Beyond pseudo-rotations in pseudo-Euclidean spaces 9780128117736, 0128117737

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces An Introduction to the Theory of Bi-gyrogroups and Bi-gyrovector Spaces

Abraham A. Ungar North Dakota State University, Department of Mathematics, Fargo, North Dakota, USA

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1800, San Diego, CA 92101-4495, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom c 2018 Elsevier Inc. All rights reserved. Copyright  No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-12-811773-6 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Candice Janco Acquisition Editor: Graham Nisbet Editorial Project Manager: Susan Ikeda Production Project Manager: Surya Narayanan Jayachandran Designer: Matthew Limbert Typeset by SPi Global, India

To my daughters Tamar and Ziva, my sons Ilan and Ofer, and my grandchildren for love and inspiration. Abraham A. Ungar

ACKNOWLEDGMENTS

The author is indebted to Nikita Barabanov for discussing and improving several topics that play an important role in the book. Abraham A. Ungar 2017

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PREFACE

This book is introductory in that it requires no previous exposure to pseudo-Euclidean spaces. What lie in the book beyond pseudo-rotations in pseudo-Euclidean spaces are the Lorentz transformations of signature (m, n), m, n ∈ N, and their parametric realization. Lorentz transformations of signature (m, n) are well known in the mathematical and physical literature. Of particular interest in the physical literature is the Lorentz transformation of signature (1, 3), which is the Lorentz transformation of Einstein’s special theory of relativity. The intended audience of the book includes graduate students and explorers who are interested in the study of Lorentz transformations in pseudo-Euclidean spaces as a powerful mathematical tool for establishing novel results in 1. non-associative algebra; 2. non-Euclidean geometry; and 3. relativity physics. In 1988 the author had realized parametrically the Lorentz transformations of signature (1, n), n ∈ N, by an orientation parameter along with a velocity parameter that turns out to be a gyrovector. Surprisingly, the parametric realization of the Lorentz transformations of signature (1, n) led to the discovery of two novel algebraic objects that possess universal significance. These are 1. the gyrogroup and 2. the gyrovector space. A gyrovector space is a gyrocommutative gyrogroup into which a scalar multiplication is incorporated. The significance of n-dimensional gyrovector spaces rests on the fact that they form the algebraic setting for n-dimensional analytic hyperbolic geometry, just as n-dimensional vector spaces form the algebraic setting for n-dimensional analytic Euclidean geometry. Analytic hyperbolic geometry is the hyperbolic geometry of Lobachevsky and Bolyai, studied analytically. The resulting study of analytic hyperbolic geometry is available in seven books that the author published during 2001–2015. The significance of results that stem from the 1988 parametric realization of the Lorentz transformations of signature (1, n), n > 1, led the author to extend the parametric realization of the Lorentz transformations of signature (m, n) to all m, n > 1. Not unexpectedly, this extension leads to the discovery of the two algebraic objects that are mentioned in the book subtitle, namely, xv

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Preface

1. the bi-gyrogroup and 2. the bi-gyrovector space. The discovery of these two algebraic objects is unfolded in the book along with the novel generalized analytic hyperbolic geometry, called bi-hyperbolic geometry of signature (m, n), to which they give rise. In the same way that 1. n-dimensional vector spaces form the algebraic setting for n-dimensional analytic Euclidean geometry and 2. n-dimensional gyrovector spaces form the algebraic setting for n-dimensional analytic hyperbolic geometry (as shown in Chap. 3); it is shown in the book that 3. bi-gyrovector spaces of signature (m, n), m, n ∈ N, form the algebraic setting for analytic bi-hyperbolic geometry of signature (m, n). In the special case when m = 1, bi-hyperbolic geometry of signature (m, n) descends to n-dimensional hyperbolic geometry. A point of bi-hyperbolic geometry of signature (m, n), m, n ∈ N, is a system of m n-dimensional subpoints, which are in geometric entanglement when m, n > 1. Hence, for instance, a triangle in bi-hyperbolic geometry of signature (m, n) is a system of m subtriangles, which are in geometric entanglement when m, n > 1. In the special case when n = 3, n-dimensional hyperbolic geometry is the geometry that underlies relativistic particle motion in Einstein’s special theory of relativity. Similarly, in the special case when n = 3, bi-hyperbolic geometry of signature (m, n) is the geometry that underlies the relativistic collective motion of a particle system that consists of m subparticles, which are in geometric entanglement. There are indications in the book that the observed geometric entanglement in bi-hyperbolic geometry of signature (m, 3), m > 1, regulates the omnipresence of entanglement of m particles in physics. In particular, it is demonstrated in the book that the existence of a universal finite speed limit poses no conflict with particle entanglement, even when the constituent particles are separated by a large distance. Abraham A. Ungar Fargo, North Dakota, USA 2017

ABOUT THE AUTHOR

Abraham A. Ungar is professor in the Department of Mathematics at North Dakota State University, since 1984. After gaining his B.Sc. in Mathematics and Physics (1965) and M.Sc. in Mathematics (1967) from the Hebrew University and Ph.D. from Tel-Aviv University in Applied Mathematics (1973), he held a postdoctoral position at the University of Toronto. His favored research areas are related to hyperbolic geometry and its applications in relativity physics, as indicated in the several books that he published since 2001. He currently serves on the editorial boards of Journal of Geometry and Symmetry in Physics; Communications in Applied Geometry; and Mathematics Interdisciplinary Research.

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CHAPTER 1

Introduction

1.1. Introduction In the context of relativity physics, pseudo-rotations are known as Lorentz transformations of signature (m, n), which are in turn, Lorentz transformations in m time dimensions and n space dimensions. A pseudo-Euclidean space Rm,n of signature (m, n), m, n ∈ N, is an (m + n)dimensional space with the pseudo-Euclidean inner product of signature (m, n). A Lorentz transformation of signature (m, n) is a special linear transformation Λ ∈ SO(m, n) in Rm,n that leaves the pseudo-Euclidean inner product invariant. It is special in the sense that the determinant of the (m + n) × (m + n) real matrix Λ is 1, and the determinant of its first m rows and columns is positive [36, p. 478]. Equivalently, it is special in the sense that it can be reached continuously from the identity transformation. Details are presented in Chap. 4. Lorentz transformations of signature (m, n) are well known in algebra. They are, however, studied in the book in a novel way that stems from their parametric realization. Indeed, what lie in the book beyond pseudo-rotations in pseudo-Euclidean spaces include the parametric realization of Lorentz transformations of signature (m, n). The space of the resulting main parameter encodes new algebraic objects called a bigyrogroup and a bi-gyrovector space. Accordingly, the book forms an introduction to the theory of these algebraic objects, demonstrating their universal significance. The parametric realization of the special relativistic Lorentz transformation of signature (1, n), n > 1, in 1988 [74] gives rise to Einstein gyrogroups and Einstein gyrovector spaces. Einstein gyrovector spaces, in turn, form the algebraic setting for analytic hyperbolic geometry, as demonstrated in Chaps. 2 and 3. In full analogy, the parametric realization of Lorentz transformations of signature (m, n), m, n > 1, gives rise in the book to Einstein bi-gyrogroups and Einstein bigyrovector spaces of signature (m, n). Einstein bi-gyrovector spaces, in turn, form the algebraic setting for analytic bi-hyperbolic geometry, as demonstrated in the book. Bihyperbolic geometry turns out to form a natural extension of the analytic hyperbolic geometry studied in [81, 84, 93, 94, 95, 96, 98]. Remarkably, unlike hyperbolic geometry, bi-hyperbolic geometry of signature (m, n), m, n > 1, involves Beyond Pseudo-rotations in Pseudo-Euclidean Spaces http://dx.doi.org/10.1016/B978-0-12-811773-6.50001-3 Copyright © 2018 Elsevier Inc. All rights reserved.

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geometric entanglement, a reminiscent of the entanglement observed in physics. The n-dimensional hyperbolic geometry of Lobachevsky and Bolyai is, in fact, a bi-hyperbolic geometry of signature (1, n), n > 1. In this sense, bi-hyperbolic geometry is a natural generalization of hyperbolic geometry from signature (1, n) to signature (m, n), m, n ∈ N.

1.2. Quantum Entanglement and Geometric Entanglement Quantum entanglement is a physical phenomenon that occurs when groups of particles interact in ways such that the quantum state of each particle cannot be described independently of the others, even when the particles are separated by a large distance. Instead, a quantum state must be described for a system of particles as a whole. Quantum entanglement is a great mystery in physics. Einstein, Podolsky, and Rosen have issued in 1935 a challenge to quantum physics, claiming that the theory is incomplete. They based their argument on the existence of the entanglement phenomenon, which in turn had been deduced to exist based on mathematical considerations of quantum systems. They, accordingly, claimed that the theory that allows for the “unreal” phenomenon of entanglement has to be incomplete [17]. Contrasting Einstein, Podolsky, and Rosen, we will see in the book that geometric entanglement is quite real within the frame of a natural extension of special relativity (1) from a theory that stems from the common Lorentz transformation of signature (1, n) (which is regulated by hyperbolic geometry) (2) to a theory that stems from the Lorentz transformation of signature (m, n), m, n ∈ N (which is regulated by bi-hyperbolic geometry). The related extension of hyperbolic geometry to bi-hyperbolic geometry, in turn, is associated with the natural emergence of the Lorentz transformation of signature (m,n), m, n > 1, into Einstein’s special relativity theory. These generalized Lorentz transformations involve m temporal dimensions and n spatial dimensions as well as a geometric entanglement. The resulting geometric entanglement shares with quantum entanglement a characteristic property. They involve particle systems in such a way that the motion of a constituent particle of a system cannot be described independently of the motion of the others, even when the constituent particles of the system are separated by a large distance. Instead, the motion of the constituent particles must be described for the system as a whole.

1.3. From Galilei to Lorentz Transformations The Galilei transformation G(v) expresses what we intuitively understand about a particle in motion. It is a linear transformation of time-space coordinates parametrized by

Introduction

a velocity parameter v,

given by the matrix

⎛ ⎞ ⎜⎜⎜v1 ⎟⎟⎟ ⎜ ⎟ v = ⎜⎜⎜⎜v2 ⎟⎟⎟⎟ ∈ R3 = R3×1 , ⎝ ⎠ v3 ⎛ ⎜⎜⎜ 1 ⎜⎜⎜v G(v) = ⎜⎜⎜⎜⎜ 1 ⎜⎜⎝v1 v1

0 1 0 0

0 0 1 0

⎞ 0⎟⎟ ⎟ 0⎟⎟⎟⎟⎟ ⎟ ∈ R4×4 , 0⎟⎟⎟⎟ ⎠ 1

(1.1)

(1.2)

where Rn is the Euclidean n-space and Rn×m is the space of all n × m real matrices, m, n ∈ N. The application of the Galilei transformation G(v) to a (1 + 3)-dimensional timespace particle ⎛ ⎞  ⎜⎜⎜⎜ t ⎟⎟⎟⎟ ⎜⎜ x ⎟⎟ t = ⎜⎜⎜⎜⎜ 1 ⎟⎟⎟⎟⎟ ∈ R4 = R4×1 , (1.3) x ⎜⎜⎝ x2 ⎟⎟⎠ x3 with position x = (x1 , x2 , x3 )t ∈ R3 (exponent t denotes transposition) at time t ∈ R yields the following result, which is intuitively clear. The Galilei transformation G(v), also called a Galilei boost, boosts the particle (t, x)t by a velocity v, moving its position x to the boosted position x + vt at time t according to the chain of equations ⎛ ⎞⎛ ⎞ ⎛ ⎞  ⎜⎜⎜⎜ 1 0 0 0⎟⎟⎟⎟ ⎜⎜⎜⎜ t ⎟⎟⎟⎟ ⎜⎜⎜⎜ t ⎟⎟⎟⎟  ⎜⎜⎜v1 1 0 0⎟⎟⎟ ⎜⎜⎜ x1 ⎟⎟⎟ ⎜⎜⎜v1 t + x1 ⎟⎟⎟ t t ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ . (1.4) = ⎜⎜⎜ G(v) ⎟⎜ ⎟ = ⎜ ⎟= x + vt x ⎜⎜⎝v2 0 1 0⎟⎟⎟⎟⎠ ⎜⎜⎜⎜⎝ x2 ⎟⎟⎟⎟⎠ ⎜⎜⎜⎜⎝v2 t + x2 ⎟⎟⎟⎟⎠ v3 0 0 1 x 3 v3 t + x 3 The velocity parameter v ∈ R3×1 of G(v) in (1.4) is a 3 × 1 real matrix. We therefore call G(v) a Galilei transformation of signature (m, n) = (1, 3), anticipating the extension to Galilei transformations of signature (m, n), for all m, n ∈ N, along with a link to Lorentz transformations of signature (m, n). Once we accept the postulates of special relativity, the Galilei transformation G(v) must be replaced by a corresponding Lorentz transformation L(v). The corresponding Lorentz transformation must descend to the Galilei transformation in the Galilean limit c → ∞. Indeed, it is shown in (6.16), p. 303, that the Lorentz transformation L(v) that corresponds to the Galilei transformation G(v) is the common Lorentz transformation

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in special relativity, which can be written as ⎞ ⎛ γ 2 v ⎟⎟⎟ ⎜⎜⎜ v (v v v ) 1 2 3 ⎟⎟⎟ ⎜⎜⎜⎜ 1+γv ⎛ ⎞ ⎛ ⎞ ⎟⎟⎟ 1 1 ⎜⎜ v v ⎜ ⎟ ⎜ ⎟ 1 1 ⎟⎟⎟ =: G(v) + 2 E(v) ∈ R4×4 . ⎜ ⎜ ⎟ ⎜ ⎟ L(v) = G(v) + 2 γv ⎜⎜ γ ⎜⎜⎜ ⎟⎟⎟ γv ⎜⎜⎜ ⎟⎟⎟ 2 v ⎟ ⎜ c c ⎜⎜⎜ 1+γ v ⎜⎜⎜v2 ⎟⎟⎟ 1+γ ⎜⎜⎜v2 ⎟⎟⎟ (v1 v2 v3 )⎟⎟⎟ v ⎝ ⎠ ⎝ v ⎝ ⎠ ⎠ v3 v3 (1.5) Here, v2 = vt v = v2 < c2 , where c is the vacuum speed of light and γv is the Lorentz gamma factor (2.3), p. 11, of special relativity. The Lorentz transformation L(v) in (1.5) is presented in its additive decomposition form as the sum of a Galilei transformation G(v) and an entanglement part, E(v), multiplied by the factor 1/c2 . The additive decomposition (1.5) of the Lorentz transformation L(v) clearly exhibits the limit lim L(v) = G(v) .

c→∞

(1.6)

Hence, as expected, the Lorentz transformation L(v) tends to its corresponding Galilei transformation G(v) in the Newtonian limit c → ∞. The extension of the additive decomposition (1.5) from v ∈ R3×1 to V ∈ Rn×m , m, n ∈ N, is presented in (5.392), p. 261, suggesting the incorporation of Lorentz transformations of signature (m, n), m, n ∈ N, into special relativity (where n = 3 in physical applications) as the transformations that act collectively on particle systems consisting of m n-dimensional subparticles. The entanglement part E(v) of the Lorentz transformation L(v) in (1.5) is responsible for the emergence of counterintuitive results into special relativity as, for instance, (i) the entanglement of time and space; (ii) time dilation; (iii) length contraction; and (iv) Thomas precession. Owing to the factor 1/c2 in (1.5), the relativistic effects that the entanglement part E(v) generates are directly noticeable only at very high speeds. Unlike the Lorentz transformation L(v) in (1.5), the Galilei transformation G(v) in (1.2) and its action in (1.4) are intuitively clear. Accordingly, it is intuitively clear how to extend the Galilei transformation from the common transformation that acts on particles individually into a transformation that acts on particles collectively. In contrast, it is not yet clear how to extend the Lorentz transformation from the common transformation that acts on particles individually into a transformation that acts on particles collectively. Fortunately, the intuitively clear extended Galilei transformations will guide us in the search for the extended Lorentz transformations that act on particles collectively, rather than individually. Remarkably, the extended Lorentz transformations that the extended Galilei transformations suggest will turn out in the book to be nothing else but the well-known Lorentz transformations of signature (m, n), m, n > 1.

Introduction

1.4. Galilei and Lorentz Transformations of Particle Systems Similarly to (1.1), let

⎞ ⎛ ⎜⎜⎜v11 v12 ⎟⎟⎟ ⎟ ⎜ V = (v1 v2 ) = ⎜⎜⎜⎜v21 v22 ⎟⎟⎟⎟ ∈ R3×2 ⎠ ⎝ v31 v32

and, similarly to (1.2), let

⎛ 0 ⎜⎜⎜ 1 ⎜⎜⎜ 0 1 ⎜⎜⎜ ⎜ G(V) = ⎜⎜⎜v11 v12 ⎜⎜⎜ ⎜⎜⎝v21 v22 v31 v32

Furthermore, similarly to (1.3), let

0 0 1 0 0

⎛ t ⎛ ⎞ ⎜⎜⎜⎜⎜ 1  ⎜⎜ t1 0 ⎟⎟ ⎜⎜ 0 ⎜⎜ ⎟⎟ ⎜⎜ T = ⎜⎜⎜⎜ 0 t2 ⎟⎟⎟⎟ = ⎜⎜⎜⎜ x11 X ⎝ ⎠ ⎜⎜⎜ x1 x2 ⎜⎝⎜ x21 x31

0 0 0 1 0

(1.7)

⎞ 0⎟⎟ ⎟ 0⎟⎟⎟⎟⎟ ⎟ 0⎟⎟⎟⎟ ∈ R5×5 . ⎟ 0⎟⎟⎟⎟ ⎠ 1

(1.8)

⎞ 0 ⎟⎟ ⎟ t2 ⎟⎟⎟⎟⎟ ⎟ x12 ⎟⎟⎟⎟ ∈ R5×2 . ⎟ x22 ⎟⎟⎟⎟ ⎠ x32

(1.9)

Emphasizing the space Rn×m = R3×2 of the parameter V ∈ R3×2 , we call G(V) in (1.8) a Galilei transformation of signature (m, n) = (2, 3). It acts in (1.10) on a system consisting of two particles that as a whole has m = 2 time dimensions and n = 3 space dimensions. Each particle has its own clock, so that the time of the particle system of m = 2 particles has 2 dimensions, being measured by two clocks. While each of the two particles has its own clock, they share the same space. Hence, the space of the particle system has n = 3 dimensions. Finally, similar to (1.4) we have ⎛ ⎞ ⎞⎛ 1 0 0 0 0⎟⎟ ⎜⎜ t1 0 ⎟⎟ ⎛ ⎞ ⎜⎜⎜⎜⎜ ⎜ ⎟ ⎟  1 0 0 0⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ 0 t2 ⎟⎟⎟⎟⎟ ⎜⎜⎜ t1 0 ⎟⎟⎟ ⎜⎜⎜ 0 T ⎜ ⎟ ⎟ ⎜ ⎟⎜ = G(V) ⎜⎜⎜⎜ 0 t2 ⎟⎟⎟⎟ = ⎜⎜⎜⎜v11 v12 1 0 0⎟⎟⎟⎟ ⎜⎜⎜⎜ x11 x12 ⎟⎟⎟⎟ G(V) X ⎝ ⎟ ⎠ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ x1 x2 ⎜⎝⎜v21 v22 0 1 0⎟⎠⎟ ⎜⎝⎜ x21 x22 ⎟⎟⎟⎠⎟ v31 v32 0 0 1 x31 x32 (1.10) ⎛ ⎞ t1 0 ⎜⎜⎜ ⎟⎟⎟ ⎛ ⎞ ⎜⎜⎜⎜ ⎟⎟⎟⎟ ⎜⎜ t1 0 t 0 ⎟⎟⎟ 2 ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟ 0 t2 ⎟⎟⎟⎟ . = ⎜⎜⎜v11 t1 + x11 v12 t2 + x12 ⎟⎟⎟ = ⎜⎜⎜ ⎜⎜⎜ ⎟⎟⎟ ⎝ ⎠ x1 + v1 t1 x2 + v2 t2 ⎜⎜⎝v21 t1 + x21 v22 t2 + x22 ⎟⎟⎠ v31 t1 + x31 v32 t2 + x32

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The generalization 1. from a Galilei transformation G(v) of signature (1, 3) that acts on the time-space coordinates of a single particle in (1.4) 2. into a Galilei transformation G(V) of signature (2, 3) that acts collectively on the time-space coordinates of a system consisting of two 3-dimensional particles in (1.10) is intuitively clear. Further extension 3. into a Galilei transformation of signature (m, n), m, n ∈ N, that acts collectively on the time-space coordinates of a system consisting of m n-dimensional particles is obvious and intuitively clear as well. It is clear from (1.10) that the collective action of a Galilei transformation G(V) of signature (2, 3) on a system consisting of two particles is equivalent to an individual action of the standard Galilei transformation G(v) of signature (1, 3) on each particle of the system. Similarly, a collective action of a Galilei transformation of signature (m, n) on a system consisting of m n-dimensional particles is equivalent to an individual action of the standard Galilei transformation of signature (1, n) on each particle of the system. Hence, seemingly, the introduction of the concept of a collective action of the Galilei transformation on particle systems is useless, giving rise to no new phenomenon. But, we are not finished. It is demonstrated in the book that the passage from Newtonian physics to special relativistic physics, guided by analogies with the additive decomposition (1.5) of the Lorentz transformation, requires the replacement of a Galilei transformation of signature (m, n) by a corresponding Lorentz transformation of same signature (m, n). Remarkably, unlike the collective action on a particle system of the Galilei transformation of signature (m, n), m, n > 1, the collective action on a particle system of the Lorentz transformation of signature (m, n), m, n > 1, is not equivalent to an individual action of the standard Lorentz transformation of signature (1, n) on each particle of the system. This non equivalence results from the presence of the entanglement phenomenon. Under a collective action of a Lorentz transformation of signature (m, n), m, n > 1, the constituents of a particle system are in entanglement. The common Lorentz transformation of signature (1, n), n ≥ 1, in special relativity (n = 3 in physical applications) introduces entanglement in space and time. In the same way, the Lorentz transformation of signature (m, n), m, n > 1, introduces entanglement in the positions and in the times of the constituents of particle systems.

1.5. Chapters of the Book The book is self-contained. The required background in the theory of Einstein gyrogroups and Einstein gyrovector spaces is presented in Chaps. 2 and 3.

Introduction

Chapter 2: Einstein Gyrogroups. This chapter presents the required background about Einstein gyrogroups. Einstein velocity addition in his special relativity theory is neither commutative nor associative. However, it is both gyrocommutative and gyroassociative. The resulting rich structure of Einstein velocity addition gives rise in this chapter to an algebraic group-like object called a gyrogroup. Einstein gyrogroups turn out to be gyrocommutative, and they are destined in the book to be extended to bi-gyrocommutative bi-gyrogroups of signature (m, n), m, n ∈ N. Chapter 3: Einstein Gyrovector Spaces. This chapter presents the required background about Einstein gyrovector spaces. Einstein addition admits a scalar multiplication, turning Einstein gyrogroups into Einstein gyrovector spaces. It is shown in this chapter that Einstein gyrovector spaces form the algebraic setting for hyperbolic geometry with spectacular gain in clarity and simplicity, just as vector spaces form the algebraic setting for Euclidean geometry. Einstein gyrovector spaces are destined in the book to be extended to bi-gyrovector spaces of signature (m, n), m, n ∈ N. The latter form in the book the algebraic setting for bi-hyperbolic geometry of signature (m, n), m, n ∈ N. Remarkably, geometric entanglement emerges in bi-hyperbolic geometry when m, n > 1. When m = 1, bi-hyperbolic geometry of signature (1, n) coincides with hyperbolic geometry in n dimensions and, accordingly, involves no geometric entanglement. Chapter 4: Bi-gyrogroups and Bi-gyrovector Spaces – P. The group SO(m, n) of Lorentz transformations of signature (m, n), m, n ∈ N, is parametrized in this chapter by two orientation parameters and a main parameter P. The space Rn×m of P is the space of all real n × m matrices. This space and its Einstein addition of signature (m, n) give rise to Einstein bi-gyrogroups of signature (m, n). The latter, along with Einstein scalar multiplication of signature (m, n), give rise to Einstein bi-gyrovector spaces of signature (m, n). When m = 1, bi-gyrogroups and bi-gyrovector spaces coincide, respectively, with gyrogroups and gyrovector spaces. When (m, n) = (1, 3), the Lorentz transformation of signature (m, n) descends to the common Lorentz transformation of Einstein’s special relativity theory, and its parameter P descends to the relativistic proper velocity, also known as a traveler’s velocity. The study of the space Rn×m of the parameter P in this chapter paves the way to the study in Chap. 5 of the space of a new parameter, V, which is the ball Rn×m of Rn×m . c Chapter 5: Bi-gyrogroups and Bi-gyrovector Spaces – V. The parameter P ∈ Rn×m of the Lorentz transformation of signature (m, n), m, n ∈ N, studied in Chap. 4, is changed in this chapter into a new parameter V ∈ Rn×m in the c-ball Rn×m of the space c c n×m R . When (m, n) = (1, 3), the Lorentz transformation of signature (m, n) descends to the common Lorentz transformation of Einstein’s special relativity theory, and its parameter V ∈ Rn×m descends to the relativistically admissible velocity, also known as c an observer’s velocity. The study of bi-gyrogroups and bi-gyrovector spaces of the parameter V in this chapter depends on the study of bi-gyrogroups and bi-gyrovector spaces of the parameter P in Chap. 4.

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Chapter 6: Applications to Time-Space of Signature (m,n). Galilei boosts are extended in this chapter to Galilei bi-boosts of signature (m, n), m, n ∈ N. A Galilei biboost of signature (m, n) is a transformation that acts collectively on the constituents of a particle system of m n-dimensional particles, where n = 3 in physical applications. A Galilei bi-boost of signature (1, n) is a Galilei boost. The collective action of a Galilei bi-boost of any signature (m, n), m, n > 1, on a particle system is intuitively clear. In contrast, the collective action of a Lorentz bi-boost of any signature (m, n), m, n > 1, on a particle system is initially unknown. Fortunately, however, guided by patterns, Galilei bi-boosts of any signature (m, n), m, n ∈ N, naturally lead in this chapter to the following remarkable result. The Lorentz transformation that corresponds to the intuitively clear Galilei bi-boost of signature (m, n), m, n ∈ N, is the Lorentz bi-boost of same signature (m, n). Results associated with Lorentz bi-boosts of signature (m, n) are explored in this chapter, leading to a link between Einstein addition of signature (m, n) and the special relativistic Einstein addition of signature (1, 3). Chapter 7: Analytic Bi-hyperbolic Geometry: The Geometry of Bi-gyrovector Spaces. Einstein gyrovector spaces, presented in Chap. 3, have been extended to Einstein bi-gyrovector spaces of signature (m, n), m, n ∈ N, and studied in Chaps. 4 – 6. In the special case when m = 1, an Einstein bi-gyrovector space of signature (m, n) coincides with the n-dimensional Einstein gyrovector space. The geometry of an Einstein bi-gyrovector space of signature (m, n) is called bihyperbolic geometry of signature (m, n). Accordingly, in the special case when m = 1, bi-hyperbolic geometry of signature (1, n) coincides with n-dimensional hyperbolic geometry. A geometric entanglement phenomenon emerges in this chapter in the study of bihyperbolic geometry of signature (m, n), m, n > 1. Figures that illustrate graphically the geometric entanglement phenomenon are presented for signature (m, n) = (3, 2), where (n = 2)-dimensional velocity curves of (m = 3) entangled particles are shown. It is graphically clear that the relativistic velocity of a subparticle of a particle system depends on the relativistic velocities of the other constituent subparticles of the system. Accordingly, it is indicated in Chaps. 6 and 7 that the geometric entanglement phenomenon in bi-hyperbolic geometry regulates the omnipresence of the entanglement phenomenon in physics.

CHAPTER 2

Einstein Gyrogroups

2.1. Introduction This chapter presents the required background about Einstein gyrogroups. Einstein’s addition law of three-dimensional relativistically admissible velocities is the corner stone [88] of Einstein’s three-vector formalism of the special theory of relativity that he founded in 1905 [15, 51]. The resulting binary operation, ⊕, called Einstein addition, is presented along with the nonassociative algebraic structures that it encodes. These algebraic structures are the gyrocommutative gyrogroup structure, studied in this chapter, and the gyrovector space structure, studied in Chap. 3. It will turn out that Einstein gyrovector spaces form the algebraic setting for the n-dimensional Cartesian-Beltrami-Klein ball model of analytic hyperbolic geometry, just as vector spaces form the algebraic setting for the standard n-dimensional Cartesian model of analytic Euclidean geometry, n ∈ N. The gyrogroup notion emerged in 1988 following the discovery of the gyroassociative laws of Einstein addition, to which the relativistic effect, known as Thomas precession, gives rise [74]. Being a new mathematical structure that emerged from relativity physics, it clearly merits extension by abstraction in an axiomatic approach [75, 76]. The resulting abstract Thomas precession is called Thomas gyration, suggesting the prefix “gyro” that we use in the book extensively to emphasize analogies shared with classical structures, laws, and operations. The resulting formalism is naturally called gyrolanguage. It is a language in which we prefix a gyro to any term that describes a concept in Euclidean geometry and in associative algebra to mean the analogous concept in hyperbolic geometry and in nonassociative algebra. Our gyroterminology thus conveys a world of meaning in an elegant and memorable fashion. The history of the gyrogroup notion begins in 1988 [74] with the parametric realization of the Lorentz group in one temporal dimension and n spatial dimensions, n ∈ N, by two parameters. The two parameters are (i) an orientation parameter On ∈ SO(n) that forms the group SO(n) under parameter composition and (ii) a relativistically admissible velocity parameter that forms a group-like algebraic object under parameter composition. It was discovered in [74] that the resulting group-like object is a gyrocommutative gyrogroup. Beyond Pseudo-rotations in Pseudo-Euclidean Spaces http://dx.doi.org/10.1016/B978-0-12-811773-6.50002-5 Copyright © 2018 Elsevier Inc. All rights reserved.

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Initially, in 1989 the author coined the terms K-loop in [77] (see also [46]) to describe what we presently call a gyrocommutative gyrogroup, as recorded in [44, pp. 169-170]. The letter K is presented in K-loop to honor Karzel and, later [80], Kikkawa for their pioneering discoveries in [42] and [45], in which some identities that are now recognized as gyrogroup identities are found. Their related identities, however, could not be tested for consistency by means of concrete examples since concrete examples were not available prior to the 1988 discovery of the relativity gyrogroup in [74]. See also [62, p. 142], [81, Remark 6.12] and [82]. The term “K-loop” that the author coined in 1989 [77] has already been in use in other senses. It was introduced in 1970 by So˘ıkis [64] and later in a different context (as a generalized Moufang loop) by Basarab [3]. The origin of the letter K in each of the K-loops of So˘ıkis and of Basarab is unknown to the author. In 1990 the structure underlying Einstein addition became known as a weakly associative commutative group [78], and in 1991 it became known as a gyrogroup in order to indicate that it is a group-like structure that stems from Thomas gyration [79]. Finally, following the discovery of non-gyrocommutative structures, which otherwise are gyrogroups, gyrogroups became gyrocommutative gyrogroups in 1997 [80], to accommodate non gyrocommutative gyrogroups as well. Hence, currently, in full analogy with groups, gyrogroups are classified into gyrocommutative and nongyrocommutative ones, as we will see in Defs. 2.13 and 2.14. Our study of gyrogroups thus begins with the study of Einstein addition, revealing its intrinsic beauty, harmony, and interdisciplinarity.

2.2. Einstein Velocity Addition Let c > 0 be any positive constant and let Rn = (Rn , +, ·) be the Euclidean n-space, n ∈ N, equipped with the common vector addition, +, and inner product, ·. The home of all n-dimensional Einsteinian velocities is the c-ball Rnc = {v ∈ Rn : v < c} .

(2.1)

The c-ball Rnc is the open ball of radius c, centered at the origin of Rn , consisting of all vectors v in Rn with magnitude v smaller than c. Einstein velocity addition is a binary operation, ⊕, in the c-ball Rnc given by the equation [81], [62, Eq. 2.9.2], [56, p. 55], [24]   1 1 γu 1 u⊕v = (u·v)u , (2.2) u+ v+ 2 c 1 + γu γu 1 + u·v c2

Einstein Gyrogroups

for all u, v ∈ Rnc , where γu is the Lorentz gamma factor, 

v2 γv = 1 − 2 c

− 1

2

≥ 1,

(2.3)

where u·v and v are the inner product and the norm in the ball, which the ball Rnc inherits from its ambient space Rn , v2 = v·v. A nonempty set with a binary operation is called a groupoid so that the pair (Rnc , ⊕) is an Einstein groupoid. The constant c > 0 represents the vacuum speed of light in special relativity theory. In the Euclidean-Newtonian limit of large c, c → ∞, the ball Rnc expands to its ambient space Rn , as we see from (2.1), and Einstein addition ⊕ in Rnc descends to the common vector addition + in Rn , as we see from (2.2) and (2.3). When the nonzero vectors u and v in the ball Rnc of Rn are parallel in Rn , uv, that is, u = λv for some λ ∈ R \ 0, Einstein addition (2.2) descends to the Einstein addition of parallel velocities, u+v u⊕v = , uv , (2.4) 1 1 + 2 u·v c which was partially confirmed experimentally by the Fizeau’s 1851 experiment [54]. Following (2.4) we have u + v (2.5) 1 1 + 2 uv c n for all u, v∈Rc . One should note that ⊕ in (2.2) and (2.4) is a binary operation between n-dimensional vectors, while ⊕ in (2.5) is a binary operation between one-dimensional vectors, that is, between scalars. The restricted Einstein addition in (2.4) and (2.5) is both commutative and associative. Accordingly, the restricted Einstein addition is a group operation, as Einstein noted in [15]; see [16, p. 142]. In contrast, Einstein made no remark about group properties of his addition (2.2) of velocities that need not be parallel. Indeed, the general Einstein addition is not a group operation but, rather, a gyrocommutative gyrogroup operation, a structure discovered more than 80 years later, in 1988 [74, 75, 79], which we will study in Sect. 2.6. Einstein addition (2.2) of relativistically admissible velocities, with n = 3, was introduced by Einstein in his 1905 paper [15] [16, p. 141] that founded the special theory of relativity, where the magnitudes of the two sides of Einstein addition (2.2) are presented. One has to remember here that the Euclidean 3-vector algebra was not so widely known in 1905 and, consequently, was not used by Einstein. Einstein calculated in [15] the behavior of the velocity components parallel and orthogonal to the u⊕v =

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

relative velocity between inertial systems, which is as close as one can get without vectors to the vectorial version (2.2) of Einstein addition. Einstein was aware of the nonassociativity of his velocity addition law of relativistically admissible velocities that need not be collinear. Accordingly, he emphasized in his 1905 paper that his velocity addition law of relativistically admissible collinear velocities forms a group operation [15, p. 907]. We naturally use the abbreviation u v = u⊕(−v) for Einstein subtraction, so that, for instance, v v = 0 and v = 0 v = −v .

(2.6)

Einstein addition and subtraction satisfy the equations (u⊕v) = u v

(2.7)

u⊕(u⊕v) = v

(2.8)

and for all u, v in the ball Rnc , in full analogy with vector addition and subtraction in Rn . Identity (2.7) is called the gyroautomorphic inverse property of Einstein addition, and Identity (2.8) is called the left cancellation law of Einstein addition. We may note that Einstein addition does not obey the naive right counterpart of the left cancellation law (2.8) since, in general, (u⊕v) v  u .

(2.9)

However, this seemingly lack of a right cancellation law of Einstein addition is remedied in (2.98), p. 34. Einstein addition and the gamma factor are related by the gamma identity,  u·v  γu⊕v = γu γv 1 + 2 , (2.10) c which can be written, equivalently, as  u·v  γ u⊕v = γu γv 1 − 2 (2.11) c for all u, v ∈ Rnc . Here, (2.11) is obtained from (2.10) by replacing u by u = −u in (2.10). We will see in the book that Einstein addition gives rise to our gyrolanguage in which we prefix a gyro to any term that describes a concept in Euclidean geometry and in associative algebra to mean the analogous concept in hyperbolic geometry and in nonassociative algebra. The prefix “gyro” stems from “gyration,” which is the mathematical abstraction of the special relativistic effect known as “Thomas precession.” In gyrolanguage, thus, Einstein addition is called a gyroaddition. The gamma identity

Einstein Gyrogroups

(2.10) of Einstein gyroaddition will be extended in (5.219), p. 223, to the bi-gamma identities of Einstein bi-gyroaddition. A useful identity that follows immediately from (2.3) is v2 v2 γv2 − 1 = 2 = . c2 c γv2

(2.12)

Einstein addition is noncommutative. Indeed, while Einstein addition is commutative under the norm, u⊕v = v⊕u ,

(2.13)

u⊕v  v⊕u ,

(2.14)

in general, u, v ∈ Rnc . Moreover, Einstein addition is also nonassociative since, in general, (u⊕v)⊕w  u⊕(v⊕w) ,

(2.15)

u, v, w ∈ Rnc . As an application of the gamma identity (2.10), we prove the Einstein gyrotriangle inequality. Theorem 2.1. (Gyrotriangle Inequality, I). u⊕v ≤ u⊕v

(2.16)

for all u, v in an Einstein groupoid (Rnc , ⊕). Proof. By the gamma identity (2.10) and by the Cauchy-Schwarz inequality [52], we have   uv γu⊕v = γu γv 1 + c2  u·v  ≥ γu γv 1 + 2 (2.17) c = γu⊕v = γu⊕v for all u, v in an Einstein groupoid (Rnc , ⊕). But γx = γx is a monotonically increasing function of x, 0 ≤ x < c. Hence (2.17) implies u⊕v ≤ u⊕v for all u, v∈Rnc .

