ൌ ‫ݔ‬ଷ ǡ ‫ ݐ‬ൌ ‫ݔ‬ସ Ǥ The above four coordinates may be written as xQ, where the suffix Q takes on the four values 1, 2, 3, 4. Consider another neighbouring point, infinitely close to the point xQ, and whose coordinates are xQ + dxQ. Thus the relation, the line element (interval): ݀‫ ݏ‬ଶ ൌ ݀‫ ݔ‬ଶ ൅ ݀‫ ݕ‬ଶ ൅ ݀‫ ݖ‬ଶ െ ܿ ଶ ݀‫ ݐ‬ଶ ǡ holds; ds2 is invariant with respect to linear transformations of the differentials dxQ In tensorial formulation the line element is written in the following form: ݀‫ ݏ‬ଶ ൌ ෍ ݀‫ݔ‬ఔ ଶ Ǥ ఔ

Thus, ds2 is the invariant distance between the two neighbouring points xQ and xQ + dxQ;and it is invariant under the Lorentz group.

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In May 1912, Einstein added a note to his March paper on the static gravitational fields (Einstein 1912b). At the end of the note he considered a material point that moves in a static gravitational field without being acted upon by external forces. He derived the equations of motion of the freely moving material point from a variation principle in its Lagrangian form: ߜ ൜න ݀‫ݐ݀ܮ‬ൠ ൌ ߜ ൜න ඥܿ ଶ ݀‫ ݐ‬ଶ െ ݀‫ ݔ‬ଶ െ ݀‫ ݕ‬ଶ െ ݀‫ ݖ‬ଶ ൠ ൌ ͲǤ Einstein explained that the variation principle provides an explanation of how the equations of motion of a material point are constructed for the dynamical gravitational field (Einstein 1912b, 458). Indeed, a year later, Einstein included in the above explanation the metric tensor. He considered a system K, in which the gravitational field was static, and another system K' in which the gravitational field was arbitrary. With respect to K', he derived the equations of motion of the freely moving material point from a variation principle and wrote the following equation: ߜ ൜න ݀‫ݏ‬ൠ ൌ ߜ ൜න ඥܿ ଶ ݀‫ ݐ‬ଶ െ ݀‫ ݔ‬ଶ െ ݀‫ ݕ‬ଶ െ ݀‫ ݖ‬ଶ ൠ ൌ ͲǤ where: ݀‫ ݏ‬ଶ ൌ ෍ ݃ఓఔ ݀‫ݔ‬ఓ ݀‫ݔ‬ఔ Ǥ ఓఔ

Einstein concluded that the quantities gPQ form a covariant tensor of the second rank, which he called the "covariant fundamental tensor" (the metric tensor). He said that the metric tensor determines the gravitational field (Einstein and Grossmann, 1913, 7-8). The metric tensor determines intervals between two neighbouring points. It relates the two neighbouring points to the space-time line element (interval) separating them. gPQ is a symmetric matrix. The metric for flat space-time is a diagonal matrix:

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0100 0100 0010 0 0 0 -1 which can be specified compactly by writing: ݃ఓఔ ൌ ݀݅ܽ݃ሺͳǡͳǡͳǡ െͳሻǤ At approximately the time when Einstein left Prague and returned to Zurich, he recognised that the gravitational field should not be described by a variable speed of light, as he had attempted to do in his 1912 coordinate-dependent theory of static gravitational fields (his papers from February 1912, "The Speed of Light and the Statics of the Gravitational Fields", and March 1912, "On the Theory of the Static Gravitational Fields"). Instead, he realised that the gravitational field is described by a metric, a symmetric tensor-field of metric gPQ (Einstein 1955, 15). Hence, in Prague, Einstein replaced the gravitational potential by the variable speed of light, and in Zurich the gravitational potential was replaced by the metric tensor gPQ. Einstein assumed that ds2 was the interval between two space-time points, or the square of the four-dimensional interval between two infinitely close space-time points. The interval that corresponds to dx1, dx2, dx3, dx4 differentials can be determined if one knows the metric tensor components gPQ that determine the gravitational field. Einstein then concluded that from the line element: ݀‫ ݏ‬ଶ ൌ ෍ ݃ఓఔ ݀‫ݔ‬ఓ ݀‫ݔ‬ఔ ǡ ఓఔ

one sees that in order to fix the physical dimensions of the quantities gPQ and xQ, ds contains the dimensions of length, as do xQ and x4 (the so-called time). He did not ascribe any physical dimension to the quantities gPQ. Starting in summer-spring 1912, Einstein undertook a long journey in the search of the correct form of the field equations for his new gravitation theory. He began collaborating with his friend from school, Marcel Grossmann, who was now a professor of mathematics in Zurich.

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Sometime upon his arrival, Einstein spoke about his concern with Grossmann and told him one day, "Grossmann, you have to help me, or I shall go crazy!" and Marcel Grossmann had managed to show him that the mathematical tools he needed, were created just in Zurich in 1869 by Christoffel in the treatise "On the Transformation of the Homogeneous differential Forms of the Second Degree", published in Volume 70 of the Journal de Crelle for Pure and Applied Mathematics (Kollros, 1955, 27). Grossmann helped him in his search for a gravitational tensor, and gradually contributed more and more mathematical tools to Einstein. Grossmann was happy to collaborate on the theory of gravitation with Einstein, but with the restriction that he would not be responsible for any statements in physics and would not assume any interpretations of physical nature. Grossmann looked through the literature, and discovered that Einstein's mathematical problem could best be solved by tensor calculus and differential geometry developed by Riemann, Ricci, Levi-Civita and Christoffel (Einstein 1955, 16; Kollros, 1955, 27). Einstein took into consideration a non-flat four dimensional space-time, i.e., a curved geometry of space-time. He realised that gravitation is described as a curvature of space-time with matter acting as the source of the curvature of space-time. He intended to represent gravitation as a curvature of space-time, and tensors define the curvature of space-time as covariant with respect to arbitrary (non-linear) transformations of the space-time coordinates; that is to say, tensors define the curvature of space-time in a form that remains unchanged under all coordinate transformations, independent of the choice of the coordinate system. A form which remains unchanged under all coordinate transformations is a representation of a generalised principle of relativity. This also represents the principle of equivalence, according to which gravitation is indistinguishable from acceleration: All physical processes in a gravitational field occur in the same way as they would without the gravitational field, if one related them to an appropriately uniformly accelerated coordinate-system. The 1907 free falling man from the roof cannot tell by any local experiment whether he is in an inertial or a noninertial frame. As we have seen this principle was founded upon a fact of experience – that of the equality of inertial and gravitational masses. In 1912, Einstein started to work with tensors and jotted gravitational equations in the Zurich Notebook, while Grossmann gradually updated him with new mathematical tools. During Einstein's first collaboration

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with Grossmann, he studied field equations – as evidenced in the Zurich Notebook – that were very close to the ones he would eventually choose when he finally presented his November 1915 general theory of relativity. These 1912 gravitation field equations had a broad covariance. However, during the winter of 1912-1913, Einstein discarded what he later (in 1915) claimed to be essentially this correct gravitational field equation. He rejected the calculations that he had done in the final pages of the Zurich Notebook, the field equations of much broader covariance, and adopted the non-covariant Entwurf field equations of 1913 with Marcel Grossmann. This Section examines Einstein's route, from the beginning of his collaboration with Grossmann, to his choice of the Entwurf field equations. The back side of the Zurich Notebook begins with Minkowski. On this page, numbered 32R, Einstein wrote the four space-time coordinates: x, y, z, ict, and x1, x2, x3, x4. He initially considered Minkowski's four-vectors, and then (on page 39L) he wrote the general four-dimensional line element; this was very likely the first time Einstein wrote this expression, because the coefficients GȝȞ of the metric tensor were written with an upper case G (Einstein 1912c, 318-321, 346, 348; Janssen, Renn, Sauer, Norton and Stachel, 2007, 503): ݀‫ ݏ‬ଶ ൌ ෍ ‫ܩ‬ఓఔ ݀‫ݔ‬ఓ ݀‫ݔ‬ఔ Ǥ ఓఔ

On the lower half of page 39L, Einstein searched for the gravitational equations, how the components of the gravitational field or the metric tensor, are generated by source masses. Einstein considered two special cases for G44 = c2: 1) In the first case c2 is a constant (c is the velocity of light) and the metric represents the Minkowski flat space-time of special relativity. The theory thus reduces to slowly moving matter. Recall that according to special relativity in inertial systems the line element: ݀‫ ݏ‬ଶ ൌ ݀‫ ݔ‬ଶ ൅ ݀‫ ݕ‬ଶ ൅ ݀‫ ݖ‬ଶ െ ܿ ଶ ݀‫ ݐ‬ଶ

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is invariant under the Lorentz group. The metric tensor in the special relativity limit, i.e. for systems with velocities that are small compared to the velocity of light is therefore diagonal (1, 1, 1, –c2). The three space components are equal to +1 and the time component is equal to –c2 or –1. One can also write it in the following way: (–1, –1, –1, c2). A free material point moves with respect to this flat metric system uniformly, in a straight line. 2) Einstein, however, also wanted to deal with systems with weak static gravitational fields. In the second case, c2 is a function of the spatial coordinates, and it represents the metrical generalisation of Einstein's 1912 theory of static gravitational theory. The metric of the form, metric tensor diagonal (– 1, – 1, – 1, c2) in which c2 is constant, represents the limit of special relativity, and not the limit of weak gravitational fields where c2 is variable and is therefore a function of the spatial coordinates, x1, x2, x3. In 1912, Einstein held a conception that, in the weak-field approximation, the spatial metric of a weak gravitational field must be flat. Empirically taking the limit as the metric tensor diagonal (– 1, – 1, – 1, c2), which could represent both the limit of special relativity and the limit of weak gravitational fields was mere coincidence that worked. Einstein expected that in the special case of a static field, and with an appropriate choice of coordinates, the gravitational field equations would reduce to his March, 1912 nonlinear static field equation (Einstein 1912c, 346, 348): ଵ

…‫ ܿ׏‬െ ሺ݃‫ܿ݀ܽݎ‬ሻଶ ൌ ߢܿ ଶ ıǤ  ଶ

Einstein thought that the gravitational field equations should reduce in the limit of weak-fields to his static gravitational field theory from 1912 and then to the Newtonian limit. He also thought the number of gravitational potentials would reduce from ten to a single potential, but he confronted the more difficult challenge – to reverse this reduction. He wanted to put the above equation into malleable form, to allow it to be one component of the 10-component metric tensor field equation. Einstein then rewrote the left hand-side of the above equation in terms of the components of the metric tensor of G44 = c2 from the special case (Janssen, Renn, Sauer, Norton and Stachel, 2007, 504-505).

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Following Ricci and Levi-Civita, Einstein represented contravariant components of a tensor by raised indices and covariant components by lowered indices. On page 41R, Einstein used the Newtonian equation of motion of a particle not subject to external forces, but constrained to move on a curved surface, to show that the trajectory traced by the particle in the surface is a geodesic. The geodesic, a curve of shortest distance, is determined by the variational principle (Einstein 1912c, 355, 357; CPAE 4, Doc. 10, 209): ߜ න ݀‫ ݏ‬ൌ Ͳቌ݀‫ ݏ‬ଶ ൌ ෍ ‫ܩ‬ఓఔ ݀‫ݔ‬ఓ ݀‫ݔ‬ఔ ǤቍǤ ఓఔ

The geodesic equation for a point particle is given by: ͳ ߲ ݀‫ݔ‬ఔ ݀‫ݔ‬௜ ݀‫ݔ‬௠ ݀ ൬݃ ൰െ ݃ ൌ ͲǤ ʹ ߲‫ݔ‬ఓ ௜௠ ݀‫ݏ݀ ݏ‬ ݀‫ ݏ‬ఓఔ ݀‫ݏ‬ This expression is not given in the notebook, but can easily be reconstructed (Janssen, Renn, Sauer, Norton and Stachel, 2007, 518, note 57). Using the integral function L as the action, a function of the coordinates and their derivatives, we evaluate the integral: න ‫߬݀ ܮ‬ǡ along a curve between two points. By using calculus of variations, we integrate along different curves (paths). This provides different values for the integral. However, we want to find a path that results in an integral being a minimum or maximum (most commonly a minimum). We now perform an infinitesimal variation of the path. L remains constant between the two end points. If the Lagrangian action does not change under small variations of the path: ߜ න ‫ ߬݀ ܮ‬ൌ Ͳǡ then we know that we have found the shortest path (geodesic line). This is the variational principle.

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On page 05R, Einstein started to use the front of his notebook for calculations, which he titled "Gravitation". By this point, Grossmann had probably provided Einstein with more mathematical tools, because the mathematical style had changed abruptly from the so-called beginning of the notebook. Einstein was searching for the stress-energy tensor TȝȞ for the gravitational field associated with a gravitation tensor he had already introduced (Einstein 1912c, 383, 385). At the top of page 05R, Einstein started with the equations of motion of a material particle in a metric field from an action principle. The action integral was the proper length of the particle's world-line. The equations of motion of the material particle are written in the form of the EulerLagrange equations. Einstein then applied this to a cloud of pressureless dust particles in the presence of a gravitational field: He generalised these results to expressions for the momentum density and the force density in the case of the cloud of dust, and identified the expression for momentum density as part of the stress-energy tensor for pressureless dust: ܶ௜௠ ൌ ߩ଴

݀‫ݔ‬௜ ݀‫ݔ‬௠ Ǥ ݀‫ݏ݀ ݏ‬

U0 is the proper density of a mass particle of dust. Subsequently, Einstein inserted this stress-energy tensor and a similar expression for the density of force acting on the cloud of dust into his initial equations of motion. On page 05R, he thus arrived at a candidate for the law of energy-momentum conservation in the presence of a gravitational field; he wrote an equation that expresses the vanishing of the covariant divergence of the stress-energy tensor TȝȞ (pressureless dust) and of the gravitational field, respectively: ߲݃ఓఔ ߲ ͳ ൫ξ‫݃ܩ‬ఓఒ ܶఓఔ ൯ െ ξ‫ܩ‬ ܶ ൌ Ͳǡ ߲‫ݔ‬ఔ ʹ ߲‫ݔ‬ఒ ఓఔ where G is determinant of gPQ. In the limit of special relativity, the metric in diagonal form has one positive and three negative values: gPQ = diagonal (– 1, – 1, – 1, 1). From this it follows that the determinant G is always negative ξെ‫ ܩ‬for a real space-time. In the Zurich Notebook Einstein wrote ξ‫ ܩ‬and not ξെ‫ܩ‬Ǥ

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The above equation expresses the requirement that the stress-energy tensor has zero divergence. Thus, conservation of energy-momentum of the matter field and gravitational field is satisfied. The equation is generally covariant (Einstein 1912c, 383, 385; Janssen, Renn, Sauer, Norton and Stachel, 2007, 516). In the next pages, Einstein becomes increasingly sophisticated – mathematically, but remains unacquainted with the Riemann tensor. What is very noticeable is Einstein's calculation, which could either be part of the field equations or play a role in their construction. However, Einstein's calculations did not yet lead to any promising candidates for the left-hand side of the field equations. They did lead to several important techniques, results and ideas that Einstein was able to put to good use upon learning about the Riemann tensor. He was now going to investigate the covariance properties of various expressions in order to find generally covariant field equations. Einstein tried to form invariant quantities from the metric tensor, and he tried to extract field equations in this way (Einstein 1912c, 386-417; Janssen, Renn, Sauer, Norton and Stachel, 2007, 523). On page 14L, Einstein finally and systematically started to explore the Riemann tensor. At the top of page 14L, he wrote on the left: "Grossmann's tensor four-rank", and next to it on the right he wrote the fully covariant form of the Riemann tensor (Einstein 1912c, 418, 420). Therefore, Grossmann very likely conveyed to Einstein the knowledge about the Riemann tensor. This signified a new stage in Einstein's search for gravitational field equations. In the course of this exploration he considered candidate field equations based on the Ricci tensor which he would come back to only on November 4, 1915. He explored the Ricci tensor for a few pages, and on page 26L he rejected his results from the pages starting at page 14L. He finally ended with limited generally covariant field equations, the Entwurf field equations (Janssen, Renn, Sauer, Norton and Stachel, 2007, 603). Abraham Pais says that the transition to Riemannian geometry must have taken place during the week prior to August 16, 1912; he dates this according to the letter that Einstein had written to Ludwig Hopf, and he adds that these conclusions are in harmony with his own recollections of a discussion with Einstein (Pais 1982, 212). Einstein told Hopf that with gravitation it was going brilliant. If he was not deceived, then he had now found the most general equations (Einstein to Hopf, August 16, 1912, CPAE 5, Doc. 416). The most general equations were the ones that

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Einstein wrote on page 14L of the Zurich Notebook. Thus the 14L page of the Zurich Notebook – in which the Riemann tensor made its first appearance – could very likely be dated to the week prior to August 16, 1912, and sometime around this date. The pages preceding the 14L page – in which Einstein was searching for candidates – could presumably be dated to the beginning of August 1912. Einstein contracted the fourth rank Riemann tensor to form the covariant form of the second rank Ricci tensor. At the bottom of the page, Einstein checked whether the Ricci tensor reduces in the weak-field approximation to the Newtonian limit. Einstein wrote this in the form of an equation in which three of its four second-order derivative terms vanish in the weakfield approximation. Below the expression he wrote that it "should vanish" (Einstein 1912c, 418, 420). The problem – how to cause the Newtonian limit to appear – was going to bring Einstein straight to pages 26L and 26R and to the limited covariant field equations. Meanwhile, Einstein tried to manipulate the Riemann tensor (Einstein 1912c, 421-429). At some stage he was frustrated with the mathematics that Grossmann had brought to him, and he wrote on page 17L that it was too complicated (Einstein 1912c, 430, 432). Generally covariant equations hold in all coordinate systems, whereas the equations of Newtonian gravitation theory do not. In the process of recovering the Poisson equation of Newtonian theory for weak-fields from a generally covariant theory, it is necessary to restrict the set of coordinate systems under consideration. This is achieved through coordinate conditions that must also be satisfied by the final solution. On page 19L, Einstein tried to recover the Newtonian limit from his generally covariant field equations with the harmonic coordinate condition (used to eliminate unwanted second order derivative terms from the Ricci tensor) (Einstein 1912c, 433-437). The Ricci tensor from page 14L appeared again, and the calculation terminated at the top of the page. The field equations that Einstein placed on page 19L in first order approximation were written again on page 19R. The left hand side of these field equations included a reduced Ricci tensor, while the right hand side included the covariant stress-energy tensor for pressureless dust multiplied by the gravitational constant N. Einstein constructed the field equations out of the Ricci tensor that satisfied the case of first order, weak-field approximation, and recovered

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the equations of the Newtonian limit (Einstein 1912c, 438-441; Janssen, Renn, Sauer, Norton and Stachel, 2007, 627). However, as he was examining the generally-covariant Ricci tensor expression to determine whether a physically acceptable field equation could be extracted from it, he assumed that he needed an additional condition, not just to recover the Poisson equation for weak-fields, but also to guarantee that the equations be compatible with the law of energymomentum conservation. On page 19R, Einstein checked energy and momentum conservation for the resulting gravitational field equations in the case of weak-fields. However, he then discovered a problem. He began by writing that for the first-order approximation our additional condition was the (linearized) harmonic coordinate condition: ෍ ߛ఑఑ ൬ʹ ఑

߲݃௜఑ ߲݃఑఑ െ ൰ ൌ ͲǤ ߲‫ݔ‬఑ ߲‫ݔ‬௜

The harmonic coordinate condition is: ߛ௜௞ ‫ؠ‬

ͳ

߲ ߲ߛ௜௞ ൬ξ‫ߛܩ‬ఓఔ ൰Ǥ ߲‫ݔ‬ఔ ξ‫ݔ߲ ܩ‬ఓ

Einstein then conjectured that the (linearized) harmonic coordinate condition could perhaps be decomposed into two extra conditions. The first condition was called by scholars the "Hertz condition", because it was later mentioned by Einstein in a letter to Paul Hertz (Einstein to Paul Hertz, August 22, 1915, CPAE 8, Doc. 111; Renn and Sauer, 2007, 184): ෍ ߛ఑఑ ఑

߲݃௜఑ ൌ ͲǤ ߲‫ݔ‬఑

This condition was related to the requirement of compatibility of the field equations and energy-momentum conservation. And the second condition:

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62

෍ ߛ఑఑ ݃఑఑ ൌ ܿ‫ݐݏ݊݋‬ǡ ఑

was related with compatibility of the field equations and Newtonian limit. Scholars call the above condition "a condition on the trace of the weakfield metric" (Janssen, Renn, Sauer, Norton and Stachel, 2007, 627). I will use this language. The trace of the metric tensor being constant (of special relativity) was incompatible with the metric tensor diagonal (– 1, – 1, – 1, c2) of a weak static gravitational field. In addition, it was incompatible with the field equations with the stress-energy tensor of matter for a cloud of pressureless dust: ܶ ൌ ߩ଴ ൌ ܿ‫ݐݏ݊݋‬Ǥ

ఓ

൫ܶ ൌ ܶఓ ൯Ǥ

The special relativity limit is achieved when gNN is equal to the following constant values: gNN = (g11, g22, g33, g44) = metric tensor diagonal (– 1, – 1, – 1, +1). Einstein wanted to satisfy the energy-momentum conservation and, at the same time, also the Newtonian limit. On page 19L, he introduced the harmonic coordinate condition to satisfy the weak-field approximation. However, the Hertz condition was troublesome for the above condition on the trace of the weak-field metric. Energy-momentum conservation was found to hold, but the two conditions into which the harmonic coordinate condition had been split contradicted each other; this meant an entanglement between the Newtonian limit and energy-momentum conservation. However, Einstein wrote at the bottom of page 19R that both the above conditions must be maintained (Einstein 1912c, 439,441; Janssen, Renn, Sauer, Norton and Stachel, 2007, 627). Einstein imposed the linearized harmonic coordinate condition. The field equations became the linearized field equations in a compact form: ᇝ݃௜௞ ൌ ߢܶఓఔǤ TPQ is the stress-energy tensor for a cloud of pressureless dust:

General Relativity between 1912 and 1916

ܶ௜௠ ൌ ߩ଴

63

݀‫ݔ‬௜ ݀‫ݔ‬௠ Ǥ ݀߬ ݀߬

The equation expressing energy-momentum conservation for pressurless dust is: ߲ܶ௜௠ ൌ ͲǤ ߲‫ݔ‬௠ Einstein needed to guarantee that the linearized field equations be compatible with the above law of energy-momentum conservation. He thus imposed the linearized Hertz condition: ߲݃௜఑ ൌ ͲǤ ߲‫ݔ‬఑ Hence, the linearized field equations and the conservation law implied: ᇝ൬

߲݃௜఑ ൰ ൌ ͲǤ ߲‫ݔ‬఑

Einstein then discovered that the combination of the harmonic coordinate condition and the Hertz condition caused a great problem: The trace of the linearized metric is constant: ݃௜௞ ൌ ߢܶ ൌ ܿ‫ݐݏ݊݋‬Ǥ ఓ

ܶ ൌ ܶఓ Ǥ For pressurless dust this means that: ݃௜௞ ൌ ߢߩ଴ ൌ ܿ‫ݐݏ݊݋‬Ǥ On page 20L Einstein again wrote the two conditions: ෍ ఑

߲݃௜఑ ൌ Ͳǡ ߲‫ݔ‬఑

the Hertz condition, and:

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෍ ݃఑఑ ൌ ܿ‫ݐݏ݊݋‬ǡ ఑

the condition on the trace of the weak-field metric. He then, however, crossed them out because the combination of these two conditions caused problems (Einstein 1912c, 442, 444). The Hertz condition was added to make sure that the divergence of the stress-energy tensor vanished in the weak-field case. However, combining of the two conditions, the harmonic and the Hertz coordinate condition, implied the necessity of eliminating any trace of the metric. Einstein used the harmonic and Hertz conditions to eliminate various terms from equations of broad covariance, and looked upon the truncated equations of severely restricted covariance, rather than upon the equations of broad covariance he started from as candidates for the fundamental field equations of his theory. Since coordinate conditions used in this manner are ubiquitous in the Zurich Notebook the scholars introduced a special name for them, "coordinate restrictions" (Norton 1984, 254). Recall that Einstein obtained on page 19L (and written again on page 19R) weak-field equations: ᇝ݃௜௞ ൌ ߢܶఓఔǤ The left hand side of these field equations included a reduced Ricci tensor and the right hand side, the covariant stress-energy tensor for pressureless dust multiplied by the gravitational constant N. By imposing the two conditions, it followed that the stress-energy tensor would have to be equal to zero. In order to avoid this problem, Einstein modified these weak-field equations and added a term with the trace of the stress-energy tensor on the right-hand side of the equations. This gave the right-hand side a trace equal to zero. With his new weak-field equations, Einstein managed to maintain the stress-energy tensor and the conservation principle. However, this solved only part of the problem caused by the combination of the harmonic coordinate and the Hertz conditions. It took care of the problem that a metric with a trace equal to zero would imply an energy-momentum tensor with a trace equal to zero; but, it did not address a second problem – that a

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metric field of the form Einstein used to represent static gravitational fields had a trace which was not equal to zero. Einstein thus crossed out his new field equations and again modified the weak-field equations from page 19L (Janssen, Renn, Sauer, Norton and Stachel, 2007, 633). Subsequently, Einstein modified the weak-field equations from page 19L by avoiding the condition that the metric had a trace which was equal to zero. The harmonic coordinate condition was imposed and the Hertz condition was removed. Einstein only imposed the harmonic coordinate condition, ensuring both the elimination of unwanted second-order derivative terms for the Ricci tensor and the vanishing of the divergence of the stress-energy tensor. With this new strategy Einstein was finally able to extract from the Ricci tensor linearized gravitational equations and recover the Newtonian limit successfully. He wrote the trace of the metric: ෍ ݃఑఑ ൌ ܷǤ ఑

Previously, when Einstein imposed the two conditions (the Hertz and the harmonic coordinate conditions), he was obliged to add a term with the trace of the stress-energy tensor on the right-hand side of the weak-field equations. Now, he was imposing only the harmonic coordinate condition, and so he added a term with the trace of the metric to the left-hand side of the weak-field equations: ͳ ο ൬݃ଵଵ െ ܷ൰ ൌ ܶଵଵ ǡο݃ଵଶ ൌ  ܶଵଶ ǡ ǥο݃ଵସ ൌ  ܶଵସ Ǥ ʹ Using the Kronecker G, these equations can be written in compact form: ͳ ᇝ ൬݃ఓఔ െ ߜఓఔ ܷ൰ ൌ ܶఓఔǡ ʹ ensuring the vanishing of the divergence of the stress-energy tensor by imposing: ͳ ߲ ൬݃ െ ߜ ܷ൰ ൌ Ͳǡ ߲‫ݔ‬఑ ఓఔ ʹ ఓఔ

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66

which is nothing but the harmonic coordinate condition. Einstein's modified weak-field equations (the penultimate field equations) had removed the need for the Hertz condition. Curiously, these weak-field equations had exactly the same form as the weak-field equations found in Einstein's final general theory of relativity of November 25, 1915. The left-hand side is the linearized version of the Einstein tensor: ଵ

ܴఓఔ െ ݃ఓఔ ܴ. ଶ

There is no indication in the notebook that Einstein tried to find the exact equations corresponding to these weak-field equations (Janssen, Renn, Sauer, Norton and Stachel, 2007, 634). Again Einstein invoked the Hertz condition, and of course arrived at the contradiction between the two coordinate conditions. This led to the troublesome additional condition for the trace of the weak-field metric, the condition for the linearized metric from page 19L (the latter was not satisfied by a metric of the form diagonal [– 1, – 1, – 1, c2]). Page 20L provides evidence of Einstein's attempt to avoid this problem by adding a trace term to the weak-field equations. However, he was then confronted by another problem: The metric of the form diagonal (– 1, – 1, – 1, c2) was no longer a solution to the modified weak-field equations. Page 21R illustrates Einstein's return to his static gravitational field equation from page 39L, and his consideration of the special case of the metric of static field. He expected to recover his March 1912 non-linear static gravitational field equation: ͳ …‫ ܿ׏‬െ ሺ݃‫ܿ݀ܽݎ‬ሻଶ ൌ ߢܿ ଶ ıǡ ʹ from his new metric tensor for the metric of the form (– 1, – 1, – 1, c2) where c2 = 1. However, the flat metric of the form (– 1, – 1, – 1, c2) represents the limit of special relativity, and the weak-field equations no longer provided a solution with a metric of this form (Einstein 1912c, 449, 346, 348). Before giving up his new field equations, Einstein wanted to re-check whether his weak-field metric was compatible with Galileo's experimental law of free fall and the equivalence principle. He arrived at the fallacious conclusion that the metric of the form diagonal (– 1, – 1, – 1, c2) was

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essential to Galileo's law of free fall (Janssen, Renn, Sauer, Norton and Stachel, 2007, 641). Recall that according to Galileo's experimental law of free fall, all bodies fall with the same acceleration in a given gravitational field. According to special relativity, the inertial mass is proportional to energy. If one mass fell differently from all others in the gravitational field then, with the help of this mass an observer in free fall (which for him locally his system is inertial) could discover that he was falling in a gravitational field. In this case, the acceleration of the falling mass would not be independent of the internal energy of the system and this violated special relativity (the inertia of energy). Therefore, Einstein concluded, according to the principle of equivalence, locally for the free falling observer and for an observer in a weak static field, the metric was represented by the form diagonal (– 1, – 1, – 1, c2). Finally, Einstein deleted the calculation of the static special case on page 21R and wrote at the bottom of the page that the special case was probably incorrect (Einstein 1912c, 447, 449). With that, Einstein gave up the modified field equations from pages 19L-20L. At this early stage, until 1915, he no longer considered field equations from the Ricci tensor with the help of the harmonic coordinate condition; nor did he examine modified weak field equations, similar to the Einstein field equations of November 1915 in linearized form (Janssen, Renn, Sauer, Norton and Stachel, 2007, 644). Einstein was not ready to give up his attempt to extract the left hand-side of the field equations from the Riemann tensor. Einstein took another approach to the problem of constructing a candidate generally covariant tensor from the Riemann curvature tensor. The Riemann curvature tensor (the Riemann-Christoffel tensor) describes the curvature in space-time geometry. The Riemann tensor has thirty-six components. If we arrange the thirty-six components in a symmetric matrix, then the Riemann tensor has only twenty independent components. If the twenty tensor components vanish in one coordinate system, they must vanish in them all. Einstein used the Ricci tensor, a contracted form of the Riemann's tensor, having only ten independent components, as the geometric description of the curvature of space-time. Einstein titled page 22R "Grossmann". Thus, perhaps at the suggestion of Grossmann, Einstein wrote another form of the Ricci tensor. This time, the Ricci tensor was in terms of the Christoffel symbols and their derivatives, rather than in terms of the metric tensor and its derivatives – as it appeared

Chapter Two

68

until then in the notebook. This was a fully covariant Ricci tensor in a form resulting from contraction of the Riemann tensor: ܶ௜௟ ൌ ෍ ఑௟

ܶ௜ ൌ

߲ ݅ߢ ߲ ݈݅ ݅ߢ ߣ݈ ݈݅ ߣߢ ቄ ቅെ ቄ ቅ ൅ ቄ ቅ ቄ ቅ െ ቄ ቅ ቄ ቅǡ ߣ ߢ ߣ ߢ ߲‫ݔ‬௟ ߢ ߲‫ݔ‬఑ ߢ

߲݈݃ξ‫ܩ‬ Ǥ ߲‫ݔ‬௜

The first term in the expansion of the Ricci tensor is itself a tensor G of the first rank. Einstein divided the Ricci tensor into two parts – a tensor of second rank: ൬

߲ܶ௜ ݈݅ െ ෍ ܶఒ ቄ ቅ൰ǡ ߣ ߲‫ݔ‬௜

and a presumed gravitational tensor: ෍ ఑௟

߲ ݈݅ ݅ߢ ߣ݈ ቄ ቅ െ ቄ ቅ ቄ ቅǤ ߣ ߢ ߲‫ݔ‬఑ ߢ

The presumed gravitation tensor was called by scholars the "November tensor". Setting the November tensor equal to the energy-momentum tensor, multiplied by the gravitational constant N, one arrives at the field equations of Einstein's first paper of November 4, 1915 (Einstein 1912c, 451, 453). Page 22L details Einstein's investigation into the behaviour of the November tensor. Einstein no longer required the harmonic coordinate condition, and he could impose the Hertz coordinate condition to eliminate all unwanted second-order derivative terms (Renn, and Sauer 2007, 219220). He mentioned that he was further rewriting the gravitation tensor. The Hertz condition also ensured that the divergence of the linearized stress-energy tensor vanished, thus satisfying energy-momentum conservation law (Einstein 1912c, 450, 452). The bottom half of page 22R details Einstein's arrival at a candidate for the left-hand side of the field equations, extracted from the November tensor by imposing the Hertz condition. However, Einstein was still unable to satisfy the conservation of energy-momentum; so, on the next page (23L) he continued with another novel method to eliminate terms with unwanted

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second-order derivatives of the metric, and by which he could extract the Newtonian limit from the Riemann tensor. He thus abandoned the Hertz condition, and wrote that it was unnecessary (Einstein 1912c, 451, 453454, 456; Janssen, Renn, Sauer, Norton and Stachel, 2007, 554, 652). Einstein already accumulated coordinate conditions – or coordinate restrictions – to eliminate the terms from his equations. Pages 19L-23L include Einstein’s extracted expressions of broad covariance from the Ricci tensor. He then truncated them by imposing additional conditions on the metric to obtain candidates for the left-hand side of the field equations that reduce to the Newtonian limit in the case of weak static fields (Janssen, Renn, Sauer, Norton and Stachel, 2007, 497). However, this model entangled the Newtonian limit and conservation of momentumenergy. From one page to the next, in the Zurich Notebook, Einstein turned from one candidate equation to another to find the suitable left-hand side of the field equations that would be compatible with energy-momentum conservation. The equations satisfied the conservation of energymomentum in the weak-field limit, but the source term – stress-energy tensor of matter – of the gravitational field was incompatible with what Einstein had obtained for the Newtonian limit (Einstein 1912c, 438456,458; Janssen, Renn, Sauer, Norton and Stachel, 2007, 183-184, 214215). On page 24R Einstein tried to extract yet another candidate for the lefthand side of the field equations. He did not extract the candidate from the Ricci tensor while imposing coordinate conditions. He established field equations while starting from the requirement of the conservation of momentum and energy. The equations could be covariant with respect to linear transformations, and they satisfied both the Newtonian limit and conservation of momentum-energy. Einstein checked this candidate using the rotation metric. He thought that his expression vanished for the rotation metric, a necessary condition for the rotation metric to be a solution of the vacuum field equations. This was a mistake that Einstein would only discover much later (in October 1915). The expression vanishing for the rotation metric may have signified to Einstein that the new field equations satisfied the relativity principle and the equivalence principle. Einstein came to believe that his expression vanished for the rotation metric because of a sign error (Einstein 1912c, 459, 461; Janssen, Renn, Sauer, Norton and Stachel, 2007, 687).

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He would do quite the same sign error and come to a similar belief with respect to the rotation metric a year later in the Einstein-Besso manuscript. There, he checked whether the rotation metric was a solution to the new Entwurf field equations he was developing (CPAE 4, Doc. 14, 41, 445, note 186). Rather than correct the above error, on pages 25L and 25R Einstein attempted to recover the new field equations from the November tensor of page 22R, which he wrote again on top of page 25L (CPAE 4, Doc. 14, 41; Janssen, Renn, Sauer, Norton and Stachel, 2007, 682). Einstein still hoped to connect the new field equations he found through energy-momentum considerations to the November tensor of page 22R. On pages 25L-25R, he explicitly tried to recover field equations along this argument and from this tensor. At some point, he rejected his efforts at recovering his new field equations found through energy-momentum considerations from the November tensor, and he abandoned general covariance. He failed to connect the November tensor to his new field equations on page 24R. Einstein then wrote in the lower left corner of page 25R the word, "impossible" (Einstein 1912c, 462-465; Janssen, Renn, Sauer, Norton and Stachel, 2007, 681, 683, 704). This brought Einstein on the very next pages (26L and 26R) straight to the field equations – the Entwurf field equations – which he also established using the same method: through energy-momentum considerations. The problem remained whether these equations were covariant enough to enable extending the principle of relativity for accelerated motion and to satisfy the equivalence principle. Under the title, "System of Equations for Matter", Einstein derived gravitational field equations of limited covariance that were not derived from the Riemann tensor. These equations spread over two facing pages, 26L and 26R, and are displayed with a neatness and order rare among the other pages of the notebook, suggesting that they were transcribed from another place (probably from Einstein's and Grossmann's joint 1913 Entwurf paper) after the result was known. Einstein ended his gravitation calculations on page 26R, with the left-hand side of the Entwurf gravitation tensor (Einstein 1912c, 466-469; CPAE 4, Doc. 10, 263). As seen from examining the Zurich Notebook, three years before November 1915, Einstein had written on page 22R the November tensor, when he considered the Ricci tensor as a possible candidate for the left

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hand side of his field equations. Einstein got so close to his November 1915 breakthrough at the end of 1912, that he even considered, on page 20L, another candidate – albeit in a linearized form – which resembles the final version of the November 25, 1915, field equation of general relativity. Einstein, therefore, first wrote down a mathematical expression close to the correct field equation and then discarded it, only to return to it more than three years later. Why did Einstein reject gravitational field equations of much broader covariance in 1912-1913, only to come back to these field equations in November 1915? Einstein believed that the special principle of relativity for uniform motion could be generalised to arbitrary motion if the field equations possessed the mathematical property of general covariance. If the principle of relativity was generalised, then the equivalence principle was satisfied. Accordingly, Einstein examined candidates for generally covariant field equations. Einstein's earlier work on static fields led him to conclude (on page 21R) that in the weak-field approximation, the spatial metric of a static gravitational field must be flat (Einstein 1912c, 447, 449). This statement appears to have led him to reject the Ricci tensor on page 22R, and fall into the trap of Entwurf limited generally covariant field equations. The Entwurf gravitational equations were thus incompatible with a general principle of relativity.  Einstein thought that the Ricci tensor should reduce in the limit to his static gravitational field theory from 1912 and then to the Newtonian limit, if the static spatial metric is flat [represented by a flat metric of the form diagonal (– 1, – 1, – 1, c2)]. This prevented the Ricci tensor from representing the gravitational potential of any distribution of matter, static or otherwise (Stachel 1989 in 2002, 304-306). Einstein's collaboration with Grossmann finally led to two joint papers: The first of these was published before the end of June 1913, and the second, almost a year later (Einstein and Grossmann 1913, 1914). Einstein wrote the physical part of the paper, and Grossmann wrote the mathematical part. Einstein and Grossmann's first joint paper entitled, "Outline of a Generalised Theory of Relativity and of a Theory of Gravitation" was called by scholars the Entwurf ("Outline") paper. In fact, Einstein himself also called this paper Entwurf, and he and Grossmann referred to the theory presented in this paper as the Entwurf theory

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(Einstein to Lorentz, 14 August, 1913, CPAE 5; Einstein and Grossmann 1914, 217). In the physical part of the 1913 Entwurf paper, Einstein explained the equivalence principle using the predecessor of the elevator thought experiment: An observer enclosed in a box can in no way decide whether the box is at rest in a static gravitational field, or whether it is in accelerated motion, maintained by forces acting on the box, in space that is free of gravitational fields (Einstein and Grossmann 1913, 3). In the Entwurf theory, the gravitational field was represented by a metric tensor, the mathematical apparatus of the theory was based on the work of Riemann, Christoffel, Ricci and Levi-Civita on differential covariants. Einstein and Grossmann first established the system of equations for material processes when the gravitational field was considered as given. These equations were covariant with respect to arbitrary substitutions of the space-time coordinates. After establishing these equations, they went on to establish the field equations of gravitation, a system of equations which were regarded as a generalisation of the Poisson equation of Newton's theory of gravitation. Einstein sought the generalisation of Poisson's equation: ‫׏‬ଶ ĭ ൌ ͶɎ ɏǤ The generalisation Einstein was seeking would likely have the form: Ȟஜ஝ ൌ ߢȣఓ஝ ǡ where Nis a constant, analogous to the Newtonian gravitation constant G, 4PQ is analogous to the source mass-density of the Poisson equation U, and *PQ is a second-rank tensor derived from the metric tensor gPQ by differential operations. These equations determined the gravitational field, provided that the material processes were given (Einstein and Grossmann 1913, 11). In line with the Newton-Poisson equation, Einstein presumed that the above equations would be of second order. However, he was later unable to find a differential expression *PQ that was a generalisation of ')and

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that proved to be a tensor with respect to arbitrary transformations. Einstein and Grossmann knew that the covariant differential tensor of second rank – the Ricci tensor – would have been the natural candidate for *PQ. However, Einstein had already checked this option in the Zurich Notebook and found that the Ricci tensor did not reduce in the weak-field limit to his static gravitational field theory from 1912; hence, in the special case of the infinitely weak, static gravitational field this tensor did not reduce to the expression '). Einstein admitted that there could be a possibility that the perfectly exact differential equations of gravitation, the field equations, would be, after all, covariant with respect to arbitrary transformations. But, he explained that given the present state of his knowledge in 1913 of the physical properties of the gravitational field, the attempt to discuss such possibilities would be premature. Therefore, given the limitation of the second order, i.e. the limitation of the adaptation to the Newton-Poisson equation in the limit of weak fields, Einstein felt he was obliged to forgo establishing gravitational equations that were covariant with respect to arbitrary transformations. Einstein decided to follow the spirit of Poisson's equation, and with heavy heart, to give up searching for generally covariant field equations – equations of gravitation that are covariant with respect to arbitrary transformations. Following the correspondence principle, the Newtonian limit, the field equations were covariant only with respect to a particular group of transformations. This group was unknown to Einstein at this stage. However, given the special theory of relativity, Einstein reasoned that it was natural to assume that the transformation group he was seeking also included the linear transformations. Although he required *PQ to be a tensor with respect to any or arbitrary transformations, he obtained an expression for a covariant tensor of the second rank with respect to the linear transformations (Einstein and Grossmann 1913, 12). In contrast to the equations for material processes, Einstein and Grossmann's derivation of the gravitational equations was assumed – in addition to the conservation laws – only upon the covariance with respect to linear transformations. Einstein felt that this issue was crucial, because of the equivalence principle. His theory depended upon this principle. His desire was that acceleration-transformations (non-linear transformations) would become permissible transformations in his theory. In this way, transformations to accelerated frames of reference would be allowed and the theory could generalise the principle of relativity for uniform motions.

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In May 1913, several weeks before the Entwurf paper was about to be published, Einstein was confident in the Entwurf theory (Einstein 1922, 47), but he still maintained doubts and was unsatisfied with the Entwurf field equations. Although he was deeply convinced of having made the right decision, he felt that he only partially succeeded in formally penetrating the problem, due to his continued lack of generally covariant field equations (Einstein to Ehrenfest, May 28, 1913, CPAE 5, Doc. 441). Subsequently he believed for a while – or persuaded himself – that generally covariant field equations were not permissible; one must restrict the covariance of the equations. He later introduced an ingenious argument – the Hole Argument – to demonstrate that generally covariant field equations were not permissible: Let there be in the four dimensional space-time a hole L (L stands for Loch, hole), in which material processes do not occur. Therefore, the components of the stress-energy tensor 7PQ vanish. The stress-energy tensor 7PQ given outside the hole, L, as requested by our assumptions, therefore also completely determines everywhere the components of the metric tensor gPQ inside the hole, L. We now imagine that, instead of the original coordinates, xQ, new coordinates x'Q are introduced of the following type: Outside of the hole, L, xQ = x'Q everywhere; inside L, however, for at least a part of L and for at least one index Q, xQ z x'Q. Einstein said that it was obvious that by means of such a substitution it could be achieved that, at least for a part of the hole, L, g'PQz gPQ. On the other hand, 7'PQ = 7PQ everywhere, namely outside of L, because outside the hole xQ = x'Q, and inside the hole, 7PQ = 0 = 7'PQ. Einstein, therefore, concluded that in the case considered, if all substitutions were allowed, then to the stress-energy tensor 7PQ belonged more than one metric tensor gPQ. However, we require that the metric tensor gPQ should be completely determined by the stress-energy tensor 7PQ, and since the above Hole Argument contradicts this requirement, then one is forced to limit the choice of the coordinate systems (that is to say, limit the covariance of the field equations) (Einstein and Grossman 1913, 260).

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The earliest version of the Hole Argument might be found at the bottom of the second page of what appears to be Besso's notes of discussions with Einstein, dated from August 1913 (what scholars call, Besso's memo) (Janssen 2007, 821). Chapter 1 (of this book) mentioned that in June 1913, Besso visited Einstein in Zurich, and they both tried to solve the new Entwurf field equations to find the perihelion advance of Mercury in the field of the static Sun, in the Einstein-Besso manuscript (discussed in detail below after presenting the Hole Argument). Before Einstein and Besso finished their joint project of calculations in June 1913, Besso had to leave Zurich and return home to Gorizia where he lived at the time. Besso visited Einstein again in August 1913. During this time they continued to work on the project, and added more pages to the Einstein-Besso manuscript. Besso wrote in his memo about the failure of realising general covariance with the gravitational Entwurf field equations. He divided space-time into regions with and without matter. Besso imagined a central mass to be surrounded by empty space and wondered whether the solution for the metric tensor was, in this case, determined uniquely for the empty region. What Besso described, could be called an inverted Hole Argument or a Hole Argument without a hole. This formulation suggests that the argument concerns the uniqueness of the metric field of the Sun, which Einstein and Besso calculated in their attempt to account for the perihelion anomaly of Mercury on the basis of the Entwurf theory.The above version of the Hole Argument found in the Besso memo could be an embryonic version of the Hole Argument that Einstein had just told Besso about during his August visit in Zurich (Janssen 2007, 821). On the other hand, the above version of the Hole Argument found in the Besso memo could have originated with Besso, and might have been Besso's idea after all. Besso could have presented it in his memo and brought this idea to the meeting with Einstein in August 1913. Then, after the meeting in August 1913, the idea could have transformed into Einstein's Hole Argument (Renn and Sauer 2007, 239-240). It appears that in August 1913 Einstein did not yet possess the Hole Argument, which he later thought would solve his problem. Two letters that he wrote to Lorentz on August 14 and 16, 1913 – at about the time of Besso's visit to Zurich – detail his dissatisfaction with the Entwurf field

76

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equations. He told Lorentz that, unfortunately, the gravitation equations do not possess the property of general covariance. Einstein felt that the theory refuted its own starting point. He then suggested a solution to his problem: Assuming the law of momentum and energy conservation, his gravitational equations were never absolutely covariant. If we restricted the choice of the reference systems, with respect to which the law of momentum and energy conservation holds, then general linear transformations remained the only right choice. Einstein did not yet mention the Hole Argument (Einstein to Lorentz, August 14, 16, 1913, CPAE 5, Doc. 467, 470). On September 9, 1913, Einstein gave a lecture in Frauenfeld, titled, "Physical Foundations of a Theory of Gravitation" (Einstein 1913a). In this lecture, Einstein could not yet deal with the fact that his gravitational field equations were not covariant with respect to arbitrary, but only covariant with respect to linear transformations. Two weeks later, on September 23, 1913, Einstein attended the eightyfifth Congress of the German Natural Scientists and Physicists in Vienna. There he presented another talk, "On the Present State of the Problem of Gravitation" concerning his Entwurf theory. He also engaged in a dispute after this talk with scientists who opposed his theory, especially Max Abraham and Gustav Mie. A text of this lecture with the discussion was published in the December volume of the Physikalische Zeitschrift. Einstein rewrote, in Section §7 of his Vienna paper, the gravitational equations he had obtained in the Entwurf 1913 paper (Einstein 1913a, 1258-1259). In regards to the field equations, there was little new in the Vienna talk. Before presenting the field equations he said that, the whole problem of gravitation would be solved satisfactorily, if one were able to find field equations covariant with respect to any arbitrary transformations that are satisfied by the metric tensor components gPQ that determine the gravitational field itself. However, he had not succeeded in solving that problem in this manner. Einstein had added a footnote, which indicated that he did arrive at some new idea. He said that in the last few days, he had found proof that such a generally covariant solution could not exist at all (Einstein 1913a, 1257). This footnote, however, appeared in the printed version of the Vienna lecture – published on December 15, 1913. Hence the footnote could be added only later, after September 1913. If it was added just before December 1913, a short time earlier, in November 1913, Einstein had an ingenious idea (Stachel, 1989, 308-309).

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Indeed on November 2, 1913, Einstein told Ludwig Hopf that he was now very happy with his gravitation theory. The fact that the gravitational equations were not generally covariant, which bothered him up till then, has proved to him now to be unavoidable. Einstein explained to Hopf that it could easily be proved that a theory with generally covariant equations could not exist (Einstein to Hopf, November 2, 1913, CPAE 5, 1913, Doc. 480); the proof was the Hole Argument. Einstein gave several formulations to the Hole Argument. The one provided above was the formulation that Einstein later added in a remark to Einstein and Grossman's 1913 Entwurf paper. One of the earliest formulations was the following: Let us formulate it in simple terms. Consider a finite portion of space-time where there is no matter. This is the hole. The stress-energy tensor vanishes inside the hole. The stressenergy tensor which is outside the hole, therefore, also determines completely everywhere the components of the metric tensor inside the hole (Einstein 1914a, 178). We now imagine the following substitution or transformation: Instead of the original coordinate system, a new coordinate system was introduced. The original coordinate system and the new coordinate system are different from one another inside the hole; but, outside and on the boundary of the hole they coincide. Inside the hole these two coordinate systems lead to two different metric fields. Outside the hole, the stress-energy tensor in the new coordinate system everywhere coincides with the stress-energy tensor of the original coordinate system. Inside the hole there is no matter and they are both zero. Hence, in the case considered, if the above substitutions are allowed, then the stress-energy tensor of the original coordinate system determines the components of two different metric tensors inside the hole: the metric tensor in the original coordinate system and in the new coordinate system. In this way, inside the hole, the metric tensor of the original coordinate system cannot be completely determined by the stress-energy tensor of this system. If the field equations are generally covariant, then inside the hole two different metric fields are produced by the same matter-field. The Hole Argument seemed to cause Einstein great satisfaction, or else he persuaded himself that he was satisfied. Having found the Hole Argument,

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Einstein spent two years after 1913 looking for a non-generally covariant formulation of gravitational field equations. Recall that in June 1913, Michele Besso visited Einstein in Zurich. Einstein and Besso both tried to solve the new Entwurf field equations to attain the perihelion advance of Mercury in the field of the static Sun in the Einstein-Besso manuscript. Actually, Einstein was already interested in the problem of Mercury's perihelion from early on in his search for a new relativistic theory of gravitation. He wrote to his close friend Konrad Habicht in 1907 that he hoped his gravitation theory would explain the anomalous advance of Mercury's perihelion (Einstein to Habicht, December 24, 1907, CPAE 5, Doc. 69). According to Kepler's laws, Mercury should move on an ellipse with the Sun in one of its focal points. The perihelion of the orbit is where the planet is closest to the Sun. This point is found on the major axis of the ellipse, its longest diameter, the line that runs through the centre and both its foci. The ellipse is not fixed in space but undergoes a slow precession. The major axis was found to slowly turn around the Sun, precess, because of the influence of other planets. The perihelion thus turned as well. This is the precession of the perihelion, and it is more pronounced the more the eccentricity is larger, for instance for Mercury which is closest to the Sun. For the perihelion of Mercury, Newtonian theory predicts (in a coordinate system at rest with respect to the Sun) a secular advance of about 570 seconds of arc per century (570"). In 1859, the French astronomer Urbain Jean Joseph Le Verrier, after working on the problem for many years, pointed out that there was a discrepancy of about 38" between the value that Newtonian theory predicts for the secular motion of Mercury's perihelion and the value that was actually observed. Le Verrier suggested that perturbations coming from an additional planet, located between the Sun and Mercury, were responsible for this discrepancy. In 1846, he had likewise predicted the existence of an additional planet to account for discrepancies between theory and observation in the case of Uranus. This planet, Neptune, was actually observed shortly afterwards, almost exactly where Le Verrier had predicted it to be. Vulcan, however, the planet to be made responsible for the discrepancy in the case of Mercury, was never found.

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In 1895, the American astronomer Simon Newcomb published a new value for the anomalous secular advance of Mercury's perihelion, based on the latest observations. He arrived at about 41". The modern value is about 43". The best explanation for the anomaly that Newcomb could find was a suggestion made a year earlier by Asaph Hall – that the gravitational force would not exactly fall off with the inverse square of the distance as in Newton's theory but slightly faster. Newton himself had already noted that any deviation from an exact inverse square law would produce a perihelion motion. Several other explanations were put forward for the anomaly, the most popular one was offered in 1906 by the German astronomer Hugo von Seeliger, who suggested the discrepancy was due to non-planetary matter (dust) between Mercury and the sun (Janssen 2002). Since the perihelion advance of Mercury could not be easily explained in the framework of Newtonian gravitation theory, it presented a good opportunity to theoretically check Einstein's newly Entwurf theory of gravitation. In the summer of 1913, Einstein and Besso searched whether Einstein's new Entwurf theory could produce the value of about 43". Besso was inducted by Einstein into the necessary calculations. In their manuscript, Einstein and Besso treated the problem in the following way: They considered the solar system as an isolated mass point (the Sun), which is far away from all other masses (galaxies and stars) in the universe. Most of the mass of the solar system is concentrated in this mass point (the Sun). The planets are mass points moving in the static gravitational field of the Sun. Einstein and Besso used an approximate method to solve the Entwurf field equations. They started with the flat Minkowski metric as the zeroth-order approximation. The Minkowski metric field components gPQ correspond to the special theory of relativity. They took the g44 component: െο݃ସସ ൌ ܿ ଶ ଴ ߢߩ଴ ǡ of the Entwurf field equation: ‫ܦ‬ఓఔ ሺ݃ሻ ൌ െߢ൫‫ݐ‬ఓఔ ൅ ܶఓఔ ൯ǡ the solution of which gave the metric field of the Sun in first-order approximation:

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80

‫ܣ‬ ݃ସସ ൌ ܿ ଶ ଴ ൬ͳ െ ൰Ǥ ‫ݎ‬ ' is the Laplacian, N is the gravitational constant, ߢ ൌ ‫ܭ‬

଼గ ௖ మబ

, U0 is the mass

density of the Sun, and c0 is the speed of light in vacuum (CPAE 4, Doc. 14, 1; "The Einstein-Besso Manuscript on the Motion of the Perihelion of Mercury", 346). They substituted the metric field of the Sun to first order in the Entwurf field equations and calculated, through successive approximations, the gravitational field of the Sun, assuming that the field of the Sun was spherically symmetric. Einstein and Besso substituted the first order solution in the Entwurf field equation and evaluated second order contributions (CPAE 4, Doc. 14, 6): ݃ସସ ൌ ܿ ଶ ଴ ቆͳ െ

‫ܣ ͵ ܣ‬ଶ ൅ ቇǤ ‫ ݎ‬ͺ ‫ݎ‬ଶ

The gravitational field of the Entwurf theory is represented by gPQ, and the perihelion advance of Mercury is calculated by first and second approximations of gPQ. In the first approximation, after applying the Entwurf theory to the static field of the Sun, the result is that the static metric remains flat. Einstein assumed weak gravitational fields and identified this with Newtonian theory. The next step was equations of motion for a mass point moving in the weak-field metric of a static Sun to second order (Euler-Lagrange equations for the action) (CPAE 4, "The Einstein-Besso Manuscript on the Motion of the Perihelion of Mercury", 349): ‫ ݏ‬ൌ න ‫ݐ݀ܮ‬ǡ with the Lagrangian: ‫ ܮ‬ൌ െ݉

݀‫ݏ‬ ǡ ݀‫ݐ‬

and: ‫ ܧ‬ൌ െ݃ସସ

݀‫ݐ‬ Ǥ ݀‫ݏ‬

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Besso derived the angular momentum conservation, which he called "the area law": ݀ ‫ݔݕ‬ሶ െ ‫ݕݔ‬ሶ ൌ ൌ ͲǤ ݀‫ݏ‬ ݀‫ݐ‬ ݀‫ݐ‬ He defined the area constant Bc0, and the area speed ݂ሶ and wrote: ʹ݂ሶ ൌ ‫ݔݕ‬ሶ െ ‫ݕݔ‬ሶ ൌ ‫ܤ‬

݀‫ݏ‬ Ǥ ݀‫ݐ‬

From the area law follows Kepler's second law (CPAE 4, Doc 14, 8; "The Einstein-Besso Manuscript on the Motion of the Perihelion of Mercury", 350). Then, equations for the angular momentum and energy conservation for the orbit of the planet were derived: ʹ݂ሶ ൌ ߮‫ݎ‬ሶ ଶ ൌ ‫ܹܤ‬ǡ‫ ܧ‬ൌ

݀‫ݏ‬ ݃ସସ Ǥܹ ൌ Ǥ ݀‫ݐ‬ ܹ

Inserting the metric field of a static Sun to second order into these equations (the equation of ʹ݂ሶ and E), and after rearrangements and additional manipulations, an additional equation was obtained representing the orbit of Mercury around the Sun (CPAE 4, Doc 14, 9): ‫ܨ‬ ‫ܣ‬ ቀͳ െ ቁ ݀‫ݎ‬ ܿ଴ ‫ݎ‬

݀߮ ൌ

ඨെߝ‫ ݎ‬ସ ൅ ሺͳ ൅ ʹߝሻ ή ‫ ܣ‬ή ‫ ݎ‬ଷ െ ൬

݀‫ݎ‬Ǥ ‫ܨ‬ଶ

‫ܨ‬ଶ

ͳͳ ଶ ‫ ܣ‬൅ ଶ ൰ ‫ ݎ‬ଶ ൅ ʹ ଶ ‫ܣ‬ଶ ή ‫ݎ‬ ͺ ܿ ଴ ܿ ଴

To find the advance of the perihelion of Mercury, this equation (dM) was integrated between the values of the radius vector r from the Sun to Mercury, between the perihelion and aphelion. After some calculations and rearrangements, the final result was obtained (CPAE, Vol. 4, Doc. 14, 14): න ݀߮ ൌ ߨ ൬ͳ ൅

‫ܣ‬ ͷ ൰ǡ ͺ ܽሺͳ െ ݁ ଶ ሻ

where a is the semi-major axis and e the eccentricity of the elliptical orbit.

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It followed that according to the Entwurf theory the field of a static Sun produced an advance of the perihelion of Mercury of: ߨ

ͷ ‫ܣ‬ ǡ Ͷ ܽሺͳ െ ݁ ଶ ሻ

per revolution: An advance of the perihelion of Mercury of 18 seconds per arc. Michel Janssen described Einstein's work with Besso. Starting from the first several pages of their manuscript, Einstein and Besso began searching for a sufficiently accurate expression for the metric field of the Sun. These pages are all in Einstein's hand, with just a few corrections of minor slips in Besso's hand. Besso takes over on the next couple of pages, deriving an equation for the perihelion motion of a planet in the metric field of the Sun. Besso's pages are not nearly as "clean" as Einstein's, because Einstein was clearly much more comfortable with the calculations than Besso. Besso's insecurity is reflected in the many deletions he made and also in the fact that he used a lot more explanatory prose than Einstein. On the next two pages, Einstein takes over again and finds an expression for the perihelion advance of a planet in the field of the Sun. Besso then rewrites this equation, making sure that it only contains quantities for which numerical values would be readily available, and some astronomical data pertaining to the Sun and Mercury along with some constants of nature. The end result is given by Einstein: 1821". The Sun, which in Newton's theory produces no perihelion motion at all, in the Entwurf EinsteinGrossmann theory, produces a perihelion motion of more than three times the size of the total perihelion motion that is observed! Fortunately, Einstein and Besso found a mistake in the numerical calculation – an error factor of 10 for the mass of the Sun. Since the perihelion motion is proportional to the square of this quantity, the final result was an error factor of 100. Several pages ahead there is a correction in Besso's hand of the erroneous value for the mass of the Sun that Einstein had used. On another page, Einstein himself corrected the old value for the mass of the Sun (which previously gave 1812'') and derived a new final result of 18".

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Even this corrected value was disappointing. Einstein and Besso therefore had to resort to additional effects in the Entwurf theory that might contribute to the perihelion advance of Mercury. In particular, they considered the effect of the rotation of the Sun. In Newtonian theory, the Sun's rotation would not produce any perihelion motion at all, but in the Einstein-Grossmann Entwurf theory (as in general relativity in its final form) it does have a small effect. However, Einstein and Besso calculated this effect to be about 1" per century, which was negligible. In fact, Einstein once again used the wrong value for the mass of the Sun, so the result should really be 0.001". In addition, the effect of the rotation of the Sun did not produce an advance, but was retrogression, of the perihelion. Before they could finish their joint project in June 1913, Besso left Zurich to return again in late August 1913. At that time, they continued to work on the project, and added a few more pages. The manuscript remained with Einstein in Zurich. In early 1914, Einstein sent it to Besso, urging his friend to finish their project. Besso added more calculations (Janssen 2002). There are two calculations in the Einstein-Besso manuscript concerning rotation. Einstein and Besso checked whether the metric field describing space and time for a rotating system was a solution for the field equations of the Entwurf theory. Einstein's answer was yes, but he later discovered that he had made a mistake in the calculations. Einstein calculated, in a first-order approximation, the metric field that a rotating shell would produce near its centre. The shell represented the distant stars according to Mach's ideas. He calculated the metric field of a uniformly rotating system; that is to say, according to the equivalence principle, this metric field was equivalent to the one in a static gravitational field. Einstein checked whether this metric was a solution for the Entwurf field equations. To find this, Einstein used the same approximation procedure he had used to calculate the field of the Sun for the perihelion advance of Mercury. In first-order approximation, the metric field for the rotating system was indeed a solution for the field equations of the Entwurf theory. Moreover, this first-order metric field had the same form as the first-order metric field for the case of the rotating shell; hence, this metric field could be interpreted as the field produced by the distant stars rotating with respect to the observer. Einstein then substituted this first-order field into the field equations in a second-order approximation and checked whether the metric field of the rotating

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observer was also a solution to the equations at this further level of approximation. He concluded that it was. Einstein, however, made some trivial sign error in this calculation. The metric field of a rotating system was not a solution of the field equations of the Entwurf theory. Einstein had not noticed the sign error, but he decided to redo the calculation. This time the Entwurf field equation was not satisfied: the metric field describing a rotating system was not a solution for the Entwurf field equations. Einstein seemed to have hesitated, and he wrote the extra term, which did not appear in the previous result, in parentheses. However, the first solution, obtained previously (while not noticing the sign error), perfectly fitted expectations of Einstein's principle of equivalence; therefore Einstein finally chose this solution upon the second one. It took Einstein two extra years to discover that the solution he had chosen was not a solution for his Entwurf field equations (Janssen 2002).

5. Einstein's 1913-1914 Polemic with Nordström: Scalar Theory versus Tensor Theory In January 1913, Gunnar Nordström agreed to a stronger version of the equality of inertial and gravitational masses and included it in his gravitation theory (Nordström 1913a). In the Einstein-Grossmann Entwurf paper Einstein then began to study Nordström's theory from the theoretical point of view and found a flaw in Nordström's new theory. He showed that Nordström's scalar theory violated the law of conservation of energy, and demonstrated that his theory could not comply with the principle of equivalence and the equality of inertial and gravitational masses due to inertia of energy, because it did not take into account the stresses (Einstein and Grossmann 1913, 20-22). Einstein thought that generally scalar gravitational theories, like Nordström's scalar gravitation theory from 1913, were inadmissible. Einstein questioned whether his ten component metric tensor field could be reduced to a single scalar gravitational potential ). Einstein's answer was negative. Einstein thought that scalar theories – such as Nordström's new version of his special relativistic, scalar gravitation theory from January 1913 (that Einstein did not mention here explicitly) – were inadmissible.

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In such a theory, one characterizes the gravitational field by a scalar, and the equation of motion of a material point is written in such a way that the four-dimensional line element is the one from special relativity. Thereafter, one proceeds in full analogy with these equations. One therefore proceeds without having to surrender special relativity. Indeed, Nordström adhered to the constancy of velocity of light from special relativity. The material process was characterized by a stress-energy tensor TPQ. But with this conception it is a scalar that determines the interaction between the gravitational field and the material process (Einstein and Grossmann 1913, 20-21). Max Laue drew Einstein's attention to a stress-energy tensor, which he presented in 1911, and Einstein called, Laue's Scalar. Laue's scalar is the trace of the stress-energy tensor. Einstein did not use this language. He rather spoke of the trace in terms of a scalar. In this book, "trace" and "scalar" are used interchangeably to describe the trace of the stress-energy tensor. If one creates a scalar theory and characterizes the material process by a stress-energy tensor, it must have the form of Laue's Scalar P: ෍ ܶఓఓ ൌ ܲǤ ఓ

This form cannot fully do justice to the principle of equivalence and the equality of inertial and gravitational masses due to inertia of energy, because – as Laue pointed out to Einstein – for a closed system (not interacting with other systems): න ܲ݀߬ ൌ න ܶସସ ݀߬Ǥ The gravity of a closed system is determined by its total energy, but the gravity of systems which are not closed would also depend on the orthogonal stresses to which the system is subjected. Einstein could not accept such a limited concept of the equality of inertial and gravitational masses due to inertia of energy, because it did not take into account the stresses. There are cases in which gravitational mass is due to the stresses in the system. He said that this gives rise to a

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consequence that seems to him unacceptable as is shown in the example of black body radiation. Einstein gave this example and showed a big flow in Nordström's scalar theory, a violation of the law of conservation of energy. If the radiation is enclosed in a massless reflecting box, then its walls experience tensile stresses, as a result of which the system – taken as a whole – possesses a gravitational mass ‫߬݀ܲ ׬‬corresponding to the energy E of the radiation. Einstein imagined that the radiation was now bounded by reflecting walls (stresses enter into the system). In this case, the gravitational mass ‫ ߬݀ܲ ׬‬of the movable system amounted only to one third of the value obtained in the case of a box moving as a whole. Thus, in order to lift the radiation against the gravitational field, one would have to apply only one third of what that one would have to apply in the previously considered case of the radiation enclosed in the box. This violates the conservation law of energy and was unacceptable to Einstein. In late June 1913, Nordström visited Einstein in Zurich, and they both discussed their gravitational theories and came to the conclusion that Einstein's above objection could be avoided ("Einstein on Gravitation and Relativity: The Collaboration with Marcel Grossmann", CPAE 4, 299). As a result of the discussion with Einstein, Nordström developed a second version of his theory. In July 1913, Nordström submitted this version of his gravitation theory to the Annalen der Physik. He corrected his theory in light of Einstein's criticism so that his new theory satisfied an equality of inertial and gravitational masses (Nordström 1913b, 538). Nordström wrote that all the referred disagreements of the theory could be removed by a very plausible setting which he owed to Max Laue and Einstein (Nordström 1913b, 533). Nordström's theory was now more natural than the Entwurf theory and more related to special relativity and to its light postulate. Moreover, Nordström's theory was also simpler than the Einstein-Grossmann Entwurf tensor theory. It was a single scalar gravitational potential theory. On the other hand, the Einstein-Grossmann theory was less appealing and more complicated. It violated the principle of special relativity – the constancy of the velocity of light postulate, and it replaced the single scalar gravitational potential of Einstein's 1912 static gravitation theory with the metric tensor field; this tensor described the gravitational field with ten independent components.

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Nordström's scalar theory became a true option for a gravitational theory, because at that time the Entwurf tensor theory remained without empirical support. The deflection of light in a gravitational field was not yet tested. Thus, a decision in favour of one or the other theory was impossible on empirical grounds. Einstein, however, thought he had a very strong reason for believing that his theory was superior to that of Nordström's. Einstein realised that Nordström's theory did not satisfy Ernst Mach's ideas: According to Nordström's theory, the inertia of bodies seemed not to have been caused by other bodies, even though it was influenced by them. The mass of a body was greater the further the other bodies were from it. This effect could not eliminate Newtonian absolute motion, because there was no symmetry between the inertia of other bodies of the universe and the single body in question. Einstein explained that his own theory eliminated the epistemological weakness of Newtonian mechanics, the absolute motion; Mach's idea of inertia having its origin in an interaction between the mass under consideration and all of the other masses. Mach objected to Newton's interpretation of his bucket experiment. With the aid of the bucket experiment Newton claimed to have proved the existence of absolute motion with respect to absolute space. The bucket experiment is elucidated below: 1. First a bucket rotates, but the water does not, its surface remaining flat 2. Then the frictional effects between the bucket and the water eventually communicate the rotation to the water. The centrifugal forces cause the water to pile up around the edges of the bucket and the surface becomes concave. The faster the water rotates the more concave the surface becomes. 3. Eventually the bucket will slow down and stop, but the water will continue rotating for a while, its surface remaining concave. 4. Finally, the water returns to rest with a flat surface. According to Newton's explanation of this experiment, the curvature of the water surface at stages 2 and 3 is formed from the centrifugal effects resulting from the rotation of the water relative to absolute space. This curvature is not directly connected with local considerations, such as the bucket's rotation, since in stage 1 the surface was flat when the bucket was rotating and in stage 3 it was curved when the bucket was at rest. Hence, inertial forces (like the centrifugal force) rise in a rotating frame, and

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indicate that it is the bucket and not the whole universe of the fixed stars that rotates. Mach reasoned that if one could fix Newton's bucket, and cause the whole universe of the fixed stars to rotate, both cases would become indiscernible one from the other. Consider Newton's bucket full of water experiment. After the bucket starts to rotate, the centrifugal forces cause the water to pile up round the edges of the bucket and the surface becomes concave. According to Newton, the surface of the water definitely shows that the water in the bucket is rotating (in an absolute circular motion) relative to absolute space. The water and the bucket are in absolute rotation. The inertial force (the centrifugal force) rises when the water rotates relative to absolute space. When its surface is concave, the water rotates with respect to absolute space (when the surface of the water is flat the frame is inertial, and does not rotate relative to absolute space). According to Mach, inertial forces have their physical origin in the masses of the universe (the fixed stars, the bucket, etc.). Mach reasoned that Newton had overlooked the fact that case number 2 did not represent the opposite of case number 1. He had forgotten to take into consideration the whole sky and universe; for when rotating, the water does not revolve with respect to the bucket alone, but also with respect to the totality of the masses in the universe. We must not only let the water revolve with respect to the resting bucket, but also with respect to the whole universe. Only then shall we present an equivalent but reverse picture. We should thus extend the two above cases to the following cases: 1. The water is fixed and the whole sky (of the fixed stars) is rotating. 2. The water is rotating and the whole sky (of the fixed stars) is fixed. Such an extension would cause the two cases to be symmetric: In both cases the surface of the water would be concave. For in case number 1, if one could fix the bucket and cause the sky with the fixed stars to rotate, the surface of the water would be concave. The bucket has very little effect on the water's rotation since its mass is so small. The fixed stars contain most of the mass in the universe and this counteracts the fact that they are a very long way away. This solution does not neutralise the centrifugal force. It will appear again in the water. Therefore, Mach suggested that, in case number 2, the centrifugal force is a consequence of the water's motion (uniform acceleration), and in case number 1, the centrifugal force should be

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understood as being an effect of the rotating sky, which is full of stars or masses. The rotating masses in the whole sky produce the centrifugal force experienced by the water. The conclusion is that we cannot know which of the two, the water or the sky, are rotating; both cases produce the same centrifugal force. Mach thus expressed a kind of equivalence principle: Both explanations (given to cases 1 and 2) lead to the same observable effect. Einstein accepted Mach's idea that we cannot know which of the two, the water or the sky, are rotating. Einstein, however, thought, in terms of fields and Mach spoke in terms of action at a distance and forces in a Newtonian mechanistic world. Einstein therefore reasoned that both cases produce the same effect because of the physical equivalence of a centrifugal field and a gravitational field. Mach also reasoned that a body in an otherwise empty universe could not be said to be in motion, since there was nothing to which the body's motion could be referred. According to Einstein, a body in an otherwise empty universe could not be said to rotate, and this statement has no meaning, since a universe with no fixed stars would have no gravitational/inertial (centrifugal) fields. Around June 1913, Einstein's Entwurf theory, however, did not satisfy Mach's ideas, because it could not explain rotation and Newton's bucket experiment. Einstein, however, was not aware of this problem. We should recall that Einstein checked whether the rotation metric was a solution of his Entwurf field equations. He thought that the metric field describing a rotating system was a solution of his vacuum field equations, and this led him to think that the Entwurf field equations did hold in rotating frames. But this was an error; Einstein did not discover this error until October 1915. Contrary to what Einstein had originally hoped for, his final gravitation theory did not fully eliminate absolute motion. Absolute motion in the context of Newton's theory means that there is an absolute distinction between inertial motion (the motion of a body on which no external forces act) and non-inertial motion (the motion of a body subject to external forces). The former motion will be represented by a straight line, the latter by a curved line. A very similar distinction holds in Einstein's general theory of relativity. There it is the distinction between the motion of a body on straightest lines in curved space-time, geodesic lines (free fall), and on non-geodesic lines in the curved space-time (Janssen 2002).

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While still in Prague, Einstein tried to find an analogy to the effect of electromagnetic induction for gravitational fields, using what he called Mach's ideas above: The entire inertia of a material point is an effect of the presence of all the other masses, which is based on a kind of interaction with the latter. All stars in the universe can be thought of as taking part in the generation of the gravitational field, because they are accelerated relative to the accelerating system K' and, therefore, induce a gravitational field just as accelerating moving electric charges induce an electric field. Approximate integration of the gravitational equations shows that such inductive effects actually occur with accelerating moving masses. The induced gravitational field that is perceived by the accelerating system K' is not produced statically, but is produced dynamically by the distant stars, through their acceleration (Einstein 1918e, 700; Einstein to Eduard Hartmann, April 27, 1917, CPAE 8, Doc. 330). Einstein's theory of the static gravitational field was like the electrostatic theory for the static electric field, and Einstein was well aware that this result would be justified once he would possess the dynamical theory of gravitation; just as in electrodynamics one had a theory for the electromagnetic field. Einstein said that this was exactly the standpoint which Ernst Mach had taken in his studies of Newton's mechanics (Einstein 1912e, 39). Now that he possessed a kind of dynamical theory of gravitation, he believed he had managed to justify Mach's ideas. In 1913 Einstein believed that Nordström's theory did not satisfy Mach's ideas and that experiment could decide between his theory and that of Nordström: Unlike the latter theory, Einstein's gravitational theory had to yield a bending of light rays near the Sun (in a gravitational field). According to Nordström there existed a redshift of spectral lines, as in Einstein's theory, but there was no bending of light rays in a gravitational field. Like its predecessor for the static gravitational field from 1911, the Entwurf theory predicted the same value for the deflection of light in a gravitational field of the Sun, 0.83 seconds of arc. Einstein, however, could not yet send an expedition to check the prediction of his theory. Since autumn 1911, he made many efforts to obtain empirical data on light deflection, first from already existing photographs and later by involving himself in the organisation of an expedition for the 1914 total solar eclipse. Luckily, the empirical verification of the light-bending effect remained elusive until 1919. In trying to push Erwin Freundlich from the Berlin Observatory to go on expedition to verify the deflection of light, Einstein told him in mid-

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August 1913 that, Nordström's January 1913 theory showed a very consistent way without the equivalence principle. Einstein told Freundlich that the investigation during the next year's solar eclipse would show which of the two conceptions corresponded to the facts. Einstein said that he had confidence in his theory, even though Nordström's scalar theory of gravitation – with no deflection of light – was more natural (Einstein to Freundlich, mid August, 1913, CPAE 5, Doc. 468; "Einstein on Gravitation and Relativity: The Collaboration with Marcel Grossmann", CPAE 4, 295). Nordström's theory could also not explain the anomalous motion of the perihelion of Mercury. Einstein's Entwurf theory predicted a perihelion advance for Mercury of 18 seconds of arc per century instead of yielding 43 seconds of arc per century. However, in late June 1913, Nordström and Besso visited Einstein in Zurich. Recall that Einstein and Besso both tried to solve the new Entwurf field equations to find the perihelion advance of Mercury in the field of the static Sun. Besso also tried to solve the field equations of Nordström's January theory (Nordström 1913a) in the Einstein-Besso manuscript, in order to see whether in a gravitational field of the Sun he could find the perihelion motion of Mercury's orbit in this field. Besso did the calculation of the perihelion motion using Nordström's theory, and the strategy followed in these calculations was the same as that followed by Einstein and Besso when they both solved the Entwurf equations to find the perihelion advance for Mercury of 18 seconds of arc per century (Janssen 2002). In June 1913, Einstein had two more visitors in Zurich. During the same month, Paul Ehrenfest visited Einstein (Klein, 1970/1985, 294). Ehrenfest met Grossmann and Besso. Einstein was thus surrounded by Grossmann, Besso and Ehrenfest while he investigated solutions to the EinsteinGrossmann Entwurf field equations, in an attempt to solve the problem of the precession of the perihelion of Mercury. Since at that time the Entwurf theory remained without empirical support, and decision in favour of one or the other theory – the Entwurf or Nordström's – was impossible on empirical grounds, Nordström's theory thus became a true option for a gravitational theory. Starting in September 1913, Einstein formulated a competing theory to his own Entwurf theory, which was based on Nordström's theory. We can call this theory the "Einstein-Nordström" theory. Einstein formulated this theory in two papers – in a talk in Vienna and in a joint paper he wrote with Hendrik Antoon Lorentz's student, Adriaan Fokker.

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In September 1913, Einstein presented the Vienna talk, "On the Present State of the Problem of Gravitation" relating to his Entwurf theory. In the Vienna 1913 talk, Einstein first formulated four basic heuristic principles that guided him in the search for any gravitational theory: 1) Satisfaction of the laws of conservation of energy and momentum. 2) Equality of inertial and gravitational mass in isolated systems. 3) Validity of the special theory of relativity; i.e., the system of equations are covariant with respect to linear substitutions (Lorentz transformations). 4) The laws of nature do not depend on the absolute magnitude of the gravitational potential/s. Subsequently, Einstein began developing the new Einstein-Nordström theory in Section §3 of the Vienna-talk paper (Einstein 1913a, 1258-1259). The new Einstein-Nordström theory had to be fully compatible with these principles, and not violate them. Eventually, the theory developed by Einstein became a serious competitor to Einstein's own EinsteinGrossmann Entwurf theory. Einstein started with the equation of motion of the mass point in a gravitational field. In Nordström's theory Minkowski's interval with c constant remains valid; the gravitational field and the gravitational potential Mare scalars. Einstein concluded that in a scalar theory, the inertia of a mass point is determined by the product mMThe smaller Mis (i.e., the more mass we pile up in the region of the mass point), the smaller the inertial resistance exerted by the particle in response to a change in its velocity becomes (Einstein 1913a, 1250-1254). Einstein then borrowed from his Entwurf theory the natural interval concept. He explained this concept as follows: Consider a transportable unit measuring rod and a transportable clock, which runs as fast such that in vacuum, light traverses a distance equal to one unit measuring rod – as measured by the clock – during one unit of time. Einstein called the fourdimensional interval ds between two infinitely close space-time points, which can be measured exactly by these measuring tools, such as in the case of special relativity, the natural four-dimensional interval dW0 of the space-time point. dW0 is defined as an invariant and thus in the case of special relativity it is equal to dW. The latter, dW, as opposed to the natural interval (Minkowski's interval) dW0, was called by Einstein, the coordinate interval, or simply the interval of the space-time point.

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In his 1913 Entwurf paper with Grossmann, Einstein explained that for given differentials, dx1, dx2, dx3, dx4, the natural interval that corresponds to these differentials can be determined only if one knows the components gPQ of the metric tensor that determine the gravitational field (Einstein and Grossmann 1913, 8-9). More specifically, Einstein first wrote the equations of motion of a material point in a gravitational field. Remember that a material point in a gravitational field moves according to the formula: Ɂ නሼ†•ሽ ൌ ͲǤ The motion of a mass point in a gravitational field is characterized by this equation and is represented by geodesics in space-time. According to Einstein, in the general theory of relativity ds plays the same role as the element of a worldline in the special theory of relativity (Einstein 1914b, 1034). Recall that ds, the line element of Minkowski's flat space-time is the following: ݀‫ ݏ‬ଶ ൌ ෍ ݀‫ݔ‬ఔ Ǥ ఔ

where x1 = x, x2 = y, x3 = z, x4 = it. The variation in the equation of motion of a material point is formed so that the coordinates xQ at the end points of integration remain unchanged. Einstein replaced the line element of Minkowski's flat space-time by the more general form of the line element. According to the above terminology of natural interval and coordinate interval, Einstein referred to the line-element below as a coordinate interval: ݀‫ ݏ‬ଶ ൌ ෍ ݃ఓఔ ݀‫ݔ‬ఓ ݀‫ݔ‬ఔ Ǥ ఓఔ

In the Entwurf theory, the ten quantities gPQ determine the gravitational field and they are functions of xQ. The special theory of relativity is still valid in the infinitesimally small local region:

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݀‫ ݏ‬ଶ ൌ ෍ ݃ఓఔ ݀‫ݔ‬ఓ ݀‫ݔ‬ఔ ൌ ෍ ݀ܺఔଶ Ǥ ఓఔ

ఔ

XQ (Q = 1, 2, 3, 4) are coordinates of space and time in the infinitely small four-dimensional local regions. They are measured with unit-measuring rods and a suitably chosen unit-clock. Consider an observer enclosed in a freely falling box. The gPQ in the box are constant, so long as their second derivatives can be neglected, and the line element of the above form is up to terms of second order (Pauli 1958, 147). Einstein called the quantity: ݀‫ ݏ‬ଶ ൌ ෍ ݀ܺఔଶ ǡ ఔ

the natural interval between two space-time points. It follows that the proper time (the time measured with a clock co-moving with the observer and the object to be measured) over a volume element is an invariant: ݀߬଴‫כ‬ଶ ൌ †• ଶ ൌ න ݀ܺଵ ݀ܺଶ ݀ܺଷ ݀ܺସ ൌ ݃݀߬ ଶ Ǥ Einstein, therefore, wrote the following relation between the proper time, which he also called the natural interval, and the coordinate interval dW, in his Entwurf theory (where g denotes the determinant of gPQthe metric tensor): ඥ݃݀߬ ൌ ݀߬଴‫ כ‬Ǥ He then rewrote this equation in the following form: ඥെ݃݀߬ ൌ ݀߬଴ Ǥ The first equation is interpreted as the relation between the proper time,݀߬଴‫ כ‬, measured by a clock at rest with the observer along his worldline, and the coordinate time ඥ݃݀߬. In the second equation dWo is imaginary. The coordinate time is the time measured with a clock that is not co-moving with the observer and the object to be measured. This time is measured with a clock moving relative to the object to be measured and is measured by some other observer.

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Einstein concluded that in the case of Nordström's theory it was possible that the natural interval dW0 would be different from the coordinate interval dW by a factor Z that was a function of the gravitational potential. However, Zwas a scalar in Nordström's theory and dependent on the scalar gravitational potential MEinstein, therefore, wrote the equation representing the relation between the natural interval and the coordinate interval, in Nordström's theory (Einstein 1913a, 1252): ߱݀߬ ൌ ݀߬଴ Ǥ Einstein then treated the material points as particles in the continuum, with coordinate volume V and natural volume V0. He considered the expressions for the law of energy-momentum conservation in relativity theory. Energy is dependent on the potential M. Einstein also took into consideration the above equation of the natural interval dependent on Z, and arrived at expressions for the stress-energy tensor TPQ (P and Q are indices running from 1 to 4), where the natural mass density is U0 = m/V0 and the gravitational force density kP in Nordström's theory is:  ஜ ൌ െɏ଴ …ɘଷ

μɔ Ǥ μšஜ

With these expressions Einstein wrote the four equations for the law of energy-momentum conservation in Nordström's theory: ෍ ఔ

μஜ஝ ൌ  ஜ ሺɊ ൌ ͳǡʹǡ͵ǡͶሻǤ μš஝

Einstein used the stress-energy tensor TPQ and wrote it in terms of Laue's scalar: ෍ ஢஢ Ǥ ஢

Einstein dealt with the equations of motion of a material point: how the gravitational field acted on matter. He answered the following question: How does matter determine the gravitational field, and the field equations? The latter were given in Nordström's theory by a scalar M. Thus, Einstein investigated a differential equation for M, in which the term entering into it would also be a scalar (Einstein 1913a, 1253).

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He created a scalar theory and characterized the material process by a stress-energy tensor TPQ that had to have the form of Laue's scalar. Accordingly, he rewrote the expression for the conservation law with Laue's scalar: ෍ ఔ

μஜ஝ ͳ μɔ ൌ ෍ ஢஢ Ǥ ɔ μšஜ μš஝ ஢

This equation represents the energy balance of any physical process, once we use for this process the stress-energy tensor TPQ. Therefore, this equation expresses the conservation law of energy-momentum in its most general form in Nordström's theory. Einstein concluded that Nordström's theory now satisfied postulate 2) the equality of inertial and gravitational masses. Subsequently, Einstein established the gravitational field equation in Nordström's theory, which was a generalisation of the Poisson equation for the Newtonian gravitational field (Einstein 1913a, 1255): െɈ ෍ ஢஢ ൌ ɔ ෍ ఛ

μɔଶ ǡ μšதଶ

where Wis taken from 1 to 4. Einstein also wrote an expression for the stress-energy tensor of the gravitational field tPQ. It then follows from the above equation and the conservation law of energy-momentum in its most general form in Nordström's theory that: ෍ ఔ

μ ൫ ൅ – ஜ஝ ൯ ൌ ͲǤ μš஝ ஜ஝

Nordström's theory now satisfied postulate 1), the laws of conservation of energy and momentum. Einstein presented another concept, natural length l0, and defined in Nordström's theory its relation to the coordinate length l, using the following method (Einstein 1913a, 1255): ݈ൌ

݈଴ ݈଴ ൌ Ǥ ߱ ܿ‫ ݐݏ݊݋‬ή ߮

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Further, in Einstein's Entwurf theory the relation between natural length l0 and coordinate length l is: ݈ൌ

݈଴ ඥെ݃

Ǥ

Likewise, recall that in Nordström's theory the relation between natural and coordinate intervals is: ߱݀߬ ൌ ݀߬଴ ǡ from which it follows: ݀߬଴ ൌ ܿ‫ ݐݏ݊݋‬ή ߮݀߬Ǥ Einstein suggested the following thought experiment: A light clock can be created by placing two mirrors at the end of a natural long length l0 – facing each other – with a light beam traversing back and forth between them in a vacuum. Einstein also suggested that a gravitational clock could be created by two masses – m1 and m2 – circling each other at a natural distance l0 under the influence of their gravitational interaction (Einstein 1913a, 1254). In Nordström's theory the natural length l0 is dependent on the potentialM. According to Einstein's definition (in a vacuum, light traverses a distance equal to one unit measuring rod during one unit of time, as measured by the clock), and in light of the above thought experiment, the above equation means that in Nordström's theory both the length and the period of the clock vary with the potential. However, Einstein explained that according to Z= const.ͼM, we find the same dependence on potential for the rate of two clocks placed in different gravitational potentials. Einstein thus confirmed postulate 4), the laws of nature do not depend on the absolute magnitude of the gravitational potential. Therefore, if we perform measurements in a certain constant gravitational potential, we will find the same laws of nature when moved to a different gravitational potential. In Einstein's Entwurf theory we find the same dependence on potential for the rate of two clocks placed in different gravitational potentials. Two clocks which are at rest with respect to each other and are going at the same rate, and then travel along timelike lines, will always go at the same rate, compared with each other; that is to say, their relative rates do not depend on their prehistories. They all therefore give the invariant ds:

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݀‫ ݏ‬ଶ ൌ ෍ ݃ఓఔ ݀‫ݔ‬ఓ ݀‫ݔ‬ఔ ǡ ఓఔ

i.e. the proper time ‫ݏ݀ ׬‬, independent of which path or timelike line has been chosen (see Chapter 3, Section 4). Einstein concluded that Nordström's scalar theory, which follows the postulate of the constancy of the velocity of light and therefore satisfied postulate 3) validity of the special theory of relativity, satisfied all the requirements for a gravitational theory that could be imposed on the basis of current experience. That is, Einstein created a new theory – the Einstein-Nordström theory – that was so brilliant, it became a competitor to his own Entwurf theory. However, this theory had a little flow from Einstein's point of view: It did not satisfy Mach's ideas. Einstein's Entwurf theory, as we have already seen, also did not satisfy Mach's ideas. In December 1913, in collaboration with Adriaan Fokker, Einstein presented the first treatment of a gravitation theory in which general covariance was strictly obeyed: He reformulated Nordström's theory in a generally covariant form (Pais 1982, 236, 487). Einstein and Fokker started their paper by stating that all previous presentations of Nordström's theory of gravitation used Minkowski's invariant. The equations of the theory were required to be covariant under linear, orthogonal space-time transformations. These conditions, imposed on the equations, restricted the theoretical possibilities for finding basic equations for the theory. Einstein then wanted to correct this defect by presenting the theory formally, in a manner similar to the presentation of the Einstein-Grossman Entwurf theory. He would do this by using the Einstein-Grossmann tools of absolute differential calculus – that is, the metric tensor. Einstein and Fokker started with the Gaussian theory of surfaces and arbitrary reference systems. They proved that if one selects preferred reference systems in which the velocity of light is constant, then in these reference systems one arrives at Nordström's theory (Einstein and Fokker 1914, 321). In other words, Einstein managed to formulate a generally covariant Nordström theory. Hence, he showed that a generally covariant formalism is presented from which Nordström's theory follows, if a single assumption is made that it is possible to choose preferred systems of reference in such a way that the velocity of light is constant.

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Einstein and Fokker presented the stress-energy tensor TPQ, the quantities of which correspond in Nordström's theory to the quantities TPQ from Einstein's Entwurf theory (TPQ represents the stress-energy components of matter and tVQ represents the stress-energy components of the gravitational field). The conservation laws in Nordström's theory take the form: ෍ ఔ

μ‰ ஜ஝ μஜ஝ ͳ ൌ ෍ ‰  Ǥ ʹ μš஝ μš஢ ஜ஝ ஜ஝ ஜ஝த

Einstein and Fokker went on to establish the gravitational field equations of Nordström's theory. The required equations were completely determined by the assumption that they were of the second order, and were a generalisation of the Poisson equation. Einstein and Fokker provided them with the form * N7 N is a constant and, in Nordström's theory, *and T are scalars.) The first represents the gravitational field and is determined by the metric tensor, gPQ. The second represents the material processes. Einstein and Fokker realised that from the studies of mathematicians of differential tensors, the only expression allowed for *is the contraction of the Riemann-Christoffel tensor (ik, im) of the fourth rank (a scalar derived from the Ricci tensor). From covariant theory for TPQ, only the following scalar related to Laue's scalar could be chosen: ஜ஝ ൌ

ͳ ඥെ݃

෍ தத Ǥ ఛ

Hence, the field equation of Nordström's theory takes the form of a scalar derived from the Ricci tensor on the left-hand side, equal to the above expression for TPQ times Non the right-hand side. This field equation was valid in the preferred reference systems – i.e., in the reference systems for which Nordström's theory was valid – and it was obtained with the tools provided by the Entwurf theory. Einstein and Fokker declared they were using the invariant theory of absolute differential calculus, and examined the formal contents of Nordström's theory. Their study clarified the relationship between Nordström's theory and the Einstein-Grossmann's Entwurf theory. Both theories satisfied the equality of inertial and gravitational mass. Einstein

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and Fokker arrived at Nordström's theory through formal considerations on the basis of the principle of the velocity of light. Einstein and Fokker concluded with the role the Riemann-Christoffel tensor played in their investigation, suggesting the idea that it should also open the way for a derivation of the Einstein-Grossmann gravitational field equations, independent of physical assumptions (Einstein and Fokker 1914, 328). Einstein followed up the question raised above in the period immediately after his work on Nordström's theory. By November 1914, Einstein thought he could derive the Einstein-Grossman field equations uniquely by purely formal considerations. However, these considerations did not yet involve the Riemann-Christoffel tensor (Stachel 1989, 320). In the Glasgow lecture of 1933, "The Origins of the General Theory of Relativity", Einstein admitted that, sometime between 1905 and 1907, he had independently developed a scalar theory very similar to Nordström's January 1913 theory. Although Einstein did not explicitly mention Nordström's name, it was evident that he was implicitly referring to Nordström's scalar theory and to a similar scalar theory. Einstein's Glasgow talk was delivered after he had completed his general theory of relativity. In light of his general theory of relativity and his novel strategy, Einstein was – in 1933 – able to explain the faults of his initial investigations for a scalar theory, between 1905 and 1907 (Einstein 1934, 250-251; 1954, 286-287). Having already fully examined Einstein's search for a scalar theory between 1905 and 1907 in Chapter 2, Section 1, the following is a summary of relevant points to Nordström's scalar theory: Einstein stated that the simplest thing was, of course, to retain the Laplacian scalar potential of gravity and to complete the Poisson equation according to special relativity. The law of motion of the mass point in a gravitational field had to be adapted to special relativity as well. However, on account of the inertia of energy, and according to the scalar theory he had tried to advance along these lines between 1905 and 1907 (actually a theory very much similar to the one advanced by Nordström in January 1913), this violated Galileo's law of free fall. He therefore rejected

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the scalar theory in favour of his own theory of 1907-1911, at the basis of which stood the equivalence principle. In the summer of 1913, Einstein realised that a scalar theory, Nordström's theory, could comply with the equality of gravitational and inertial mass, and the conservation of energy-momentum. He again tried to advance a scalar theory in the Vienna talk. Einstein believed that Nordström's theory was simpler than his own Entwurf theory, in which the field equations were not generally covariant. In 1913-1914 Einstein thought that Nordström's second theory, especially the recasting given to it by Einstein, was simpler than his own Entwurf theory. Nordström's scalar theory was indeed simpler. In such a theory, one characterized the gravitational field by a single scalar gravitational potential. In Einstein's Entwurf theory, the gravitational field was represented by a ten component metric tensor field. In Nordström's theory, one proceeded without having to give up special relativity. In Einstein's Entwurf theory one had to give up the constancy of light postulate and claim that light was deflected in a gravitational field. In Nordström's theory, the stress-energy tensor was a scalar, and Einstein was even able to demonstrate that this theory satisfied the equality of inertial and gravitational mass. Finally, in 1914 after failing to do so for his own Entwurf theory, Einstein and Fokker reformulated Nordström's theory in a generally covariant form. Nordström's theory was, therefore, a true option for a gravitation theory. However, after arriving at the generally covariant field equations and presenting the final general theory of relativity, Nordström converted to Einstein's general theory of relativity. Einstein, on the other hand, realised that the path he had chosen, was the most simple and natural one, and whose underlying assumptions would appear to have been secured experimentally (Einstein 1916a, 777).

6. Einstein's Polemic with Gustav Mie: Matter and Gravitation Gustav Mie, another competitor of Einstein, also responded to Minkowski's theory and Einstein's 1911 gravitation theory. In 1912, Mie published an electromagnetic theory of matter that was heavily inspired by Minkowski's four-dimensional space-time formalism (Pyenson 1985, 184,

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185). Mie investigated a connection between gravitation and matter, between electrical and gravitational effects. Mie's 1912 theory of matter and gravitation thus had a great deal in common with competing theories, due to Abraham and Nordström. Like Nordström, Mie retained an invariant speed of light and upheld the strict validity of the principle of relativity. This set his approach apart from Abraham's work, in which he renounced the constancy of the speed of light and retained the validity of the principle of relativity. Einstein's equivalence principle was later a target of Mie's criticisms – since Mie's commitment to retaining the framework of special relativity implied that in his theory, inertial and gravitational mass would not be exactly equal (Smeenk and Martin 2007, 631). Mie's 1912 theory of matter and gravitation sprung from his previous explorations of the field concept of electrodynamics and his study of the new electromagnetic view of nature. In 1910 he wrote a textbook on electrodynamics, and based his analysis on the electromagnetic ether. Mie addressed the constitution of the electromagnetic ether and indicated how molecules, atoms, and electrons were singularities in this ether (Pyenson, 1985, 184). He implemented this very idea in his 1912 theory of matter. Einstein later – when he developed a unified field theory – also represented matter particles as singularities in the field (see Chapter 2, Section 14). Mie's starting point was that neither Maxwell’s equations (the laws of electrodynamics) nor those of mechanics could be valid in the interior of the electron. Recent developments in quantum theory required the formulation of new equations to account for the phenomena that took place inside the atom. Mie's theory was envisioned as a massive project, a theory of matter and ether that would lead to a generalised principle of relativity. Its aim was to present the phenomenon of gravitation as a necessary consequence of his theory of matter. Mie wrote at the beginning of his paper that his immediate goals were to explain the existence of the indivisible electron and to view the actuality of gravitation as in a necessary connection with the existence of matter. He believed one had to start with this, for electric and gravitational effects were surely the most direct expression of those forces upon which the very existence of matter rests (Mie, 1912, 512).

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Mie's basic assumption was that electric and magnetic fields occur also in the interior of electrons. According to this view, electrons and accordingly the smallest particles of matter in general are not different in nature from the world ether; they are not foreign bodies in the ether, but are only locations where the ether has taken on a particular state, which we designate by the term electric charge. However, the enormous intensity of the field and charge-states at the location itself that we designate as the electron implies that here the electromagnetic Maxwell equations are no longer valid. In Mie's theory, the electron is not a particle in the ether with a sharp boundary, but consists of a nucleus with a continuous transition into an atmosphere of electric charge that extends to infinity, and which becomes so extraordinarily dilute already quite close to the nucleus that it cannot be experimentally detected in any way. An atom is an agglomeration of a larger number of electrons glued together by a relatively dilute charge of opposite sign. Atoms are probably surrounded by more substantial atmospheres, which however are still so dilute that they do not cause noticeable electric fields, but which presumably are asserted in gravitational effects (Mie, 1912, 512-513). Mie thought that with the aid of this conception he could demonstrate the general validity of the principle of relativity, and also that the states of the ether (that is electric field, magnetic field, electric charge and charge current) sufficed completely to describe all phenomena of the material world (Mie 1912, 513). To paraphrase Einstein, although Mie's theory provided a fine formal framework, it was not clear how to fill it with physical content (Smeenk and Martin 2007, 631). Nonetheless, Mie was quite pleased with himself and he wrote to the astronomer Karl Schwarzschild in 1912 that his theory of matter was in complete agreement with the relativity principle and reproduces all known facts about gravitation. Mie's theory of matter, however, was based on Minkowski's space-time formalism and, like Minkowski's work, was essentially untestable (Pyenson 1985, 185). It should be mentioned that, in terms of the further development of gravitational theories, Mie's influence on David Hilbert was more significant than his own theory (Smeenk and Martin 2007, 631). Indeed Mie's theory of matter inspired Hilbert when, in 1914, he studied Einstein's general theory of relativity. On September 23, 1913, Einstein presented his Vienna talk on his Entwurf theory. In a discussion after this talk, he engaged in a dispute with Mie and

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others. Mie claimed he presented a theory, which followed Einstein's special relativity, as preferable over Einstein's Entwurf theory. Mie said that his theory better generalised the special principle of relativity within a comprehensive theory of matter; moreover, it complied with the theory of gravitation proposed by Nordström. Actually, Mie's theory leads us to conclude, rather, that he was following Fin de Siècle physicists. Indeed Mie accepted the special theory of relativity and made full use of its concepts and mathematical methods. Yet, Mie fully agreed with the physical view, the electromagnetic view, of his predecessors – that ultimately, the world consists of structures in an electromagnetic ether. Mie's view was thus basically the same as that of earlier physicists such as Joseph Larmor, Wilhelm Wien and Lorentz (Kragh 1999, 117). In 1914, Einstein's Entwurf theory was again severely attacked in the philosophical journal Scientia by Abraham and Mie. Between 1913 and 1914, Einstein felt that the path he followed was reasonable for the excellent reason, that he presented an ingenious Hole Argument, according to which generally covariant field equations were unacceptable or uninteresting. Although he had not succeeded in setting up a relation between the metric tensor components gPQ and the stress-energy tensor in a generally covariant form, he felt his colleagues – Max Abraham, Gustav Mie, and Gunnar Nordström – were setting up a lethal trap for his theory because of this problem (Einstein, 1914a). They had touched a sensitive spot: Einstein's failure to find generally covariant field equations. His failure to find generally covariant field equations was due, they were alleging, to deficiency in his gravitation theory. Einstein sensed that the majority of his professional colleagues were sceptical towards his new gravitation theory, and they were opposed to it because of the lack of general covariance of its gravitational field equations (Einstein 1914c, 337). He knew, however, that he was on the right track: He was guided by the equivalence principle, the equality of inertial and gravitational mass, and Mach's ideas. Although at that time, emission theories also predicted some sort of bending of light, Einstein knew that his new metric gravitational theory of relativity was in the right direction. He knew this even though he had not succeeded in setting up a relation between gPQ and the stress-energy tensorin a generally covariant form.

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Of course this last mathematical dark spot was caught by his rivals and they pounced on it. They needled him but, Einstein had an ingenious answer in the form of the Hole Argument. Did he really believe in the Hole Argument between 1913 and 1914? He definitely believed that his new gravitational theory – the Entwurf theory – was the correct physical scaffold or framework.

7. 1914 Collaboration with Grossmann and Final Entwurf Theory In 1914, Einstein invoked a new consideration in two papers – in a joint paper with Grossmann, "Covariance Properties of the Field Equations of the Theory of Gravitation Based on the General Theory of Relativity", and in his comprehensive review paper on the Entwurf theory, "The Formal Foundation of the General Theory of Relativity". Einstein presented a new consideration according to which the metric field components that characterize the gravitational field could not be completely determined by generally-covariant equations. From the gravitation Entwurf field equations and the conservation law he obtained four coordinate conditions by which he could restrict the coordinate system. He reasoned that the gravitation equations hold for every reference system that is adapted to the four coordinate conditions; that is to say, the gravitation field equations hold for adapted coordinate systems. Nonetheless, the covariance of the field equations was far-reaching in these adapted coordinate systems. Einstein proved that his gravitational tensor was a covariant tensor for adapted coordinate systems, and coordinate transformations between two adapted coordinate systems were arbitrary non-linear transformations (Einstein to Besso, March 10, 1914, CPAE 5, Doc. 514; Einstein and Grossmann 1914, 224; Einstein, 1914b). Consider the gravitation Entwurf field equations: ఓ஝ ሺ‰ሻ ൌ െț൫ఓ஝ ൅ –ఓ஝ ൯Ǥ DPQ (g) is given by: ஜ஝ ሺ‰ሻ ൌ  ෍ ஑ஒ

μ‰ ஜத μ‰ ஝஡ μ‰ ஜ஝ μ ቆɀ஑ஒ ඥെ݃ ቇ െ ෍ ߛఈఉ ɀத஡ Ǥ μšஒ μš஑ μšஒ ඥെ݃ μš஑ ఈఉத஡ ͳ

ή

These field equations show that the stress-energy tensor tPQ of the gravitational field acts as a field generator in the same way as the

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covariant stress-energy tensor TPQ of the material process. On the left hand-side we find a differential expression formed from the components of the metric tensor gPQ, which Einstein emphasised (in previous writings and talks) has to be covariant with respect to linear transformations. Pais wrote, "Then the trouble began" (Pais 1982, 221). Now Einstein was going to try to solve this trouble with the adapted coordinate systems. From the above Entwurf field equations and from the conservation law: ෍ ஝

μ ൫ ൅ – ஜ஝ ൯ ൌ Ͳǡ μš஝ ஜ஝

it follows that: ෍ ஑ஒ

μ‰ ஜ஝ μ ቆɀ஑ஒ ඥെ݃ ቇ ൌ ͲǤ μšஒ ඥെ݃ μš஑ ͳ

ή

Einstein wrote these in short: BV = 0. These are four, third-order equations for the gPQ, which can be conceived as the conditions for the special choice of the adapted coordinate systems. Einstein and Grossmann presented a variational formalism from which the Entwurf field equations and energy momentum conservation law were derived in an elegant method. They demonstrated that the Entwurf field equations were derived by means of the variational principle, so that the Entwurf gravitational tensor was a covariant tensor for adapted coordinate systems (Einstein and Grossmann 1914, 219-220). Hence, Einstein and Grossmann believed they had discovered a simple mathematical approach (the variational formalism) that justified the Hole Argument and the restriction by the conditions BV = 0 to coordinate transformations between adapted coordinate systems. In their 1914 paper, Einstein and Grossmann thanked the mathematician Paul Bernays for having suggested this simplification by introducing the variational principle to them (Einstein and Grossmann 1914, 219). In the 1914 review paper, "The Formal Foundation of the General Theory of Relativity", Einstein presented an improved and elaborated version of the variational formalism from his March 1914 paper with Grossmann (Einstein and Grossmann 1914, 219-220).

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Einstein tried to obtain the Entwurf field equations from a variational principle. Einstein's action integral was the following: න ‫ܮ‬ඥെ݃ ݀߬Ǥ He offered a method in terms of invariant integrals. According to Einstein, physical invariance (an integral with the same form in any arbitrary coordinate system) is intermingled with mathematical invariance. Einstein's integrand was: ‫ܮ‬ඥെ݃Ǥ The square root of the negative determinant of the metric tensor ඥെ݃is a scalar. ඥെ݃݀߬ is also a scalar and an invariant. If we change coordinates, the integral will be of the same form due to theඥെ݃݀߬. Includingඥെ݃݀߬ in the integral gives an invariant expression valid for any arbitrary coordinate system. The integral: න ‫ܮ‬ඥെ݃݀߬ǡ is thus invariant under an arbitrary change of coordinates (i.e., it has the same form in any arbitrary coordinate system). In 1914 Einstein varied the gravitational field of gPQ by an infinitely small amount, so that gPQ are replaced by gPQ + GgPQ, where the GgPQ vanish at the boundaries of a certain region of space-time. L becomes LGLIn the 1914 Entwurf theory the above integral was invariant under limitation to adapted coordinate systems. We could set: े ൌ ‫ܮ‬ඥെ݃ǡ where े is the Lagrangian density and L is a scalar. However, in 1914, Einstein did not follow this path.