(2.18) 

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

The gyrotriangle inequality (2.16) will be extended from Einstein gyrovector spaces (Rnc , ⊕, ⊗) to Einstein bi-gyrovector spaces (Rn×m c , ⊕E , ⊗) of any signature (m, n), m, n ∈ N, in (5.558), p. 294. Remark 2.2. (Einstein Addition Domain Extension). Einstein addition u⊕v in (2.2) involves the gamma factor γu of u, while it is free of the gamma factor γv of v. Hence, unlike u, which must be restricted to the ball Rnc in order to insure the reality of a gamma factor, v need not be restricted to the ball. Hence, the domain of v can be extended from the ball Rnc to its ambient space Rn . Moreover, also the gamma identity (2.10) remains valid for all u ∈ Rnc and v ∈ Rn under appropriate choice of the square root of negative numbers. If 1 + u·v/c = 0, then u⊕v is undefined, and, by (2.10), γu⊕v = 0, so that u⊕v = ∞. Remark 2.3. (Generalized Einstein Addition). Einstein addition ⊕ in (2.2) will be generalized to the so-called Einstein addition ⊕E = ⊕E,(m,n),c of signature (m, n), m, n ∈ N, in the ball Rn×m c . Einstein addition ⊕ in (2.2) will, then, be recognized in Sect. 5.17 as the Einstein addition ⊕E,(1,n),c of signature (1, n) in the ball Rn×1 = Rnc , that is, ⊕ = c ⊕E = ⊕E,(1,n),c .

2.3. Einstein Addition with Respect to Cartesian Coordinates Ren´e Descartes and Pierre de Fermat revolutionized the study of Euclidean geometry with their introduction of Cartesian coordinate systems. Significant outcomes were the rise of analytic Euclidean geometry and the introduction of Euclidean vector spaces as an appropriate framework for Euclidean geometry. Accordingly, we face the task of revolutionizing the study of hyperbolic geometry with the introduction of Cartesian coordinate systems, giving rise to analytic hyperbolic geometry and the introduction of hyperbolic vector spaces as an appropriate framework for hyperbolic geometry. The resulting hyperbolic vector spaces are Einstein gyrovector spaces whose gyrovector addition is given by Einstein (velocity) addition. Einstein addition with respect to Cartesian coordinates is therefore required. Like any physical law, Einstein velocity addition law (2.2) is coordinate independent. Indeed, it is presented in (2.2) in terms of vectors, noting that one of the great advantages of vectors is their ability to express results independent of any coordinate system. However, in order to generate numerical and graphical demonstrations of laws in physics and results in analytic geometry, we need coordinates. Accordingly, we introduce Cartesian coordinates into the Euclidean n-space Rn and its ball Rnc , with respect to which we generate graphical presentations. Introducing the Cartesian coordinate

Einstein Gyrogroups

system Σ into Rn and Rnc , each point P ∈ Rn is given by an n-tuple P = (x1 , x2 , . . . , xn ),

x12 + x22 + · · · + xn2 < ∞ ,

(2.19)

of real numbers, which are the coordinates, or components, of P with respect to Σ. Similarly, each point P ∈ Rnc is given by an n-tuple P = (x1 , x2 , . . . , xn ),

x12 + x22 + · · · + xn2 < c2 ,

(2.20)

of real numbers, which are the coordinates, or components, of P with respect to Σ. Equipped with a Cartesian coordinate system Σ and its standard vector addition given by component addition, along with its resulting scalar multiplication, Rn forms the standard Cartesian model of n-dimensional Euclidean geometry. In full analogy, equipped with a Cartesian coordinate system Σ and its Einstein addition, along with its resulting scalar multiplication (to be studied in Sect. 3.3, p. 67), the ball Rnc forms the Cartesian-Beltrami-Klein ball model of n-dimensional hyperbolic geometry (as we will see in Chap. 3, particularly, in (3.39) – (3.40), p. 72). As an illustrative example, we present below the Einstein velocity addition law (2.2) in R3c with respect to a Cartesian coordinate system. Let R3c be the c-ball of the Euclidean 3-space, equipped with a Cartesian coordinate system Σ, ⎛ ⎞ ⎧⎛ ⎞ ⎫ ⎪ ⎪ ⎜⎜⎜ x1 ⎟⎟⎟ ⎜⎜⎜ x1 ⎟⎟⎟  ⎪ ⎪ ⎪ ⎪ ⎨⎜⎜ ⎟⎟ ⎬ ⎟⎟⎟ 3 ⎜ 2 2 2 ⎜ ⎜ ⎜ ⎟ x x = ∈ R . (2.21) R3c = ⎪ : x + x + x < c ⎪ 2⎟ 2⎟   ⎜ ⎜ ⎪ ⎪ 1 2 3 ⎜ ⎜ ⎟ ⎟ ⎝ ⎠ ⎪ ⎪ ⎩⎝ x ⎠ ⎭ x3 3 Accordingly, each point of the ball is represented by its coordinates (x1 , x2 , x3 )t (exponent t denotes transposition) with respect to Σ, satisfying the condition x12 + x22 + x32 < c2 . Furthermore, let u, v, w ∈ R3c be three points in R3c ⊂ R3 given by their coordinates with respect to Σ, ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜⎜⎜v1 ⎟⎟⎟ ⎜⎜⎜w1 ⎟⎟⎟ ⎜⎜⎜u1 ⎟⎟⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ v = ⎜⎜⎜⎜v2 ⎟⎟⎟⎟ , w = ⎜⎜⎜⎜w2 ⎟⎟⎟⎟ , (2.22) u = ⎜⎜⎜⎜u2 ⎟⎟⎟⎟ , ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ u3 v3 w3 where w = u⊕v .

(2.23)

The dot (inner) product of u and v is given in Σ by the equation u·v = u1 v1 + u2 v2 + u3 v3 ,

(2.24)

and the squared norm v2 = v·v of v is given by the equation v2 = v21 + v22 + v23 .

(2.25)

15

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Hence, it follows from the coordinate-free, vector representation (2.2) of Einstein addition that the coordinate Einstein addition (2.23) with respect to the Cartesian coordinate system Σ takes the form ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜⎜⎜u1 ⎟⎟⎟ ⎜⎜⎜v1 ⎟⎟⎟ ⎜⎜⎜w1 ⎟⎟⎟ 1 ⎜⎜⎜ ⎟⎟⎟ ⎜ ⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎝w2 ⎟⎟⎠ = ⎜⎜⎝u2 ⎟⎟⎠ ⊕ ⎜⎜⎜⎜⎝v2 ⎟⎟⎟⎟⎠ = u 1 v 1 + u2 v 2 + u3 v 3 1+ w3 u3 v3 c2 (2.26) ⎛ ⎞ ⎛ ⎞⎫ ⎧ ⎪ ⎪ u v ⎜ ⎟ ⎜ ⎟ 1 1 ⎪ ⎪ ⎜ ⎟ ⎜ ⎟ ⎪ ⎜⎜ ⎟⎟ 1 ⎜⎜⎜ ⎟⎟⎟⎪ 1 γu ⎬ ⎨ ⎜v ⎟⎪ , [1 + 2 × ⎪ (u1 v1 + u2 v2 + u3 v3 )] ⎜⎜⎜⎜u2 ⎟⎟⎟⎟ + ⎪ ⎪ ⎪ ⎝ ⎠ γu ⎜⎜⎝ 2 ⎟⎟⎠⎪ c 1 + γu ⎩ u v ⎭ 3

3

where γu = 

1 u2 + u22 + u23 1− 1 c2

.

(2.27)

Note that (i) γu is real if and only if u < c, (ii) γu = ∞ if and only if u = c, and (iii) γu is purely imaginary if and only if u > c. The three components of Einstein addition (2.23) are w1 , w2 , and w3 in (2.26). For a two-dimensional illustration of Einstein addition (2.26) one may impose the condition u3 = v3 = 0, implying w3 = 0. An illustrative example in two dimensions is presented in Example 2.4. In the Newtonian-Euclidean limit, c → ∞, the ball R3c expands to the Euclidean 3-space R3 , and Einstein addition (2.26) reduces to the common vector addition in R3 , ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜⎜⎜w1 ⎟⎟⎟ ⎜⎜⎜u1 ⎟⎟⎟ ⎜⎜⎜v1 ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ (2.28) ⎜⎜⎝w2 ⎟⎟⎠ = ⎜⎜⎝u2 ⎟⎟⎠ + ⎜⎜⎝v2 ⎟⎟⎠ . w3 u3 v3 We will find that Einstein addition plays in the Cartesian model of the BeltramiKlein ball model of hyperbolic geometry the same role that vector addition plays in the Cartesian model of Euclidean geometry. Suggestively, the Cartesian-Beltrami-Klein ball model of hyperbolic geometry is also known as the relativistic velocity model [1, 2]. Vector equations and identities are represented by coordinate-free expressions, like Einstein addition in (2.2). For numerical and graphical presentations, however, these must be converted into a coordinate dependent form relative to a Cartesian coordinate system that must be introduced. The latter, in turn, can be presented relative to Cartesian coordinates numerically and graphically, as we do in the generation of figures. In general, Cartesian coordinates are not shown in figures. For the sake of demonstration, however, they are shown in Figs. 3.3 and 3.4, p. 77.

Einstein Gyrogroups

Example 2.4. As an illustrative example of a two-dimensional Einstein addition with respect to a Cartesian coordinate system, we employ (2.26) to calculate the elegant result of the Einstein sum (0, b)t ⊕(x, b)t in the relativistic velocity plane        x x 2 2  2 2 ∈ R :   = x + y < c (2.29) Rc =  y  y of two-dimensional relativistically admissible velocities, equipped with the Cartesian coordinate system Σ = (x, y). Following (2.26) we have         x 0 x 0 ⊕ = ⊕ b −b b b =

1 1−

b2 s2

⎧  ⎫   ⎪ ⎪ ⎪ γb b2 0 1 x⎪ ⎨ ⎬ (1 − ) + ⎪ ⎪ ⎪ 2 ⎩ ⎭ 1 + γb c −b γb b ⎪

=

γb2

⎧    ⎫ ⎪ ⎪ ⎪ γb γb2 − 1 0 1 x⎪ ⎬ ⎨ + (1 − ) ⎪ ⎪ ⎪ 2 ⎭ ⎩ −b 1 + γb γb γb b ⎪

=

γb2

⎧    ⎫ ⎪ ⎪ ⎪ 1 x⎪ ⎨1 0 ⎬ + ⎪ ⎪ ⎪ ⎩ γ −b ⎭ b⎪ γ b b

(2.30)

  x , = γb 0 so that

      0 x  b ⊕ b  = γb |x| .

(2.31)

2.4. Einstein Addition vs. Vector Addition Vector addition, +, in Rn is both commutative and associative, satisfying u+v=v+u u + (v + w) = (u + v) + w

Commutative Law Associative Law (2.32)

for all u, v, w ∈ Rn . In contrast, Einstein addition, ⊕, in Rnc is neither commutative nor associative. Gyrations gyr[u, v] ∈ Aut(R3c , ⊕), u, v ∈ R3c , are defined in terms of Einstein addition by the equation

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

gyr[u, v]w = (u⊕v)⊕{u⊕(v⊕w)}

(2.33)

for all u, v, w ∈ R3c . Equation (2.33) presents the application to w of the gyration gyr[u, v] generated by u and v. Gyrations turn out, in general, to be nontrivial automorphisms of the Einstein groupoid (R3c , ⊕). An automorphism of a groupoid (S , ⊕) is a bijective map f of S onto itself that respects the binary operation, that is, f (a⊕b) = f (a)⊕ f (b) for all a, b ∈ S . The set of all automorphisms of a groupoid (S , ⊕) forms a group, denoted by Aut(S , ⊕), where the group operation is given by automorphism composition. To emphasize that the gyrations of an Einstein groupoid (R3c , ⊕) are automorphisms of the groupoid, gyrations are also called gyroautomorphisms. A gyration gyr[u, v], u, v ∈ R3c , is trivial if gyr[u, v]w = w for all w ∈ R3c . Thus, for instance, the gyrations gyr[0, v], gyr[v, v], and gyr[v, v] are trivial for all v ∈ R3c , as we see from (2.33). More generally, gyrations gyr[u, v] are trivial when u, v ∈ Rnc ⊂ Rn are parallel in Rn . Possessing their own rich structure, gyrations measure the extent to which Einstein addition deviates from commutativity and associativity as we see from the following list of identities [81, 84, 93]: u⊕v = gyr[u, v](v⊕u) u⊕(v⊕w) = (u⊕v)⊕gyr[u, v]w (u⊕v)⊕w = u⊕(v⊕gyr[v, u]w) gyr[u⊕v, v] = gyr[u, v] gyr[u, v⊕u] = gyr[u, v] gyr[ u, v] = gyr[u, v]

Gyrocommutative Law Left Gyroassociative Law Right Gyroassociative Law Gyration Left Reduction Property Gyration Right Reduction Property Gyration Even Property

(gyr[u, v])−1 = gyr[v, u]

Gyration Inversion Law (2.34)

for all u, v, w ∈ Rnc . It is clear from (2.34) that the departure of Einstein addition, ⊕, from commutativity and associativity is strictly controlled by gyrations. The reduction properties in (2.34) present important gyration identities. One of them, the left reduction property, will soon demonstrate its power and elegance in solving the gyrogroup equation x⊕a = b in (2.87) and (2.93), p. 32. The elegant properties of Einstein gyroaddition and gyrations in (2.34) give rise to a new mathematical formalism and its associated gyrolanguage. These properties will be extended and, then, verified. They will be extended in Chap. 5 to Einstein bi-gyroaddition and bi-gyrations of signature (m, n), m, n ∈ N, where the special sig-

Einstein Gyrogroups

nature (m, n) = (1, 3) corresponds to Einstein’s special relativity theory in m = 1 time dimension and n = 3 space dimensions.

2.5. Gyrations Owing to its nonassociativity, Einstein addition gives rise in (2.33) to gyrations, gyr[u, v] : Rnc → Rnc ,

(2.35)

of an Einstein groupoid (Rnc , ⊕) for any u, v ∈ Rnc . Gyrations, in turn, regulate Einstein addition, ⊕, endowing it with the rich structure of a gyrocommutative gyrogroup, as we will see in Sect. 2.6, and a gyrovector space, as we will see in Sect. 3.3. In the formal approach to gyrogroups in Def. 2.13, p. 22, the left reduction property is elevated to the Reduction Axiom. The gyration left reduction axiom is also known as the left loop property. The more revealing term, reduction axiom, was coined by F. Chatelin in [9] since it triggers remarkable reduction in complexity as, for instance, in (2.93), p. 32. Gyrations are defined in (2.33) in terms of Einstein addition. An explicit presentation of the gyrations of Einstein groupoids (Rnc , ⊕) in terms of vector addition rather than Einstein addition is given by the equation gyr[u, v]w = w +

Au + Bv , D

(2.36)

where A=− + B=−

γu2 1 1 (γv − 1)(u·w) + 2 γu γv (v·w) 2 c (γu + 1) c γu2 γv2 2 (u·v)(v·w) c4 (γu + 1)(γv + 1)

(2.37)

1 γv {γ (γ + 1)(u·w) + (γu − 1)γv (v·w)} c2 γv + 1 u v

D = γu γv (1 +

u·v ) + 1 = γu⊕v + 1 ≥ 2 c2

for all u, v, w ∈ Rnc . Remark 2.5. (Gyration Domain Extension). The domain of u, v∈Rnc ⊂ Rn in (2.36) – (2.37) is restricted to Rnc in order to insure the reality of the gamma factors of u and v in (2.37). However, while the expressions in (2.36) – (2.37) involve gamma factors of u and v, they involve no gamma factors of w. Hence, the domain of w in (2.36) – (2.37) can be extended from Rnc to Rn . Indeed, extending in (2.36) – (2.37) the domain of w

19

20

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

from Rnc to Rn , gyrations gyr[u, v] are expanded from maps of Rnc to linear maps of Rn for any u, v ∈ Rnc , gyr[u, v] : Rn → Rn . In each of the three special cases when (i) u = 0, or (ii) v = 0, or (iii) u and v are parallel in Rn , uv, we have Au + Bv = 0 so that gyr[u, v] is trivial. Thus, we have gyr[0, v]w = w gyr[u, 0]w = w gyr[u, v]w = w,

(2.38) uv ,

Rnc ,

for all u, v ∈ where uv in the third equation, and for all w ∈ Rn . It follows from (2.36) by straightforward algebra that gyr[v, u](gyr[u, v]w) = w

(2.39)

for all u, v ∈ Rnc , w ∈ Rn , or equivalently, gyr[v, u]gyr[u, v] = I

(2.40)

for all u, v ∈ Rnc , where I denotes the trivial map, also called the identity map. Hence, gyrations are invertible linear maps of Rn , the inverse, gyr−1 [u, v], of gyr[u, v] being gyr[v, u]. We thus have the gyration inversion property gyr−1 [u, v] = gyr[v, u]

(2.41)

for all u, v ∈ Rnc . Gyrations keep the inner product of elements of the ball Rnc invariant, that is, gyr[u, v]a·gyr[u, v]b = a·b

(2.42)

for all a, b, u, v ∈ Rnc . Hence, gyr[u, v] is an isometry of Rnc , keeping the norm of elements of the ball Rnc invariant, gyr[u, v]w = w .

(2.43)

Accordingly, gyr[u, v] represents a rotation of the ball Rnc about its origin for any u, v ∈ Rnc . The invertible map gyr[u, v] of Rnc respects Einstein addition in Rnc , gyr[u, v](a⊕b) = gyr[u, v]a⊕gyr[u, v]b

(2.44)

for all a, b, u, v ∈ Rnc , so that gyr[u, v] is an automorphism of the Einstein groupoid (Rnc , ⊕). Example 2.6. As an example that illustrates the use of the invariance of the norm

Einstein Gyrogroups

under gyrations, we note that  u⊕v = u v =  v⊕u .

(2.45)

Indeed, we have the following chain of equations, which are numbered for subsequent derivation: (1)

  u⊕v ===  ( u⊕v) (2)

 === u v (3)

 === gyr[u, v]( v⊕u)

(2.46)

(4)

 ===  v⊕u for all u, v ∈ Rnc . Derivation of the numbered equalities in (2.46): (1) Follows from the result that w = −w, so that  w =  − w = w for all w ∈ Rnc . (2) Follows from the automorphic inverse property (2.7), p. 12, of Einstein addition. (3) Follows from the gyrocommutative law of Einstein addition. (4) Follows from the result that, by (2.43), gyrations keep the norm invariant.

2.6. From Einstein Velocity Addition to Gyrogroups Guided by analogies with groups, the key features of Einstein groupoids (Rnc , ⊕), n = 1, 2, 3, . . ., suggest the formal gyrogroup definition in which gyrogroups form a most natural generalization of groups. Accordingly, definitions related to groups are presented in order to set the stage for the definition of gyrogroups. Definition 2.7. (Binary Operations). A binary operation + in a set S is a function + : S ×S → S . We use the notation a + b to denote +(a, b) for any a, b ∈ S . Definition 2.8. (Groupoids, Automorphisms). A groupoid (S , +) is a nonempty set, S , with a binary operation, +. An automorphism φ of a groupoid (S , +) is a bijective self-map of S which respects its groupoid operation, that is, φ(a + b) = φ(a) + φ(b) for all a, b ∈ S . Definition 2.9. (Groups). A groupoid (G, +) is a group if its binary operation satisfies the following axioms. In G there is at least one element, 0, called a left identity, satisfying (G1) 0+a=a

21

22

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

for all a ∈ G. There is an element 0∈G satisfying axiom (G1) such that for each a∈G there is an element −a∈G, called a left inverse of a, satisfying (G2) −a + a = 0 . Moreover, the binary operation obeys the associative law (G3) (a + b) + c = a + (b + c) for all a, b, c ∈ G. Groups split up into commutative and noncommutative ones. Definition 2.10. (Commutative Groups). A group (G, +) is commutative if its binary operation obeys the commutative law (G6) a+b=b+a for all a, b ∈ G. Definition 2.11. (Subgroups). A subset H of a group (G, +) is a subgroup of G if it is nonempty, and H is closed under group compositions and inverses in G, that is, x, y ∈ H implies x + y ∈ H and −x ∈ H. Theorem 2.12. (The Subgroup Criterion). A subset H of a group (G, +) is a subgroup of G if and only if (i) H is nonempty and (ii) x, y ∈ H implies x − y ∈ H. For a proof of the Subgroup Criterion see any book on group theory. Definition 2.13. (Gyrogroups). A groupoid (G, ⊕) is a gyrogroup if its binary operation satisfies the following axioms. In G there is at least one element, 0, called a left identity, satisfying (G1) 0⊕a = a for all a ∈ G. There is an element 0 ∈ G satisfying axiom (G1) such that for each a ∈ G there is an element a ∈ G, called a left inverse of a, satisfying (G2) a⊕a = 0 . Moreover, for any a, b, c ∈ G there exists a unique element gyr[a, b]c ∈ G such that the binary operation obeys the left gyroassociative law (G3) a⊕(b⊕c) = (a⊕b)⊕gyr[a, b]c . The map gyr[a, b] : G → G given by c → gyr[a, b]c is an automorphism of the groupoid (G, ⊕), that is, (G4) gyr[a, b] ∈ Aut(G, ⊕) , and the automorphism gyr[a, b] of G is called the gyroautomorphism, or the gyration, of G generated by a, b ∈ G. The operator gyr : G × G → Aut(G, ⊕) is called the

Einstein Gyrogroups

gyrator of G. Finally, the gyroautomorphism gyr[a, b] generated by any a, b ∈ G possesses the left reduction property (G5) gyr[a, b] = gyr[a⊕b, b] , called the reduction axiom. The gyrogroup axioms (G1) – (G5) in Definition 2.13 are classified into three classes: 1. The first pair of axioms, (G1) and (G2), is a reminiscent of the group axioms. 2. The last pair of axioms, (G4) and (G5), presents the gyrator axioms. 3. The middle axiom, (G3), is a hybrid axiom linking the two pairs of axioms in (1) and (2). As in group theory, we use the notation a b = a⊕( b) in gyrogroup theory as well. In full analogy with groups, gyrogroups split up into gyrocommutative and nongyrocommutative ones. Definition 2.14. (Gyrocommutative Gyrogroups). A gyrogroup (G, ⊕) is gyrocommutative if its binary operation obeys the gyrocommutative law (G6) a ⊕ b = gyr[a, b](b ⊕ a) for all a, b ∈ G. The abstract gyrocommutative gyrogroup is an algebraic structure tailor made to suit Einstein velocity addition of relativistically admissible velocities. Indeed, the Einstein groupoid (Rnc , ⊕) is a gyrocommutative gyrogroup. Gyrogroups, both gyrocommutative and nongyrocommutative, abound in group theory as shown in [25] and [26]. A finite, non-gyrocommutative gyrogroup of order 16, K16 , is presented in [81, Figs. 2.1-2.2, p. 41]. Einstein addition in the real ball Rnc can straightforwardly be extended to the complex ball Cnc , giving rise to the complex Einstein groupoid (Cnc , ⊕), studied in [63, 83, 92, 23].

2.7. Gyrogroup Cooperation (Coaddition) Our plan to capture analogies with groups dictates the introduction into the abstract gyrogroup (G, ⊕) a second binary operation, , called the gyrogroup cooperation, or coaddition. Definition 2.15. (Gyrogroup Cooperation (Coaddition)). Let (G, ⊕) be a gyrogroup. The gyrogroup cooperation (or, coaddition), , is a second binary operation in G related to the gyrogroup operation (or, addition), ⊕, by the equation a  b = a⊕gyr[a, b]b

(2.47)

23

24

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

for all a, b ∈ G. Naturally, we use the notation a  b = a  ( b) where b = −b, so that a  b = a gyr[a, b]b .

(2.48)

The gyrogroup cooperation is commutative if and only if the gyrogroup operation is gyrocommutative, as we will see in Theorem 2.44, p. 50. Hence, in particular, Einstein coaddition  is commutative since Einstein addition ⊕ is gyrocommutative. Indeed, let us calculate, as a concrete example of (2.47), the Einstein coaddition . By substituting into (2.47) both 1. Einstein addition in (2.2), p. 10, and 2. Einstein gyration gyr[u, v]w in (2.33), p. 18, lengthy, but straightforward, algebra (that can be handled easily by employing a computer algebra system like Mathematica) reveals the following important result: Einstein coaddition  is given explicitly by the equation uv=

γu2

+

γv2

γ u + γv + γu γv (1 +

u·v ) s2

−1

(γu u + γv v)

γ u + γv v = 2⊗ u γu + γv

(2.49)

for all u, v ∈ Rnc where, by definition, 2⊗v = v⊕v. Einstein coaddition (2.49) of two summands is extended to k summands, k ≥ 2, in [98, Eqs. (6.23) and (6.84)]. Owing to its commutativity, Einstein coaddition proves useful in a new method to process color images via mathematical morphology [7]. In fact, with the appearance of ´ Einstein coaddition, Emile Borel’s dream to create a commutative version of Einstein addition came true [100].

2.8. First Gyrogroup Properties While it is clear how to define a right identity and a right inverse in a gyrogroup, the existence of such elements is not presumed. Indeed, the existence of a unique identity and a unique inverse, both left and right, is a consequence of the gyrogroup axioms, as the following theorem shows, along with other immediate, important results in gyrogroup theory. Theorem 2.16. (First Gyrogroup Properties). Let (G, ⊕) be a gyrogroup. For any elements a, b, c, x ∈ G we have: 1. If a⊕b = a⊕c, then b = c (general left cancellation law; see Item (9)).

Einstein Gyrogroups

2. 3. 4. 5. 6. 7. 8. 9.

gyr[0, a] = I for any left identity 0 in G. gyr[x, a] = I for any left inverse x of a in G. gyr[a, a] = I There is a left identity which is a right identity. There is only one left identity. Every left inverse is a right inverse. There is only one left inverse, a, of a, and ( a) = a. The Left Cancellation Law: a⊕(a⊕b) = b .

(2.50)

gyr[a, b]x = (a⊕b)⊕{a⊕(b⊕x)}.

(2.51)

10. The Gyrator Identity:

11. gyr[a, b]0 = 0 . 12. gyr[a, b]( x) = gyr[a, b]x . 13. gyr[a, 0] = I . Proof. The proof of each item of the theorem follows: 1. Let x be a left inverse of a corresponding to a left identity, 0, in G. We have x⊕(a⊕b) = x⊕(a⊕c), implying (x⊕a)⊕gyr[x, a]b = (x⊕a)⊕gyr[x, a]c

(2.52)

by left gyroassociativity. Since 0 is a left identity, gyr[x, a]b = gyr[x, a]c. Since automorphisms are bijective, b = c. 2. By left gyroassociativity we have for any left identity 0 of G a⊕x = 0⊕(a⊕x) = (0⊕a)⊕gyr[0, a]x = a⊕gyr[0, a]x .

(2.53)

Hence, by Item (1) we have x = gyr[0, a]x for all x ∈ G so that gyr[0, a] = I, I being the trivial (identity) map. 3. By the left reduction property and by Item (2) we have gyr[x, a] = gyr[x⊕a, a] = gyr[0, a] = I .

(2.54)

4. Follows from an application of the left reduction property and Item (2). Thus, I = gyr[0, a] = gyr[0⊕a, a] = gyr[a, a] .

(2.55)

5. Let x be a left inverse of a corresponding to a left identity, 0, of G. Then, by left gyroassociativity and Item (3), x⊕(a⊕0) = (x⊕a)⊕gyr[x, a]0 = 0⊕0 = 0 = x⊕a . Hence, by (1), a⊕0 = a for all a ∈ G so that 0 is a right identity.

(2.56)

25

26

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

6. Suppose 0 and 0∗ are two left identities, one of which, say 0, is also a right identity. Then, 0 = 0∗ ⊕0 = 0∗ . 7. Let x be a left inverse of a. Then, x⊕(a⊕x) = (x⊕a)⊕gyr[x, a]x = 0⊕x = x = x⊕0 ,

(2.57)

by left gyroassociativity, (G2) of Def. 2.13 and Items (3), (5), (6). By Item (1) we have a⊕x = 0 so that x is a right inverse of a. 8. Suppose x and y are left inverses of a. By Item (7), they are also right inverses, so a⊕x = 0 = a⊕y. By Item (1), x = y. Let a be the resulting unique inverse of a. Then, a⊕a = 0 so that the inverse ( a) of a is a. 9. By left gyroassociativity and by 3 we have a⊕(a⊕b) = ( a⊕a)⊕gyr[ a, a]b = b .

(2.58)

10. By an application of the left cancellation law in Item (9) to the left gyroassociative law (G3) in Def. 2.13 we obtain the result in Item (10). 11. We obtain Item (11) from Item (10) with x = 0. 12. Since gyr[a, b] is an automorphism of (G, ⊕) we have from Item (11) gyr[a, b]( x)⊕gyr[a, b]x = gyr[a, b]( x⊕x) = gyr[a, b]0 = 0

(2.59)

and hence the result. 13. We obtain Item (13) from Item (10) with b = 0, and a left cancellation, Item (9). 

2.9. Elements of Gyrogroup Theory Einstein gyrogroups (G, ⊕) possess the gyroautomorphic inverse property, according to which (a⊕b) = a b for all a, b ∈ G. In general, however, (a⊕b)  a b in some gyrogroups. Hence, the following theorem is interesting. Theorem 2.17. (Gyrosum Inversion Law). For any two elements a, b of a gyrogroup (G, ⊕) we have the gyrosum inversion law (a⊕b) = gyr[a, b]( b a) .

(2.60)

Proof. By the gyrator identity in Theorem 2.16(10) and a left cancellation, given in Theorem 2.16(9), we have gyr[a, b]( b a) = (a⊕b)⊕(a⊕(b⊕( b a))) = (a⊕b)⊕(a a) = (a⊕b) .

(2.61) 

Einstein Gyrogroups

Theorem 2.18. For any two elements, a and b, of a gyrogroup (G, ⊕), we have gyr[a, b]b = { (a⊕b)⊕a} gyr[a, b]b = (a b)⊕a .

(2.62)

Proof. The first identity in (2.62) follows from Theorem 2.16(10) with x = b, and Theorem 2.16(12), and the second part of Theorem 2.16(8). The second identity in (2.62) follows from the first one by replacing b by b, noting Theorem 2.16(12).  A nested gyroautomorphism is a gyration generated by points that depend on another gyration. Thus, for instance, some gyrations in (2.63) – (2.65) are nested. Theorem 2.19. Any three elements a, b, c of a gyrogroup (G, ⊕) satisfy the nested gyroautomorphism identities gyr[a, b⊕c]gyr[b, c] = gyr[a⊕b, gyr[a, b]c]gyr[a, b]

(2.63)

gyr[a⊕b, gyr[a, b]b]gyr[a, b] = I

(2.64)

gyr[a, gyr[a, b]b]gyr[a, b] = I

(2.65)

and the gyroautomorphism product identities gyr[ a, a⊕b]gyr[a, b] = I

(2.66)

gyr[b, a⊕b]gyr[a, b] = I .

(2.67)

Proof. By two successive applications of the left gyroassociative law in two different ways, we obtain the following two chains of equations for all a, b, c, x ∈ G: a⊕(b⊕(c⊕x)) = a⊕((b⊕c)⊕gyr[b, c]x) = (a⊕(b⊕c))⊕gyr[a, b⊕c]gyr[b, c]x

(2.68)

and a⊕(b⊕(c⊕x)) = (a⊕b)⊕gyr[a, b](c⊕x) = (a⊕b)⊕(gyr[a, b]c⊕gyr[a, b]x) = ((a⊕b)⊕gyr[a, b]c)⊕gyr[a⊕b, gyr[a, b]c]gyr[a, b]x

(2.69)

= (a⊕(b⊕c))⊕gyr[a⊕b, gyr[a, b]c]gyr[a, b]x . By comparing the extreme right-hand sides of these two chains of equations, and by

27

28

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

employing the left cancellation law, Theorem 2.16(1), we obtain the identity gyr[a, b⊕c]gyr[b, c]x = gyr[a⊕b, gyr[a, b]c]gyr[a, b]x

(2.70)

for all x ∈ G, thus verifying (2.63). In the special case when c = b, (2.63) reduces to (2.64), noting that the left-hand side of (2.63) becomes trivial owing to Items (2) and (3) of Theorem 2.16. Identity (2.65) results from the following chain of equations, which are numbered for subsequent derivation: (1)

 I === gyr[a⊕b, gyr[a, b]b]gyr[a, b] (2)

 === gyr[(a⊕b) gyr[a, b]b, gyr[a, b]b]gyr[a, b] (3)

 === gyr[a⊕(b b), gyr[a, b]b]gyr[a, b]

(2.71)

(4)

 === gyr[a, gyr[a, b]b]gyr[a, b] . Derivation of the numbered equalities in (3.37): (1) Follows from (2.64). (2) Follows from (1) by the left reduction property. (3) Follows from (2) by the left gyroassociative law. Indeed, an application of the left gyroassociative law to the first entry of the left gyration in (3) gives the first entry of the left gyration in (2), that is, a⊕(b b) = (a⊕b) gyr[a, b]b. (4) Follows from (3) immediately, since b b = 0. To verify (2.66) we consider the special case of (2.63) when b = a, obtaining gyr[a, a⊕c]gyr[ a, c] = gyr[0, gyr[a, a]c]gyr[a, a] = I

(2.72)

where the second identity in (2.72) follows from Items (2) and (3) of Theorem 2.16. Replacing a by a and c by b in (2.72) we obtain (2.66). Finally, (2.67) is derived from (2.66) by an application of the left reduction property to the first gyroautomorphism in (2.66) followed by a left cancellation, Theorem 2.16(9). Accordingly, I = gyr[ a, a⊕b]gyr[a, b] = gyr[ a⊕(a⊕b), a⊕b]gyr[a, b]

(2.73)

= gyr[b, a⊕b]gyr[a, b] , as desired.



The nested gyroautomorphism identity (2.65) in Theorem 2.19 allows the equation

Einstein Gyrogroups

that defines the coaddition  to be dualized with its corresponding equation in which the roles of the binary operations  and ⊕ are interchanged, as shown in the following theorem. Theorem 2.20. (Operation-Cooperation Duality Symmetry). Let (G, ⊕) be a gyrogroup with cooperation  given in Def. 2.15, p. 23, by the equation a  b = a⊕gyr[a, b]b .

(2.74)

a⊕b = a  gyr[a, b]b

(2.75)

Then, for all a, b ∈ G. Proof. Let a and b be any two elements of G. By (2.74) and (2.65) we have a  gyr[a, b]b = a⊕gyr[a, gyr[a, b]b]gyr[a, b]b = a⊕b ,

(2.76) 

thus verifying (2.75).

Identities (2.74) – (2.75) exhibit a duality symmetry between the binary operations ⊕ and , associated with a duality symmetry between the gyrations gyr[a, b] and gyr[a, b]. A remarkable duality symmetry solely between the gyrations gyr[a, b] and gyr[a, b] is presented in [81, Theorem 4.22, p. 129]. We naturally use the notation a  b = a  ( b)

(2.77)

in a gyrogroup (G, ⊕), so that, by (2.77), (2.74), and Theorem 2.16(12), a  b = a  ( b) = a⊕gyr[a, b]( b)

(2.78)

= a gyr[a, b]b and, hence, a  a = a a = 0 ,

(2.79)

as it should. Identity (2.79), in turn, implies the equality between the inverses of a ∈ G with respect to ⊕ and , a = a for all a ∈ G.

(2.80)

29

30

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Theorem 2.21. Let (G, ⊕) be a gyrogroup. Then, ( a⊕b)⊕gyr[ a, b]( b⊕c) = a⊕c

(2.81)

for all a, b, c ∈ G. Proof. By the left gyroassociative law and the left cancellation law, and using the notation d = b⊕c, we have ( a⊕b)⊕gyr[ a, b]( b⊕c) = ( a⊕b)⊕gyr[ a, b]d = a⊕(b⊕d) = a⊕(b⊕( b⊕c)) = a⊕c .

(2.82)

 Theorem 2.22. (Left Gyrotranslation Theorem, I). Let (G, ⊕) be a gyrogroup. Then, ( a⊕b)⊕( a⊕c) = gyr[ a, b]( b⊕c)

(2.83)

for all a, b, c ∈ G. Proof. Identity (2.83) is a rearrangement of (2.81) obtained by left gyroadding the term ( a⊕b) to both sides of (2.81) followed by a left cancellation.  The importance of Identity (2.83) is indicated by the analogy it shares with its group counterpart, −(−a + b) + (−a + c) = −b + c in any group with group operation +. The identity of Theorem 2.21 can readily be generalized to any number of terms as, for instance, ( a⊕b)⊕gyr[ a, b]{( b⊕c)⊕gyr[ b, c]( c⊕d)} = a⊕d ,

(2.84)

which generalizes the obvious group identity (−a + b) + (−b + c) + (−c + d) = −a + d in any group with group operation +.

(2.85)

Einstein Gyrogroups

2.10. The Two Basic Gyrogroup Equations The two basic equations of gyrogroup theory are a⊕x = b

(2.86)

x⊕a = b ,

(2.87)

and a, b, x ∈ G, each for the unknown x in a gyrogroup (G, ⊕). The following theorem asserts that each of the two basic equations possesses a unique solution. The existence of unique solutions, in turn, gives rise in Sect. 2.11 to the basic gyrogroup cancellation laws. Theorem 2.23. (The Two Basic Gyrogroup Equations). Let (G, ⊕) be a gyrogroup, and let a, b ∈ G. The unique solution of the equation a⊕x = b

(2.88)

x = a⊕b

(2.89)

in G for the unknown x is

and the unique solution of the equation x⊕a = b

(2.90)

x = b  a.

(2.91)

in G for the unknown x is

Proof. Let x be a solution of the first basic equation, (2.86). Then, we have by (2.86) and the left cancellation law, Theorem 2.16(9), p. 25, a⊕b = a⊕(a⊕x) = x .

(2.92)

Hence, if a solution x of (2.86) exists then it must be given by x = a⊕b, as we see from (2.92). Conversely, x = a⊕b is, indeed, a solution of (2.86) as we see by substituting x = a⊕b into (2.86) and applying the left cancellation law in Theorem 2.16(9). Hence, the gyrogroup equation (2.86) possesses the unique solution x = a⊕b. The solution of the second basic gyrogroup equation, (2.87), is quite different from that of the first, (2.86), owing to the noncommutativity of the gyrogroup operation. Let x be a solution of (2.87). Then, we have the following chain of equations, which

31

32

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

are numbered for subsequent derivation: (1)

 x === x⊕0 (2)

 === x⊕(a a) (3)

 === (x⊕a)⊕gyr[x, a]( a) (4)

 === (x⊕a) gyr[x, a]a

(2.93)

(5)

 === (x⊕a) gyr[x⊕a, a]a (6)

 === b gyr[b, a]a (7)

 ===

b  a.