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The term: ඥെ݃݀߬ǡ in Einstein's action integral is the proper time dW0 (Eigenzeit, eigentime). In Section §6 of his 1914 review paper, Einstein derived the relation between proper time, dW0, and coordinate time, dW. Firstly, he wrote an equation representing the metric tensor – the "fundamental tensor", as he put it, and the equation: ෍ ݃ఓఙ ݃ఔఙ ൌ ߜఓఔ ǡ ఙ

where, ߜఓఔ is the Kronecker delta defined by: ߜఓఔ ൌ ͳ‹ˆߤ ൌ ߥǡ ߜఓఔ ൌ Ͳ‹ˆߤ ് ߥǡ and according to the multiplication theorem for determinants: อ෍ ݃ఓఙ ݃ఔఙ อ ൌ หߜఓఔ ห ൌ ͳǤ ఙ

From this it follows: ห݃ఓఔ ห ή ȁ݃ఓఔ ȁ ൌ ͳǤ As already stated, Einstein called the line-element: ݀‫ ݏ‬ଶ ൌ ෍ ݃ఓఔ ݀‫ݔ‬ఓ ݀‫ݔ‬ఔ ǡ ఓఔ

the coordinate interval (dW) between two space-time points, and the quantity: ݀‫ ݏ‬ଶ ൌ ෍ ݀ܺఔଶ ǡ ఔ

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the natural interval (dW0) between two space-time points. .

The special theory of relativity is valid in the infinitesimally small local region of space-time: ݀‫ ݏ‬ଶ ൌ ෍ ݃ఓఔ ݀‫ݔ‬ఓ ݀‫ݔ‬ఔ ൌ ෍ ݀ܺఔଶ Ǥ ఓఔ

XQ (Q = 1, 2, 3, 4) are coordinates of space and time in the infinitely small four dimensional local regions of space-time. It follows that the proper time is an invariant: ݀߬଴‫כ‬ଶ ൌ †• ଶ ൌ න ݀ܺଵ ݀ܺଶ ݀ܺଷ ݀ܺସ ൌ ݃݀߬ ଶ ǡ and the integral: ݀߬଴‫ כ‬ൌ න ݀ܺఔ ǡ is also an invariant – completely independent of the choice of coordinates. Einstein wrote: ݀ܺఙ ൌ ෍ ߙఙఓ ݀‫ݔ‬ఓ ǡ ఓ

by which the invariant is: ݀߬଴‫ כ‬ൌ หߙఙఓ ห݀߬ǡ where: ݀߬ ൌ න ݀‫ݔ‬ఔǡ  xQ (Q = 1, 2, 3, 4). Einstein concluded that: ݃ఓఔ ൌ ෍ ߙఙఓ ߙఙఔ ǡ ఙ

and, according to the multiplication theorem for determinants:

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ଶ

ห݃ఓఔ ห ൌ อ෍ ߙఙఓ ߙఙఔ อ ൌ หߙఓఔ ห Ǥ ఙ

Einstein wrote: ห݃ఓఔ ห ൌ ݃ for short. According to: ݀‫ ݏ‬ଶ ൌ ෍ ݃ఓఔ ݀‫ݔ‬ఓ ݀‫ݔ‬ఔ ൌ ෍ ݀ܺఔଶ ǡ ఓఔ

(valid in the infinitesimally small local region of space-time), the dXVcorrespond to the coordinates in special relativity: Three of these are real values and one is imaginary (i.e., the dX4). Consequently, dW in the equation: ݀߬଴‫ כ‬ൌ หߙఙఓ ห݀߬ is imaginary. This is a relation between the proper time and the coordinate time. However, in special relativity, the determinant g with a real value time coordinate is negative, because gPQ is diagonal (– 1, – 1, – 1, 1). ඥ݃ is therefore also imaginary. In order to avoid imaginary coordinates, Einstein put: ͳ ݀߬଴ ൌ න ݀ܺఔ ǡ ݅ and rewrote the above invariant: ݀߬଴‫ כ‬ൌ ȁ݃ȁ݀߬ǡ in the following form: ݀߬଴‫ כ‬ൌ ඥ݃݀߬Ǥ He finally wrote this invariant as (Einstein, 1914b, pp. 1039-1042):

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݀߬଴ ൌ ඥെ݃݀߬Ǥ According to the principle of equivalence, we can choose a locally free falling box where ඥെ݃݀߬ ൌ Ͳ. Then, at this local box, the laws of special relativity are valid. Einstein started with the Lagrangian L of the contravariant fundamental tensor gPQ and its first derivatives߲݃ஜ஝ ൗ߲‫ݔ‬஢ , where the latter are called ఓఔ ݃ఙ for short.He then wrote the following equation for the action integral J (Einstein 1914b, 1069-1070): ‫ ܬ‬ൌ ‫ܮ ׬‬ඥെ݃ ݀߬Ǥ He considered the change ' due to infinitesimal transformation of the invariant ݀߬଴ ൌ ඥെ݃݀߬ at some point of space-time, and found: ȟඥെ݃݀߬ ൌ Ͳ. If the gravitational field of gPQ is varied by an infinitely small amount, so that the metric tensor components gPQ are replaced by gPQ + GgPQ, where the GgPQ shall vanish at the boundaries ofan infinitesimally small local region of space-time, then the Lagrangian function, L becomes LGLand the action J becomes J + GJ. Einstein varied the gPQ infinitesimally and rewrote the equation ‫ ܬ‬ൌ ‫ܮ ׬‬ඥെ݃݀߬ in the following form: μඥെ‰ μ μඥെ‰ Ɂ ൌ න ݀߬ ෍ Ɂ‰ ஜ஝ ൝ െ෍ ቆ ቇൡǤ μš஢ μ‰ ஜ஝ μ‰ ஜ஝ ஢ ఓఔ

஢

He deduced the existence of ten components, which have tensorial character if we limit ourselves to adapted coordinate systems (Einstein 1914b, 1073). Einstein proved that under limitation to adapted coordinate systems this expression was an invariant (Einstein 1914b, 1071; see Chapter 2, Section 8). He defined:

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ॄஜ஝ ൌ

μඥെ‰ μ μඥെ‰ െ෍ ቆ ቇǡ ஜ஝ μš஢ μ‰ ஜ஝ μ‰ ஢ ஢

and concluded that the above integral divided byඥെ‰ is also an invariant: ͳ ඥെ‰

෍ Ɂ‰ ஜ஝ ॄஜ஝ Ǥ

Einstein called the tensor density ॄஜ஝ a gravitation tensor and concluded that under limitation to adapted coordinate systems and substitutions between them: ॄஜ஝ ඥെ‰ is a covariant tensor; and ॄஜ஝ is a covariant tensor density. Einstein derived his Entwurf field equations from a variational principle. The field equations represent a correlation between the gravitation tensor ॄಔಕ

ǡ and the stress-energy tensor density ॎఔఛ :

ඥି୥

ॄఓఔ ൌ ߢॎఔఛ Ǥ The stress-energy tensor density ॎఔఛ is a symmetrical tensor of rank two. It relates the stress components with the mass sources and energy and momentum densities. Einstein derived ten equations for the metric, which have tensorial character, when we limit ourselves to adapted coordinate systems. He combined the stress and mass-energy-momentum four-vector density and developed a product tensor having ten independent components like the metric tensor (Einstein 1914b, 1056). Einstein discovered that the following Lagrangian: ൌ

μ‰ த஡ μ‰ த஡ ͳ ෍ ‰ ஑ஒ ǡ Ͷ μš஑ μšஒ ஑ஒத஡

led to the Entwurf field equations that were valid for adapted coordinate systems. According to Einstein, without using the physics (the Entwurf theory of gravitation), we have arrived at field equations using variational

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formalism, valid for adapted coordinate systems (Einstein 1914b, 1076). Einstein thus showed that mathematical variational formalism led to the Entwurf field equations that were valid for adapted coordinate systems. This demonstration, he thought, lent support to his gravitation tensor ॄಔಕ

. Contrary to what Einstein thought, however, this was wrong.

ඥି୥

He wrote the Entwurf field equations usingॄఓఔ : ෍ ݃ఔఛ ቆ ఛఈ

μඥെ‰ μ μඥെ‰ െ ቇ ൌ ߢॎఔఙ Ǥ μšఈ μ‰ ఙఛ μ‰ ఙఛ ఈ

He then wrote the field equations in the following way: ෍ ஑த

μඥെ‰ μඥെ‰ μඥെ‰ μ ቆ‰ ఔத ቇ ൅ ෍ ቆെ‰ ఔத െ ‰ ఔఛ ቇ ൌ െɈॎఔఙ Ǥ ఈ ஢த μš஑ μ‰ ஢த μ‰ μ‰ ஢த ஑ ஑ த஑

For adapted coordinate systems, the following coordinate condition is satisfied (the divergence of the first term on the left-hand side of the above equation is null): ஜ ൌ ෍ ஑தఔ

μඥെ‰ μ μ ቆ‰ ఔத ቇ ൌ ͲǤ μšఔ μš஢ μ‰ ஢த ஑

We thus obtain the law of conservation of energy-momentum as a consequence of the Entwurf field equations, valid for adapted coordinate systems: μඥെ‰ μඥെ‰ μ ͳ ൝ॎఔ ൅ ෍ ቆെ‰ ఔத െ ‰ ఔఛ ቇൡ ൌ ͲǤ ఈ μš஑ ఙ Ɉ μ‰ ఙఛ μ‰ ஢த ஑ ஑த

The divergence of the stress-energy tensor ॎఔఙ vanishes for all adapted coordinate systems, and the same applies, according to the field equations ॄఓఔ ൌ Ɉॎఓఔ – to the gravitation tensor

ॄಔಕ

Ǥ

ඥି୥

Einstein chose the following Lagrangian:

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ൌ

μ‰ த஡ μ‰ த஡ ͳ ෍ ‰ ஑ஒ Ǥ Ͷ μš஑ μšஒ ஑ஒத஡

The same Lagrangian can be written in the following way (due to the components of the gravitational field): ஡

ஜ

 ൌ ෍ ‰ ஑ஒ Ȟஜத Ȟ஡஝ ǡ ஜ஡த஝

where: ͳ ߲݃ఙఛ ஝ ൌ ෍ ݃ఔఛ ǡ Ȟ஢ஒ ʹ ߲‫ݔ‬ఉ ఛ

are the components of the gravitational field. With the equation for the above Lagrangian, Einstein rewrote the above field equations in the following form: ෍ ஑ஒ

μ ஝ ൫ඥെ‰݃ఈఉ Ȟ஢ஒ ൯ ൌ െߢሺॎఔఙ ൅ ‫ݐ‬ఙఔ ሻǡ μš ஑

where: – ஝஢ ൌ

ͳ ඥെ‰ ஡ ஜ ஡ ஜ ෍ ൬‰ ஝த Ȟஜ஢ Ȟ஡த െ Ɂ஝஢ ‰ ததᇱ Ȟஜத Ȟ஡தᇱ ൰ ǡ ʹ ߢ ஜ஡ததᇱ

ఔ has tensorial character isthe energy tensor of the gravitational field. ‫ݐ‬ఙ only under linear transformations.

Einstein found that ෍ ఔ

߲ሺॎఔఙ ൅ ‫ݐ‬ఙఔ ሻ ൌ Ͳǡ ߲‫ݔ‬ఔ

the law of conservation of energy-momentum holds (Einstein 1914b, 1073-1077).

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In Section §12 of his 1914 review paper, "The Formal Foundation of the General Theory of Relativity", Einstein presented an improved and elaborated version of the Hole Argument (Einstein 1914b, 1066-1067). Consider a finite part of space-time (a hole) where material processes do not occur. Physical events are completely determined if the metric field components gPQ are functions of xQ with respect to a coordinate system K. The totality of the gPQ is symbolically denoted by G(x). Consider the gravitational field with respect to a new coordinate system K' and the totality of the g'PQ (xQ') is symbolically denoted by G'(x'). G'(x') and G(x) describe the same gravitational field outside the hole, but smoothly differ one from another inside the hole. We now make the following substitution: We replace the coordinate x'Q by the coordinate xQ in the functions g'PQ for the region inside the hole. We thus obtain: G'(x). Hence, G'(x) also describes a gravitational field with respect to the coordinate system K inside the hole. If we assume now that the differential equations of the gravitational field are generally covariant, then they are satisfied for G'(x') (with respect to K') if they are satisfied by G(x) with respect to K. They are then also satisfied with respect to K by G'(x). With respect to K, there then exist two different solutions G(x) and G'(x), which are different from one another everywhere outside the hole; nonetheless, at the boundary and inside the hole both solutions coincide, i.e., what is happening in the hole cannot be determined uniquely by generally-covariant differential equations for the gravitational field. Subsequently, Einstein demonstrated that the Hole Argument was compatible with the restriction of the coordinate systems to adapted coordinate systems. He considered a finite portion of space-time (a hole) and the coordinate system K, and imagined a series of infinitely related coordinate systems K', K''. Einstein showed that among these systems there are systems adapted to the gravitational field; that is to say, the four coordinate conditions BV = 0 (that are a direct consequence of the Entwurf gravitational field equations) hold for these systems, in which two metric field components g'PQand gPQ are not functions of the same xQ(these are the adapted coordinate systems). It is a sufficient condition that the coordinate system is adapted to the gravitational field and that the covariance of the field equations is far-reaching in these adapted coordinate systems (Einstein, 1914b, 1070-1071). Einstein knew that the problem of gravitation was finding differential equations for the metric tensor components, which were invariant under

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non-linear coordinate transformations. Such differential equations and only those should be taken into consideration as the field equations of the gravitational field. The gravitation tensor in his field equations had to be covariant, in a form that remains unchanged under all coordinate transformations – i.e., independent of the choice of the coordinate system. The Entwurf field equations only remained unchanged under coordinate transformations between adapted coordinate systems. To put it differently, on account of the adapted coordinate systems, Einstein expected that there existed acceleration transformations of varied kinds, which transform the equations to themselves (e.g. also rotation), so that the principle of equivalence would be preserved, even to an unexpectedly large extent (Einstein and Grossmann 1914, 219). Restriction under the four coordinate conditions would apparently solve the problem of physical equivalence of a centrifugal field and a gravitational field, and the Entwurf field equations would apparently satisfy the relativity principle and the equivalence principle. So, the adapted coordinate systems not only justified the restricted covariance of the field equations and Einstein's Entwurf gravitational tensor, but even supplied reasoning for why generally covariant field equations would be unacceptable. Einstein stubbornly adhered to his adapted coordinate systems, and would not hear any criticism. In fact, he had a propensity for clinging to ideas (e.g. the adapted coordinate systems and later the static cosmological model, see Chapter 3, Section 2), even after they had been mathematically discredited. Einstein's competition with his colleagues went hand-in-hand with his stubbornness and refusal to accept criticism of the heuristic principles underlying his theory. Einstein's collaboration with his friends Grossmann and Besso, on the one hand, and his competition against his opponents Abraham, Nordström, Mie and David Hilbert (see Chapter 2, Section 9) on the other, led to several breakthroughs in his general theory of relativity. Einstein's competition with his rivals forced him to triumph over his colleagues and reconsider his proofs, demonstrations and findings in light of the criticism of those who challenged his theories. This took place when Einstein was already insecure about his 1912 coordinate-dependent static gravitational theory, and later about his 1913-1914 Entwurf gravitation theory, which came after the first theory.

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In 1914, Einstein firmly believed that his heuristic guiding principles (equivalence principle, Mach's ideas [principle], relativity principle, energy-momentum conservation principle, correspondence principle) rendered the underlying scaffold of his gravitational theory immune to refutation. This led him to refuse to admit he had made mistakes, and to overlook minor calculation errors. An example for this is the calculation of the rotation metric in the Einstein-Besso Manuscript, which eventually became crucial after he finally realised that his adapted coordinate systems were faulty.

8. Einstein's Polemic with Tullio Levi-Civita on the Entwurf Theory Until October 1915, Einstein did not realise that the adapted coordinate systems were inadmissible. He proved that his gravitational tensor was a covariant tensor for adapted coordinate systems. In his 1914 paper, "The Formal Foundation of the General Theory of Relativity", Einstein explained the formal steps that led to his choice of adapted coordinate systems. He presented a theorem and provided proof that it supplies the formal basis for his belief that if the coordinate system is an adapted coordinate system, then the gravitational tensor is a covariant tensor. Recall from Chapter 2, Section 7 that he started with the Lagrangian function L of the contravariant fundamental tensor gPQ and its first ஜ஝ derivatives߲݃ஜ஝ ൗ߲‫ݔ‬஢ ൌ ‰ ஢ . He then wrote the following equation for the integral J; the integral is extended over a finite part 6 of space-time: ‫ ܬ‬ൌ ‫ܮ ׬‬ඥെ݃ ݀߬Ǥ Consider a coordinate system, K1. Einstein asked for the change 'J of J when going from the system K1 to another system K2, which is infinitesimally different from K1 (Einstein 1914b, 1069). He considered the change ' due to the infinitesimal transformation of ݀߬଴ ൌ ඥെ݃݀߬ at some point of space-time, and found: ȟඥെ݃݀߬ ൌ Ͳ. ఓఔ

He then obtained an expression for 'Lin terms of'gPQ and ȟ݃ఙ :

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1) Einstein first referred to the transformation law: ‫ܣ‬ఓఔᇱ ൌ ෍ ఈஒ

μšԢஜ μšԢ஝ ఈఉ ‫ ܣ‬ǡ μš஑ μšஒ

He took the following relations into account:

'gPQ = gPQ' – gPQ, 'xP= x'P – xP, expressed the 'gPQ in terms of 'xP, and obtained: ȟ‰ ஜ஝ ൌ ෍ ቆ‰ ஜ஑ ఈ

μȟšஜ μȟš஝ ൅ ‰ ஝஑ ቇǤ μš஑ μš஑ ఓఔ

He then obtained an additional expression for ȟ݃ఙ . 2) L must be chosen in such a way that it is invariant under linear transformations. Einstein therefore obtained an expression for'J: μඥെ‰ μଶ ȟšஜ ͳ ȟ ൌ න †ɒ ෍ Ǥ ஜ஝ ʹ μ‰ ஢ μš஢ μš஑ ఓఔఙఈ

From this, by partial integration, he acquired: ͳ ȟ ൌ න †ɒ ෍ሺ ȟšஜ ஜ ሻ ൅ Ǥ ʹ ఓ

Einstein wrote for BP: ஜ ൌ ෍ ఈ஢஝

μඥെ‰ μଶ ቆ‰ ஝஑ ஜ஝ ቇǡ μš஢ μš஑ μ‰ ஢

He also wrote an additional expression for F. This enabled Einstein to restrict the coordinate systems (Einstein 1914b, 1070). He considered the finite portion6 and the coordinate system K. He imagined a series of infinitely related coordinate systems K', K'' and so on, so that 'xP and

ப୼୶ಔ  ப୼୶ಉ

vanished at the boundaries. For every infinitesimal

coordinate transformation between neighbouring coordinate systems of the

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total coordinate systems K, K', K'',… we obtain: F = 0, and the equation for 'J becomes: ͳ ȟ ൌ ෍ න †ɒ ȟšஜ ஜ Ǥ ʹ ఓ

Among the above systems K', K'' and so on, Einstein chose coordinate systems adapted to the gravitational field. What characterizes these systems? The equations BP = 0 hold for these adapted coordinate systems. This is then a sufficient condition for the coordinate system adapted to the gravitational field. In Section §14 of his 1914 review paper, Einstein proved a theorem that supplied the formal basis for the claim that if the coordinate system was an adapted coordinate system, then the gravitational tensor was indeed a covariant tensor. If the gravitational field of gPQ is varied by an infinitely small amount, so that gPQ are replaced by gPQ + GgPQ, where the GgPQ shall vanish at the boundaries of 6, then L becomes LGLand the action J becomes J + GJ. Then the equation: '^GJ} = 0 always holds whichever way the GgPQ might be chosen, provided the coordinate systems (K1 and K2) are adapted coordinate systems with respect to the unvaried gravitational field. This means that under the restriction to adapted coordinate systems, GJ is an invariant. Einstein thought he proved his theorem and deduced the existence of ten components, which have tensorial character, when we limit ourselves to adapted coordinate systems (Einstein 1914b, 1071, 1073). After varying infinitesimally the gPQ, Einstein rewrote equation: ‫ ܬ‬ൌ ‫ܮ ׬‬ඥെ݃ ݀߬ in the following form: μඥെ‰ μ μඥെ‰ Ɂ ൌ න ݀߬ ෍ Ɂ‰ ஜ஝ ൝ െ෍ ቆ ቇൡǤ ஜ஝ μš஢ μ‰ ஜ஝ μ‰ ஢ ఓఔ

஢

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Einstein proved that under limitation to adapted coordinate systems GJ was invariant. The quantity in the brackets of the above equation is the gravitation tensor ॄஜ஝ : ॄஜ஝ ൌ

μඥെ‰ μ μඥെ‰ െ෍ ቆ ቇǤ ஜ஝ μš஢ μ‰ ஜ஝ μ‰ ஢ ஢

Einstein demonstrated that it is also an invariant: ͳ ඥെ‰

෍ Ɂ‰ ஜ஝ ॄஜ஝ Ǥ

He thus rewrote: μඥെ‰ μ μඥെ‰ Ɂ ൌ න ݀߬ ෍ Ɂ‰ ஜ஝ ൝ െ෍ ቆ ቇൡǡ ஜ஝ μš஢ μ‰ ஜ஝ μ‰ ஢ ఓఔ

஢

in the following form: Ɂ ൌ න ݀߬ ෍ Ɂ‰ ஜ஝ ॄஜ஝ Ǥ ఓఔ

According to Einstein,GJ is invariant. Einstein thus concluded: ॄஜ஝ ඥെ‰ under limitation to adapted coordinate systems, and transformations between them, is a covariant tensor, and ॄஜ஝ is a covariant tensor density (Einstein 1914b, 1073). Tullio Levi-Civita, one of the founders of tensor calculus, objected to this major element in the Entwurf theory, which reflected its global problem: Its field equations were restricted to an adapted coordinate system. LeviCivita did not directly question the limited covariance properties of Einstein's Entwurf field equations; instead, he attacked Einstein's proof of the covariance properties of his Entwurf gravitation tensor: 

General Relativity between 1912 and 1916

ॄஜ஝ ඥെ‰

121

Ǥ

In an exchange of letters and postcards that began in March 1915 and ended in May 1915, Levi-Civita presented his objections to Einstein's above proof: Levi-Civita could not accept that Einstein's Entwurf gravitational tensor

ॄಔಕ

was a covariant tensor for adapted coordinate

ඥି୥

systems (within a theory which was limited in its covariance). In his correspondence with Einstein he demonstrated to the latter that his gravitational tensor

ॄಔಕ

could not be a covariant tensor for adapted

ඥି୥

coordinate systems. Einstein tried to find ways to save his proof by answering Levi-Civita's quandaries and demonstrations. In most of the letters he repeated the same arguments over and over again. Einstein found it hard to give up his proof that his gravitational tensor

ॄಔಕ

was a covariant tensor for adapted

ඥି୥

coordinate systems. However, Levi-Civita gradually showed Einstein that he attempted the impossible (Einstein to Levi-Civita, March 5, 17, 20, 26, 28, April 2, 8 14, 20, May 5, 1915, CPAE 8, Doc. 60, 62, 64, 66, 67, 69, 71, 75, 77, 80;Cattani and De Maria, 1989, 176). Levi-Civita did not agree with Einstein's view that under limitation to adapted coordinate systems: Ɂ ൌ න ݀߬ ෍ Ɂ‰ ஜ஝ ॄஜ஝ ǡ ఓఔ

is an invariant. Einstein referred to his 1914 review paper and to the proof from this paper. He intended to show by a new strategy that, under limitation to adapted coordinate systems, GJ was an invariant, and therefore the gravitation tensor

ॄಔಕ

was a covariant tensor.

ඥି୥

Einstein rewrote the above equation in the following form:

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122

ॄஜ஝ Ɂ ൌ ෍ ቊ න ඥെ‰Ɂ‰ ஜ஝ ݀߬ቋ Ǥ െ‰ ඥ ஜ஝ He then obtained the following equation: ‫ܣ‬ఓఔ ൌ න ඥെ݃ ߜ݃ఓఔ ݀߬Ǥ He obtained this equation in the following manner: He started with the contravariant metric tensor gPQ. It transforms according to: ‫ܣ‬ఓఔᇱ ൌ ෍ ఈஒ

μšԢஜ μšԢ஝ ఈఉ ‫ ܣ‬ǡ μš஑ μšஒ

in the following way: ݃ᇱఘఙ ൌ ෍

μšԢ஡ μšԢ஢ ఓఔ ݃ Ǥ μšஜ μš஝

The variation GgPQ (a tensor) transforms according to the same equation: ߜ݃ᇱఘఙ ൌ ෍

μšԢ஡ μšԢ஢ ఓఔ ߜ݃ Ǥ μšஜ μš஝

Einstein multiplied this equation by ඥെ݃Ԣ݀߬ ᇱ ൌ ඥെ݃݀߬ and integrated over 6 and obtained:  μšԢ஡ μšԢ஢ ఓఔ ߜ݃ Ǥ න ඥെ‰ԢɁ‰Ԣఘఙ ݀߬Ԣ ൌ න ඥെ‰ †ɒ ෍ μšஜ μš஝ He then re-wrote this equation in the following short form: ‫ܣ‬ఓఔ ൌ න ඥെ݃ ߜ݃ఓఔ ݀߬Ǥ He inserted this equation into: ॄஜ஝ Ɂ ൌ ෍ ቊ න ඥെ‰Ɂ‰ ஜ஝ ݀߬ቋ Ǥ െ‰ ඥ ஜ஝

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123

This gives: ॄஜ஝ ఓఔ ‫ ܣ‬ቋǤ Ɂ ൌ ෍ ቊ ඥെ‰ Einstein arrived at the following conclusion: Since GJ is an invariant and APQ is a contravariant tensor with independent chosen components, then: ॄஜ஝ ඥെ‰ is a covariant tensor (Einstein to Levi-Civita, March 5, 17, 1915, CPAE 8, Doc. 60, 62). Levi-Civita corrected Einstein's above equation to: Ɂ ൌ ෍ ஜ஝

ͳ ඥെ‰

ॄஜ஝ ‫ܣ‬ఓఔ ൅ ɂ ൌ ‹˜ƒ”‹ƒ–ǡ

where, H is an infinitesimal quantity of higher order. Levi-Civita explained that Einstein's equation was not a tensor. Einstein did not agree with Levi-Civita. He informed the latter that for "justified transformations", namely transformations between adapted coordinate systems: ෍

ͳ ඥെ‰

ॄஜ஝ ‫ܣ‬ఓఔ ൌ ෍

ͳ ඥെ‰Ԣ

ॄԢఙఛ ‫ܣ‬ᇱఙఛ Ǥ

Einstein again wrote the transformation law: ‫ܣ‬ᇱఙఛ ൌ ෍

μšԢ஢ μšԢத ఓఔ ‫ ܣ‬ǡ μšஜ μš஝

inserted it in the above equation, and obtained: ෍ ఓఔ

ॄஜ஝ ඥെ‰

‫ܣ‬ఓఔ ൌ ෍ ஢தஜ஝

ॄԢ஢த μšԢ஢ μšԢத ఓఔ ‫ ܣ‬ǡ ඥെ‰Ԣ μšஜ μš஝

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and: ॄஜ஝ ඥെ‰

ൌ෍ ஢த

μšԢ஢ μšԢத ॄԢ஢த Ǥ μšஜ μš஝ ඥെ‰Ԣ

Einstein concluded that

ॄಔಕ ඥି୥

has a tensor character.

Suppose we replace Einstein's equation: ෍

ͳ ඥെ‰

ॄஜ஝ ‫ܣ‬ఓఔ ǡ

by Levi-Civita's equation: ෍ ஜ஝

ͳ ඥെ‰

ॄஜ஝ ‫ܣ‬ఓఔ ൅ ɂǤ

Einstein showed that the end result (

ॄȝȞ

ඥି୥

is a tensor) would not change

(Einstein to Levi-Civita, March 26, 1915, CPAE 8, Doc. 66). The great mathematician, Levi-Civita, had a much better command of these mathematical matters than Einstein. However, Einstein was stubborn and after thorough considerations, he believed he had maintained his proof. Thus, Einstein persistently maintained his adapted coordinate systems. Einstein thought that if he managed to demonstrate that under the limitation to adapted coordinate systems GJ was an invariant, he could prove that his gravitation tensor was a covariant tensor for adapted coordinate systems; and then the principle of general relativity would be valid for these adapted coordinate systems. Einstein confused invariance with covariance and mathematical formal invariance. He thought that mathematical formal invariance meant that the laws of physics would be valid in all adapted systems of coordinates, just as special relativistic invariance meant that the laws of physics would be valid in all inertial systems of reference. Levi-Civita could not accept that under limitation to adapted coordinate systems, GJ was invariant, because he was speaking about formal invariance. Therefore, he eventually demonstrated to Einstein that his Lagrangian was not a formal invariant under justified

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transformations, and his Entwurf field equations were not covariant under these transformations either (Cattani 1993, 208). Levi-Civita intended to demonstrate to Einstein that what the latter had considered a covariant gravitation tensor was incorrect. Einstein, however, did not concede from the most important proof of his 1914 Entwurf theory. He explained to Levi-Civita: Strangely enough, the proof was not refuted for the following reason. The proof failed in precisely that special case (demonstration) that Levi-Civita had treated. Einstein even tried to demonstrate that Levi-Civita's example did not constitute a refutation of his stated proof. However, Levi-Civita was not persuaded. Einstein remarked that he did not quite understand Levi-Civita's objection; he did not understand why the conclusion drawn from his proof of the covariance of his gravitation tensor ought not to apply. Einstein told Levi-Civita that in the calculus of variations it is always done in the way in which he had done it in his Entwurf theory. Einstein tried to demonstrate to Levi-Civita the mathematical reasoning behind his proof, but at the same time his confidence was somewhat shaken. He was not altogether sure that his proof was now truly solid, and begged Levi-Civita to inform him of his opinion of the proof upon reconsideration. Levi-Civita was not satisfied with Einstein's stubbornness, but the latter ignored Levi-Civita's criticism and attempted to show that the conclusion drawn from his proof was correct. So Einstein stubbornly adhered to his adapted coordinate systems, and would not hear any criticism, even when a mathematical flaw was found in his derivation. This incidence was to reoccur later, during his correspondence with Levi-Civita on the topic of the components representing the energy of the gravitational field ‫ߥߪݐ‬. ‫ߥߪݐ‬ was defined by Einstein as a pseudo-tensor having a tensorial character only under linear transformations. Levi-Civita criticised Einstein's use of the ‫ ߥߪݐ‬pseudo-tensor for the energy components of the gravitational field (See Chapter 3, Section 1). This incidence was to reoccur later also during Einstein's correspondence with Willem de Sitter on the issue of the cosmological model and Much's principle (See Chapter 3, Section 2). Einstein intended to illustrate, again and again, by different strategies that the conclusion arrived at from his equations was correct, and his Entwurf

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126

gravitational tensor

ॄಔಕ

was a covariant tensor. Einstein still hoped that

ඥି୥

Levi-Civita would not find any significant holes in his proof. Unfortunately, Einstein was disappointed. Levi-Civita remained unsatisfied despite each new letter received containing Einstein's demonstrations. Each time, Einstein received a response from Levi-Civita with a counterargument that disproved the tensorial character of his Entwurf gravitational tensor

ॄಔಕ

.

ඥି୥

Einstein, however, remained unconvinced by Levi-Civita's criticism and, using his usual sense of humour mixed with cynicism and honesty, told Levi-Civita that in his letters he first flattered him (Einstein) kindly, to prevent him from making a dour face upon reading his (Levi-Civita's) new objections. Desperate, Einstein could not quite accept Levi-Civita's new objections and repeatedly endeavoured to prove that for any substitutions between adapted coordinate systems his gravitation tensor

ॄಔಕ

had a tensor

ඥି୥

character. Suddenly he realised that there was a little problem with his derivation, but he inspected that a suggested modification was needed for his covariance proof. Nonetheless, he stubbornly clung to what had remained from his proof, and he now reasoned that it did not fail generally, but only in certain, special cases. Einstein, therefore, believed that it proved nothing about the validity of his proof in general. Hence, his Entwurf gravitational tensor

ॄಔಕ

was covariant and it could be used as a gravitation tensor.

ඥି୥

There was even a moment in which Einstein regained full confidence in his Entwurf gravitational tensor

ॄಔಕ

. Full of courage, he admitted to Levi-

ඥି୥

Civita, through the deeper considerations to which the latter interesting letters have led him, that he became even more firmly convinced that the proof of the tensor character of the Entwurf gravitation tensor

ॄಔಕ

was

ඥି୥

correct in principle. Einstein, however, was eventually most disappointed when Levi-Civita was still unsatisfied. Although Einstein realised that Levi-Civita still objected to his claims, he still hoped that his new derivations and

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127

manipulations sent to Levi-Civitawould finally convince him. Levi Civita, however, was as stubborn in his objections to Einstein's proof as Einstein was in his adherence to this proof.  On April 11, Einstein sent Levi-Civita a postcard. Einstein offered LeviCivita a special case for the claim that

ॄಔಕ ඥି୥

was a tensor. This special case

involved setting L = const. The coordinate condition BP = 0 is then satisfied identically, and L is invariant under arbitrary, and thus also linear substitutions.

ॄಔಕ ඥି୥

must, in this case, be a tensor under arbitrary

substitutions. Einstein gave the example L = const. to refute Levi-Civita's objection that

ॄಔಕ ඥି୥

was not a tensor (Einstein to Levi-Civita, April 11 and

21, 1915, CPAE, Vol. 8, Doc. 74, 78).  However, L = constant was incompatible with the Lagrangian from Einstein's 1914 review paper, "The Formal Foundation of the General Theory of Relativity" (Einstein 1914b, 1076): ൌ

μ‰ த஡ μ‰ த஡ ͳ ෍ ‰ ஑ஒ Ǥ Ͷ μš஑ μšஒ ஑ஒத஡

According to Einstein, this was the only Lagrangian function that led to the Entwurf field equations. L = constant did not lead to the Entwurf field equations (Cattani and De Maria 1989, 192). Finally, Levi-Civita convinced Einstein about the fault in his arguments. Einstein admitted that his proof was incomplete. He was incredibly disappointed, and – desperately – suggested to Levi-Civita a formulation for a possible new gravitation tensor, one that could not solve the problem of the relativity of motion. Einstein was obliged to admit that his demonstration was incomplete, and admitted the impossibility of proving the tensorial character of his Entwurf gravitation tensor

ॄಔಕ

within a

ඥି୥

limited-covariant theory. Only in spring 1916, long after Einstein had abandoned the 1914 Entwurf theory, did he finally understand the main problem with his 1914 gravitational tensor

ॄಔಕ

. In autumn 1915, the brilliant Göttingen

ඥି୥

mathematician, David Hilbert, found the central flaw in Einstein's 1914 derivation. In March 1916, Einstein sent Hilbert a letter admitting that the

Chapter Two

128

error he had found in his paper of 1914 was now completely clear to him (Einstein to Hilbert, November 7, 1915 and March 30 1916, CPAE 8, Doc. 136, 207). Einstein left Berlin for Zurich at the beginning of September 1915 for three weeks. He stayed with his friend Heinrich Zangger (Einstein to Hertz, August 22, 1915, CPAE 8, Doc. 111; Fölsing 1997, 365). We have seen that Einstein checked whether the metric field describing a rotating system was a solution for his Entwurf field equations. On pages 41-42 of the Einstein-Besso manuscript Einstein wrote, in a coordinate system rotating counterclockwise with respect to the z axis of an inertial coordinate system, a Minkowski metric field similar to the one below: –1

0

0

Zy

0

–1

0

Zx

0

0

–1

0

Zy Zx

0

1 – Z2(x2 + y2)

g14 = Zyg24 = Zx, The Entwurf field equation: DPQ(g) +NtPQ = 0, D44(g) = –'g44 – 4Z2. Einstein wrote a solution for his equations that has the form of the g44 – the 44 component of the Minkowski metric in rotating coordinates – but included a sign error on page 41: –'g44 – 4Z + 2Z = 0, 'g44 = –2Z, and, – 4Z + 2Z = –2Z

General Relativity between 1912 and 1916

129

Thus, because of this minor error, from 1913-1915, he thought that the metric field describing a rotating system was a solution for the Entwurf field equations. The solution Einstein wrote for the g44 was: g44 = 1 –Z(x y . Einstein asked himself whether the g44 found above was the same as the g44 one would obtain through direct transformation of the Minkowski metric to the rotating coordinate system. According to the above equation Einstein arrived at the conclusion that it was, and next to the solution he wrote: "correct". Even though next to the solution he wrote that it was correct, he decided to redo the calculation. On page 42 he put: –'g44 = (1/4)(4Z  and thus: – 4Z + 2Z = (1/4)(4Z  Upon revisiting the calculation, he found that his Entwurf field equation was not satisfied: The metric field describing a rotating system was not a solution of the Entwurf field equations. Einstein obtained a solution of the form for the g44: g44 = 1 – Z(x y .  Einstein did not know what to do. He decided to write the above expression in the following form: g44 = 1 – !Z(x y .  However, the solution he first obtained on page 41 from the Entwurf field equations (while not noticing the sign error): g44 = 1 –Z(x y , showed that the metric field describing a rotating system was a solution for the Entwurf field equations, and it perfectly fitted expectations of the

Chapter Two

130

heuristic guiding principle of his theory, the principle of equivalence. Therefore, Einstein finally chose it. Upon returning to Berlin, he realised that the metric field describing a rotating system was not a solution to the Entwurf field equations. On October 1, 1915 in a letter to Otto Naumann, Einstein redid the above calculation and found that the Entwurf field equations were not satisfied (CPAE 4, Doc. 14, 41, 42, notes 184, 186, 193; Einstein to Naumann, October 1, 1915, CPAE 8, Doc. 124, note 5): –'g44 – 2Z2 – (1/4)(4Z2)z 0 , –'g44 = 4Z This realisation could have stimulated what Einstein had already discovered earlier that, Mercury's perihelion motion was too small, namely 18 seconds of arc per century. For small angular velocities, the metric field of a rotating system has the same general form as the metric field of the Sun – considered by Einstein and Besso in their 1913 manuscript for calculating the perihelion of Mercury. Thus, Einstein could use the approximation procedure used to find the metric field of the Sun to second order to find the metric in a rotating coordinate system (CPAE 4, Doc. 14, 41, note 185). He could also do the opposite; refer from his calculation of the rotating coordinate system about the case of the metric field of the Sun and Mercury's perihelion. However, Einstein was unable to do this in September 1915, because his solution for the perihelion of Mercury suffered from the same fault as the solution for a rotating coordinate system: The perihelion advance for Mercury was too small (18 seconds of arc per century), and the metric field of a rotating system was actually not a solution for the Entwurf field equations (Einstein to Freundlich, September 30, 1915, CPAE 8, Doc. 123). In October 1915, Einstein also admitted he had chosen a problematic Lagrangian function: ൌ

μ‰ த஡ μ‰ த஡ ͳ ෍ ‰ ஑ஒ Ǥ Ͷ μš஑ μšஒ ஑ஒத஡

In his 1914 review paper, Einstein explained that the ten field equations determine the ten functions gPQ, but the gPQ must also satisfy the four equations BP = 0, because the coordinate system is an adapted one.

General Relativity between 1912 and 1916

131

According to Einstein, the selection of the above Lagrangian could (formally) be supported by this restriction. He provided a proof that the above Lagrangian is uniquely selected by the requirement that it is invariant for adapted coordinate systems. Einstein believed that the variation of the Lagrangian uniquely led to the Entwurf field equations. Therefore, Einstein thought he had demonstrated that his Entwurf field equations were the only equations that were invariant for adapted coordinate systems. However, he discovered that this was an error. In fact the selection of L was not dependent on the above restriction; and L need not be limited. Thus covariance with respect to adapted coordinate systems was a flop (Einstein to Lorentz, October 12, 1915; Einstein to Sommerfeld, November 28, 1915, CPAE 8, Doc. 129, 153). Only in October 1915 did Einstein understand that his 1914 variational formalism did not uniquely lead to the Entwurf field equations (valid for adapted coordinate systems). This choice of L led to covariance with respect to adapted coordinate systems. Einstein recognised that introducing the adapted coordinate system was incorrect, and that general covariance must be demanded. He, therefore, finally understood that his Entwurf gravitational field equations were problematic, and gave up his 1914 Entwurf field equations.

9. Einstein's 1915 Competition with David Hilbert and General Relativity Already in the summer of 1915, David Hilbert invited Einstein to Göttingen to talk about his 1914 Entwurf gravitation theory that he presented in his review paper, "The Formal Foundation of the General Theory of Relativity". Einstein held six, two-hour lectures in Göttingen on his gravitation theory, where he thought he completely convinced Hilbert and his mathematician friends (CPAE 8, note 5, 146). In his first November 1915 paper on general relativity, presented on November 4, 1915, to the mathematical-physical class of the Prussian academy, Einstein gradually expanded the range of the covariance of his gravitation field equations (Einstein 1915a). Every week he expanded the covariance a little further until, on November 25, he reached his fully, generally covariant field equations (Einstein 1915d).

Chapter Two

132

During October 1915, Einstein realised that the key to the solution of his problem lay in the equation between the natural interval (Minkowski's interval) and the coordinate interval (the interval of general relativity) of his 1914 Entwurf theory. He adopted the determinant in this equation ඥെ݃ ൌ 1 as a postulate and started to use what is called unimodular transformations. This finally led him to general covariance. Recall that Einstein wrote the relation between the natural interval and the coordinate interval: ඥെ݃݀߬ ൌ ݀߬଴ Ǥ This may also be written in the following way for the relation between two intervals: ݀߬ ᇱ ൌ

߲ሺ‫ݔ‬ଵᇱ ǥ ‫ݔ‬ସᇱ ሻ ݀߬Ǥ ߲ሺ‫ݔ‬ଵᇱ ǥ ‫ݔ‬ସᇱ ሻ

Einstein recognised that ඥെ݃ ൌ ͳ, or the determinant ห݃ఙఓ ห could be equated to 1: ߲ሺ‫ݔ‬ଵᇱ ǥ ‫ݔ‬ସᇱ ሻ ൌ ͳǤ ߲ሺ‫ݔ‬ଵᇱ ǥ ‫ݔ‬ସᇱ ሻ So that only substitutions of determinant 1 are permitted. Therefore: ݀߬ ᇱ ൌ ݀߬Ǥ Thus, Einstein recognised that the above determinant is equal to 1, and the allowed transformations are unimodular transformations. A transformation is unimodular if the determinant of the metric tensor is equal to 1. Consider again the line element, the interval of general relativity: ݀‫ ݏ‬ଶ ൌ ෍ ݃ఓఔ ݀‫ݔ‬ఓ ݀‫ݔ‬ఔ ǡ ఓఔ

and the proper time or natural interval dW0:

General Relativity between 1912 and 1916

133

ඥെ݃݀߬ ൌ ݀߬଴ Ǥ In the weak-field approximation (according to Einstein's 1907 and 1911 studies): ݃ ൌͳ൅

Ȱ Ǥ ܿଶ

We obtain in the weak-field case: ݀߬଴  ൎ ඥ݃ସସ ݀߬ ൌ ൬ͳ ൅

Ȱ ൰ ݀߬Ǥ ܿଶ

The weak field approximation coincides with the special relativistic approximation only when ) = 0. Recall that, in the Zurich Notebook Einstein was not quite aware of this subtlety (see Chapter 2, Section 4). For the special relativistic approximation, consider Einstein's 1907 famous thought experiment: a man falling from a roof. Let dW0 be proper time or the natural interval in the local reference system of the free falling man from the roof (where special relativity applies): ݀߬଴ ൌ ඥെ݃݀߬. Let us return to Einstein's definition of the March 1912 equivalence principle applicable to local systems (Einstein 1912b, 455-456): Experiments in a sufficiently small free falling system, over a sufficiently short time interval, give results that are indistinguishable from those of the same experiments done in an inertial frame in which special relativity applies. Consider the following thought experiment: Imagine two systems. One system is Einstein's 1907 imaginary man falling from the roof in a gravitational field. In the other system there is a man at rest in a gravitational field. Consider the instantaneous local inertial system (local free-falling frame of reference): Imagine the man at the moment he starts to fall from the roof. At this infinitesimally initial moment, he is locally at rest. Both men in both systems are thus at rest at this very moment. The worldlines of these men are comprised of the time intervals according to the equation: ඥ݃ସସ Ԣ݀߬Ԣ ൌ ඥ݃ସସ ݀߬.

134

Chapter Two

ඥ݃ସସ represents the component of the metric tensor measured for the free falling man. He is in a local free-falling frame of reference (inertial system) and therefore locally: ඥ݃ସସ ൌ ͳǤ In the local free-falling frame of reference all the diagonal components of the Minkowski flat metric tensor are constants and are equal to 1, and the off diagonal components are zero. Therefore, we arrive at the equation: ݀߬ ൌ ඥ݃ସସ Ԣ݀߬ԢǤ On November 4, 1915, Einstein was guided by the equivalence principle and postulated that only substitutions of determinant 1 are permitted, and by postulating this he arrived at: ݀߬ ൌ ݀߬ ᇱ Ǥ In Einstein's Entwurf theory ඥെ݃ occurred in the most basic formulas. Tensor components frequently occurred multiplied byඥ݃ and ඥെ݃: ‫ܣ‬ఙ ඥ݃ ൌ िఙ ǡ ‫ܣ‬ఔఙ ඥ݃ ൌ िఓఔ Ǥ Therefore, Einstein wanted a particular name for this kind of tensor, and he called them volume tensors or V-tensors for short; in modern terminology tensor densities. When multiplied by dW they simply represented tensors, because ඥെ݃݀߬ is a scalar. In 1914, Einstein claimed that in the general theory of relativity, there were tensors of different character: covariant, contravariant, mixed, tensor densities – and a choice should be made. He wrote that any tensor could be obtained from another tensor of another character by multiplying it with the fundamental tensor, gPQ, or with ඥെ݃. On November 4, 1915, Einstein explained that in a covariant theory, in which only substitutions of determinant 1 are allowed, the ඥെ݃ factor no longer occurs in the basic formulas and this simplifies the calculations. For instance, Einstein represented the stress-energy tensor TVQ by the mixed tensor density ॎఔఙ (Einstein 1914b, 1053-1056). The generally

General Relativity between 1912 and 1916

135

covariant equations that represented energy-momentum conservation in the 1914 Entwurf theory were therefore: ෍ ఔ

߲݃ఓఔ ఔ ߲ॎఔఙ  ͳ ൌ ෍ ݃ఛఓ ॎ Ǥ ʹ ߲‫ݔ‬ఔ ߲‫ݔ‬ఙ ఙ ఓఛఔ

In a theory, in which only substitutions of determinant 1 are allowed, the ඥെ݃ factor no longer occurs in the above equation. Hence, ܶఙఔ ൌ ॎఔఙ ǡ there is no distinction between tensor densities and tensors, and tensor densities of the Entwurf theory are equal to the tensors. Einstein thus rewrote the above Entwurf equation with ordinary tensors only: ෍ ఔ

߲݃ఓఔ ఔ ߲ܶఙఔ ͳ ൌ ෍ ݃ఛఓ ܶ Ǥ ߲‫ݔ‬ఔ ʹ ߲‫ݔ‬ఙ ఛ ఓఛఔ

ܶఙఔ is an ordinary tensor (not a tensor density) (Einstein 1915, 782).This simplified the equations of the theory. In his 1914 Entwurf theory Einstein wrote the Riemann covariant tensor in the following way: ‫ܩ‬୧୩୪୫ ൌ ඥ݃ߜ୧୩୪୫ ǡ

Giklm is the Kronecker delta and it is +1 or –1 depending on the iklm and their equality to 1234 by an even or odd permutation of the indices. From the tensor Giklm one can form a contravariant tensor:

୧୩୪୫ ൌ

ͳ ඥ݃

ߜ୧୩୪୫ Ǥ

We see that Giklm z Giklm (Einstein 1914b, 1042-1043). In gravitation Einstein was interested in tensors of rank two, the Ricci tensor Gim, which could be obtained from the Riemann tensor by contraction, or multiplication with gPQ: ‫ܩ‬୧୫ ൌ ෍ ݃௞௟ ሺ݅݇ǡ ݈݉ሻǤ ௞௟

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However, since Giklm z Giklm, Einstein did not pursue the Riemann tensor of rank four and the Ricci tensor for constructing the field equations of his Entwurf theory. In October 1915, Einstein postulated ඥ݃ ൌ ͳǤ This solved the above problem and Giklm Giklm. He thus understood that the Ricci tensor was suitable for constructing covariant field equations and the gravitational tensor under the constraint to transformations with determinant equal to 1. He noted that the derivation of this tensor was better obtained from a different form of the Riemann tensor than the above, the RiemannChristoffel tensor (ik, lm) – as he had done three years earlier in the Zurich Notebook in 1912 (Einstein 1915a, 782): ‫ܩ‬୧୫ ൌ ෍ ݃௞ఘ ሼ݅ߩǡ ݈݉ሽ ൌ ఘ

߲ ݅݉ ߲ ݈݅ ݅݉ ߩ݈ ݈݅ ߩ݉ ቄ ቅെ ቄ ቅ ൅ ෍ ൤൜ ൠ ቄ ቅ െ ൜ ൠ ቄ ቅ൨Ǥ ߩ ߢ ߩ ߢ ߲‫ݔ‬௟ ߢ ߲‫ݔ‬௠ ߢ ఘ

Contracting the Riemann tensor results in the Ricci tensor: ‫ܩ‬௜௠ ൌ ሼ݈݅ǡ ݅݉ሽ ൌ ܴ௜௠ ൅ ܵ௜௠ Ǥ This division was indeed already implicitly obtained by Einstein on page 22R of his Zurich Notebook (Einstein 1912c, 451, 453). In 1915, Einstein equated Rim to the second term in the above expression of Gim, like he had done three years earlier on page 22R: ܴ୧୫ ൌ െ

߲ ݅݉ ݈݅ ߩ݉ ቄ ቅ ൅ ෍ ൜ ൠ ቄ ቅǡ ߩ ݈ ߲‫ݔ‬௟ ݈ ఘ

and the first and third terms to: ܵ୧୫ ൌ

߲ ݈݅ ݅݉ ߩ݈ ቄ ቅ െ ൜ ൠ ቄ ቅǤ ߩ ߲‫ݔ‬௠ ݈ ݈

Under the constraint to transformations with determinant equal to 1, Gim, Rim and Sim were all tensors. Hence, in the November 4, 1915 paper Einstein noted that the tensor Rim was of utmost importance for gravitation

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(Einstein 1915a, 781-782). It actually replaced Einstein's problematic 1914 Entwurf gravitation tensor

ॄഋഌ ξି௚

Ǥ

In Section §3 of the November 4, 1915 paper Einstein derived the new field equations. He wrote the general form of the field equations with his new gravitation tensor Rim: ܴஜ஝ ൌ െߢܶஜ஝ Ǥ Einstein noted that these equations were covariant with respect to arbitrary transformations of a determinant equal to 1 (Einstein 1915a, 783). Hence, these were not yet generally covariant, but only covariant for transformations satisfying the postulateඥ݃ ൌ ͳ. With this statement, Einstein knew he was presenting the first paper and this was only the beginning; he was very likely already working on the second part of the theory, expanding covariance. With the new components of the gravitational field (in terms of the Christoffel symbols): ߲݃ఓఈ ߲݃ఔఈ ߲݃ఓఔ ͳ ߤߥ ߤߥ ஢ ൌ െ ቄ ቅ ൌ െ ෍ ݃ఙఈ ቄ ቅ ൌ െ ෍ ݃ఙఈ ቆ ൅ െ ቇǡ Ȟஜ஝ ߪ ߪ ߲‫ݔ‬ఔ ߲‫ݔ‬ఓ ߲‫ݔ‬ఈ ʹ ఈ

ఈ

the generally covariant equations that represent energy-momentum conservation take the form: ෍ ఈ

߲ܶఙఈ ஑ ஒ ൌ െ ෍ Ȟ஢ஒ Ȟ஑ Ǥ ߲‫ݔ‬ఈ ఈఉ

ܶఙఔ denotes the stress-energy tensor. If there is no gravitational field, then gPQ = const., no external forces acting, and: ෍ ఈ

߲ܶఙఈ ൌ ͲǤ ߲‫ݔ‬ఈ

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These energy-momentum conservation equations transform into the original equations of energy-momentum conservation law for the special theory of relativity when:  ݃ఓఔ ൌ ݀݅ܽ݃ሺെͳǡ െͳǡ െͳǡͳሻ.  When the gravitational field exists, i.e., when the gPQare not constant, one has to demand that for the material system and associated gravitational field combined, the constancy of the total momentum and total energy of matter plus gravitational field should be expressed. A complex quantity designated by ‫ݐ‬ఙఔ should exist for the energy of the gravitational field, such that the following equations apply for the conservation law for the total energy-momentum: ෍ ఔ

߲ሺܶఙఔ ൅ ‫ݐ‬ఙఔ ሻ ൌ ͲǤ ߲‫ݔ‬ఔ

The components representing the energy and momentum stored in the gravitational field ‫ݐ‬ఙఔ are: ͳ ஑ ஒ ஑ ఔ Ȟ஝஑ െ ෍ ݃ఓఔ Ȟஜ஢ Ȟ஝஑ Ǥ – ఔ஢ ൌ ߜఙఔ ෍ ݃ఓఔ Ȟஜஒ ʹ ఓఔఈఉ

ఓఔఈ

– ఔ஢ is a pseudo-tensor or, as Einstein put it, the energy tensor of the gravitational field has tensorial character only under linear transformations (Einstein 1915a, 784). According to the the new components of the gravitational field, the field equations in full form are: ෍ ఈ

஑ ߲Ȟஜఔ ஑ ஒ ൅ ෍ Ȟஜஒ Ȟ஝஑ ൌ െߢܶఓఔ Ǥ ߲‫ݔ‬ఈ ఈఉ

These equations are non-linear. On the left-hand side of the equation RPQ includes the metric tensor and its derivatives; on the right-hand side – the sources determine these variables, the stress-energy tensor. The equations ஢ are non-linear because ofȞஜ஝ ǡ which are defined by the components of the gravitational field.

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On November 7, 1915, Einstein sent David Hilbert the proofs of the November 4 paper. Since Hilbert had found a mistake in Einstein's 1914 paper, Einstein wished him to look at his new work (Einstein to Hilbert, November 7, 1915, CPAE 8, Doc. 136). Remember that later in March, 1916, Einstein sent a letter to Hilbert in which he explained the mistake. By November 10, 1915, Hilbert probably answered Einstein's letter, telling him about his system of electromagnetic theory of matter and the unified theory of gravitation and electromagnetism. In his first general relativity paper (the November 4, 1915, paper), Einstein once again used a variational formalism in his gravitation theory (Einstein 1915a, 784). Einstein started from the action and the variation of this action according to the variational principle: Ɂ ൜න †ɒൠ ൌ ͲǤ He then wrote the 1914 Lagrangian. However, the new tensor L consisted ஑ ஒ of the second term Ȟஜஒ Ȟ஝஑ of RPQ multiplied with gVW: ஒ

஑  ൌ ෍ ‰ ஢த Ȟ஢ஒ Ȟఛ஑ ǡ ఙఛఈఉ

and the new components of the gravitational field:5 ߲݃ఓఈ ߲݃ఔఈ ߲݃ఓఔ ͳ த ൌ െ ݃ఛఈ ቆ ൅ െ ቇǤ Ȟஜ஝ ʹ ߲‫ݔ‬ఔ ߲‫ݔ‬ఓ ߲‫ݔ‬ఈ He then wrote the variation: ஒ

ஜஒ

஑ ஑ Ɂ ൌ െȞஜஒ Ȟఔ஑ Ɂ‰ ஜ஝ െ Ȟஜஒ Ɂ‰ ఈ ǡ

  5

I will use the "Einstein summation convention". In 1916 Einstein introduced the Einstein summation convention (Einstein 1916, 781). The Einstein summation convention, tells us that whenever we have an expression with a repeated index, we can omit the sign of summation and implicitly sum over that index from 1 to n.

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which gives: μ ஢ ஜ஝ ൌ Ȟஜ஝ ǡ μ‰ ஢

μ ஑ ஒ ൌ െȞஜஒ Ȟఔ஑ ǡ μ‰ ஜ஝ ஜ஝

where, ‰ ஢ ൌ

డ௚ഋഌ డ௫഑

Ǥ

He then obtained: μ μ ߲ ቆ ቇ െ ஜ஝ ൌ ͲǤ μ‰ ߲‫ݔ‬ఈ μ‰ ஜஒ ఈ Inserting the penultimate equations into these equations leads to the vacuum field equations: ஢ ߲Ȟஜ஝ ஢ ஒ ൅ Ȟஜఉ Ȟఔ஑ ൌ ͲǤ ߲‫ݔ‬ఈ

Einstein showed that the vacuum field equations could be written in the above Lagrangian form. However, in the November 4, 1915, paper, "On the General Theory of Relativity", Einstein wrote the field equations with the sources (the stress-energy tensor) on the right-hand side: ߲ μ μ ቆ ቇ െ ஜ஝ ൌ െߢஜ஝ Ǥ μ‰ ߲‫ݔ‬ఈ μ‰ ஜஒ ఈ Remember that Einstein adopted ඥെ݃ ൌ 1 as a postulate. These equations can be written in the following way: ஢ ߲Ȟஜ஝ ஢ ஒ ൅ Ȟஜఉ Ȟఔ஑ ൌ െɈஜ஝ Ǥ ߲‫ݔ‬ఈ ஜ஝

Einstein multiplied the above equations by ‰ ஢ with summation over the indices P and Qand obtained (Einstein 1915a, 784): μ ߲ ஜ஝ μ ஜ஝ ቆ‰ ቇെ ൌ െɈஜ஝ ‰ ஢ ǡ ߲‫ݔ‬ ߲‫ݔ‬ఈ ஢ μ‰ ஜ஝ ஢ ఈ

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He defined: ஜ஝

െʹɈ– ఈ஢ ൌ ‰ ஢

μ ఈ ஜ஝ െ ߜఙ Ǥ μ‰ ఈ

This equation expresses the law of conservation of momentum and energy for the gravitational field. The quantities – ఈ஢ are the energy components of the gravitational field. The energy-momentum conservation law for the stress-energy tensor of matter reads: 

߲ܶఙఒ ͳ ߲݃ఓఔ ൌെ ܶ Ǥ ʹ ߲‫ݔ‬ఙ ఓఔ ߲‫ݔ‬ఒ

The three above equations led to conservation of energy-momentum (Einstein 1915a, 784): ߲ ൫ܶ ఒ ൅ – ஛஢ ൯ ൌ ͲǤ ߲‫ݔ‬ఒ ఙ Recall that: μ ஢ ஜ஝ ൌ Ȟஜ஝ Ǥ μ‰ ஢ With this equation the energy components of the gravitational field become: ͳ ஑ ஒ ஑ ஒ Ȟ஝஑ െ ݃ఓఔ Ȟஜஒ Ȟ஝ఙ Ǥ Ɉ– ఈ஢ ൌ ߜఙఈ ݃ఓఔ Ȟஜஒ ʹ Einstein had already developed the above variational method in his 1914 Entwurf review paper, "The Formal Foundation of the General Theory of Relativity" (see Chapter 2, Section 8, in which the topic had already been stated): He derived the Entwurf field equations from a variational principle, but he had not yet achieved general covariance (Einstein 1914b, 1071-1074). In 1915, Einstein adopted ඥെ݃ ൌ 1 as a postulate, and

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corrected and adjusted his Entwurf variational formalism accordingly. Between 1913 and 1914, Einstein had thus discovered the basic tools of his 1915 general theory of relativity (Weinstein 2012a 32-42, 2012b 2125). Einstein then tried to obtain the Newtonian limit in his November 4, 1915, paper. He attempted this by contracting the Ricci tensor in his field equations. However, in doing so he obtained the following equation:

෍ ஑ஒ

߲Ž‘‰ඥെ‰ μ ቆ݃ఈఉ ቇ ൌ െߢ ෍ ܶఙఙ Ǥ μš஑ ߲‫ݔ‬ఉ ఙ

According to this equation, it is impossible to choose a coordinate system in whichඥെ‰ ൌ ͳ, for if we chose such a coordinate system, then log 1 = 0 and the trace of the stress-energy tensor vanishes. Einstein called this "the scalar of the energy tensor σఙ ܶఙఙ is set to zero" (Einstein 1915a, 785). I will use Einstein's language. If σఙ ܶఙఙ ൌ Ͳ, why do we have to choose a coordinate system in whichඥെ‰ ൌ ͳ? A week later, on November 11, 1915, the above equation led Einstein back to the mathematical-physical class of the Prussian academy. The class gathered again to hear a correction to Einstein's November 4 paper. In the previous paper, Einstein obtained field equations covariant with respect to transformations of determinant equal to 1 (unimodular transformations). Einstein then obtained the above problematic equation and the impossibility to choose a coordinate system in whichඥെ‰ ൌ ͳǡ for then σఙ ܶఙఙ ൌ ͲǤ He thus set to solve this problem by dropping the November 4 postulate of determinant ඥെ݃ ൌ 1 and adopting it as a coordinate condition. In addition, Einstein supposed we reduce matter to electromagnetic processes, and also assume that gravitational fields could be related to matter; that is, gravitational fields form an important constituent of matter. ఓ In that case, the scalar or trace σఓ ܶఓ of the stress-energy tensor of matter ܶఓఒ vanishes in an electromagnetic field, but it differs from zero for matter proper (gravitational fields). ఓ

Instead of σఓ ܶఓ as the stress-energy tensor to represent matter, Einstein ఓ ఓ used σ൫ܶఓ ൅ ‫ݐ‬ఓ ൯, which is composed of two contributions –

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electromagnetic + gravitational fields. Einstein assumed that ‫ݐ‬ఓ is due to gravitational fields, and it was part of matter. This combination of the ఓ stress-energy tensor enabled him to include σ ‫ݐ‬ఓ that could be positive, ఓ while in general the whole expression vanished, and σఓ ܶఓ actually ఓ vanished everywhere. He thus added the coordinate condition: σ ܶఓ ൌ Ͳ (Einstein 1915b, 799-800). This hypothesis allowed Einstein to take the last step and write the field equations of gravitation in a general covariant form (Einstein 1915b): ‫ܩ‬ఓఔ ൌ െߢܶఓఔ Ǥ This is the Ricci tensor. In the November 4, 1915 paper, Einstein started with the Ricci tensor: ‫ܩ‬௜௠ ൌ ሼ݈݅ǡ ݅݉ሽ ൌ ܴ௜௠ ൅ ܵ௜௠ ǡ and wrote the general form of the field equations with the gravitation tensor RPQ: ܴஜ஝ ൌ െߢܶஜ஝ Ǥ Einstein added to the above field equation the limiting condition (Einstein 1915b, 801): ඥെ݃ ൌ ͳ. The generally covariant field equations (the penultimate field equations) do not lead to a contradiction only if the coordinate condition: σ ܶఓఓ ൌ Ͳapplies. ఓ

Einstein imposed two conditions: the conditionσ ܶఓ ൌ Ͳ and the determinant conditionඥെ݃ ൌ ͳ, and his November 4 field equation ܴஜ஝ ൌ െߢܶஜ஝ and the November 11 field equation ‫ܩ‬ఓఔ ൌ െߢܶఓఔ were both covariant. Einstein chose the generally covariant Ricci tensor ‫ܩ‬ఓఔ ൌ െߢܶఓఔ and this led him to write the field equations in a general covariant form. These field equations were not limited to unimodular

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transformations (transformations with determinant equal to 1); they were not covariant for transformations satisfying the postulateඥ݃ ൌ ͳ. These field equations were generally covariant. However, Einstein added to the above field equation the limiting conditionඥെ݃ ൌ ͳ. They thus applied to all systems of coordinates for which ඥെ݃ ൌ ͳǤ Consider: ‫ܩ‬ఓఔ ൌ െߢܶఓఔ Ǥ When the gravitational potential is obtained in the 44th component of the metric tensor: g44 = 1 + )/c2. This is Einstein's 1907-1912 static gravitational theory factor. Indeed, in the weak field approximation, the equation between the 44th components of the stress-energy tensor and the gravitation tensor is equivalent to the Newtonian equation: G44 = – T44. When, )= 0, for low velocities, we arrive at the special relativity limit. The vacuum field equations with the gravitational field components are: ෍ ఈ

஢ ߲Ȟஜ஝ ஢ ஒ ൅ ෍ Ȟஜఉ Ȟఔ஑ ൌ ͲǤ ߲‫ݔ‬ఈ ఈఉ

The following day, November 12, 1915, Einstein wrote to Hilbert. He first thanked him for his kind response, sent sometime between November 8 and 10 to his own letter from November 7. Einstein reported about the progress and his main finding to which he very likely arrived at as a result of reconsideration of his equations from the November 4, 1915, paper (Einstein to Hilbert, November 12, 1915, CPAE 8, Doc. 139). Einstein received a very prompt response from Hilbert. On November 13, 1915 Hilbert replied to Einstein's letter and told him that he had already been fully immersed in his problem. Hilbert said that he was going to present his new findings before the mathematical society of Göttingen and he invited Einstein to attend. He told Einstein that as far as he understood his November 4, 1915 paper, Einstein's solution was entirely different from his own. Indeed on Tuesday, November 16, Hilbert presented his talk

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"The Fundamental Equations of Physics" to the Mathematical Society of Göttingen (Hilbert to Einstein, November 13, 1915, CPAE 8, Doc. 140). On November 15, 1915, Einstein replied to Hilbert and showed little sign of being willing to come at the moment to Göttingen, but he was enormously curious about Hilbert's findings. He asked Hilbert to please send him, if possible, a proof copy of his investigation in order to satisfy his impatience (Einstein to Hilbert, November 15, 1915, CPAE 8, Doc. 144). On November 16, Hilbert perhaps sent a copy of the lecture he had given on the subject or else a copy of a manuscript of the paper he presented five days later, on November 20, to the Royal Society in Göttingen. By this stage, after receiving Hilbert's system, Einstein began losing his patience. On November 18, 1915, he replied to Hilbert telling him that his given system agrees – as far as he can see – exactly with what he found in the last few weeks and has presented to the Prussian Academy (Einstein to Hilbert, November 18, 1915, CPAE 8, Doc. 148). Hilbert may have put enormous effort of work and thought into his system, but according to Einstein he did not reinvent the wheel. In the November 18 letter to Hilbert, Einstein explained that he was presenting on that very day to the Academy a paper in which he derived quantitatively out of general relativity, without any guiding hypothesis, the perihelion motion of Mercury discovered by Le Verrier. This was not achieved until then by any gravitational theory (Einstein 1915c). Recall that von Seeliger explained, with the guiding dust hypothesis, the perihelion motion of Mercury (Seeliger 1906). Einstein was looking for the equation of a point moving along the geodesic line in the gravitational field of the Sun. When Einstein worked with Besso in the Einstein-Besso manuscript, they considered the Sun in the solar system as an isolated mass, which is far away from other masses in the universe. Most of the mass of the solar system is concentrated in the Sun – more than 0.9998 of the total mass of the solar system. One can treat the planets, the masses of which are negligible compared to the Sun, as mass points moving in the static gravitational field of the Sun. Inside the solar system one can neglect the static gravitational potential of the planets and deal only with the gravitational potential of the Sun.

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Consider a planet, a point with negligible mass, which moves in the static gravitational field of a body of spherical symmetry, in a great distance from this central mass. In a very great distance from this central mass the gravitational field is so weak that it is not felt and we arrive back at the Minkowski flat metric. These are the conditions that Einstein imposed on the gravitational field of the Sun in 1915, and they are very similar to the ones he had considered with Besso in 1913. The gravitational field of the Sun in a vacuum satisfies the following field equations (with a coordinate conditionඥെ݃ ൌ ͳሻ obtained by Einstein in the addendum of November 11, 1915 (Einstein 1915c, 832): ෍ ఈ

஢ ߲Ȟஜ஝ ஢ ஒ ൅ ෍ Ȟஜఉ Ȟఔ஑ ൌ ͲǤ ߲‫ݔ‬ఈ ఈఉ

Reminder: The left-hand side of the equation is the Ricci tensor, and it includes the metric tensor and its derivatives. These equations are non஢ linear because of Ȟஜ஝ , which are defined by the components of the gravitational field: ஢ Ȟஜ஝ ൌቄ

߲݃ఓఉ ߲݃ఔఉ ߲݃ఓఔ ͳ ߤߥ ൅ െ ቇǤ ቅ ൌ ෍ ݃ఈఉ ቆ ߙ ߲‫ݔ‬ఔ ߲‫ݔ‬ఓ ߲‫ݔ‬ఈ ʹ ఉ

These are the Christoffel symbols of the second kind, so that: ቄ

ߤߥ ߤߥ ቅ ൌ ෍ ݃ఙఈ ቂ ቃǤ ߙ ߪ ఙ

The Christoffel symbols of the first kind are: ቂ

ͳ ߲݃ఓఙ ߲݃ఔఙ ߲݃ఓఔ ߤߥ ൅ െ ቇǤ ቃൌ ቆ ߪ ߲‫ݔ‬ఓ ߲‫ݔ‬ఙ ʹ ߲‫ݔ‬ఔ

Einstein started from the 0th approximation:

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gPQ corresponds to the special theory of relativity, to a flat Minkowski metric. Einstein succinctly wrote the flat Minkowski metric in the following form (Einstein 1915c, 832): ݃ఘఙ ൌ ߜఘఙ ǡ ݃ఘସ ൌ ݃ସఘ ൌ Ͳǡ ݃ସସ ൌ ͳǤ Here U and V signify indices, 1, 2, 3; the Kronecker delta GUV is equal to 1 or 0 when U V or UzV, respectively. He then assumed that gPQ differ from the values given in this equation by an amount that is small compared to 1. He treated this deviation as a small change of first order. He used the vacuum field equations for calculation through successive approximations of the gravitational fieldof the Sun up to quantities of the nth order.  He calculated the metric field of the Sun using the vacuum field equations in a first order approximation, substituted the result of this calculation back into the field equations, obtained second-order approximation and solved the field equations to obtain more accurate approximations for the metric field of the Sun. The metric field gPQ (the solution) has the following properties (which are conditions on the gravitational field of the Sun): The solution is static; all components of the solution are independent of the time coordinate; the solution gPQ is spherically symmetric about the origin of the coordinate system; and at infinity the gPQ tend to the values of the Minkowski flat metric of special relativity given by the 0th approximation. To first order, the vacuum field equations and the above conditions were satisfied by the following metric (Einstein 1915c, 833): ߙ ߙ ݃ఘఙ ൌ †‹ƒ‰ ቀെ ቂͳ ൅ ቃ ǡ െͳ െ ͳǡͳ െ ቁǡ ‫ݎ‬ ‫ݎ‬ the following solution: ݃ఘఙ ൌ െߜఘఙ െ Ƚ

š஡ š஢ ߙ ǡ ݃ସସ ൌ ͳ െ Ǥ ‫ݎ‬ ”ଷ

The gUV tends to the Minkowski flat metric, ‫ ݎ‬ൌ ඥ‫ݔ‬ଵ ଶ ൅ ‫ݔ‬ଶ ଶ ൅ ‫ݔ‬ଷ ଶ ǡ D = 2GM/c2.

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Subsequently, Einstein obtained the value for the components of the gravitational field of the static Sun to the second-order approximation. He then wrote equations of motion for a point mass moving in the gravitational field of the Sun. A planet in a free fall in the gravitational field of the Sun moves on a geodesic line according to the geodesic equation (Einstein 1915c, 835): ݀‫ݔ‬ఙ ݀‫ݔ‬ఔఛ ݀ ଶ ‫ݔ‬஝ ஝ ൌ ෍ Ȟ஢த Ǥ ଶ ݀‫ݏ‬ ݀‫ݏ݀ ݏ‬ ఙఛ

The point mass moves on a geodesic line under the influence of the gravitational field of the Sun, which is determined by the Christoffel ஝ ). symbols of the second kind (by components of the metric tensorȞ஢த Einstein calculated the equations of the geodesic lines in this space and compared them with the Newtonian equations of the orbits of the planets in the solar system: ݀ ଶ ‫ݔ‬஝ ஝ ሺɋ ൌ ͳǡʹǡ͵ሻǡ ൌ Ȟସସ ݀‫ ݏ‬ଶ and in polar coordinates r and I(where )is the potential) (Einstein 1915c, 837): ݀ ଶ ‫ݔ‬஝ μȰ ߙ ݀ଶ ߶ ൌ ǡȰ ൌ ቆͳ ൅ ቇǡ μ‫ݔ‬஝ ʹ‫ݎ‬ ݀‫ ݏ‬ଶ ݀‫ ݏ‬ଶ Einstein investigated any correspondence between general relativity and the Newtonian theory. In Newtonian theory, the gravitational attraction is a central force, and all planets move in a constant plane around the Sun. With polar coordinates, the motion of this plane is dependent on the distance r of the planet from the centre, and the angle I between the line that connects the planet to the centre and a line that is chosen arbitrarily. One can obtain the orbit equation, and the distance r of the planet from the centre as a function of the distance of the planet from the Sun at any given angle I. The solution of the Newtonian orbit equation is the equation of an ellipse – an orbit in the plane – and its eccentricity determines the characteristic of the elliptic orbit.

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Einstein could thus proceed exactly as one would do in the Newtonian case. He wrote the relativistic law of conservation of energy for the orbit of the planet and obtained the equation for the orbit in the plane of the planet. This equation, written in a more familiar form, follows (Einstein 1915c, 837-838): ݀ ͳ ଶ ʹ‫ܯܩʹ ܧ‬ ͳ ʹ‫ܯܩ‬ ൤ ൬ ൰൨ ൌ ଶ ൅ ଶ ଶ െ ଶ ൅  ଶ ଷ ǡ ݀߶ ‫ݎ‬ ‫ܮ‬ ܿ ‫ݎ ݎܮ‬ ܿ ‫ݎ‬ where, ‫ ܮ‬ൌ ‫ݎ‬ଶ

݀߶ ǡ ݀‫ݏ‬

ͳ ‫ ܧ‬ൌ ‫ݑ‬ଶ ൅ Ȱǡ ʹ

ƒ†—ଶ ൌ

݀‫ݔ‬ఙ ݀‫ݔ‬ఛ Ǥ ݀‫ݏ݀ ݏ‬

Einstein integrated the equation for the orbit of the planet and calculated the angle Ibetween the radius vector from the Sun to the planet between the perihelion and the aphelion of the elliptical orbit of the planet. The result closely corresponded to the Newtonian equation that arises from the orbit equation by omission of ʹ‫ܯܩ‬ൗܿ ଶ ‫ ݎ‬ଷ (or െʹȰൗܿ ଶ ‫)ݎ‬. Einstein, therefore, considered the integral without this additional term, and he added to it the integral perturbing the Newtonian classical integral as a result of the presence of the term ʹ‫ܯܩ‬ൗܿ ଶ ‫ ݎ‬ଷ . Put differently, in Einstein's theory the geodesic equation leads to an orbit equation. The geodesic equation led Einstein to a relativistic equation of the orbit. Einstein discovered that the difference between the Newtonian orbit equation and the relativistic orbit equation was an additional term ʹ‫ܯܩ‬ൗ ܿ ଶ ‫ ݎ‬ଷ that appears in the relativistic equation. He, therefore, first treated the Newtonian solution to this equation as a first approximation, and then checked the size of the correction that resulted from adding the additional term ʹ‫ܯܩ‬ൗܿ ଶ ‫ ݎ‬ଷ . He integrated the Newtonian orbit equation first. The Newtonian solution to the Newtonian orbit equation describes an ellipse of a planet, for which the direction of the major axis and the perihelion should both stay fixed. Subsequently, he added the perturbation of the additional term ʹ‫ܯܩ‬ൗ ܿ ଶ ‫ ݎ‬ଷ to this solution in order to see whether the turning of the perihelion of Mercury resulted from this additional term in the relativistic

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equation. If this was indeed the result, then the precession of the perihelion would be the result of a relativistic effect; this was the first triumph of Einstein's new theory. Einstein's solution was an ellipse that had a major axis that was not constant, and rotated. This caused a precession of the perihelion. Einstein wrote the solution to the relativistic equation of the orbit from his November 18 paper, an advance of the perihelion of 43 seconds of arc per century. On November 18, 1915, Einstein presented his paper, "Explanation of the Perihelion Motion of Mercury from the General Theory of Relativity", to the Prussian Academy (Einstein 1915c). During this talk, he reported that the perihelion motion of Mercury is explained by finding approximate solutions to his November 11, 1915 field equations. Einstein had attempted to obtain a solution, without considering whether or not there was only one possible unique solution. For Mercury, astronomers found an advance of the perihelion of approximately 45 seconds of arc per century. If Einstein arrived at this result for the advance of the perihelion of Mercury, then his method of using an approximate rather than an exact and unique solution could not be criticised (Einstein 1915c, 832). The day afterwards, on November 19, 1915, Hilbert sent a polite letter in which he congratulated Einstein on overcoming the perihelion motion, and in which he stated "If I could calculate as rapidly as you do". Einstein did not, in fact, calculate the result that rapidly. A week after November 11, on the coming Thursday, he presented his work on the perihelion motion of Mercury. However, recall that the basic calculation had already been undertaken two years earlier with Besso in the Einstein-Besso manuscript. Einstein transferred the basic framework of the calculation from the Einstein-Besso manuscript and corrected it according to his November 11 generally covariant field equations. However, Einstein did not acknowledge his earlier work with Besso, and did not mention Besso's name in his 1915 paper explaining the anomalous precession of Mercury. Although Einstein simply considered Besso his sounding board, an acknowledgment would still have been appropriate. After all, what Einstein had achieved during that week in November was simply a recalculation of the work he had undertaken with Besso in June 1913, using his new field equations instead of the Entwurf equations (Hilbert to Einstein, November 19, 1915, CPAE 8, Doc. 149; Janssen 2002). 

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Indeed, before Leopold Infeld and Einstein had published one of their papers, the former suggested to the latter that he should look up the literature to quote previous scientists who had worked on the same subject before them. Laughing loudly, Einstein remarked: "Oh Yes. Do it by all means. Already I have sinned too often in this respect" (Infeld 1941, 277). Einstein felt so much joy at the result that the field equations yielded the correct perihelion motion of Mercury. He was beside himself for a few days in joyous excitement. Later Einstein told Adrian Fokker that this discovery had given him palpitations of the heart. What he told Wander Johannes de Hass was even more profoundly significant: "When he saw that his calculations agreed with the unexplained astronomical observations, he had the feeling that something actually snapped in him" (Pais 1982, 253; Einstein to Ehrenfest, January 17, 1916, CPAE 8, Doc. 182). Between November 11 and November 18, 1915, Einstein benefitted from interactions with his rival, Hilbert. First, in 1915, Hilbert worked on the same problem as Einstein and thus based his research on Einstein's 1914 Entwurf review paper and November 4, 1915, paper. Hilbert, however, also inspired Einstein in his November 11 and 18, 1915, works. In the correction paper from November 11, 1915, Einstein adopted a new approach related to an electromagnetic theory of matter. He made new assumptions on electromagnetic and gravitational matter, and it seems he was influenced from the current discussions of the electrodynamic worldview with Hilbert. His feeling that he was now competing against Hilbert could have driven him to relate his findings to an electromagnetic theory of matter. In his November 4 paper, Einstein postulated that in a theory of covariants, only substitutions of determinant 1 were allowed (a theory covariant under unimodular transformations). Einstein obtained field equations covariant with respect to transformations of determinant equal to 1 (covariant under unimodular transformations). However, he finally found that it was impossible to choose a coordinate system in which only substitutions of determinant 1 were allowed, because in such a coordinate system, the scalar (trace) of the energy tensor was set to zero (Einstein, 1915a, 785). A week later, Einsteindropped his November 4 postulate and adopted it as a coordinate condition that allowed him to take the last step in writing the field equations of gravitation in a general covariant form.

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He seemed to have discussed this matter with the brilliant mathematician Hilbert before November 11, 1915, because he used Hilbert's theory of electromagnetic matter to solve the November 4 problem of the trace of the energy tensor that was set to zero. Einstein solved this problem by dropping the November 4 postulate of determinant 1 and adopting it as a coordinate condition (Einstein, 1915b, 799-800). In the addendum of the November 11 paper, Einstein added a condition relating to the vanishing stress-energy tensor of matter, which follows from the setting ඥെ‰ ൌ ͳǤ Einstein related this condition to an electromagnetic theory of matter (Renn and Stachel 2007, 904). Reminder: The trace of the stress-energy tensor of matter vanishes in an electromagnetic field, but differs from zero for matter proper. Einstein considered, as the simplest special case, an incompressible fluid. For this case, he stated, the trace of the stress-energy tensor does not vanish. Suppose we reduce matter to electromagnetic processes, and also assume that gravitational fields could be related to matter; that is, assume that gravitational fields form an important constituent of matter. In such a theory the trace of the energy tensor would also vanish. Now the stressenergy tensor representing matter is composed of two contributions: electromagnetic + gravitational fields. The combination of the stressenergy tensor enabled Einstein to include the energy tensor due to the gravitational field that could be positive. Hence, in general the whole expression vanished, and the trace of the stress-energy tensor of all material processes actually vanishes everywhere. Einstein thus added this as a coordinate condition. This hypothesis allowed Einstein to take the last step and write the field equations of gravitation in a general covariant form (Einstein, 1915b, 799-800). It is reasonable to assume that Einstein's arrival at the above solution was influenced by Hilbert's possible lost letter (sent between November 8 and 10, 1915). A week later, the success of Einstein's calculation of the perihelion of Mercury, in the November 18, 1915, paper, was based on his November 11 theory. Indeed this latter theory implemented the assumption of an electromagnetic origin of matter. Einstein definitely favoured his November 11, 1915 theory over that of November 4. Thus, when writing the perihelion paper, Einstein was still influenced by Hilbert's electromagnetic theory of matter (Renn and Stachel 2007, 907). Einstein had very busy weeks working on the problem of the perihelion of Mercury. Sometime between November 11 and November 25 he was able to resolve the final difficulties in his theory. It probably took him just a

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couple of days to arrive at the November 25 generally covariant field equations. He expanded his November 11 vacuum generally covariant field equations, and presented the final version. The November 25, 1915 field equations emerged after Einstein struggled for three years with tensor calculus. Three days later, he wrote Arnold Sommerfeld that he had immortalised the errors in this struggle in the Prussian Academy papers (Einstein to Sommerfeld, November 28, 1915, CPAE 8, Doc. 153). Einstein incidentally wrote "Verevigt" instead of "Verewigt", which means immortalised. Hence not only had Einstein immortalised his errors in the 1914 Entwurf academy paper (Einstein 1914b), he also immortalised a typo-error related to this very error in his letter to Sommerfeld. Recall that on November 16, Hilbert had probably sent Einstein a copy of his lecture or a copy of a manuscript of a paper he intended to present on November 20 to the Royal Society in Göttingen. On November 18, 1915, Einstein replied to Hilbert telling him that his system agrees exactly with what he (Einstein) had found in his November works. We can ask questions of priority: When did Einstein get the idea to expand his November 11 equations? Was it before he received Hilbert's system, or afterwards? Did Hilbert's system induce Einstein's arrival at the final solution of November 25? Hilbert ended his November 19, 1915, letter to Einstein by asking the latter to keep him up to date on his latest advances. However, he did not tell Einstein about a particularly important talk he planned to give the day afterwards. On November 20, Hilbert presented a paper, "The Foundations of Physics", including his version on the gravitational field equations of general relativity to the Göttingen Academy of Sciences. Five days later on November 25, Einstein presented his version of the gravitational field equations to the Prussian Academy. On November 26, 1915, a day after Einstein presented the final version of the field equations, he wrote to his close friend Heinrich Zangger that only one colleague had really understood his general theory of relativity, and he was seeking to clearly nostrify (borrow, yearn to possess) it. Einstein mentioned that this was Abraham's expression. Einstein's achievements were darkened, by bitter, personal rivalry with Hilbert. He sensed that in his personal experience he had hardly came to know the wretchedness of mankind better than through his theory and everything connected to it (Einstein to Zangger, November 26, 1915, CPAE 8, Doc. 152).

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Chapter Two

Reminder: In 1912, Max Abraham had blamed Einstein's theory of relativity and Einstein as well for borrowing some equations from him. Abraham accused Einstein of borrowing expressions from his new gravitation theory. Nonetheless, Abraham needed Einstein's result of the mass of energy principle for his theory of gravitation. It was Abraham who corrected his theory according to Einstein's ideas, and not the other way round. After blaming Einstein, however, and because of his resentment toward the theory of relativity, Abraham could not accept this state of affairs. He thus found an original solution to the particularly important question: Who actually arrived at the idea of the mass of energy? In August 1912, Einstein explained this unpleasant experience to Ludwig Hopf. He told Hopf that Abraham slaughtered him along with the theory of relativity in two massive attacks, and wrote down the only correct theory of gravitation under the nostrification of his results. Einstein told Hopf that Abraham noted that the knowledge of the mass of energy came from Robert Mayer (Einstein to Hopf, August 16, 1912, CPAE 5, Doc. 416). Although, in 1912 Abraham blamed Einstein for nostrifying the mass energy equivalence from Mayer, and the results based on this finding from him (Abraham), Einstein invented the expression nostrification. Furthermore, the mass of energy indeed came from Einstein (see Chapter 2, Section 2). Was Hilbert's "System" a catalyst for Einstein's arrival at the final solution of November 25? On this day, Einstein presented his short paper, "The Field Equations of Gravitation", to the mathematical physical class of the Prussian Academy of sciences (Einstein 1915d). What happened between November 18 and 25 that brought Einstein back to the Prussian Academy with the final version of his field equations? Einstein explained that he had recently discovered his ability to manage without the November 11 ఓ hypothesis about the trace of the stress-energy tensor of matter, σ ܶఓ ൌ Ͳ, merely by inserting it into the field equations in a slightly different way. He mentioned that the field equations for vacuum, onto which he based the explanation of the Mercury perihelion, remain unaffected by this modification. He briefly summarised his previous two papers of November 4 and 11, and historically explained the way his field equations evolved. Finally, he wrote the new field equations of November 25. After November 18, 1915, Einstein, seemed to have departed from Hilbert's electromagnetic theory of matter. According to Einstein, however, Hilbert was seeking to clearly nostrify his November 25 general theory of relativity.

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Einstein's biographer, Albrecht Fölsing, explained that the nostrified colleague – Hilbert – was excited during the summer, a few days before Einstein had published the November 25 field equations. Einstein presented his field equations on November 25, 1915, but six days earlier, on November 20, Hilbert had derived the identical field equations for which Einstein had been searching such a long time. Fölsing asked about a possible draft that Hilbert sent Einstein before November 18: Could Einstein, casting his eye over this paper, have discovered the term which was still lacking in his own equations, and thus nostrify Hilbert? (Fölsing, 1993, 421; 1997, 375-376). Fölsing took Einstein's phraseology nostrifiziert and turned it against him. By apparent contrast, historical evidence supports a scenario according to which Einstein discovered his November 25 field equations by "casting his eye over" his own works of November 4, 1915, and his Entwurf review paper of 1914. Einstein explained that the key to his solution was his realisation that the Christoffel symbols were to be regarded as the natural expression of the gravitational field components. Once he realised this, then the November 25 field equation was very simple. He had already basically possessed the November 4 equations three years before, presenting them in 1912 together with Grossmann, who had brought to his attention the Riemann tensor. Since he had not recognised the formal importance of the Christoffel symbols, he could not obtain a clear overview of the problem and he therefore "fell into the jungle" of the Entwurf field equations (Einstein to Sommerfeld, November 28, 1915, and to Lorentz, January, 1 1916, CPAE 8, Doc. 153, 177). What did Einstein mean here? In October-November 1914 (corrected on November 4, 1915), Einstein had written an equation for the conservation of energy-momentum for material processes: ෍ ఔ

߲݃ఓఔ ఔ ߲ܶఙఔ ͳ ൌ ෍ ݃ఛఓ ܶ ǡ ߲‫ݔ‬ఔ ʹ ߲‫ݔ‬ఙ ఛ ఓఛఔ

(gPQ is the metric tensor and ܶఙఔ denotes the stress-energy tensor). This equation of conservation led him to view the quantities:

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156

߲݃ఓఔ ͳ ෍ ݃ఛఓ ǡ ʹ ߲‫ݔ‬ఙ ఓ

as the natural expression for the components of the gravitational field, even though in view of the formulas of the absolute differential calculus, it was better to introduce the Christoffel symbols as the natural expression for the components of the gravitational field: ቄ

ߤߥ ߤߥ ቅ ൌ ෍ ݃ఙఛ ቂ ቃǡ ߪ ߬ ఙ

where: ቂ

ͳ ߲݃ఓఙ ߲݃ఔఙ ߲݃ఓఔ ߤߥ ൅ െ ቇǤ ቃൌ ቆ ߪ ߲‫ݔ‬ఓ ߲‫ݔ‬ఙ ʹ ߲‫ݔ‬ఔ

By November 4, 1915, Einstein found it advantageous to use Christoffel symbols for the components of the gravitational field (Einstein 1915a, 782783): īıȝȞ ൌ െ ቄ

߲݃ఓఈ ߲݃ఔఈ ߲݃ఓఔ ͳ ߤߥ ߤߥ ൅ െ ቇǤ ቅ ൌ െ ෍ ݃ఙఈ ቂ ቃ ൌ െ ෍ ݃ఙఈ ቆ ߪ ߪ ߲‫ݔ‬ఔ ߲‫ݔ‬ఓ ߲‫ݔ‬ఈ ʹ ఈ

ఈ

Einstein had also shown that a material point in a gravitational field moves on a geodesic line in space-time: ݀‫ݔ‬ఓ ݀‫ݔ‬ఔ ݀ ଶ ‫ݔ‬ı ൌ ෍ īıȝȞ ǡ ݀‫ ݏ‬ଶ ݀‫ݏ݀ ݏ‬ ఓఔ

the equation of which was written in terms of the Christoffel symbols. Subsequently, Einstein wrote the 1914 equation for the conservation of energy-momentum for material processes in the following form: ෍ ఒ

߲ܶఙఒ ൌ െ ෍ īȝఙȞ ȝȞ Ǥ ߲‫ݔ‬ఒ ఓఔ

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Einstein then proceeded to the field equations (Einstein 1915a, 783). Once having the components of the gravitational field, Einstein was able to write the new 1915 field equations of the general theory of relativity. For gravitation, Einstein was interested in the Ricci tensor; contracting the Riemann tensor results in the Ricci tensor. Einstein considered the tensor Rim, and wrote the general form of the field equations RPQ= NTPQ, which are covariant with respect to arbitrary transformations of a determinant equal to 1. In the final 1915 general relativity paper, presented on November 25, 1915, Einstein first wrote the field equations in the following form (Einstein 1915d, 845): ͳ ‫ܩ‬௜௠ ൌ െߢ ൬ܶ௜௠ െ ݃௜௠ ܶ൰ǡ ʹ with: ෍ ݃ఘఙ ܶఘఙ ൌ ෍ ܶఙఙ ൌ ܶǤ ఘఙ

ఙ

T is the trace of the stress-energy tensor of matter. Einstein imposed the coordinate condition ඥെ݃ ൌ ͳ (he selected unimodular coordinates) and replaced the above equations by: ͳ ܴ௜௠ ൌ െߢ ൬ܶ௜௠ െ ݃௜௠ ܶ൰Ǥ ʹ Finally, Einstein probably found the final form of the generally covariant field equations of November 25, 1915, by manipulating his own (November 4, 1915) equations. Einstein then realised he could demonstrate that his field equations satisfied the conservation of momentum-energy. He stated in the final November 25, 1915, paper that these reasons motivated him to introduce the second term on the right-hand side of his new field equations. Einstein argued that his stress-energy tensor Tim satisfied the conservation of energy-momentum for material processes written above (Einstein, 1915b, 845):

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158

෍ ఒ

߲ܶఙఒ ൌ െ ෍ īȝఙȞ ȝȞ Ǥ ߲‫ݔ‬ఒ ఓఔ

Consider the 1914 conservation of energy-momentum for material processes: ෍ ఒ

߲ܶఙఒ ͳ ߲݃ఓఔ ൌെ ෍ ܶ Ǥ ʹ ߲‫ݔ‬ఒ ߲‫ݔ‬ఙ ఓఔ ఓఔ

The most interesting point of this result is that, perhaps, the term on the right-hand side of this equation times gPQ, already written in 1914 and in the November 4, 1915, paper (Einstein 1915a, 782, 784), could have provided Einstein with the idea for the second term on the right-hand side (1/2 gimT) of his November 25, 1915, field equations. Hence, the equation of energy-momentum conservation, which led Einstein in 1914 to a natural expression for the components of the gravitational field, could have also brought Einstein to the natural expression of the final form of his November 25, 1915, field equations. The term on the right-hand side of this equation: ߲݃ఓఔ ఔ ͳ ෍ ݃ఛఓ ܶ ǡ ʹ ߲‫ݔ‬ఙ ఛ ఓ

could, however, have been a heuristic guide for Einstein for the second term on the right-hand side (1/2 gimT) of his November 25 field equations: ͳ ܴ௜௠ ൌ െߢ ൬ܶ௜௠ െ ݃௜௠ ܶ൰Ǥ ʹ We may rightly conclude that the energy-momentum conservation principle did, after all, play a crucial role in Einstein's search for gravitational field equations before and after 1915. In contrast to Hilbert, Einstein searched for a gravitational field equation that would satisfy some heuristic requirements. Let us see how the additional term on the right hand side of the November 25, 1915, field equations involving the trace of the energy-momentum tensor could have sprung from Einstein's November 4, 1915, field equations.

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Einstein multiplied the November 4, 1915, field equation: ஢ ߲Ȟஜ஝ ஢ ஒ ൅ Ȟஜఉ Ȟఔ஑ ൌ െɈஜ஝ ǡ ߲‫ݔ‬ఈ

by gQV and summed over Q (Einstein 1915a, 785): ݃ఔఙ

஑ μȞஜ஝ હ ઺ૃ െ ࢍࣇો ડૅ઺ ડૅહ ൌ െߢ݃ఔ஢ ܶఓఔ ൌ െߢܶஜఛ Ǥ μš஑

The second term on the left-hand side above is the second term on the right-hand side of the components of the energy of the gravitational field: ͳ ஑ ஒ હ ઺ Ɉ– ఈ஢ ൌ ߜఙఈ ࢍࣆࣇ ડૄ઺ ડૅહ െ ݃ఓఔ Ȟஜஒ Ȟ஝ఙ Ǥ ʹ Einstein, therefore, combined the two and obtained, ͳ μ ஑ ૃ ઺ ൫݃ఙఉ Ȟஜஒ ൯ െ ߜఙఈ ࢍࣆࣇ ડૄ઺ ડૅૃ ൌ െߢ൫ܶஜఛ ൅ –ఓ஛ ൯Ǥ ʹ μš஑ In a letter to Paul Ehrenfest, and later in his 1916 paper, "The Foundation of the General Theory of Relativity", Einstein defined: ஒ

஛ ݃ఓఔ Ȟஜஒ Ȟ஝஛ ൌ ߢ‫ݐ‬Ǥ ஒ

ଵ

஛ or: ݃ఓఔ Ȟஜஒ Ȟ஝஛ ൌ െߢ ቀ–ఓఙ െ ߜఓఙ –ቁ. ଶ

Inserting this in the above November 4, 1915 equation we get: 

ͳ μ ஑ ൫݃ఙఉ Ȟஜஒ ൯ െ ߜஜఈ ߢ– ൌ െߢ൫ܶஜఙ ൅ –ఓఙ ൯Ǥ ʹ μš஑

Einstein rewrote this equation in a form valid for matter-free gravitational fields (Einstein 1916a, 802): 

μ ͳ ஑ ൫݃ఙఉ Ȟஜஒ ൯ ൌ െߢ ൬–ఓఙ െ ߜఓఙ –൰ ǡ μš஑ ʹ

ඥെ݃ ൌ ͳǤ

Here, the November 4, 1915, postulate is adopted as a coordinate condition ඥെ݃ ൌ ͳǤ

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160

Einstein was then able to rewrite this equation as follows (Einstein 1916a, 807): 

μ ͳ ஑ ൫݃ఙఉ Ȟஜஒ ൯ ൌ െߢ ൤൫–ఓఙ ൅ ܶஜఙ ൯ െ ߜஜఈ ሺ– ൅ ‫܂‬ሻ൨ ǡ ʹ μš஑

ඥെ݃ ൌ ͳǤ

Einstein gave an interpretation of this equation to Ehrenfest. He argued that this equation was interesting, because it showed the source of the gravitation field was determined solely by the sum ܶ஢ఔ ൅– ఔఙ (Einstein to Ehrenfest, January 24 or later, 1916, CPAE 8, Doc. 185). A comparison between the two above equations reveals, from the above derivation, that sometime before November 25, 1915, Einstein added to: ͳ μ ஑ ൫݃ఙఉ Ȟஜஒ ൯ ൌ െߢ ൬–ఓఙ െ ߜఓఙ –൰ǡ ʹ μš஑ ૚

the term: െ ࢾࢻૄ ࢀ, and this enabled him to write: ૛

μ ͳ ஑ ൫݃ఙఉ Ȟஜஒ ൯ ൌ െߢ ൤൫–ఓఙ ൅ ܶஜఙ ൯ െ ߜஜఈ ሺ– ൅ ‫܂‬ሻ൨Ǥ μš஑ ʹ The added term just mentioned, though more than just an additional term, ஜ is Laue's scalar  ൌ ܶஜ (Einstein 1916a, 808). This was quite the same term Einstein and Fokker had inserted in December 1913 into the field equations of Nordström's theory when they developed their own generally covariant version of his theory (Einstein and Fokker 1914, 328; see Chapter 2, Section 5). Einstein's work on Nordström's theory and his polemic and competition with Nordström could have inspired him to search for the additional term on the right-hand side involving the trace of the energy-momentum tensor of his November 25, 1915 field equations. Taking into account: 

μ ͳ ஑ ൫݃ఙఉ Ȟஜஒ ൯ ൌ െߢ ൬–ఓఙ െ ߜఓఙ –൰ǡ μš஑ ʹ

and: ஢ ߲Ȟஜ஝ ஢ ஒ ൅ Ȟஜఉ Ȟఔ஑ ൌ Ͳǡ ߲‫ݔ‬ఈ

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and multiplying: ͳ μ ஑ ൫݃ఙఉ Ȟஜஒ ൯ ൌ െߢ ൤൫–ఓఙ ൅ ܶஜఙ ൯ െ ߜஜఈ ሺ– ൅ ‫܂‬ሻ൨ ǡ ʹ μš஑

ඥെ݃ ൌ ͳ

by gPQ leads to the field equations of November 25, 1915, which are valid for coordinate system ඥെ݃ ൌ ͳ (Einstein 1916a, 808): 

஑ μȞஜ஝ ͳ ஑ ஒ ൅ Ȟஜஒ Ȟ஝஑ ൌ െߢ ൬ܶఓఔ െ ݃ఓఔ ܶ൰ ǡ ʹ μš஑

ඥെ݃ ൌ ͳǤ

Findings of other historians regarding Hilbert's November 20 paper further support the arguments and findings described above. Accordingly, it is very unlikely that Einstein poached anything pertaining to the November 25 field equations from Hilbert's paper, or the summary sent to him. Indeed, the situation was quite the reverse: The question is, whether Hilbert might have taken something from Einstein's November 25 paper, because it was quite possible for him to do so. Hilbert submitted his paper to the Göttingen Academy of Sciences on November 20, 1915. Einstein's paper in which he gave the final form of his generally covariant field equations was submitted to the Prussian Academy of Sciences on November 25, 1915. Hilbert very likely sent to Einstein (on November 16?) – before November 18 – a summary of his November 20 work (Stachel 1999, 357-358). The November 20, 1915, proofs bear a printer's date stamp, "6 December 1915". The paper was not yet published (Corry, Renn and Stachel 1997, 340; 2004). According to Fölsing, the November 20, 1915, proofs of Hilbert’s paper are equivalent to Hilbert's printed paper and thus contain an equivalent version of Einstein's November 25 field equations. The November 20, 1915, paper, however, did not contain a generally covariant theory. And thus represent Hilbert's states of work submitted on November 20. Hilbert started correcting his proofs only on December 6, 1915. Einstein's November 25, 1915, paper was published on December 2, 1915. Hilbert's paper was only published on March 31, 1916, and he had plenty of time to correct his November 20, 1915, paper according to Einstein's published work of December 2, 1915. Hilbert indeed rewrote his November 20 paper, sometime between December 1915 and March 1916 (Stachel 1999, 358).

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There are, however, differences between the November 20 paper and the printed version from March 1916. In the November 20 proofs, Hilbert based his assertion on a slightly more sophisticated version of Einstein's Hole Argument against general covariance (after Einstein had silently dropped it), which he would eventually drop later when he would publish his paper in March 1916. In addition, in Hilbert's proofs of November 20, the gravitational field equations do not appear explicitly. In the published version of March 1916 – after Einstein had published the final form of his field equations – the expression equivalent in form to Einstein's November 25 field equations is written down explicitly. Knowledge of Einstein's result may have been crucial to Hilbert's introduction of the second term in his equation, which was equivalent in form to the term in Einstein's equation. Thus, Einstein's papers helped Hilbert in putting his November 20 paper in a malleable form, containing generally covariant field equations. Finally, Hilbert later supplemented his reference to the gravitational potentials gPQ with the handwritten phrase, "first introduced by Einstein". In the March 1916 printed version of his November 20 paper, Hilbert added a reference to Einstein's November 25 paper and wrote that the differential equations of gravitation that result are, as it seemed to him, in agreement with the theory of general relativity established by Einstein in his last papers (Stachel 1999, 359; Corry, Renn and Stachel 1997, 344, 359; 2004). Arthur Stanley Eddington, sitting in Britain and receiving German papers through colleagues in the Netherlands due to the British blockade in World War I, mentioned Hilbert's November 20, 1915, paper on an equal footing to Einstein's theory of general relativity. In 1919, in the preface to the first edition of his report to the Physical Society of London, Eddington wrote that he had made extensive use of Einstein's 1916 review article, "The Foundation of the General Theory of Relativity" (Einstein 1916a) and Willem de Sitter's papers on cosmology (de Sitter 1917a, 1917b). Subsequently, he explained (Eddington 1919a, preface, vi): "Other important papers on the subject, most of which have been drawn on to some extent, are —" H HILBERT. "Die Grundlagen der Physik, Cf Göttingen Nachrichten," 1915, Nov. 20. (Hilbert 1915). Eddington also mentioned Schwarzschild's 1916 paper (Schwarzschild 1916a), Lorentz's paper (Lorentz 1916b), Einstein's cosmological constant paper (Einstein 1917b) and Levi-Civita's contribution from 1917 (Levi-Civita 1917).

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On the other hand, Lorentz said on February 26, 1916, in a meeting at the Royal Netherlands Academy of Arts and Sciences, perhaps following Einstein's explanations to him during one of his visits to Leiden (Lorentz 1916a, 1389/1341): "In pursuance of his important researches on gravitation, Einstein has recently attained the aim which he had constantly kept in view; he has succeeded in establishing equations whose form is not changed by an arbitrarily chosen change of the system of coordinates. Shortly afterwards, working out an idea that had been expressed already in one of Einstein's papers, Hilbert has shown the use that may be made of a variation law that may be regarded as Hamilton's principle in a suitably generalised form".