Derivation of the numbered equalities in (2.93): (1) Follows from the existence of a unique identity element, 0, in the gyrogroup (G, ⊕) by Theorem 2.16, p. 24. (2) Follows from the existence of a unique inverse element a of a in the gyrogroup (G, ⊕) by Theorem 2.16. (3) Follows from (2) by the left gyroassociative law in Axiom (G3) of gyrogroups in Def. 2.13, p. 22. (4) Follows from (3) by Theorem 2.16(12). (5) Follows from (4) by the left reduction property (G5) of gyrogroups in Def. 2.13. (6) Follows from (5) by the assumption that x is a solution of (2.87). (7) Follows from (6) by (2.78). Hence, if a solution x of (2.87) exists then it must be given by x = b  a, as we see from (2.93). Conversely, x = b  a is, indeed, a solution of (2.87), as we see from the following

Einstein Gyrogroups

chain of equations: (1)

 x⊕a === (b  a)⊕a (2)

 === (b gyr[b, a]a)⊕a (3)

 === (b gyr[b, a]a)⊕gyr[b, gyr[b, a]a]gyr[b, a]a (4)

 === b⊕( gyr[b, a]a⊕gyr[b, a]a)

(2.94)

(5)

 === b⊕0 (6)

 === b . Derivation of the numbered equalities in (2.94): (1) Follows from the assumption that x = b  a. (2) Follows from (1) by (2.78). (3) Follows from (2) by Identity (2.65) of Theorem 2.19, according to which the gyration product applied to a in (3) is trivial, that is, gyr[b, gyr[b, a]]gyr[b, a] = I. (4) Follows from (3) by the left gyroassociative law. Indeed, an application of the left gyroassociative law to (4) results in (3). (5) Follows from (4) since gyr[b, a]a is the unique inverse of gyr[b, a]a. (6) Follows from (5) since 0 is the unique identity element of the gyrogroup (G, ⊕).  Let (G, ⊕) be a gyrogroup, and let a ∈ G. The maps λa and ρa of G, given by λa : G → G,

λa : g → a⊕g ,

ρa : G → G,

ρa : g → g⊕a ,

(2.95)

are called, respectively, a left gyrotranslation of G by a and a right gyrotranslation of G by a. Theorem 2.23 asserts that each of these transformations of G is bijective, that is, it maps G onto itself in a one-to-one manner.

2.11. The Basic Gyrogroup Cancellation Laws The basic cancellation laws of gyrogroup theory are obtained in this section from the basic equations of gyrogroups solved in Sect. 2.10. Substituting the solution (2.89) into its equation (2.88) we obtain the left cancellation law a⊕( a⊕b) = b

(2.96)

33

34

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

for all a, b ∈ G, already verified in Theorem 2.16(9). Similarly, substituting the solution (2.91) into its equation (2.90) we obtain the first right cancellation law (b  a)⊕a = b

(2.97)

for all a, b ∈ G. The latter can be dualized, obtaining the second right cancellation law (b a)  a = b

(2.98)

for all a, b ∈ G. Indeed, (2.98) results from the following chain of equations: b = b⊕0 = b⊕( a⊕a) = (b a)⊕gyr[b, a]a

(2.99)

= (b a)⊕gyr[b a, a]a = (b a)  a , where we employ the left gyroassociative law, the left reduction property, and the definition of the gyrogroup cooperation. Identities (2.96) – (2.98) form the three basic cancellation laws of gyrogroup theory.

2.12. Automorphisms and Gyroautomorphisms In this section we find that automorphisms of a gyrogroup commute with its gyroautomorphisms in a special, interesting way. Theorem 2.24. For any two elements a, b of a gyrogroup (G, ⊕) and any automorphism A of (G, ⊕), A ∈ Aut(G, ⊕), Agyr[a, b] = gyr[Aa, Ab]A .

(2.100)

Proof. For any three elements a, b, x ∈ (G, ⊕) and any automorphism A ∈ Aut(G, ⊕) we have by the left gyroassociative law (Aa⊕Ab)⊕Agyr[a, b]x = A((a⊕b)⊕gyr[a, b]x) = A(a⊕(b⊕x)) = Aa⊕(Ab⊕Ax)

(2.101)

= (Aa⊕Ab)⊕gyr[Aa, Ab]Ax . Hence, by a left cancellation, Agyr[a, b]x = gyr[Aa, Ab]Ax

(2.102)

Einstein Gyrogroups

for all x ∈ G, implying (2.100).



Theorem 2.25. Let a, b be any two elements of a gyrogroup (G, ⊕) and let A ∈ Aut(G) be an automorphism of G. Then, gyr[a, b] = gyr[Aa, Ab]

(2.103)

if and only if the automorphisms A and gyr[a, b] are commutative. Proof. If gyr[Aa, Ab] = gyr[a, b] then, by Theorem 2.24, the automorphisms gyr[a, b] and A commute, Agyr[a, b] = gyr[Aa, Ab]A = gyr[a, b]A. Conversely, if gyr[a, b] and A commute then, by Theorem 2.24, gyr[Aa, Ab] = Agyr[a, b]A−1 = gyr[a, b]AA−1 = gyr[a, b].  As a simple, but useful, consequence of Theorem 2.25 we note the elegant identity gyr[gyr[a, b]a, gyr[a, b]b] = gyr[a, b] .

(2.104)

Theorem 2.26. A gyrogroup (G, ⊕) and its associated cogyrogroup (G, ) possess the same automorphism group, Aut(G, ) = Aut(G, ⊕) .

(2.105)

Proof. Let τ ∈ Aut(G, ⊕). Then, by Theorem 2.24 τ(a  b) = τ(a⊕gyr[a, b]b) = τa⊕τgyr[a, b]b = τa⊕gyr[τa, τb]τb = τa  τb ,

(2.106)

so that τ ∈ Aut(G, ), implying Aut(G, )  Aut(G, ⊕) .

(2.107)

Conversely, let τ ∈ Aut(G, ). Then, 0⊕τ0 = τ0 = τ(0  0) = τ0  τ0 = τ0⊕τ0,

(2.108)

τ0 = 0 .

(2.109)

0 = τ0 = τ(a  a) = τ(a  (a)) = τa  τ(a),

(2.110)

implying

Hence,

35

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

implying that τ(a) is the unique inverse of τa in (G, ), τ(a) = τa ,

(2.111)

so that, finally, τ(a  b) = τ(a  (b)) = τa  τ(b) = τa  (τb) = τa  τb .

(2.112)

Owing to the second right cancellation law (2.98), and (2.112), we have τa = τ((a⊕b)  b) = τ(a⊕b)  τb ,

(2.113)

so that by the first right cancellation law (2.97), τ(a⊕b) = τa⊕τb .

(2.114)

Hence, τ ∈ Aut(G, ⊕), implying Aut(G, )  Aut(G, ⊕) ,

(2.115)

so that, by (2.107) and (2.115), Aut(G, ) = Aut(G, ⊕) ,

(2.116) 

as desired.

Theorem 2.26 augments the duality symmetry that Einstein addition and coaddition share in Theorem 2.20.

2.13. Gyrosemidirect Product Definition 2.27. (Gyroautomorphism Groups, Gyrosemidirect Product). Let G = (G, ⊕) be a gyrogroup, and let Aut(G) = Aut(G, ⊕) be the automorphism group of G. A gyroautomorphism group, Aut0 (G), of G is any subgroup of Aut(G) containing all the gyroautomorphisms gyr[a, b] of G, a, b ∈ G. The gyrosemidirect product group G×Aut0 (G)

(2.117)

of a gyrogroup G and any gyroautomorphism group, Aut0 (G), of G is a group of pairs (x, X), where x ∈ G and X ∈ Aut0 (G), with operation given by the gyrosemidirect product (x, X)(y, Y) = (x⊕Xy, gyr[x, Xy]XY) .

(2.118)

It is anticipated in Def. 2.27 that the gyrosemidirect product set (2.117) of a gyrogroup and any one of its gyroautomorphism groups is a set that forms a group with group operation given by the gyrosemidirect product (2.118). The following theorem asserts that this is indeed the case.

Einstein Gyrogroups

Theorem 2.28. Let (G, ⊕) be a gyrogroup, and let Aut0 (G, ⊕) be a gyroautomorphism group of G. Then, the gyrosemidirect product G×Aut0 (G) is a group, with group operation given by the gyrosemidirect product (2.118). Proof. We will show that the set G×Aut0 (G) with its binary operation (2.118) satisfies the group axioms. (i) Existence of a left identity: A left identity element of G×Aut0 (G) is the pair (0, I), where 0 ∈ G is the identity element of G, and I ∈ Aut0 (G) is the identity automorphism of G. Indeed, (0, I)(a, A) = (0⊕Ia, gyr[0, Ia]IA) = (a, A) ,

(2.119)

noting that the gyration in (2.119) is trivial by Theorem 2.16(2). (ii) Existence of a left inverse: Let A−1 ∈ Aut0 (G) be the inverse automorphism of A ∈ Aut0 (G). Then, by the gyrosemidirect product (2.118) we have ( A−1 a, A−1 )(a, A) = ( A−1 a⊕A−1 a, gyr[ A−1 a, A−1 a]A−1 A) = (0, I)

(2.120)

Hence, a left inverse of (a, A) ∈ G×Aut0 (G) is the pair ( A−1 a, A−1 ), (a, A)−1 = ( A−1 a, A−1 ) .

(2.121)

(iii) Validity of the associative law: We have to show that the successive products in (2.122) and in (2.123) are equal. On the one hand, we have (a1 , A1 )((a2 , A2 )(a3 , A3 )) = (a1 , A1 )(a2 ⊕A2 a3 , gyr[a2 , A2 a3 ]A2 A3 ) = (a1 ⊕A1 (a2 ⊕A2 a3 ), gyr[a1 , A1 (a2 ⊕A2 a3 )]A1 gyr[a2 , A2 a3 ]A2 A3 )

(2.122)

= (a1 ⊕(A1 a2 ⊕A1 A2 a3 ), gyr[a1 , A1 a2 ⊕A1 A2 a3 ]gyr[A1 a2 , A1 A2 a3 ]A1 A2 A3 ) , where we employ the gyrosemidirect product (2.118) and the commuting relation (2.100). On the other hand, we have ((a1 , A1 )(a2 , A2 ))(a3 , A3 ) = (a1 ⊕A1 a2 , gyr[a1 , A1 a2 ]A1 A2 )(a3 , A3 ) = ((a1 ⊕A1 a2 )⊕gyr[a1 , A1 a2 ]A1 A2 a3 ,

(2.123)

gyr[a1 ⊕A1 a2 , gyr[a1 , A1 a2 ]A1 A2 a3 ]gyr[a1 , A1 a2 ]A1 A2 A3 ) , where we employ (2.118). In order to show that the gyrosemidirect products in (2.122) and (2.123) are equal,

37

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

using the notation a1 = a A1 a2 = b

(2.124)

A1 A2 A3 = c , we have to establish the identity (a⊕(b⊕c), gyr[a, b⊕c]gyr[b, c]A1 A2 A3 ) = ((a⊕b)⊕gyr[a, b]c, gyr[a⊕b, gyr[a, b]c]gyr[a, b]A1 A2 A3 ) .

(2.125)

This identity between two pairs is equivalent to the two identities between their corresponding entries, a⊕(b⊕c) = (a⊕b)⊕gyr[a, b]c gyr[a, b⊕c]gyr[b, c] = gyr[a⊕b, gyr[a, b]c]gyr[a, b] .

(2.126)

The first identity in (2.126) is valid, being the left gyroassociative law, and the second identity in (2.126) is valid by (2.63), p. 27.  Instructively, a second proof of Theorem 2.28, in which we employ the subgroup criterion in Theorem 2.12, p. 22, is given below. Proof. A one-to-one map of a set Q1 onto a set Q2 is said to be bijective and, accordingly, the map is said to be a bijection. The set of all bijections of a set Q onto itself forms a group under bijection composition. Let S be the group of all bijections of the set G onto itself under bijection composition. Let each element (a, A) ∈ S 0 := G×Aut0 (G)

(2.127)

act bijectively on the gyrogroup (G, ⊕) according to the equation (a, A)g = a⊕Ag .

(2.128)

The unique inverse of (a, A) in S 0 = G×Aut0 (G) is, by (2.121), (a, A)−1 = ( A−1 a, A−1 ) .

(2.129)

Being a set of special bijections of G onto itself, given by (2.128), S 0 is a subset of the group S , S 0 ⊂ S . Employing the subgroup criterion in Theorem 2.12, p. 22, we will show that, under bijection composition, S 0 is a subgroup of the group S . Two successive bijections (a, A), (b, B) ∈ S 0 of G are equivalent to a single bijection (c, C) ∈ S 0 according to the following chain of equations. Employing successively

Einstein Gyrogroups

the bijection (2.128) along with the left gyroassociative law we have (a, A)(b, B)g = (a, A)(b⊕Bg) = a⊕A(b⊕Bg) = a⊕(Ab⊕ABg) = (a⊕Ab)⊕gyr[a, Ab]ABg

(2.130)

= (a⊕Ab, gyr[a, Ab]AB)g =: (c, C)g for all g ∈ G, (a, A), (b, B) ∈ S 0 . It follows from (2.130) that bijection composition in S 0 is given by the gyrosemidirect product, (2.118), (a, A)(b, B) = (a⊕Ab, gyr[a, Ab]AB) .

(2.131)

Finally, for any (a, A), (b, B) ∈ S 0 we have by (2.129) and (2.131) (a, A)(b, B)−1 = (a, A)( B−1 b, B−1 ) = (a AB−1 b, gyr[a, AB−1 b]AB−1 )

(2.132)

∈ S0 . Hence, by the subgroup criterion in Theorem 2.12, p. 22, the subset S 0 of the group S of all bijections of G onto itself is a subgroup under bijection composition. But, bijection composition in S 0 is given by the gyrosemidirect product (2.131). Hence, as desired, the set S 0 = G×Aut0 (G) with composition given by the gyrosemidirect product (2.131) forms a group.  The gyrosemidirect product group enables problems in gyrogroups to be converted to the group setting, thus gaining access to the powerful group theoretic techniques. Illustrative examples for the use of gyrosemidirect product groups are provided by the proof of each of the two Theorems 2.29 and 2.30. Theorem 2.29. Let (G, ⊕) be a gyrogroup, let a, b ∈ G be any two elements of G, and let Y ∈ Aut(G) be any automorphism of (G, ⊕). Then, the unique solution of the automorphism equation Y = gyr[b, Xa]X

(2.133)

for the unknown automorphism X ∈ Aut(G) is X = gyr[b, Ya]Y .

(2.134)

39

40

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Proof. Let X be a solution of (2.133), and let x ∈ G be given by the equation x = b  Xa ,

(2.135)

so that, by a right cancellation, (2.97), b = x⊕Xa. Then, we have the following gyrosemidirect product: (x, X)(a, I) = (x⊕Xa, gyr[x, Xa]X) = (x⊕Xa, gyr[x⊕Xa, Xa]X) = (b, gyr[b, Xa]X)

(2.136)

= (b, Y) , so that (x, X) = (b, Y)(a, I)−1 = (b, Y)( a, I)

(2.137)

= (b⊕Ya, gyr[b, Ya]Y) . Comparing the second entries of the extreme sides of (2.137) yields (2.134). Hence, if a solution X of (2.133) exists, then it must be given by (2.134). Conversely, the automorphism X in (2.134) is, indeed, a solution of (2.133) as we see by substituting X from (2.134) into the right-hand side of (2.133), and employing the nested gyration identity (2.65), p. 27. Indeed, by (2.134) and (2.65), gyr[b, Xa]X = gyr[b, gyr[b, Ya]Ya]gyr[b, Ya]Y =Y,

(2.138) 

as desired.

The gyrosemidirect product will be extended to the bi-gyrosemidirect product in Sect. 7.13.

2.14. Basic Gyration Properties Gyrations are destined to be extended in the book into bi-gyrations. Hence, the study of gyration properties is instructive. Theorem 2.30. (Gyrosum Inversion, Gyroautomorphism Inversion). For any two elements a, b of a gyrogroup (G, ⊕) we have the gyrosum inversion law (a⊕b) = gyr[a, b]( b a)

(2.139)

Einstein Gyrogroups

and the gyroautomorphism inversion law gyr−1 [a, b] = gyr[ b, a] .

(2.140)

Proof. Let Aut0 (G) be any gyroautomorphism group of (G, ⊕), and let G×Aut0 (G) be the gyrosemidirect product of the gyrogroup G and the group Aut0 (G) according to Def. 2.27. Being a group, the product of two elements of the gyrosemidirect product group G×Aut0 (G) has a unique inverse. This inverse can be calculated in two different ways. On the one hand, the inverse of the left-hand side of the gyrosemidirect product (a, I)(b, I) = (a⊕b, gyr[a, b])

(2.141)

in G×Aut0 (G) is (b, I)−1 (a, I)−1 = ( b, I)( a, I) = ( b a, gyr[ b, a]) .

(2.142)

On the other hand, the inverse of the right-hand side of the product (2.141) is, by (2.129), ( gyr−1 [a, b](a⊕b), gyr−1 [a, b])

(2.143)

for all a, b ∈ G. Comparing corresponding entries in (2.142) and (2.143) yields b a = gyr−1 [a, b](a⊕b)

(2.144)

gyr[ b, a] = gyr−1 [a, b] .

(2.145)

and Eliminating gyr−1 [a, b] between (2.144) and (2.145), we have b a = gyr[ b, a](a⊕b) .

(2.146)

Renaming (a, b) as ( b, a), (2.146) becomes a⊕b = gyr[a, b]( b a) . Identities (2.147) and (2.145) complete the proof.

(2.147) 

Instructively, the gyrosum inversion law (2.139) is verified here as a by-product along with the gyroautomorphism inversion law (2.140) in Theorem 2.30 in terms of the gyrosemidirect product group. A direct proof of (2.139) is, however, simpler as we see in Theorem 2.17, p. 26.

41

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Theorem 2.31. Let (G, ⊕) be a gyrogroup. Then, for all a, b ∈ G gyr−1 [a, b] = gyr[a, gyr[a, b]b]

(2.148)

gyr−1 [a, b] = gyr[ a, a⊕b]

(2.149)

gyr−1 [a, b] = gyr[b, a⊕b]

(2.150)

gyr[a, b] = gyr[b, b a]

(2.151)

gyr[a, b] = gyr[ a, b a]

(2.152)

gyr[a, b] = gyr[ (a⊕b), a] .

(2.153)

Proof. Identity (2.148) follows from (2.65). Identity (2.149) follows from (2.66). Identity (2.150) results from an application to (2.149) of the left reduction property followed by a left cancellation. Identity (2.151) follows from the gyroautomorphism inversion law (2.140) and from (2.149), gyr[a, b] = gyr−1 [ b, a] = gyr[b, b a] .

(2.154)

Identity (2.152) follows from an application, to the right-hand side of (2.151), of the left reduction property followed by a left cancellation. Identity (2.153) follows by inverting (2.149) by means of the gyroautomorphism inversion law (2.140).  Theorem 2.32. (Gyration Inversion Law; Gyration Even Property). The gyroautomorphisms of any gyrogroup (G, ⊕) obey the gyration inversion law gyr−1 [a, b] = gyr[b, a]

(2.155)

and possess the gyration even property gyr[ a, b] = gyr[a, b] ,

(2.156)

satisfying the four mutually equivalent nested gyroautomorphism identities gyr[b, gyr[b, a]a] = gyr[a, b] gyr[b, gyr[b, a]a] = gyr[a, b] gyr[ gyr[a, b]b, a] = gyr[a, b] gyr[gyr[a, b]b, a] = gyr[a, b] for all a, b ∈ G.

(2.157)

Einstein Gyrogroups

Proof. By the left reduction property and (2.150) we have gyr−1 [a⊕b, b] = gyr−1 [a, b] = gyr[b, a⊕b]

(2.158)

for all a, b ∈ G. Let us substitute a = c  b into (2.158), so that by a right cancellation a⊕b = c, obtaining the identity gyr−1 [c, b] = gyr[b, c]

(2.159)

for all c, b ∈ G. Renaming c in (2.159) as a, we obtain (2.155), as desired. Identity (2.156) results from (2.140) and (2.155), gyr[ a, b] = gyr−1 [b, a] = gyr[a, b] .

(2.160)

Finally, the first identity in (2.157) follows from (2.148) and (2.155). By means of the gyroautomorphism inversion law (2.155), the third identity in (2.157) is equivalent to the first one. The second (fourth) identity in (2.157) follows from the first (third) by replacing a by a (or, alternatively, by replacing b by b), noting that gyrations are even by (2.156).  The gyration inversion law (2.155) enables the result in Theorem 2.16(13), p. 25, to be extended in the following theorem. Theorem 2.33. For any element a of a gyrogroup (G, ⊕) we have gyr[a, 0] = gyr[0, a] = I

(2.161)

Proof. The gyration identity gyr[a, 0] = I is the result in Theorem 2.16(13). Its counterpart gyration identity gyr[0, a] = I follows from the gyration inversion law (2.155), gyr[0, a] = gyr−1 [a, 0] = I −1 = I ,

(2.162) 

as desired

The left gyroassociative law and the left reduction property of gyrogroups admit right counterparts, presented in the following theorem. Theorem 2.34. For any three elements a, b, and c of a gyrogroup (G, ⊕) we have (i) (ii)

(a⊕b)⊕c = a⊕(b⊕gyr[b, a]c) gyr[a, b] = gyr[a, b⊕a]

Right Gyroassociative Law, Right Reduction Property.

43

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Proof. The right gyroassociative law follows from the left gyroassociative law and the gyration inversion law (2.155) of gyroautomorphisms, a⊕(b⊕gyr[b, a]c) = (a⊕b)⊕gyr[a, b]gyr[b, a]c = (a⊕b)⊕c .

(2.163)

The right reduction property results from (2.150) and the gyration inversion law (2.155), gyr[b, a⊕b] = gyr−1 [a, b] = gyr[b, a] .

(2.164) 

Theorem 2.35. (A Mixed Gyroassociative Law). Let (G, ⊕) be a gyrogroup. Then, (a  b)⊕c = a⊕gyr[a, b](b⊕c)

(2.165)

for all a, b, c ∈ G. Proof. Whenever convenient we use the notation ga,b = gyr[a, b], etc. By (2.78), the right gyroassociative law, and the second nested gyroautomorphism identity in (2.157), we have (a  b)⊕c = (a ga,b b)⊕c = a⊕( ga,b b⊕gyr[ ga,b b, a]c) = a⊕( ga,b b⊕gyr[a, b]c) = a⊕gyr[a, b]( b⊕c) , which gives the desired identity when b is replaced by b.

(2.166)



The right cancellation law allows the reduction property to be dualized in the following theorem. Theorem 2.36. (Coreduction Property - Left and Right). Let (G, ⊕) be a gyrogroup. Then, gyr[a, b] = gyr[a  b, b] gyr[a, b] = gyr[a, b  a] for all a, b ∈ G.

Left Coreduction Property Right Coreduction Property

Einstein Gyrogroups

Proof. The proof follows from an application of the left and the right reduction property followed by a right cancellation, gyr[a  b, b] = gyr[(a  b)⊕b, b] = gyr[a, b] , gyr[a, b  a] = gyr[a, (b  a)⊕a] = gyr[a, b] .

(2.167)

 A right and a left reduction give rise to the identities in the following theorem. Theorem 2.37. Let (G, ⊕) be a gyrogroup. Then, gyr[a⊕b, a] = gyr[a, b] gyr[ a, a⊕b] = gyr[b, a]

(2.168)

for all a, b ∈ G. Proof. By a right reduction, a left cancellation, and a left reduction we have gyr[a⊕b, a] = gyr[a⊕b, a⊕(a⊕b)] = gyr[a⊕b, b] = gyr[a, b] ,

(2.169)

thus verifying the first identity in (2.168). The second identity in (2.168) follows from the first one by gyroautomorphism inversion, (2.155).  In general, (a⊕b)  a b in a gyrogroup (G, ⊕). In fact, we have (a⊕b) = a b for all a, b ∈ G if and only if the gyrogroup (G, ⊕) is gyrocommutative, as we see from Theorem 2.43, p. 49. In contrast, the gyrogroup cooperation  possesses the cogyroautomorphic inverse property (a  b) = ( b)  ( a), as the following theorem asserts. Theorem 2.38. (Cogyroautomorphic Inverse Property). Any gyrogroup (G, ⊕) possesses the cogyroautomorphic inverse property, (a  b) = ( b)  ( a)

(2.170)

for any a, b ∈ G. Proof. We verify (2.170) in the following chain of equations, which are numbered for

45

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

subsequent derivation: (1)

 a  b === a⊕gyr[a, b]b (2)

 === gyr[a, gyr[a, b]b]{ gyr[a, b]b a} (3)

 === gyr[a, gyr[a, b]b]{ ( gyr[a, b]b a)} (4)

 === gyr[a, gyr[a, b]( b)]{ ( gyr[a, b]b a)} (5)

 === gyr−1 [a, b]{ ( gyr[a, b]b a)}

(2.171)

(6)

 === ( b gyr−1 [a, b]a) (7)

 === { b gyr[b, a]a} (8)

 === {( b)  ( a)} . Inverting both extreme sides of (2.171) we obtain the desired identity (2.170). Derivation of the numbered equalities in (2.171): (1) (2) (3) (4) (5) (6) (7) (8)

Follows from Def. 2.15, p. 23, of the gyrogroup cooperation . Follows from (1) by the gyrosum inversion, (2.139). Follows from (2) by Theorem 2.16(12) applied to the term {. . .} in (2). Follows from (3) by Theorem 2.16(12) applied to b, that is, gyr[a, b]b = gyr[a, b]( b). Follows from (4) by Identity (2.148) of Theorem 2.31. Follows from (5) by distributing the gyroautomorphism gyr−1 [a, b] over each of the two terms in {. . .}. Follows from (6) by the gyroautomorphism inversion law (2.140). Follows from (7) by Def. 2.15, p. 23, of the gyrogroup cooperation . 

Theorem 2.39. Let (G, ⊕) be a gyrogroup. Then, a⊕{( a⊕b)⊕a} = b  a

(2.172)

for all a, b ∈ G. Proof. The proof rests on the following chain of equations, which are numbered for

Einstein Gyrogroups

subsequent explanation: (1)

 a⊕{( a⊕b)⊕a} === {a⊕( a⊕b)}⊕gyr[a, a⊕b]a (2)

 === b⊕gyr[b, a⊕b]a (3)

 === b⊕gyr[b, a]a

(2.173)

(4)

 === b  a . The derivation of the equalities in (2.173): (1) Follows from the left gyroassociative law. (2) Follows from (1) by a left cancellation, and by a left reduction followed by a left cancellation. (3) Follows from (2) by a right reduction, that is, an application of the right reduction property to (3) gives (2). (4) Follows from (3) by Def. 2.15, p. 23, of the gyrogroup cooperation . 

2.15. An Advanced Gyrogroup Equation As an example, we present in Theorem 2.40 an advanced gyrogroup equation and its unique solution. The equation is advanced in the sense that its unknown appears in the equation both directly and indirectly in the argument of a gyration. Theorem 2.40. Let c = gyr[b, x]x

(2.174)

be an equation for the unknown x in a gyrogroup (G, ⊕). The unique solution of (2.174) is x = ( b (c  b)) .

(2.175)

Proof. If a solution x to the gyrogroup equation (2.174) exists then by (2.174) and by the second identity in (2.62), p. 27, c = gyr[b, x]x = (b x)⊕b .

(2.176)

Applying a right cancellation to the extreme sides of (2.176) we obtain (b x) = c  b

(2.177)

47

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

or, equivalently, by gyro-sign inversion, b x = (c  b) ,

(2.178)

x = b (c  b) ,

(2.179)

x = ( b (c  b)) .

(2.180)

so that, by a left cancellation

implying, by gyro-sign inversion,

Hence, if a solution x to (2.174) exists, it must be given by (2.180). Conversely, x given by (2.180) is a solution of (2.174). Indeed, the substitution of x of (2.180) into the right-hand side of (2.176) results in c, as we see from the following chain of equations, which are numbered for subsequent derivation: (1)

 gyr[b, x]x === gyr[b, b (c  b)]{ b (c  b)} (2)

 === {b⊕( b (c  b))}⊕b (3)

 === gyr[ {b⊕( b (c  b))}, b]( b⊕{b⊕( b (c  b))}) (4)

 === gyr[b⊕( b (c  b)), b]{ b (c  b)} (5)

 === gyr[b, b (c  b)]{ b (c  b)}

(2.181)

(6)

 === gyr[ (c  b), b]{ b (c  b)} (7)

 === gyr[c  b, b]{ b (c  b)} (8)

 === (c  b)⊕b (9)

 === c . Derivation of the numbered equalities in (2.181): (1) (2) (3) (4) (5) (6)

Follows from the substitution of x from (2.180), and from Theorem 2.16(12). Follows from (1) by the first identity in (2.62), p. 27. Follows from (2) by the gyrosum inversion law (2.60), p. 26. Follows from (3) by the gyration even property and by a left cancellation. Follows from (4) by the first identity in (2.168). Follows from (5) by the second identity in (2.168).

Einstein Gyrogroups

(7) Follows from (6) by the gyration even property. (8) Follows from (7) by the gyrosum inversion law (2.60), p. 26. (9) Follows from (8) by a right cancellation.  Corollary 2.41. Let (G, ⊕) be a gyrogroup. The map a → c = gyr[b, a]a

(2.182)

of G onto itself is bijective so that when a runs over all the elements of G its image, c, runs over all the elements of G as well, for any given element b ∈ G. Proof. It follows immediately from Theorem 2.40 that, for any given b ∈ G, the map (2.182) maps G onto itself bijectively and hence the result of the Corollary. 

2.16. Gyrocommutative Gyrogroups Definition 2.42. (Gyroautomorphic Inverse Property). A gyrogroup (G, ⊕) possesses the gyroautomorphic inverse property if for all a, b ∈ G, (a⊕b) = a b .

(2.183)

Theorem 2.43. (Gyroautomorphic Inverse Property). A gyrogroup is gyrocommutative if and only if it possesses the gyroautomorphic inverse property. Proof. Let (G, ⊕) be a gyrogroup possessing the gyroautomorphic inverse property. Then, the gyrosum inversion law (2.60), p. 26, specializes, by means of Theorem 2.16(12), p. 24, to the gyrocommutative law (G6) in Def. 2.14, p. 23, a⊕b = gyr[a, b]( b a) = gyr[a, b]{ ( b a)}

(2.184)

= gyr[a, b](b⊕a) for all a, b ∈ G. Conversely, if the gyrocommutative law is valid then by Theorem 2.16(12) and the gyrosum inversion law, (2.60), p. 26, we have gyr[a, b]{ ( b a)} = gyr[a, b]( b a) = a⊕b = gyr[a, b](b⊕a) ,

(2.185)

so that by eliminating the gyroautomorphism gyr[a, b] on both extreme sides of (2.185)

49

50

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

and inverting the gyro-sign we recover the gyroautomorphic inverse property, (b⊕a) = b a

(2.186)

for all a, b ∈ G.



Theorem 2.44. The gyrogroup cooperation  of a gyrogroup (G, ⊕) is commutative if and only if the gyrogroup (G, ⊕) is gyrocommutative. Proof. For any a, b ∈ G we have, by equality (7) of the chain of equations (2.171) in the proof of Theorem 2.38, p. 46, a  b = ( b gyr[b, a]a) .

(2.187)

b  a = b⊕gyr[b, a]a .

(2.188)

ab=ba

(2.189)

( b c) = b⊕c

(2.190)

c = gyr[b, a]a ,

(2.191)

But by definition,

Hence for all a, b ∈ G if and only if for all a, b ∈ G, where

as we see from (2.187) and (2.188). But the map of G that takes a to c in (2.191), a → gyr[b, a]a = c ,

(2.192)

for any given b ∈ G is bijective, by Corollary 2.41, p. 49. Hence, the commutative relation (2.189) for  holds for all a, b ∈ G if and only if (2.190) holds for all b, c ∈ G. The latter, in turn, is the gyroautomorphic inverse property that, by Theorem 2.43, is equivalent to the gyrocommutativity of the gyrogroup (G, ⊕). Hence, (2.189) holds for all a, b ∈ G if and only if the gyrogroup (G, ⊕) is gyrocommutative.  Theorem 2.45. Let (G, ⊕) be a gyrocommutative gyrogroup. Then, gyr[a, b]gyr[b⊕a, c] = gyr[a, b⊕c]gyr[b, c]

(2.193)

for all a, b, c ∈ G. Proof. Using the notation ga,b = gyr[a, b] whenever convenient, we have by Theorem

Einstein Gyrogroups

2.24, p. 34, by the gyrocommutative law, and by Identity (2.63), p. 27 gyr[a, b]gyr[b⊕a, c] = gyr[ga,b (b⊕a), ga,b c]gyr[a, b] = gyr[a⊕b, gyr[a, b]c]gyr[a, b]

(2.194)

= gyr[a, b⊕c]gyr[b, c] , 

as desired.

Theorem 2.46. Let a, b, c ∈ G be any three elements of a gyrocommutative gyrogroup (G, ⊕), and let d ∈ G be determined by the “gyroparallelogram condition” (see (3.55), p. 82) d = (b  c) a .

(2.195)

Then, the elements a, b, c, and d satisfy the telescopic gyration identity gyr[a, b]gyr[b, c]gyr[c, d] = gyr[a, d]

(2.196)

for all a, b, c ∈ G. Proof. By Identity (2.193), along with an application of the right and the left reduction property, we have gyr[a , b ⊕a ]gyr[b ⊕a , c ] = gyr[a , b ⊕c ]gyr[b ⊕c , c ] ,

(2.197)

for all a , b , c , ∈ G. Let a = c c = a

(2.198)

b = b ⊕a , so that, by the third equation in (2.198) and by (2.195) we have b ⊕c = (b  a )⊕c = (b  c) a = d .

(2.199)

Then, (2.197), expressed in terms of a, b, c, d in (2.198) – (2.199), takes the form gyr[ c, b]gyr[b, a] = gyr[ c, d]gyr[d, a] .

(2.200)

Inverting both sides of (2.200) by means of the gyration inversion law (2.155), p. 42, and the gyration even property (2.156), p. 42, we obtain the identity gyr[a, b]gyr[b, c] = gyr[a, d]gyr[d, c] ,

(2.201)

from which the telescopic gyration identity (2.196) follows immediately, by a gyration inversion and the gyration even property. 

51

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Theorem 2.47. The gyroparallelogram condition (2.195), d = (b  c) a,

(2.202)

in a gyrocommutative gyrogroup (G, ⊕) is equivalent to the identity c⊕d = gyr[c, b](b a) .

(2.203)

Proof. In a gyrocommutative gyrogroup (G, ⊕) the gyroparallelogram condition (2.202) implies the following chain of equations, which are numbered for subsequent derivation: (1)

 d === (b  c) a (2)

 === (c  b) gyr[b, c]gyr[c, b]a (3)

 === (c  b) gyr[c, gyr[c, b]b]gyr[c, b]a (4)

 === (c⊕gyr[c, b]b) gyr[c, gyr[c, b]b]gyr[c, b]a

(2.204)

(5)

 === c⊕(gyr[c, b]b gyr[c, b]a) (6)

 === c⊕gyr[c, b](b a) for all a, b, c ∈ G. Derivation of the numbered equalities in (2.204): (1) This is the gyroparallelogram condition (2.202). (2) Follows from (1): (i) b  c = c  b, since the gyrogroup cooperation  in a gyrocommutative gyrogroup is commutative, by Theorem 2.44, p. 50, and (ii) gyr[b, c]gyr[c, b] = I, since, by gyration inversion along with the gyration even property in (2.155) – (2.156), p. 42, the gyration product applied to a in (2) is trivial. (3) Follows from (2) by the second nested gyration identity in (2.157), p. 42. (4) Follows from (3) by Def. 2.15, p. 23, of the gyrogroup cooperation . (5) Follows from (4) by the left gyroassociative law. Indeed, an application of the left gyroassociative law to (5) results in (4). (6) Follows from (5) since gyroautomorphisms respect their gyrogroup operation. Finally, (2.203) follows from (2.204) by a left cancellation, moving c from the extreme right-hand side of (2.204) to its extreme left-hand side. 

Einstein Gyrogroups

Theorem 2.48. Let (G, ⊕) be a gyrocommutative gyrogroup. Then, gyr[a, b]{b⊕(a⊕c)} = (a⊕b)⊕c

(2.205)

for all a, b, c ∈ G. Proof. By the left gyroassociative law and by the gyrocommutative law we have the chain of equations b⊕(a⊕c) = (b⊕a)⊕gyr[b, a]c = gyr[b, a](a⊕b)⊕gyr[b, a]c

(2.206)

= gyr[b, a]{(a⊕b)⊕c} , from which (2.205) is derived by the gyroautomorphism inversion law (2.140), p. 41.  The special case of Theorem 2.48 corresponding to c = a gives rise to a new cancellation law in gyrocommutative gyrogroups, called the left-right cancellation law.

Theorem 2.49. (Left-Right Cancellation Law). Let (G, ⊕) be a gyrocommutative gyrogroup. Then, (a⊕b) a = gyr[a, b]b

(2.207)

for all a, b, c ∈ G. Proof. Identity (2.207) follows from (2.62), p. 27, and the gyroautomorphic inverse property (2.183), p. 49. Alternatively, Identity (2.207) is equivalent to the special case of (2.206) when c = a.  The left-right cancellation law (2.207) is not a complete cancellation since the echo of the “canceled” a remains in the argument of the involved gyroautomorphism. Theorem 2.50. Let (G, ⊕) be a gyrocommutative gyrogroup. Then, a⊕{( a⊕b)⊕a} = a  b

(2.208)

for all a, b ∈ G. Proof. The proof of (2.208) follows immediately from Theorem 2.39, p. 46, and Theorem 2.44, p. 50. 