Lorentz referred, in a footnote, to Hilbert's paper (1915): "D. HILBERT, Die Grundlagen der Physik I, Göttinger Nachrichten, Math.-phys. Klasse. Nov. 19l5". During World War I Einstein travelled to the Netherlands, where he could explain to Lorentz his theory, and the difference between his theory and Hilbert's achievements. However, he did not travel to Britain. Eddington's text was proofread by Willem de Sitter (Eddington 1919a, preface, vi), also living in the Netherlands, but it seems that de Sitter mainly proofread the parts concerning cosmological considerations. When Einstein developed a unified field theory, he realised from 1927 onwards that the only equations of gravitation, which follow without ambiguity from the fundamental assumptions of the general theory of relativity, are the vacuum field equations. He thought it was important to know whether these equations alone are capable of determining the motion of bodies. Einstein wanted to show that the vacuum field equations are in fact sufficient to determine the motion of matter. In this respect, one could say in a word that in his unified field theory Einstein gave up the stressenergy tensor in the November 25, 1915 field equations (see Chapter 2, Section 14). David Hilbert, however, deserves priority for the following important achievement in general relativity. He was the first to obtain the field equations in an axiomatic framework based on the extensive use of a metric variational principle using a Lagrangian density. Einstein, however, was rather critical of Hilbert's axiomatic method and wrote to Hermann Weyl that Hilbert's Ansatz about matter appears to be childish. Einstein could not accept (in 1916) the mixture of well founded

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considerations originating from the postulate of general relativity with unfounded, risky hypotheses about the structure of the electron and matter (unification of gravitation and electromagnetism). He concluded by saying that Hilbert's axiomatic method might be of little use in general relativity (Einstein to Weyl, November 23, 1916, CPAE 8, Doc. 278). In his November 4, 1915, paper and his review paper of 1916, "The Foundation of the General Theory of Relativity", Einstein derived the set of field equations, obtaining them from a metric variational principle. However, the treatment was unsatisfactory, since the coordinate condition ඥെ݃ ൌ ͳ was assumed. Einstein apologised for not developing generally covariant field equations in his paper, but only field equations covariant with respect to a coordinate system in which ඥെ݃ ൌ ͳǤ In the manuscript of the 1916 review paper, "The Foundation of the General Theory of Relativity", this apology appears in a note on page 40a (a sheet that is half written, and the bottom half is empty), and added after the following equation (Einstein 1916a, 815; 1916d, 40a): ͳ ıȞ ൌ െ ıĮ ȞĮ ൅ įȞı Įȕ Įȕ Ǥ Ͷ where ஢஝ represents the energy components of the electromagnetic field. Sheet 40a appears in the manuscript after sheet 40, which includes the beginning of Part E and Section §21 dealing with Newton's theory. In the manuscript, Einstein mentions that the note is an addition to part D and he signs both pages 40 and 40a on the top with a red star (Einstein 1916d, 40a; Weinstein 2012c, 47-48). The apology: Einstein wrote that the most general laws for the gravitational field and matter have been derived, for a coordinate system for whichඥെ݃ ൌ ͳ. By this, a considerable simplification of the formulas and calculations have been achieved, without having to omit the requirement of general covariance – because our equations were found, through specialization of a coordinate system, from generally covariant equations. Einstein still wondered whether the field equations could be formulated (without assuming ඥെ݃ ൌ ͳሻ so that we would arrive at conservation of energy and momentum. He then noted that he had found that both are, in

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fact, the case. However, he decided not to communicate these comprehensive considerations in the 1916 review paper, "The Foundation of the General Theory of Relativity", because they did not contribute anything objectively new (Einstein 1916a, 815). The comprehensive considerations that Einstein had decided not to communicate in the 1916 review paper were contained in a five-page appendix to part D: "Appendix: Formulation of the Theory on the Basis of a Variational Principle". The editors of the CPAE identified the five-page manuscript as originally intended for Einstein's 1916 review paper (Einstein 1916a). In his November 20, 1915, paper (published later in March 1916) on the foundations of physics, Hilbert was the first to obtain Einstein's November 25 field equations in an axiomatic framework based on the extensive use of a metric variational principle using a Lagrangian density. Hilbert actually formulated the field equations without assuming ඥെ݃ ൌ ͳǤOn November 20, 1915, Hilbert was the first to write the Ricci scalar (the curvature scalar) as the Lagrangian density. When Einstein wrote on page 40a the above equation for ܶఙఔ ǡthe energy components of the electromagnetic field, he probably had in mind Hilbert's paper (unification of gravitation and electromagnetism). Therefore, Hilbert deserves priority for his achievement of deriving Einstein's field equations by a variational principle with a Lagrangian density. He also took into consideration the interaction with electromagnetism in the framework of Mie's theory. Hilbert wrote the following equation for the action integral (Hilbert 1915, 30, 38): න ‫ܮ‬ඥ݃ ݀߱ǡ where, ห݃ఓఔ ห ൌ ݃ and ݀߱ ൌ ݀߱௜ ሺ݅ ൌ ͳǡʹǡ͵ǡͶሻǤ L is a Lagrangian density. Hilbert divided the Lagrangian density L into two parts: L = K + M. The first part belongs to the gravitational field K: ‫ ܭ‬ൌ ෍ ݃ఓఔ ‫ܭ‬ఓఔ ǡ ఓఔ

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where, KPQ is the Ricci tensor and K is the Ricci scalar. The second part is the matter Lagrangian M, which depends on the components of the metric tensor and the components of the electromagnetic potential, and on their derivatives. Hilbert arrived at the following field equations: െ

ͳ μඥ‰ ͳ ൌ ‫ܭ‬ஜ஝ െ ݃ஜ஝ ‫ܭ‬Ǥ ஜ஝ ʹ ඥ‰ μ‰

He probably only added the gravitational field equations in terms of the ଵ Einstein tensor ‫ܭ‬ஜ஝ െ ݃ஜ஝ ‫ ܭ‬after having read Einstein's paper of ଶ November 25, 1915, which contains an equivalent form of the field equations given above. Hilbert modified his field equations of November 20, 1915: െ

ͳ μඥ‰ ൌ ‫ܭ‬ஜ஝ ǡ ஜ஝ ඥ‰ μ‰ ଵ

by replacing KPQ with ‫ܭ‬ஜ஝ െ ݃ஜ஝ ‫( ܭ‬Hilbert, 1915, 40; Corry, Renn, and ଶ Stachel 1997, 343). The question (deriving the field equations by a variational principle) was then reexamined by Lorentz in a series of papers, which appeared in 1916, shortly after Einstein's and Hilbert's papers. In particular in the third paper, Lorentz applied his general apparatus to the correct Hilbert Lagrangian density coupled with several kinds of matter terms having a greater generality than that considered by Hilbert in his paper (Lorentz 1916b). As a consequence, Einstein revisited the problem in his short paper, "Hamilton's Principle and the General Theory of Relativity", presented at the session of the Prussian Academy of Sciences on October 26, 1916. In this paper he presented the "comprehensive considerations" as promised in his 1916 review paper, "The Foundation of the General Theory of Relativity", and so he succeeded in relinquishing the restrictive unimodular coordinate conditions ඥെ݃ ൌ ͳ assumed earlier in this paper. He therefore tried to treat the variational principle without the restriction ඥെ݃ ൌ ͳǤ

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Einstein made, especially in contrast to Hilbert, as few restrictive assumptions as possible about the constitution of matter (that is to say, at this stage, he did not refer to the unification of gravity and electromagnetism). He noted in his paper that the purely metric variational approach to general relativity had reached a self-consistent, covariant, and satisfactory formulation, thanks to the efforts of Hilbert and Lorentz (Einstein 1916b, 1111; Ferraris and Ferancaviglia 1982, 246-248). Following Hilbert, Einstein exchanged his action integral:‫ܮ ׬‬ඥെ݃ ݀߬ with an action integral: ‫߬݀ े ׬‬ǡ where: ‫ܮ‬ൌ

े ඥെ݃

Ǥ

He divides the Lagrangian density े into two parts, one belonging to the gravitational field and the other to matter: े ൌ ॄ ൅ ैǤ ॄ depends on the components of the metric tensor and on their derivatives, and ै depends on the components of the metric tensor, and on the components of matter and their derivatives. Einstein then derived the field equations satisfied by ेǤ Since the line element: ݀‫ ݏ‬ଶ ൌ ෍ ݃ఓఔ ݀‫ݔ‬ఓ ݀‫ݔ‬ఔ ఓఔ

is invariant, ඥെ݃݀߬ is also an invariant. The functions: ‫ܮ‬ൌ

े ඥെ݃

ǡ‫ ܩ‬ൌ

ॄ ඥെ݃

ƒ†‫ ܯ‬ൌ

ै ඥെ݃

ǡ

are invariants under arbitrary transformations of the space-time coordinates.

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Hence, ‫ ߬݀ े ׬‬is also an invariant. The variation of this action according to the variational principle: න े ݀߬ ൌ Ͳǡ leads to generally covariant field equations (Einstein 1916b, 1111-1113). Reminder: In March 1916, Einstein wrote a five-page appendix, "Appendix: Formulation of the Theory on the Basis of a Variational Principle". The editors of the CPAE identified the five-page appendix as originally intended for Einstein's 1916 review paper (Einstein 1916a). Einstein began composing the manuscript as Section §14, but then thought to change it into an appendix. He ultimately decided to remove it altogether before he submitted the review paper on March 20, 1916. What motivated Einstein to do so remains unclear. An intriguing possibility, however, suggests itself in light of the challenge posed by Hilbert's first paper of November 20, 1915, on "The Foundations of Physics" (Rowe 2001, 408). Below is the following suggestion: In the five-page appendix, Einstein ఛ formulated ॄ in terms of the Riemann curvature tensor ‫ܤ‬ఓఔఛ . The Riemann curvature tensor is an invariant (Einstein 1916e). The Riemann curvature tensor, he reasoned, was the natural candidate for the gravitational part of ॄ Ǥ the Lagrangian density ‫ ܩ‬ൌ ξି௚

In his November 20, 1915, paper Hilbert illustrated that the Riemann ఛ allows the following invariant: curvature tensor ‫ܤ‬ఓఔఛ ఛ ‫ ܭ‬ൌ ݃ఓఔ ‫ܤ‬ఓఔఛ Ǥ

K is the Ricci scalar (i.e., the scalar of the Riemann curvature tensor). ॄ must be equal to the Ricci Einstein, like Hilbert, realised that ‫ ܩ‬ൌ ξି௚

scalar K. Einstein thus wrote the action in the following form: ͳ െ න ‫ ܭ‬ඥെ݃݀߬ ൌ න ॄ ݀߬ ൅ ‫ܨ‬ǡ ߢ

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௖ర

where,N is a constant ( , G is the gravitational constant and c the ଵ଺గீ velocity of light), and F is an additional integral that depends on the components of the metric tensor and on their derivatives (Einstein 1916e, 342-343; Einstein 1916b, 1113). Today, the above equation is called the "Einstein-Hilbert action". Einstein wrote an expression for ॄ in terms of a simplified version of the Ricci scalar K. However, he then discovered that in the above expression‫ ߬݀ ॄ ׬‬൅ ‫ܨ‬ǡ –Š‡‹–‡‰”ƒŽ ‫ ߬݀ ॄ ׬‬was not an invariant. The solution to the problem was an equation from Einstein's 1916 review paper (Einstein 1916a, 817): 

μ ͳ Į ቀ݃ఙఉ īȝȕ ቁ ൌ െߢ ൤൫–ఓఙ ൅ ܶȝఙ ൯ െ ߜȝఈ ሺ– ൅ ሻ൨ ǡ ʹ μšĮ

ඥെ݃ ൌ ͳǤ

–ఓఙ ൅ ܶஜఙ are the energy components of matter and of gravitational field. In light of this equation, the variation of the Hilbert-Einstein action leads to GF = 0. Einstein wrote the variation of the Einstein-Hilbert action (without F) in the following way: ͳ ߜ ൜න ‫ ܭ‬ඥെ݃݀߬ൠ ൌ ߜ ൜න ॄ ݀߬ൠǤ ߢ Since K is an invariant, the left-hand side of the equation is invariant, and the right-hand side is invariant like the left-hand side. ‫ ߬݀ ॄ ׬‬is now invariant. Einstein then wrote the action ‫ ߬݀ ॄ ׬‬and the variational principle (Einstein 1916e, 343-345): ߜ ൜න ॄ݀߬ൠ ൌ ͲǤ We vary with respect to the metric tensor: μ μॄ μॄ ቆ ஜ஝ ቇ െ ஜ஝ ቋ ݀߬Ǥ න Ɂ‰ ஜ஝ ቊ μš஢ μ‰ ஢ μ‰ This expression is also an invariant.

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Compare this equation to Einstein's 1914 Entwurf equation (Einstein 1914b, 1070; see Chapter 2, Section 7): μඥെ‰ μ μඥെ‰ Ɂ ൌ න ݀߬ ෍ Ɂ‰ ஜ஝ ൝ െ෍ ቆ ቇൡǤ ஜ஝ μš஢ μ‰ ஜ஝ μ‰ ஢ ఓఔ

஢

This expression is "an invariant" under limitation to adapted coordinate systems. Einstein considered the change ' due to infinitesimal transformation at some point of space-time, and found: ȟ‫ ܨ‬ൌ Ͳ. Further, due to the invariance of K and ඥെ݃݀߬ǣ ȟ ൜න ‫ ܭ‬ඥെ݃݀߬ൠ ൌ Ͳǡ

ȟ ൜න ॄ݀߬ൠ ൌ ͲǤ

The square root of the determinant of the metric, ඥെ݃, is related to the scalar of the Riemann curvature tensor, K. Hence, Einstein connected between the volume element ඥെ݃݀߬ and the curvature of space-time. Einstein employed Hilbert's notation K for the Ricci scalar and not the notation R and T. These symbols appear in Einstein's review paper, "The Foundation of the General Theory of Relativity" (Einstein 1916a). In his paper, "Hamilton's Principle and the General Theory of Relativity", Einstein mentioned in footnote number 1 Hilbert's paper of November 20, 1915 (Einstein 1916b, 1112). He therefore read Hilbert's paper of November 20, 1915. Hilbert defined the stress-energy tensor in the following way (Hilbert 1915, 50): െ

ͳ μඥ‰ ൌ ܶஜ஝ Ǥ ஜ஝ ඥ‰ μ‰

Consider the Einstein-Hilbert action. The variation of the Einstein-Hilbert action leads to:

General Relativity between 1912 and 1916

െ

171

ͳ μඥ‰ ͳ ͳ ൌ ൬‫ܭ‬ஜ஝ െ ݃ஜ஝ ‫ܭ‬൰ǡ ஜ஝ ߢ ʹ ඥ‰ μ‰

and finally to: ͳ ‫ܭ‬ஜ஝ െ ݃ஜ஝ ‫ ܭ‬ൌ ߢܶஜ஝ Ǥ ʹ Although Hilbert had not included these final field equations in his paper of November 20, 1915, this derivation suggests that between November 20, 1915, and March 1916, Einstein arrived at the conclusion that the Einstein-Hilbert action leads to the above field equations. In March 1918, before publishing his book Space-Time-Matter, Weyl instructed his publisher to send Hilbert the proofs of his book. Hilbert looked carefully at the proofs of Weyl's book but noticed that the latter did not even mention his first Göttingen paper from November 20, 1915, "Foundations of Physics". Though Weyl mentioned profusely Einstein's works on general relativity, no mention was made of Hilbert's paper. Hilbert also stressed that the use of the Riemann curvature in the Lagrangian integral (the Einstein-Hilbert action) and the separation of the Lagrangian density into G and M, which Weyl presented in his book, stemmed from Hilbert alone. Moreover, Hilbert emphasised that the whole presentation of Gustav Mie's theory was precisely that which Hilbert gave for the first time in his November 20, 1015 paper on the foundations of physics. Hilbert maintained that Einstein's earlier work (Einstein's November 1915 papers) on his "theory of gravitation appeared at the same time as mine (namely in November 1915). Einstein's other papers, in particular… on Hamilton's principle appeared however much later" (Rowe 2003, 66). Indeed a year before in 1917, Weyl had written in his paper "On the Theory of Gravitation", that Hilbert, following Mie's 1913 theory in a more general way than Lorentz and Einstein (in 1916), showed that the gravitation equations could be derived by a Lagrangian principle (Weyl 1917, 118). Weyl's book went to press and he added a few citations and brief remarks on Hilbert's November 20, 1915 paper, but these remained shadowy

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features of his book compared with his own contributions and, of course, Einstein's. Weyl's 1918 animus against Hilbert's views revealed the influence of Einstein's authority in physics even on Hilbert's protégés (Rowe 2003, 66; see Chapter 3, Section 4).

10. Einstein Answers Paul Ehrenfest's Queries: 1916 General Relativity Einstein used to travel on a regular basis to his close friend Paul Ehrenfest, who worked at the University of Leiden. They corresponded often, and exchanged views on physics matters. Already in 1907, Ehrenfest posed to Einstein the first query about the theory of special relativity. Ehrenfest caused Einstein to rethink the foundations of his then new theory of relativity. Einstein's response to Ehrenfest's query was important for the demarcation between a theory of relativity and Lorentz's ether-based theory. In 1909, Ehrenfest was again unsatisfied. The issue was the Lorentz contraction and rigid bodies that cannot really exist in special relativity (Ehrenfest 1909, 918). A rigid body is a body in which the distance between its parts remains the same. According to special relativity, no signal can travel faster than light. Consider a rigid rod at rest. If we apply a force at one end, by definition of being rigid, the whole rod must start moving at the same instant. But the information about the force being applied cannot be transmitted simultaneously to the other end, because it cannot be transmitted with infinite velocity (travel faster than light). Hence, rigid bodies cannot exist in special relativity. Ehrenfest considered a relativistic rigid cylinder with radius R. It is gradually set into rotation around its axis until it reaches a state of constant rotation with angular velocity. As measured by an observer at rest, the radius of the rotating cylinder is R'. Then R' has to fulfil the following two contradictory conditions: 1. When the cylinder is moving, an observer at rest measures the circumference of the cylinder to be smaller due to the Lorentz contraction of lengths relative to its rest length. This implies that R' < R. 2. If one considers each element along the radius of the cylinder, then the instantaneous velocity of each element is directed perpendicular to the radius. Hence, the elements of a radius cannot show any contraction relative to its rest length. This means that: R' = R.

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Ehrenfest had pointed out that a uniformly rotating rigid disc would be a paradoxical object in special relativity; since, on setting it into motion its circumference would undergo a contraction whereas its radius would remain uncontracted. This is "Ehrenfest's paradox": How can R' < R and at the same time R' = R? Einstein published nothing directly on this question over the next few years; but, as mentioned before, his first published reference to the rigidly rotating disc occurred in the first of two papers on static gravitational fields from February 1912.In his February 1912 paper on static gravitation theory, "The Speed of Light and the Statics of the Gravitational Fields", Einstein referred to the rotating disc in Section §1 (Einstein 1912a, 356; see Chapter 2, Section 2). Later, Einstein explained why it was impossible for a rigid disc in a state of rest to gradually set into rotation around its axis. A rigid circular disc at rest would have to snap when set into rotation, because of the Lorentz shortening of the tangential periphery and the non-shortening of the radial direction. Similarly, a rigid disc in rotation, made by casting, would have to shatter as a result of the inverse changes in length if one attempted to bring it to a state of rest (Einstein to Joseph Petzoldt, August 19, 1919, CPAE 9, Doc. 93). This led Einstein to consider, in his 1912 paper on the static gravitational field, a system K in a state of uniform rotation (a uniformly rotating disc) in the direction of its x-coordinate, which is observed from a nonaccelerated system. The origin of K possessed no velocity. Einstein thus considered a rigid body already in a state of uniform rotation observed from an inertial system (Einstein 1912a, 356). The system K is equivalent to a system at rest (K') in which there exists a certain kind of mass-free static gravitational field. By theoretical consideration of the processes that take place relative to K, we are informed as to the course of processes in K'. Special measurements are performed by means of measuring rods. When these are compared to one another in a state of rest, at the same location in K, they possess the same length. Hence, the laws of Euclidean geometry must hold for the lengths measured so, and for the relations between the coordinates and for other lengths; but they most probably do not hold in a uniformly rotating system in which, owing to the Lorentz contraction, the ratio of the circumference to the diameter would have to be different from

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S. Einstein claimed that the measuring rod, as well as the coordinate axes, were to be conceived as rigid bodies. This was permitted despite the fact that, according to the special theory of relativity, the rigid body could not really exist (Einstein 1912a, 356). Einstein explained the rotating disc problem very succinctly. We take a great number of small measuring rods (all equal to each other) and place them end-to-end across the diameter and circumference of the uniformly rotating disc. From the point of view of a system at rest, all the measuring rods on the circumference are subject to the Lorentz contraction. An observer in the system at rest concludes that in the uniformly rotating disc the ratio of the circumference to the diameter is different from S. Thus, Einstein indicated that Euclidean geometry could not be applied in the rotating disc. If we adhere to Euclidean geometry, then we arrive at a contradiction because the same measuring rod produces what looks like two different lengths due to the Lorentz contraction. From the point of view of a system at rest, the diameter of the uniformly rotating disc would apparently be measured with uncontracted measuring bodies, but the circumference would seem to be longer because the measuring bodies would be contracted. The possibility of Euclidean geometry on a uniformly rotating disc is thus unacceptable. Neither the measurement of the circumference, nor any other feature of the rotating disc could be said to be in accordance with the Euclidean geometry, for the Euclidean geometry could not be applied in the rotating disc. Thus, at one blow the rotating disc ostensibly destroyed the basic merits of the laws of Euclidean geometry, which did not hold anymore for the lengths so measured. Thinking about Ehrenfest's paradox and Born's rigidity, and taking into consideration the principle of equivalence, by 1912 Einstein had already understood that a connection between nonEuclidean geometry and gravitation existed (even though he did not explicitly talk about it in his 1912 paper on static gravitational fields). Einstein claimed that the measuring rod and the coordinate axes were to be conceived as rigid bodies. This was permitted despite the fact that, according to the special theory of relativity, the rigid body could not really exist. Even though rigid bodies were not permitted in relativity theory, Einstein still referred to measuring rods as rigid bodies in a coordinatedependent (static) gravitational theory. He thought about extremely small measuring rigid rods in terms of rods whose shape and size remained

General Relativity between 1912 and 1916

175

unchanged in the system K; and the coordinate axes were rigid structures that served as standards to which the system was referred. Hence, Einstein believed he could certainly claim that the measuring rod and the coordinate axes were to be conceived as rigid bodies. As stated above, Einstein also explained that it was impossible for a rigid disc in a state of rest to gradually set into rotation around its axis, and impossible for a rigid disc in rotation to be brought to a state of rest. However, Einstein's disc was already in a state of rotation, and was neither brought to rest from rotation nor set into rotation from rest. Even within the framework of special and general relativity, Einstein spoke in terms of measurements performed by means of rigid measuring rods and clocks. In the Entwurf theory, Einstein believed it was obvious that one could use measuring rods and clocks in much the same way as one could do so in the special theory of relativity. Einstein thus pursued the physical meaning – the measurability – of the space-time quantities (Einstein and Grossmann 1913, 8-9). In his 1913 Entwurf theory, Einstein stated that the line element ds possesses a physical meaning that is independent of the chosen reference system. He assumed that in special relativity, ds was the naturally measured interval between two space-time points, or the square of the four-dimensional interval between two infinitely close space-time points. It is measured by means of a rigid body that is not accelerated in a system, and which is introduced by means of linear transformations with respect to the immediate vicinity system of a pointin space-time. ds is also measured by means of unit measuring rods and clocks at rest relative to the rigid body. The natural interval that corresponds to the differentials of fourcoordinates of the point can be determined only if one knows the components of the metric tensor that determine the gravitational field (Einstein and Grossmann 1913, 8-9). Only in November 1915, did Einstein discover that in general relativity one could not use measuring rods and clocks in the same way as in special relativity. This is because time and space were deprived of the last trace of objective reality. Thus, space and time coordinates have no physical meaning in general relativity (Einstein 1915c, 831). Nonetheless, in 1919, Einstein still referred to the line element ds as possessing a physical meaning. He spoke in terms of measurements

176

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performed by means of rigid measuring rods and clocks (see Chapter 3, Section 4). In December 1915, Ehrenfest posed a new query. This time, Ehrenfest questioned Einstein about the Hole Argument from Section §12 of his 1914 Entwurf review article. This query was probably the most annoying of all the queries that Ehrenfest had posed Einstein over the years. Between 1913 and 1914 Einstein adhered to the Hole Argument against general covariance. Then, Einstein quietly dropped this argument in October 1915 and failed to mention it in his November 1915 papers. After the November 1915 papers, Einstein explained to Ehrenfest and Besso the problems with the Hole Argument against general covariance, an argument he supported so strongly for over two years (Einstein to Ehrenfest, December 26, 1915, and to Besso January 3, 1916, CPAE 8, Doc. 173, 178). Einstein had to clarify the problem of the Hole Argument against general covariance. He explained to Ehrenfest that in Section §12 of his 1914 paper of the previous year, everything was correct up to the end of the third paragraph. The correction then, he continued, should be that the reference system has no meaning and that the realisation of two gravitational fields in the same region of the continuum (hole) is impossible. Einstein maintained that a new consideration should replace §12 the Point Coincidence Argument, and presented it in a new 1916 review paper, "The Foundation of the General Theory of Relativity". The Hole Argument was replaced by the following consideration: Nothing is physically real but the totality of space-time point coincidences. If, for example, all physical events were to be built up from the motions of material points alone, then the meetings of these points, namely the points of intersection of the world lines, would be the only real things, the only things observable in principle. Intersection of the world lines prevents the possibility of assigning two values to one and the same point in space-time. These points of intersection are naturally preserved during all coordinate transformations. It is therefore most natural to demand of the laws that they determine no more than the totality of these space-time coincidences. This is already attained through the use of generally covariant equations. A week later, Einstein also gave his friend Besso the same explanation he provided to Ehrenfest (Einstein to Ehrenfest, December 26, 1915, and to Besso, January 3, 1916, CPAE 8, Doc. 173, 178).

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Thus, Einstein avoided the Hole Argument quite naturally by maintaining that there is no meaning to space-time measurements: All points are indistinguishable from one another. According to the 1914 Hole Argument, considering generally covariant field equations, at the boundary of the hole and inside it, more than one metric tensor gPQbelongs to the stress-energy tensor 7PQ. In the language of the 1914 Hole Argument, with respect to K there are two possible solutions, which differ from one another. Nevertheless, at the boundary of the hole and inside it, the two solutions coincide. However, according to the Point Coincidence Argument, there is no difference between these solutions, and they should both represent the same gravitational field, because all events consist only of the motion of material points; the points of intersection of the world lines are the only real things, and we are only dealing with the coincidences of the space-time points. Consider the two solutions inside the hole. If we take the first solution and think of it as material points moving in space-time, then we are only dealing with the meeting (intersection) of the two material points. If we take the second solution and think of it, too, as material points moving in space-time, this solution, according to Einstein's Point Coincidence Argument, is also reduced to the meeting of two material points. The two points are indistinguishable, because there is no reason to prefer one to the other: both are considered space-time points, intersections of world lines. Intersection of the world lines prevents the possibility of assigning different values to one and the same point in space-time (Einstein to Ehrenfest, December 26, 1915, CPAE 8, Doc. 173; Einstein 1916a, 776777). Einstein realised that general relativity may be regarded as a theory in which space and time cannot be defined in such a way that spatial coordinate differences are directly measured by the unit measuring rod, and time by a standard clock. Moreover, in the general theory of relativity, the method of laying coordinates in the space-time continuum (in a definite manner) breaks down, and coordinate systems cannot be adapted to the four-dimensional space. This seems to be the greatest breakthrough made by Einstein about space-time measurements. Unlike general relativity, in special relativity, coordinates of space and time have direct physical meaning. In classical mechanics and in special relativity, space-time measurements of four-coordinates are made with rods and standard clocks. With these, we define lengths and times in all inertial reference frames. The notion of coordinates and measurements in

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classical mechanics and in special relativity presuppose the validity of Euclidean geometry. Also, according to the special theory of relativity, the laws of geometry are directly interpreted as laws relating to the possible relative positions (at rest) of solid bodies, and the laws of kinematics are to be interpreted as laws that describe the relationships of measuring bodies and clocks. The general theory of relativity cannot adhere to this simple physical interpretation of space and time (Einstein 1916a, 770, 774). These conclusions brought Einstein to a formulation of a principle of general covariance: If we cannot depend on space and time measurements in general relativity, then we must regard all imaginable systems of coordinates, on principle, as equally suitable for the description of nature. The argument that supports this principle of general covariance is the Point Coincidence Argument, which replaces the Hole Argument. Since all our physical experience can be ultimately reduced to point coincidences, there is no immediate reason for preferring certain systems of coordinates to others, i.e., we arrive at the requirement of general covariance (Einstein 1916a, 775-776).  Ehrenfest posed more queries to Einstein, and in an exchange of letters Einstein gradually and patiently answered them all; and, while doing so, he developed the basis of a new review article. Finally, Ehrenfest's queries led Einstein to reconsider his November 4, 1915 field equations, and to add another term to these equations, and consequently to re-derive the November 25, field equations. This was the first formulation of the 1916 General Theory of Relativity (Einstein 1916a). In section §3 of the 1916 review paper, before presenting the Point Coincidence Argument, Einstein dealt with the uniformly rotating disc problem. Note that, in November 1915, Einstein had already realised that coordinates of space and time have no direct physical meaning in general relativity, because time and space are deprived of the last trace of objective reality. Fairly soon after this, Einstein reconsidered the 1912 rotating disc thought experiment in his 1916 review paper, "The Foundation of the General Theory of Relativity" (Einstein 1916a, 774775). Einstein considered two systems, a Galilean system K, and the other K', which is in uniform rotation relative to K. He then demonstrated by the rotating disc thought experiment that we are unable to define coordinates in K' properly, and Euclidean geometry breaks down for K'. He concluded that we are also unable to properly define time by clocks at rest in K'. The

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coordinates of space and time, therefore, have no direct physical meaning with respect to K'. The origin of both systems, as well as their axes of z, permanently coincide one with the other. The circle around the origin in the x, y plane of K is regarded at the same time as a circle in the x', y' plane of K'. In section §3 of Einstein's 1916 review paper, the uniformly rigid rotating body was geometrized and became a circle. Einstein imagined that the circumference and diameter of this circle were measured with a unit measuring rod (infinitely small relative to the radius of the circle), and he formed the quotient of the two results. If the experiment is performed with a measuring rod at rest relative to K, the quotient will be S. With a measuring rod at rest relative to K', the quotient will be greater than S. This can be seen, if the whole process of measurement is viewed from the system K, taking into consideration that the periphery undergoes a Lorentz contraction, while the measuring rod applied to the radius does not. An observer in a Galilean system K is an inertial observer, and therefore, he uses special relativity to measure the circumference and diameter of the circle. In 1909, Ehrenfest suggested that, in a uniformly rotating disc, the Lorentz contraction causes the circumference to be shorter than a circumference in a disc at rest. Naturally, if we consider Einstein's explanation, we find quite the opposite, of course. The measurement is performed from the Galilean system K. The circumference of the uniformly rotating disc, therefore, appears longer as measured by an observer in K, because this observer would need more rods to cover the circumference of the disc, and each of these rods suffers a Lorentz contraction according to special relativity (Einstein 1916a, 774-775). Subsequently, Einstein treated time measurements. However, he did not explicitly discuss this matter in his 1912 paper, "The Speed of Light and the Statics of the Gravitational Fields". In his 1907 paper, "On the Relativity Principle and the Conclusions Drawn from It" and his 1911 paper, "On the Influence of Gravitation on the Propagation of Light", Einstein considered a uniformly accelerated reference system K' accelerated relative to a Galilean system K in the direction of its z-axis. The system K' is instantaneously at rest relative to the system K when the clocks of the latter read the time t = 0. The clocks of the former are then synchronised to read the time W. Two clocks are

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placed at opposite ends of system K': Clock U1 is placed at the top of the system K' and the second clock U2 is placed at the bottom of the system K'. Einstein found that in the system K' the clock U1 runs faster in the next time element than the identical clock U2 reading the time W. According to the 1907 equivalence principle, time measurements in a uniformly accelerated reference system are also valid for a coordinate system in which a homogeneous gravitational field is acting. Einstein concluded that if we measure time at a point (at a higher gravitational potential) with one clock U, we must measure the time at the origin (at a lower gravitational potential) with a clock that runs more slowly than the clock U, if we compare it with the clock U in the same place. Hence, for an observer located somewhere in space, his clock runs faster than an identical clock located at the coordinate origin near a massive object. Clocks, and more generally any physical processes, move more slowly the lower the gravitational potential at the position of the process taking place (Einstein 1907, 458; 1911, 906). In 1916 Einstein greatly expanded the above thought experiment. He imagined two clocks of identical constitution placed one at the origin of coordinates and the other at the periphery of the uniformly rotating circle K'. Both clocks are observed from the Galilean system K. According to special relativity (time dilation), judged from K, the clock at the periphery of the circle runs more slowly than the other clock at the origin, because the clock at the circumference is in motion and the latter at the origin is at rest.An observer located at the origin of K' and capable of observing the clock at the circumference by means of light, sees the periphery clock lagging behind the clock beside him. He will interpret this observation as showing that the clock at the periphery "really" runs more slowly than the clock at the origin. He will thus define time in such a way that the rate of the clock depends upon its location (Einstein 1916a, 775). Einstein arrived at two important results: If we regard the process of measurement in system K, the measuring-rod applied to the periphery undergoes a Lorentzian contraction (while the one applied along the radius does not), and the clock at the periphery of the circle runs more slowly than the other clock at the origin. Einstein concluded: We cannot properly define coordinates in K'. The length measurements have no direct meaning for K'. We are unable to properly define time by clocks at rest in K' either. The coordinates of space

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and time, therefore, have no direct physical meaning with respect to K' (Einstein 1916a, 775). At the end of his review paper, "The Foundation of the General Theory of Relativity", Einstein treated the rotating disc thought experiment as a gravitational problem. To begin with consider the first order approximate solution (to the vacuum field equations of November 18, 1915) (Einstein 1915c, 833; see Chapter 2, Section 9): ݃ఘఙ ൌ െߜఘఙ െ Ƚ

š஡ š஢ ߙ ǡ ݃ସସ ൌ ͳ െ Ǥ ଷ ‫ݎ‬ ”

According to the equivalence principle, a system in uniform rotation, the disc, rotating around the origin, and relative to the Galilean system, is equivalent to a system at rest in which there exists a certain kind of static gravitational field produced by a mass point at the origin of the coordinates. The mass point produces a gravitational field that is calculated from the field equations by successive approximations. The above solution is the first order approximation. Einstein showed that the unit measuring rod appears a little shortened with respect to the coordinate system by the presence of the gravitational field of the mass point, if it is laid in the radial direction. As to the length of a measuring rod in the tangential direction, the gravitational field of the mass point has no influence on the length of a rod (Einstein 1916a, 819): Einstein considered the line element: ݀‫ ݏ‬ଶ ൌ ෍ ݃ఓఔ ݀‫ݔ‬ఓ ݀‫ݔ‬ఔ ǡ ఓఔ

and small unit-measuring rods laid on the x-axis. Therefore for a unit measuring rod laid parallel to the x-axis the differentials dx2 = dx3 = dx4 = 0 and we set: ds2 = – 1. The first order approximate solution to the vacuum field equations gives: ߙ ʹ‫ܯܩ‬ ݃ଵଵ ൌ െ ቀͳ ൅ ቁ ൌ െ ൬ͳ ൅ ଶ ൰Ǥ ‫ݎ‬ ܿ ‫ݎ‬

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G – gravitational constant, M – mass of the mass point at rest located at the origin of the coordinate system. The first order approximate solution leads to: െͳ ൌ ݃ଵଵ ݀‫ݔ‬ଵଶ ൌ െ ൬ͳ ൅

ʹ‫ܯܩ‬ ൰ ݀‫ݔ‬ଵଶ Ǥ ܿଶ‫ݎ‬

Both these equations yield: ݀‫ ݔ‬ൌ ͳ െ

‫ܯܩ‬ Ȱ ߙ ൌ ͳ െ ଶ ൌ ͳ ൅ ଶǤ ܿ ‫ݎ‬ … ʹ‫ݎ‬

where )is the gravitational potential. 

11. The Third Prediction of General Relativity: Gravitational RedShift In Appendix III of his popular book, Relativity: The Special and General Theory, Einstein derived gravitational redshift as a consequence of the disc thought experiment (Einstein 1920, Relativity: The Special and General Theory, Appendix III). He extended his 1907-1911 derivation from uniformly accelerated systems to a system K', which is in rotation with respect to a Galilean system K. In the derivation Einstein used only the equivalence principle. In the system K', the rotating disc, clocks of identical construction, and which are considered at rest with respect to the disc, run at rates which are dependent on the positions of the clocks (whether they are positioned in the origin or the periphery of the disc). In the disc thought experiment Einstein examined this dependence qualitatively. In order to obtain redshift of spectral lines he examined this dependence quantitatively. Consider a clock, which is situated at a distance r from the centre of the disc, and has a velocity, relative to the Galilean system K, which is given by ‫ ݒ‬ൌ ߱‫ ݎ‬where Z represents the angular velocity of rotation of the disc K' with respect to K. If Q0, represents the number of ticks of the clock per unit time (the rate of the clock) relative to K when the clock is at rest, then the rate of the clock (Q) when it is moving relative to K with a velocity ‫ݒ‬, but at rest with respect to the disc, is given by:

General Relativity between 1912 and 1916

ߥ ൌ ߥ଴ ඨͳ െ

183

‫ݒ‬ଶ ǡ ܿଶ

or with sufficient accuracy by: ߥ ൌ ߥ଴ ቆͳ െ

ͳ ‫ݒ‬ଶ ቇǤ ʹ ܿଶ

Einstein wrote this expression in the following form: ߥ ൌ ߥ଴ ቆͳ െ

ͳ ߱ଶ ‫ ݎ‬ଶ ቇǤ ܿଶ ʹ

Einstein concluded from this expression that two clocks of identical construction will run at different rates when situated at different distances from the centre of the disc. This result is also valid from the standpoint of an observer who is rotating with the disc. Einstein represented the difference between the centrifugal force at the position of the clock and at the centre of the rotating disc as a difference of potential ). This is the work, considered negatively, which must be performed on the unit of mass against the centrifugal force in order to transport it from the position of the clock on the rotating disc to the centre of the disc. Einstein, therefore, obtained: Ȱൌെ

߱ଶ ‫ ݎ‬ଶ Ǥ ʹ

Inserting this result into the above equation we get: ߥ ൌ ߥ଴ ൬ͳ ൅

Ȱ ൰Ǥ ܿଶ

Now, according to the equivalence principle, as judged from the disc, an observer who is rotating with the disc is in a gravitational field of potential ), hence the above result generally holds for gravitational fields.

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Einstein explained that we can regard an atom which is emitting spectral lines as a clock. The atom absorbs or emits light of a frequency which is dependent on the potential of the gravitational field in which it is situated. Gravitational redshift is therefore a form of time dilation. In 1911 (as shown later below) Einstein derived the above equation representing gravitational redshift using the Doppler Effect; and in special relativity the Doppler principle is related to time dilation (Einstein 1911, 903-905). As already mentioned, Einstein re-examined the uniformly rotating disc thought experiment at the end of his 1916 review paper, "The Foundation of the General Theory of Relativity". He used the first order approximate solution to his vacuum field equations from the November 18, 1915 perihelion of Mercury paper to show the effect of the presence of the gravitational field of a mass point on unit measuring rods. He concluded that Euclidean geometry does not hold even to first-order approximation in the gravitational field. He now proceeded to examine clock rates. He again started from the line element (Einstein 1916a, 820): ݀‫ ݏ‬ଶ ൌ ݃ఓఔ ݀‫ݔ‬ఓ ݀‫ݔ‬ఔ Ǥ He considered the rate of a unit clock, which is arranged at rest in a static gravitational field. For the clock period we set: ds = 1; dx1 = dx2 = dx3 = 0. Therefore: ͳ ൌ ݃ସସ ݀‫ݔ‬ସଶ Ǥ Consider, ݀‫ݔ‬ସ ൌ

ͳെ

ͳ ඥ݃ସସ

ൌ

ͳ ඥͳ ൅ ሺ݃ସସ െ ͳሻ

ൌͳെ

݃ସସ െ ͳ Ǥ ʹ

݃ସସ െ ͳ ߔ ൌ ͳ ൅ ଶ Ǥ … ʹ

Einstein obtained the above result not by means of the heuristic equivalence principle but rather by using the metric tensor and the line element, and arrived at the same factor he had obtained by assuming the heuristic equivalence principle.

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Einstein concluded that the clock runs more slowly if it is placed near masses. It follows that the spectral lines of light reaching us from the surface of large stars must appear displaced towards the red end of the spectrum. It is not surprising that Einstein derived the redshift using the line element. Recall that the line element of general relativity may be written in the following form: ݀‫ ݏ‬ଶ ൌ ݃݀߬ ଶ Ǥ Einstein wrote a relation between the proper time (the time measured with a clock comoving with the observer and the object to be measured) and the coordinate time (the time measured with a clock which is not comoving with the observer and the object to be measured): ඥ݃݀߬ ൌ ݀߬଴ Ǥ This expression is very similar to the time dilation formula from special relativity: ඨͳ െ

‫ݒ‬ଶ ݀߬ ൌ ݀߬଴ Ǥ ܿଶ

By 1919, two of the predictions of general relativity, light deflection near the Sun and the perihelion of Mercury, were considered confirmed. In the eclipse expedition of 1919 Eddington and Charles Rundle Davidson went to find whether they could verify Einstein's prediction of the deflection of starlight in the gravitational field of the Sun. Eddington and his assistant went to the island of Principe off the coast of Africa while Davidson and his assistant went to Sobral in North Brazil. In presenting their observations to the Royal Society of London in November 1919, the conclusion was that they verified Einstein's prediction of deflection at the Sun's limb to very good accuracy. Nevertheless, people still criticised and disputed the results of Eddington's experiments, and further observations of Einstein's prediction were made at subsequent eclipses (see Chapter 3, Section 3). The third prediction, that of the redshift, still demanded confirmation. However, until the 1920s all efforts to confirm Einstein's redshift

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prediction gave negative results; experimenters could not find the predicted redshift in the Sun's spectrum. Eddington and Davidson wrote that Einstein's theory predicts a displacement of the red of the spectral lines on the Sun and experimental studies found that this displacement was not confirmed. Eddington and Davidson said that if this disagreement would be taken as final it would necessitate considerable modifications of Einstein's general theory of relativity (Dyson, Eddington and Davidson 1920, 292). Physicists tried to escape the problem of getting negative results in experiments performed to verify Einstein's redshift. They suggested compensations and all sorts of explanations and additions to Einstein's theory, and even queried the importance of the redshift as evidence for verifying the general theory of relativity. Einstein was willing to abandon the whole theory of general relativity because the redshift effect was derived by the heuristic equivalence principle, and if the redshift did not exist, then it meant that the equivalence principle, on which the general theory of relativity theory was based, was a shaky principle. Einstein's general theory of relativity is a principle theory, based on the equivalence principle and the principle of general covariance. Einstein, therefore, could not modify his theory without completely abandoning it first and then creating a new theory at the basis of which were new underlying principles. Einstein explained this to Eddington in his letter thanking him for his experiments. On December 15, 1919 Einstein congratulated Eddington on the happy success of his (Principe) expedition but told Eddington that he was convinced that the redshift of spectrum lines was an absolutely compelling consequence of the general theory of relativity. If it were proved that this effect did not exist in nature, then Einstein was willing to abandon the whole theory (Einstein to Eddington, December 15, 1919, CPAE 9, Doc. 216). In 1920 Einstein told Eddington that two young physicists, Leonhard Grebe and Albert Bachem, found evidence for redshift in the spectrum of the Sun that was in agreement with his predictions in general relativity. Grebe and Bachem were also able to explain why previous experimenters were unable to confirm redshift. However, Eddington replied that although the results looked convincing, Charles Edward St John (who was previously getting negative results in his experiments to verify Einstein's redshift) was still getting negative

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results, and he expected that spectroscopists for the time being would be divided on this question as to what the result would really be (Einstein to Eddington, February 2, 1920, Eddington to Einstein, March 15, 1920, CPAE 9, Doc. 293, 353). Finally the first test to agree with Einstein's predictions of redshift was performed in 1925 by Walter Adams at Mount Wilson. Previous unsuccessful experiments were focusing on displacement of the spectrum of the Sun in order to interpret these observations in terms of gravitational redshift. Adams attempted to detect the gravitational redshift in the spectrum of the companion Sirius A and B (a binary star system). Eddington had come to believe that white dwarf stars like Sirius B are very dense, and he hoped to verify his prediction with the measurements of a high gravitational redshift. The mass of Sirius B, Eddington estimated, was 0.85 solar masses, and its radius only 19,600 km, making Sirius B larger than planet Earth but smaller than planet Uranus. The estimated density was very high, hence the gravitational field extremely strong, and this would create a pronounced gravitational redshift as compared to the much smaller gravitational redshift one would expect to measure for the Sun. Since Sirius is a binary star, however, the Doppler redshift due to motion could be eliminated and the only remaining component was the gravitational redshift measured by taking the difference between the redshifts of Sirius A and Sirius B. Eddington estimated that the world's largest telescope (which was at Mount Wilson), might suffice and he wrote to Adams at Mount Wilson and asked him whether he could perform the measurements. Eddington apparently had thought only to verify his prediction of high density, but Adams realised that here was a possible opportunity to complete Einstein's third test of general relativity by finding gravitational redshift. Adams indeed found a redshift that was very nearly the predicted gravitational redshift in Einstein's theory. Eddington, however, wanted Adams's gravitational redshift observations reproduced by another observer and apparent confirmation from another independent observer. In 1928 Joseph Moore of the Lick Observatory confirmed Adams's results. The results of Adams and Moore, however, were finally considered inconclusive because astronomers had not yet studied white dwarfs such as Sirius B and therefore claimed that the results could have been

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contaminated and could just as well have been caused by other natural phenomena and not by gravitational redshift of Sirius (Hetherington 1980, 248). On May 12, 1952 the elderly Einstein wrote Max Born that even if the deflection of light, the perihelion motion of Mercury and redshift were unknown, the gravitation equations would still be convincing (Einstein to Born, May 12, 1952, Einstein and Born, letter 99). Einstein was no longer willing to abandon the whole theory of general relativity because of predictions derived from his theory. The reason Einstein changed his mind was his involvement with unified field theory, which was based on his general relativity (See Chapter 2, Section 14 and Chapter 3, Sections 1 and 10). It was not until after Einstein's death, in 1959, that the first conclusive redshift experiment was performed directly on the Earth in Harvard's Jefferson Laboratory by physicists Robert Pound and Glen Rebka. They verified gravitational redshift by means of the Mössbauer effect (discovered a year earlier by Rudolf Mössbauer), and following Einstein's 1911 procedure presented in his 1911 paper, "On the Influence of Gravitation on the Propagation of Light". In section §3 of this 1911 paper Einstein returned to the Doppler principle, which he had discussed in the 1905 relativity paper in section §7. In special relativity the Doppler principle is related to time dilation (Einstein 1911, 903-906). Einstein considers two systems S1 and S2, each provided with measuring instruments and situated on the z-axis of the system K in a homogeneous gravitational field (S1 is near the origin of the axes). They are separated by a distance h from each other. The gravitational potential in S2 is greater than that in S1 by J·h.

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Einstein uses the principle of equivalence of K and K', and in place of the system K in a homogeneous gravitational field, he considers a gravitational-free system K', which moves with uniform acceleration in the direction of positive z, and with the z-axis, to which the systems S1 and S2 are connected. Consider the radiation emitted from S2 towards S1 in the uniformly accelerated system K'. At its arrival at S1 its frequency is no longer Q2 with respect to S1's clock, but has a greater frequency Q1 so that to first approximation: ߥଵ ൌ ɋଶ ൬ͳ ൅

ɀŠ ൰Ǥ …ଶ

The radiation is emitted from S2 towards S1 in the system K'. On the arrival of the radiation at S1, then K' has the velocity Jh/c2 with respect to a nonaccelerating system K0. Einstein is now guided by the principle of equivalence of the systems K' and K. The above equation should also hold good for the system K in a uniform gravitational field. If at a certain gravitational potential at S2 a ray of light is emitted towards S1 with a frequency Q2 – compared with a clock located in S2 – it will have the frequency Q1 at its arrival in S1, compared with a clock located in S1. Einstein then substitutes for Jh the gravitational potential )of S2, while the gravitational potential in S1 is 0, and he assumes that what he has obtained for a homogenous gravitational field also holds for other forms of fields. He thus obtains: ߥଵ ൌ ɋଶ ൬ͳ ൅

Ȱ ൰Ǥ …ଶ

From this equation Einstein deduced the gravitational redshift effect. Suppose S2 is located in the Sun and emitting from there light which is reaching the Earth (S1). We measure the frequency of the arriving light. ) is the (negative) gravitational potential difference between the surfaces of the Sun and the Earth. Thus the spectral lines of sunlight towards the Earth must be somewhat shifted toward the red. Einstein computed the amount from the above equation: – )/c2 = 2·10-6. S2 is a source that emits radiation and thus vibrates with a certain frequency. Einstein said that this was also a definition of a clock. He mentioned another consequence of the above equation: If we measure time

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in S1 with a clock U, we must measure the time in S2 with a clock that runs 1 + )/c2 times slower than the clock U, for when measured by such a clock the frequency of the ray of light considered above, at its emission, is in S2: ɋଶ ൬ͳ ൅

Ȱ ൰ǡ …ଶ

equal to the frequency Q1 of the same ray of light on its arrival in S1. Hence: –ଵ ൌ – ଶ ൬ͳ ൅

Ȱ ൰Ǥ …ଶ

Pound and his graduate student Rebka performed an experiment that actually reproduced Einstein's above mentioned thought experiment from 1911, by which he established the gravitational redshift effect. Pound and Rebka were able to reproduce Einstein's 1911 thought experiment by means of the Mössbauer effect which was only discovered in 1958, after Einstein's death in 1955. They proposed, in 1959 and again in 1960 measuring the effect (originally hypothesised by Einstein in his 1911 paper) of gravitational potential on the apparent frequency of electromagnetic radiation. For this end they used the energy of J rays emitted and absorbed in solids, as discovered by Mössbauer (Pound and Rebka 1960, 337): An atom transitions from an excited state to its ground state and emits a J ray photon with a characteristic energy and frequency. When the same atom in its ground state encounters a J ray photon with the same characteristic frequency and energy, it absorbs the photon and will transition to the excited state. If the J ray photon's frequency and energy differ by a small amount, the atom cannot absorb it. According to Einstein's general relativity, when the J ray photon travels through a gravitational field, its frequency and energy will change due to the gravitational redshift. As a result, the receiving atom can no longer absorb the J ray. However, if the emitting atom moves with just the right speed relative to the receiving atom (i.e. there is no velocity difference between the emitting atom and the receiving atom), there is a Doppler shift and it exactly cancels out the gravitational redshift. This way the receiving atom

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can now absorb the photon as before. If there is any velocity difference between emitter and absorber, the absorber will not be able to absorb the J ray photon. This Doppler shift is therefore the measure of the gravitational redshift. The Jefferson Laboratory on the Harvard campus contains an h = 22.5meter-high vertical enclosed isolated tower extending from the roof to a basement room. Pound and Rebka placed an absorber A on top of the Jefferson Laboratory Tower and an emitter E on an hydraulic movable platform in the basement. The source in the basement (iron 57 isotopes), emitted 14.4 keV J ray photons from the bottom of the tower, near the Earth's surface of gravitational potential ). The J ray photons from the bottom were absorbed at the absorber placed at the top of the tower. Pound and Rebka raised the movable platform in the basement as the source emitted the J ray photons. The emitting source thus moved with just the right speed relative to the absorber. This produced a blue Doppler shift that exactly compensated for the gravitational redshift. The receiving atom could now absorb the J ray photons as before and these could be detected. The gravitational potential at the absorber A is: Ȱ ൅ οߔ, where οߔ ൌ ݄݃ǡ and g is the acceleration due to gravity. By measuring the detection rate of the J ray photons at the absorber as they moved the platform of the emitter up and down slightly, Pound and Rebka could find the velocity difference (the Doppler shift, the blue shift) between emitter E and absorber A that cancels out the gravitational redshift. Thus when, the frequency of theJ ray photons measured at E (the emitter) is Q1, and the frequency of theJ ray photons measured at A (the absorber) is Q2. Then: οߔ ݄݃ ߥଶ െ ߥଵ οߥ ൌ ൌ െ ଶ ൌ െ ଶǤ ߥ ܿ ܿ ߥଵ Pound and Rebka reproduced the experiment with the source at the top of the tower and the absorber at the bottom to verify the results. The emitter was placed on top of the Jefferson Laboratory Tower and the absorber in the basement. The movable platform of the emitter was again moved up

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and down as the source emitted the J ray photons. The former result was verified with this set-up. According to the principle of equivalence, a J photon emitted by the emitter on the top of the laboratory is in free fall. It is therefore locally in an inertial system, and the light is blue shifted according to the Doppler Effect. Since Pound and Rebka are at rest in the laboratory system in a gravitational field, therefore the light frequency of the J photon is gravitationally red shifted rather than blue Doppler shifted. Or, as in Einstein's 1911 equation: οߔ ݄݃ ߥଵ ൌ ͳ ൅ ଶ ൌ ͳ ൅ ଶǤ ߥଶ ܿ ܿ For h = 22.5 m, Pound and Rebka obtained in the case of one-way height difference (Pound and Rebka 1960, 340): ݄݃ οߥ ൌ െ ଶ ൌ െʹǤͷ͸ ή ͳͲିଵହ ǡ ܿ ߥ And for two-way height difference: ݄݃ οߥ ൌ െʹ ଶ ൌ െͷǤͳ͵ ή ͳͲିଵହ Ǥ ܿ ߥ The calculated theoretical value, from the 1911 equivalence principle general theory of relativity, using Pound and Rebka's values (h = 22.5 meter, iron isotope 57Fe, etc. is: െͶǤͻʹ ή ͳͲିଵହ Ǥ

12. Erich Kretschmann's Critiques of Einstein's Point Coincidence Argument It seems likely that Einstein got immediate inspiration for the Point Coincidence Argument (to escape his earlier Hole Argument) from Erich Kretschmann's paper, "On Determining the Principle of the Legitimate Reference System of any Theory of Relativity". The paper seems to anticipate essential elements of the Point Coincidence Argument found in Einstein's 1916 review article, "The Foundation of the General Theory of Relativity".