53

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Theorem 2.51. Let (G, ⊕) be a gyrocommutative gyrogroup. Then, a  (a⊕b) = a⊕(b⊕a)

(2.209)

for all a, b ∈ G. Proof. By a left cancellation and Theorem 2.50 we have a⊕(b⊕a) = a⊕({ a⊕(a⊕b)}⊕a) = a  (a⊕b) ,

(2.210) 

as desired.

Theorem 2.52. (Left Gyrotranslation Theorem, II). Let (G, ⊕) be a gyrocommutative gyrogroup. For all a, b, c ∈ G, (a⊕b)⊕(a⊕c) = gyr[a, b]( b⊕c) (a⊕b) (a⊕c) = gyr[a, b](b c) .

(2.211)

Proof. The first identity in (2.211) follows from the Left Gyrotranslation Theorem 2.22, p. 30, with a replaced by a. Hence, it is valid in nongyrocommutative gyrogroups as well. The second identity in (2.211) follows from the first by the gyroautomorphic inverse property of gyrocommutative gyrogroups, presented in Theorem 2.43, p. 49. Hence, it is valid in gyrocommutative gyrogroups.  The following theorem gives an elegant gyration identity in which the product of three telescopic gyrations is equivalent to a single gyration. Theorem 2.53. Let a, b, c ∈ G be any three elements of a gyrocommutative gyrogroup (G, ⊕). Then, gyr[ a⊕b, a c] = gyr[a, b]gyr[b, c]gyr[c, a] .

(2.212)

Proof. By Theorem 2.24, p. 34, and by the gyrocommutative law we have gyr[a, b]gyr[b⊕a, c] = gyr[gyr[a, b](b⊕a), gyr[a, b]c]gyr[a, b] = gyr[a⊕b, gyr[a, b]c]gyr[a, b] .

(2.213)

Hence, Identity (2.63) in Theorem 2.19, p. 27, can be written as gyr[a, b⊕c]gyr[b, c] = gyr[a, b]gyr[b⊕a, c] .

(2.214)

Einstein Gyrogroups

By gyroautomorphism inversion, the latter can be written as gyr[a, b⊕c] = gyr[a, b]gyr[b⊕a, c]gyr[c, b] .

(2.215)

Using the notation b⊕a = d, which implies a = b⊕d, Identity (2.215) becomes, by means of Theorem 2.37, p. 45, gyr[ b⊕d, b⊕c] = gyr[ b⊕d, b]gyr[d, c]gyr[c, b] = gyr[ b, d]gyr[d, c]gyr[c, b] .

(2.216)

Renaming the elements b, c, d ∈ G, (b, c, d) → ( a, c, b), (2.216) becomes gyr[a b, a⊕c] = gyr[a, b]gyr[ b, c]gyr[c, a] .

(2.217)

By means of the gyroautomorphic inverse property, Theorem 2.43, p. 49, and the gyration even property (2.156) in Theorem 2.32, p. 42, Identity (2.217) can be written, finally, in the desired form (2.212).  The special case of Theorem 2.53 when c = b is interesting, giving rise to the following theorem. Theorem 2.54. Let (G, ⊕) be a gyrocommutative gyrogroup. Then, gyr[a, b] = gyr[ a⊕b, a⊕b]gyr[a, b] .

(2.218)

Proof. Owing to the gyration inversion law in Theorem 2.32, p. 42, Identity (2.212) can be written as gyr[ a⊕b, a c]gyr[ a, c] = gyr[a, b]gyr[b, c] ,

(2.219)

from which the result (2.218) of the theorem follows in the special case when c = b by applying the gyration even property.  Identity (2.218) is interesting since it relates the gyration gyr[a, b] to the gyration gyr[a, b]. Furthermore, it gives rise, by gyration inversion and the gyration even property, to the following elegant gyration identity in which the product of two gyrations is equivalent to a single gyration: gyr[ a, b]gyr[b, a] = gyr[ a⊕b, a⊕b] .

(2.220)

As an application of Theorem 2.53 we present theorem 2.55. This theorem, the proof of which involves a long chain of gyrocommutative gyrogroup identities, possesses an important special case about the hyperbolic parallelogram, which is presented in Corollary 2.56.

55

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Theorem 2.55. Let (G, ⊕) be a gyrocommutative gyrogroup. Then, (x⊕a)  (x⊕b) = x⊕{(a  b)⊕x}

(2.221)

for all a, b, x ∈ G. Proof. The proof of Theorem 2.55 is given by the following chain of equations, which are numbered for subsequent derivation: (1)

 (x⊕a)  (x⊕b) === (x⊕a)⊕gyr[x⊕a, (x⊕b)](x⊕b) (2)

 === (x⊕a)⊕gyr[x⊕a, x b](x⊕b) (3)

 === x⊕{a⊕gyr[a, x]gyr[x⊕a, x b](x⊕b)} (4)

 === x⊕{a⊕gyr[a, x]gyr[ x, a]gyr[a, b]gyr[b, x](x⊕b)} (5)

 === x⊕{a⊕gyr[a, x]gyr[x, a]gyr[a, b](b⊕x)} (6)

 === x⊕{a⊕gyr[a, b](b⊕x)}

(2.222)

(7)

 === x⊕{a⊕(gyr[a, b]b⊕gyr[a, b]x)} (8)

 === x⊕{(a⊕gyr[a, b]b)⊕gyr[a, gyr[a, b]b]gyr[a, b]x} (9)

 === x⊕{(a⊕gyr[a, b]b)⊕gyr[b, a]gyr[a, b]x} (10)

 === x⊕{(a⊕gyr[a, b]b)⊕x} (11)

 === x⊕{(a  b)⊕x} . Derivation of the numbered equalities in (2.222): Follows from Def. 2.15, p. 23, of the gyrogroup cooperation . Follows from (1) by the gyroautomorphic inverse property, Theorem 2.43, p. 49. Follows from (2) by the right gyroassociative law. Follows from (3) by Identity (2.212) of Theorem 2.53, thus providing an elegant example for an application of that theorem. (5) Follows from (4) by the gyration even property, and by the gyrocommutative law. (6) Follows from (5) by the gyration inversion law (2.155), p. 42. (7) Follows from (6) by expanding the gyration application term by term. (1) (2) (3) (4)

Einstein Gyrogroups

(8) Follows from (7) by the left gyroassociative law. (9) Follows from (8) by the second identity in (2.157), p. 42. (10) Follows from (9) by the gyration even property and the gyration inversion law in Theorem 2.32, p. 42, implying gyr[b, a]gyr[a, b] = I. (11) Follows from (10) by Def. 2.15, p. 23, of .  Renaming the elements x, a, b ∈ G of Identity (2.221) of Theorem 2.55 as a, b, c, respectively, we obtain the parallelogram addition law in Corollary 2.56 of Theorem 2.55. Corollary 2.56. (Gyroparallelogram Addition Law). Let a, b, c be any three elements of a gyrocommutative gyrogroup (G, ⊕), and let d ∈ G be given by the gyroparallelogram condition, (2.195), d = (b  c) a .

(2.223)

Then, the points a, b, d, c ∈ G satisfy the gyroparallelogram addition law ( a⊕b)  ( a⊕c) = ( a⊕d) .

(2.224)

The term gyroparallelogram addition law in Corollary 2.56 is justified in Sect. 3.10 in terms of analogies it shares with the common parallelogram addition law in Euclidean geometry. Theorem 2.57 presents an interesting “coincidence” in which the expression (b  c) a in a gyrocommutative gyrogroup is gyrocovariant in the sense that the expression and its arguments, a, b, and c are covariant (vary together) under left gyrotranslations and under automorphisms of the gyrogroup. The hyperbolic geometric significance of identities similar to (2.226) – (2.227) of Theorem 2.57 is emphasized in Sect. 3.14 in the context of general gyrocovariance under the gyromotions of Einstein gyrovector spaces. Theorem 2.57. Let a, b, c∈G be any three elements of a gyrocommutative gyrogroup (G, ⊕), and let T : G3 → G be a map given by the equation T (a, b, c) = (b  c) a .

(2.225)

Then, T (a, b, c) is gyrocovariant under left gyrotranslations, that is, x⊕T (a, b, c) = T (x⊕a, x⊕b, x⊕c)

(2.226)

for all x∈G, and under automorphisms, that is, τT (a, b, c) = T (τa, τb, τc)

(2.227)

57

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

for all τ∈Aut(G, ⊕). Proof. The proof of (2.227) follows from Identities (2.106) and (2.114), p. 36. To prove (2.226) we have to verify the identity x⊕{(b  c) a} = {(x⊕b)  (x⊕c)} (x⊕a) .

(2.228)

The proof of (2.228) is presented in the following chain of equalities, which are numbered for subsequent explanation: {(x⊕b)  (x⊕c)} (x⊕a) (1)

 === (x⊕b)⊕gyr[x⊕b, (x⊕c)]{(x⊕c) (x⊕a)} (2)

 === (x⊕b)⊕gyr[x⊕b, (x⊕c)]gyr[x, c](c a) (3)

 === x⊕{b⊕gyr[b, x]gyr[x⊕b, (x⊕c)]gyr[x, c](c a)} (4)

 === x⊕{b⊕gyr[b, x]gyr[x, b]gyr[b, c]gyr[c, x]gyr[x, c](c a)} (5)

 === x⊕{b⊕gyr[b, c](c a)}

(2.229)

(6)

 === x⊕{b⊕(gyr[b, c]c gyr[b, c]a)} (7)

 === x⊕{(b⊕gyr[b, c]c) gyr[b, gyr[b, c]c]gyr[b, c]a} (8)

 === x⊕{(b⊕gyr[b, c]c) gyr[c, b]gyr[b, c]a} (9)

 === x⊕{(b  c) a} , as desired. The derivation of the equalities in (2.229): (1) Follows from the Mixed Gyroassociative Law (2.165), p. 44. (2) Follows from the Left Gyrotranslation Theorem 2.52. Since we use the second identity in Theorem 2.52 rather than the first identity, gyrocommutativity must be assumed. (3) Is derived by right gyroassociativity. (4) Follows from Theorem 2.53 and the gyroautomorphism even property (2.156). (5) Is derived by gyroautomorphism inversion, Theorem 2.32, p. 42. (6) Is derived by automorphism expansion. (7) Is derived by left gyroassociativity.

Einstein Gyrogroups

(8) Follows from the second nested gyration identity (2.157), p. 42. (9) Follows from the cooperation definition, Def. 2.15, p. 23, and gyroautomorphism inversion.  Relationships between some gyrocommutative gyrogroups, known as M¨obius gyrogroups [95], and Clifford algebra are studied in [19, 20, 22]. More about the abstract gyrogroup is found, for instance, in [84, 93], [25, 26], [67, 68, 69, 70, 71], and [18, 48].

59

CHAPTER 3

Einstein Gyrovector Spaces

3.1. The Abstract Gyrovector Space This chapter presents the required background about Einstein gyrovector spaces. Definition 3.1. (Real Inner Product Vector Spaces). A real inner product vector space (V, +, ·) (vector space, in short) is a real vector space together with a map V×V → R,

(u, v) → u·v ,

(3.1)

called a real inner product, satisfying the following properties for all u, v, w ∈ V and r ∈ R: (1) v·v ≥ 0, with equality if, and only if, v = 0 (2) u·v = v·u (3) (u + v)·w = u·w + v·w (4) (ru)·v = r(u·v) The norm v ≥ 0 of v ∈ V is given by the equation v2 = v·v. Note that the properties of vector spaces imply (i) the Cauchy-Schwarz inequality |u·v| ≤ uv

(3.2)

for all u, v∈V and (ii) the positive definiteness of the inner product, according to which u·v = 0 for all u∈V implies v = 0 [52]. Definition 3.2. (Real Inner Product Gyrovector Spaces). A real inner product gyrovector space (G, ⊕, ⊗) (gyrovector space, in short) is a gyrocommutative gyrogroup (G, ⊕) that obeys the following axioms: (1)

G is a subset of a real inner product vector space V called the ambient space of G, G ⊂ V, from which it inherits its inner product, ·, and norm, ·, which are invariant under gyroautomorphisms, that is,

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces http://dx.doi.org/10.1016/B978-0-12-811773-6.50003-7 Copyright © 2018 Elsevier Inc. All rights reserved.

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(V1)

gyr[u, v]a·gyr[u, v]b = a·b

Inner Product Gyroinvariance

for all points a, b, u, v ∈ G. (2)

(V2) (V3) (V4) (V5) (V6) (V7) (3) (V8)

G admits a scalar multiplication, ⊗, possessing the following properties. For all scalars (real numbers) r, r1 , r2 ∈ R and all points a ∈ G: 1⊗a = a (r1 + r2 )⊗a = r1 ⊗a⊕r2 ⊗a (r1 r2 )⊗a = r1 ⊗(r2 ⊗a) a |r|⊗a = , a  0, r  0 r⊗a a gyr[u, v](r⊗a) = r⊗gyr[u, v]a gyr[r1 ⊗v, r2 ⊗v] = I

Identity Scalar Multiplication Scalar Distributive Law Scalar Associative Law Scaling Property Gyroautomorphism Property Identity Gyroautomorphism.

Real, one-dimensional vector space structure (G, ⊕, ⊗) for the set G of one-dimensional “vectors” (see, for instance, [8]) G = {±a : a ∈ G} ⊂ R

Vector Space

with vector addition ⊕ and scalar multiplication ⊗, such that for all r ∈ R and a, b ∈ G, (V9) r⊗a = |r|⊗a (V10) a⊕b ≤ a⊕b

Homogeneity Property Gyrotriangle Inequality.

Remark 3.3. We use the notation (r1 ⊗a)⊕(r2 ⊗b) = r1 ⊗a⊕r2 ⊗b, and a⊗r = r⊗a. Our ambiguous use of ⊕ and ⊗ in Def. 3.2 as interrelated operations in the gyrovector space (G, ⊕, ⊗) and in its associated vector space (G, ⊕, ⊗) should raise no confusion since the sets in which these operations operate are always clear from the context. These operations in the former (gyrovector space (G, ⊕, ⊗)) are nonassociative-nondistributive gyrovector space operations, and in the latter (vector space (G, ⊕, ⊗)) are associativedistributive vector space operations. Additionally, the gyroaddition ⊕ is gyrocommutative in the former and commutative in the latter. Note that in the vector space (G, ⊕, ⊗) gyrations are trivial so that, in G, the two binary operations  and ⊕ coalesce, that is,  = ⊕,

in G .

(3.3)

Hence, as an illustrative example, while in general a⊕b  a  b, we have a⊕b = a  b .

(3.4)

Einstein Gyrovector Spaces

Moreover, following (3.4) and (2.49), p. 24, we have the elegant identity a⊕b = 2⊗

γa a + γb b γa + γb

,

(3.5)

noting that γa = γa , etc. Identity (3.5) proves useful in the study of the gyroellipse in [98, Chap. 12]. While each of the operations ⊕ and ⊗ has distinct interpretations in the gyrovector space G and in the vector space G, they are related to one another by the gyrovector space axioms (V9) and (V10). The analogies that conventions about the ambiguous use of ⊕ and ⊗ in G and G share with similar vector space conventions are obvious. Indeed, in vector spaces we use the same notation, +, for the addition operation between vectors and between their magnitudes, and same notation for the scalar multiplication between two scalars and between a scalar and a vector. In full analogy, in gyrovector spaces we use the same notation, ⊕, for the gyroaddition operation between gyrovectors and between their magnitudes, in (V10), and the same notation, ⊗, for the scalar gyromultiplication between two scalars and between a scalar and a gyrovector, in (V9). Owing to Axiom (V7), the gyrosum r1 ⊗a⊕r2 ⊗a⊕ . . . ⊕rn ⊗a , rk ∈ R, k = 1, 2, 3, . . . , n, a ∈ G, is commutative and associative, so that the terms rk ⊗a can be grouped in any way. This result enables the scalar distributive law (V3) to be extended to any number of terms. Immediate consequences of the gyrovector space axioms are presented in the following theorem. Theorem 3.4. Let (G, ⊕, ⊗) be a gyrovector space whose ambient vector space is V, and let 0, 0, and 0V be the neutral elements of (R, +), (G, ⊕), and (V, +), respectively. Then, for all n ∈ N, r ∈ R, and a ∈ G, (1) (2) (3) (4) (5) (6) (7) (8)

0⊗a = 0 n⊗a = a⊕ . . . ⊕a (n terms) (−r)⊗a = (r⊗a) =: r⊗a r⊗( a) = r⊗a r⊗0 = 0  a = a 0 = 0V r⊗a = 0 ⇐⇒ (r = 0 or a = 0)

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Proof. (1) Follows from the scalar distributive law (V3), r⊗a = (r + 0)⊗a = r⊗a⊕0⊗a ,

(3.6)

so that, by a left cancellation, 0⊗a = (r⊗a)⊕(r⊗a) = 0. (2)

Follows from (V2), and the scalar distributive law (V3). Indeed, with “. . .” signifying “n terms,” we have a⊕ . . . ⊕a = 1⊗a⊕ . . . ⊕1⊗a = (1 + . . . + 1)⊗a = n⊗a .

(3.7)

(3) This equation results from Item (1) and the scalar distributive law (V3), 0 = 0⊗a = (r − r)⊗a = r⊗a⊕(−r)⊗a ,

(3.8)

implying (r⊗a) = (−r)⊗a. (4) This equation results from the identity scalar multiplication (V2), Item (3), and the scalar associative law (V4), r⊗( a) = r⊗( (1⊗a)) = r⊗((−1)⊗a) = (r(−1))⊗a = (−r)⊗a = r⊗a .

(3.9)

(5) Follows from Item (1), (V4), (V3), (3), r⊗0 = r⊗(0⊗a) = r⊗((1 − 1)⊗a) = (r(1 − 1))⊗a = (r − r)⊗a = r⊗a⊕(−r)⊗a = r⊗a⊕( (r⊗a)) = r⊗a r⊗a = 0.

(3.10)

(6) Follows from Item (3), the homogeneity property (V9), and (V2),  a = (−1)⊗a = | − 1|⊗a = 1⊗a = 1⊗a = a .

(3.11)

Einstein Gyrovector Spaces

(7) This equation results from Item (5), (V9), and (V8) as follows: 0 = 2⊗0 = 2⊗0 = 0⊕0 ,

(3.12)

implying 0 = 0 0 = 0 in the vector space (G, ⊕, ⊗). This equation, 0 = 0, is valid in the vector space V as well, where it implies 0 = 0V . (8) This equation results from the following considerations. Suppose r⊗a = 0, but r  0. Then, by (V2), (V4) and Item (5) we have a = 1⊗a = (1/r)⊗(r⊗a) = (1/r)⊗0 = 0 ,

(3.13) 

as desired.

Clearly, in the special case when all the gyrations of a gyrovector space are trivial, the gyrovector space reduces to a vector space. In general, gyroaddition does not distribute with scalar multiplication, r⊗(a⊕b)  r⊗a⊕r⊗b .

(3.14)

However, gyrovector spaces possess a weak distributive law, called the monodistributive law, presented in the following theorem. Theorem 3.5. (Monodistributive Law). A gyrovector space (G, ⊕, ⊗) possesses the monodistributive law r⊗(r1 ⊗a⊕r2 ⊗a) = r⊗(r1 ⊗a)⊕r⊗(r2 ⊗a)

(3.15)

for all r, r1 , r2 ∈R and a∈G. Proof. The proof follows from (V3) and (V4), r⊗(r1 ⊗a⊕r2 ⊗a) = r⊗{(r1 + r2 )⊗a} = (r(r1 + r2 ))⊗a = (rr1 + rr2 )⊗a = (rr1 )⊗a⊕(rr2 )⊗a = r⊗(r1 ⊗a)⊕r⊗(r2 ⊗a) .

(3.16)



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3.2. Einstein Special Relativistic Scalar Multiplication Einstein addition admits a scalar multiplication, giving rise to Einstein gyrovector spaces. Let k⊗v = v⊕v . . . ⊕v (k terms, k = 1, 2, 3, . . .) be the Einstein addition, (2.2), p. 10, of k copies of v ∈ Rnc , defined inductively as (k + 1)⊗v = v⊕(k⊗v), for any v ∈ Rnc . Then,

 k  v 1+ − 1− c k⊗v = c  k  v 1+ + 1− c

1⊗v = v , v c v c

(3.17)

k v k v .

(3.18)

The definition of scalar multiplication in an Einstein gyrovector space requires analytically continuing k off the positive integers, thus obtaining the following definition.

Definition 3.6. (Einstein Scalar Multiplication; Einstein Gyrovector Spaces). An Einstein gyrovector space (Rnc , ⊕, ⊗) is an Einstein gyrogroup (Rnc , ⊕) endowed with a scalar multiplication ⊗ given by  r  r v v 1+ − 1− c c v v v r⊗v = c  = c tanh(r tanh−1 ) , (3.19) r  r c v v v v + 1− 1+ c c where r is any scalar, r ∈ R, v ∈ Rnc , v  0, and r⊗0 = 0, and with which we use the notation v⊗r = r⊗v. The scalar multiplication (3.19) in gyrovector spaces will be extended in Theorem 5.86, p. 263, to a scalar multiplication in bi-gyrovector spaces. Example 3.7. (Einstein Half). In the special case when r = 1/2, (3.19) descends to 1 2 ⊗v

=

γv v, 1 + γv

(3.20)

so that, as expected, 1 1 2 ⊗v⊕ 2 ⊗v

=

γv γ v⊕ v v = v . 1 + γv 1 + γv

(3.21)

Einstein Gyrovector Spaces

Having Einstein half in hand, we can recast the relativistic kinetic energy of moving objects into a form that captures analogies with its classical counterpart. The relativistic kinetic energy Krel of an object with rest mass m that moves uniformly with velocity v relative to an inertial frame Σ0 is given by the equation [87] Krel = c2 m(γv − 1) ,

(3.22)

where c is the vacuum speed of light. We manipulate (3.22) in the following chain of equations, some of which are numbered for subsequent explanation: (1)

 Krel === c2 m(γv − 1) = (2)

 ===

γv2 2 γ +1 γv2 2 γv2 − 1 c m(γv − 1) v 2 = cm 2 γv + 1 γv γv + 1 γv

γ v γv2 mv2 = v ·mγv v γv + 1 γv + 1

(3.23)

(3)

 1 === ( 2 ⊗v)·(mγv v) , mγv being the velocity-dependent relativistic mass of the moving object relative to Σ0 .

Derivation of the numbered equalities in (3.23): (1) Follows from (3.22). (2) Follows from (2.12), p. 13. (3) Follows from Einstein half (3.20). The relativistic kinetic energy Krel in (3.23), Krel = ( 12 ⊗v)·(mγv v) ,

(3.24)

appears as the inner product of a “relativistic half-velocity” and a corresponding relativistic momentum, in full analogy with the classical kinetic energy Kcls , Kcls = 12 mv2 = ( 12 v)·(mv) ,

(3.25)

which appears as the inner product of a “classical half-velocity” and a corresponding classical momentum. The ability of Einstein scalar multiplication to capture analogies between modern and classical results thus emerges.

3.3. Einstein Gyrovector Spaces Einstein gyrovector spaces turn out to be concrete realizations of the abstract gyrovector space in Def. 3.2. In fact, Def. 3.2 is guided by analogies with vector spaces and is

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motivated by key features of Einstein addition and scalar multiplication that give rise to axioms. Theorem 3.8. Einstein gyrogroups (Rnc , ⊕) with Einstein scalar multiplication ⊗ in Rnc form gyrovector spaces (Rnc , ⊕, ⊗), n = 1, 2, 3, . . .. Proof. We will show that any Einstein gyrovector space (Rnc , ⊕, ⊗) satisfies the gyrovector space axioms in Def. 3.2, p. 61. (V1) Inner Product Gyroinvariance: Axiom (V1) is satisfied as verified in (2.42), p. 20. (V2) Identity Scalar Multiplication: Let a∈Rnc . If a = 0 then 1⊗0 = 0 by definition, so that, in this case, Axiom (V2) is satisfied. For a  0 we have 1⊗a = c tanh(tanh−1

a a ) = a, c a

(3.26)

thus verifying Axiom (V2) of gyrovector spaces. (V3) Scalar Distributive Law: Let r1 , r2 ∈R. If a = 0 then (r1 + r2 )⊗a = 0 as well as r1 ⊗a⊕r2 ⊗a = 0⊕0 = 0, so that in this case, Axiom (V3) is satisfied. For a  0 we have, by (3.19), the addition formula of the hyperbolic tangent function, and (2.4), p. 11, (r1 + r2 )⊗a = c tanh([r1 + r2 ] tanh−1 = c tanh(r1 tanh−1 =c =

a a ) c a

a a a + r2 tanh−1 ) c c a

a −1 a c ) + tanh(r2 tanh c ) −1 a tanh(r1 tanh−1 a c ) tanh(r2 tanh c )

tanh(r1 tanh−1 1+

a a

(3.27)

r1 ⊗a + r2 ⊗a 1 + c12 r1 ⊗ar2 ⊗a

= r1 ⊗a⊕r2 ⊗a , thus verifying Axiom (V3) of gyrovector spaces. (V4) Scalar Associative Law: For a = 0 the validity of Axiom (V4) of gyrovector spaces is obvious. Assuming a  0, let us use the notation b = r2 ⊗a = c tanh(r2 tanh−1

a a ) , c a

(3.28)

Einstein Gyrovector Spaces

so that b a = tanh(r2 tanh−1 ) c c

(3.29)

a b = . b a

(3.30)

and, hence,

Then, r1 ⊗(r2 ⊗a) = r1 ⊗b = c tanh(r1 tanh−1

b b ) c b

= c tanh(r1 tanh−1 (tanh(r2 tanh−1 = c tanh(r1 r2 tanh−1

a a )) c a

(3.31)

a a ) c a

= (r1 r2 )⊗a . (V5) Scaling Property: It follows from the definition of Einstein scalar multiplication that a a |r|⊗a = c tanh(|r| tanh−1 ) c a |r|⊗a = c tanh(|r| tanh−1

a ) c

(3.32)

a )| c a ) = c tanh(|r| tanh−1 c for r∈R and a  0. Axiom (V5) follows immediately from the first and the third identity in (3.32). (V6) Gyroautomorphism Property: If v = 0 then Axiom (V6) clearly holds. Let Rn be the ambient vector space of the ball Rnc , and let φ : Rn → Rn be an invertible linear map that keeps invariant the inner product and, hence, the norm in Rn . Furthermore, let v ∈ Rnc ⊂ Rn , v  0. Then, by (3.19), r⊗a = |c tanh(r tanh−1

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r  ⎛  v ⎜⎜⎜ − 1− ⎜⎜⎜ 1 + c ⎜⎜⎜ φ(r⊗v) = φ ⎜⎜⎜c  r  ⎜⎜⎜ v ⎝ 1+ + 1− c 

r  v 1+ − 1− c = c r  v 1+ + 1− c

r ⎞ v ⎟⎟⎟ ⎟ c v ⎟⎟⎟⎟⎟ ⎟⎟ r v v ⎟⎟⎟⎟ ⎠ c

r v c φ(v) r v v c

(3.33)



r  r φ(v) φ(v) 1+ − 1− c c φ(v) = c r  r φ(v) φ(v) φ(v) 1+ + 1− c c = r⊗φ(v) , r∈R, v∈Rnc . It should be noted that while φ is a self-map of Rn , it is restricted in (3.33) to the ball Rnc ⊂ Rn . Any gyration of Rnc is expandable to a linear map of Rn , as explained in Remark 2.5, p. 19. In the special case when the map φ of Rn is a gyration of Rn , φ(a) = gyr[u, v]a for a ∈ Rn and arbitrary, but fixed, u, v ∈ Rnc , Identity (3.33) reduces to Axiom (V6). (V7) Identity Gyroautomorphism: If either r1 = 0 or r2 = 0 or a = 0 then Axiom (V7) holds by Theorem 2.33, p. 43, noting that 0⊗a = r⊗0 = 0. Assuming r1  0, r2  0, and a  0, the vectors r1 ⊗a and r2 ⊗a are parallel in Rn . Hence, by (2.38), p. 20, the gyration gyr[r1 ⊗a, r2 ⊗a] is trivial, thus verifying Axiom (V7). (V8) Vector Space: The vector space (G, ⊕, ⊗) is well known, studied in [8]. (V9) Homogeneity Property: Axiom (V9) follows immediately from the second and the third identities in (3.32). (V10) Gyrotriangle Inequality: Axiom (V10) is Theorem 2.1, p. 13.  Since scalar multiplication in Einstein gyrovector spaces does not distribute over Einstein addition, the following theorem about 2⊗(u⊕v) proves useful. Theorem 3.9. (The Two-Sum Identity). Let u, v be any two points of an Einstein

Einstein Gyrovector Spaces

gyrovector space (Rnc , ⊕, ⊗). Then, 2⊗(u⊕v) = u⊕(2⊗v⊕u) .

(3.34)

Proof. Employing the right gyroassociative law in (2.34), the identity gyr[v, v] = I, Theorem 2.16(4), the left gyroassociative law, and the gyrocommutative law in (2.34) we have the chain of equations u⊕(2⊗v⊕u) = u⊕((v⊕v)⊕u) = u⊕(v⊕(v⊕gyr[v, v]u)) = u⊕(v⊕(v⊕u)) = (u⊕v)⊕gyr[u, v](v⊕u) = (u⊕v)⊕(u⊕v) = 2⊗(u⊕v) ,

(3.35)



as desired.

As an application of Theorem 3.9 we prove the following theorem, which is related to the gyrovector space gyromidpoint concept. Theorem 3.10. Let u, v∈Rnc be any two points of an Einstein gyrovector space (Rnc , ⊕, ⊗). Then, u⊕( u⊕v)⊗ 12 = 12 ⊗(u  v) .

(3.36)

Proof. The proof is given by the following chain of equations, which are numbered for subsequent derivation: (1)

 2⊗{u⊕( u⊕v)⊗ 12 } === u⊕{( u⊕v)⊕u} (2)

 === {u⊕( u⊕v)}⊕gyr[u, u⊕v]u (3)

 === v⊕gyr[u, u⊕v]u (4)

 === v⊕gyr[v, u]u (5)

 === v  u (6)

 === u  v .

(3.37)

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The result in (3.37) implies (3.36) by the scalar associative law (V4) of gyrovector spaces in Def. 3.2 and in Theorem 3.8. Derivation of the numbered equalities in (3.37): (1) Follows from the Two-Sum Identity in Theorem 3.9 and the scalar associative law (V4) of gyrovector spaces. (2) Follows from (1) by the left gyroassociative law. (3) Follows from (2) by a left cancellation. (4) Follows from (3) by applying successively the gyration left and right reduction property. (5) Follows from (4) by Def. 2.15, p. 23, of the gyrogroup cooperation . (6) Follows from (5) by the commutativity of Einstein coaddition  according to (2.49), p. 24. 

3.4. Einstein Addition and Differential Geometry Einstein distance function, d(u, v), in an Einstein gyrovector space (Rnc , ⊕, ⊗) is given in terms of Einstein addition by the equation d(u, v) =  u⊕v = u v ,

(3.38)

u, v ∈ Rnc . We call it a gyrodistance function in order to emphasize the analogies it shares with its Euclidean counterpart, the distance function u − v in Rn . Among these analogies is the gyrotriangle inequality in Theorem 2.1, p. 13, and in Theorem 3.22, p. 93. In a two-dimensional Einstein gyrovector space (R2c , ⊕, ⊗) the squared gyrodistance between a point x ∈ R2c and an infinitesimally nearby point x + dx ∈ R2c , dx = (dx1 , dx2 ), is defined by the equation [93, Sect. 7.5] [84, Sect. 7.5] ds2 = (x + dx) x2 = Edx12 + 2Fdx1 dx2 + Gdx22 + . . . ,

(3.39)

where, if we use the notation r2 = x12 + x22 , E = c2 F = c2

c2 − x22 (c2 − r2 )2 x1 x2 − r2 )2

(c2

c2 − x12 G=c 2 . (c − r2 )2 2

(3.40)

Einstein Gyrovector Spaces

The triple (g11 , g12 , g22 ) = (E, F, G) along with g21 = g12 is known in differential geometry as the metric tensor gi j [47]. It turns out to be the metric tensor of the Beltrami-Klein disc model of hyperbolic geometry [53, p. 220]. Hence, ds2 in (3.39) – (3.40) is the Riemannian line element of the Beltrami-Klein disc model of hyperbolic geometry, linked to Einstein velocity addition (2.2), p. 10, and to Einstein gyrodistance function (3.38) [85]. The link between Einstein gyrovector spaces and the Beltrami-Klein ball model of hyperbolic geometry, already noted by Fock [24, p. 39], has thus been established in (3.38) – (3.40) in two dimensions. Note that in (3.39) vector addition, +, is used to describe an infinitesimally nearby point, while gyrovector subtraction, , is used to describe a gyrodistance. The extension of the link to higher dimensions is presented in [81, Sect. 9, Chap. 3], [93, Sect. 7.5], [84, Sect. 7.5], and [85]. For a brief account of the history of linking Einstein’s velocity addition law with hyperbolic geometry see [61, p. 943]. In Sects. 3.5 – 3.6 we will consider the transition from (i) Euclidean lines in the standard, Cartesian model of the Euclidean geometry of Euclidean vector spaces Rn into (ii) hyperbolic lines, called gyrolines, in the Cartesian-Beltrami-Klein model of the hyperbolic geometry of Einstein gyrovector spaces Rnc .

3.5. Euclidean Lines In this section we set the stage for the introduction of hyperbolic lines in Sect. 3.6. We introduce Cartesian coordinates into Rn in the usual way in order to specify uniquely each point P of the Euclidean n-space Rn by an n-tuple of real numbers, called the coordinates, or components, of P. Cartesian coordinates provide a method of indicating the position of points and rendering graphs on a two-dimensional Euclidean plane R2 and in a three-dimensional Euclidean space R3 . As an example, Fig. 3.1 presents a Euclidean plane R2 equipped with an unseen Cartesian coordinate system Σ. The position of points A and B and their midpoint MAB with respect to Σ are shown. The missing Cartesian coordinates in Fig. 3.1 are shown in Fig. 3.3. The set of all points Leuc AB (t) = A + (−A + B)t ,

(3.41)

t ∈ R, forms a Euclidean line. The segment AB on this line, corresponding to 0 ≤ t ≤ 1, and a generic point P on the segment, are shown in Fig. 3.1. The midpoint MA,B of the points A and B is obtained from (3.41) by selecting t = 1/2, MA,B = A + (−A + B) 12 ,

(3.42)

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B

MA,B P A

MA,B = A + (−A + B) 12 d(A, B) = A − B

d(A, MA,B ) = d(B, MA,B ) d(A, P) + d(P, B) = d(A, B) Leuc AB (t) = A + (−A + B)t −∞ ≤ t ≤ ∞

Figure 3.1 The Euclidean line. The line A + (−A + B)t, t ∈ R, in a Euclidean plane is shown. The points A and B correspond to the parameter values t = 0 and t = 1, respectively. The point P is a generic point on the line through the points A and B lying between these points. The sum, +, of the distance from A to P and from P to B equals the distance from A to B. Accordingly, distance along a line is additive. The point MA,B is the midpoint of the points A and B, corresponding to t = 1/2.

shown in Fig. 3.1. Being collinear, the points A, P, and B, where P lies between A and B, obey the triangle equality d(A, P) + d(P, B) = d(A, B) ,

(3.43)

d(A, B) =  − A + B

(3.44)

where is the (Euclidean) distance function, which measures the distance from A to B in Rn . The triangle equality (3.43) asserts that distance along a line is additive. Fig. 3.1 demonstrates the use of the standard Cartesian model of Euclidean geometry for graphical presentations. In a fully analogous way, Fig. 3.2 demonstrates the use of the Cartesian-Beltrami-Klein model of hyperbolic geometry, as we will see in Sect. 3.6. The introduction in (2.19), p. 15, of Cartesian coordinates (x1 , x2 , . . . , xn ), x12 + 2 x2 + . . . + xn2 < ∞, into the Euclidean n-space Rn , along with the use of the common vector addition and scalar multiplication in Cartesian coordinates, enables Euclidean geometry to be studied analytically. As an illustrative example, the analytic study of lines in Euclidean geometry is indicated in Fig. 3.1.

Einstein Gyrovector Spaces

Analogously, the introduction in (2.20), p. 15, of Cartesian coordinates (x1 , x2 , . . . , xn ), x12 + x22 + . . . + xn2 < c2 , into the c-ball Rnc of the Euclidean n-space Rn , along with the common Einstein addition and scalar multiplication in Cartesian coordinates, presented in Sect. 2.3, p. 14, enables hyperbolic geometry to be studied analytically. As an illustrative example, the analytic study of gyrolines in hyperbolic geometry is indicated in Fig. 3.2. Indeed, Figs. 3.3 and 3.4 of Sect. 3.6 and Figs. 3.5 and 3.6 of Sect. 3.7 indicate the way we study analytic hyperbolic geometry, where we are guided by analogies with analytic Euclidean geometry.