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Kretschmann's paper appeared on December 21, 1915, in the Annalen der Physik, five days before the earliest of the surviving letters in which Einstein presented for the first time his version of the Point Coincidence Argument to Paul Ehrenfest. Einstein, therefore, very likely read Kretschmann's paper when it appeared and elaborated the ideas on coincidences to explain a few days later to Ehrenfest how to evade his 1914 Hole Argument (Howard and Norton 1993, 53-54; Kretschmann 1915, 914-916; Einstein 1916, 776-777). In another publication of his from 1917, Kretschmann cited this publication from 1915 for further details on the Point Coincidence Argument, referring to Einstein's 1916 version of the Argument for the purpose of introducing the term "Coincidence" to replace another term he had used in 1915, "topological", in relation to his argument (Howard and Norton 1993, 53; Kretschmann 1917, 575-576; Norton 1995, 108). According to Kretschmann, the forms in which different authors expressed the theory of relativity, and especially the forms in which Einstein "recently" expressed his postulate of general relativity, admit the following interpretation: Physical laws satisfy the relativity postulate if the equations that represent it are covariant with respect to the group of spatio-temporal coordinate transformations associated with that postulate. Kretschmann maintained that if one accepts this interpretation, and (according to his analysis of 1915) "all physical observations consist in the determination of purely topological relations ('coincidences') between objects of spatio-temporal perception", then from this it indeed follows that no coordinate system is privileged. However, there is an additional conclusion that follows from this: By means of a purely mathematical reformulation of the equations of the theory, and even with mathematical complications that follow this reformulation, any physical theory can be brought into agreement with any arbitrary relativity postulate, and this without modifying the theory's content that can be tested by observation. In his 1916 review article Einstein used the Point Coincidence Argument to establish the requirement of general covariance. Kretschmann uses here Einstein's 1916 Point Coincidence Argument to conclude that any theory could be put into a form satisfying Einstein's principle of general covariance. Since general covariance has no physical content and is a purely mathematical formulation of the theory, then using the methods of

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tensor calculus of Ricci and Levi Civita, any physical system of equations could be brought into generally covariant formulation without any alteration of its observationally testable content. This is obvious if we accept Einstein's 1916 Point Coincidence Argument. Kretschmann actually argues that Einstein's connection between general covariance (a purely mathematical property) and the general principle of relativity (a physical principle) is problematic, since there must be something more to Einstein's general relativity principle than just the mathematical property of general covariance (Norton 1995, 108). As a result of Kretschmann's 1917 criticism, in 1918 Einstein returned to the foundations of his general theory of relativity and redefined his principles on which he had based himself. In his paper, "Principles of the General Theory of Relativity", he wrote that his theory rested on three principles that are not independent of each other: the relativity principle, the equivalence principle (inertia and weight are identical in nature) and Mach's principle. He formulated the principle of relativity in terms of the Point Coincidence Argument: the laws of nature are merely statements about space-time coincidences. Therefore they find their natural expression in generally covariant equations. He added Mach's principle to the list (Einstein 1918, 241-242). Formulating the principle of relativity in terms of the Point Coincidence Argument did not escape Kretschmann's criticism, namely that on the basis of Einstein's 1916 Point Coincidence Argument his general covariance has no physical content. Einstein then replied to Kretschmann that general covariance had physical content if it was supplemented by the requirement of simplicity, which caused the general principle of relativity to be in this sense a heuristic guide. A generally covariant formulation of general relativity was simpler than a generally covariant formulation of Newtonian gravitation theory. In fact, Einstein believed that although a generally covariant formulation of the latter could be possible in theory, in practice this would turn out to be unworkable. Einstein believed that his formulation of the laws of nature in general relativity was the most natural and simplest path to be taken. He used to speak about God creating the world, which represented the simplest possible world. Leopold Infeld remarked that Einstein used the word God more often than a Catholic priest. In Princeton, in Fine Hall, there was a

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room usually kept closed but opened when a distinguished visitor was entertained. Engraved on the fireplace was Einstein's sentence "Raffiniert ist der Herrgott aber boshaft ist er nicht", which someone in Fine Hall translated into American English: "God is slick but He ain't mean". He told Infeld, for instance: "God does not care about our mathematical difficulties; He integrates empirically." He asked: "Could God have created the world in this way?" Or: "Is this mathematical structure worthy of God?" He also said: "God created the world." This phrase, so often repeated by Einstein, represented his religious feeling that laws of nature could be formulated simply and in the simplest form (Infeld 1941, 267, 271, 279). Einstein also believed that in light of Mach's ideas, his simpler formulation of the laws of nature in general relativity eliminated absolute motion and absolute space, while the Newtonian gravitation theory (e.g. the bucket experiment) suffered from the defect of retaining absolute motion and absolute space. In the 1920s, however, a generally covariant formulation of Newtonian theory was derived, though one that was much more complex than the generally covariant formulation of Einstein's 1916 general theory of relativity. In 1923 and 1924 Élie Joseph Cartan, in attempts to unify gravitation with electromagnetism, formulated Newtonian theory of gravity as a geometric-dynamical theory and provided a generally covariant formulation of Newtonian gravity in space-time. The first major task he set himself was to geometrize classical field theories – electromagnetism and Newtonian gravity – in terms of an affine connection. Classical field theories and general relativity all assume the existence of an affine connection in space-time (Cartan 1923, 1924; see Malament 2012, 256-280). On a four-dimensional manifold two geometrical objects are introduced, the metric tensor and a symmetric affine connection. In general relativity, the metric tensor describes through the line element ds the behavior of clocks (the chronometry) and measuring rods (the geometry) of spacetime. We combine the one-dimensional chronometrical and the threedimensional geometrical structures into the chrono-geometrical structure of space-time. In the four-dimensional version of Newtonian gravitation theory the chronometry is represented by a scalar field on the four-dimensional

196

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manifold, called "absolute Newtonian time", and the geometry by a degenerate metric field of rank 3. In the space-time formulation of Newton's theory, proper time is not defined. Cartan's new theory is named Newton-Cartan space-time or the NewtonCartan theory of gravity. It is thus possible to redo Newtonian gravity as a theory of curved space-time. Furthermore, the Newton-Cartan theory is written in a generally covariant form as in Einstein's theory of general relativity. In the Principia, Newton defines the inertial mass as being proportional to the mass of the body. He also wrote that the gravitational mass is proportional to the weight, as he had found by experiments on pendulums (Newton 1726, Book I, 9). He therefore realised that there was an (empirical) equality between inertial and gravitational mass. For Newton, however, this connection was accidental whereas Einstein said that Galileo's law of free fall could be viewed as Newton's equality between inertial and gravitational mass but for him the relation was not accidental (see Chapter 2, Section 1). In Cartan's formulation of Newtonian gravitation theory, the equality between inertial and gravitational mass is taken as a principle and this was later named the weak equivalence principle: In a gravitational field all local freely falling objects are fully equivalent. Free material particles follow the geodesic curve and a tangent vector field is parallel transported along a geodesic curve. This is parallel transport of a tangent vector in an affine connection on a manifold. Consider a freely falling particle moving in a gravitational field. We look first for the equation of the trajectory traced by the particle, and then express this equation in geometrical language. A particle in free fall moves along a geodesic in space-time (in 3+1 space-time in Newtonian theory). The geodesic equation can be written in terms of the affine connection. In 1907 Einstein imagined an observer freely falling from the roof of a house; for the observer there is locally no gravitational field during the fall. If the observer releases any bodies they remain, relative to him, in a state of rest or uniform motion regardless of their particular chemical and

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physical nature. The observer is therefore justified in interpreting his state as being locally at rest. Hence a freely falling system can always be chosen locally, thus enabling us to formulate the law of inertia: We define local inertial systems as freely falling systems in which the components of the inertio-gravitational field vanish. Material particles in free fall move along geodesic lines in flat affine space, that is to say we assume a flat affine connection (a flat derivative operator ‫׏‬ୟ ). This definition, however, seems problematic in several respects. Excluding the action of gravity does not seem consonant with the weak principle of equivalence, which has its roots in the equality between inertial and gravitational mass. The inertial systems thus defined are local inertial systems. One cannot distinguish Newtonian inertial systems (material particles in free fall following geodesic lines in curved non-flat spacetime) from local inertial systems (material particles in free fall following geodesic lines in flat space). We cannot, therefore, locally separate gravity from inertia. Gravity and inertia are described by a single inertiogravitational field. We incorporate an inertio-gravitational field into the geometric formulation of Newtonian gravity theory. We cannot separate a flat affine connection of space-time ‫׏‬ୟ from a field describing gravitation (a gravitational potential)). More precisely, Cartan solved the problem in the following way: He found a more general dynamical affine connection which represents both inertia and gravity. He introduced a Newtonian classical space-time: A fourdimensional manifold M endowed with a Euclidean special metric hab = diag (0, 1, 1, 1) and absolute time tab = tatb = diag (1, 0, 0, 0), i.e. special and temporal metrics signatures (0, +, +, +) and (+, 0, 0, 0). We additionally require that tab and hab be orthogonal (orthogonality): ݄௔௕ – ୠ – ୡ ൌ Ͳ (replacing tab with tb). The special and temporal metrics determine the connection ‫׏‬ୟ ǡthrough the condition of metric compatibility:

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‫׏‬ୟ – ୠ – ୡ ൌ ‫׏‬ୟ ݄௔௕ ൌ Ͳǡ which requires that the parallel transport of a vector should preserve its length. Our concern is with geometrizing Newtonian theory. The Newtonian law of inertia is given by: ݀ଶ ‫ݔ‬ ൌ ͲǤ ݀‫ ݐ‬ଶ However, what about inertia and gravity in Newton's universe? The weak equivalence principle says that in a gravitational field all local freely falling objects are fully equivalent: ݉௜

݀ଶ ‫ݔ‬ ൌ െ݉௚ ‫׏‬ĭǤ ݀‫ ݐ‬ଶ

Since, mi = mg, the above equation reduces to: ݀ଶ ‫ݔ‬ ൌ െ‫׏‬ĭǡ ݀‫ ݐ‬ଶ where ) is the scalar potential. Suppose now that we make an attempt to geometrize the above equations: we shall reformulate the equations in terms of the metric and connection. The acceleration of any massive particle, i.e. its deviation from a geodesic trajectory, is determined by the forces acting on it (other than gravity). If the particle has mass, and the vector field Fa on the manifold represents the vector sum of the various (non-gravitational) forces acting on the particle, then the particle's four-acceleration ߦ ௔ ‫׏‬ୟ ߦ ௕ (where [a is the fourvelocity) satisfies: ‫ ܨ‬௔ ൌ ݉ߦ ௔ ‫׏‬ୟ ߦ ௕ Ǥ Fa is also determined by the relation:

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‫ ܨ‬௔ ൌ െ݄݉௔௕ ‫׏‬ୟ ߔǤ Now invoking the weak equivalence principle these last equations finally become: ߦ ௔ ‫׏‬ୟ ߦ ௕ ൌ െ݄௔௕ ‫׏‬ୟ ߔǤ Consider the geometrized formulation of the above equations of motion for material particle trajectories (time-like geodesics) in terms of a non-flat connection ‫׏‬ୟୠୡ and Newtonian gravitational potential): ݀ଶ ‫ݔ‬ ݀‫ݔ‬௕ ݀‫ݔ‬௖ െ ‫׏‬ୟୠୡ ൌ Ͳǡ ଶ ݀‫ݏ‬ ݀‫ݏ݀ ݏ‬ where ‫׏‬ୟୠୡ denotes a non-flat connection on a four-dimensional manifold M with Euclidean special metric hab and absolute time tatb: ‫׏‬ୟୠୡ ൌ ݄௔௕ ‫׏‬ୟ ‫׏‬ୠ ĭ– ୟ – ୠ Ǥ The connection satisfies: ‫׏‬ୟ – ୠ – ୡ ൌ ‫׏‬ୟ ݄௔௕ ൌ Ͳand: ݄௔௕ – ୠ – ୡ ൌ Ͳ. Furthermore, an affine geodesic is the curve along which tangent vectors are parallel transported: ݀ଶ ‫ݔ‬ ݀‫ݔ‬௔ ݀‫ݔ‬௕ ݀‫ݔ‬௖ െ ‫׏‬ୟୠୡ ൌߣ Ǥ ݀‫ ݏ‬ଶ ݀‫ݏ݀ ݏ‬ ݀‫ݏ‬ IfO vanishes, then the parameter is an affine parameter and the affine geodesic equation reduces to: ݀ଶ ‫ݔ‬ ݀‫ݔ‬௕ ݀‫ݔ‬௖ െ ‫׏‬ୟୠୡ ൌ ͲǤ ଶ ݀‫ݏ‬ ݀‫ݏ݀ ݏ‬

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Absolute time tab is chosen to be an affine parameter, which represents the time read by clocks moving along all time-like affine geodesics (the compatibility condition): ‫׏‬ୟ – ୠ – ୡ ൌ ‫׏‬ୟ ݄௔௕ ൌ ͲǤ Once again, we shall reformulate the above geodesic equation in terms of the four-velocity and connection: ߦ ௔ ‫׏‬ୟୠୡ ߦ ௕ ൌ ͲǤ Although ‫׏‬ୟୠୡ is non-flat (curved), it is still not the metric connection. The Christoffel symbols do not serve as the components of the connection ‫׏‬ୟୠୡ and the connection is not written in terms of the metric tensor components. Rather than thinking of particles being attracted by forces, one thinks of them as moving along geodesic lines in curved, four-dimensional spacetime. Therefore gravity is not a force anymore but is interpreted geometrically and should be considered as curvature of space-time. The above non-flat affine connection ‫׏‬ୟୠୡ defines the amount of curvature of geodesic lines. Let us now define the total mass-momentum field Tab associated with the matter field. We require that the conservation condition: ‫׏‬ୟ ܶ ௔௕ ൌ Ͳ holds at all points in M (the four-dimensional manifold). To find the field equations of the Newton-Cartan theory, it is useful to define the Newtonian field equations similarly to the Einstein field equations: Field equations relate chrono-geometrical quantities and the non-flat connection to the matter distribution. Given the Newtonian Poisson equation: ‫׏‬ଶ ĭ ൌ ͶɎ ɏǡ

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whereUis the mass density and G the gravitational constant, the Poisson equation is now replaced with a generalised Poisson equation:  ௔௕ ൌ ͶɎɏ– ୟ – ୠ ǡ where:  ௔௕ ൌ ݄௔௕ ‫׏‬ୟ ‫׏‬ୠ ĭ– ୟ – ୠ  ൌ ݄௔௕ ‫׏‬ଶ ĭ– ୟ – ୠ is the Ricci tensor. From the above point of view, the Ricci tensor Rab is defined in terms of the non-flat connection. I shall only add that by means of these expressions (orthogonality and compatibility): ‫׏‬ୟ – ୠ – ୡ ൌ ‫׏‬ୟ ݄௔௕ ൌ Ͳǡ ݄௔௕ – ୠ – ୡ ൌ Ͳ, the Ricci tensor becomes:  ௔௕ ൌ ͲǤ At the same time the Poisson equation becomes:  ௔௕ ൌ ͶɎɏ– ୟ – ୠ ൌ ͲǤ By examination of the above equations, it will readily be seen that this is only true if space-time is flat. However, Cartan's new affine connection ‫׏‬ୟୠୡ is non-flat and cannot break up into a Newtonian flat affine connection ‫׏‬ୟ and a gravitational potential ). In the local inertial system, which means in a freely falling system, we cannot separate gravity from inertia. This embodies the inertiogravitational field. Accordingly, in the Newtonian limit, although we obtain Poisson's equation and space-time is flat, Cartan's affine connection

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‫׏‬ୟୠୡ remains non-flat, that is to say, the Ricci tensor is expressed in terms of Cartan's non-flat connection. Special relativity describes space-time without gravity but according to general relativity, locally, space-time is not the gravity-free Minkowski flat space-time. The four-dimensional formulation of the Newtonian Poisson field equation:  ௔௕ ൌ ͶɎɏ– ୟ – ୠ is a precise definition of the Newtonian limit of Einstein's field equations. The metric tensor in the special relativity limit is diagonal (1, 1, 1, –c2). The three spatial components are equal to +1 and the temporal component is equal to –1. A free material point moves with respect to this flat metric system uniformly, in a straight line. In 1912, Einstein held a conception that, locally, space-time is represented by the Minkowski flat metric, i.e. in the weak-field approximation, the spatial metric of a weak gravitational field must be flat. He thus mixed between the metric tensor diagonal (– 1, – 1, – 1, c2), the limit of special relativity, and the limit of weak static gravitational fields diagonal (– 1, – 1, – 1, c2). In weak static gravitational fields, c2 is a function of the spatial coordinates, x1, x2, x3. Empirically taking the limit as the metric tensor diagonal (– 1, – 1, – 1, c2), which could represent both the limit of special relativity and the limit of weak gravitational fields was mere coincidence that worked. Einstein did not possess the affine connection within his mathematical toolbox. He thus chose a pathway, which followed the mathematicalphysical knowledge of his day. In the absence of the affine approach, Einstein obtained the Newtonian results by using the special relativistic limit diag (1, 1, 1, -1), the weak field approximation diag (1, 1, 1, c2) and by assuming that space was flat (1, 1, 1). As seen from examining the Zurich Notebook, three years before November 1915, Einstein had written on page 22R the November tensor, when he considered the Ricci covariant tensor as a possible candidate for the left hand side of his field equations. Einstein got so close to his November 1915 breakthrough at the end of 1912, that he even considered, on page 20L, another candidate – albeit in a linearized form – which resembles the final version of the November 25, 1915, field equation of

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general relativity. Einstein, therefore, first wrote down a mathematical expression close to the correct field equation and then discarded it, only to return to it more than three years later. Einstein rejected gravitational field equations of much broader covariance in 1912-1913, only to come back to these field equations in November 1915. He came close to the final form of the November 1915 field equations of his general relativity theory, but retreated because of his earlier work on static fields. Einstein's earlier work on static fields led him to conclude (on page 21R) that in the weak-field approximation, the spatial metric of a static gravitational field must be flat (Einstein 1912c, 447, 449). This statement appears to have led him to reject the Ricci tensor on page 22R, and fall into the trap of Entwurf limited generally covariant field equations. Or as Einstein later put it, he abandoned the generally covariant field equations with "heavy heart" and began to search for non-generally covariant field equations. Einstein thought that the Ricci tensor should reduce in the limit to his static gravitational field theory from 1912 and then to the Newtonian limit, if the static spatial metric is flat. This prevented the Ricci tensor from representing the gravitational potential of any distribution of matter, static or otherwise (see Chapter 2, Section 4). Stachel claims that Einstein had good reason for his long-held intuition that, in Newtonian theory space (as opposed to space-time) should be flat. When Einstein solved the Entwurf field equations with Besso in 1913 in order to find the perihelion motion of Mercury, he started with the flat Minkowski space-time as the zeroth approximation. In the first approximation, after applying the Entwurf theory, to the static field of the Sun, the result was that the static metric remained flat. Einstein assumed weak gravitational fields and identified this with Newtonian theory. On the other hand, almost the same scheme, solved two years later using the November 11, 1915 vacuum generally covariant field equations, gave in the first approximation non-flat metric (Stachel 1989, 304-306). In winter 1916, after presenting the final general theory of relativity, Einstein told Besso that Grossmann and he believed that the conservation equations were not fulfilled, and that Newton's law did not come out in the first approximation. Einstein also told Besso that he was now surprised by the occurrence of non-flat components g11 – g33. Einstein tried to solve the November 11, 1915 vacuum field equations to attain the perihelion advance of Mercury in the field of the static Sun. He

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was looking for the equation of Mercury moving along the geodesic line in the static gravitational field of the Sun. In a very great distance from the Sun the gravitational field is so weak that it is not felt and we arrive back at the Minkowski flat metric. Einstein thus started from the 0th approximation. gPQ corresponds to a flat Minkowski metric. He then calculated the metric field of the Sun using the vacuum field equations in a first order approximation. He told Besso that he was surprised by the occurrence of non-flat components of the first order metric, g11 – g33, i.e. he discovered that the static spatial metric need not be flat (Einstein and Besso letter 12.1, 60-62): ݀‫ ݏ‬ଶ ൌ ݃ఓఔ ݀‫ݔ‬ఓ ݀‫ݔ‬ఔ ൌ ݃଴଴ ܿ ଶ ݀‫ ݐ‬ଶ െ ݃ଵଵ ݀‫ ݎ‬ଶ െ݃ଷଷ ݀߮ ଶ ǡ (in polar coordinates r and M) and Mercury moves along a geodesic curve that depends on non-flat components of the metric, g11 – g33. In 1913, the basic calculation of Mercury's perihelion had already been undertaken two years earlier by Einstein and Besso (with a final disappointing result of 18") in the Einstein-Besso manuscript. In November 1915, Einstein transferred the basic framework of the calculation from the Einstein-Besso manuscript and corrected it according to his generally covariant vacuum field equations (see Chapter 2, Section 9). Judged from the historical point of view of his time, Einstein lacked the appropriate mathematical tools to correctly taking the Newtonian limit of general relativity. Actually with hindsight the story is more complicated. What was eventually mere coincidence for Einstein would later turn to be a consequence derived by new mathematical tools, the affine connection, which was invented after Einstein had arrived at generally covariant field equations. Later, however, it was claimed that one could not properly take the Newtonian limit of general relativity without the concept of an affine connection, and the corresponding affine reformulation of Newtonian theory. Indeed, the problem of correctly taking the Newtonian limit of general relativity only began to be solved in 1927. In the absence of the affine approach, more-or-less heuristic detours through the weak field, special-relativistic (i.e. fast motion) limit followed by a slow motion approximation basically out of step with the special-relativistic approach, had to be used to obtain the desired Newtonian results. In 1912 Einstein originally thought that he knew the form of the weak field metric in the static case. Had Einstein known about the connection representation of the

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inertio-gravitational field, he would have been able to see that the spatial metric can go to a flat Newtonian limit, while the Newtonian connection remains non-flat without violating the compatibility conditions between metric and connection (Stachel 2007b, 438-439). According to Einstein, the general theory of relativity assumes its simplest form when expressed in a generally covariant form. Of two theoretical systems compatible with experience, the one that is simpler and more transparent from the standpoint of differential calculus is to be preferred. Suppose we possess a generally covariant formulation of both theories – Einstein's general relativity and Newtonian non-relativistic theory. For the time being, we should still prefer Einstein's field equation over Newton's when dealing with strong gravitational fields, not only in view of its simplicity, but also in view of the fact that Einstein's four classic predictions have already been established experimentally, and further experimental tests have been performed and have verified even more predictions of the theory. Einstein's general theory of relativity, moreover, provides new insights into understanding the nature of space-time, hence giving us some feasible reason to prefer general relativity over Newtonian physics. There is something special about Einstein's general theory of relativity. Einstein explained that "it is characteristic of Newtonian physics that it has to ascribe independent and real existence to space and time", i.e. Newtonian physics assumes a fixed, non-dynamical background space-time structure. General relativity, unlike Newtonian physics and even special relativity, is a background-independent theory. In general relativity, "there is no such thing as an empty space, i.e. a space without field". Furthermore, "spacetime does not claim existence on its own, but only as a structural quality of the field" (Einstein 1952, 155, 176). Space-time is dynamical, and ceases to exist when a singularity is reached. Hence Einstein's general theory of relativity is a dynamical background-independent theory. In general relativity, the behaviour of measuring rods and clocks (chronogeometry, i.e. in general the metric) is determined by the inertiogravitational field. Both the chrono-geometrical and the interiogravitational structures are dynamical fields. Wherever there is a chronogeometric structure there is always also an affine inertio-gravitational structure. Chrono-geometrical and inertio-gravitational structures obey field equations coupling them to each other and to all other physical

206

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processes. Thus physical processes do not take place in space-time. Spacetime is just an aspect of the totality of physical processes. In Newtonian physics, however, the measurement of time and space is unaffected by the presence of an inertio-gravitational field. We thus define compatibility of chronometry and geometry with the inertio-gravitational field (Stachel 2007b, 429). Accordingly, given that the special metric hab is interpreted as Euclidean three-dimensional space and tab as absolute time, it seems that the Newton-Cartan theory involving absolute structures is not consistent with the action-reaction principle.

13. Einstein and Mach's Ideas In 1914, Einstein's Entwurf theory was severely attacked in the philosophical journal Scientia by Abraham (Abraham 1914a, 23; 1914b) and Mie (1914b) – who both strived to protect the 1905 special relativity theory and the principle of the constancy of the velocity of light; they thus attacked Einstein for giving up this principle in his new gravitation theory. Mie attempted to develop a (classical) unified field theory to solve the problems of gravitation (See Chapter 2, Section 6). Einstein attacked Mie and claimed that his theory violated the principle of equivalence, especially the equality of inertial and gravitational masses. Mie replied that for Einstein the equality between the two masses was essentially important, but since the Entwurf theory admitted only general linear transformations, it had nothing to do with accelerated motions and with a generalised principle of relativity. And therefore, the equivalence principle did not hold in Einstein's Entwurf theory either; furthermore, Mie proved that Einstein's theory could only guarantee the equality of inertial and gravitational masses by making inconsistent assumptions and auxiliary assumptions containing contradictions. Hence, Mie recommended renouncing the equality between inertial and gravitational mass and by this to adhere to the original special theory of relativity with its constant light postulate (Mie 1914a, 1914b; Smeenk and Martin, 624-625, 631). A year earlier, in 1913, Mie and others attacked Einstein's Entwurf theory of gravitation, but Einstein reasoned that his Entwurf theory removed an epistemological defect, which was inherent not only in the special theory of relativity, but also in the Galilean mechanics, and had been emphasised especially by Ernest Mach: Mach's idea, according to which inertia has its origin in an interaction between a mass point under consideration and all of the other mass points. Although Einstein could not deal with the

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mathematical defect of general covariance, he felt he managed for the time being to remove this epistemological defect; and especially he thought he found that the metric field describing a rotating system was a solution of the Entwurf field equations. Einstein thus thought his gravitational equations indicated inertia of relativity (Einstein 1913a, 1262-1263). In 1915-1916 Einstein's field equations were already generally covariant and he came back to Mach's ideas. In 1916 Einstein replied to his critics Abraham, Mie and others who were attacking his theory of gravitation for violating the light postulate. He first explained why the principle of the constancy of the velocity of light in vacuum had to be modified in general relativity. He then gave a thought experiment the conclusion of which was a support of Mach's viewpoint. Consider K' to be moving with uniformly translated acceleration with respect to K. Einstein explained that it was obvious from experience that the principle of the constancy of the velocity of light in vacuum had to be modified. Since we easily recognize that the path of a ray of light with respect to K' had in general to be curved, if with respect to K light was propagated in a straight line with a certain constant velocity. In his 1916 review article Einstein clarified the demarcation between the special and general theories of relativity, and the relation between them. Light rays that moved in straight lines signified an affiliation with Euclidean geometry (Einstein 1916, 773). In the same 1914 volume of Scientia, in which Einstein was severely attacked by Abraham and Mie, Einstein published a paper in which he proposed a thought experiment, originally suggested by Newton in the Principia, the two globes thought experiment (Newton 1726, 18). Einstein's intention was to explain Mach's ideas. In his gravitation theory Einstein considered two systems: one of them K, in uniform translation motion, and conforming to a Galilei-Newtonian coordinate system, and the other K', in uniform rotation relative to K. Centrifugal forces act on the masses at rest relative to K', while they do not act upon the masses which are at rest relative to K. According to Newton, the rotation of K' had to be regarded as "absolute", and thus one could not consider K' as "at rest" like K. Einstein maintained, however, that this argument – as shown particularly by Ernst Mach – was not valid. The existence of these centrifugal forces did not necessarily require the motion of K'. We could just as well derive them from the

208

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averaged rotational movement with respect to K' of distant masses in the environment and thereby treat K' as "at rest". At that time Einstein was under the impression that the above argument spoke in favour of his new Entwurf theory of general relativity: the centrifugal force, which acts under given conditions upon a body, is determined by precisely the same natural constant as the effect of the gravitational field, so that we have no means to distinguish a centrifugal field from a gravitational field. Einstein thus interpreted the rotating system K' as at rest and the centrifugal field as a gravitational field, and he introduced the inertio-gravitational field. The breakup into inertia and gravitation is relative to the acceleration (Einstein 1914b, 1031-1032). In the two globes thought experiment Einstein considered two masses, and did not say anything about their shape. It was implicitly assumed that these masses were symmetric, and the problem was whether the Newtonian explanation applied, or rather the Machian one (Einstein 1914c, 344). In outer space there are two masses floating at a great distance from all celestial bodies. The masses are close enough to be able to exert mutual influence on each other. An observer follows the motions of both bodies by constantly sighting in the direction of the line connecting the two masses toward the vault of the fixed stars. He assumes that the line of sight traces a closed line on the visible vault of the fixed stars, which does not change its position with respect to the visible vault of the fixed stars. If the observer possesses natural intelligence but has learned neither geometry nor mechanics, he will conclude that his masses perform a motion that is, at least in part, causally determined by the system of fixed stars. Hence, the laws according to which masses move in his environment are determined by the fixed stars. These are Machian ideas. A Newtonian man will say that the motion of the masses has nothing to do with the heaven of the fixed-stars; on the contrary, by the law of mechanics it is completely independent of the other masses. There is a space R in which these laws are valid and are such that the masses remain continually in one place in this space. But the system of fixed stars cannot rotate in this space because otherwise it would be torn apart by tremendous centrifugal forces. It is therefore necessarily at rest, and this is why the plane in which the mass is moving always passes by the same fixed stars. On his part the Machian observer thinks of the space R as a very subtle net of bodies to which other things are referred. We can then get in addition to

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R, a second such network R' that rotates relative to R and for which the Newtonian equations are at the same time also valid relative to R'. The Newtonian man denies this, but the Machian reasons that the masses cannot know with respect to which of the spaces R or R', should they be moving according to Newtonian laws, and cannot recognize after which space or spaces they will have to follow. In 1916 Einstein adapted this 1914 thought experiment in order to extend the special principle of relativity and presented the two masses as nonsymmetric masses. This way special relativity no longer applied to the two non-symmetric masses (one inertial and the other non-inertial). Einstein added to the 1914 description that the masses were fluid, which enabled him to present two apparently non-symmetric bodies of different shapes but of the same size and nature. He considered that the surfaces of both bodies (S1 and S2) are measured by means of measuring rods (relatively at rest), and it followed that the surface of S1 was a sphere, and that of S2 was an ellipsoid of revolution. The 1916 masses hover freely at such a great distance from each other that one takes into consideration only the gravitational forces arising from the interaction of different parts of the same body. They are measured by observers that are at rest with respect to the bodies; an observer who is at rest on either mass judges the other mass as rotating with constant angular velocity (Einstein 1916, 771). This was the first part of the thought experiment. The first part of the two globes thought experiment appears to be an extension of the magnet and conductor thought experiment from Einstein's relativity paper of 1905: If the conductor is at rest in the ether and the magnet is moved with a given velocity, a certain electric current is induced in the conductor. If the magnet is at rest, and the conductor moves with the same relative velocity, a current of the same magnitude and direction flows in the conductor. However, the ether theory gives a different explanation for the origin of this current in the two cases. In the first case an electric field is supposed to be created in the ether by the motion of the magnet relative to it (Faraday’s induction law). In the second case, no such electric field is supposed to be present since the magnet is at rest in the ether, but the current results from the motion of the conductor through the static magnetic field [Lorentz’s force law, F =q(v x B) in modern terms]. Einstein claimed that this asymmetry of the explanation is foreign to the phenomenon.

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In 1920 Einstein indeed explained that when in 1907, he was working on his review paper on the special relativity theory, "On the Relativity Principle and the Conclusions Drawn from It", and also tried to modify Newton's theory of gravitation, there came to him the happiest thought of his life. He realised that the gravitational field is considered to be and has only a relative existence like the electric field generated by magnetoelectric induction (the magnet and conductor thought experiment) (Einstein 1920a, CPAE 7, Doc. 31, 265). After presenting the 1905 magnet and conductor thought experiment in 1905 Einstein said that examples similar to the magnet and conductor experiment led to the conjecture that the phenomena of electrodynamics as well as those of mechanics possess no properties corresponding to the idea of absolute rest (Einstein 1905a, 891). The globes thought experiment was intended to demonstrate that this could be extended to accelerated motions and to the theory of gravitation using Mach's principle (still not defined as a principle in 1916). Einstein tried to solve the problem with the two apparently non-symmetric fluid masses S1 and S2 in much the same way as he had done with the magnet and conductor thought experiment. He was guided by Mach's ideas and discussed the asymmetry problem: We ask: What is the reason why body S1 and body S2 behave differently? Newtonian mechanics does not give a satisfactory answer to this question. The laws of mechanics apply to the space R1, with respect to which the body S1 is at rest, but not to the space R2, with respect to which the body S2 is at rest. But the legitimate space R1, thus introduced, is a merely factitious cause, and not a thing that can be observed. It is also clear that Newton's mechanics demands that the factitious cause R1 is responsible for the behaviour of the bodies S1 and S2. A satisfactory answer to the question raised can only be the following one: the physical system consisting of S1 and S2 reveals within itself no imaginable cause to which the differing behaviour of S1 and S2 can be referred. The cause must therefore lie outside this system. We have to admit that the general laws of motion, which in particular determine the shapes of S1 and S2, must be such that the mechanical behaviour of S1 and S2 is also conditioned, in quite essential respects, by distant masses, which we have not included in the system under consideration.

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Einstein's two globes thought experiment may be regarded as the twobody problem in general relativity: A configuration consisting of two spherical bodies. In empty space the two fluid masses S1 and S2 are floating at a great distance from all celestial bodies. The masses are close enough to be able to exert mutual influence on each other. An observer follows the motions of both bodies by constantly sighting in the direction of the line connecting the two masses.  One takes into consideration only the gravitational forces arising from the interaction of different parts of the same body; an observer who is at rest on either mass judges the other mass as rotating with constant angular velocity. Consider a spherical closed (finite) static universe, devoid of matter whose material mass is concentrated in just two spherical bodies S1 and S2 on opposite sides of the world. Much later, when he was working on cosmological problems, Einstein demonstrated that, for an equilibrium configuration consisting of two spherical bodies at rest, there was a singularity on the line connecting the two bodies – in the empty space between the two bodies. Consequently, time stands still in the empty space between the two masses and this signifies there are other masses distributed between the two bodies (Einstein 1922d). Therefore, according to Mach, the mechanical behaviour of S1 and S2 is also conditioned by other masses (see Chapter 2, Section 14 below and Chapter 3, section 4). The conclusion from the two globes thought experiment was an extension of the postulate of relativity to the following principle of general relativity: The laws of physics must apply to systems of reference in any kind of motion (Einstein 1916, 771-772).

14. Einstein's Reaction to Karl Schwarzschild's Solution It cannot be denied that, Einstein's solution to the problem of the anomalous precession of Mercury's perihelion was open to objections. Already in the year 1906, Hugo von Seeliger suggested an accurate explanation for the anomalous advance of the perihelion of Mercury (Seeliger 1906). According to Seeliger, the perturbations to Mercury's orbit are accounted for by attraction of other matter, a distribution of nonplanetary matter (dust), which must be very near the Sun. Seeliger was an uncompromising opponent of Einstein's theory of relativity and he was evidently a man not easy moved in his opinions. Indeed in his later years

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he seems to have had little sympathy with the new ideas growing up in physics and astronomy.6 In 1915, Erwin Freundlich published a criticism of Seeliger's dust hypothesis. It was already clear that Seeliger could not accept Freundlich's opinion and he published a bona fide aggressive response (Seeliger 1916). Writing to Arnold Sommerfeld, Einstein added a note saying that his colleague Seeliger has a horrible temperament. Einstein preferred not to speak of the refutation of Seeliger's theory of the perihelion motion of Mercury. According to Einstein, Seeliger's work might be described as noticing an open door and hitting it until it is shattered (Einstein to Sommerfeld, December 9, 1915, February 2, 1916, CPAE 8, Doc. 161, 186). I shall dwell no further upon details of Seeliger's dust hypothesis and various alternative nonrelativistic explanations for the anomalous advance of the perihelion of Mercury. It should be pointed out, however, that compared with Seeliger's explanation, Einstein's solution has the advantage that no arbitrary masses are needed (Pauli 1958, 169). I shall only mention two alternative explanations. In 1916, Einstein and the Göttingen geophysicist Emil Wiechert participated in a colloquium concerning the perihelion of Mercury. Wiechert sought to explain Einstein's effects (anomalous advance of the perihelion of Mercury, deflection of light and red shift) through a unified ether theory of gravity and electrodynamics. By 1916, he was confident that the perihelion advance of Mercury could be explained by an ether theory of gravity and electrodynamics (Wiechert 1916). To be clear, Wiechert's vision was not Einstein's cup of tea and he put him in a very bad light at the colloquium (Einstein to Besso, after December 6, 1916, CPAE 8, Doc. 283a). Part of Einstein's difficulty stemmed from Wiechert's classical notion of the "ether". In fact, from Einstein's point of view, there were other reasons for objecting to Wiechert's account of the anomalous advance of the perihelion of Mercury. Such a proposed theory – in contrast to Einstein's general relativistic derivation – constitutes a unified theory of gravity and electrodynamics. It is certainly clear that this development did not fit the scheme that we can identify as Einstein's 1916 Weltanschauung.

  6

"Obituary Notice: Associate: Seeliger, Hugo von." Monthly Notices of the Royal Astronomical Society 85, 316-319.

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We read in a paper by Ludwig Silbserstein on the perihelion of Mercury, dated 1917, that Einstein gave "the formula (G)" for the anomalous advance of the perihelion of Mercury but "with all respect due to Einstein, identically the same formula was given eighteen years earlier" in 1898 by Paul Gerber, "whose investigations, entirely independent of any relativity, seems to have passed unobserved". Gerber, however, had deduced his formula from an untenable theory, or at least from one which had not been based upon well-established general principles (Silberstein 1917, 503504). From Silberstein's point of view, there was absolutely no difference, between Gerber's non-relativistic formula and Einstein's relativistic one (which according to Silberstein was very far from being well established). Both led to the same result. In fact, notwithstanding its mathematical elegance, Einstein's general relativity certainly offered many serious difficulties in its very foundations, while its predictions had not been verified. Furthermore, the fact that Einstein's theory gave Gerber's formula did not seem to Silberstein to be decisive in its favour. Under such circumstances, a method of improving Newton's law of gravitation seemed to Silberstein a great deal safer than that based on relativity. Silberstein was a native of Poland who moved to England. On the one hand, like Max Abraham and Seeliger, Silberstein became antagonistic to Einstein's theory of relativity, criticised it, and entered into a controversial debate with Einstein between 1933 and 1936. On the other hand, he wrote books on relativity and explained the theory. For instance, in 1914 Silberstein wrote a book on the theory of relativity consisting of special and general relativity up to 1914. He updated the book in 1924, and even included cosmology (Einstein's modification of his field equations with the cosmological constant) (Silberstein, 1914). Ironically, Silberstein was one of the most prolific physicists to publish papers or books dealing with general relativity in British journals and publishing houses (he could be considered the second after Eddington) (Sánchez-Ron 1992, 71). Nonetheless, in 1917 Silberstein adopted Seeliger's hypothesis and less than a year later, he even published an alternative theory of gravitation, which he called "General Relativity without the Equivalence Hypothesis" (Silberstein 1918). Silberstein's proposal is, of course, a direct support of anti-relativity view at the expense of Einstein's Weltanschauung: he not only rejected the equivalence principle, but also Mach's principle.

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In 1920, Silberstein was the most vocal of those who saw the negative redshift results as prima facie falsifiers of general relativity. According to Silberstein, one reason why the negative redshift results were so important was that they threatened to undermine the support given by the detection of the bending of light. He even suggested that the deflection of light by the Sun can be viewed as a reason to resurrect the very ether which Einstein's special theory of relativity had seemed to bury. To strengthen his argument, Silberstein quoted a remark of Einstein's which had appeared in the Times of London for November 28, 1919: "If any deduction from it [general relativity] proves untenable, it must be given up. A modification of it seems impossible without destruction of the whole" (Earman and Glymour 1980b, 199). Einstein's field equations are non-linear partial differential equations of the second rank. This complicated system of equations cannot be solved in the general case, but can be solved in particular simple situations. The first to offer an exact solution to Einstein's November 18 (11), 1915 field equations was Karl Schwarzschild, the director of the Astrophysical Observatory in Potsdam. On December 22, 1915 Schwarzschild wrote to Einstein from the Russian front and set out to rework Einstein's calculation in his November 18, 1915 paper of the Mercury perihelion problem. In November 18, 1915 Einstein had reported to the Prussian Academy that the perihelion motion of Mercury was explained by his new general theory of relativity: Einstein found approximate solutions to his November 11, 1915 field equations. He started from the zeroth-order approximation, in which gPQ corresponds to the flat Minkowski metric, and then wrote the solution to first-order approximation that satisfied his November 11, 1915 vacuum field equations (see Chapter 2, Section 9): Schwarzschild first responded to Einstein's solution for the first order approximation and found another first order approximate solution. Schwarzschild told Einstein that if several approximate solutions could be found the problem would then be physically undetermined. Subsequently, Schwarzschild presented a complete solution. He said he realised that there was only one line element: ߙ ݀‫ ݎ‬ଶ ଶ ଶ ଶ ଶ ݀‫ ݏ‬ଶ ൌ ቀͳ െ ቁ ݀‫ ݐ‬ଶ െ ߙ െ ‫ ݎ‬ሺ݀ߴ ൅ •‹ ߴ݀߮ ሻǡ ‫ݎ‬ ͳെ ‫ݎ‬

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where: ߙ ൌ ʹ‫ ݎ‬ൌ

ʹ‫݉ܩ‬ ǡ ܿଶ

that satisfies the four conditions imposed by Einstein on the gravitational field of the Sun – the solution is static, spherically symmetric, gr4 = g4r and at special infinity the metric is flat – as well as Einstein's field equations from the November 18, 1915 paper. Schwarzschild considered a body in which the origin of the coordinates is its geometric centre. If one assumes isotropy of space and a static solution that does not change with time, then there exists spherical symmetry around the centre; and one can work with a system of spherical coordinates r, ߴ, ij. The symmetry of the solution means that the variables are independent of the angular coordinates ߴ, ij. Since the solution is static, there is no dependence on time, and thus only r, the distance from the centre, is an independent variable. According to the four conditions imposed by Einstein, as ‫ ݎ‬՜ λǣ ݀‫ ݏ‬ଶ ൌ ܿ ଶ ݀‫ ݐ‬ଶ െ ݀‫ ݎ‬ଶ െ ‫ ݎ‬ଶ ሺ݀ߴ ଶ ൅ •‹ଶ ߴ݀߮ ଶ ሻǡ and we obtain the Minkowski line element in spherical symmetric form. A mathematical singularity is seen to occur at the origin of the Schwarzschild line element when r = 0. This singularity cannot be removed by coordinate transformations. However, consider the classical Poisson's equation for gravity: ‫׏‬ଶ ĭ ൌ ͶɎ ɏǡ the solution of which is the Newtonian potential: ܷൌെ

‫݉ܩ‬ Ǥ ‫ݎ‬

U depends on the mass m of the mass point that produces the gravitational field, and G is a universal gravitational constant. This solution also has a singularity at r = 0 (Bergmann 1942, 202-203). Recall that Newtonian physics ascribes independent and real existence to

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space and time and assumes a fixed, non-dynamical background spacetime structure. General relativity, unlike Newtonian physics and even special relativity, is a dynamical background-independent theory. In general relativity, there is no such thing as an empty space, a space without field. Furthermore, space-time does not claim existence on its own, but only as a structural quality of the field (Einstein 1952, 155, 176; see Chapter 2, Section 12). Space-time is dynamical, and ceases to exist when a singularity r = 0 is reached. Schwarzschild told Einstein that his line element has only one singularity at the origin where r = 0. He meant to say that his line element had only one real or true singularity that could not be removed. In addition, Schwarzschild's r, ߴ, ij, were not "allowed" coordinates, with which the field equations could be formed, because these spherical coordinates did not satisfy the coordinate condition from Einstein's November 11 and 18, 1915 papers: ඥെ݃ ൌ ͳǤIndeed, for Schwarzschild's line element,ඥെ݃ ൌ ‫ ݎ‬ଶ •‹ ߴǡand thus the determinant differs from 1. From Einstein's point of view, Schwarzschild chose the non-allowed coordinates and a mathematical singularity was seen to occur in his solution. Nonetheless, Schwarzschild explained to Einstein that although his coordinates did not satisfyඥെ݃ ൌ ͳ, his line element gave the best fit in spherical coordinates, and with his line element the equation of Mercury's orbit remained exactly as Einstein had obtained in November 18, 1915 for the first-order approximation (Schwarzschild to Einstein, 22 December 1915, CPAE 8, Doc. 169, note 5). On December 29, 1915 Einstein replied to Schwarzschild's December 22 letter (in which the latter had sent him his solution) telling him that his calculation provided a very interesting unique solution to the problem. Subsequently, Schwarzschild sent Einstein a manuscript in which he derived his solution of Einstein's November 18, 1915 field equations for the field of a single mass (Einstein to Schwarzschild, 29 December 1915, CPAE 8, Doc. 176). Schwarzschild was troubled by his coordinates not satisfying ඥെ݃ ൌ ͳ and tried to find a way to avoid it. Sitting at the Russian front, he found a "simple trick" that allowed him to avoid the problem with his coordinates not satisfying ඥെ݃ ൌ ͳǤThis can be seen by writing his identical line element in different coordinates. This exactly constitutes a shining

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example of Schwarzschild's mathematical virtues and virtuoso, which Einstein lacked. It meant that the problem in Schwarzschild's line element was due to the spherical coordinates and not to the Schwarzschild metric itself.Returning back to the "standard" spherical coordinates, we arrive at the exact solution to Einstein's problem, and to the problem that Schwarzschild's line element does not satisfy ඥെ݃ ൌ ͳ(Schwarzschild 1916a, 191). Einstein received the manuscript at the beginning of January 1916 and examined it with great interest. He told Schwarzschild that he had not expected that one could so easily formulate a rigorous solution to the problem. In mid-January 1916, Einstein delivered Schwarzschild's paper before the Prussian Academy with a few words of explanation (Einstein to Schwarzschild, January 9, 1916, CPAE 8, Doc. 181). Schwarzschild's paper, "On the Gravitational Field of a Point-Mass according to Einstein's Theory" was published a month later (Schwarzschild 1916a). When Schwarzschild calculated the gravitational field of a mass point and arrived at the exact solution which contained a singularity, he solved Einstein's vacuum field equations of November 11, 1915 or November 18. Schwarzschild considered a mass point moving along the geodesic line in the gravitational field, according to his solution, and sought the equation of motion for that point. After reading Einstein's November 25 paper, which extended his November 11 (18) field equations, Schwarzschild decided to approach the problem of the singularity in his line element. Acquainting himself with Einstein's energy tensor, Schwarzschild extended his calculation from the gravitational field of a mass point to the gravitational field of an incompressible homogeneous and isotropic fluid sphere. Using Einstein's new field equations from his November 25 paper, Schwarzschild found an exact solution for the interior of the incompressible fluid sphere. This solution deals with the space inside the mass sphere.He started his paper from Einstein's November 25, 1915 field equations. Schwarzschild calculated the gravitational field of a homogeneous sphere of finite radius, consisting of incompressible fluid – a model of an ideal star. He opined that the addition of incompressible fluid was necessary because according to the theory of relativity, gravitation depends not only on mass but also on energy (Schwarzschild 1916b, 424).

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The Einstein tensor GPQ (ܴఓఔ െ ݃ఓఔ ) vanishes where there is no matter ଶ (outside an incompressible fluid sphere). The components of the energymomentum tensor of an incompressible fluid inside the sphere are determined in terms of the pressure p and density U of the fluid. Considering a state of equilibrium for the incompressible fluid sphere, Schwarzschild wrote the line-element for the region inside the sphere initially at rest (before starting to collapse). He also wrote an expression for the pressure of the (star) sphere inside this region, determined by the constant density U0 of the sphere of fluid, and related to the radius vector r (Schwarzschild 1916b, 424, 430-431). In his new investigation, Schwarzschild then showed that his line element (Schwarzschild 1916b, 431): ߙ ݀‫ ݎ‬ଶ ଶ ଶ ଶ ଶ ݀‫ ݏ‬ଶ ൌ ቀͳ െ ቁ ݀‫ ݐ‬ଶ െ ߙ െ ‫ ݎ‬ሺ݀ߴ ൅ •‹ ߴ݀߮ ሻ ‫ݎ‬ ͳെ ‫ݎ‬ is valid for the exterior of an incompressible fluid sphere, with regard to the space outside the sphere where: r is related to the radius vector of the sphere and the density U of the fluid; andDis related to the constant density U0 of the sphere of fluid r3ext = r3int + U.and also:D3 =U for a mass point, for the outside of the sphere is dependent on the density of the sphere. Finally, Schwarzschild noted that as the velocity of collapse of the sphere increases, at constant mass and growing density, there is transition to a smaller-than-before radius of the sphere, and the sphere releases radiant energy. At the centre of the sphere (rint = 0) the velocity of light and the density become infinite. Schwarzschild calculated that when the velocity of fall reaches 8/9 times the velocity of light, this is the upper limit for the given concentration. Beyond this limit of concentration, a sphere of incompressible fluid collapses and cannot exist (Schwarzschild 1916b, 433-434). Schwarzschild calculated the pressure of the sphere measured from within a collapsing sphere to its surface and the pressure of the sphere measured from outside the collapsing sphere. He concluded that the line-element for the space outside the sphere and the line element for the space inside the sphere coincide at the boundary of the collapsing sphere. This happens at

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the Schwarzschild limit, when the radius is a little greater than r = 2GM. Beyond the Schwarzschild limit (r = 2GM and r < 2GM), the pressure and the density become infinite at the centre of the sphere. Schwarzschild thought that he had thus set a limit, the Schwarzschild limit, beyond which the incompressible fluid sphere could not exist. Schwarzschild ended his paper by saying that for an external observer, as follows from his equations, a sphere of a gravitational mass cannot have a radius, measured from outside, whose numerical value is less than r =2GM. If it does have a radius r 2GM, or more precisely, a size which is bigger than the Schwarzschild limit. Schwarzschild and Einstein thought that it was meanigless to speak of what occurs beyond the Schwarzschild Singularity, because what occurs beyond this area has no physical meaning. Therefore, when we speak about the Schwarzschild Singularity we mean r = 2GM, and not r = 0. Outside the sphere the value of r is always r > 2GM, whereas the singularity r = 0 is inside r < 2GM and remains unreachable. Schwarzschild thought he had managed to solve the problem with the singularity r = 0 because beyond the limit he had set, (r < 2GM), the density and pressure were calculated to be infinite at the centre of the sphere (Schwarzschild 1916b, 434). Schwarzschild presented these results to Einstein on February 6, 1916, and attached to the letter a paper containing his new investigation, "On the Gravitational Field of a Sphere of Incompressible Fluid, According to Einstein's Theory" (Schwarzschild to Einstein, 6 February 1916, CPAE 8, Doc. 188; Schwarzschild 1916b). Einstein presented this paper to the Prussian Academy on February 24, 1916. In March 1916, Einstein submitted to the Annalen der Physik his review article on the general theory of relativity, "The Foundation of the General Theory of Relativity". The paper was published two months later, in May 1916. The 1916 review article was written after Schwarzschild had found the complete exact exterior and interior solutions to Einstein's November 18, and November 25, 1915 field equations. Einstein preferred in his 1916 paper to write his November 18, 1915 approximate solution upon the Schwarzschild exact exterior solution with the mathematical singularity that occurs at its origin (Einstein 1916, 789, 801).

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In 1920 Eddington, commenting on the Schwarzschild singularity, said that we start with a large value of r. By-and-by we descend towards the point where r = 2GM. But by its definition, from there onwards we arrive at a singularity, so that, however large the measured interval may be, the change in the radius is zero. Eddington continued that we can go on shifting the measuring-rod through its own length time after time, but the change in radius is still zero; that is to say, we do not reduce r. There is a magic circle which no measurement can bring us inside r = 2GM. It would not be unnatural should we picture something obstructing our closer approach to the singularity (Eddington 1920, 98). In 1922 Einstein was much troubled by the Schwarzschild singularity. The astronomer-physicist, Charles Nordmann, described lectures and three discussion sessions given by Einstein at the Collège de France during his visit to Paris in spring 1922. The second discussion session of Wednesday, April 5, 1922, opened with special relativity and was followed by questions concerning general relativity that were raised. Jacques Hadamard, a celestial mechanics professor at the Collège de France, opened the discussion session with a question relating to what Einstein would soon call the "Hadamard catastrophe" (Nordmann 1922a, 155-156). Hadamard asked Einstein about the Schwarzschild solution and its practical relevance for astronomy; he was much concerned with the singularity in Schwarzschild's solution. He posed before Einstein the query: if the radius term in Schwarzschild's solution becomes zero, that is to say, what if this term becomes infinite (singular), then the Schwarzschild formula no longer makes sense, or at least one could ask, what is its physical meaning? Hadamard was intrigued by the singularity at r = 0 in the Schwarzschild solution (the quantity that becomes infinite). He wondered what would actually happen in reality if mathematically this singularity could really become infinite in our world. He asked Einstein: Could this practically and physically happen in our world? Though it may not happen in our solar system but might certainly be possible in the universe, for instance, a star could become infinitely more massive than our Sun. According to Nordmann's account, this question embarrassed Einstein. He said that if the radius term could really become zero or infinite (be singular) somewhere in the universe, then it would be an unimaginable disaster for his general theory of relativity.

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Einstein said that it would be very difficult to say a priori what would happen physically under such circumstances because in that case r = 0, and the Schwarzschild formula ceases to apply. If this Schwarzschild singularity could actually be applicable in reality, Einstein considered that it would be a catastrophe, and jokingly called it the "Hadamard catastrophe". Hence, according to Einstein, the Schwarzschild singularity at r = 0 characterized a catastrophic region. Einstein did not think that the "Hadamard catastrophe" was feasible and he did not want to think about the possible physical effects of such an event. On December 7, 1921 a year before coming to Paris, Einstein told Paul Painlevé that the quantity r in itself has no physical meaning and that coordinates do not have any physical significance. They do not represent the result of a measurement but only the conclusions, accessible by the elimination of coordinates that may negate the claim for an objective significance. In 1922 Einstein informed Nordmann that he wanted to circumvent the misfortune that the Hadamard catastrophe represented for the theory. Nordmann mentioned that although at that time (in 1922) there were some known stars much larger than the Sun, astronomers were aware that even several of the largest stars, whose masses they were able to determine observationally, were never sufficiently massive; each individual mass was not sufficiently greater than the mass of the Sun in order to cause a catastrophe. Nordmann explained that according to Eddington's approach to the problem of stellar structure, when the mass of stars tended to increase gradually the internal temperature of the stars increased greatly. As a result, the radiation produced tended to throw outwards any new addition of matter in order to balance the gravitational attraction effect. Nordmann, therefore, concluded that it would be in the very nature of things that an insurmountable limit would be reached in the mass increase of a star. A star could never grow that much greater than the mass of our Sun. Therefore, the very physics of things would prevent the "Hadamard catastrophe" from ever happening because sufficiently massive stars could not be produced. In the following session of April 7, Einstein brought up the result of a calculation he had made concerning this topic: He came back with a calculation concerning the mass of a star, the volume of which increases

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indefinitely but its density stays constant. Einstein demonstrated that the pressure at the centre of the mass becomes infinite, well before it has grown massive enough to have reached the Schwarzschild radius (r =2GM) and consequently the "Hadamard catastrophe" conditions. When these conditions are met, clocks run at zero speed, nothing goes on, it would be like death; and therefore there could be no possible change capable of bringing the star towards the "Hadamard catastrophe". Einstein's above mentioned example is exactly Schwarzschild's 1916 sphere of a finite radius consisting of incompressible fluid, which is determined by its constant density and varying volume. Recall that Schwarzschild thought he had managed to solve the singularity r = 0 problem by setting a limit (the Schwarzschild limit) to the radius (a little bigger than r = 2GM) and claiming that beyond this limit the density and pressure were calculated to be infinite at the centre of the sphere (Schwarzschild 1916b, 434). Hadamard seemed to be satisfied with this answer and believed it would prevent the catastrophe from ever happening, thus changing it from a possibility to an impossibility. Between 1935 and 1936, Einstein was again occupied with the Schwarzschild solution and the singularity within it while working in Princeton on the unified field theory and, with his assistant Nathan Rosen, on the theory of the Einstein-Rosen bridges. He informed Leopold Infeld (who replaced Rosen) about his theory of "bridges" and the difficulties he and his collaborator Rosen had encountered while developing that theory during a whole year of tedious work (Infeld 1941, 259). When Infeld first met Einstein in Princeton in 1936, Einstein asked him: "Do you speak German?" and when he replied: "Yes", Einstein said: "Perhaps I can tell you on what I am working", and then proceeded to explain the basic ideas underlying his unified field theory. Infeld continued, "Quietly he took a piece of chalk, went to the blackboard and started to deliver a perfect lecture". And later, "Keeping a dead pipe in his mouth, he formed his sentences perfectly". Infeld listened carefully and understood everything. Einstein explained about basic problems. He had always been interested in basic problems. He once told Infeld, "I am really more a philosopher than a physicist" (Infeld 1941, 258). Infeld believed he could express some of Einstein's ideas on unified field theory in simple

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language (Infeld 1941, 254-259). These are the basic philosophical ideas underlying Einstein's and Rosen's 1936 "bridges" paper. Einstein explained to Infeld that there are two fundamental concepts in the development of physics: field and matter. The mechanical point of view is based upon the belief that we can explain all phenomena by assuming particles and simple forces acting on them. The forces acting on particles depend only upon distances. This mechanical point of view reduces complicated phenomena to a simpler model. This serves as a means to explain a mechanical picture from which the phenomenon could be deduced. In this sense, the nineteenth century physicist considered the universe as a great and complicated mechanical system. However, the simple concepts of particles and forces were insufficient to explain all phenomena of nature, and a new idea was invoked: the basic concept of our description is that of a field, which characterizes changes in space spread in time through all space. If we see an object, a mechanistic physicist will claim the object consists of small particles held together by forces. The field physicist would say we can look upon the object as a portion of space where the field is very intense (where the energy is especially dense). According to the mechanistic physicist the object is localised at a point in space (a particle). The field physicist claims the field is everywhere, but it diminishes outside this portion so rapidly that our senses are aware of it only in that portion of space (matter is understood as a concentration of the field). Einstein explained that basically three philosophical approaches to essentially physical problems were possible: the mechanistic (to reduce everything to particles and forces acting upon them and depending only on the distances); the field view (to reduce everything to field concepts concerning continuous changes in time and space); and the dualistic view (to assume the existence of both matter and field). The past generation believed in the first possibility. None of the present generation believes in it anymore. The majority of physicists accept the third (quantum) view, assuming both matter and field. Einstein himself followed the second field view: he strived to form a consistent picture based only on field concepts, to develop a pure field theory and to create what appears as matter out of field. Einstein was almost alone in his belief that he could construct a representation of matter (elementary particles) from the equations of the gravitational field.

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Newton's theory of gravitation treats the problem of gravitation in terms of the mechanical viewpoint. Moreover, his theory caused this view to spread over all physics. An aspect that Einstein worked on in the 1930s was the following: In those years Einstein saw in his general theory of relativity a system that fitted the gravitational problem into the field frame. By formulating his equations for the gravitation field he did for gravitation theory what Faraday and Maxwell did for the theory of electricity. Einstein was continually seeking for a unified field theory that would exclude singularities in the field, and used the methods of general relativity to account for atomic and electrical phenomena. Einstein believed that singularities had to be totally excluded in a final unified field theory. He thought it was unreasonable to introduce into a unified field theory points and world lines for which the field equations did not hold (Einstein 1956, 164). However, the Schwarzschild solution for the spherically symmetric static gravitational field does have a singularity. In the Einstein-Rosen bridges paper of 1935, Einstein and his young assistant Nathan Rosen claimed that writers had suggested that material particles might be considered to be singularities of the field. At that time in 1935, Einstein and Rosen (that is to say Einstein) could definitely not accept this point of view because a singularity actually nullifies the laws of general relativity. Einstein and Rosen thought they had found a way, in the context of the Einstein-Rosen bridges (discussed below), to exclude singularities from the theory, and at the same time material particles would not have to be represented as singularities of the field. This was simply an ideal situation, killing two birds with one stone: the field equations of general relativity no longer possessed the solution containing the Schwarzschild singularity, and the unwanted assumption of particles represented as singularities of the gravitational field was eliminated. In 1909 and 1917, as well as 1927 with his assistant Jacob Grommer Einstein had worked on the suggestion that material particles might be considered singularities of the field. Already in 1909 Einstein maintained that it was not out of the question that the entire energy of the electromagnetic field might be viewed as localised in singularity points (light quanta), exactly like in the old theory of action at a distance (Einstein 1909, 499). In 1917 Einstein considered the question whether an electron is to be treated as a singular point, and whether true singularities

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are actually admissible as being of great interest in the physical description (Einstein to Hermann Weyl, January 3, CPAE 8, Doc. 285). Einstein already changed his mind in his 1922 book The Meaning of Relativity and provided a strong argument against a representation of matter or energy as singularities, within the context of Maxwell's electromagnetic theory. He said that an attempt had been made to consider the charged particles as proper singularities, but in his opinion this meant abandoning a real understanding of the structure of matter (Einstein 1922b, 53). In 1927, in his paper with his assistant Grommer, Einstein admitted that all his attempts of recent years to explain the elementary particles of matter by means of continuous fields had failed, and he suspected that he lacked the correct concept of material particles. He also understood that his gravitational equations for empty space, or vacuum field equations, did not have solutions that were free of singularities. Because of this and all his failed attempts at explaining the elementary particles of matter by means of fields, he had been forced to consider elementary particles (matter) as singularity points or world lines in the field. On his part, Einstein was led to a different approach in which there were no field variables other than in the gravitational and electromagnetic fields (with the possible acceptance of the cosmological term). Einstein assumed that singular world lines existed, and that the law of motion of the singularities is completely determined by the field equations and the character of the singularities, without the necessity of additional assumptions (Einstein and Grommer 1927, 4; Pais 1982, 290). In the Einstein-Rosen bridges paper of 1935, Einstein negated the possibility that particles were represented as singularities of the gravitational field because of his polemic with Ludwig Silberstein. Einstein said that Silberstein confirmed in his presence that one could not accept the possibility that material particles might be considered singularities of the gPQ field (Einstein and Rosen 1935, 73). In December 1933, Silberstein thought he had demonstrated that general relativity was problematic. He constructed, for the vacuum field equations for the two-body problem, an exact static solution with two singularity points that lie on the line connecting these two points. The singularities were located at the positions of the mass centres of the two material bodies.

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Silberstein concluded that this solution was inadmissible physically and contradicted experience. According to his equations the two bodies in his solution were at rest and were not accelerated towards each other; these were non-allowed results and therefore Silberstein thought that Einstein's field equations should be modified together with his general theory of relativity. Before submitting his results as a paper to the Physical Review, Silberstein communicated them to Einstein (EA 21-074 in Havas 1993, 106). This prompted Einstein's remark, in his paper with Rosen in 1935, that matter particles could not be represented as singularities in the field (Einstein and Rosen 1935, 73). However, Einstein quickly changed his mind after the "bridges" paper. A year later, Einstein relegated this strict demand and was willing to admit singularities of this kind, and accepted singularities as representing material particles. In December 1935, Einstein objected to Silberstein's results, and found a lacuna in his solution claiming it had an additional singularity on the line connecting the two singularities (EA 21-076 in Havas 1993, 107).7 Indeed, Silberstein's solution contained singularities to which Einstein objected in a final field theory. Since Silberstein's solution was found to be erroneous, Einstein returned to his previous idea that material particles could be considered as singularities of the field in general relativity. Afterwards, the controversy between Silberstein and Einstein continued with more strident tones (Einstein and Rosen 1936). In March 1936, Einstein wrote to Silberstein about his advice to him to withdraw publication. In addition, he said that the newspaper contained the idiotic claim that he had revised his general theory of relativity because of Silberstein's earlier letters to him. By his efforts Silberstein had made it necessary for Einstein (and Rosen) to correct his errors publicly in The Physical Review letter. In response Silberstein wrote to "Sweet Mr. Einstein", and the Silberstein solution was never mentioned again (Einstein and Rosen 1936; EA 21-087, 21-088 in Havas 1993, 110-113).

  7

Einstein had given the very same explanation thirteen years earlier, in 1922. In 1921, Erich Trefftz constructed an exact static spherically symmetric solution for Einstein's vacuum field equations with the cosmological term (see Chapter 3, Section 4). The Trefftz metric represents a model for the two body problem. In a comment on Trefftz's paper, Einstein identified a problem with the Trefftz line element. In 1922 Einstein demonstrated that the Trefftz exact solution contained a true singularity in the empty space between the two bodies (Einstein 1922d, 448).

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In 1936-1937 Einstein demonstrated that exact gravitational waves could not be described without introducing singularities into the components of the metric describing the wave. One could, however, deal with these singularities by a change of coordinates. The so-called Einstein-Rosen metric could be transformed from space-time suitable for representing plane gravitational waves to cylindrical coordinates. The singularity was then located at the origin of the cylindrical axis. Hence, the material source of the cylindrical waves at the origin represented the singularity (see Chapter 3, Section 1). Starting in 1938, Einstein and his assistants, Infeld and Banesh Hoffmann, elaborated the above idea (matter representing singularity). Einstein considered gravitational equations in which some specific energymomentum tensor for matter had been assumed. When he started to think about unified field theories he realised, however, that such an energymomentum tensor must be regarded as a temporary device for representing the structure of matter, and its entry into the field equations makes it impossible to determine how far the results obtained are independent of the particular assumption made concerning the constitution of matter. Einstein realised that the only equations of gravitation, which follow without ambiguity from the fundamental assumptions of the general theory of relativity, are the field equations for empty space (the vacuum field equations). He thought it was important to know whether these equations alone are capable of determining the motion of bodies. Einstein wanted to show that the gravitational field equations for empty space are in fact sufficient to determine the motion of matter represented as point singularities of the field. He explained that representing matter by singularities was a simplifying assumption, and he still wished to represent matter in terms of a field theory in which singularities are excluded. The representation of matter by means of singularities did not enable the field equations to fix the sign of mass. Einstein thus had to fix it by convention and claim that the interaction between two bodies is always an attraction and not a repulsion. He thought that the field equations could lead to a positive mass and not to a negative one only in a theory that provides a representation of matter free from singularities. In this respect he referred to his Einstein-Rosen bridges paper from 1935 (Einstein, Infeld and Hoffmann 1938, 65-66; Einstein and Infeld 1949, 209-210; Infeld 1950, 1075; Einstein and Rosen 1935).

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Thus Einstein was well aware of the fact that he had tried the impossible, to have his cake and eat it too, because his wished-for target was a unified field theory the equations of which had solutions free from singularities but, at the same time, this theory resorted to the good old trick of getting rid of the singularities by representing matter by singularities of the field. The singularities haunted Einstein; the first singularity that seemed to trouble him was the Schwarzschild singularity. This is the methodological background necessary to understand the 1935 Einstein-Rosen bridges theory. Einstein and Rosen (that is Einstein) were trying to permanently dismiss the Schwarzschild singularity and adhere to the fundamental principle that singularities of the field are to be excluded. Einstein explained that one of the imperfections of the general theory of relativity was that as a field theory it was not complete in the following sense: it represented the motion of particles by the geodesic equation: ݀ ଶ ‫ݔ‬஝ ݀‫ݔ‬ఙ ݀‫ݔ‬ఔఛ ஝ ൌ ෍ Ȟ஢த Ǥ ݀‫ ݏ‬ଶ ݀‫ݏ݀ ݏ‬ ఙఛ

A mass point moves on a geodesic line under the influence of a gravitational field. However, a complete field theory implements only fields and not the concepts of particle and motion. These must not exist independently of the field but must be treated as part of it. That is the reason why material particles could not be considered singularities of the field. Einstein wanted to demonstrate that the field equations for empty space are sufficient to determine the motion of mass points. In 1935 Einstein could not accept singularities as representing material particles because these are not primary entities. Einstein attempted to present a satisfactory treatment that accomplishes a unification of gravitation and electromagnetism. For this unification, or as he called it the combined problem, he needed a description of a particle without singularity. In 1935, Einstein and Rosen showed that it was possible to do this in a natural way and they proposed the following new solution (Einstein and Rosen 1935, 76). The gravitational field is entirely described by the metric tensor gPQ. In the ߤߥ ஢ components of the gravitational field Ȟஜ஝ ൌ െ ቄ ቅ there appears a ߪ

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determinant of gPQ, g = | gPQ |, which should nowhere vanish. In order for g not to vanish one has to replace the vacuum field equations: ஢ ߲Ȟஜ஝ ஢ ஒ ൅ Ȟஜఉ Ȟఔ஑ ൌ  ஜ஝ ൌ Ͳǡ ߲‫ݔ‬ఈ

by: ‰ ଶ  ஜ஝ ൌ ͲǤ These field equations have the centrally symmetric solution presented by Schwarzschild: ݀‫ ݏ‬ଶ ൌ ൬ͳ െ

ʹ݉ ݀‫ ݎ‬ଶ ൰ ݀‫ ݐ‬ଶ െ െ ‫ ݎ‬ଶ ሺ݀ߴ ଶ ൅ •‹ଶ ߴ݀߮ ଶ ሻǤ ʹ݉ ‫ݎ‬ ͳെ ‫ݎ‬

At r = 2m we have the so-called Schwarzschild singularity. If one introduces in place of this r a new variable according to the equation: u2 = r – 2m, one then obtains for ds2 the expression: ݀‫ ݏ‬ଶ ൌ

‫ݑ‬ଶ ݀‫ ݐ‬ଶ െ Ͷሺ‫ݑ‬ଶ ൅ ʹ݉ሻ݀‫ݑ‬ଶ ‫ݑ‬ଶ ൅ ʹ݉ െ ሺ‫ݑ‬ଶ ൅ ʹ݉ሻଶ ሺ݀ߴ ଶ ൅ •‹ଶ ߴ݀߮ ଶ ሻǤ

The Schwarzschild solution thus becomes a regular solution, free from singularities for all finite points. The above solution is interpreted in the following way: The fourdimensional space is described mathematically by two congruent sheets corresponding to u > 0 and u < 0, that are joined by a hyperplane r = 2m or u = 0 in which the metric tensor gPQ vanishes and the g, the determinant of the gPQ, vanishes. Einstein and Rosen called such a connection between the two sheets a "bridge" (known today as the Einstein-Rosen bridge). The bridge, which

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connects the two sheets, is spatially finite and characterizes or describes the presence of an electrically neutral elementary particle. With this conception Einstein thought he could represent an elementary particle by using only the field equations. The two sheets can also be interpreted as each corresponding to the same physical space (Einstein 1936a, 379-380). Einstein explains that his new field theory enables one to understand the atomistic character of matter as well as the fact that there can be no particles of negative mass. If one had started from the Schwarzschild solution with negative mass m one would have then been unable to make the solution regular, free from singularities, by introducing a new variable u instead of r. Hence, no Einstein-Rosen bridge that corresponds to a particle of negative mass is possible (Einstein and Rosen 1935, 75). Einstein and Rosen added an electromagnetic field to their solution, so that it could also represent a charged particle. According to Einstein general relativity was not complete as a field theory: the law of motion of a particle (described without singularity) is given by the equation of the geodesic. A complete field theory considers only fields. The particles do not exist independently of the field. If several particles are present, this case corresponds to several bridges. Einstein did not succeed in finding a multi-bridge solution of his modified field equations, and could not describe the whole field without introducing singularities. And indeed, Einstein soon decided to abandon the bridge as a non-singular particle model. In 1962, Robert Fuller and John Archibald Wheeler were occupied with causality violation of the Einstein-Rosen bridge, or a "wormhole" (as Wheeler named it), with the possibility of sending signals through such a shortened route in space-time (Fuller and Wheeler 1962, 919). They asked: Can a disturbance start from a point on the lower space and get through to a point on the upper space? It might appear that the answer is plainly yes, because the geometry of the Schwarzschild throat is perfectly regular and light propagates there just as it does anywhere else. Consider a light signal, travelling through the throat of the wormhole, i.e. the connection between the two mouths of the wormhole, which could be a short distance, and arriving at the other side. Fuller and Wheeler were troubled by the apparent possibility that the test particle, or the ray of light, could pass from one point in space to another point in space, distanced perhaps extremely far away, in a negligible interval of time. Such rapid communication from place A to another place B (a particle or a light ray,

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passing through such an Einstein-Rosen bridge) violates elementary principles of causality, according to which a light signal cannot exceed the speed of light. Such a signaling process, viewed from an inertial system moving at a sufficiently great but still allowable speed (v 0, to which Einstein objected, but a few months later, de Sitter read Einstein's 1931 paper and changed his mind. He adopted Einstein's above-mentioned new line element, and on account of the expansion of the universe he studied the non-static Einstein solution of the field equations with constant density. De Sitter explained that observations made estimations about the distribution of nebulae (galaxies) in the universe. It appeared that they were distributed evenly at least in our neighbourhood. They were also observed to be roughly of the same size. This of course did not correspond to de Sitter's empty universe (the density of matter in space is equal to zero and the density of energy [the cosmological constant] influences the curvature of space-time). This, however, brought astronomers of the day to estimate the density of matter in space. They estimated that the density was constant. There was only one possible solution possessing constant density, the Einstein universe. However, the Einstein world was static and de Sitter searched for a non-static world with both constant density and expansion, which could explain Hubble's observations that the spectra of the celestial bodies are displaced to the red corresponding to receding velocity that is increasing with the distance and is proportional to it (de Sitter 1931b, 308, 313). In a paper from August 7, 1931, de Sitter wrote the following (De Sitter 1931a, 141). The non-static solutions of the field equations of the general theory of relativity, of which the line element is: ݀‫ ݏ‬ଶ ൌ െܴଶ ݀ߪ ଶ ൅ ܿ ଶ ‫ ݐ‬ଶ ǡ R being a function of t alone and dV2 being the line-element of a threedimensional space of constant curvature with unit radius, have been investigated by Friedmann in 1922 and independently by Lemaître in 1927, and have attracted general attention during the last year or so. Einstein in his 1931 paper expressed his preference for the particular solution of this kind corresponding to the value O = 0 of the cosmological constant. "This solution belongs to a family of solutions which were not included in my discussion in B. A. N 193" (De Sitter, 1930a). In the 1930 paper (B. A. N number 193) de Sitter indeed had not studied the non-static solution, but de Sitter's 1931 research led to the Einstein-de Sitter universe. A few months later in 1932, in a joint paper and in what seemed as a final end to the good old competing static models, Einstein

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and de Sitter presented the Einstein-de Sitter suggestion following Einstein's lead without the O-term. They wrote that historically the cosmological term containing the cosmological constant O was introduced into the field equations in order to enable them to account theoretically for the existence of a finite mean density in a static universe. It now appeared to them that in the dynamic case this end could be reached without the introduction of the cosmological constant O (Einstein and de Sitter 1932, 213). De Sitter received a copy of Otto Heckmann's 1931 publication and then, when both he and Einstein were visiting Mount Wilson Observatory on January 25, 1932, they wrote the 1932 joint paper and mentioned Heckmann. They said that in a recent note in the Göttinger Nachrichten Heckmann has pointed out that the non-static solutions of the field equations of the general theory of relativity with constant density do not necessarily imply a positive curvature (k) of three-dimensional space, but that this curvature may also be negative or zero. Einstein and de Sitter mentioned that there was no direct observational evidence for the curvature, the only directly observed data being the mean density and the expansion of the universe, which latter proved that the actual universe corresponds to the non-static case. They therefore concluded that it was clear that from the direct data of observation they could derive neither the sign nor the value of the curvature. Einstein and de Sitter asked whether it was possible to represent the observed facts without introducing a curvature at all (Einstein and de Sitter, 1932, 213; Heckmann 1931; Kerzberg 1989, 348). If the field equations were those given by Einstein's theory, the FriedmanLemaître cosmological models resulted; the simplest of these was the Einstein-de Sitter model (McCrea 1985, 34): the model was the lineelement given by Einstein in 1931 without the introduction of O and without introducing a curvature at all. If we suppose the curvature to be zero we neglect the pressure. Einstein and de Sitter derived the coefficient ଵ of expansion ݄ଶ ൌ ߢߩwhich depends on the measured redshifts. And ଷ

the numerical value of U happens to coincide exactly with the upper limit for the density adopted by de Sitter in his August 1931 paper (Einstein and de Sitter 1932, 213-214; De Sitter 1931a, 142). In 1935 Robertson published three major papers under the title "Kinematics and World-Structure". In the first paper, Robertson

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demonstrated that the general theory of relativity finds that "the cosmological speculations" may, on assigning appropriate coordinates and time W to each event, "be based on any Riemannian map whose invariant metric ds2 is of the form" (the Einstein-de Sitter metric): ݀‫ ݏ‬ଶ ൌ ݀߬ ଶ െ

ܴሺ߬ሻଶ ଶ ݀‫ ݑ‬ǡ ܿଶ

where: ݀‫ݑ‬ଶ ൌ ൫݀‫ ݎ‬ଶ ൅ ‫ ݎ‬ଶ ሺ݀ߠ ଶ ൅ •‹ଶ ߠ ݀߮ ଶ ሻ൯ǡ and R(W)/c is an arbitrary function of W. Generally, du2 defines any threespace of constant Riemannian curvature k (which may, without loss of generality, be restricted to the values k = í1, 0 +1). We thus have here a wide range of possibilities, corresponding to the three possible values of k and the choice of the arbitrary function R(W) – although the condition adopted above requires that R(0) = 0 (Robertson 1935, 285-286). Robertson's 1929 paper "On the Foundations of Relativistic Cosmology" and Robertson's 1935 paper, together with Arthur Geoffrey Walker's 1937 paper "On Milne's Theory of World-Structure", gave rise to what is now known as the Friedmann-Robertson-Walker space-time metric (Walker 1937; Hitchin 2006, 417). In 1929 Robertson found the following line element (in spherical coordinates r, T, M) (Robertson 1929, 826): ݀‫ ݏ‬ଶ ൌ ݀‫ ݐ‬ଶ െ ܴሺ‫ݐ‬ሻଶ ൭

݀‫ ݎ‬ଶ ൅ ‫ ݎ‬ଶ ሺ݀ߠ ଶ ൅ •‹ଶ ߠ ݀߮ ଶ ሻ൱ǡ ͳ െ ݇‫ ݎ‬ଶ

where k = 1/R02 and the curvature radius is R. The above metric is "an invariant Riemannian metric of precisely the form and generality of that on which the general relativistic theory of cosmology is based, and in terms of which all given elements can be interpreted in the same way as in the relativistic theory" (Robertson 1935, 284). We shorten the term between brackets as: ݀ȳଶ ൌ ሺ݀ߠ ଶ ൅ •‹ଶ ߠ ݀߮ ଶ ሻǡ

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and rewrite the above line element in the following form: ݀‫ ݏ‬ଶ ൌ ݀‫ ݐ‬ଶ െ ܴሺ‫ݐ‬ሻଶ ቆ

݀‫ ݎ‬ଶ ൅ ‫ ݎ‬ଶ ݀ȳଶ ቇǡ ͳ െ ݇‫ ݎ‬ଶ

and put: ݀ȭଶ ൌ

݀‫ ݎ‬ଶ ൅ ‫ ݎ‬ଶ ݀ȳଶ ǡ ͳ െ ݇‫ ݎ‬ଶ

from which it follows: ݀‫ ݏ‬ଶ ൌ ݀‫ ݐ‬ଶ െ ܴሺ‫ݐ‬ሻଶ ݀ȭଶ Ǥ (The velocity of light c is set to 1).

13. Einstein's Reaction to Lemaître's Big Bang Model At the time, on the other side of the Atlantic, Eddington was still intensely interested in the cosmological constant O > 0. According to Eddington, the effect of the cosmological constant was to introduce cosmical repulsion (provided O> 0) into the universe, and this could be looked upon as the cause of the expansion of the universe. The outcome of observations was therefore apparently most impressively to vindicate Einstein's introduction of the cosmological constant. Eddington's universe has no "beginning"; it arises, as already stated, from the perturbed Einstein static universe with no beginning; the Einstein static universe is unstable against a slight increase in the radius of curvature which leads to unlimited expansion. The rate of expansion is at first very slow, but then the model tends asymptotically to the de Sitter empty universe. Eddington submitted the following paper (received on August 11, 1931), "On the Value of the Cosmical Constant", to The Proceedings of the Royal Society of London. He began his paper by saying that the cosmological term Orepresents a scattering force which tends to make all very remote bodies recede from one another; this phenomenon is the basis of the theories of de Sitter and Lemaître concerning expansion of the universe. If the observed recession of the spiral nebulae (i.e. galaxies) is a manifestation of this effect, the value of the cosmological constant Ocan be found from the astronomical observations. Eddington found the value for the cosmological constant: O = 9.79 . 10-55, which gave a speed of

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recession of the spiral nebulae of 528 km. per sec. per megaparsec. The observed speed according to Hubble was 465 km. per sec. per megaparsec (Eddington 1931a, 605, 615). On March 21, 1931 Nature published a paper by Eddington, "The End of the World: from the Standpoint of Mathematical Physics", in which he repeated his view that the universe had no beginning. In May 1931 Lemaître responded to Eddington's article and proposed the first modern history of the universe having a clear beginning. Lemaître suggested a model with point-source-creation in which the cosmological constant played an important role; the cosmological constant and the initial velocity of expansion were adjusted so that there was a stage in the expansion approximating to the Einstein static universe. Later stages in the expansion were then the same as in the Lemaître-Eddington model (Hoyle 1948, 375). However, because of Einstein's authority and since he withdrew the cosmological constant when dealing with dynamical models, the use of the cosmological constant was generally out of favour; hence, at the time Lemaître's ideas were not given much attention. Since 1933 cosmologists have regarded his work as the beginning of big-bang cosmology. Taking a cosmological constant O > 0, Lemaître found an expansion factor satisfying his equations for the relativistic homogeneous isotropic expanding cosmological model. The model had a singularity at time zero followed by rapid expansion, this being decelerated by self-gravitation leading to near-stagnation in the vicinity of the Einstein static state, independent of time, if the value of the cosmological constant O was suitably chosen, until the onset of accelerated expansion under cosmic repulsion. Lemaître pictured the very early universe as a "primeval atom", a cosmic atomic nucleus, with the big bang as its spontaneous radioactive decay. Thus the very early universe would have been dominated by high-energy particles producing a homogenous early universe. Cosmic rays were inferred to be the most energetic relict particles from the decay, so that they constituted background radiation for the model. Thus the very early universe would have been dominated by high-energy particles producing a homogeneous early universe (Lemaître 1931c; Eddington 1931b; Luminet 2011; McCrea 1988, 55-56).

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It was probably the second time that Lemaître felt that his ideas had been rejected because of a misunderstanding. André Deprit, a student of Lemaître, later noted that for the past fifteen years, since his memorable address to the British Association in 1931, Lemaître had been put on the defensive. The Big Bang (primeval atom) theory "had been held in suspicion by most astronomers, not the least by Einstein, if only for the reason that it was proposed by a Catholic priest and seconded by a devout Quaker [Eddington], hence highly suspect of concordism" (Deprit 1983, 387). Actually, Einstein and Eddington both objected to the Schwarzschild singularity (see Chapter 2, Section 14), and since Lemaître's primeval atom model had a singularity at time zero followed by rapid expansion, they also objected to this model. Indeed Lemaître tried many times to convince Eddington of the Big Bang Theory but Eddington remained firm in his objection to the Big Bang. People indeed had difficulties in accepting the Big Bang theory; in January 1933, Lemaître and Einstein came to California (California Institute of Technology) for a series of seminars, where they met Hubble and others and discussed recent developments in cosmology. Lemaître gave seminars on the expanding universe and on the Big Bang (primeval atom) theory. Journalists reported about the lectures. One of them, who was a New York Times reporter who had interviewed Lemaître, wrote that Lemaître told his audience: "There is no conflict between religion and science".15 In California Einstein learned that Adolf Hitler had been appointed Chancellor of the German republic on January 30, 1933. In March Einstein and his wife Elsa left the United States and Einstein decided to sever his ties with Germany completely. Upon his arrival in Antwerp (Belgium), he vowed not to return to Germany. He went to the German Embassy in Brussels to surrender his passport and renounce his German citizenship; he also resigned from his positions at the Prussian Academy of Sciences and at the University of Berlin. The Nazi authorities demanded that the academy expel Einstein, and in an official communiqué the academy announced Einstein's resignation "without regret". Meanwhile, Lemaître and Théophile Ernest de Donder were fully immersed in organising a series of six seminars to be given by Einstein in

 

15 Aikman, Duncan (1933). "Lemaître Follows Two Paths to Truth", New York Times Sunday Magazine, February 19, 3. 

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Brussels.16 The organisation of the lectures took a great deal of effort because adequate security measures had to be provided. Einstein's wife asked for police protection, afraid that the Nazis might assassinate him. Einstein delivered lectures in French on May 3, 6 and 10. Before the session of May 13, 1933, at which de Donder presented his research, Einstein announced that the next seminar would be given by Lemaître, "who has interesting things to tell us". On May 17, Lemaître gave the seminar and was interrupted several times by Einstein talking to himself but in a loud voice and exclaiming that it was "very beautiful, very beautiful indeed" (Stern 1999, 153; Deprit 1983, 375-376).  Although Eddington tried to push on in Lemaître's direction, in 1933 de Sitter was already confused and perhaps following Einstein's lead reasoned that astronomical observations gave no means whatsoever to decide which of these possible solutions corresponded to the actual universe. He thought the choice must, as Eddington said, depend on aesthetic considerations (De Sitter 1933, 630). Einstein, however, had already made the aesthetic choice: he had dropped the cosmological constant. On January 23, 1932, The Brisbane Courier reported: Sir Arthur Eddington went on to explain that light can no longer go round the world in finite time. The circumference of the world is expanding, he said, and light is like a runner on an expanding track with the winning post receding faster than he can run. In the early days light and other radiation went round and found the world until it was absorbed. The merry-go-round lasted during the very early stages of expansion. But when the world had expanded to 1.003 times its original radius the bell rang for the last lap. Light waves then running will make just one more circuit. Those which started later will never get round ("The Expanding Universe." The Brisbane Courier, 19).

 

16 De Donder and Einstein corresponded from June to July 1916 (de Donder to Einstein and Einstein to de Donder, CPAE 8, Doc. 228, 230, 232, 234, 236). In the first letter to Einstein de Donder made a priority claim. De Donder wanted Einstein to acknowledge his priority regarding the derivation of the field equations from a variational principle. De Donder told Einstein that he was very pleased to know that Einstein's field equations are equivalent to de Donder's field equations (derived from a variational principle). This was especially the case because Einstein derived his field equations much later. Einstein replied to de Donder and spoke about their lack of agreement for the excellent reason, that he probably suffered qualms whenever he heard the words equivalent to (or nostrification).

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Eddington died in 1944 and left a finished draft of most of a book he had been writing, with brief notes of the intended contents of the rest. He clearly meant this work to supersede most of his previous writings on the foundations of physics. The trouble was that nobody could understand it! Some people could follow what Eddington claimed to have done, and some could follow his mathematics, but none could see how everything fitted together. However, such was Eddington's prestige as well as his success in deriving apparently from nowhere uncannily accurate values for all known fundamental constants of physics, that his colleagues thought he must surely be right, if only they could grasp his arguments. Hubble's own value of about 560 km s-1 Mpc-1 for the Hubble constant was available, and Eddington claimed to have derived a theoretical value of 572.36. In 1947 it was found that Eddington had made a mistake and had overlooked a factor in his calculations that multiplied his value by 4/9 (McCrea 1974, 66). After little more than two decades of cosmological work Eddington's colleagues thought he must be surely right about the fundamental constants of nature; but what about another constant, the cosmological constant? Lemaître contributed two articles marking Einstein's seventieth birthday. In July 1949 the Reviews of Modern Physics devoted an issue to a celebration of Einstein's seventieth birthday. It contained articles about Einstein's achievements by Einstein's friends and colleagues. Lemaître's contribution was a technical paper entitled, "Cosmological Application of Relativity", in which he explained in much detail the "Expanding Space – Friedmann's equation", and his 1931 new idea of cosmic rays. Lemaître then mentioned in a short passage the central issue that had long played an important role in Einstein's discussion with de Sitter. He wrote that the de Sitter singularity, like the Schwarzschild singularity, is an artificial singularity not of the field but of the coordinates introduced to describe this field (Lemaître 1949a, 361, 365-366). This mention was made many years after the 1917 polemic with de Sitter; doubtless the elderly Einstein had almost forgotten about it, until Lemaître possibly reminded him about some of the sweetest memories that were now perhaps too amusing to be consigned to oblivion. By doing so Lemaître, however, rubbed salt into Einstein's wounds, for recall that ten years earlier, in 1939, Einstein had objected to the Schwarzschild singularity. He had claimed that there was no way a Schwarzschild singularity could ever be possible and had said that the Schwarzschild singularity did not exist in physical reality and did not appear for the

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reason that matter could not be concentrated arbitrarily (Einstein 1939, 922, 936). In 1946 Paul Arthur Schilpp a professor of philosophy, dedicated a book to Einstein that was later published in 1949 to celebrate Einstein's seventieth Birthday. Schilpp had edited a series of books about great living philosophers, and he wanted to edit a book on Einstein as well. Each book was devoted to a single person and this one contained Einstein's specially written autobiography, followed by a series of essays by authorities evaluating and criticising his work. These essays were then answered by the philosopher himself. Lemaître dedicated a popular essay to the cosmological constant. With the aid of quotations from Eddington's 1933 book, The Expanding Universe (Eddington 1933), and Einstein's works, Lemaître tried to persuade the reader that the structure of Einstein's field equations quite naturally allowed for the presence of a second constant besides the gravitational one, the cosmological constant. Lemaître explained that the history of science provided many instances of discoveries which had been made for reasons which were no longer considered satisfactory. It could be that the discovery of the cosmological constant was such a case. The reasons which were no longer considered satisfactory were probably Einstein's static cosmological universe. Lemaître intended to demonstrate that there were other empirical reasons to maintain the cosmological constant in a dynamical universe (Lemaître 1949b, 442-443). As previously mentioned, Einstein himself wrote answers in Schilpp's book to the series of essays by scientists evaluating and criticising his work. In his reply to Lemaître, Einstein did not agree that when dealing with an expanding universe, a dynamical universe, there were any reasons to maintain the cosmological constant. He explained that models of the expanding universe could be obtained without any mention of the cosmological constant, and he proposed that the cosmological term be again dropped from the theory of general relativity. In his reply to the article by Lemaître Einstein specifically explained why he had dropped the cosmological constant in the dynamical case. He commented that, Lemaître's arguments did not appear to him to be sufficiently convincing in view of the present state of their knowledge. The introduction of the cosmological constant implied a considerable renunciation of the logical simplicity of his 1915-1916 general theory of relativity, a renunciation which appeared to him unavoidable in 1917 only so long as he had no reason to doubt the essentially static nature of space. However, after

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Hubble's discovery of the expansion of the universe, and since Friedmann's discovery that the field equations with no cosmological constant involve the possibility of the existence of an average (positive) density of matter in an expanding universe, it appeared to Einstein that the introduction of such a constant, from the theoretical standpoint, was unjustified (Einstein 1949, 684-685; McCrea 1971, 142). Einstein's philosophical standpoint was to choose the logically simplest field equations in light of experimental discoveries (the existence of constant density of matter in an expanding universe). The field equations with no cosmological constant allowed this possibility and this was the simplest solution, which Einstein therefore chose. In 1946 (1949), it seemed almost pointless to Einstein to explain once again the obvious conclusion that the cosmological constant was superfluous. Indeed, Einstein's main object in the 1916 general theory of relativity had been to develop a theory in which the chosen path leading into it would be the natural one psychologically, and its underlying assumptions would appear to have been secured experimentally. The expanding universe model could be achieved without the cosmological constant. However, Einstein still hoped that his theory would eliminate the epistemological weakness of Newtonian mechanics, the absolute space from the laws of physics. As regards Mach's ideas, in the 1940s Einstein was not sure anymore that his theory eliminated the epistemological weakness of absolute space. The elderly Einstein could not say whether he was inspired by Mach in the same way as could the young Einstein who was inspired by Mach's ideas when creating the theory of relativity. In 1948 Michele Besso (who recommended Mach to Einstein in 1897) asked Einstein about Mach's influence on his thinking. Einstein replied that, frankly, it was not clear to him how far Mach's works had influenced his own work. Einstein was obviously not sure about this (Einstein to Besso, January 6, 1948, in Einstein and Besso 1971, Letter 153). Mach's ideas (economy of thought) seem to have still haunted Einstein in 1950 when speaking about Mach's principle of economy (Einstein 1950, 13). Nevertheless, the most devout adherent of Mach's principle had to say in the year before his death that in his opinion one should no longer speak at all of Mach's principle. And Einstein explained further that Mach's principle dated back to days when it was thought that matter was the only physical entity. He said that he was well aware of the fact that he had also

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been influenced by this obsession for a long time (Einstein to Felix Pirani, February 2, 1954, EA 17-447 in Jammer 2000, 150-151).

14. Einstein's Interaction with George Gamow: Cosmological Constant is the Biggest Blunder George Gamow reported about discussing cosmological problems with Einstein in his 1970 posthumously published autobiography My World Line (Gamow 1970, 44): "Thus, Einstein's original gravity equation was correct, and changing it was a mistake. Much later, when I was discussing cosmological problems with Einstein, he remarked that the introduction of the cosmological term was the biggest blunder he ever made in his life. But this 'blunder', rejected by Einstein, is still used by cosmologists even today, and the cosmological constant demoted by the Greek letter ȁ rears its ugly head again and again and again."

Scholars explain that "biggest blunder" did not constitute an undertaking on the part of Gamow to quote Einstein, but that he stringed two words together to represent his memory of a conversation with Einstein. Gamow indeed said: "when I was discussing cosmological problems with Einstein, he remarked…" [my emphasis]. Biggest blunder could be attributed to Einstein by Gamow. Note that Gamow does not purport to be quoting Einstein directly. Gamow says in a Scientific American article entitled "The Evolutionary Universe", published in September 1956: "Einstein remarked to me many years ago that the cosmic repulsion idea was the biggest blunder he had made in his entire life". The account given in Gamow's autobiography My World Line is similar. The existing evidence is insufficient to decide whether Einstein himself used the word "blunder" or whether this was Gamow's own embellishment (Gamow 1956; Earman 2001, 189). Mario Livio raises the following question: Did Einstein actually say, "biggest blunder?" Livio could find no documentation for Einstein saying that the cosmological term was the biggest blunder he had ever made in his life. Instead, he claims, all references eventually lead back only to Gamow, who reported Einstein's use of the phrase in the two sources: My World Line and the 1956 Scientific American article (Livio 2013, 232, 236-237, 241).

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It is noteworthy that already in 1999 John Stachel remarked that, "The comment ['biggest blunder'] doesn't appear in Einstein's writings" (Harvey and Schucking 2000, 726). We shall soon see that this is not quite true. So far as Einstein's meeting with Gamow is concerned, the evidence indicates that they met together during World War II (more than fifteen years after Einstein dropped the cosmological constant) and shortly afterwards. Gamow had reported somewhat sketchily in My World Line about his consultation work for the armed forces of the United States during World War II. He said that it would have been, of course, natural for him to work on nuclear explosions, but he was not cleared for such work until 1948, after Hiroshima. The reason was presumably his Russian origin and the story he had freely told his friends of having been a colonel in the field artillery of the Red Army around the age of twenty. Hence, he was very happy when he was offered a consultantship in the Division of High Explosives of the US Navy Bureau of Ordnance Department. A more interesting activity during that time was his periodic contact with Einstein, who served as a consultant for the High Explosives Division. Accepting this consultantship, Einstein stated that because of his advanced age he would be unable to travel periodically from Princeton to Washington, D.C. and back, and that somebody had to come to his home in Princeton bringing the problems with him. Since Gamow happened to have known Einstein earlier, on non-military grounds, he was selected to carry out this job. Thus on every other Friday he took a morning train to Princeton, carrying a briefcase, tightly packed with confidential and secret Navy projects to Einstein. After the business part of the visit was over, Gamow and Einstein had lunch either at Einstein's home or at the cafeteria of the Institute for Advanced Study, which was not far away, and the conversation would turn to the problems of astrophysics and cosmology. In Einstein's study there were always numerous sheets of paper scattered over his desk and on a nearby table, and Gamow saw that they were covered with tensor formulae which seemed to be related to the unified field theory, but Einstein never spoke about that with Gamow. However, Gamow said that in discussing purely physical and astronomical problems Einstein was very refreshing, and his mind was as sharp as ever (Gamow 1970; 1979).

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There was quite a flutter of surprise when Livio came across a small article published in 1986 by Stephen Brunauer entitled "Einstein and the Navy". Brunauer was the scientist who had recruited both Einstein and Gamow to the Navy. In that article, Brunauer described the entire episode in detail. When explaining Gamow's precise role, Brunauer revealed that Gamow, in later years, gave the impression that he was the Navy's liaison man with Einstein, that he visited every two weeks, and that the professor listened but made no contribution of his own – but this was all false. The greatest frequency of visits was Brunauer's, and that was about every two months. Livio mentions that Brunauer visited Einstein at Princeton on May 16, 1943. Clearly, Livio says, Gamow exaggerated his relationship with the famous physicist (Livio 2013, 235-236). It is not surprising that presumably Gamow, and of course other people, exaggerated their friendship with Einstein. Einstein's fame led people who knew him to write about their personal acquaintance with him. Einstein had a friend, a Hungarian Jewish physician named János Plesch. In his autobiography Plesch described how he began collecting people both as patients and as friends. If we can believe his autobiography, he knew everybody and above all Albert Einstein (Plesch 1949, 101; Bernstein 2004), who told him many valuable typical Einstein anecdotes. How do we know whether Plesch's reports are authentic or not? We cross-reference them with letters and unpublished talks, etc. written by Einstein himself. Gamow had a well established reputation as a jokester and had been given to hyperbole. Fred Hoyle wrote that Gamow would shout from the other end: "The elements were made in less time than you could cook a dish of duck and roast potatoes" (Harvey and Schucking 2000, 726; Hoyle 1992, 197). However, Einstein also had a great sense of humour. The young and especially the elderly Einstein sometimes played tricks on biographers, reporters, and people by telling them leg-pull stories when they annoyed him with long personal questions. Already in 1919, Einstein told a New York Times correspondent (who came to his Berlin home to interview him) about the 1907 man falling from the roof thought experiment. The correspondent transformed the thought experiment from Bern to Berlin and into a realistic amusing story. Presumably Einstein told the correspondent the story in such a way that the latter did not notice that Einstein was pulling his leg. Nevertheless, the reader can rest assured that the imaginary man fell from an imaginary roof in Bern and not in Berlin. The correspondent wrote that it was from Einstein's lofty library in Berlin, in which the conversation with him took

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place, that his host had observed years ago a man dropping from a neighbouring roof – luckily on a pile of soft rubbish – and escaping almost without injury. "This man told Dr. Einstein that in falling he experienced no sensation commonly considered as the effect of gravity which, according to Newton's theory, would pull him down violently toward the earth" ("Einstein Expounds his New Theory", New York Times 1919, December 3). Gamow in turn fooled Einstein on his seventieth birthday. Recall that in July 1949 the Reviews of Modern Physics devoted an issue to a celebration of Einstein's seventieth birthday. It contained articles about Einstein's achievements written by his friends and colleagues. Gamow contributed the paper, "On Relativistic Cosmology", and in it he used his literary gifts to present a lively and humorous account of his theory. He wrote about "the neutron-capture theory of the origin of atomic species recently developed by Alpher, Bethe, Gamow, and Delter". In the footnote the references were: G. Gamow, Phys. Rev. 70, 572 (1946); Alpher, Bethe, and Gamow, Phys. Rev. 73, 803 (1948); R. A. Alpher, Phys. Rev. 74, 1577 (1948); R.A. Alpher and R. C. Herman, Phys. Rev. 74, 1737 (1948) (Gamow 1949, 369). The 1948 ĮȕȖ paper was created by Ralph Alpher under the supervision of Gamow and was published on April 1, 1948 in Physical Review. April fool's day was Gamow's favourite publication date. Gamow noticed that if he were to add as a co-author Hans Bethe, then the three names would be Alpher, Bethe, Gamow, like alpha beta gamma, even though Hans Bethe, who agreed to have his name included, had nothing to do with that paper (Livio 2013, 167-168). But in his 1949 paper on Einstein's seventieth birthday occasion, "On Relativistic Cosmology", Gamow gave Einstein a birthday present: he wrote about the DEJG theory, "Deltor", Robert C. Herman, corresponding to the fourth letter in the Greek alphabet. When referring to Robert Herman, Gamow wrote: "R. C. Herman, who stubbornly refuses to change his name to Delter". Einstein owned Gamow's humorous popular science books in his personal library in Princeton. For instance, one can find there two of Gamow's popular books kept in the Einstein Archives: Gamow, George, Mr. Tompkins in Wonderland: or stories of c, G, and h [the three constants of physics: the velocity of light, the Gravitational constant and the Planck constant], illustrated by John Hookham, 1940, New York: Macmillan.

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Gamow, George, One, two, three… infinity: facts & speculations of science, illustrated by the author, 1947, New York: Viking Press. The first lines of One, Two, Three… Infinity consist of the verse (Gamow 1947): "There was a young fellow from Trinity, Who took the square root of infinity. But the number of digits, Gave him the fidgets; He dropped Math and took up Divinity."

Einstein also invented verses about funny and amusing situations, and about ridiculous incidents. For instance, on the occasion of his fiftieth birthday (14.3.1929) he wrote a witty verse to his friends (Seelig 1954, 215-216): 17 Every one showed today, You all came from far away; Great is the loving, Cards all so touching; Blessed have you wished me, So thoughtful and kindly; For an elderly man and wan, All's sweetness and fun; What a beautiful day, Everything's going my way, The scroungers donate a madrigal to play. My heart beats like a drum, The day is nearly over, I complain to you my lover, After all you have done, Your love shines like the sun.

An essential point at issue is Gamow's attitude towards the cosmological constant in the years 1949-1952. In his above-mentioned paper "On Relativistic Cosmology", published in the 1949 special volume of Reviews of Modern Physics celebrating Einstein's seventieth birthday, Gamow briefly reviewed a then-known problem in cosmology, and thereafter dealt with a possible solution to this matter. According to Hubble's measurements of the rate of uniform expansion we assume that it must have started 1.8 x 109 years ago. At that epoch the

  17

My translation into English of Einstein's verse.

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material of the universe must have been in a state of very high density and correspondingly high temperature. However, the study of rocks indicates that the solid crust of the earth must have existed for at least 2.5 x 109 years, so that the age of the universe would be closer to 3 x 109 years. Since this discrepancy is certainly beyond the limits of errors in astronomical measurements, Gamow emphasised that it appears that we have here the first serious disagreement between the conclusions of relativistic cosmology and the observed facts. This disagreement, however, can be resolved by considering the effect of the cosmological term in the general equation of the expanding universe (Gamow 1949, 376). In 1952 Gamow claimed that when the fact that the universe was not static but rapidly expanding was recognised, the introduction of the cosmological constant became superfluous. However, this constant could still be of some help in cosmology, even though the primary reason for its introduction had vanished (Gamow 1952, 25-26). How could the cosmological constant be of help in cosmology? Gamow called attention to the all-important question of the disagreement between the conclusions of relativistic cosmology and the geological observed facts. He explained that Lemaître introduced the cosmological constant because the presence of such a force would make the universe expand with ever-increasing velocity and shift the position of the zero point in time. In effect if the expansion process is accelerational, the recession velocities of the neighbouring galaxies would have been smaller in the past than today, so that the date of the beginning would be shifted back in time. Assuming such a small numerical value as 10-33 sec-1 for the cosmological constant, one could bring Hubble's original value into agreement with the geological estimate (Gamow 1952, 29-30). Einstein, however, held that Lemaître was founding too much upon the field equations with the cosmological constant (Einstein 1949, 684-685). And there was a twist to the tale, because after 1952 Gamow quietly dropped the cosmological constant and never returned to it (Kragh 2005, 181). In that case, it seems reasonable to presume that Einstein might have influenced Gamow to drop the cosmological constant. Lastly, remember that Gamow wrote in 1956 that Einstein had remarked to him "many years ago that the cosmic repulsion idea was the biggest blunder he had made in his entire life" [my italics] (Gamow 1956, 66-67).

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It seems reasonable to the present writer that although Einstein and Gamow had met occasionally during World War II and shortly afterwards, Einstein had probably spoken about the cosmological constant with him in the early 1940s. After 1952, Gamow dropped the cosmological constant and subsequently, in 1956, wrote his memory of the meeting with Einstein. Who thus made the blunder? Was it Gamow reporting his memories of his own experience of dropping the cosmological constant? One may never know exactly what Einstein told Gamow. It is quite obvious that Einstein and Gamow met sometime in the early 1940s during the war. We cannot say a priori anything further about the contents of these meetings beyond Gamow's reports. Remember that in 1970 Gamow reported that: "the cosmological constant demoted by the Greek letter ȁ rears its ugly head again and again and again". We notice firstly that in 1947 Einstein referred to the modified field equations with the ȁ term as being repulsive, disgusting or ugly. On September 26, 1947, Einstein wrote to Lemaître in response to his letter from July 30, 1947 informing him that since he had introduced the cosmological ȁ term, he had always had a bad conscience. But at that time he could see no other possibility to deal with the fact of the existence of a finite mean density of matter. He found it very ugly indeed that the field equations of gravitation should comprise two logically independent terms connected by addition. About the justification of such feelings concerning logical simplicity it was difficult to argue. Einstein could not help feeling it strongly and was unable to believe that such an ugly thing should be realised in nature (Einstein to Lemaitre, 26 September 1947, AE 15-085 in Kragh 1996, 54). We have no assurance that anything of this sort was ever said when Einstein met Gamow, but it might perhaps be possible to interpret Gamow's above excerpt as describing how Einstein felt towards the cosmological term. After all, without the cosmological term ȁ general relativity involves no general constants and is a logical simple theory. Withȁit acquires a constant that at first sight seems to be different from all other constants of physics (McCrea 1971, 146). In 1922 Einstein wrote to Max Born: "I too committed a monumental blunder some time ago". Einstein referred to his experiment on the emission of light with positive rays, and said that one must not take it to heart too seriously. He added: "Death alone can save one from making

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blunders" (Einstein to Born, undated (sometime in spring 1922), Einstein and Born 1969, letter 42).18 More than 40 years later, in 1965, Max Born commented on this letter (Einstein and Born, Born's Comment to letter 42, 1969, 103; 1971, 71):19 "Here Einstein admits that the considerations which led him to the positive-ray experiments were wrong: 'a monumental blunder' [a "capital blunder" in Born's German original comment]. I should add that now (1965), when I read through the old letters again, I could not understand Einstein's observation at all and found it untenable before I had finished reading".

It is likely that when Einstein met Gamow he formulated his views in his native German, and perhaps he told Gamow that suggesting his cosmological constant was a "blunder"; or else Einstein told Gamow that when he (Gamow) suggested the cosmological constant in 1949 to explain the discrepancy between Hubble's findings and geological findings it was a blunder. Einstein perhaps told Gamow that the cosmological constant was a "capital blunder"; he might have told Gamow that the ȁ term was a "monumental blunder", and Gamow could have embellished Einstein's words to become the famous aperçu: "biggest blunderhe ever made in his life".