3.6. Gyrolines – The Hyperbolic Lines Hyperbolic points are called gyropoints and, in full analogy, hyperbolic lines are called gyrolines. Thus, for instance, A and B in Fig. 3.1 are points in the Euclidean plane R2 . They are joined by a line with a point P lying between A and B, and with the midpoint MA,B between A and B. In contrast, A and B in Fig. 3.2 are gyropoints in the hyperbolic plane R2c . They are joined by a gyroline with a gyropoint P lying between A and B and with the gyromidpoint MA,B between A and B. In general, when c → ∞ a hyperbolic plane R2c tends to the Euclidean plane R2 , a gyropoint tends to a corresponding point and a gyroline tends to a corresponding line. Let A, B ∈ Rnc be two distinct gyropoints of an Einstein gyrovector space (Rnc , ⊕, ⊗), shown in Fig. 3.2, and let t ∈ R be a real parameter. Then, in full analogy with the Euclidean line (3.41), shown in Fig. 3.1, the graph of the set of all gyropoints Lhyp AB (t) = A⊕( A⊕B)⊗t ,

(3.45)

t ∈ R, in the Einstein gyrovector space (Rnc , ⊕, ⊗) is a chord of the ball Rnc . This chord is a geodesic line of the Cartesian-Beltrami-Klein ball model of hyperbolic geometry, shown in Fig. 3.2 for n = 2. The missing Cartesian coordinates for the hyperbolic disc in Fig. 3.2 are shown in Fig. 3.4. The gyromidpoint MA,B of the gyropoints A and B is obtained from (3.45) by selecting t = 1/2, MA,B = A⊕( A⊕B)⊗ 12 = 12 ⊗(A  B) ,

(3.46)

shown in Fig. 3.2. The second identity in (3.46) follows from (3.37), indicating the ability of Einstein coaddition to capture analogies with classical results. The geodesic line (3.45) is the unique gyroline that passes through the gyropoints A and B. It passes through the gyropoint A when t = 0 and, owing to the left

75

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

B

EB

MA,B P

MA,B = A⊕( A⊕B)⊗ 12 d(A, B) = A B

A

EA

d(A, MA,B ) = d(B, MA,B ) d(A, P)⊕d(P, B) = d(A, B) Lhyp AB (t) = A⊕( A⊕B)⊗t −∞ ≤ t ≤ ∞

Figure 3.2 Gyroline, the hyperbolic line. This figure is the hyperbolic counterpart of Fig. 3.1. The analogies between lines and gyrolines, as illustrated in Figs. 3.1 and 3.2, are obvious. The gyroline n Lhyp AB approaches the boundary of Rc at its boundary gyropoints E A and E B . The boundary gyropoints hyp of the gyroline LAB are determined by the gyropoints A and B as shown in [98, Sect. 5.9].

cancellation law, (2.50), p. 25, it passes through the gyropoint B when t = 1. Furthermore, it passes through the gyromidpoint MA,B of A and B when t = 1/2. Accordingly, the gyrosegment that joins the gyropoints A and B in Fig. 3.2 is obtained from the gyroline (3.45) with 0 ≤ t ≤ 1. Each gyropoint of (3.45) with 0 < t < 1 is said to lie between A and B. Thus, for instance, the gyropoint P in Fig. 3.2 lies between the gyropoints A and B. As such, the gyropoints A, P, and B obey the gyrotriangle equality according to which d(A, P)⊕d(P.B) = d(A, B) ,

(3.47)

where, in full analogy with Euclidean geometry, d(A, B) =  A⊕B

(3.48)

is the hyperbolic distance function, called the gyrodistance function, which measures

Einstein Gyrovector Spaces

B

B

mA,B

mA,B P

mA,B = A + (−A +

B) 12

P

d(A, B) = A − B

A

d(A, mA,B ) = d(B, mA,B )

d(A, B) = A B d(A, mA,B ) = d(B, mA,B )

A

d(A, P)⊕d(P, B) = d(A, B)

d(A, P) + d(P, B) = d(A, B) Leuc AB (t)

mA,B = A⊕( A⊕B)⊗ 12

= A + (−A + B)t

Lhyp AB (t)

−∞ ≤ t ≤ ∞

= A⊕( A⊕B)⊗t

−∞ ≤ t ≤ ∞

Figure 3.3 The Cartesian coordinate system for the Euclidean plane R2 , (x1 , x2 ), x12 + x22 < ∞, unseen in Fig. 3.1, is shown here. The points A and B are given, with respect to these Cartesian coordinates by the equations A = (−0.60, −0.15) and B = (0.18, 0.80).

Figure 3.4 The Cartesian coordinate system that the c-disc R2c inherits from its Euclidean plane R2 , (x1 , x2 ), x12 + x22 < c2 , unseen in Fig. 3.2, is shown here. The gyropoints A and B are given, with respect to these Cartesian coordinates by c = 1, A = (−0.60, −0.15), and B = (0.18, 0.80).

the gyrodistance from A to B in Rnc . The gyrotriangle equality (3.47) asserts that gyrodistance along a gyroline is gyroadditive. The gyropoints in Fig. 3.2 are drawn with respect to an unseen Cartesian coordinate system. The missing Cartesian coordinates for the hyperbolic disc in Fig. 3.2 are shown in Fig. 3.4. The concept of gyrolines will be extended in Sect. 7.4 to the concept of bi-gyrolines of signature (m, n), m, n ∈ N, in such a way that bi-gyrolines of signature (1.n) are n-dimensional gyrolines.

3.7. Gyroangles – The Hyperbolic Angles The analogies between lines and gyrolines suggest corresponding analogies between angles and gyroangles. In full analogy with the notions of distance and angle, the notion of the gyroangle is deduced from the notion of the gyrodistance. Let O, A, and B be any three distinct gyropoints in an Einstein gyrovector space (Rnc , ⊕, ⊗). The resulting gyrosegments OA and OB that emanate from the gyropoint O form a gyroangle α = ∠AOB with gyrovertex O, as shown in Figs. 3.5 and 3.6 for n = 2. Following the analogies between gyrolines and lines, the radian measure of gyroangle α is, suggestively, given by the equation cos α =

O⊕A O⊕B · .  O⊕A  O⊕B

(3.49)

Here, in gyrotrigonometry, ( O⊕A)/ O⊕A and ( O⊕B)/ O⊕B are unit

77

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

B A

A

B

C

A B α O cos α =

α = ∠AOB = ∠A OB

=

cos α = cos β =

cos γ =

0≤α≤π O⊕A · O⊕B  O⊕A  O⊕B

α β

O⊕A · O⊕B  O⊕A  O⊕B

O⊕B · O⊕C  O⊕B  O⊕C

O⊕A · O⊕C  O⊕A  O⊕C

O

O⊕A O⊕B  O⊕A   O⊕B 

·

Figure 3.5 Gyroangles share remarkable analogies with angles, allowing the use of the elementary trigonometric functions cos, sin, etc., in gyrotrigonometry as well. Let A and B be any two gyropoints different from O, lying arbitrarily on the gyrosegments OA and OB, respectively, that emanate from a common gyropoint O in an Einstein gyrovector space (Rnc , ⊕, ⊗) as shown here for n = 2. The measure of the gyroangle α formed by the two gyrosegments OA and OB or, equivalently, formed by the two gyrosegments OA and OB is given by cosα, as shown here. In full analogy with angles, the measure of gyroangle α is independent of the choice of A and B .

γ =α+β Figure 3.6 Let A and C be two distinct gyropoints, let O be a gyropoint not on gyroline AC , and let B be a gyropoint between A and C in an Einstein gyrovector space (Rnc , ⊕, ⊗). Furthermore, let α = ∠AOB and β = ∠BOC be the two adjacent gyroangles that the three gyrosegments OA, OB, and OC form, and let γ be their composite gyroangle, formed by gyrosegments OA and OC . Then, γ = α + β, demonstrating that, like angles, gyroangles are additive. We call ( O⊕A)/ O⊕A a unit gyrovector. When applied to an inner product of unit gyrovectors, the common cosine function of trigonometry becomes the gyrocosine function of gyrotrigonometry.

gyrovectors, and cos is the common cosine function of trigonometry, which we assign to the inner product of unit gyrovectors as opposed to unit vectors in trigonometry. Accordingly, contrasting vector spaces, in the context of gyrovector spaces we refer the function “cosine” of trigonometry to as the function “gyrocosine” of gyrotrigonometry. Similarly, all the other elementary trigonometric functions and their interrelations survive unimpaired in their transition from the common trigonometry in Euclidean spaces Rn to a corresponding gyrotrigonometry in Einstein gyrovector space Rnc , as shown in [93, Chap. 12]. The center 0 = (0, . . . , 0) ∈ Rnc of the ball Rnc = (Rnc , ⊕, ⊗) is conformal (to Euclidean geometry) in the sense that the measure of any gyroangle with gyrovertex 0 is equal to the measure of its Euclidean counterpart. Indeed, if O = 0 then (3.49) descends to A B cos α = · , (3.50) A B

Einstein Gyrovector Spaces

−C + D = −A + B C 

−B + D = −A + C

v=

−A

+C

D

A

u=

−A



D

+ −A = w MABDC

The Parallelogram Condition : D = B + C − A

MAD = 12 (A + D) MBC = 12 (A + C)

+B



B

MABDC =

A+B+C+D 4

MABDC = MAD = MBC (−A + B) + (−A + C) = −A + D u+v=w Figure 3.7 The Euclidean parallelogram and its addition law in a Euclidean vector plane (R2 , +, ·). The diagonals AD and BC of parallelogram ABDC intersect each other at their midpoints. The midpoints of the diagonals AD and BC are, respectively, MAD and MBC , each of which coincides with the parallelogram centroid MABDC . This figure sets the stage for its hyperbolic counterpart, shown in Figs. 3.8 and 3.9.

which is indistinguishable from its Euclidean counterpart. Unlike Euclidean geometry, where the differences −O + A and A − O are equal, in general, the gyrodifferences O⊕A and A O are distinct. The presence of the gyrodifferences O⊕A and O⊕B in the gyroangle definition (3.49), rather then A O and B O, is dictated by the condition that gyroangles are gyroinvariant, that is, invariant under gyromotions, as stated in [84, Theorem 8.6, p. 241]. Being invariant under the gyromotions of Rnc , which are left gyrotranslations and rotations about the origin, gyroangles are geometric objects of the hyperbolic geometry of the Einstein gyrovector space (Rnc , ⊕, ⊗).

3.8. The Parallelogram Law We present in this section the well-known Euclidean parallelogram together with its vector addition law in order to set the stage for the gyroparallelogram and its

79

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

gyrovector addition law in Sects. 3.9 and 3.10. The resulting vector addition law, known as the parallelogram law, is illustrated in Fig. 3.7. Definition 3.11. (Parallelograms). Let A, B, and C be any three noncollinear points in a Euclidean space Rn . Then, the four points A, B, C, D in Rn are the vertices of the parallelogram ABDC, ordered either clockwise or counterclockwise, shown in Fig. 3.7, if D satisfies the parallelogram condition D = B+C − A.

(3.51)

The two vertices in each of the pairs (A, D) and (B, C) are said to be opposite to one another. The segments of adjacent vertices, AB, BD, DC, and CA, are the sides of the parallelogram. The segments AD and BC that join opposite vertices of the parallelogram ABDC are the diagonals of the parallelogram. The center MABDC of the parallelogram ABDC, shown in Fig. 3.7, is the midpoint of each of its two diagonals, so that MABDC = MAD = MBC . The definition of a degenerate parallelogram ABDC is the same, but with one exception: the set {A, B, C} of points is collinear. Let A, B, C ∈ Rn be three noncollinear points, and let u = −A + B v = −A + C

(3.52)

be two vectors in Rn that possess the same tail, A, as shown in Fig. 3.7. Furthermore, let D be a point of Rn given by the parallelogram condition (3.51). Then, the quadrangle ABDC (quadrangle is a term coined by Coxeter and Greitzer, recommending against the commonly used alternative term quadrilateral, [11, p. 52]) is a parallelogram in Euclidean geometry, shown in Fig. 3.7. As such, its two diagonals, AD and BC, intersect at their midpoints, that is, the midpoints of AD and BC coincide, 1 2 (A

+ D) = 12 (B + C) .

(3.53)

Clearly, the midpoint equality (3.53) is equivalent to the parallelogram condition (3.51). The vector addition of the vectors u and v that generate the parallelogram ABDC, according to (3.52), gives the vector w by the parallelogram addition law w := −A + D = (−A + B) + (−A + C) = u + v .

(3.54)

Here, by definition, w is the vector formed by the diagonal AD of the parallelogram ABDC, as shown in Fig. 3.7.

Einstein Gyrovector Spaces

D C

C⊕D = gyr[C, B]gyr[B, A]( A⊕B) B⊕D = gyr[B, C]gyr[C, A]( A⊕C)

The Gyroparallelogram Condition : D = (B  C) A MABDC

A B

MAD =

γA A+γD D γA +γD

MBC =

γB B+γC C γB +γC

MABDC =

γA A+γB B+γC C+γD D γA +γB +γC +γD

MABDC = MAD = MBC ( A⊕B)  ( A⊕C) = A⊕D

Figure 3.8 The Einstein gyroparallelogram and its addition law in an Einstein gyrovector space (Rnc , ⊕, ⊗). This figure is identical to Fig. 3.9 with one exception. Here gyropoints are denoted by capital italic letters, A, B, C, D, paving the way to Fig. 3.9 where gyrodifferences of gyropoints are recognized as gyrovectors, denoted by bold roman lowercase letters u, v, w. A gyroparallelogram ABDC is characterized by Theorem 3.14 as the gyroquadrangle the two gyrodiagonals, AD and BC , of which are concurrent at their gyromidpoints MAD and MBC . This gyropoint of concurrency is the gyroparallelogram gyrocentroid MABDC . The gyromidpoints MAD and MBC and the gyrocentroid MABDC , together with their gyrobarycentric representations with respect to the set S = {A, B, C, D} of the gyroparallelogram gyrovertices, share obvious analogies with their Euclidean counterparts, to which they tend in the Newtonian-Euclidean limit, c → ∞.

3.9. Einstein Gyroparallelograms Guided by analogies with parallelograms, the definition of gyroparallelograms follows. Definition 3.12. (Gyroparallelograms). Let A, B, and C be three non-gyrocollinear gyropoints in an Einstein gyrovector space (Rnc , ⊕, ⊗). Then, the four gyropoints A, B, C, and D in Rnc are the gyrovertices of the gyroparallelogram ABDC, ordered either clockwise or counterclockwise, shown in Fig. 3.8, if D satisfies the gyroparallelogram

81

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

condition D = (B  C) A .

(3.55)

The two gyrovertices in each of the pairs (A, D) and (B, C) are said to be opposite to each other. The gyrosegments of adjacent gyrovertices, AB, BD, DC, and CA, are the gyrosides of the gyroparallelogram. The gyrosegments AD and BC that join opposite gyrovertices of the gyroparallelogram ABDC are the gyrodiagonals of the gyroparallelogram. The gyrocentroid MABDC of the gyroparallelogram ABDC, shown in Fig. 3.8, is the gyromidpoint of each of its two gyrodiagonals, so that MABDC = MAD = MBC (See Theorem 3.13). The definition of a degenerate gyroparallelogram ABDC is the same, with one exception: the set {A, B, C} of gyropoints is gyrocollinear. By Def. 3.27, p. 98, and Theorem 2.57, p. 58, the gyroparallelogram condition (3.55) is gyrocovariant. Hence, the gyroparallelogram is a hyperbolic geometric object. In what seemingly sounds like a contradiction in terms, in Def. 3.12 we have translated the concept of the Euclidean parallelogram into hyperbolic geometry where the parallel postulate is denied. The resulting gyroparallelogram shares remarkable analogies with its Euclidean counterpart, giving rise to the gyroparallelogram law of gyrovector addition, which is commutative and fully analogous to the common parallelogram law of vector addition in Euclidean geometry. A gyroparallelogram in an Einstein gyrovector plane, that is, in the Cartesian-Beltrami-Klein disc model of hyperbolic geometry, is presented in Figs. 3.8 and 3.9. Theorem 3.13. (Gyroparallelogram Symmetries). Every gyrovertex of the gyroparallelogram ABDC, shown in Fig. 3.8, satisfies the gyroparallelogram condition, (3.55), that is, A = (B  C) D B = (A  D) C C = (A  D) B D = (B  C) A .

(3.56)

Furthermore, the two gyrodiagonals of a gyroparallelogram are concurrent, the concurrency gyropoint being the gyromidpoint of each of the two gyrodiagonals. Proof. The last equation in (3.56) follows from the gyroparallelogram condition (3.55) in Def. 3.12 of the gyroparallelogram. By the right cancellation law (2.98), p. 34, this

Einstein Gyrovector Spaces

equation is equivalent to the equation AD = BC.

(3.57)

By (2.49), p. 24, the coaddition  in Einstein gyrovector spaces is commutative. Hence, by the right cancellation law (2.97), p. 34, the equation in (3.57) is equivalent to each of the equations in (3.56), thus verifying (3.56). Equation (3.57) is equivalent to the gyroparallelogram gyromidpoint condition 1 2 ⊗(A

 D) = 12 ⊗(B  C) .

(3.58)

By (3.46), p. 75, the left- and the right-hand side of (3.58) are, respectively, the gyromidpoint of the gyrodiagonal AD and the gyrodiagonal BC. Hence, (3.58) implies that the gyromidpoints of the two gyrodiagonals of the gyroparallelogram coincide.  A gyroquadrangle is a not self-intersecting gyroconvex gyropolygon with four gyrosides and four gyrovertices in an Einstein gyrovector space. The following theorem characterizes gyroquadrangles which are gyroparallelograms. Theorem 3.14. A gyroquadrangle ABDC in an Einstein gyrovector space is a gyroparallelogram if and only if its gyrodiagonals AD and BC intersect at their gyromidpoints. Proof. The gyrodiagonals AD and BC of a gyroquadrangle ABDC, shown in Fig. 3.8, in an Einstein gyrovector space intersect at their gyromidpoints if and only if (3.58) is satisfied. The latter, in turn, is satisfied if and only if the gyroparallelogram condition (3.55) is satisfied, as explained in the proof of Theorem 3.13. Finally, by Def. 3.12, the gyroparallelogram condition is satisfied if and only if the gyroquadrangle ABDC is a gyroparallelogram.  In view of Theorem 3.14, a gyroparallelogram is a gyroconvex gyroquadrangle the two gyrodiagonals of which are concurrent at their gyromidpoints. This gyropoint of concurrency, shown in Figs. 3.8 and 3.9, is the gyroparallelogram gyrocentroid.

3.10. The Gyroparallelogram Law The gyroparallelogram law of gyrovector addition gives a commutative binary operation between gyrovectors in Einstein gyrovector spaces, shown in Fig. 3.9. As demonstrated in Theorem 3.15, it is fully analogous to the common parallelogram law of vector addition in Euclidean geometry [89].

83

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

D





C

=

A

⊕C

w

A

u=

⊕ A

D

C⊕D = gyr[C, B]gyr[B, A]( A⊕B) B⊕D = gyr[B, C]gyr[C, A]( A⊕C)

The Gyroparallelogram Condition : D = (B  C) A

MABDC

v=

84

A⊕

B



B

MAD =

γA A+γD D γA +γD

= 12 ⊗(A  D)

MBC =

γB B+γC C γB +γC

= 12 ⊗(A  C)

MABDC =

γA A+γB B+γC C+γD D γA +γB +γC +γD

MABDC = MAD = MBC ( A⊕B)  ( A⊕C) = A⊕D uv=w Figure 3.9 Einstein gyroparallelogram law of gyrovector addition. Let A, B, C ∈ Rnc be any three gyropoints of an Einstein gyrovector space (Rnc , ⊕, ⊗), giving rise to the two gyrovectors u = A⊕B and v = A⊕C . Furthermore, let D be a gyropoint of the gyrovector space such that ABDC is a gyroparallelogram, that is, D = (B  C) A by Def. 3.12 of the gyroparallelogram. Then, Einstein coaddition of gyrovectors u and v, u  v = w, expresses the gyroparallelogram law, where w = A⊕D. Einstein coaddition, , thus gives rise to the gyroparallelogram addition law of Einsteinian velocities, which is commutative and fully analogous to the parallelogram addition law of Newtonian velocities. Einsteinian velocities are, thus, gyrovectors that add according to the gyroparallelogram law just as Newtonian velocities are vectors that add according to the parallelogram law. Like vectors, a gyrovector A⊕B in an Einstein gyrovector space (Rnc , ⊕, ⊗), n = 2, 3, is described graphically as a straight arrow from the tail A to the head B.

Theorem 3.15. (The Gyroparallelogram (Addition) Law). Let ABDC be a gyroparallelogram in an Einstein gyrovector space (Rnc , ⊕, ⊗), n ∈ N. Then, ( A⊕B)  ( A⊕C) = A⊕D .

(3.59)

Proof. By Corollary 2.56, p. 57, and by the gyroparallelogram condition (3.55) we have (See Fig. 3.9 for n = 2) ( A⊕B)  ( A⊕C) = A⊕{(B  C) A} = A⊕D .

(3.60) 

Einstein Gyrovector Spaces

A gyroparallelogram ABDC in an Einstein gyrovector plane (R2c , ⊕, ⊗) is shown in Fig. 3.9, along with its gyroparallelogram addition law. The following theorem establishes a relationship between opposite gyrosides of the gyroparallelogram. Theorem 3.16. Opposite gyrosides of a gyroparallelogram ABDC in an Einstein gyrovector space (Rnc , ⊕, ⊗), shown in Figs. 3.8 and 3.9, are equal modulo gyrations, C⊕D = gyr[C, B]gyr[B, A]( A⊕B) = gyr[C, B](B A) B⊕D = gyr[B, C]gyr[C, A]( A⊕C) = gyr[B, C](C A)

(3.61)

and, equivalently, C⊕D = gyr[C, B]( B⊕A) C⊕A = gyr[C, B]( B⊕D) .

(3.62)

Two opposite gyrosides of a gyroparallelogram are congruent, having equal gyrolengths,  A⊕B =  C⊕D  A⊕C =  B⊕D .

(3.63)

Proof. By Theorem 2.21, p. 30, we have the gyrogroup identity A⊕D = ( A⊕C)⊕gyr[ A, C]( C⊕D) ,

(3.64)

and by Theorem 3.15, noting the definition of the gyrogroup cooperation , we have A⊕D = ( A⊕C)  ( A⊕B) = ( A⊕C)⊕gyr[ A⊕C, A B]( A⊕B)

(3.65)

for gyroparallelogram ABDC. Comparing the right-hand side of (3.64) and the extreme right-hand side of (3.65), and employing a left cancellation we have gyr[ A⊕C, A B]( A⊕B) = gyr[ A, C]( C⊕D) .

(3.66)

Identity (3.66) can be written in terms of Identity (2.212), p. 54, as gyr[A, C]gyr[C, B]gyr[B, A]( A⊕B) = gyr[ A, C]( C⊕D) ,

(3.67)

which is reducible to the first identity in (3.61) by eliminating gyr[A, C] on both sides of (3.67). Similarly, interchanging B and C, one can verify the second identity in (3.61). The equivalence between (3.61) and (3.62) follows from the gyroautomorphic

85

86

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

inverse property and a gyration inversion. Finally, (3.63) follows from (3.61) since gyrations preserve the gyrolength.



The Newtonian counterpart of the gyroparallelogram addition law (3.59) gives the parallelogram addition law (−A + B) + (−A + C) = −A + D

(Euclidean Geometry) ,

(3.68)

where A, B, D, C are the vertices of a Euclidean parallelogram, where −A + B, −A + C, and −A + D are three Euclidean vectors that emanate from the point A, and where the point D satisfies the parallelogram condition D = B+C −A

(Euclidean Geometry)

(3.69)

in Rn , which is analogous to the gyroparallelogram condition (3.55) in Rnc . The parallelogram condition (3.69) is equivalent to the parallelogram midpoint condition 1 2 (A

+ D) = 12 (B + C)

(Euclidean Geometry) ,

(3.70)

according to which the diagonals AD and BC of the parallelogram ABDC in Rn intersect each other at their midpoints. This equivalence is in full analogy with the equivalence between the gyroparallelogram condition (3.55) and its associated gyroparallelogram gyromidpoint condition (3.58). Experimental evidence that supports the physical significance of Einstein gyroparallelogram velocity addition law (3.59) is provided by the relativistic interpretation of the cosmological stellar aberration phenomenon, as explained in detail in [93, Chap. 13]. In this evidence, one of the velocities involved is the speed of light in empty space, assumed to be c. According to M. Hausner [37, fn. p. 48], it seems that there is no experimental, or even theoretical, evidence for the parallelogram velocity addition law. Einstein gyroparallelogram addition law  of collinear velocities coincides with Einstein addition law ⊕ of collinear velocities. An experimental evidence that supports Einstein’s velocity addition law for collinear velocities is provided by the famous Fizeau’s 1851 experiment [54]. For non collinear velocities, however, Einstein addition law is noncommutative, but gyrocommutative, while Einstein gyroparallelogram addition law is commutative, so that in general, ⊕  . Along with the above-mentioned experimental support of Einstein gyroparallelogram addition law that the cosmological stellar aberration phenomenon offers, there is no experimental evidence that supports the validity of Einstein addition law of noncollinear velocities [41, 58]. Einstein was aware of the result that his velocity addition law satisfies the Newtonian law of velocity parallelogram only to a first approximation. He thus noted in

Einstein Gyrovector Spaces

1905 that “Das Gesetz vom Parallelogramm der Geschwindigkeiten gilt also nach unserer Theorie nur in erster Ann¨aherung.” A. Einstein [15] [Thus the law of velocity parallelogram is valid according to our theory only to a first approximation.] Einsteinian velocities are regulated by hyperbolic geometry and its gyrovector space structure, just as Newtonian velocities are regulated by Euclidean geometry and its vector space structure. Accordingly, Einsteinian velocities obey the gyroparallelogram addition law of gyrovectors, just as Newtonian velocities obey the parallelogram addition law of vectors. The gyroparallelogram law (3.59) of gyrovector addition in hyperbolic geometry, shown in Fig. 3.9, is analogous to the parallelogram law of vector addition in Euclidean geometry shown in Fig. 3.7, and is given by the coaddition of gyrovectors. Gyrovectors v, in turn, are defined in terms of Einstein gyrodifferences as, for instance, v = A⊕B. The asymmetry in A and B of the gyrodifference A⊕B in Einstein gyrovector spaces is as natural as the asymmetry in A and B of the difference −A + B in Euclidean spaces. Similarly, the symmetry in A⊕B and A⊕C of the gyrovector coaddition ( A⊕B)  ( A⊕C) in Einstein gyrovector spaces is as natural as the analogous symmetry of the vector addition (−A + B) + (−A + C) in Euclidean spaces. Remarkably, in order to capture analogies between Newtonian and Einsteinian velocity compositions we must employ both the gyrocommutative operation ⊕ and the commutative cooperation  of Einstein gyrovector spaces. Along the analogies, a remarkable disanalogy emerges as well. Newtonian velocity addition, +, and the parallelogram addition law, +, of Newtonian velocities coincide. In contrast, Einstein velocity addition, ⊕, and the gyroparallelogram addition law, , of Einsteinian velocities do not coincide. The reason is clear: Owing to the presence of gyrations, Einstein velocity addition, ⊕, is in general noncommutative, while the gyroparallelogram addition, , is commutative. We thus see that 1. the hyperbolic-relativistic analog of the Euclidean-Newtonian difference −A + B is the gyrodifference A⊕B, where both −A + B and A⊕B are asymmetric in A and B, while 2. the hyperbolic-relativistic analog of the Euclidean-Newtonian sum A + B is the cogyrosum A  B, where both A + B and A  B are symmetric in A and B. Remarkably, both gyroaddition and cogyroaddition in Einstein gyrovector spaces descend to the usual vector addition in the associative case, but several standard

87

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properties of vector addition tend to split between these two operations in gyrovector spaces. Thus, both turn out to be useful in our task to capture analogies with classical results. An important point in case is the interplay of the two operations in the study of gyroparallelograms and gyromidpoints in gyrovector spaces. The gyroparallelogram and its gyroparallelogram law will be extended in Sect. 7.11 to the bi-gyroparallelogram and its bi-gyroparallelogram law.

3.11. Euclidean Isometries In this section and in Sect. 3.12 we present well-known results about Euclidean isometries and Euclidean motions in order to set the stage for the introduction of hyperbolic isometries (gyroisometries) and motions (gyromotions) in Sects. 3.13 and 3.14. The Euclidean distance function (distance, in short) in Rn , d(A, B) =  − A + B ,

(3.71)

A, B ∈ Rn , gives the distance between any two points A and B. It possesses the following properties: 1. 2. 3. 4. 5.

d(A, B) = d(B, A), d(A, B) ≥ 0, d(A, B) = 0 if and only if A = B, d(A, B) ≤d(A,C)+d(C, B) (the triangle inequality), d(A, B) =d(A,C)+d(C, B) (the triangle equality, for C lying between A and B),

for all A, B, C ∈ Rn . Definition 3.17. (Isometries). A map φ : Rn → Rn is a Euclidean isometry of Rn (isometry, in short) if it preserves the distance between any two points of Rn , that is, if d(φA, φB) = d(A, B)

(3.72)

for all A, B ∈ Rn . An isometry is injective (one-to-one into). Indeed, if A, B ∈ Rn are two distinct points, A  B, then 0   − A + B =  − φA + φB ,

(3.73)

so that φA  φB. We will now characterize the isometries of Rn , following which we will find that isometries are surjective (onto). For any X ∈ Rn , a translation of Rn by X is the map λX : Rn → Rn given by λX A = X + A

(3.74)

Einstein Gyrovector Spaces

for all A ∈ Rn . Theorem 3.18. (Translational Isometries). Translations of a Euclidean space Rn are isometries. Proof. The proof is trivial, but we present it in order to set the stage for the gyrocounterpart Theorem 3.24, p. 94, of this theorem. Let λX , X ∈ Rn , be a translation of a Euclidean space Rn . Then, λX is an isometry of the space, as we see from the following obvious chain of equations:  − λX A + λX B =  − (X + A) + (X + B) =  − A + B .

(3.75) 

Theorem 3.19. (Isometry Characterization [60, p. 19]). Let φ : Rn → Rn be a map of Rn . Then, the following are equivalent: 1. The map φ is an isometry. 2. The map φ preserves the distance between points. 3. The map φ is of the form φX = A + RX ,

(3.76)

where R ∈ O(n) is an n × n orthogonal matrix (that is, Rt R = RRt = I is the identity matrix) and A = φO ∈ Rn , O = (0, . . . , 0) being the origin of Rn . Proof. By definition, Item (1) implies Item (2) of the Theorem. Suppose that φ preserves the distance between any two points of Rn , and let R : Rn → Rn be the map given by RX = φX − φO .

(3.77)

Then, RO = O, and R also preserves the distance. Indeed, for all A, B ∈ Rn  − RA + RB =  − (φA − φO) + (φB − φO) =  − φA + φB =  − A + B . (3.78) Hence, R preserves the norm, RX =  − RO + RX =  − O + X = X .

(3.79)

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Consequently, R is orthogonal, R ∈ O(n). Indeed, for all X, Y ∈ Rn we have X − Y2 = (X − Y)·(X − Y) = X·X − X·Y − Y·X + Y·Y

(3.80)

= X2 + Y2 − 2X·Y , so that 2RX·RY = RX2 + RY2 − RX − RY2 = X2 + Y2 − X − Y2

(3.81)

= 2X·Y . Thus, following (3.77), there is an orthogonal n × n matrix R such that φX = φO + RX ,

(3.82)

and so (2) implies (3). If φ is of the form (3.76) then φ is the composite of an orthogonal transformation followed by a translation, and so φ is an isometry. Thus, (3) implies (1), as desired.  Following Theorem 3.19, it is now clear that isometries of Rn are surjective (onto), the inverse of isometry A + RX being (A + RX)−1 = −Rt A + Rt X .

(3.83)

Theorem 3.20. (Isometry Unique Decomposition). Let φ be an isometry of Rn . Then, it possesses the decomposition φX = A + RX ,

(3.84)

where A ∈ Rn and R ∈ O(n) are unique. Proof. By Theorem 3.19, φX possesses a decomposition (3.84). Let φX = A1 + R1 X φX = A2 + R2 X

(3.85)

be two decompositions of φX, X ∈ Rn . For X = O we have R1 O = R2 O = O, implying A1 = A2 . The latter, in turn, implies R1 = R2 , and the proof is complete  Let R be an orthogonal matrix. As RRt = I, we have that (detR)2 = 1, so that

Einstein Gyrovector Spaces

detR = ±1. If detR = 1, then R represents a rotation of Rn about its origin. The set of all rotations R in O(n) is a subgroup SO(n) ⊂ O(n) called the special orthogonal group. Accordingly, SO(n) is the group of all n × n orthogonal matrices with determinant 1. The set of all isometries φX = A + RX of Rn , A, X ∈ Rn , R ∈ O(n), forms a group called the isometry group of Rn . Following [4, p. 416], 1. the isometries φX = A + RX of Rn with detR = 1 are called direct isometries, or motions, of Rn and 2. the isometries φX = A + RX of Rn with detR = −1 are called opposite isometries. The motions of Rn , studied in Sect. 3.12, form a subgroup of the isometry group of Rn .

3.12. The Group of Euclidean Motions The Euclidean group of motions of Rn is the direct isometry group. It consists of the (i) commutative group of all translations of Rn and (ii) the group of all rotations of Rn about its origin. A rotation R of Rn about its origin is an element of the group SO(n) of all n×n orthogonal matrices with determinant 1. The rotation of A ∈ Rn by R ∈ SO(n) is RA. The map R ∈ SO(n) is a linear map of Rn that keeps the inner product invariant, that is, R(A + B) = RA + RB RA·RB = A·B

(3.86)

for all A, B ∈ Rn and all R ∈ SO(n). The Euclidean group of motions is the semidirect product group Rn × SO(n)

(3.87)

of the Euclidean commutative group Rn = (Rn , +) and the rotation group SO(n). It is a group of pairs (X, R), X ∈ (Rn , +), R ∈ SO(n), acting isometrically on Rn according to the equation (X, R)A = X + RA

(3.88)

for all A ∈ Rn . Each pair (X, R) ∈ Rn × SO(n), accordingly, represents a rotation of Rn followed by a translation of Rn . The group operation of the semidirect product group (3.87) is given by action composition. Accordingly, let (X1 , R1 ) and (X2 , R2 ) be any two elements of the semidirect product group Rn × SO(n). Their successive applications to A ∈ Rn is equivalent to a single application to A, as shown in the following chain of equations (3.89), in which

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we employ the associative law of vector addition, +, in Rn : (X1 , R1 )(X2 , R2 )A = (X1 , R1 )(X2 + R2 A) = X1 + R1 (X2 + R2 A) = X1 + (R1 X2 + R1 R2 A) = (X1 + R1 X2 ) + R1 R2 A = (X1 + R1 X2 , R1 R2 )A

(3.89)

for all A ∈ Rn . It follows from (3.89) that the group operation of the semidirect product group (3.87) is given by the semidirect product (X1 , R1 )(X2 , R2 ) = (X1 + R1 X2 , R1 R2 )

(3.90)

for any (X1 , R1 ), (X2 , R2 ) ∈ Rn × SO(n). Definition 3.21. (Covariance). A map T : (Rn )k → Rn

(3.91)

from k copies of Rn into Rn is covariant (with respect to the motions of Rn ) if its image T (A1 , A2 , . . . , Ak ) co-varies (that is, varies together) with its preimage points A1 , A2 , . . . , Ak under the motions of Rn , that is, if X + T (A1 , . . . , Ak ) = T (X + A1 , . . . , X + Ak ) RT (A1 , . . . , Ak ) = T (RA1 , . . . , RAk )

(3.92)

for all X ∈ Rn and all R ∈ SO(n). In particular, the first equation in (3.92) represents covariance with respect to (or, under) translations, and the second equation in (3.92) represents covariance with respect to (or, under) rotations. The importance of covariance under the motions of a geometry was first recognized by Felix Klein (1849–1924) in his Erlangen Program, the traditional professor’s inaugural speech that he gave at the University of Erlangen in 1872. The thesis that Klein published in Erlangen in 1872 is that a geometry is a system of definitions and theorems that express properties invariant under a given group of transformations called motions. The Euclidean motions of Euclidean geometry are described in this section, and the hyperbolic motions of hyperbolic geometry are described in Sect. 3.14. It turns out that the Euclidean and the hyperbolic motions share remarkable analogies.

Einstein Gyrovector Spaces

3.13. Gyroisometries – The Hyperbolic Isometries In this section we are guided by analogies with the Euclidean isometries studied in Sect. 3.11. The hyperbolic counterpart of the Euclidean distance function d(A, B) in Rn , given by (3.71), is the gyrodistance function d(A, B) in an Einstein gyrovector space Rnc = (Rnc , ⊕, ⊗), given by d(A, B) =  A⊕B ,

(3.93)

A, B ∈ Rnc , giving the gyrodistance between any two gyropoints A and B. It should always be clear from the context whether d(A, B) is the distance function in Rn or the gyrodistance function in Rnc . Like the distance function, the gyrodistance function possesses the following properties: 1. 2. 3. 4. 5.

d(A, B) = d(B, A), d(A, B) ≥ 0, d(A, B) = 0 if and only if A = B, d(A, B) ≤ d(A, C)⊕d(C, B) (The gyrotriangle inequality), d(A, B) = d(A, C)⊕d(C, B) (The gyrotriangle equality, for C lying between A and B),

for all A, B, C ∈ Rnc . A proof of the gyrotriangle inequality in Item (4) is presented in the following theorem. Theorem 3.22. (Gyrotriangle Inequality, II). The gyrodistance function of an Einstein gyrovector space (Rnc , ⊕, ⊗) obeys the gyrotriangle inequality  A⊕C ≤  A⊕B⊕ B⊕C

(3.94)

for all A, B, C∈Rnc . Proof. By Theorem 2.21, p. 30, we have A⊕C = ( A⊕B)⊕gyr[ A, B]( B⊕C).

(3.95)

Hence, by the gyrotriangle inequality (2.16) in Theorem 2.1, p. 13, and by the invariance of the norm under gyrations, we have  A⊕C = ( A⊕B)⊕gyr[ A, B]( B⊕C) ≤  A⊕B⊕gyr[ A, B]( B⊕C) =  A⊕B⊕ B⊕C .

(3.96) 

The gyrotriangle inequality (3.94) will be extended from Einstein gyrovector spaces (Rnc , ⊕, ⊗) to Einstein bi-gyrovector spaces (Rn×m c , ⊕E , ⊗) in (5.566), p. 295.

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Definition 3.23. (Gyroisometries). A map φ : Rnc → Rnc is a gyroisometry of Rnc if it preserves the gyrodistance between any two gyropoints of Rnc , that is, if d(φA, φB) = d(A, B)

(3.97)

for all A, B ∈ Rnc . A gyroisometry is injective (one-to-one into). Indeed, if A, B ∈ Rnc are two distinct gyropoints, A  B, then 0   A⊕B =  φA⊕φB ,

(3.98)

so that φA  φB. We will now characterize the gyroisometries of Rnc , following which we will find that gyroisometries are surjective (onto). For any X ∈ Rnc , a left gyrotranslation of Rnc by X is the map λX : Rnc → Rnc given by λX A = X⊕A

(3.99)

for all A ∈ Rnc . Theorem 3.24. (Left Gyrotranslational Gyroisometries). Left gyrotranslations of an Einstein gyrovector space are gyroisometries. Proof. Let λX , X ∈ Rnc , be a left gyrotranslation by X of an Einstein gyrovector space (Rnc , ⊕, ⊗). Then, λX is a gyroisometry of the space, as we see from the following chain of equations, which are numbered for subsequent derivation: (1)

  λX A⊕λX B ===  (X⊕A)⊕(X⊕B) (2)

 === gyr[X, A]( A⊕B)

(3.100)

(3)

 ===  A⊕B for all A, B, X ∈ Rnc . Derivation of the numbered equalities in (3.100): (1) Follows from (3.99). (2) Follows from Item (1) by the Left Gyrotranslation Theorem 2.22, p. 30. (3) Follows from Item (2) by the norm invariance under gyrations, (2.43), p. 20. 