15. Einstein, Gödel and Backward Time Travel In January 1940 Kurt Gödel left Austria and immigrated to America, to the newly founded Institute for Advanced Study, located in Princeton University's Fine Hall. Princeton University mathematician, Oskar Morgenstein, told Bruno Kreisky on October 25, 1965 that Einstein had often told him that in the late years of his life he had continually sought Gödel's company in order to have discussions with him, and that he once

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"Auch ich habe vor einiger Zeit einen monumentalen Bock geschossen [...] Gegen das Böcke-Schießen hilft nur der Tod".  19 "Hier gesteht Einstein, daß seine Überlegung, die zu dem Kanalstrahlenexperiment führte, falsch sei; ein kapitaler Bock. Dazu muß ich sagen, daß ich jetzt (1965), als ich die alten Briefe wieder las, Einsteins Betrachtung überhaupt nicht verstand und sofort für unhaltbar hielt, ehe ich noch weitergelesen hatte". 

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came to the Institute merely to have the privilege of being able to walk home with Gödel (Wang 1987, 31). There was a problem still not solved in relativistic cosmology: why do the observed galaxies rotate and where does this rotation come from? This problem led George Gamow in 1946 to make the following suggestion published in a letter to Nature (Gamow 1946): the rotation of galaxies is explained by a cosmological model: the galaxies originated out of a rotating universe. Gamow suggested that all matter in the visible universe might be in a status of general rotation around some centre located far beyond the reach of our telescopes, the rotation though being extremely small. Gamow wrote to Einstein that he did not think anybody ever tried to obtain a line-element for such an anisotropic rotating expanding universe, and he wondered whether Einstein had ever thought about it. Einstein did not find the idea meaningful and Gamow seemed to have abandoned it. But Gödel soon found the kind of solutions Gamow had asked for (Kragh 1996, 109-110). It appears that Gödel's many discussions with Einstein led him to become seriously involved in general relativity. Presumably, it was from Einstein that he learnt the basic principles of the subject, but he developed for himself his own highly original approach to a relativistic cosmology (Penrose 2007, 2). In his walks back home from the institute with Gödel, Einstein may have told him about Gamow's suggestion regarding a rotating universe, and that Gamow replied he did not think that anybody ever tried to obtain a line element for such a universe. Between 1946 and 1947, Gödel proposed just this line element, an exact solution to Einstein's field equations, and he presented it to the volume published on the occasion of Einstein's seventieth birthday (Gödel 1949a, 560-561). In the paper presented in the special issue of Reviews of Modern Physics dedicated to Einstein on his seventieth birthday, Gödel published an exact solution to Einstein's field equations with a cosmological term that was not equal to zero, but had a negative value. Gödel's solution was a static and not an expanding universe, with non-vanishing density of matter distributed uniformly; and it was a singularity-free solution to Einstein's field equations.

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Since the density and pressure in Gödel's world are the same everywhere and this world therefore does not expand, Gödel's solution has much in common with Einstein's static universe. Gödel's universe is spatially homogeneous (it looks the same when viewed from different positions) and isotropic (it is equal in all directions), but it indeed models a rotating universe. All matter in this universe, galaxies and nebulas, slowly rotate with respect to the "compass of inertia". The "compass of inertia" is the direction of rotation of all global matter distribution in the universe. Since the universe is static, all matter in this universe rotates with constant angular momentum. Gödel's universe, in which matter has large-scale rotation, clearly violates Mach's principle. According to Mach's principle, rotation motion can only be a relative motion of one piece of matter with respect to another one. Imagine a universe containing one body, and numerous fixed stars surrounding it. If the body is rotating with a certain angular velocity and the fixed stars are at rest, apparent centrifugal forces arise by the relative motion with respect to the mass of the fixed stars. If one could fix the body and cause the whole sky of the fixed stars to rotate with this same angular velocity in the opposite direction, the two cases would become indiscernible the one from the other. Then according to Mach an equal centrifugal force should appear in the body. In Gödel's universe, however, this Machian condition is explicitly violated. The angular velocity of all matter (rotation of the galaxies in the universe) is everywhere relative to the "compass of inertia". Hence an observer sees all nearby observers around him rotating and the whole universe around him is also rotating in the same direction. Gödel's static universe had the further property that there was no absolute time coordinate; it did not admit a global notion of time and simultaneity. It was not possible to assign a time coordinate t to each space-time point in such a way that t always increased if one moved in a positive timelike direction. An observer in Gödel's static universe could accelerate and travel forward in time and arrive at his starting point, or else travel on a path that could bring him backwards in time into his own past. However, in his paper Gödel admitted that his static world could not be a realistic model of the universe since it did not contain Hubble's expansion of the universe (Gödel 1949b, 447). Normally, a timelike curve in space-time is an open line that runs from past to future and describes the history of a particle from past to future.

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Gödel found that Einstein's theory of general relativity allows the existence of closed timelike curves (CTCs), paths through space-time that, if followed, allow an observer to travel into the starting point of his/her voyage or to his/her past and thus to interact with his/her former self. By making (a long-term) round trip on a rocket ship in a sufficiently wide curve, it is possible in Gödel's static world to travel into any region of the past, present, and future, and back again, in exactly the same way that makes it possible in other worlds to travel to distant parts of space. A closed timelike curve in Minkowski's flat space-time is a closed worldline. The worldline starts as a timelike worldline inside the light cone. It then loops back on itself and goes outside of the light cone and becomes a spacelike worldline, and then it enters again into the light cone and becomes timelike. This means that it would not be possible to go on a closed timelike curve because traversing such a curve means travelling faster than light, which is forbidden in special relativity. That is the reason why there are no closed timelike curves in the Minkowski space-time of special relativity. However, in the general theory of relativity this is not the case. In general relativity, it is possible for a certain space-time to have closed timelike curves that will always remain inside the light cone and will thus be only timelike. In Gödel's universe the rotation tilts the light conesso that they are tilted more and more in the direction of rotation as one moves away from the "compass of inertia". Hence, the timelike curve stays always inside the light-cones and loops back to the past as we observe it. This means that it would be possible to travel on such a closed timelike curve in a slower-than-light velocity and thus to travel into one's own past. Gödel was aware of the fact that this state of affairs seemed to imply an absurdity. For it enabled the following paradox (later called the grandfather paradox). Gödel elaborated it to Einstein: it enabled a person to travel into the near past of those places where he himself had previously lived. There he would find himself at some earlier period of his life where he could now do something to himself, which, from his memory, he knows had not previously happened to him. Gödel did not think this proved the impossibility of his static world; but the grandfather paradox and similar contradictions, however, presupposed the actual feasibility of the journey into one's own past.

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In order to prove the impossibility of his world, because of such paradoxes regarding the possibility of a journey into one's own past, Gödel resorted to engineering limitations. He showed that practically, backward time travel was impossible. Therefore, Gödel's world cannot be excluded a priori on theoretical grounds, but can only be excluded practically: achieving backward time travel is beyond our possibility because of technical limitations. In a footnote Gödel calculates the velocities which would be necessary in order to complete the voyage backwards in time in a reasonable length of time. He explains that he bases his calculation on the mean density of matter equal to that observed in the universe. He says that the time traveller would have to move at least as fast as almost 71% of the speed of light, and that if his rocket ship could transform matter completely into energy, then the weight of the fuel for the rocket ship would be greater than that of the rocket by a factor of 1022 divided by the square of the duration of the trip (in years, in rocket time, as measured by the traveller) (Gödel 1949a, 561). A trip to the past in Gödel's universe would require a time machine looking something like Dr Who's telephone booth attached to a fuel tank the size of several hundred trillion ocean liners (Nahin 1993, 251). Gödel's above calculation demonstrated that the closed timelike curves in his universe were not geodesics. Gödel provided in the above calculation a measure of the accelerations and the rocket action that would be involved in backwards time travel. Later it was shown that backwards time travel was impossible not because of engineering or technical limitations; it was impossible because it led to inconsistencies within the physics of general relativity. Already in 1918, Hermann Weyl thought that it was not impossible for a world line of a body in his system, although it had a timelike direction at every point, to return to the neighbourhood of a point which it had already once passed through. Weyl said that the result would be a spectral image of the world; more fearful than anything the weird fantasies the German writer (of horror fantasies) Ernst Theodor Wilhelm Hoffmann had ever conjured up. Weyl thought that in actual fact the very considerable fluctuations of the metric tensor components gPQ that would be necessary to produce this effect did not occur in the region of the world in which we live (Weyl 1918a, 236; 1922, 274).

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Gödel published an essay in Schilpp's 1949 volume celebrating Einstein's seventieth birthday, and in the essay answering all the scholars participating in the volume, Einstein replied to Gödel's above-mentioned claims, explaining that this problem had already disturbed him at the time when he had been engaged in building up the general theory of relativity, without his having succeeded in clarifying it (Einstein 1949, 687). Indeed in his 1914 review article on the Entwurf theory Einstein brought up a deep-reaching question of fundamental significance while admitting he was unable to answer it. In special relativity, every line that describes the motion of a material point, i.e. every timelike line is necessarily nonclosed, the reason being that for such a timelike line dx4 never vanishes (it is inside the light-cone, not a CTC). Einstein, was therefore troubled by this possibility. He said that in general relativity it was theoretically possible to imagine a point moving on such a four-dimensional curve, which is almost a closed one. In this case one and the same material point could exist in a small area of space-time in several seemingly mutually independent representations. Einstein thought that although this runs counter to his imagination, he could not demonstrate that his then-1914 general theory of relativity (the Entwurf theory) excluded the occurrence of these closed paths (Einstein 1914b, 1079). Einstein later answered Gödel's claims more specifically in Schilpp's 1949 volume (Einstein 1949, 687-688). He thought that the problem presented itself as follows. Suppose P is a point inside a light cone. B and A are two points on a timelike world line and B is before P, and A is after P.

We send or telegraph a light signal from B to A through the point P. We cannot send a signal from A to B through P because there is no free choice for the direction of the arrow of time. This is an irreversible process, a

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process which is connected with the growth of entropy. Hence, the asymmetrical character of time is secured by thermodynamics. If, therefore, B and A are two, sufficiently neighbouring world-points, then the assertion B is before A makes physical sense. But this assertion does not make sense anymore, if these points A and B, connected by the timelike line B-P-A, are arbitrarily separated at a great distance from each other, and if these points are connected by closed timelike lines, and if these closed timelike lines exist. In that case the distinction earlier-later is abandoned for world-points which lie far apart in a cosmological sense, and those paradoxes, put forward by Gödel, arise regarding the direction of the causal connection. In the previous case, where B and A were two sufficiently neighbouring world-points, an observer on the timelike world-line A-P-B could send a light signal from B to A; since the velocity of light is finite and constant, and A and B lie far apart in a cosmological sense, it takes a long time for the signal from B to reach A and in the meanwhile the world does not remain static but changes, and is therefore a dynamic world. Einstein, therefore, concluded that the distinction earlier-later had to be abandoned. In other words, Einstein did not believe that closed timelike lines existed in physical reality, and he did not believe that a static solution with a cosmological constant could exist. Einstein said that it would be interesting to verify whether Gödel's cosmological solution of the gravitational equations, with a non-vanishing cosmological constant, would not be excluded on physical grounds (Einstein 1949, 687). The reason why Einstein could not ascribe physical significance to closed timelike curves, or CTCs, was that they supplied a counterexample to the regular order of cause and effect. Even though CTCs did not contain any mathematical contradiction, they conflicted with Einstein's physical intuition and with his experience to such an extent that it seemed to him a sufficient reason to prove their impossibility. Gödel's solution and Einstein's response to it began the discussion on time travel into the past in Einstein's classical general theory of relativity.

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Subsequently, in 1950 Gödel gave a lecture in the International Congress of Mathematicians and there he presented exact solutions representing rotating and non-static spatially expanding and homogeneous universes. Unlike his 1949 static universe, which did not permit redshift, Gödel said that it could be proved that his new solutions included redshift that for small distances increased linearly with the distance, and this implied an expansion. Because of the expansion there is no constant density of matter and dust in his rotating universe. For sufficiently great distances, there must be more galaxies in one half of the sky than in the other half (Gödel 1950, 176). Gödel said that his non-static expanding rotating universe contained no closed timelike curves. This way Gödel got rid of the backwards time travel anomaly, and demonstrated that the new solution contained the precise necessary and sufficient condition for the nonexistence of closed timelike curves. Hence, t always increases if one moves along a timelike line in its positive direction (Gödel 1950, 178-179). Finally Gödel managed to get rid of his closed timelike curves in a particular instance. There exist rotating stationary homogeneous solutions with finite space, in which no closed timelike lines exist, and have the cosmological term > 0; in particular, also such solutions as arbitrarily differ little from Einstein's static universe (Gödel 1950, 181). But there still remained his static world with the cosmological term < 0 with closed timelike lines. Gödel also found rotating solutions with no cosmological constant (i.e. cosmological constant equal to zero) (Gödel 1950, 180). Gödel's non-static expanding rotating universe solutions could represent realistic models of the universe; but of course since they were rotating universes they still violated Mach's principle. Recall that Einstein gave up the cosmological constant and Mach's principle at that time.



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INDEX

Abraham, Max, x, 3, 37-40, 44-47, 49, 76, 102, 104, 116, 154, 206  Attacks Einstein's Entwurf theory, see Einstein.  Corrects own theory, see Einstein  Criticises Einstein's Static fields theory, see Einstein First theory of gravity, 38 Gravitational waves, 40 Polemic with Einstein, see Einstein Adams, Walter, S., 187, 352 Gravitational redshift test, 187188 Adler, Friedrich, 5-6 Affine, 195-205, 345-347 Affine connection, xiv-xv, 195205, 345-347  Affine geodesic, 199-200, 346 Affine parameter, 199-200 Alice in Wonderland, 321, 328 Alpher, Ralph, 369. Annalen der Physik, xiv, 7, 13-14, 40-41, 43, 86, 193, 219, 267  Average null energy condition, 232 Bach (Förster), Rudolf, 318 Bachem, Albert, 186 Background-independent theory, xv, 205, 216 Backward time travel, 233-234, 375-379 Bargmann, Valentine, 235, 239  Beckmann, Ernst, 20  Bergmann, Peter, xvi, 235, 239  Bern 1955 Jubilee Conference, xvi Bernays, Paul, 106

Besso, Michele, 2-3, 15, 18, 21, 44, 75, 78- 83  Einstein-Besso manuscript, see Einstein Besso and Einstein, perihelion of Mercury, see Einstein Besso and Einstein, perihelion of Mercury, (Nordström's theory), see Einstein Memo of, 75  Bethe, Hans, 369 Big bang primeval atom theory, see Lemaître  Black hole, xii, xiii, 237-238, 316  Bondi, Hermann, 353-354  Born, Max, 174, 188, 242-243, 260, 372-373  Gravitational waves query, see Einstein Brownian motion, 3, 6 Brunauer, Stephen, 368 Campbell, William Wallace, 300302, 305 Eclipse expedition, 300-302 Light deflection, 300-302 Cartan, Élie Joseph, xiii, xiv, 195196, 201-202, 346-347, 349-350  Newton-Cartan correspondence principle, 201-205 Cartan-Newton theory, xiv, 195201, 206 Ricci tensor, 201-202 Unified field theory, see Einstein Chandrasekhar, Subrahmanyan, 237-238  Christoffel, Elwin Bruno, 11, 54, 72 

406 Chrono-geometrical structure, xv, 195, 200, 206  Clock synchronisation, 4, 30, 179180, 233-234  Closed timelike curves, 234, 376. Cohn, Bernard, 6 Collège de France, xii, xiii, 220, 350.  Constant light velocity of waves, 242-243, 244, 246, 259, 281 Cortie, Aloysius Laurence, 292  Cosmic rays, 360, 363  Cosmological constant, 306, 311313, 316-322, 326, 331, 336, 341, 344, 353, 356, 359-360, 362-364, 371 Cosmological constant (Friedmann models), 331, 336, 352-353, 355-356 Cosmological redshift, 236, 336338, 343, 353, 357, 380  Cottingham, Edwin T., 293  Curie, Marie, 9  Curtis, Heber D., 300-301 Dark matter, 326, 330  Davidson, Charles Rundle, 185, 291, 293-294, 295  Eclipse expedition of, 185, 293294  Gravitational redshift, 185 De Donder, Théophile Ernest, 362  Priority claim, see Einstein. De Hass, Wander Johannes, 151  De Sitter, Willem, 125, 162-163, 242, 244, 247-251, 253, 256, 267-270, 274-280, 283, 285286, 305, 321-322, 324-325, 327-330, 339, 351, 354-357, 362-363  Boundary conditions polemic, see Einstein  Cosmological model polemic, see Einstein De Sitter–Schwarzschild solution, xi-xii, 316-319

Index Gravitational waves polemic, see Einstein  Einstein-de Sitter world, see Einstein. Einstein and mass equator, see Einstein Einstein and mass equator (de Sitter's world), see Einstein Einstein's objections to empty world of, see Einstein  Empty world, 274-276 Expanding balloon thought experiment, 351-352 Spectral shift effect, 279-280, 319-322, 329, 335, 339-340 Temporal term, mass equator (horizon), ix-x, 279, 281284, 318-319, 328-330, 334335  World-matter in Einstein's world, see Einstein  Deprit, André, 361  Doppler effect, 184, 187-188, 190192, 321-322, 335, 337, 352, 355  Dr Who time machine, 377  Dyson, Sir Frank, 291, 298 Eddington, Sir Arthur Stanley, xixii, 162-163, 185-187, 213, 220221, 237-238, 258-260, 280281, 285-286, 291-301, 304305, 327-331, 334-335, 340344, 355-356, 359-364  Death of, 363-364 Eclipse expedition, 294-297  Gravitational lensing, 304-305  Gravitational redshift, 185-187  Gravitational waves, 258-260  Light deflection, 185, 254, 286, 292, 294, 296-298, 300, 304  Literary gifts of, 327-328 Lemaître-Eddington cosmological model, 334335, 340-344, 359-360

General Relativity Conflict and Rivalries Einstein's possible objection to Lemaître-Eddington model, see Einstein Objection to the Big bang, 359361 Quadrupole formula, 258 Einstein's 1916 papers received by, 162, 285-286 Schwarzschild singularity, 220, 237-238, 329-330 Silberstein understands general relativity, 298 Speed of thought, 260  Theory of stellar structure, 221  Translation of Lemaître's paper requested by, 342-343  Three people understand general relativity, see Silberstein Ehrenfest, Paul, 6, 15, 21-22, 24, 91, 159, 172-174, 176-179, 193, 250, 253, 285, 322-323, 332.  Paradox of, 172-174, 179 Hole argument and Point Coincidence Argument, see Einstein  Einstein, Albert, Abraham corrects theory of, 3839  Abraham and Nordström criticism of, 49 Adapted coordinate systems by, 105-107, 111-117, 119-121, 123-126, 130-131, 170 Adopted metric tensor by, 11, 53, 55  Appendix to 1916 general relativity paper, 165, 168 A previous solution reconfirmed by ,1915, 70-71, 136, 153, 155, 202-203 Authority in physics, xi-xii, 171-172, 237-238, 263, 282283, 335-336, 354, 360-361 Benefits received from Hilbert, 151-152 

407

Besso perihelion of Mercury, 15, 18, 75, 78-83, 91, 130, 150  Besso perihelion of Mercury (Nordström's theory), 91  Biggest blunder of, 366-367, 372-373 Cartan priority claim acknowledged by, 349-350  Cartan unified field theory noted by, 346-350  Christoffel symbols, 155  Clock paradox noted by, 35-37  Competition with Hilbert, 3, 116, 139, 144-145, 150, 153  Components of the gravitational field (1914), 99, 114, 156 Components of the gravitational field (1915), 137-139, 141, 146, 148, 156-158. Conservation of momentum and energy (Entwurf field equations), 76, 92, 106, 112, 113, 115, 117, 135  160, 164, 247, 253, 257-258 Conservation of momentum and energy (1915 field equations), 137-138, 141, 155-158, Conservation of momentum and energy (1917 field equations), 273 Conservation of momentum and energy (Nordström's theory) by, 84-86, 92-96, 101  Conservation of momentum and energy (static gravitational fields), 42 Conservation of momentum and energy (Zurich Notebook), 12, 58-59, 61-64, 68-70 Constancy of the velocity of light, 9, 22-23, 46, 49, 8586, 98, 100, 206-207, 240, 242-243, 370

Index

408 Contraction of lengths (special relativity), 41-42, 172-174, 179-181  Coordinate-dependent theory, 8, 32, 53, 117, 174  Copernicus (Planck on Einstein), 6  Correspondence principle (Zurich Notebook), 56, 6062, 65, 69, 71 Cosmological constant, 272-278 Cosmological constant paper, 270-273 Cosmological constant renounced by, 333-334, 344, 352-357, 362, 364-365, 371 Cosmological model (static world) by, 273-277, 280, 318, 322, 329, 331-333, 338, 341, 343- 344, 353357, 359-360, 364 Deflection of light (1911), 8, 37,  90, 290, 292, 300 Deflection of light (1915), 288290, 292, 297-298, 302  Deflection of light (1916), 290292, 297-298  Deflection of light near the Sun, 2, 8-9, 23-25, 32, 37, 41, 87, 90-91, 101, 104, 185, 188, 212, 214, 240, 254, 286-305 De Sitter mass equator (horizon) by, 282-284 Distant parallelism (tele parallelism) by, xiii-xiv, 347-350 Eddington's results presented by Lorentz to, 297  Einstein-Besso manuscript, 15, 70, 75, 78, 79-84, 91, 117, 128-130, 145-146, 150, 204, 254.  Einstein-de Sitter world, 356358 



Einstein-Hilbert action, 168171, 313-314 Einstein Hilbert paper (November 20, 1915), 170 Einstein-Fokker, EinsteinNordström theory, 21, 98101 Einstein–Podolsky–Rosen (EPR argument), 238-239 Einstein-Nordström theory (1913), 91-92, 95-98  Einstein-Rosen bridge, 222, 229-230, 238  Einstein-Rosen bridge (Schwarzschild nontraversable wormhole), 230232 Einstein (Rosen) gravitational waves paper (1937), 261, 264-265  Einstein-Rosen gravitational waves submission paper (1936), 261-264  Einstein-Straus vacuole, SwissCheese model, 236-237  Einstein tensor, 218, 311 Ehrenfest, Hole argument and Point Coincidence Argument, 176-178, 193  Elevator thought experiment, 72, 94, 111, 287-288  Entwurf 1913 gravitation theory and Einstein-Grossmann field equations, 12, 15, 18, 22, 25, 55, 59, 70-71, 75-76, 78-80, 82-84, 86-87, 89-94, 97-99, 101, 103-105  Entwurf gravitation field equations, 113-114 Entwurf gravitation tensor, 120  Entwurf gravitation tensor (with flaw found by Hilbert), 127128 Entwurf Lagrangian, 114, 127, 139 

General Relativity Conflict and Rivalries Entwurf problematic Lagrangian, 127, 130-131  Entwurf Einstein-Grossmann paper 1913)), 11, 15, 70-72, 74, 76-77, 84, 86, 93  Entwurf Einstein-Grossmann paper (1914), 99, 101, 103, 105-106, 116-120  Entwurf review paper (1914), 24, 105-116, 170  Entwurf prediction of deflection of light, 90 Entwurf 1913 theory attacked by Abraham, Nordström and Mie (in 1914), 104, 206 Entwurf 1913 theory attacked by Abraham (in Vienna), 76, 104  Entwurf 1913 theory attacked by Mie (in Vienna), 103-104 Equality of inertial and gravitational mass by, 2, 8, 29, 32-33, 54, 84-86, 92, 96, 99, 101, 104, 196-197, 206, 288  Explanation of basic ideas to Infeld by, 222-224 Frauenfeld talk, 19, 76  Friedmann's model disputed by, 332-333, 352 General relativity manuscript (1916), 164 Galileo's principle of free-fall by, 2, 8, 28-29, 32-33, 46, 48, 66-67, 100, 196  Gamow plays tricks on, 369 General relativity theory (1911), 7-8, 32-34, 37-38, 41-42, 45, 48, 101, 133, 179-180, 182, 184, 188-190, 240, 286-288, 290-291 Glasgow talk by, 3, 100 Geodesic line in Zurich Notebook, 57 Geodesic line equation by, 148, 156, 228

409

Gödel's static universe time travel. 378-379 Göttingen lectures on Entwurf gravitation theory, 131 Gravitational lensing 1936 paper, 303-305  Gravitational lensing, Mandl and Einstein, 303-304 Gravitational redshift 2, 31-32, 90, 182, 184-191, 214, 240, 278, 298-299  Gravitational time dilation 1907, 30-32, 38, 179  Gravitational time dilation 1911, 41, 180, 189-190, 240  Gravitational time dilation 1912, 41  Gravitational time dilation 1913-1914, Entwurf theory, 97 Gravitational time dilation 1916, 184  Gravitational time dilation (de Sitter's world), 282-283  Physical meaning to space-time measurements and interval, 175, 177-181, 308-309  Gravitational waves 1916 Addendum, 248-250 Gravitational waves do not exist (1936), 227, 236, 260, 262 Gravitational waves do not exist (1938 Einstein–Infeld– Hoffmann theory), 266  Gravitational waves, (1913) initial thoughts, 242-243  Gravitational waves (1916) theory, 244-248, 251 Gravitational waves ((1918 theory, 256-258 "Grossmann you have to help me", 54 "Happiest thought of my life," 2, 29, 44, 54, 133, 196, 210, 368-369

Index

410 Harmonic coordinate condition (Zurich Notebook), 60-68  Harmonic coordinate condition (gravitational waves), 245, 250, 260 Hertz coordinate condition, (Zurich Notebook), 61-66, 68, 69 Heuristic guiding principles by, x, 43, 93, 116-117, 129-130, 158, 184, 186, 194, 204, 252- 253,286, 299 Hilbert's axiomatic method, 163-164 Hilbert congratulates Einstein on perihelion motion calculation, 150 Hilbert's possible influence on, 151-152, 154 Hilbert to Einstein: "your solution is entirely different from mine", 144-145 Hilbert receives proofs on November 4, 1915 paper by, 139 Hole Argument, 11, 19, 74-77, 104-106, 115, 162, 176-179, 192-193, 248 Hubble's discovery, 353-356, 365 Hubble's 1931 visit to, 352 Humble genius, vii, ix, 239, 261-262 Inertio-gravitational field, xvxvi, 44, 197, 201, 205-206, 208, 324, 345-346 Infeld and gravitational waves by, 262-264 Influence of Hilbert's field equations on, 153, 155-162 Inspiring ideas presented by Mie, 310 "I will a little tink", x ,239 Lagrangian (1915), 139 

Lagrangian (1916) (density), 167-168. Laue's objections to 1911 relativity paper of, 34-35, 37  Laue's scalar, 85, 96, 99, 160  Lemaître's use of the cosmological constant, 364365, 371  Levi-Civita in Princeton, 261262 Line element, general relativity by, 52-53, 55, 93, 108, 132, 167, 181, 185, 307 Line element, special relativity by, 50-51, 55, 93 Line element (natural interval) by, 93-94, 109 Literary gifts of, 370  Mach's Ether by, 322-325, 330  Mach's ideas by, 5, 83, 87, 8990, 98, 104, 117, 195, 206207, 267, 270, 272, 274275, 323-325, 333 Mach's ideas, analogy to electromagnetic induction by, 90 Mach's ideas and de Sitter's objections to, 269-270, 275276  Mach's principle by, 194, 210, 214, 274, 276, 282, 284, 299, 311, 317, 322, 326, 354, 366, 375, 380  Magnet and conductor experiment, 209-210 Manuscript (1912) on the special theory of relativity, 50 Mass-energy equivalence, 28, 32, 34, 42, 47-48, 67, 84-85, 100, 154, 288 Mercury's perihelion precession problem, introduction, 78-79  Mercury's perihelion precession problem (Entwurf theory)

General Relativity Conflict and Rivalries by, 15, 18, 75, 78-83, 91, 130, 150  Mercury's perihelion precession problem (1915) by, 146150, 243-244  Natural interval, 92-95, 109, 132-133, 175 Newton-Cartan theory, 202-203  Newton of the Day, LeviCivita's reference to, 252 Newton's two globes thought experiment by, 207-211  Nobel Prize in Physics (1921), 305 Nordström converts to Einstein's theory, 101, 253-254 Nordström theory's deficiency by, 87, 90, 98  Nostrify by, 47, 153-155, 362  November 4, 1915 field equations, 136-138, 140, 158  November 11, 1915 field equations, 143-144  November 25, 1915 field equations, 157-161, 273, 276, 311  November 4, 1915 paper, 11, 131-132, 134, 136-138, 139142, 156  November 11, 1915 paper, 142144 November 18, 1915 paper, 146150 November 25, 1915 paper, xi, 157  November tensor, 59, 68, 136  Objections to de Sitter empty world by, 282-284 Objection to Weyl unified field theory by, 307-310 Particles represented as singularities, 224-228, 230231, 236-237, 265-266  Patent office, 2-4, 9, 13, 29 

411

Personal rivalry with Hilbert, 153  Photoelectric effect, 305 Physical Review affair, 261 Point Coincidence Argument, 176-178 Polemic with Abraham, 3, 40, 44-47, 49, 116, 153-154  Polemic with Levi-Civita on Entwurf gravitational tensor, 120-127 Polemic with Levi-Civita on pseudo tensor, 125, 251253, 256-257 Polemic with Nordström on pseudo tensor, 253-256 Polemic with Silberstein, 225227  Polemic with de Sitter on gravitational waves, 244, 247-251, 256  Polemic with de Sitter on boundary conditions, 267270  Polemic with de Sitter on cosmological model, 276278 Possible objection to LemaîtreEddington model by, 344  Principle of equivalence (1907), 2-3, 30-32, 44, 101, 180, 290  Principle of equivalence (1911), 8, 32-34, 37, 41, 101, 104, 117, 134, 184, 186, 189, 192, 286-288, 290  Principle of equivalence (1912) 9, 43-44, 46, 54, 66-67, 111, 240 Principle of equivalence (1914), 44, 197, 201, 205-206, 208, 324, 345-346  Principle of equivalence (1918), 194, 282, 299  Principle of equivalence and elevator experiment, 72 

Index

412 Principle of equivalence and rotating systems, 41, 83-84, 130, 133, 174, 181-183  Principle of equivalence conflict with Entwurft theory by, 73, 116-117  Principle of equivalence conflict with Nordström first theory by, 84-85, 91 Principle of equivalence criticism of, 34-35, 36-37 Principle of equivalence, polemic with Mie, 102, 206  Principle of equivalence, Silberstein against Einstein, 213-214  Principle of general covariance, 178, 186, 193-194, 207, 282, 299 Prussian Academy 1914 paper error immortalised, 153 Pseudo tensor, energy of the gravitational field (1914), 114 Pseudo tensor, energy of the gravitational field (1915), 138, 251  Quadrupole formula, 258  Relativistic mirror clock experiment, 97  Relativity paper (1907), 2, 2932, 179-180 Relativity principle (general), 54, 71, 194, 206, 268, 323, 330 Relativity principle (special), 3, 13-14, 22, 29, 32, 38, 46-48, 70-71, 73, 102-104, 209, 268 Relativity principle (1916), general covariance, Kretschmann's criticism of, 193-194, 282 Relativity principle (1918), Point Coincidence Argument, 194, 282 

Renouncing German citizenship, 361 Reply to Born on gravitational waves, 242-243 Rigid body, 41, 173-176  Robertson's advice on gravitational waves to, 261265 Rotating system not solution of Entwurf field equations, 8384, 89, 128-130 Special relativity paper (1905), xv, 13-14 Schwarzschild Singularity (Hadamard catastrophe), xii, 220-222 Schwarzschild Singularity Einstein-Rosen bridge, 222, 224, 228-231, 260 Schwarzschild Singularity 1939, 234-235, 363-364 Scratch Notebook, 43, 303  Seeliger interaction with, 212 Seventieth birthday, 363-364, 369-370, 374, 378 Silberstein's antagonism to Einstein's theory of relativity, 213-214, 226, 298-299  Static gravitational field theory, 43 Static gravitational first field theory, 8, 40-42 Static gravitational first field theory with flaw, 42-43 Static gravitational second field theory, 8, 43-44 Steady-state model, 353-345  Stubbornness of, 116-117, 124127, 253-254  Stubbornness of a mule, Straus' reference to, 236 Summation convention with, 139 

General Relativity Conflict and Rivalries Tour with Weizmann to US, 325-326 Tricks played on biographers and journalists by, 368-369 Two-body problem in general relativity by, xi, 211, 226, 317-319 Unified field theory, xiv, 102, 163, 188, 222-224, 238-239, 261, 266, 310, 344, 346, 349, 367 Unified field theory (1919), 310-313, 316-317 Unified field theory (1928), 347 Unified field theory (1929), xiiixiv, 348-349  Uniformly rotating disc (1912), 8, 41-42, 173-175  Uniformly rotating disc (1916), 178-184, 339  Unimodular coordinates by, 142-144, 166, 216, 243-244, 247-251, 254, 256  Unimodular transformations by, 132, 134-137, 142, 151-152, 157  "Uses the word God more often than a religion man", 194195 Vacuum field equations (1915), 140, 147, 163, 181, 184, 203-204, 214, 217, 227, 229, 243, 265-266, 288, 313-314, 316 Vacuum field equations (1919), x, 225-226, 312, 316-318  Variable velocity of light as a function of the place, 8-9, 11, 31, 37-38, 40, 45-49, 53, 239-240, 286, 289  Variational formalism (1914), 106-107, 111-114, 117-120, 130-131, 141-142, 170 Variational formalism (1915), 139-141, 164 

413

Variational formalism (1916), 165-170, 314  Variational formalism, priority claim by de Donder against, 362 Vienna talk, 19, 76, 91-92, 101, 103-104, 242-243  Visit to Eddington, 344 Weyl's 1918 book Space-TimeMatter, xii, 306-307, 309, 314  Weyl's influence on Einstein's 1919 unified field theory, 310 Weyl-Trefftz line element problem identified by, 226, 317 Wiechert's explanation of the perihelion of Mercury, 212213 World-matter, 269-270, 275278, 284, 323, 326

 Zurich Notebook, 11, 54-71, 133, 136, 202-203  Einstein, Hermann, 17 Electrical factory of, 17 Einstein, Rudolf, 17 Ether, 19th century, x, 6, 102-104, 172, 209-210, 212, 214, 291, 323, 325  Euclidean geometry, 9, 41-42, 5051, 173-174, 178, 184, 207 Evershed, John, 299 Exotic matter, 232-233 Expanding universe, 236-237, 281, 331-339, 341, 343, 351-354, 356, 360-365, 371, 374, 380 Faraday, Michael, 224 Induction law, 209  Feynman, Richard, 266 Fisher, Emil, 24 

414 Fokker, Adriaan, 21-22, 91, 98-101, 151, 160, 253, 256  Einstein-Fokker, EinsteinNordström theory, see Einstein Frank, Philipp, 4-5, 7  Freundlich, Erwin, 8, 21-24, 90-91, 212, 300 Criticism of Seeliger's dust hypothesis, 212 Deflection of light, 23, 90 Eclipse expedition, 23-24, 90, 300  Friedmann, Aleksandr, ix, 331-334, 336, 340-343, 352-356, 358, 363, 365  Einstein's objection to Friedmann's model, see Einstein Friedmann's Model, 331-332, 355 Friedmann-Robertson-Walker space-time metric, see Robertson Fuller, Robert, 230-231 Galileo principle of free fall, see Einstein Gamow, George, 366-374 Alphabeta-gamma theory, 369 Age of the universe, 371 A jokester, 368 Cosmological constant/term is the biggest blunder", see Einstein Literary gifts, 370 Tricks played on Einstein by, see Einstein Gerber, Paul, 213 German University of Prague (KarlFerdinand University), 4-7, 13 Gnehm, Robert, 10, 19  Gödel, Kurt, 373-380 Compass of inertia, 375-376

Index Einstein's reply to Gödel's backward time travel, see Einstein  Impossibility of backward time travel, 377 Non-static rotating universe, 380  Static rotating universe and backward time travel, 375376 Static rotating universe (negative cosmological constant), 374-376  Static rotating universe (positive cosmological constant), 380 Static rotating universe violates Mach's principle, 375  Gold, Thomas, 353-354  Göttingen Academy of Sciences, 145, 153, 161  Grandfather paradox, 376 Grebe, Leonhard, 186 Grommer, Jacob, 224-225  Grossmann, Marcel, vii, 9-12, 15, 22, 50, 53-55, 58-60, 67, 70-73, 82-86, 91-93, 98-100, 105-107, 116, 155, 203, 252, 290  Einstein's solution anteceded by, see Einstein  1913 Entwurf gravitation theory and Einstein-Grossmann field equations, see Einstein  1913 Entwurf EinsteinGrossmann paper, see Einstein 1914 Entwurf EinsteinGrossmann paper, see Einstein "Grossmann you have to help me", see Einstein On Zurich Notebook, see Einstein Haber, Fritz, 12, 15, 17-18, 20-22  Habicht, Konrad, 78 Hadamard, Jacques, xii-xiii, 220222, 349 

General Relativity Conflict and Rivalries Hadamard Catastrophe, see Einstein Hall, Asaph, 79 Heckmann, Otto, 357 Herman, Robert C., 369 Hertz, Paul, 61 Hilbert, David, x, xii, 3, 103, 116, 127-128, 131, 139, 144-145, 150-154, 161-172, 306, 310, 313-314  Axiomatic method, 163-165 Competition with Einstein, see Einstein Einstein probably read paper of, see Einstein Einstein told "Your solution is entirely different from mine" by, see Einstein Einstein's action together with, see Einstein Einstein's benefit from interactions with, see Einstein Einstein's perihelion motion calculation congratulated by, see Einstein Einstein's criticism of axiomatic method of, see Einstein Einstein's final solution possibly induced by field equations of, see Einstein Einstein's proofs of November 4, 1915 paper received by, 139 Flaw found in Entwurf 1914 gravitation tensor by, see Einstein Mie's influence on, 103 Paper of November 20, 1915 by, xi, 3, 103, 153, 155, 161171, 306  Personal rivalry with Einstein, see Einstein Possible influence on Einstein by, see Einstein

415

Theory of gravitation and electromagnetism, 310 Variational formalism, 165-171, 313-314 Hoffmann, Banesh, 227, 239, 265266  Non-existence of gravitational waves, (Einstein–Infeld– Hoffmann (Einstein–Infeld– Hoffmann 1938 theory), see Einstein Hoffmann, Ernst Theodor Wilhelm, 377 Hole Argument, see Einstein Hopf, Ludwig, 47, 59, 77, 154  Hoyle, Fred, 353, 368  Hubble, Edwin Powell, 236, 337344, 350-356, 360-361, 363, 365, 371, 373, 375  Discrepancy between geological findings and those of, 371 Einstein's visit to, see Einstein Hubble's law, 337 Ronald Angel jingle, 350-351 Support for de Sitter spectral shift effect, 338-340 Huygens, principle, 287-290 Infeld, Leopold, 151, 194-195, 222223, 227-228, 239, 261-166, 287  Gravitational waves do not exist (Einstein–Infeld–Hoffmann 1938 theory), see Einstein Einstein's gravitational waves, 262-264  Einstein's unified field theory explained to, see Einstein Institute for Advanced Studies, Princeton (Fine Hall), 194, 222, 235, 239, 261-263, 303, 367369, 373  Jaumann, Gustav, 5 Jeans, James, 291

416 Kaiser Wilhelm Institute for Physics, 12, 20, 25 Kaiser Wilhelm Institute of Physical Chemistry and Electrochemistry, 12, 16, 22  Kaiser Wilhelm Society, 12, 21  Kapteyn, Jacobus Cornelius, 326, 330 Dark matter, 326, 330 Karl-Ferdinand University, see German University of Prague Kepler's laws, 78, 81, 299  Klein, Felix, ix, 283-284, 314  Kleiner, Alfred, 5  Kollros, Louis, 14  Koppel, Leopold, 15-16, 21  Kreisky, Bruno, 374 Kretschmann, Erich, 192-194, 282 Criticism of Einstein's principles, see Einstein Krüß, Hugo, 12 Lampa, Anton, 4-7  Lanczos, Kornel, 335 Clarification of de Sitter's model, 35 Larmor, Joseph, 104 Laue von, Max, 20, 34-35, 37, 8586  Scalar, see Einstein  Objections to Einstein's paper, see Einstein Le Verrier, Urbain Jean Joseph, 78, 145  Lemaître, George Abbé, xi, 334338, 340-344, 351, 355-357, 359-364, 371  Big bang primeval atom theory, 284, 359-361 On de Sitter's model 1925, 334335, 338 Eddington's reaction to big bang theory of, see Eddington Einstein's criticism of the cosmological constant of, see Einstein

Index Einstein's reaction to big bang theory of, see Einstein Einstein on cosmological model of, see Einstein Lemaître-Eddington model, see Eddington Lemaître's model, 336-337 Levi-Civita, Tullio, x, xiv, 11, 54, 57, 72, 117, 120-121, 123-127, 162, 242, 251-253, 256-257, 261-262, 313, 344-347  Einstein described as Newton of the Day by, see Einstein Levi-Civita connection, 345347 Meeting Einstein in Princeton, see Einstein Parallel transport introduced by, 344, 347-348 Polemic with Einstein on Entwurf gravitational tensor, see Einstein Polemic with Einstein on pseudo tensor, see Einstein Lorentz, Hendrik Antoon, 6-7, 2122, 75-76, 91, 104, 162-163, 166-167, 171-172, 245, 253, 285, 297, 313, 323-324, 332  Contraction of lengths, see Einstein Einstein informed about Eddington's results by, see Einstein Electron theory, 6, 172 Lorentz transformations, 38-39, 41, 46, 51, 56, 92, 325  Variational formalism, 162, 166-167, 171 Löwenthal, Elsa, 17-18, 23, 361 Low water mark of general relativity xvi, 305 

 Mach, Ernst, 5, 87-90, 207-208, 211, 267, 269, 323, 329, 354, 365, 375  Death in 1916 of, 267

General Relativity Conflict and Rivalries Mach's ideas, see Einstein Mach's ideas, analogy to the effect of electromagnetic induction, see Einstein Mach's principle, see Einstein Mach's principle of economy of thought, 365  Mach on Newton's bucket experiment, 87-89 Mandl, Rudi W., 303-304 Gravitational lensing, see Einstein Mariü, Mileva, 4, 22  Marx, Erich, 50 Mathematical Society of Göttingen, 144-145 Maxwell, James Clerk, 224 Electromagnetic theory and equations, xi, 6, 102-103, 139, 225, 245, 262, 266, 306, 310, 311, 347-348  Electromagnetic waves, 40, 244, 246, 257, 262 Mayer, Robert, 47, 154  McCrea, William Hunter, 343 McVittie, George C., 341-342  Mie, Gustav, viii, 101-104, 116, 165, 171, 206-207, 256, 310, 313  Classical unified field theory, 76, 102-103, 165, 171 Einstein inspired by Mie's ideas, see Einstein 1913 Entwurf theory attacked in Vienna by, see Einstein 1913 Entwurf theory attacked in 1914 by, see Einstein Fin de Siècle theory, 104  Inspires Hilbert, 165, 171, 313 Polemic with Einstein on principle of equivalence, see Einstein Minkowski, Hermann, 9, 38-39, 47, 49, 51, 101  Minkowski's four-dimensional space-time formalism, 9, 38-

417

40, 45, 47-50, 55, 102-103, 268, 273-274, 276, 278  Flat metric space-time, 51, 55, 79, 93, 134, 146-147, 202204, 214-215, 236-237, 243246, 250, 257, 259, 267269, 272-273, 289, 299, 338, 276  Moore, Joseph, 187 Morgenstein, Oskar, 374 Mössbauer, Rudolf, 188 Mössbauer effect, 188, 190 Nernst, Walther, 12-21 Neumann, Carl, 270-271 Neumann-Seeliger paradox, 270-271 Newcomb, Simon, 79 Newton, Isaac, 29, 79, 87-88, 196, 207-208, 252, 299  Bucket experiment, 87-89, 195, 288 Newton-Cartan theory, see Cartan Newton-Cartan theory, correspondence principle, see Cartan Newton, Opticks, 292 Newton, Principia, 29, 196, 207  Newtonian absolute motion, 87, 89, 195, 288  Newtonian absolute space, 8788, 195, 205, 268, 274, 278, 288, 365 Newtonian absolute time, 197, 199-200, 206, 268, 273-274, 330  Newton's deflection of light, 292, 298  Newton of the Day (Levi-Civita on Einstein), see Einstein Newtonian limit, Einstein's correspondence principle, see Einstein Newtonian orbit equation, 148149

418 Newtonian mechanics, xiv-xv, 57, 87, 89-90, 148-149, 197198, 208-210, 216, 243, 278, 365  Newtonian theory of gravity, xiv-xv, 23, 28, 36, 40, 60, 72-73, 78-80, 82-83, 87, 96, 144, 164, 194-196, 200, 202-206, 210, 213, 215-216, 224, 266, 270-272, 286, 300, 305, 307, 348, 369  Post-Newtonian approximation, 266 Two globes thought experiment, 207-208 Non-Euclidean geometry, 6, 9, 174, 292 Nordmann, Charles, 220-221  Nordström, Gunnar, viii, 21, 47-49, 84-91, 100-102, 104, 116, 242, 253-256  Attacks 1913 Entwurf theory, 104 Converts to Einstein's theory, see Einstein Criticises Einstein's 1912 theory of static fields, see Einstein Finds problem with Einstein's 1915 pseudo tensor, see Einstein  First scalar theory, 47-49, 84-86, 100  First scalar theory violates conservation of momentum and energy, see Einstein  Nordström-Einstein theory, see Einstein Nordström's theory true option for gravitational theory, 8687, 91  Nordström's theory not satisfying Mach's ideas, see Einstein Nordström's theory, perihelion of Mercury, see Einstein

Index Principle of equivalence conflicts with first theory of, see Einstein Second scalar theory, 86-91, 98, 101 Painlevé, Paul, 221  Pais, Abraham, 15, 59, 106, 298 Palatini variational method, 347 Parallel transport, xiv, 196, 198-199, 344-348 Physical Society, 162, 286  Pick, Georg, 7 Planck, Max, 6, 12-23, 233  Copernicus (Planck on Einstein), see Einstein. Plesch, János, 368 Poincaré, Henri, 9 Point Coincidence Argument, see Einstein  Post-Newtonian approximation, see Newton Pound, Robert, 188, 190-192  Pound and Rebka redshift experiment, 191-192  Prussian Academy of Sciences, 1216, 18-22, 131, 142, 145, 150, 153-154, 161, 166, 214, 217, 219, 243, 248, 256, 307, 361 Quantum foam, 233 Quantum gravity, 232-233 Quantum of light hypothesis, 3-4, 6, 13-14, 102, 225  Quantum mechanics, 224, 239, 341 Rebka, Glen, 188, 190-192 Pound and Rebka redshift experiment, see Pound  Ricci, Gregorio Curbastro, 11, 54, 57, 72, 194  Ricci scalar, 165, 168-170, 314  Ricci tensor, 11-12, 59-61, 6465, 67-73, 99, 135-136, 142143, 146, 157, 166, 201203, 244, 314 

General Relativity Conflict and Rivalries Ricci tensor in Newton-Cartan theory, see Cartan Riemann, Georg Friedrich Bernhard, 11, 54, 59-60  Riemann-Christoffel tensor, 12, 67-70, 99-100, 136, 157, 168, 272 Riemann tensor, 59, 67, 72, 135136  Robertson, Harvey Percy, 235-236, 238, 261, 263-265, 338, 358  Robertson on gravitational waves, see Einstein  Friedmann-Robertson-Walker spacetime metric, 358-359 Robertson on Schwarzschild's singularity, 235-236 Rosen, Nathan, 222-224, 226-228, 230, 238-239, 260-262, 264-265  Einstein–Podolsky–Rosen) EPR) argument, see Einstein Einstein-Rosen bridge, see Einstein Einstein-Rosen bridge (Schwarzschild nontraversable wormhole), see Einstein Einstein (Rosen) 1937 paper on gravitational waves, see Einstein Einstein-Rosen 1936 submission paper on gravitational waves, see Einstein Physical Review affair, see Einstein. Royal Astronomical Society, xii, 237, 281, 286, 298, 301, 342343, 351  Royal Society of London, 185, 298299  Rubens, Heinrich, 12-14, 19-21 Schilpp, Paul Arthur, 364, 378 Schmidt, Friedrich, 20 Schrödinger, Erwin, 239, 256 

419

Schwarzschild, Karl, 103, 162, 211, 214-219, 222, 250, 313 Black hole, x, 238  Einstein on Schwarzschild Singularity (1939), see Einstein Exact exterior solution, xi-xii, 215-216, 218-220, 229-230, 236-237, 254, 313-315, 329 Exact exterior solution and unimodular coordinates, 216-217, 254-255 Exact interior solution, xi-xii, 218-220, 222, 236-237, 314319, 329. Limit, 219, 222 Non-traversable wormhole, see Einstein Radius, 219-220, 222, 229-230, 234-235, 238, 315-316, 329  Singularity, xii-xiii, 219-220, 224, 228-231, 234-236, 238, 260, 315, 329, 344, 361, 363-364. Singularity (Einstein-Rosen bridge), see Einstein Singularity (Hadamard catastrophe), see Einstein  Seeliger, von Hugo Hans Ritter, 79, 145, 211-213, 270-271  Antagonistic to Einstein's theory of relativity, 213 Criticism of Freundlich, 212 Dust hypothesis, 79, 145, 211212 Einstein on Seeliger, see Einstein  Neumann-Seeliger paradox, see Neumann Silberstein adopts hypothesis of, see Silberstein  Shapiro, Irwin, 240 Shapiro delay, 38, 240  Silberstein, Ludwig, 213, 225-227, 298-299, 320-321 

Index

420 Adopts Seeliger's hypothesis, 213 Against Einstein's principle of equivalence, see Einstein Averse to Einstein's theory of relativity, see Einstein  Averse to deflection of light, 214, 298-299 Averse to de Sitter spectral shift effect, 320-321 Averse to gravitational redshift, 214, 299 Averse to perihelion of Mercury, 213, 299 Polemic with Einstein on twobody problem, see Einstein Three persons understand general relativity, 298 Slipher, Vesto Melvin, 278, 280, 319, 322, 337  Smart, William Marshall, 342-343  Sommerfeld, Arnold, 153, 212, 309  St. John, Charles Edward, 186, 299  Stark, Johannes, 29-30 Stodola, Aurel, 3 Straus, Ernst Gabor, 236-237 Einstein-Straus vacuole, SwissCheese model, see Einstein  Stubbornness of a mule (Straus on Einstein), see Einstein Swiss Federal Polytechnic, 3, 9-13, 19-20  Thomson, Sir Joseph John, 298  Thorne, Kip, 232-233  Time machine, 232, 377 Tolman, Richard, 279, 338  Torsion, 345-348 Trefftz, Erich, xi, 226, 316-317 Problem with Weyl-Trefftz line element, see Einstein. Weyl-Trefftz line element, see Weyl Trumpler, Robert J., 301-302  Twin paradox, 234 University of Berlin, 16, 18, 361



University of Zurich, 3-5, 13  Vant Hoff, Jacobus, 18 Vibert Douglas, Alice, 327 Walker, Arthur Geoffrey, 358 Friedmann-Robertson-Walker space-time metric, see Roberston. Warburg, Emile, 12-14, 19-21 Weak field Weak field linearized approximation, 243 Weak field static approximation, 243 Weak-field approximation 56, 60, 62, 71, 133, 202-203  Weak principle of equivalence, 196199  Weiss, Pierre, 9  Weizmann, Chaim, 325 US tour with Einstein, see 1921  Einstein Wells, Herbert George, 328 Weyl, Hermann, xi-xii, xiv, 163, 171-172, 259, 283, 306-310, 313-322, 327, 340-341, 344345, 377  Affine connection, 344-345 Backward time travel, 377 de Sitter spectral shift effect, 340  Einstein's 1918 reaction to Space-Time-Matter, see Einstein Einstein identifies problem with unified field theory of, see Einstein Einstein's objections to unified field theory of, see Einstein Einstein's 1919 unified field theory influenced by, see Einstein Favouring de Sitter's empty universe, 283, 319 Re-derivation of Schwarzschild solutions, 313-315



General Relativity Conflict and Rivalries Space-Time-Matter book by, xi, 171-172, 306, 309, 314, 318, 320  Three types of gravitational waves, 259 Unified field theory of, xiii-xiv, 306-310 Variational formalism, 313-315 Weyl-Bach two-body problem, 319 Weyl-Trefftz line element, x-xi, 316-318 Weyl's Principle, 319-320 Wheeler, John Archibald, 230-231, 238

421

Wiechert, Emil, 212 Einstein's criticism of, see Einstein Wien, Wilhem, 40, 42, 44, 104, 267  Wormhole, 230-234  Non-traversable wormhole, 231232 Traversable wormhole, 232-234 Zangger, Heinrich, 2-3, 10, 21, 44, 128, 153 Zurich Polytechnic, see Swiss Federal Polytechnic Zurheilen, Walther, 24