Einstein Gyrovector Spaces

Theorem 3.25. (Gyroisometry Characterization). Let φ : Rnc → Rnc be a map of Rnc . Then, the following are equivalent: 1. The map φ is a gyroisometry. 2. The map φ preserves the gyrodistance between gyropoints. 3. The map φ is of the form φX = A⊕RX ,

(3.101)

where R ∈ O(n) is an n×n orthogonal matrix (that is, Rt R = RRt = I is the n×n identity matrix) and A = φO ∈ Rnc , O = (0, . . . , 0) being the origin of Rnc . Proof. By definition, Item (1) implies Item (2) of the Theorem. Suppose that φ preserves the gyrodistance between any two gyropoints of Rnc ,  φA⊕φB =  A⊕B ,

(3.102)

and let R : Rnc → Rnc be the map given by RX = φO⊕φX .

(3.103)

Then, RO = O and, by the left cancellation law, φX = φO⊕RX .

(3.104)

Furthermore, R also preserves the gyrodistance. Indeed, for all X, Y ∈ Rnc we have the following chain of equations, which are numbered for subsequent explanation: (1)

  RX⊕RY ===  ( φO⊕φX)⊕( φO⊕φY) (2)

 === gyr[ φO, φX]( φX⊕φY) (3)

 ===  φX⊕φY

(3.105)

(4)

 ===  X⊕Y . Derivation of the numbered equalities in (3.105): (1) (2) (3) (4)

Follows from (3.103). Follows from Item (1) by the Left Gyrotranslation Theorem 2.22, p. 30. Follows from Item (2) by the invariance of the norm under gyrations, (2.43), p. 20. Follows from Item (3) by Assumption (3.102). The map R preserves the norm since, by (3.105), RX =  RO⊕RX =  O⊕X = X .

(3.106)

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Moreover, R preserves the inner product as well. Indeed, by the gamma identity (2.11), p. 12, in Rnc and by (3.105)(3.106), and noting that γA = γA for all A ∈ Rnc , we have the following chain of equations: γX γY (1 −

X·Y ) = γ X⊕Y = γ RX⊕RY c2 RX·RY = γRX γRY (1 − ) c2 RX·RY = γX γY (1 − ), c2

(3.107)

implying RX·RY = X·Y,

(3.108)

as desired, so that R is orthogonal, R∈O(n). Thus, following (3.103), there is an orthogonal n × n matrix R∈O(n) such that φX = φO⊕RX ,

(3.109)

and so Item (2) implies Item (3) of the Theorem. If φ is of the form (3.101) then φ is the composite of an orthogonal transformation followed by a left gyrotranslation, and so φ is a gyroisometry. Thus, Item (3) implies Item (1) of the Theorem, and the proof is complete.  Following Theorem 3.25, it is now clear that gyroisometries of Rnc are surjective (onto), the inverse of gyroisometry A⊕RX being (A⊕RX)−1 = Rt A⊕Rt X .

(3.110)

Theorem 3.26. (Gyroisometry Unique Decomposition). Let φ be a gyroisometry of Rnc . Then, it possesses the decomposition φX = A⊕RX,

(3.111)

where A ∈ Rnc and R ∈ O(n) are unique. Proof. By Theorem 3.25, φX possesses a decomposition (3.111). Let φX = A1 ⊕R1 X φX = A2 ⊕R2 X

(3.112)

be two decompositions of φX, for all X ∈ Rnc . For X = O we have R1 O = R2 O = O, implying A1 = A2 . The latter, in turn, implies R1 = R2 , and the proof is complete. 

Einstein Gyrovector Spaces

Let R be an orthogonal matrix. As RRt = I, we have that (detR)2 = 1, so that detR = ±1. If detR = 1, then R represents a rotation of Rnc about its origin. The set of all rotations R in O(n) is a subgroup SO(n) ⊂ O(n) called the special orthogonal group. Accordingly, SO(n) is the group of all n × n orthogonal matrices with determinant 1. In full analogy with isometries, the set of all gyroisometries φX = A⊕RX of Rnc , A, X ∈ Rnc , R ∈ O(n), forms a group called the gyroisometry group of Rnc . Accordingly, by analogy with isometries, 1. the gyroisometries φX = A⊕RX of Rnc with detR = 1 are called direct gyroisometries, or motions, of Rnc and 2. the gyroisometries φX = A⊕RX of Rnc with detR = −1 are called opposite gyroisometries. In gyrolanguage, the motions of Rnc are called gyromotions. They form a subgroup of the gyroisometry group of Rnc , studied in Sect. 3.14.

3.14. Gyromotions – The Motions of Hyperbolic Geometry The group of gyromotions of an Einstein gyrovector space Rnc = (Rnc , ⊕, ⊗), n ∈ N, is the direct gyroisometry group of Rnc . It consists of the gyrocommutative gyrogroup of all left gyrotranslations of Rnc and the group SO(n) of all rotations of Rnc about its origin. A rotation R of Rnc about its origin is an element of the group SO(n) of all n×n orthogonal matrices with determinant 1. The rotation of a gyropoint A∈Rnc by R ∈ SO(n) is the gyropoint RA∈Rnc . The map R∈SO(n) is a gyrolinear map of Rnc that respects Einstein addition and keeps the inner product invariant, that is, R(A⊕B) = RA⊕RB RA·RB = A·B

(3.113)

for all A, B ∈ Rnc and all R ∈ SO(n), in full analogy with (3.86), p. 91. The group of gyromotions of Rnc possesses the gyrosemidirect product group structure, studied in Sect. 2.13. It is the gyrosemidirect product group Rnc × SO(n)

(3.114)

of the Einstein gyrocommutative gyrogroup Rnc = (Rnc , ⊕) and the rotation group SO(n). More specifically, it is a group of pairs (X, R), X ∈ (Rnc , ⊕), R ∈ SO(n), acting gyroisometrically on Rnc according to the equation (X, R)A = X⊕RA

(3.115)

for all A ∈ Rnc . Each pair (X, R) ∈ Rnc × SO(n), accordingly, represents a rotation of Rnc followed by a left gyrotranslation of Rnc .

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The group operation of the gyrosemidirect product group (3.114) is given by action composition. Accordingly, let (X1 , R1 ) and (X2 , R2 ) be any two elements of the gyrosemidirect product group Rnc × SO(n). Their successive applications to A ∈ Rnc is equivalent to a single application to A, as shown in the following chain of equations (3.116), in which we employ the left gyroassociative law of Einstein addition, ⊕, in Rnc : (X1 , R1 )(X2 , R2 )A = (X1 , R1 )(X2 ⊕R2 A) = X1 ⊕R1 (X2 ⊕R2 A) = X1 ⊕(R1 X2 ⊕R1 R2 A) = (X1 ⊕R1 X2 )⊕gyr[X1 , R1 X2 ]R1 R2 A = (X1 ⊕R1 X2 , gyr[X1 , R1 X2 ]R1 R2 )A

(3.116)

for all A ∈ Rnc . It follows from (3.116) that the group operation of the gyrosemidirect product group (3.114) is given by the gyrosemidirect product (X1 , R1 )(X2 , R2 ) = (X1 ⊕R1 X2 , gyr[X1 , R1 X2 ]R1 R2 )

(3.117)

for any (X1 , R1 ), (X2 , R2 ) ∈ Rnc × SO(n). Gyrocovariance with respect to gyromotions is formalized in the following definition. Definition 3.27. (Gyrocovariance). A map T : (Rnc )k → Rnc

(3.118)

from k copies of Rnc = (Rnc , ⊕, ⊗) into Rnc is gyrocovariant (with respect to the gyromotions of Rnc ) if its image T (A1 , A2 , . . . , Ak ) co-varies (that is, varies together) with its preimage gyropoints A1 , A2 , . . . , Ak under the gyromotions of Rnc , that is, if X⊕T (A1 , . . . , Ak ) = T (X⊕A1 , . . . , X⊕Ak ) RT (A1 , . . . , Ak ) = T (RA1 , . . . , RAk )

(3.119)

for all X ∈ Rnc and all R ∈ SO(n). In particular, the first equation in (3.119) represents gyrocovariance with respect to (or, under) left gyrotranslations, and the second equation in (3.119) represents gyrocovariance with respect to (or, under) rotations. As an important example, gyrobarycentric coordinates are gyrocovariant, as explained in [98, Theorem 5.14, p. 135] and [84, Sect. 10.20]. All the gyrotriangle gyrocenters presented in [95, 96] are determined in terms of their gyrobarycentric coordinates with respect to their reference gyrotriangles. Hence, these gyrotriangle

Einstein Gyrovector Spaces

gyrocenters are gyrocovariant and, as such, possess geometric significance in the sense of Felix Klein’s Erlangen program. Following the presentation in Chaps. 2 and 3 of the required background about Einstein gyrogroups and gyrovector spaces, we are now in the position to study the theory of Einstein bi-gyrogroups and bi-gyrovector spaces.

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CHAPTER 4

Bi-gyrogroups and Bi-gyrovector Spaces – P

4.1. Introduction Let Rn×m be the set of all n × m real matrices. In this chapter, pseudo-rotations with a parameter P ∈ Rn×m in pseudo-Euclidean spaces lead to two novel algebraic structures called a bi-gyrogroup and a bi-gyrovector space. In the context of relativity physics, these pseudo-rotations are known as Lorentz transformations of signature (m, n), which are, in turn, Lorentz transformations in m time-like dimensions and n space-like dimensions. The common Lorentz transformation in special relativity is the Lorentz transformation of signature (1, n), where n = 3 in physical applications. A pseudo-Euclidean space Rm,n of signature (m, n), m, n ∈ N, is an (m + n)dimensional space with the pseudo-Euclidean inner product of signature (m, n). A Lorentz transformation of signature (m, n) is a special linear transformation Λ ∈ S O(m, n) in Rm,n that leaves the pseudo-Euclidean inner product invariant. It is special in the sense that the determinant of the (m + n) × (m + n) real matrix Λ is 1, and the determinant of its first m rows and columns is positive [36, p. 478]. Equivalently, it is special in the sense that it can be reached continuously from the identity transformation. The group SO(m, n) of all Lorentz transformations of signature (m, n) is also known as the special pseudo-orthogonal group. A Lorentz transformation without rotations is called a boost. In general, two successive boosts are not equivalent to a boost. Rather, they are equivalent to a boost associated with two rotations, called a left rotation and a right rotation, or collectively a bi-rotation. The two rotations of a bi-rotation are generally nontrivial if both m > 1 and n > 1. The special case when m = 1 and n > 1 was studied in 1988 in [74], resulting in the discovery of two novel algebraic objects that became known as a gyrogroup and a gyrovector space [81]. Subsequent study of gyrovector spaces reveals in [81, 84, 93, 94, 96, 95, 98] that gyrovector spaces form the algebraic setting for hyperbolic geometry, just as vector spaces form the algebraic setting for Euclidean geometry. The aim of this chapter is to extend the study of the parametric realization of the Lorentz group in [74] from m = 1 to m ≥ 1 and to reveal the resulting new Beyond Pseudo-rotations in Pseudo-Euclidean Spaces http://dx.doi.org/10.1016/B978-0-12-811773-6.50004-9 Copyright © 2018 Elsevier Inc. All rights reserved.

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algebraic objects, bi-gyrogroups and bi-gyrovector spaces. In order to emphasize that when m > 1 and n > 1 a successive application of two boosts generates a bi-rotation, a Lorentz boost of signature (m, n), m, n > 1, is called a bi-boost of signature (m, n). The composition law of two bi-boosts gives rise to a bi-gyrocommutative bi-gyrogroup operation, just as the composition law of two boosts gives rise in [74] to a gyrocommutative gyrogroup operation, as demonstrated in [81]. Accordingly, a bi-gyrogroup of signature (m, n), m, n ∈ N, is a group-like structure that specializes to a gyrogroup when either m = 1 or n = 1. It is shown in Theorem 4.12, p. 118, that a Lorentz transformation Λ of signature (m, n) possesses the unique parametrization Λ = Λ(P, On , Om ), where 1. P ∈ Rn×m is any real n × m matrix; where 2. On ∈ S O(n) is any n × n special orthogonal matrix, taking P into On P (that is a left rotation); and, similarly, where 3. Om ∈ S O(m) is any m × m special orthogonal matrix, taking P into POm (that is a right rotation). In the special case when m = 1, the Lorentz transformation of signature (m, n) specializes to the Lorentz transformation of special relativity theory (n = 3 in physical applications), where the parameter P is a vector that represents relativistic proper velocities (as indicated in Example 5.33, p. 213). The parametrization of the Lorentz transformation Λ ∈ SO(m, n) enables in Theorem 4.31, p. 139, the Lorentz transformation composition (or, product) law to be expressed in terms of parameter composition. Under the resulting parameter composition, the parameter On of Λ, called a left rotation (of P ∈ Rn×m ), forms a group. The group that the left rotations form is the special orthogonal group S O(n). Similarly, under the parameter composition, the parameter Om of Λ, called a right rotation (of P ∈ Rn×m ), forms a group. The group that the right rotations form is the special orthogonal group S O(m). The pair (On , Om ) ∈ S O(n) × S O(m) is called a bi-rotation, taking P ∈ Rn×m into On POm ∈ Rn×m . Contrasting the left and right rotation parameters, the parameter P does not form a group under parameter composition. Rather, it forms a novel algebraic object, called a bi-gyrocommutative bi-gyrogroup. A bi-gyrocommutative bi-gyrogroup is a grouplike structure that generalizes the gyrocommutative gyrogroup. The latter, in turn, is a group-like structure that forms a natural generalization of the commutative group. The concept of the gyrogroup emerged from the 1988 study of the parametrization of the Lorentz group in [74]. Presently, the gyrogroup concept plays a universal computational role, which extends far beyond the domain of special relativity, as noted by Chatelin in [10, p. 523] and in references therein and as evidenced, for instance, from [12, 19, 21, 22, 23, 40], [67, 68, 69, 70, 71], and [43, 86, 97, 101]. In a similar way, the concept of the bi-gyrogroup emerges in this chapter from the study of

Bi-gyrogroups and Bi-gyrovector Spaces – P

the parametrization of the Lorentz group S O(m, n), m, n ∈ N. Hence, like gyrogroups, bi-gyrogroups are capable of playing a universal computational role that extends far beyond the domain of Lorentz transformations in pseudo-Euclidean spaces of any signature (m, n). The introduction of a scalar multiplication into the bi-gyrogroups that underlie the space Rn×m , m, n ∈ N, turns them into bi-gyrovector spaces that underlie Rn×m and that generalize the gyrovector space studied in Chap. 3. The main goal of this chapter is, accordingly, to uncover the bi-gyrogroup and the bi-gyrovector space structures that underlie the space Rn×m , which are finally summarized in Sect. 4.28.

4.2. Pseudo-Euclidean Spaces and Pseudo-Rotations The study of pseudo-Euclidean spaces and pseudo-rotations is important in linear algebra [35]. A pseudo-Euclidean space Rm,n of signature (m, n), m, n ∈ N, is an (m + n)dimensional space with an orthogonal basis ei , i = 1, . . . , m + n,

where

ei ·e j = i δi j ,

(4.1)

⎧ ⎪ ⎪ ⎨+1, i = 1, . . . , m i = ⎪ ⎪ ⎩− 12 , i = m + 1, . . . , m + n , c

(4.2)

where c is an arbitrarily fixed positive constant. Without loss of generality one may select c = 1. However, we prefer to consider c as a free parameter in order to allow limits as c → ∞ to be available, where the modern and new descend to the classical and familiar. An illustrative point in case is the Lorentz transformation of special relativity theory, which descends to the Galilei transformation when the vacuum speed of light c tends to infinity. A pseudo-Euclidean space Rm,n of signature (m, n) is equipped with an inner product of signature (m, n). The inner product x·y of two vectors x, y ∈ Rm,n , x= y=

m+n  i=1 m+n 

xi ei (4.3) yi ei ,

i=1

is x·y =

m+n  i=1

i xi yi =

m  i=1

xi yi −

m+n 1  xi yi . c2 i=m+1

(4.4)

Let Im be the m × m identity matrix, and let η be the (m + n) × (m + n) diagonal

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

matrix

⎞ ⎛ ⎜⎜⎜ Im 0m,n ⎟⎟⎟ ⎟⎟⎠ , η = ⎜⎜⎝ 0n,m − c12 In

(4.5)

where 0m,n is the m × n zero matrix. Then, the matrix representation of the inner product (4.4) is x·y = xt ηy, where x and y are the column vectors ⎞ ⎛ ⎜⎜⎜ x1 ⎟⎟⎟ ⎜⎜⎜ x ⎟⎟⎟ ⎜ 2 ⎟ x = ⎜⎜⎜⎜⎜ .. ⎟⎟⎟⎟⎟ ⎜⎜⎜ . ⎟⎟⎟ ⎠ ⎝ xm+n

and

(4.6) ⎛ ⎜⎜⎜ ⎜⎜⎜ ⎜ y = ⎜⎜⎜⎜⎜ ⎜⎜⎜ ⎝

y1 y2 .. . ym+n

⎞ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟ ⎠

(4.7)

and exponent t denotes transposition. Let Λ be an (m + n) × (m + n) matrix that leaves the inner product (4.6) invariant. Then, for all x, y ∈ Rm,n , (Λx)t ηΛy = xt ηy ,

(4.8)

implying xt Λt ηΛy = xt ηy, so that [49, p. 193] Λt ηΛ = η .

(4.9)

The determinant of the matrix equation (4.9) yields (detΛ)2 = 1 ,

(4.10)

noting that det(Λt ηΛ) = (detΛt )(detη)(detΛ) and detΛt = detΛ. Hence, detΛ = ±1. The special transformations Λ that can be reached continuously from the identity transformation of Rm,n constitute the special pseudo-orthogonal group S O(m, n), called the group of pseudo-rotations of signature (m, n). A pseudo-rotation of signature (m, n) is also called a Lorentz transformation of signature (m, n). The Lorentz transformation of signature (1, 3) turns out to be the common homogeneous, proper, orthochronous Lorentz transformation of Einstein’s special theory of relativity [93]. The formal definition of SO(m, n) follows. Definition 4.1. (SO(m,n)). Let m, n ∈ N be two positive integers. A linear transformation Λ of the pseudo-Euclidean space Rm,n is called a pseudo-rotation of signature (m, n), or a Lorentz transformation of signature (m, n), if it leaves the inner product (4.4) invariant, and if it can be reached continuously from the identity transformation of Rm,n . The group of all Lorentz transformations of signature (m, n) is denoted by SOc (m, n).

Bi-gyrogroups and Bi-gyrovector Spaces – P

When there is no need to emphasize that SOc (m, n) depends on c we omit the index c, using the notation SOc (m, n) = SO(m, n). If Λ ∈ SO(m, n) then its determinant is equal to 1, and the determinant of its first m rows and columns is positive [36, p. 478].

4.3. Matrix Representation of SO(m,n) Let Rm×n be the set of all m × n real matrices, m.n ∈ N. Following Norbert Dragon [13], in order to parametrize the special pseudo-orthogonal group S O(m, n), we partition each (m + n) × (m + n) matrix representation of Λ ∈ S O(m, n) into four blocks consisting of the submatrices (i) A ∈ Rm×m , (ii) Aˆ ∈ Rn×n , (iii) B ∈ Rn×m , and (iv) Bˆ ∈ Rm×n , so that 

A Bˆ . (4.11) Λ= cB cAˆ By means of (4.11) and (4.5), the matrix equation (4.9) takes the form ⎞ ⎞

t  ⎛ ⎛ ⎜⎜⎜ Im 0m,n ⎟⎟⎟ A Bˆ 0m,n ⎟⎟⎟ A cBt ⎜⎜⎜⎜ Im ⎟⎟ ⎟⎟⎠ = ⎜⎜⎝ ⎜ Bˆ t cAˆ t ⎝0n,m − 12 In ⎠ cB cAˆ 0n,m − c12 In c or, equivalently,

⎞  ⎛ ⎜⎜⎜ Im 0m,n ⎟⎟⎟ At A − Bt B At Bˆ − Bt Aˆ ⎟⎟⎠ , = ⎜⎜⎝ Bˆ t A − Aˆ t B Bˆ t Bˆ − Aˆ t Aˆ 0n,m − c12 In

(4.12)

(4.13)

implying At A = Im + Bt B Aˆ t Aˆ = c−2 In + Bˆ t Bˆ

(4.14)

A Bˆ = B Aˆ . t

t

The symmetric matrix Bt B is diagonalizable by an orthogonal matrix with nonnegative diagonal elements [38, pp. 171, 396-398, 402]. Hence, the eigenvalues of Im + Bt B are not smaller than 1, so that (detA)2 = det(Im + Bt B) ≥ 1. This, in turn, ˆ 2 = det(c−2 In + Bˆ Bˆ t ) ≥ 1/c2n , so that Aˆ is implies that A is invertible. Similarly, (detA) invertible. An invertible m × m real matrix A can be uniquely decomposed into the product of a special orthogonal matrix O ∈ S O(m), Ot = O−1 , detO = 1, and an m × m positive definite symmetric matrix S , S t = S , with positive eigenvalues [27, p. 286], A = OS .

(4.15)

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Following (4.15) we have At A = S t Ot OS = S 2

(4.16)

with positive eigenvalues λi > 0, i = 1, . . . , m. Hence, √ (4.17) S = At A √ has the positive eigenvalues λi and the same eigenvectors as S 2 . The matrix S given by (4.17) satisfies (4.15) since AS −1 is orthogonal, as it should be, by (4.15). Indeed, (AS −1 )t AS −1 = (S −1 )t At AS −1 = S −1 S 2 S −1 = Im .

(4.18)

Similarly, Aˆ ∈ Rn×n is invertible and possesses the decomposition cAˆ = Oˆ Sˆ ,

(4.19)

where Oˆ ∈ S O(n) is an orthogonal matrix and Sˆ is an n × n positive definite (and hence, symmetric) matrix. By means of (4.15) and (4.19), the block matrix (4.11) possesses the decomposition

  O 0m,n S Pˆ Λ= , (4.20) 0n,m Oˆ P Sˆ where the submatrices P ∈ Rn×m and Pˆ ∈ Rm×n are to be determined by (4.20) and (4.11) in (4.22). Following (4.20) and (4.11), along with (4.15) and (4.19), we have 

 OS OPˆ A Bˆ , (4.21) Λ= ˆ = cB cAˆ OP Oˆ Sˆ ˆ = cB and OPˆ = B, ˆ that is so that OP P = c Oˆ −1 B Pˆ = O−1 Bˆ .

(4.22)

In (4.20), S and Sˆ are invertible symmetric matrices, and O ∈ SO(m) and Oˆ ∈ SO(n) are orthogonal matrices with determinant 1. By means of (4.9) and (4.20) we have the matrix equation      

t O 0m,n S Pˆ S Pt Ot 0m,n Im Im 0m,n 0m,n = . (4.23) 0n,m −c−2 In Pˆ t Sˆ t 0n,m Oˆ t 0n,m −c−2 In 0n,m Oˆ P Sˆ Noting that

    Ot 0m,n Im O 0m,n Im 0m,n 0m,n = , 0n,m −c−2 In 0n,m Oˆ t 0n,m −c−2 In 0n,m Oˆ

(4.24)

Bi-gyrogroups and Bi-gyrovector Spaces – P

the matrix equation (4.23) yields    

t S Pt Im Im S Pˆ 0m,n 0m,n = , 0n,m −c−2 In Pˆ t Sˆ t 0n,m −c−2 In P Sˆ so that

and hence

(4.25)

   S t −c−2 Pt S Pˆ Im 0m,n = 0n,m −c−2 In Pˆ t −c−2 Sˆ t P Sˆ

(4.26)

  Im 0m,n S t S − c−2 Pt P S t Pˆ − c−2 Pt Sˆ = . 0n,m −c−2 In Pˆ t S − c−2 Sˆ t P Pˆ t Pˆ − c−2 Sˆ t Sˆ

(4.27)



Noting that S t S = S 2 and Sˆ t Sˆ = Sˆ 2 , (4.27) yields the equations 1 t PP c2 Sˆ 2 = In + c2 Pˆ t Pˆ (4.28) 1 S t Pˆ = 2 Pt Sˆ . c Formalizing the main results obtained in (4.11) – (4.28), we have the following Lemma. S 2 = Im +

Lemma 4.2. If Λ ∈ SO(m, n) then Λ possesses the representation  

O 0m,n S Pˆ , Λ= 0n,m Oˆ P Sˆ where O ∈ SO(m), Oˆ ∈ SO(n), P ∈ Rn×m , and Pˆ ∈ Rm×n , and where 1 S 2 = I m + 2 Pt P c Sˆ 2 = In + c2 Pˆ t Pˆ 1 S t Pˆ = 2 Pt Sˆ . c

(4.29)

(4.30)

We will now show that if S , Sˆ , P, and Pˆ are related by (4.30) then Pˆ = Pt /c2 . Noting that the matrix S is symmetric, the third and the first equations in (4.30) imply 1 Pˆ = 2 S −1 Pt Sˆ c 1 S −2 = (Im + 2 Pt P)−1 . c

(4.31)

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Inserting (4.31) into the second equation in (4.30), Sˆ 2 = In + c2 Pˆ t Pˆ 1 1 = In + c2 ( 2 S −1 Pt Sˆ )t 2 S −1 Pt Sˆ c c 1 = In + 2 Sˆ t P(S −1 )t S −1 Pt Sˆ c 1 = In + 2 Sˆ PS −2 Pt Sˆ c 1 1 = In + 2 Sˆ P(Im + 2 Pt P)−1 Pt Sˆ . c c Multiplying both extreme sides of (4.32) by Sˆ −2 , we have 1 1 In = Sˆ −2 + 2 P(Im + 2 Pt P)−1 Pt . c c

(4.32)

(4.33)

Let ω be an eigenvector of the matrix c−2 PPt , and let λ be its associated eigenvalue, 1 ω. PPtω = λω c2

(4.34)

If Ptω is not zero, Ptω  0, then it is an eigenvector of c−2 Pt P with the same eigenvalue λ, 1 t (P P)Ptω = λPtω . (4.35) c2 Adding Ptω to both sides of (4.35) we have (Im +

1 t P P)Ptω = (1 + λ)Ptω , c2

(4.36)

so that 1 1 Ptω = (Im + 2 Pt P)−1 Ptω . 1+λ c Multiplying both sides of (4.37) by P, we have 1 1 PPtω = P(Im + 2 Pt P)−1 Ptω , 1+λ c so that, by means of (4.34),

(4.37)

(4.38)

1 t −1 t λ P P) P ω = c2 ω (4.39) 2 1+λ c for any eigenvector ω of PPt for which Pt ω  0. Equation (4.39) remains valid also when Ptω = 0 since in this case λ = 0 by (4.34). P(Im +

Bi-gyrogroups and Bi-gyrovector Spaces – P

By means of (4.33) and (4.39) we have ω = Sˆ −2ω +

1 1 λ ω, P(Im + 2 Pt P)−1 Ptω = Sˆ −2ω + 2 1+λ c c

(4.40)

so that Sˆ −2ω =

1 ω 1+λ

(4.41)

and, by (4.34), 1 1 ω = Inω + λω ω = Inω + 2 PPtω = (In + 2 PPt )ω ω (4.42) Sˆ 2ω = (1 + λ)ω c c for any eigenvector ω of PPt . The eigenvectors ω constitute a basis of Rn [27, Theorem 5, p. 273]. Hence, it follows from (4.42) that 1 (4.43) Sˆ 2 = In + 2 PPt c and, hence,  (4.44) Sˆ = In + c−2 PPt . Following (4.30) – (4.31) and (4.44) we have  −1 1  1 Pˆ = 2 S −1 Pt Sˆ = 2 Im + c−2 Pt P Pt In + c−2 PPt . c c

(4.45)

Employing (4.45) and the eigenvectors ω of PPt , we will show in (4.50) that Pˆ = Pt /c2 . As in (4.34), ω is an eigenvector of the matrix c−2 PPt , Pt ω  0, with its associated eigenvalue λ > 0, implying (4.37). Following (4.37), the matrix (Im + c−2 Pt P)−1 possesses an eigenvector Ptω with its associated eigenvalue 1/(1 + λ). Hence, the matrix  −1 −2 t I√ possesses the same eigenvector Ptω with its associated eigenvalue m+c PP 1/ 1 + λ,  −1 1 Im + c−2 Pt P Ptω = √ (4.46) Pt ω . 1+λ ω = (1 + λ)ω ω, Similarly, the matrix In + c−2 PPt satisfies the equation (In + c−2 PPt )ω so that itpossesses an eigenvector ω with its associated eigenvalue 1 + λ. Hence, the matrix In + c−2 PPt possesses the same eigenvector ω with its associated eigenvalue √ 1 + λ,  √ In + c−2 PPt ω = 1 + λ ω . (4.47)

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Hence, by (4.47) and (4.46),    √ −1 −1 Im + c−2 Pt P Pt In + c−2 PPt ω = 1 + λ Im + c−2 Pt P Ptω

(4.48)

= Pt ω for any eigenvector ω of PPt for which Ptω  0. Equation (4.48) remains valid also for ω with Ptω = 0 since in this case λ = 0 by (4.34) and, hence by (4.47),  In + c−2 PPtω = ω . The eigenvectors ω constitute a basis of Rn [27, Theorem 5, p. 273]. Hence, it follows from (4.48) that   −1 Im + c−2 Pt P Pt In + c−2 PPt = Pt , (4.49) so that, by (4.45) and (4.49), 1 Pˆ = 2 Pt . (4.50) c Formalizing the main results obtained in (4.30) – (4.50), we have the following Lemma. Lemma 4.3. If S , Sˆ , P, and Pˆ are related by (4.30) then   Pt In + c−2 PPt = Im + c−2 Pt PPt

(4.51)

and 1 Pˆ = 2 Pt . c

(4.52)

Combining Lemmas 4.2 and 4.3 we obtain the following Lemma. Lemma 4.4. If Λ ∈ SO(m, n) then Λ possesses the representation ⎞ ⎞⎛ ⎛ ⎟⎟⎟ c−2 Pt ⎜⎜⎜ O 0m,n ⎟⎟⎟ ⎜⎜⎜ Im + c−2 Pt P ⎟⎠⎟ ⎜⎜⎜ ⎟⎟⎠⎟ , Λ = ⎜⎝⎜  ⎝ 0n,m Oˆ P In + c−2 PPt where O ∈ SO(m), Oˆ ∈ SO(n) and P ∈ Rn×m . Furthermore, Identity (4.51) of Lemma 4.3 yields the following Lemma.

(4.53)

Bi-gyrogroups and Bi-gyrovector Spaces – P

Lemma 4.5. The following commuting relations hold for all P ∈ Rn×m ,   P Im + c−2 Pt P = In + c−2 PPt P   Pt In + c−2 PPt = Im + c−2 Pt P Pt   PPt In + c−2 PPt = In + c−2 PPt PPt   Pt P Im + c−2 Pt P = Im + c−2 Pt P Pt P .

(4.54) (4.55) (4.56) (4.57)

Proof. The commuting relation (4.55) follows from (4.49). The commuting relation (4.54) is obtained from (4.55) by matrix transposition. The commuting relation (4.57) is obtained by successive applications of (4.55) and (4.54). Finally, the commuting relation (4.56) is obtained by successive applications of (4.54) and (4.55). 

4.4. Parametric Realization of SO(m,n) The block orthogonal matrix in (4.53) can be uniquely resolved as a commuting product of two orthogonal block matrices,

   Om 0m,n Om 0m,n Im 0m,n = . (4.58) 0n,m On 0n,m In 0n,m On The first and the second orthogonal matrices on the right-hand side of (4.58) represent, respectively, (i) a right rotation Om ∈ S O(m) of Rn×m , Om : P → POm ; and (ii) a left rotation On ∈ S O(n) of Rn×m , On : P → On P. Hence, the orthogonal matrix on the left-hand side of (4.58), which represents the composition of the rotations Om and On , is said to be a bi-rotation of the pseudo-Euclidean space Rm,n about its origin. By means of (4.58), (4.53) can be written as ⎞ ⎛ ⎞⎛ ⎞⎛ 1 t ⎟⎟⎟ ⎜⎜⎜ Om 0m,n ⎟⎟⎟ ⎜⎜⎜ Im 0m,n ⎟⎟⎟ ⎜⎜⎜ Im + c−2 Pt P 2P c ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟⎟ . (4.59) Λ = ⎜⎜⎜⎝  ⎜ ⎠⎝ ⎠ ⎠ −2 t 0n,m In 0n,m On ⎝ P In + c PP Lemma 4.6. The commuting relations   Im + c−2 Pt POm = Om Im + c−2 (POm )t (POm )   On In + c−2 PPt = In + c−2 (On P)(On P)t On hold for all P ∈ Rn×m , Om ∈ S O(m), and On ∈ S O(n).

(4.60)

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Proof. Im + c−2 Pt P = Om Im Otm + c−2 Om Otm Pt POm Otm = Om (Im + c−2 Otm Pt POm )Otm   = Om Im + c−2 Otm Pt POm Im + c−2 Otm Pt POm Otm   = Om Im + c−2 Otm Pt POm Otm Om Im + c−2 Otm Pt POm Otm  = (Om Im + c−2 Otm Pt POm Otm )2 . Hence,



 Im + c−2 Pt P = Om Im + c−2 Otm Pt POm Otm ,

(4.61)

(4.62)

thus proving the first matrix identity in (4.60). Similarly, In + c−2 PPt = Otn In On + c−2 Otn On PPt Otn On = Otn (In + c−2 On PPt Otn )On   = Otn In + c−2 On PPt Otn In + c−2 On PPt Otn On   = Otn In + c−2 On PPt Otn On Otn In + c−2 On PPt Otn On  = (Otn In + c−2 On PPt Otn On )2 . Hence,



 In + c−2 PPt = Otn In + c−2 On PPt Otn On ,

thus proving the second matrix identity in (4.60). Lemma 4.7. The commuting relations   −1 −1 Im + c−2 Pt P Om = Om Im + c−2 (POm )t (POm )   −1 −1 On In + c−2 PPt = In + c−2 (On P)(On P)t On

(4.63)

(4.64) 

(4.65)

hold for all P ∈ Rn×m , Om ∈ S O(m), and On ∈ S O(n). Proof. Inverting the first matrix identity in (4.60), we have the matrix identity   −1 −1 Im + c−2 Pt P = Im + c−2 (POm )t (POm ) O−1 (4.66) O−1 m m , which implies the first equation in the Lemma. Similarly, inverting the second matrix identity in (4.60), we obtain a matrix identity that implies the second equation in the Lemma. 

Bi-gyrogroups and Bi-gyrovector Spaces – P

Lemma 4.8. The commuting relation ⎞⎛ ⎛ ⎜⎜⎜ Om 0m,n ⎟⎟⎟ ⎜⎜⎜ Im + c−2 (POm )t (POm ) ⎟⎟⎟ ⎜⎜⎜ ⎜⎜⎜ ⎠⎜ ⎝ 0n,m In ⎝ POm ⎛ ⎜⎜⎜ Im + c−2 Pt P = ⎜⎜⎜⎜⎝ P



⎞ ⎟⎟⎟ ⎟⎟⎟ ⎟ t⎠

1 (POm )t c2

In + c−2 (POm )(POm )

⎞⎛ ⎞ ⎟⎟⎟ ⎜⎜ Om 0m,n ⎟⎟ ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎟⎠  ⎟⎠ ⎜⎝ In + c−2 PPt 0n,m In

(4.67)

1 t P c2

holds for all P ∈ Rn×m and Om ∈ S O(m). Proof. Let J1 and J2 denote the left-hand side and the right-hand side of (4.67), respectively, so that we have to prove that J1 = J2 . Clearly, ⎞ ⎛ 1 t ⎟⎟⎟ ⎜⎜⎜ Im + c−2 Pt P Om 2P c ⎟⎟⎟ (4.68) J2 = ⎜⎜⎜⎜⎝  ⎟⎠ −2 t POm In + c PP and, by means of the first commuting relation in Lemma 4.6, ⎞ ⎛  1 t ⎟⎟⎟ ⎜⎜⎜Om Im + c−2 (POm )t (POm ) O (PO ) m m 2 c ⎟⎟⎟ J1 = ⎜⎜⎜⎜⎝  ⎟⎠ POm In + c−2 (POm )(POm )t ⎛ ⎜⎜⎜ Im + c−2 Pt P Om = ⎜⎜⎜⎜⎝ POm



1 t P c2

In + c−2 PP

⎞ ⎟⎟⎟ ⎟⎟⎟ . ⎟ t⎠

Hence, J1 = J2 , and the proof is complete.

(4.69)



Lemma 4.9. The commuting relation ⎞⎛ ⎛ ⎞ 1 t ⎟⎟⎟ ⎜⎜ Im 0m,n ⎟⎟ ⎜⎜⎜ Im + c−2 (On P)t (On P) (O P) n 2 c ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎜⎜⎜ ⎟⎠  ⎟⎠ ⎜⎝ ⎜⎝ On P In + c−2 (On P)(On P)t 0n,m On ⎛ ⎞⎛ ⎜⎜⎜ Im 0m,n ⎟⎟⎟ ⎜⎜⎜ Im + c−2 Pt P ⎟⎟⎟ ⎜⎜⎜ = ⎜⎜⎜⎝ ⎠⎜ 0n,m On ⎝ P



1 t P c2

In + c−2 PP

⎞ ⎟⎟⎟ ⎟⎟⎟ ⎟ t⎠

(4.70)

holds for all P ∈ Rn×m and On ∈ S O(n). Proof. Let J3 and J4 denote the left-hand side and the right-hand side of (4.70),

113

114

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

respectively. Clearly,

⎛ ⎞ 1 t ⎜⎜⎜ Im + c−2 Pt P ⎟⎟⎟ 2P c ⎟⎟⎟ J4 = ⎜⎜⎜⎜⎝  ⎟⎠ −2 t On P On In + c PP

(4.71)

and, by means of the second commuting relation in Lemma 4.6, ⎛ ⎞ 1 ⎜⎜⎜ Im + c−2 (On P)t (On P) ⎟⎟⎟ (On P)t On c2 ⎜ ⎟⎟⎟ J3 = ⎜⎜⎜⎝  ⎟⎠ On P In + c−2 (On P)(On P)t On (4.72)

⎛ ⎞ ⎜⎜⎜ Im + c−2 Pt P ⎟⎟⎟ Pt ⎟⎟⎟ . = ⎜⎜⎜⎜⎝  ⎟⎠ −2 t On P On In + c PP Hence, J3 = J4 , and the proof is complete.



By (4.59) and Lemma 4.9, ⎛ ⎞⎛ ⎞⎛ ⎜⎜⎜ Om 0m,n ⎟⎟⎟ ⎜⎜⎜ Im 0m,n ⎟⎟⎟ ⎜⎜⎜ Im + c−2 Pt P ⎟⎟⎟⎠ ⎜⎜⎜⎝ ⎟⎟⎟⎠ ⎜⎜⎜⎜ Λ = ⎜⎜⎜⎝ 0n,m In 0n,m On ⎝ P ⎛ ⎞⎛ ⎜⎜⎜ Om 0m,n ⎟⎟⎟ ⎜⎜⎜ Im + c−2 (On P)t (On P) ⎟⎟⎟ ⎜⎜⎜ = ⎜⎜⎜⎝ ⎠⎜ 0n,m In ⎝ On P



1 t P c2

In + c−2 PP

⎞ ⎟⎟⎟ ⎟⎟⎟ ⎟ t⎠

⎞⎛ ⎞ ⎟⎟⎟ ⎜⎜ Im 0m,n ⎟⎟ ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎟⎠ ,  ⎟⎠ ⎜⎝ −2 t In + c (On P)(On P) 0n,m On (4.73) 1 (On P)t c2

where P, Om , and On are generic elements of Rn×m , S O(m), and S O(n), respectively, forming the three matrix parameters that determine Λ ∈ S O(m, n). The matrix parameter P of Λ in (4.59) is a generic element of Rn×m , and the orthogonal matrix On ∈ S O(n) maps Rn×m onto itself bijectively, On : P → On P. Hence, the generic element On P ∈ Rn×m in (4.73) can, equivalently, be replaced by the generic element P ∈ Rn×m , thus obtaining from (4.73) the parametric representation of the generic Lorentz transformation Λ, ⎞⎛ ⎛ ⎞⎛ ⎞ 1 t ⎟⎟⎟ ⎜⎜ Im 0m,n ⎟⎟ ⎜⎜⎜ Om 0m,n ⎟⎟⎟ ⎜⎜⎜ Im + c−2 Pt P P 2 c ⎜ ⎟⎟⎟ ⎜ ⎟ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎠⎟ ⎜⎜⎜ (4.74) Λ = ⎜⎜⎝⎜  ⎝ ⎠⎟ , ⎝ ⎠ −2 t 0n,m In 0 O n,m n P In + c PP called the bi-gyration decomposition of Λ. As in (4.73), P, Om , and On in (4.74) are generic elements of Rn×m , S O(m), and S O(n), respectively. Hence, if Λ ∈ SO(m.n) then Λ possesses the bi-gyration decomposition (4.74). This important result will be formalized in Theorem 4.12, p. 118.

Bi-gyrogroups and Bi-gyrovector Spaces – P

The generic Lorentz transformation matrix Λ of order (m + n) × (m + n) is expressed in (4.74) as the product of the following three matrices in (4.75) – (4.77): The bi-boost: The (m + n) × (m + n) matrix B(P), ⎛ ⎞ 1 t ⎜⎜⎜ Im + c−2 Pt P ⎟⎟⎟ P c2 ⎜ ⎟⎟⎟ ∈ SO(m, n) ⊂ R(m+n)×(m+n) , ⎜ B(P) := ⎜⎜⎝ (4.75)  ⎟⎠ P In + c−2 PPt is parametrized by P ∈ Rn×m , m, n ∈ N (as in [13, p. 129]). In order to emphasize that B(P) is associated in (4.74) with a bi-rotation (Om , On ), we call it a bi-boost. If m = 1 and n = 3, the bi-boost descends to the common boost of a Lorentz transformation in special relativity theory, studied for instance in [74, 81, 93, 96]. The right rotation: The (m + n) × (m + n) block orthogonal matrix 

Om 0m,n ρ(Om ) := ∈ R(m+n)×(m+n) (4.76) 0n,m In is parametrized by Om ∈ S O(m). For m > 1, Om is an m × m orthogonal matrix, destined to be right-applied to the n × m matrices P, P → POm . Hence, ρ(Om ) is called a right rotation of the bi-boost B(P). The left rotation: The (m + n) × (m + n) block orthogonal matrix

 Im 0m,n λ(On ) := (4.77) ∈ R(m+n)×(m+n) 0n,m On is parametrized by On ∈ S O(n). For n > 1, On is an n × n orthogonal matrix, destined to be left-applied to the n × m matrices P, P → On P. Hence, λ(On ) is called a left rotation of the bi-boost B(P). A left and a right rotation are called collectively a bi-rotation. Suggestively, the term bi-boost emphasizes that the generic bi-boost is associated with a generic birotation (On , Om ) ∈ S O(n) × S O(m). With the notation in (4.75) – (4.77), the results of Lemma 4.9 and Lemma 4.8 can be written as commuting relations between bi-boosts and left and right rotations, as the following lemma asserts. Lemma 4.10. The commuting relations λ(On )B(P) = B(On P)λ(On ) B(P)ρ(Om ) = ρ(Om )B(POm ) λ(On )B(P)ρ(Om ) = ρ(Om )B(On POm )λ(On ) hold for any P ∈ Rn×m , Om ∈ S O(m), and On ∈ S O(n).

(4.78)

115

116

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Proof. The first matrix identity in (4.78) is the result of Lemma 4.9, expressed in the notation in (4.75) – (4.77). Similarly, the second matrix identity in (4.78) is the result of Lemma 4.8, expressed in the notation in (4.75) – (4.77). The third matrix identity in (4.78) follows from the first and second matrix identities in (4.78), noting that λ(On ) and ρ(Om ) commute. 

4.5. Bi-boosts We show in this section straightforwardly that, as expected, B(P) ∈ SO(m, n) for any P ∈ Rn×m . Let m, n ∈ N be two positive integers, and let ⎛ ⎞ ⎛ ⎞ ⎜⎜⎜ t1 ⎟⎟⎟ ⎜⎜⎜ x1 ⎟⎟⎟ ⎜⎜⎜ . ⎟⎟⎟ ⎜⎜ ⎟⎟ m t = ⎜⎜⎜ .. ⎟⎟⎟ ∈ R , x = ⎜⎜⎜⎜ ... ⎟⎟⎟⎟ ∈ Rn , (4.79) ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ tm xn so that

 t = (t1 , . . . , tm , x1 , . . . , xn )t ∈ Rm,n x

(4.80)

is a generic point of the pseudo-Euclidean space Rm,n . Following (4.4) – (4.7), the inner product of signature (m, n) in Rm,n is given by  

  t 1 t t Im t2 0m,n t1 (4.81) · 2 := 1 = t1 ·t2 − 2 x1 ·x2 x1 x2 x1 0n,m −c−2 In x2 c for all (t1 , x1 )t , (t2 , x2 )t ∈ Rm,n , where t1 ·t2 = tt1 t2 and x1 ·x2 = xt1 x2 are the standard inner product in Rm and Rn , respectively. We use the convenient notation  bm := Im + c−2 Pt P ∈ Rm×m (4.82)  n×n −2 t bn := In + c PP ∈ R , P ∈ Rn×m , so that, by (4.75),

⎛ ⎜⎜bm B(P) = ⎜⎜⎜⎝ P

⎞

1 t⎟ P⎟ c2 ⎟ ⎟

bn

⎟⎠ ∈ R(m+n)×(m+n) .

(4.83)

Let P1 , P2 ∈ Rn×m . It is clear from (4.83) that B(P1 ) = B(P2 ) if and only if P1 = P2 , so that the relationship between bi-boosts B(P) ∈ SO(m, n) and their parameter P ∈ Rn×m is bijective.

Bi-gyrogroups and Bi-gyrovector Spaces – P

Theorem 4.11. Let m, n ∈ N be two positive integers. Then, B(P) ∈ SO(m, n)

(4.84)

for any P ∈ Rn×m . Proof. Clearly, B(0n,m ) = Im+n is the identity element of SO(m, n), so that B(P) can be reached continuously from the identity transformation. Hence, by Def. 4.1, it remains to show that B(P) leaves the Pseudo-Euclidean inner product (4.81) invariant, that is,



   t1 t t t B(P) ·B(P) 2 = 1 · 2 . (4.85) x1 x2 x1 x2 We have

⎞ ⎞  ⎛

 ⎛ ⎜⎜⎜bm t + c−2 Pt x⎟⎟⎟ ⎜⎜⎜bm c−2 Pt ⎟⎟⎟ t t ⎟⎟⎠ . ⎟⎟⎠ = ⎜⎜⎝ B(P) = ⎜⎜⎝ x x P bn Pt + bn x

(4.86)

Hence, by (4.86), (4.81), and the commuting relations (4.54) and (4.55) we obtain the following chain of equations that verifies (4.85):



 t1 t Bc (P) ·B (P) 2 x1 c x2 ⎞t ⎛ ⎞⎛ ⎞ ⎛ ⎜⎜⎜bm t1 + c−2 Pt x1 ⎟⎟⎟ ⎜⎜⎜ Im 0m,n ⎟⎟⎟ ⎜⎜⎜bm t2 + c−2 Pt x2 ⎟⎟⎟ ⎜ ⎟ ⎜ ⎟ ⎟⎟⎠ ⎜ = ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ Pt1 + bn x1 0n,m −c−2 In Pt2 + bn x2 ⎛ ⎞ ⎜⎜⎜bm t2 + c−2 Pt x2 ⎟⎟⎟ t −2 t −2 t t t = (t1 bm + c x1 P, − c (t1 P + x1 bn )) ⎜⎜⎝ ⎟⎟⎠ Pt2 + bn x2 = tt1 b2m t2 + c−2 tt1 bm Pt x2 + c−2 xt1 Pbm t2 + c−4 xt1 PPt x2 − c−2 (tt1 Pt Pt2 + tt1 Pt bn x2 + xt1 bn Pt2 + xt1 b2n x2 )

(4.87)

= tt1 (Im + c−2 Pt P)t2 + c−2 tt1 bm Pt x2 + c−2 xt1 Pbm t2 + c−4 xt1 PPt x2 − c−2 (tt1 Pt Pt2 + tt1 Pt bn x2 + xt1 bn Pt2 + xt1 (In + c−2 PPt )x2 ) = tt1 (Im + c−2 Pt P)t2 + c−2 tt1 bm Pt x2 + c−2 xt1 Pbm t2 − c−2 (tt1 Pt Pt2 + tt1 bm Pt x2 + xt1 Pbm t2 + xt1 x2 ) = tt1 t2 − c−2 xt1 x2

  t t = 1 · 2 . x1 x2 

117

118

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

4.6. Lorentz Transformation Decomposition Theorem 4.12. (Lorentz Transformation Bi-gyration Decomposition). Let m, n ∈ N be two positive integers and let Λ ∈ R(m+n)×(m+n) . Then, Λ ∈ S O(m, n) if and only if Λ possesses the bi-gyration decomposition ⎞⎛ ⎛ ⎞⎛ ⎞ 1 t ⎟⎟⎟ ⎜⎜ Im 0m,n ⎟⎟ ⎜⎜⎜ Om 0m,n ⎟⎟⎟ ⎜⎜⎜ Im + c−2 Pt P 2P c ⎜ ⎟⎟⎟ ⎜ ⎟ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟⎠ ⎜⎜⎜ (4.88) Λ = ⎜⎜⎜⎝ ⎟⎠ ,  ⎝ ⎝ ⎠ 0n,m In P In + c−2 PPt 0n,m On where Om ∈ SO(m), On ∈ SO(n) and P ∈ Rn×m . Proof. If Λ ∈ SO(m, n) then Λ possesses the bi-gyration decomposition (4.88), as shown in the derivation of (4.74). Conversely, if Λ possesses the matrix decomposition (4.88), then Λ ∈ SO(m, n), as we show below. The matrix decomposition (4.88) is given by Λ = ρ(Om )B(P)λ(On ) where ρ(Om ) and λ(On ) are given by (4.76) – (4.77), and B(P) is given by (4.75). Clearly, both ρ(Om ) and λ(On ) leave the inner product (4.81) invariant and can be reached continuously from the identity transformation. Hence, by Def. 4.1, ρ(Om ), λ(On ) ∈ SO(m, n). Also B(P) ∈ SO(m, n) by Theorem 4.11. Hence, the product of these matrices is an element of the group SO(m, n), that is, Λ = ρ(Om )B(P)λ(On ) ∈ SO(m, n).  It is convenient to write the bi-gyration decomposition (4.88) parametrically, in a column notation, as ⎛ ⎞ ⎜⎜⎜ P ⎟⎟⎟ ⎜ ⎟ (4.89) Λ = Λ(Om , P, On ) = ρ(Om )B(P)λ(On ) = ⎜⎜⎜⎜ On ⎟⎟⎟⎟ , ⎝ ⎠ Om so that a product of two Lorentz matrices is written as a product between two column triples. Thus, for instance, the product (or, composition) of the two Lorentz transformations Λ1 = Λ(On,1 , P1 , Om,1 ) and Λ2 = Λ(On,2 , P2 , Om,2 ) is written as ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎜⎜⎜ P1 ⎟⎟⎟ ⎜⎜⎜ P2 ⎟⎟⎟ ⎜⎜⎜ P12 ⎟⎟⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎜ Λ1 Λ2 = ⎜⎜⎜⎜ On,1 ⎟⎟⎟⎟ ⎜⎜⎜⎜ On,2 ⎟⎟⎟⎟ = ⎜⎜⎜⎜ On,12 ⎟⎟⎟⎟ =: Λ12 , (4.90) ⎠⎝ ⎠ ⎝ ⎠ ⎝ Om,1 Om,2 Om,12 where the composite parameters P12 =: P1 ⊕P2 , On,12 , and Om,12 of the composite Lorentz transformation Λ12 are to be determined in Sect. 4.8 in terms of the parameters of Λ1 and Λ2 . The Lorentz transformation product law, written in column notation, will be presented in Theorem 4.31, p. 139, following the study of associated special left and right

Bi-gyrogroups and Bi-gyrovector Spaces – P

automorphisms, called left and right gyrations or, collectively, bi-gyrations. Theorem 4.13. (Lorentz Transformation Polar Decomposition). Let m, n ∈ N be two positive integers and let Λ ∈ R(m+n)×(m+n) . Then, Λ ∈ S O(m, n) if and only if Λ possesses the polar decomposition ⎛ ⎞⎛ ⎞⎛ ⎞ 1 t ⎜⎜⎜ Im + c−2 Pt P ⎟⎟⎟ ⎜⎜ Om 0m,n ⎟⎟ ⎜⎜ Im 0m,n ⎟⎟ P 2 c ⎜ ⎜ ⎟ ⎟⎟⎟ ⎜ ⎟ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ (4.91) Λ = ⎜⎜⎜⎝  ⎝ ⎝ ⎠ ⎠⎟ , ⎠ −2 t 0 I 0 O n,m n n,m n P In + c PP where Om ∈ SO(m), On ∈ SO(n) and P ∈ Rn×m . Proof. We establish the proof by demonstrating that (4.91) is equivalent to (4.88), so that Theorem 4.13 follows from Theorem 4.12. Replacing the generic element P ∈ Rn×m by On P ∈ Rn×m , where On ∈ SO(n), and noting Lemma 4.9, the matrix decomposition (4.91) can be written as ⎞⎛ ⎛ ⎞⎛ ⎞ 1 t ⎟⎟⎟ ⎜⎜ Im 0m,n ⎟⎟ ⎜⎜ Om 0m,n ⎟⎟ ⎜⎜⎜ Im + c−2 (On P)t (On P) 2 (On P) c ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎜⎜⎜  ⎠⎟ ⎝⎜ ⎠⎟ ⎠⎟ ⎝⎜0 ⎝⎜ −2 t On 0n,m In On P In + c (On P)(On P) n,m ⎛ ⎞⎛ ⎜⎜⎜ Im 0m,n ⎟⎟⎟ ⎜⎜⎜ Im + c−2 Pt P ⎟⎟⎟ ⎜⎜⎜ = ⎜⎜⎜⎝ ⎠⎜ 0n,m On ⎝ P

⎞⎛ ⎞ ⎟⎟⎟ ⎜⎜ Om 0m,n ⎟⎟ ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎟⎠ .  ⎟⎠ ⎜⎝ In + c−2 PPt 0n,m In

(4.92)

1 t P c2

Hence, the representations (4.91) and (4.88) of Λ are equivalent, as desired.



4.7. Inverse Lorentz Transformation Theorem 4.14. (The Inverse Bi-boost). The inverse of the bi-boost B(P), P ∈ Rn×m , is B(−P), B(P)−1 = B(−P) .

(4.93)

Proof. By Lemma 4.5 along with the notation in (4.82) we have the commuting relations Pt bn = bm Pt bn P = Pbm .

(4.94)

119

120

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Hence, by (4.83), (4.82), and (4.94), ⎞⎛ ⎞ ⎛ ⎜⎜⎜bm c−2 Pt ⎟⎟⎟ ⎜⎜⎜ bm −c−2 Pt ⎟⎟⎟ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ B(P)B(−P) = ⎜⎜⎝ P bn −P bn ⎛ 2 ⎞ ⎜⎜⎜bm − c−2 Pt P −c−2 (−bm Pt + Pt bn )⎟⎟⎟ ⎟⎟⎠ = ⎜⎜⎝ −c−2 PPt + b2n Pbm − bn P

(4.95)

⎛ ⎞ ⎜⎜⎜ Im 0m,n ⎟⎟⎟ ⎟⎟⎠ = Im+n , = ⎜⎜⎝ 0n,m In 

as desired.

A Lorentz transformation matrix Λ of order (m + n) × (m + n), m, n ≥ 2, involves the bi-rotation (λ(On ), ρ(Om )), as shown in (4.89). Bi-boosts are Lorentz transformations without bi-rotations, that is, by (4.89), bi-boosts B(P) are given by ⎛ ⎞ ⎜⎜⎜ P ⎟⎟⎟ ⎜ ⎟ (4.96) B(P) = ρ(Im )B(P)λ(In ) = Λ(Im , P, In ) = ⎜⎜⎜⎜ In ⎟⎟⎟⎟ , ⎝ ⎠ Im for any P ∈ Rn×m , where Om = Im and On = In are trivial rotations. Rewriting (4.93) in the column notation (4.89), we have ⎛ ⎞−1 ⎛ ⎞ ⎜⎜⎜ P ⎟⎟⎟ ⎜⎜⎜−P⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜ ⎟ ⎜⎜⎝ In ⎟⎟⎠ = ⎜⎜⎜⎜⎝ In ⎟⎟⎟⎟⎠ , Im Im

(4.97)

so that, accordingly,

⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎜⎜⎜ P ⎟⎟⎟ ⎜⎜⎜−P⎟⎟⎟ ⎜⎜⎜0n,m ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟ B(P)B(−P) = ⎜⎜ In ⎟⎟ ⎜⎜ In ⎟⎟ = ⎜⎜ In ⎟⎟⎟⎟ = B(0n,m ) = Im+n , ⎝ ⎠⎝ ⎠ ⎝ ⎠ Im Im Im

(4.98)

(0n,m , In , Im )t being the identity Lorentz transformation of signature (m, n). Theorem 4.15. (The Inverse Lorentz Transformation). The inverse of a Lorentz transformation Λ = (P, On , Om )t is given by the equation ⎛ ⎞−1 ⎛ t ⎞ ⎜⎜⎜−On POtm ⎟⎟⎟ ⎜⎜⎜ P ⎟⎟⎟ ⎜⎜⎜⎜ On ⎟⎟⎟⎟ = ⎜⎜⎜⎜ Ot ⎟⎟⎟⎟ . (4.99) n ⎜⎝ ⎜⎝ ⎟⎠ ⎟⎠ Om Otm

Bi-gyrogroups and Bi-gyrovector Spaces – P

Proof. The proof is given by the following chain of equations, which are numbered for subsequent explanation: (1)

−1

Λ(Om , P, On )

 === {ρ(Om )B(P)λ(On )}−1 (2)

 === λ(Otn )B(−P)ρ(Otm ) (3)

 === B(−Otn P)λ(Otn )ρ(Otm )

(4.100)

(4)

 === B(−Otn P)ρ(Otm )λ(Otn ) (5)

 === ρ(Otm )B(−Otn POtm )λ(Otn ) . Derivation of the numbered equalities in (4.100): (1) (2) (3) (4) (5)

Follows by (4.89). is obvious, noting (4.93). Follows from (2) by the first matrix identity in (4.78). Follows from (3) by commuting λ(Otn ) and ρ(Otm ), noting (4.76) – (4.77). Follows from (4) by the second matrix identity in (4.78). Rewriting the extreme sides of (4.100) in the column notation (4.89) yields (4.99). 

4.8. Bi-boost Parameter Composition Definition 4.16. (Bi-gyroaddition, Bi-gyrogroupoid). Let Λ = B(P1 )B(P2 ) be a Lorentz transformation given by the product of two bi-boosts parametrized by P1 , P2 ∈ Rn×m . Then, the main parameter, P12 , of Λ is said to be the composition of P1 and P2 , P12 = P1 ⊕P2 ,

(4.101)

giving rise to a binary operation, ⊕, called bi-gyroaddition, in the space Rn×m of all n × m real matrices. Being a groupoid of the parameter P ∈ Rn×m , the resulting groupoid (Rn×m , ⊕) is called the parameter bi-gyrogroupoid. Furthermore, the bi-rotation (Om , On ) ∈ SO(m) × SO(n) associated with the biboost composition Λ = B(P1 )B(P2 ) is called the bi-gyration generated by P1 and P2 , denoted by (Om , On ) = (rgyr[P1 , P2 ], lgyr[P1 , P2 ]) ∈ SO(m) × SO(n). The two components of the bi-gyration generated by P1 and P2 are the right and the left gyration generated by P1 and P2 . Following Def. 4.16 we will study the bi-boost composition law in order to reveal

121

122

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

the elegant, rich algebraic structure of the parameter bi-gyrogroupoid (Rn×m , ⊕). In general, the product of two bi-boosts is not a bi-boost. However, the product of two bi-boosts is an element of the Lorentz group S O(m, n) and, hence, by Theorem 4.12, can be parametrized, as shown in (4.89). Following (4.75), let ⎛ ⎞ ⎛ 1 t ⎜⎜⎜ Im + c−2 Pt Pk ⎟⎟⎟ ⎜ BR 1 t⎞ P 2 k ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ m,Pk c2 Pk ⎟⎟⎟⎟ c k ⎟⎟⎟ = ⎜⎜⎝ ⎟⎟ ∈ S O(m, n) , (4.102) B(Pk ) = ⎜⎜⎜  L ⎠ ⎟ ⎝⎜ Pk Bn,P Pk In + c−2 Pk Ptk ⎠ k Pk ∈ Rn×m , k = 1, 2, be two bi-boosts, where we use the notation  BRm,P = Im + c−2 Pt P L Bn,P

=



(4.103) In +

c−2 PPt

Then, on the one hand we have ⎞⎛ ⎛ R ⎜⎜⎜ Bm,P1 c12 Pt1 ⎟⎟⎟ ⎜⎜⎜ BRm,P2 ⎟⎟⎟ ⎜⎜⎜ Λ := B(P1 )B(P2 ) = ⎜⎜⎜⎝ L ⎠⎝ P1 Bn,P1 P2

1 t⎞ P ⎟⎟ c2 2 ⎟ ⎟ L Bn,P 2

⎛1 t ⎜⎜⎜ c2 P1 P2 + BRm,P1 BRm,P2 = ⎜⎜⎜⎝ L P2 P1 BRm,P2 + Bn,P 1 ⎛ ⎜⎜⎜E11 =: ⎜⎜⎜⎝ E21

.

⎟⎟⎠

⎞ L + Pt1 Bn,P )⎟⎟ 2 ⎟ ⎟⎟⎟ ⎠ 1 t L L P P + Bn,P1 Bn,P2 c2 1 2

1 (BRm,P1 Pt2 c2

(4.104)

⎞ 1 E ⎟⎟ c2 12 ⎟ ⎟ E22

⎟⎟⎠ .

On the other hand, Λ = B(P1 )B(P2 ) is an element of SO(m, n) and, as such, it possesses the decomposition (4.89), Λ = B(P1 )B(P2 ) = ρ(Om )B(P12 )λ(On ) .

(4.105)

The parameters Om ∈ SO(m), On ∈ SO(n), and P12 ∈ Rn×m in (4.105) are determined by P1 and P2 , as we will see later. Accordingly, following Def. 4.16 we introduce into (4.105) the notation P12 =: P1 ⊕P2 ∈ Rn×m Om =: rgyr[P1 , P2 ] ∈ SO(m) On =: lgyr[P1 , P2 ] ∈ SO(n) , where ⊕ is a binary operation in Rn×m .

(4.106)

Bi-gyrogroups and Bi-gyrovector Spaces – P

Clearly, rgyr[P1 , P2 ]t = rgyr−1 [P1 , P2 ] (4.107)

lgyr[P1 , P2 ]t = lgyr−1 [P1 , P2 ] t −1 for any P1 , P2 ∈ Rn×m , since Otm = O−1 m and On = On .

Remark 4.17. We will, thus, see in this section that the Bi-gyration Decomposition Theorem 4.12, p. 118, gives rise to a binary operation, ⊕, and to a bi-gyrator, gyr = (lgyr, rgyr), in the space Rn×m of the parameter P. (Compare with Remark 5.38, p. 220). We will see from (4.109) – (4.111) that the composition P1 ⊕P2 of P1 and P2 in R is associated with a right gyration, rgyr[P1 , P2 ] ∈ SO(m), and a left gyration, lgyr[P1 , P2 ] ∈ SO(n). These gyrations are generated by P1 and P2 as a by-product of the composition P1 ⊕P2 of P1 and P2 , and will turn out to be important automorphisms of the groupoid (Rn×m , ⊕). By means of (4.105) and (4.106) we have the following unique decomposition of a bi-boost product: n×m

Λ = B(P1 )B(P2 ) = ρ(rgyr[P1 , P2 ])B(P1 ⊕P2 )λ(lgyr[P1 , P2 ]) ⎛ ⎞⎛ ⎜⎜⎜rgyr[P1 , P2 ] 0m,n ⎟⎟⎟ ⎜⎜⎜ BRm,P1 ⊕P2 ⎜ ⎟⎟⎟ ⎜⎜⎜ = ⎜⎜⎝ ⎠⎝ P1 ⊕P2 0n,m In ⎛ ⎜⎜⎜rgyr[P1 , P2 ]BRm,P1 ⊕P2 =: ⎜⎜⎜⎝ P1 ⊕P2

E = 11 E21

1 E c2 12 E22

⎞⎛ 1 (P1 ⊕P2 )t ⎟⎟⎟ ⎜⎜⎜ Im c2 ⎟⎜ L Bn,P 1 ⊕P2

⎟⎟⎠ ⎜⎜⎝

⎞ ⎟⎟⎟ ⎟⎟⎟ ⎠

0m,n

0n,m lgyr[P1 , P2 ]

⎞ 1 rgyr[P1 , P2 ](P1 ⊕P2 )t lgyr[P1 , P2 ]⎟⎟⎟ c2 ⎟ L Bn,P lgyr[P1 , P2 ] 1 ⊕P2

(4.108)

⎟⎟⎠

 ∈ S O(m, n) .

Comparing the blocks E21 in (4.104) and (4.108) we have, by (4.103),   P1 ⊕P2 = P1 Im + c−2 Pt2 P2 + In + c−2 P1 Pt1 P2 .

(4.109)

Comparing the blocks E11 in (4.104) and (4.108) we have   −1 . rgyr[P1 , P2 ] = c−2 Pt1 P2 + BRm,P1 BRm,P2 BRm,P1 ⊕P2

(4.110)

123

124

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Comparing the blocks E22 in (4.104) and (4.108) we have  −1   L −2 t L L lgyr[P1 , P2 ] = Bn,P P P + B B c . 1 ⊕P n,P n,P 2 1 2 1 2

(4.111)

Comparing the blocks E12 in (4.104) and (4.108) we have, by (4.103),   t t t t −2 rgyr[P1 , P2 ](P1 ⊕P2 ) lgyr[P1 , P2 ] = Im + c P1 P1 P2 + P1 In + c−2 P2 Pt2 (4.112)

(2)

(1)

 t t  === P1 ⊕P2 === (P2 ⊕P1 )t . The equation marked by (1) in (4.112) follows immediately from (4.109), replacing P1 , P2 ∈ Rn×m by Pt1 , Pt2 ∈ Rm×n . The equation marked by (2) in (4.112) is derived from (4.109) in the following straightforward chain of equations:    t (P2 ⊕P1 )t = P2 Im + c−2 Pt1 P1 + In + c−2 P2 Pt2 P1 =



Im + c−2 Pt1 P1 Pt2 + Pt1

= (Pt1 )





In + c−2 (Pt2 )t Pt2 +

In + c−2 P2 Pt2



(4.113)

Im + c−2 (Pt1 )(Pt1 )t (Pt2 ) = Pt1 ⊕Pt2 .

Formalizing results in (4.109) – (4.113), we obtain the following theorem. Theorem 4.18. (Bi-gyroaddition and Bi-gyration). The bi-gyroaddition and the bigyration in the parameter bi-gyrogroupoid (Rn×m , ⊕) are given by the equations   P1 ⊕P2 = P1 Im + c−2 Pt2 P2 + In + c−2 P1 Pt1 P2 (P1 ⊕P2 )t = Pt2 ⊕Pt1     rgyr[P1 , P2 ] = c−2 Pt1 P2 + Im + c−2 Pt1 P1 Im + c−2 Pt2 P2 ×



Im +



c−2 (P

1 ⊕P2

)t (P

1 ⊕P2 )

−1 −1

In + c−2 (P1 ⊕P2 )(P1 ⊕P2 )t     × c−2 P1 Pt2 + In + c−2 P1 Pt1 In + c−2 P2 Pt2

lgyr[P1 , P2 ] =

for all P1 , P2 ∈ Rn×m .

(4.114)

Bi-gyrogroups and Bi-gyrovector Spaces – P

Interestingly, in the limit of large c, c → ∞, the binary operation, ⊕, in Rn×m descends to the common matrix addition, +, in Rn×m , and gyrations become trivial, that is, limc→∞ rgyr[P1 , P2 ] = Im and limc→∞ lgyr[P1 , P2 ] = In for all P1 , P2 ∈ Rn×m . Corollary 4.19 follows immediately from Theorem 4.18. Corollary 4.19. (Trivial Bi-gyrations). lgyr[0n,m , P] = lgyr[P, 0n,m ] = In lgyr[P, P] = lgyr[P, P] = In rgyr[0n,m , P] = rgyr[P, 0n,m ] = Im

(4.115)

rgyr[P, P] = lgyr[P, P] = Im for all P ∈ Rn×m . The trivial bi-gyration lgyr[P, P] = In rgyr[P, P] = Im

(4.116)

for all P ∈ Rn×m cannot be derived immediately from Theorem 4.18. It will, therefore, be derived in (4.165) and (4.163) and formalized in Theorem 4.26, p. 135. The bi-boost product in (4.105) – (4.106), written in the column notation, takes the elegant form ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎜⎜⎜P1 ⎟⎟⎟ ⎜⎜⎜P2 ⎟⎟⎟ ⎜⎜⎜ P1 ⊕P2 ⎟⎟⎟ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ (4.117) B(P1 )B(P2 ) = ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ = ⎜⎜⎜⎜⎜lgyr[P1 , P2 ]⎟⎟⎟⎟⎟ , ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ Im Im rgyr[P1 , P2 ] for all P1 , P2 ∈ Rn×m , as we see from (4.108). When P1 = P and P2 = −P, (4.117) specializes to ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎜⎜⎜ P ⎟⎟⎟ ⎜⎜⎜−P⎟⎟⎟ ⎜⎜⎜ P⊕(−P) ⎟⎟⎟ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ B(P)B(−P) = ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ = ⎜⎜⎜⎜⎜lgyr[P, −P]⎟⎟⎟⎟⎟ , ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ rgyr[P, −P] Im Im

(4.118)

for all P ∈ Rn×m . But, the left-hand side of (4.118) is also determined in (4.98), implying the identities P⊕(−P) = 0n,m lgyr[P, −P] = In rgyr[P, −P] = Im .

(4.119)

125

126

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

The first equation in (4.119) implies that −P = P

(4.120)

is the inverse, P, of P with respect to the binary operation ⊕ in Rn×m . Hence, we use the notations −P and P interchangeably. Furthermore, we naturally use the notation P1 ⊕(−P2 ) = P1 ⊕(P2 ) = P1 P2 and rewrite (4.119) as PP = 0n,m lgyr[P, P] = In rgyr[P, P] = Im , for all P ∈ Rn×m , in agreement with (4.115). Similarly, we rewrite (4.97) as ⎛ ⎞−1 ⎛ ⎞ ⎜⎜⎜ P ⎟⎟⎟ ⎜⎜⎜P⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ I = ⎜⎜⎝ n ⎟⎟⎠ ⎜⎜⎝ In ⎟⎟⎠ , Im Im

(4.121)

(4.122)

that is, B(P)−1 = B(P)

(4.123)

for all P ∈ Rn×m . The first equation in (4.114) implies that (−P1 )⊕(−P2 ) = −(P1 ⊕P2 ) .

(4.124)

Hence, following (4.120), and by (4.124), the bi-gyroaddition ⊕ obeys the gyroautomorphic inverse property (P1 ⊕P2 ) = P1 P2 ,

(4.125)

for all P1 , P2 ∈ Rn×m . It follows from the gyroautomorphic inverse property (4.125) and from (4.114) that bi-gyrations are even, that is, lgyr[−P1 , −P2 ] = lgyr[P1 , P2 ] rgyr[−P1 , −P2 ] = lgyr[P1 , P2 ]

(4.126)

lgyr[P1 , P2 ] = lgyr[P1 , P2 ] rgyr[P1 , P2 ] = lgyr[P1 , P2 ] .

(4.127)

or, equivalently,

As an immediate, interesting application of the unique bi-boost product decomposition (4.108) we present the following theorem.

Bi-gyrogroups and Bi-gyrovector Spaces – P

Theorem 4.20. Let P1 , P2 , P3 ∈ Rn×m . Then, P3 = P1 ⊕P2

(4.128)

and lgyr[P1 , P2 ] = In (4.129)

rgyr[P1 , P2 ] = Im if and only if B(P1 )B(P2 ) = B(P3 ) .

(4.130)

Proof. Following (4.108), the bi-boost product B(P1 )B(P2 ) possesses the unique decomposition B(P1 )B(P2 ) = ρ(rgyr[P1 , P2 ])B(P1 ⊕P2 )λ(lgyr[P1 , P2 ]) .

(4.131)

If P1 , P2 , and P3 satisfy (4.130) then the uniqueness of the decomposition of the bi-boost product B(P1 )B(P2 ) in (4.131) implies (4.128) and (4.129). Conversely, if P1 , P2 , and P3 satisfy (4.128) and (4.129) then (4.130) follows from (4.131). 

4.9. On the Block Entries of the Bi-boost Product In this section we explore relationships between the block entries Ei j , i, j = 1, 2, of the bi-boost product B(P1 )B(P2 ), P1 , P2 ∈ Rn×m , in (4.104) and (4.108). Theorem 4.21. Let P1 , P2 ∈ Rn×m , m, n ∈ N, and let Ei j , i, j = 1, 2, be given by (4.104) and (4.108), that is,   E11 = c−2 Pt1 P2 + Im + c−2 Pt1 P1 Im + c−2 Pt2 P2 E12 =



E21 = P1

Im + c−2 Pt1 P1 Pt2 + Pt1 

Im +

c−2 Pt2 P2

E11 = c−2 P1 Pt2 +



+





In + c−2 P2 Pt2 = Pt1 ⊕Pt2 (4.132)

In +

In + c−2 P1 Pt1

c−2 P



t 1 P1

P2 = P1 ⊕P2

In + c−2 P2 Pt2 .

Then, (P1 ⊕P2 )t = Pt2 ⊕Pt1

(4.133)

127

128

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

and t t E11 = Im + c−2 E21 E21 E11 t t E22 E22 = In + c−2 E21 E21 .

(4.134)

Proof. Identity (4.133) is obvious from the second and the third identity in (4.132). The two identities in (4.134) follow straightforwardly from (4.132) by means of the commuting relations in (4.54) – (4.55), p. 111.  By means of (4.114), (4.132), and (4.134) we have the following elegant equations that determine left and right gyrations:   −1 −1 t t rgyr[P1 , P2 ] = E11 Im + c−2 E21 E21 = E11 E11 E11 (4.135)   −1 −1 t t lgyr[P1 , P2 ] = In + c−2 E21 E21 E22 = E22 E22 E22 .

4.10. Bi-gyration Exclusion Property The following two lemmas are useful in the proof of Theorem 4.24 about the bigyration exclusion property. Lemma 4.22. Let A, B ∈ Rn×n be positive definite matrices. Then, the eigenvalues of the matrix AB are positive. Proof. Let ABx = λx for some nonzero vector x ∈ Rn and λ ∈ R, and let √ −1 y = A x. Then,

The matrix

√ √

√

√ AB A y = λy .

AB A is positive definite since √ √ yt AB A y = xt Bx > 0

for all y ∈ Rn . Hence λ > 0, so that each eigenvalue of AB is positive.

(4.136)

(4.137)

(4.138)

(4.139) 

Bi-gyrogroups and Bi-gyrovector Spaces – P

Lemma 4.23. Let A ∈ Rn×m . Then,

 Ax < Im + At A x

(4.140)

for any nonzero vector x ∈ Rm , and

 At x < In + AAt x

(4.141)

xt At Ax < xt (Im + At A)x,

(4.142)

  (Ax)t Ax < ( Im + At A x)t Im + At A x .

(4.143)

 Ax 2 < Im + At A x 2 ,

(4.144)

for any nonzero vector x ∈ Rn . Proof. Clearly,

so that, equivalently,

Hence,

thus proving (4.140). Inequality (4.141) follows from Inequality (4.140) by replacing A ∈ Rn×m by At ∈ m×n R and, accordingly, replacing x ∈ Rm by x ∈ Rn . 

Theorem 4.24. (Bi-gyration Exclusion Property). rgyr[P1 , P2 ]  −Im lgyr[P1 , P2 ]  −In

(4.145)

for any P1 , P2 ∈ Rn×m . Proof. With no loss of generality we assume c = 1. Seeking a contradiction, we assume that there exist P1 , P2 ∈ Rn×m such that rgyr[P1 , P2 ] = −In . Then, by the third equation in (4.114),    Pt1 P2 + Im + Pt1 P1 Im + Pt2 P2 = − Im + (P1 ⊕P2 )t (P1 ⊕P2 ) .

(4.146)

The right-hand side of (4.146) is a negative definite matrix. Hence, in order to obtain a contradiction, it is sufficient to show that the left-hand side of (4.146) is not negative definite.

129

130

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Applying Lemma 4.22 to the positive definite matrices  −1 A = Im + Pt1 P1 B=



(4.147) Im + Pt2 P2 ,

let x ∈ Rm be an eigenvector of the matrix AB with a positive eigenvalue λ, so that  −1  Im + Pt1 P1 Im + Pt2 P2 x = λx . (4.148) Then,



Im + Pt2 P2 x = λ



Im + Pt1 P1 x .

(4.149)

Hence, by (4.149),    t t t x Im + P1 P1 Im + P2 P2 x = λ Im + Pt1 P1 x 2 = =

 

Im + Pt1 P1 x λ Im + Pt1 P1 x





Im + Pt1 P1 x

(4.150)

Im + Pt2 P2 x .

By means of Cauchy inequality and Lemma 4.23 we have   |xt Pt1 P2 x| ≤ P1 x P2 x < Im + Pt1 P1 x Im + Pt2 P2 x .

(4.151)

By means of (4.150) and (4.151) we have   xt (Pt1 P2 + Im + Pt1 P1 Im + Pt2 P2 ) x > 0 .

(4.152)

Hence, the matrix on the left-hand side of (4.146) is not negative definite, thus obtaining the desired contradiction. Hence, the proof of the first exclusion property in (4.145) is complete. The proof of the second exclusion property in (4.145) is similar. 

4.11. Automorphisms of the Parameter Bi-gyrogroupoid Left and right rotations turn out to be left and right automorphisms of the parameter bigyrogroupoid (Rn×m , ⊕). We recall that a groupoid, (S , +), is a nonempty set, S , with a binary operation, +. A left automorphism of a groupoid (S , +) is a bijection f of S , f : S → S , s → f s, that respects the binary operation, that is, f (s1 + s2 ) = f s1 + f s2 . Similarly, a right automorphism of a groupoid (S , +) is a bijection f of S , f : S → S , s → s f , that respects the binary operation, that is, (s1 + s2 ) f = s1 f + s2 f . The need

Bi-gyrogroups and Bi-gyrovector Spaces – P

to distinguish between left and right automorphisms of the bi-gyrogroupoid (Rn×m , ⊕) is obvious in Theorem 4.25. Any rotation Theorem 4.25. (Left and Right Automorphisms of (Rn×m , ⊕)). On ∈ S O(n) is a left automorphism of the parameter bi-gyrogroupoid (Rn×m , ⊕), and any rotation Om ∈ S O(m) is a right automorphism of the parameter bi-gyrogroupoid (Rn×m , ⊕), that is, On (P1 ⊕P2 ) = On P1 ⊕On P2 (P1 ⊕P2 )Om = P1 Om ⊕P2 Om On (P1 ⊕P2 )Om = On P1 Om ⊕On P2 Om

(4.153)

for all P1 , P2 ∈ Rn×m , On ∈ S O(n) and Om ∈ S O(m). Proof. By the first equation in (4.114) and the second equation in (4.60), p. 111,   On (P1 ⊕P2 ) = On (P1 Im + c−2 Pt2 P2 + In + c−2 P1 Pt1 P2 )   = On P1 Im + c−2 (On P2 )t (On P2 ) + On In + c−2 P1 Pt1 P2 (4.154)   = On P1 Im + c−2 (On P2 )t (On P2 ) + In + c−2 (On P1 )(On P1 )t On P2 = On P1 ⊕On P2 , thus proving the first identity in (4.153). Similarly, by the first equation in (4.114) and the first equation in (4.60),   t −2 (P1 ⊕P2 )Om = (P1 Im + c P2 P2 + In + c−2 P1 Pt1 P2 )Om   = P1 Im + c−2 Pt2 P2 Om + In + c−2 (P1 Om )(P1 Om )t P2 Om (4.155)   = P1 Om Im + c−2 (P2 Om )t (P2 Om ) + In + c−2 (P1 Om )(P1 Om )t P2 Om = P1 Om ⊕P2 Om , thus proving the second identity in (4.153). The third identity in (4.153) follows immediately from the first two identities in (4.153).  By (4.106) and Theorem 4.25, left gyrations, lgyr[P1 , P2 ] ∈ SO(n), and right gyrations, rgyr[P1 , P2 ] ∈ SO(m), P1 , P2 ∈ Rn×m , are left and right automorphisms of the gyrogroupoid (Rn×m , ⊕), respectively. Hence, left and right gyrations are also called left and right gyroautomorphisms of (Rn×m , ⊕) or, collectively, bi-gyroautomorphisms of the parameter bi-gyrogroupoid (Rn×m , ⊕).

131

132

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Since −P = P, we clearly have the identities On (P) = On P (P)Om = POm

(4.156)

On (P)Om = On POm for all P ∈ Rn×m , On ∈ S O(n) and Om ∈ S O(m).

4.12. Squared Bi-boosts We are now in the position to determine the parameters of the squared bi-boost, enabling us to prove Theorem 4.26, p. 135. If we use the convenient notation (4.82),  bm := Im + c−2 Pt P ∈ Rm×m (4.157)  bn := In + c−2 PPt ∈ Rn×n , P ∈ Rn×m , then, by (4.75),

⎞ ⎛ ⎜⎜⎜bm c−2 Pt ⎟⎟⎟ ⎟⎟⎠ ∈ R(m+n)×(m+n) . B(P) = ⎜⎜⎝ P bn

(4.158)

The squared bi-boost B(P) leads to the following chain of equations, which are numbered for subsequent explanation: ⎞⎛ ⎞ ⎛ (1) ⎜⎜bm c−2 Pt ⎟⎟⎟ ⎜⎜⎜bm c−2 Pt ⎟⎟⎟ 2  ⎜ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ B(P) === ⎜⎜⎝ P bn P bn ⎞ ⎛ 2 (2) −2 t −2 t t  ⎜⎜⎜⎜bm + c P P c (bm P + P bn )⎟⎟⎟⎟ === ⎜⎝ ⎟⎠ b2n + c−2 PPt Pbm + bn P (4.159) ⎞ ⎛ (3) −2 t c−2 2Pt bn ⎟⎟⎟  ⎜⎜⎜⎜Im + c 2P P ⎟⎟⎠ === ⎜⎝ 2bn P In + c−2 2PPt ⎞ ⎛ (4) −2  ⎜⎜⎜⎜E11 c E12 ⎟⎟⎟⎟ === : ⎜⎝ ⎟⎠ . E21 E22 Derivation of the numbered equalities in (4.159): (1) (2) (3) (4)

This equation follows from (4.158). Follows from Item (1) by block matrix multiplication. Results from (4.157) and the commuting relations (4.55) and (4.54). This equation defines Ei j , i, j = 1, 2. Comparing the blocks Ei j in (4.159) and in (4.108), p. 123, with P1 = P2 =: P, we

Bi-gyrogroups and Bi-gyrovector Spaces – P

have P⊕P = E21 rgyr[P, P] = E11 lgyr[P, P] =





t Im + c−2 E21 E21

t In + c−2 E21 E21

−1

(4.160)

−1

E22

rgyr[P, P](P⊕P)t lgyr[P, P] = E12 , where Ei j are given by Item (4) of (4.159). Following the first equation in (4.160) and the definition of E21 in Item (4) of (4.159), and (4.54), we have the equations E21 = P⊕P = 2bn P = 2Pbm .

(4.161)

In order to derive Identity (4.163), let us consider the following chain of equations, some of which are numbered for subsequent explanation: (1)

 E11 === Im + c−2 2Pt P  1 === Im + c−2 4Pt P + 4(c−2 Pt P)2 2  1 === Im + 4(Im + c−2 Pt P)c−2 Pt P 2  1 === Im + c−2 4b2m Pt P 2

(4.162)

(2) 1   === Im + c−2 4Pt b2n P 2  1 === Im + c−2 2(bn P)t 2bn P 2 (3)   t === Im + c−2 E21 E21 .

Derivation of the numbered equalities in (4.162): (1) This equation follows from the definition of E11 in Item (4) of (4.159). (2) This equation is obtained from its predecessor by two successive applications of the commuting relation (4.55). (3) Follows from (4.161). Substituting E11 from (4.162) into the second equation in (4.160) we see that the right gyration generated by P and P is trivial, rgyr[P, P] = Im

(4.163)

133

134

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

for all P ∈ Rn×m . Similarly to (4.162), in order to derive Identity (4.165), let us consider the following chain of equations, some of which are numbered for subsequent explanation: (1)

 E22 === In + c−2 2PPt  1 === In + 4c−2 PPt + 4(c−2 PPt )2 2  1 === In + 4(In + c−2 PPt )c−2 PPt 2  1 === In + c−2 4b2n PPt 2

(4.164)

(2) 1   === In + c−2 4Pb2m Pt 2  1 === In + c−2 2Pbm 2(Pbm )t 2 (3)   t In + c−2 E21 E21 . ===

Derivation of the numbered equalities in (4.164): (1) This equation follows from the definition of E22 in Item (4) of (4.159). (2) This equation is obtained from its predecessor by two successive applications of the commuting relation (4.54). (3) Follows from (4.161). Substituting E22 from (4.164) into the third equation in (4.160) we see that the left gyration generated by P and P is trivial, lgyr[P, P] = In

(4.165)

for all P ∈ Rn×m . It follows from (4.160) – (4.165) that E21 = P⊕P E12 = (P⊕P)t  E11 = Im + c−2 (P⊕P)t (P⊕P)  E22 = In + c−2 (P⊕P)(P⊕P)t .

(4.166)

Hence, by the extreme sides of (4.159) B(P)2 = B(P⊕P) ,

(4.167)

Bi-gyrogroups and Bi-gyrovector Spaces – P

so that the square of a bi-boost is, again, a bi-boost. As a by-product of squaring the bi-boost, we have obtained the results in (4.163) and (4.165), which we formalize in the following theorem. Theorem 4.26. (A Trivial Bi-gyration). lgyr[P, P] = In rgyr[P, P] = Im

(4.168)

for all P ∈ Rn×m .

4.13. Commuting Relations Between Bi-gyrations and Bi-rotations Bi-gyrations (lgyr[P1 , P2 ], rgyr[P1 , P2 ]) ∈ S O(n) × S O(m) and bi-rotations (On , Om ) ∈ S O(n) × S O(m) commute in a special, interesting way stated in the following theorem. Theorem 4.27. (Bi-gyration – bi-rotation Commuting Relation). On lgyr[P1 , P2 ] = lgyr[On P1 , On P2 ]On

(4.169)

rgyr[P1 , P2 ]Om = Om rgyr[P1 Om , P2 Om ]

(4.170)

and for all P1 , P2 ∈ R

n×m

, On ∈ S O(n) and Om ∈ S O(m).

Proof. The matrix identity (4.169) is proved in the following chain of equations, which are numbered for subsequent explanation: (1)  −1  On lgyr[P1 , P2 ] === On In + c−2 (P1 ⊕P2 )(P1 ⊕P2 )t     −2 t t t −2 −2 × c P 1 P 2 + In + c P 1 P 1 I n + c P 2 P 2 (2) −1   In + c−2 (On P1 ⊕On P2 )(On P1 ⊕On P2 )t On ===     × c−2 P1 Pt2 + In + c−2 P1 Pt1 In + c−2 P2 Pt2 (3) −1   In + c−2 (On P1 ⊕On P2 )(On P1 ⊕On P2 )t ===     −2 t t t −2 −2 × c On P1 P2 + On In + c P1 P1 In + c P2 P2

(4.171)

135

136

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

(4) −1   In + c−2 (On P1 ⊕On P2 )(On P1 ⊕On P2 )t ===     × c−2 (On P1 )(On P2 )t + In + c−2 (On P1 )(On P1 )t In + c−2 (On P2 )(On P2 )t On (5)

 === lgyr[On P1 , On P2 ]On . Derivation of the numbered equalities in (4.171): (1) (2) (3) (4)

This equation follows from the third equation in (4.114), p. 124. Follows from Item (1) by Lemma 4.7, p. 112, and Theorem 4.25, p. 131. Follows from Item (2) by the linearity of On . Follows from Item (3) by the obvious matrix identity On P1 Pt2 = (On P1 )(On P2 )t On , and from Lemma 4.6, p. 111, and by the linearity of On . (5) Follows from Item (4) by the third equation in (4.114). The proof of the matrix identity (4.170) in (4.172) is similar to the proof of the matrix identity (4.169) in (4.171): (1)      rgyr[P1 , P2 ]Om === c−2 Pt1 P2 + Im + c−2 Pt1 P1 Im + c−2 Pt2 P2  −1 × Im + c−2 (P1 ⊕P2 )t (P1 ⊕P2 ) Om (2)      === c−2 Pt1 P2 + Im + c−2 Pt1 P1 Im + c−2 Pt2 P2 Om  −1 × Im + c−2 (P1 Om ⊕P2 Om )t (P1 Om ⊕P2 Om ) (3)      −2 t t t −2 −2 === c P1 P2 Om + Im + c P1 P1 Im + c P2 P2 Om  −1 × Im + c−2 (P1 Om ⊕P2 Om )t (P1 Om ⊕P2 Om ) (4)   −2 === c Om (P1 Om )t (P2 Om )    + Om Im + c−2 (P1 Om )t (P1 Om ) Im + c−2 (P2 Om )t (P2 Om )  −1 × Im + c−2 (P1 Om ⊕P2 Om )t (P1 Om ⊕P2 Om ) (5)

 === Om rgyr[P1 Om , P2 Om ] . Derivation of the numbered equalities in (4.172): (1) This equation follows from the fourth equation in (4.114). (2) Follows from Item (1) by Lemma 4.7, p. 112, and Theorem 4.25, p.131.

(4.172)

Bi-gyrogroups and Bi-gyrovector Spaces – P

(3) Follows from Item (2) by the linearity of Om . (4) Follows from Item (3) by the obvious matrix identity Pt1 P2 Om = Om (P1 Om )t (P2 Om ), and from Lemma 4.6, p. 111. (5) Follows from Item (4) by the linearity of Om and by the fourth equation in (4.114).  The following corollary results immediately from Theorem 4.27. Corollary 4.28. Let P1 , P2 ∈ Rn×m , On ∈ S O(n), and Om ∈ S O(m). Then, lgyr[On P1 , On P2 ] = lgyr[P1 , P2 ]

(4.173)

if and only if On and lgyr[P1 , P2 ] commute, that is, On lgyr[P1 , P2 ] = lgyr[P1 , P2 ]On . Similarly, rgyr[P1 Om , P2 Om ] = rgyr[P1 , P2 ]

(4.174)

if and only if Om and rgyr[P1 , P2 ] commute, that is, Om rgyr[P1 , P2 ] = rgyr[P1 , P2 ]Om . Example 4.29. The left (right) gyration lgyr[P1 , P2 ] (rgyr[P1 , P2 ]) commutes with itself. Hence, by Corollary 4.28, lgyr[lgyr[P1 , P2 ]P1 , lgyr[P1 , P2 ]P2 ] = lgyr[P1 , P2 ] rgyr[P1 rgyr[P1 , P2 ], P2 rgyr[P1 , P2 ]] = rgyr[P1 , P2 ] .

(4.175)

Left gyrations are invariant under parameter right rotations Om ∈ S O(m), and right gyrations are invariant under parameter left rotations On ∈ S O(n), as the following theorem asserts. Theorem 4.30. (Bi-gyration Invariance Relation). lgyr[P1 Om , P2 Om ] = lgyr[P1 , P2 ]

(4.176)

rgyr[On P1 , On P2 ] = rgyr[P1 , P2 ]

(4.177)

for all P1 , P2 ∈ Rn×m , On ∈ S O(n), and Om ∈ S O(m). Proof. The proof follows straightforwardly from the third and the fourth equations in (4.114), p. 124, and from Theorem 4.25, p. 131, noting that (P1 Om )(P2 Om )t = P1 Pt2 and (On P1 )t (On P2 ) = Pt1 P2 for all P1 , P2 ∈ Rn×m , On ∈ S O(n), and Om ∈ S O(m). 

137

138

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

4.14. Product of Lorentz Transformations Let Λ1 , Λ2 ∈ SO(m, n) be two Lorentz transformations of signature (m, n), m, n ∈ N, parametrized by the main parameter, P ∈ Rn×m , along with the orientation parameters, Om ∈ SO(m) and On ∈ SO(n). Then, According to (4.89), Λ1 = Λ(On,1 , P1 , Om,1 ) = ρ(Om,1 )B(P1 )λ(On,1 ) = (P1 , On,1 , Om,1 )t Λ2 = Λ(On,2 , P2 , Om,2 ) = ρ(Om,2 )B(P2 )λ(On,2 ) = (P2 , On,2 , Om,2 )t .

(4.178)

The product Λ1 Λ2 of Λ1 and Λ2 is obtained in the following chain of equations, which are numbered for subsequent explanation: (1)

 Λ1 Λ2 === ρ(Om,1 )B(P1 )λ(On,1 )ρ(Om,2 )B(P2 )λ(On,2 ) (2)

 === ρ(Om,1 )B(P1 )ρ(Om,2 )λ(On,1 )B(P2 )λ(On,2 ) (3)

 === ρ(Om,1 )ρ(Om,2 )B(P1 Om,2 )B(On,1 P2 )λ(On,1 )λ(On,2 ) (4)

 === ρ(Om,1 Om,2 )B(P1 Om,2 )B(On,1 P2 )λ(On,1 On,2 ) (5)

 === ρ(Om,1 Om,2 ) × ρ(rgyr[P1 Om,2 , On,1 P2 ])B(P1 Om,2 ⊕On,1 P2 )λ(lgyr[P1 Om,2 , On,1 P2 ]) × λ(On,1 On,2 ) (6)

 === ρ(Om,1 Om,2 rgyr[P1 Om,2 , On,1 P2 ]) × B(P1 Om,2 ⊕On,1 P2 ) × λ(lgyr[P1 Om,2 , On,1 P2 ]On,1 On,2 ) . (4.179) Derivation of the numbered equalities in (4.179): (1) (2) (3) (4)

This equation follows from (4.178). Follows from (1) since λ(On,1 ) and ρ(Om,2 ) commute. Follows from (2) by Lemma 4.10, p. 115. Follows from (3) by the obvious matrix identities ρ(Om,1 )ρ(Om,2 ) = ρ(Om,1 Om,2 ) and λ(On,1 )λ(On,2 ) = λ(On,1 On,2 ). (5) Follows from (4) by the bi-boost composition law (4.117), p. 125. (6) Obvious (Similar to the argument in Item (4)). In the column notation (4.89), the result of (4.179) gives the product law of Lorentz transformations in the following theorem.

Bi-gyrogroups and Bi-gyrovector Spaces – P

Theorem 4.31. (Lorentz Transformation Product Law, P). The product of two Lorentz transformations Λ1 = (P1 , On,1 , Om,1 )t and Λ2 = (P2 , On,2 , Om,2 )t of signature (m, n), m, n ∈ N, in the Lorentz group SO(m, n) is given by ⎞ ⎞⎛ ⎞ ⎛ ⎛ P1 Om,2 ⊕On,1 P2 ⎟⎟⎟ ⎜⎜⎜ P1 ⎟⎟⎟ ⎜⎜⎜ P2 ⎟⎟⎟ ⎜⎜⎜ ⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎜⎜⎜ ⎜ ⎜ ⎟ ⎟ ⎜ (4.180) Λ1 Λ2 = ⎜⎜⎜ On,1 ⎟⎟⎟ ⎜⎜⎜ On,2 ⎟⎟⎟ = ⎜⎜⎜ lgyr[P1 Om,2 , On,1 P2 ]On,1 On,2 ⎟⎟⎟⎟⎟ , ⎟⎠ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ ⎜⎝ Om,1 Om,2 Om,1 Om,2 rgyr[P1 Om,2 , On,1 P2 ] along with the identity Λ0 ∈ SO(m, n),

⎞ ⎛ ⎜⎜⎜0n,m ⎟⎟⎟ ⎟⎟ ⎜⎜ Λ0 = ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ . ⎟⎠ ⎜⎝ Im

(4.181)

The Lorentz transformation product law (4.180) – (4.181) is a parametric representation of the matrix product in (4.179). As such, the Lorentz transformation product law is associative. We will see in the sequel that the associativity of the Lorentz transformation product law determines the bi-gyrocommutativity and the bi-gyroassociativity of the binary operation ⊕ in Rn×m . Example 4.32. In the special case when P1 = P2 = 0n,m and Om,1 = Om,2 = Im , the parameter composition law (4.180) yields the equation ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎜⎜⎜ 0n,m ⎟⎟⎟ ⎜⎜⎜ 0n,m ⎟⎟⎟ ⎜⎜⎜ 0n,m ⎟⎟⎟ ⎜⎜⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ (4.182) ⎜⎜⎜On,1 ⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜On,2 ⎟⎟⎟⎟⎟ = ⎜⎜⎜⎜⎜On,1 On,2 ⎟⎟⎟⎟⎟ , ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ Im Im Im demonstrating that under the parameter composition law (4.180) the parameter On forms the special orthogonal group S O(n). Example 4.33. In the special case when P1 = P2 = 0n,m and On,1 = On,2 = In , the parameter composition law (4.180) yields the equation ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎜⎜⎜ 0n,m ⎟⎟⎟ ⎜⎜⎜ 0n,m ⎟⎟⎟ ⎜⎜⎜ 0n,m ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟ ⎜⎜⎜ In ⎟⎟⎟ ⎜⎜⎜ In ⎟⎟⎟ = ⎜⎜⎜ In ⎟⎟⎟⎟⎟ , (4.183) ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ Om,1 Om,2 Om,1 Om,2 demonstrating that under the parameter composition law (4.180) the parameter Om forms the special orthogonal group S O(m).

139

140

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Example 4.34. In the special case when On,1 = On,2 = In and Om,1 = Om,2 = Im the parameter composition law (4.180) yields the equation ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎜⎜⎜P1 ⎟⎟⎟ ⎜⎜⎜P2 ⎟⎟⎟ ⎜⎜⎜ P1 ⊕P2 ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟ ⎜⎜⎜ In ⎟⎟⎟ ⎜⎜⎜ In ⎟⎟⎟ = ⎜⎜⎜lgyr[P1 , P2 ]⎟⎟⎟⎟⎟ . (4.184) ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ Im Im rgyr[P1 , P2 ] Clearly, under the parameter composition law (4.180) the parameter P ∈ Rn×m does not form a group. Indeed, following the parametrization of the Lorentz group S O(m, n) in (4.178), we face the task of determining the composition law of the parameter P ∈ Rn×m along with the resulting group-like structure of the parameter set Rn×m . We will find in the sequel that the group-like structure of Rn×m that results from the composition law of the parameter P is a natural generalization of the gyrocommutative gyrogroup structure, called a bi-gyrocommutative bi-gyrogroup. The Lorentz transformation product (4.180) represents matrix multiplication. As such, it is associative and, clearly, its inverse obeys the identity −1 (Λ1 Λ2 )−1 = Λ−1 2 Λ1 .

(4.185)

4.15. The Bi-gyrocommutative Law in Bi-gyrogroupoids Bi-boosts are Lorentz transformations without bi-rotations. Let B(Pk ) = (Pk , In , Im )t ,

(4.186)

Pk ∈ Rn×m , k = 1, 2, be two bi-boosts in R(m+n)×(m+n) . Then, by (4.180) with On,1 = On,2 = In and Om,1 = Om,2 = Im (or by (4.117)), and by (4.99) with On = lgyr[P1 , P2 ] and Om = rgyr[P1 , P2 ], ⎧⎛ ⎞ ⎛ ⎞⎫−1 ⎛ ⎞−1 ⎪ ⎜⎜⎜P1 ⎟⎟⎟ ⎜⎜⎜P2 ⎟⎟⎟⎪ ⎜⎜⎜ P1 ⊕P2 ⎟⎟⎟ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎟⎟ ⎜⎜ ⎪ ⎬ ⎨⎜⎜⎜⎜ ⎟⎟⎟⎟ ⎜⎜⎜⎜ ⎟⎟⎟⎟⎪ = ⎜⎜⎜⎜⎜lgyr[P1 , P2 ]⎟⎟⎟⎟⎟ (B(P1 )B(P2 ))−1 = ⎪ ⎜⎜⎜ In ⎟⎟⎟ ⎜⎜⎜ In ⎟⎟⎟⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎜ ⎟⎪ ⎟⎠ ⎜⎝ ⎪ ⎪ ⎪ ⎭ ⎩⎝ Im ⎠ ⎝ Im ⎠⎪ rgyr[P1 , P2 ] (4.187) ⎞ ⎛ −1 −1 ⎜⎜⎜−lgyr [P1 , P2 ](P1 ⊕P2 )rgyr [P1 , P2 ]⎟⎟⎟ ⎟⎟⎟ ⎜⎜ ⎟⎟⎟ . = ⎜⎜⎜⎜⎜ lgyr−1 [P1 , P2 ] ⎟⎟⎠ ⎜⎝ rgyr−1 [P1 , P2 ] Here lgyr−1 [P1 , P2 ] = (lgyr[P1 , P2 ])−1 and, similarly, rgyr−1 [P1 , P2 ] = (rgyr[P1 , P2 ])−1 .

Bi-gyrogroups and Bi-gyrovector Spaces – P

Calculating (B(P1 )B(P2 ))−1 in a different way, as indicated in (4.185), yields ⎞⎛ ⎞ ⎛ ⎞−1 ⎛ ⎞−1 ⎛ ⎜⎜⎜P2 ⎟⎟⎟ ⎜⎜⎜P1 ⎟⎟⎟ ⎜⎜⎜−P2 ⎟⎟⎟ ⎜⎜⎜−P1 ⎟⎟⎟ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ (B(P1 )B(P2 ))−1 = B(P2 )−1 B(P1 )−1 = ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ = ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ Im Im Im Im (4.188) ⎛ ⎞ ⎜⎜⎜⎜ (−P2 )⊕(−P1 ) ⎟⎟⎟⎟ ⎜ ⎟ = ⎜⎜⎜⎜⎜lgyr[−P2 , −P1 ]⎟⎟⎟⎟⎟ . ⎜⎝ ⎟⎠ rgyr[−P2 , −P1 ] Hence, the extreme right-hand sides of (4.187) – (4.188) are equal, implying the equality of their corresponding entries, giving rise to the three equations in (4.189) – (4.190). The second and third entries of the extreme right-hand sides of (4.187) – (4.188), along with the even property (4.126) of bi-gyrations, imply the bi-gyration inversion law, lgyr−1 [P1 , P2 ] = lgyr[−P2 , −P1 ] = lgyr[P2 , P1 ] rgyr−1 [P1 , P2 ] = rgyr[−P2 , −P1 ] = rgyr[P2 , P1 ]

(4.189)

for all P1 , P2 ∈ Rn×m . The first entry of the extreme right-hand sides of (4.187) – (4.188), along with (4.189) and the gyroautomorphic inverse property (4.125), yields (−P2 )⊕(−P1 ) = −lgyr−1 [P1 , P2 ](P1 ⊕P2 )rgyr−1 [P1 , P2 ] = −lgyr[−P2 , −P1 ](P1 ⊕P2 )rgyr[−P2 , −P1 ] = lgyr[−P2 , −P1 ]{−(P1 ⊕P2 )}rgyr[−P2 , −P1 ] = lgyr[−P2 , −P1 ]{(−P1 )⊕(−P2 )}rgyr[−P2 , −P1 ] ,

(4.190)

for all P1 , P2 ∈ Rn×m . Renaming −P1 and −P2 as P2 and P1 , the extreme sides of (4.190) give the bigyrocommutative law of the bi-gyroaddition ⊕, P1 ⊕P2 = lgyr[P1 , P2 ](P2 ⊕P1 )rgyr[P1 , P2 ] ,

(4.191)

for all P1 , P2 ∈ Rn×m . Instructively, a short derivation of the bi-gyrocommutative law (4.191) of ⊕ is presented below. Transposing the extreme sides of (4.112), p. 124, noting that by (4.189) lgyr[P1 , P2 ]t = lgyr−1 [P1 , P2 ] = lgyr[P2 , P1 ] rgyr[P1 , P2 ]t = rgyr−1 [P1 , P2 ] = rgyr[P2 , P1 ] ,

(4.192)

141

142

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

and renaming the pair (P1 , P2 ) as (P2 , P1 ), we obtain the matrix identity P1 ⊕P2 = lgyr[P1 , P2 ](P2 ⊕P1 )rgyr[P1 , P2 ] .

(4.193)

for all P1 , P2 ∈ Rn×m . The matrix identity in (4.193) and in (4.191) gives the bi-gyrocommutative law of the binary operation ⊕ in Rn×m , according to which P1 ⊕P2 equals P2 ⊕P1 bi-gyrated by the bi-gyration (lgyr[P1 , P2 ], rgyr[P1 , P2 ]) generated by P1 and P2 , for all P1 , P2 ∈ Rn×m . Formalizing the result in (4.191) and in (4.193) we obtain the bi-gyrocommutative law in the following theorem. Theorem 4.35. (Bi-gyrocommutative Law in (Rn×m , ⊕)). The binary operation ⊕ in Rn×m possesses the bi-gyrocommutative law P1 ⊕P2 = lgyr[P1 , P2 ](P2 ⊕P1 )rgyr[P1 , P2 ]

(4.194)

for all P1 , P2 ∈ Rn×m . When m = 1, right gyrations are trivial, rgyr[P1 , P2 ] = Im = 1. Hence, in the special case when m = 1, the bi-gyrocommutative law (4.194) of bi-gyrogroup theory descends to the gyrocommutative law of gyrogroup theory, presented in Def. 2.14, p. 23, and studied, for instance, in [81, 84, 93, 94, 96, 95, 98]. By means of the bi-gyrocommutative law (4.194) and Theorem 4.20, p. 127, we immediately obtain the following theorem. Theorem 4.36. Let P1 , P2 , P3 ∈ Rn×m . Then, P3 = P1 ⊕P2 = P2 ⊕P1

(4.195)

B(P3 ) = B(P1 )B(P2 ) = B(P2 )B(P1 ) .

(4.196)

if and only if

Proof. By the bi-gyrocommutative law (4.194), the condition P1 ⊕P2 = P2 ⊕P1 in (4.195) is equivalent to the condition (4.129) in Theorem 4.20. Hence, the present theorem follows immediately from Theorem 4.20.  Formalizing the results in (4.192) we obtain the Bi-gyration inversion law in the following theorem.

Bi-gyrogroups and Bi-gyrovector Spaces – P

Theorem 4.37. (Bi-gyration Inversion Law). The bi-gyrogroupoid (Rn×m , ⊕) possesses the left gyration inversion law and the right gyration inversion law, lgyr−1 [P1 , P2 ] = lgyr[P2 , P1 ] rgyr−1 [P1 , P2 ] = rgyr[P2 , P1 ] ,

(4.197)

for all P1 , P2 ∈ Rn×m . Identities (4.197) form the inversive symmetric property of bi-gyrations.

4.16. The Bi-gyroassociative Law in Bi-gyrogroupoids Matrix multiplication is associative. Hence (Λ1 Λ2 )Λ3 = Λ1 (Λ2 Λ3 ) .

(4.198)

Let P1 , P2 , P3 ∈ Rn×m . On the one hand, by (4.117) and (4.180), ⎧⎛ ⎞ ⎛ ⎞⎫ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎪ ⎜⎜⎜P3 ⎟⎟⎟ ⎜⎜⎜ P1 ⊕P2 ⎟⎟⎟ ⎜⎜⎜P3 ⎟⎟⎟ ⎜⎜⎜P1 ⎟⎟⎟ ⎜⎜⎜P2 ⎟⎟⎟⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎟⎟ ⎜⎜ ⎟⎟ ⎪ ⎬ ⎜⎜⎜⎜ ⎟⎟⎟⎟ ⎜⎜⎜⎜ ⎨⎜⎜⎜⎜ ⎟⎟⎟⎟ ⎜⎜⎜⎜ ⎟⎟⎟⎟⎪ (B(P1 )B(P2 ))B(P3 ) = ⎪ ⎜⎜⎜ In ⎟⎟⎟ = ⎜⎜⎜lgyr[P1 , P2 ]⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ ⎜⎜⎜ In ⎟⎟⎟ ⎜⎜⎜ In ⎟⎟⎟⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟⎪ ⎟⎜ ⎟ ⎪ ⎪ ⎪ ⎭ ⎝ Im ⎠ ⎝rgyr[P1 , P2 ]⎠ ⎝ Im ⎠ ⎩⎝ Im ⎠ ⎝ Im ⎠⎪ ⎛ ⎞ (P1 ⊕P2 )⊕lgyr[P1 , P2 ]P3 ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟ = ⎜⎜⎜⎜ lgyr[P1 ⊕P2 , lgyr[P1 , P2 ]P3 ]lgyr[P1 , P2 ] ⎟⎟⎟⎟⎟ . ⎜⎝ ⎟⎠ rgyr[P1 , P2 ]rgyr[P1 ⊕P2 , lgyr[P1 , P2 ]P3 ]

(4.199)

On the other hand, similarly, by (4.117) and (4.180), ⎞ ⎛ ⎞ ⎧⎛ ⎞ ⎛ ⎞⎫ ⎛ ⎞ ⎛ ⎜⎜⎜P2 ⎟⎟⎟ ⎜⎜⎜P3 ⎟⎟⎟⎪ ⎜⎜⎜P1 ⎟⎟⎟ ⎜⎜⎜ P2 ⊕P3 ⎟⎟⎟ ⎜⎜⎜P1 ⎟⎟⎟ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎜ ⎜ ⎜ ⎜ ⎟⎟⎟ ⎟ ⎟ ⎟ ⎜⎜⎜ ⎟⎟⎟ ⎪ ⎪ ⎪ ⎨⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟⎬ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ I I I lgyr[P , P ] = B(P1 )(B(P2 )B(P3 )) = ⎜⎜⎜⎜ In ⎟⎟⎟⎟ ⎪ ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎪ n n n 2 3 ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠⎪ ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ ⎪ ⎪ ⎜⎝ ⎟⎠ ⎪ ⎪ ⎪ ⎪ ⎪ Im ⎩ Im Im ⎭ Im rgyr[P2 , P3 ] ⎛ ⎞ P1 rgyr[P2 , P3 ]⊕(P2 ⊕P3 ) ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟ = ⎜⎜⎜⎜ lgyr[P1 rgyr[P2 , P3 ], P2 ⊕P3 ]lgyr[P2 , P3 ] ⎟⎟⎟⎟⎟ . ⎜⎝ ⎟⎠ rgyr[P2 , P3 ]rgyr[P1 rgyr[P2 , P3 ], P2 ⊕P3 ]

(4.200)

Hence, by (4.198) – (4.200), corresponding entries of the extreme right-hand sides of (4.199) and (4.200) are equal, giving rise to the bi-gyroassociative law (P1 ⊕P2 )⊕lgyr[P1 , P2 ]P3 = P1 rgyr[P2 , P3 ]⊕(P2 ⊕P3 )

(4.201)

143

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

and to the bi-gyration identities lgyr[P1 ⊕P2 , lgyr[P1 , P2 ]P3 ]lgyr[P1 , P2 ] = lgyr[P1 rgyr[P2 , P3 ], P2 ⊕P3 ]lgyr[P2 , P3 ] rgyr[P1 , P2 ]rgyr[P1 ⊕P2 , lgyr[P1 , P2 ]P3 ] = rgyr[P2 , P3 ]rgyr[P1 rgyr[P2 , P3 ], P2 ⊕P3 ] (4.202) for all P1 , P2 , P3 ∈ Rn×m . Formalizing the result in (4.201) we obtain the following theorem. Theorem 4.38. (Bi-gyroassociative Law in (Rn×m , ⊕)). The bi-gyroaddition ⊕ in Rn×m possesses the bi-gyroassociative law (P1 ⊕P2 )⊕lgyr[P1 , P2 ]P3 = P1 rgyr[P2 , P3 ]⊕(P2 ⊕P3 )

(4.203)

for all P1 , P2 , P3 ∈ Rn×m . Note that in the bi-gyroassociative law (4.203), P1 and P2 are grouped together on the left-hand side, while P2 and P3 are grouped together on the right-hand side. When m = 1, right gyrations are trivial, rgyr[P1 , P2 ] = Im=1 = 1. Hence, in the special case when m = 1, the bi-gyroassociative law (4.203) descends to the gyroassociative law of gyrogroup theory presented in Def. 2.13, p. 22, and studied, for instance, in [81, 84, 93, 94, 96, 95, 98]. The bi-gyroassociative law gives rise to the left and the right cancellation laws in the following theorem. Theorem 4.39. (Left and Right Cancellation Laws in (Rn×m , ⊕)). gyrogroupoid (Rn×m , ⊕) possesses the left and the right cancellation laws

The bi-

P2 = P1 rgyr[P1 , P2 ]⊕(P1 ⊕P2 )

(4.204)

P1 = (P1 ⊕P2 )lgyr[P1 , P2 ]P2

(4.205)

and for all P1 , P2 ∈ Rn×m . Proof. The left cancellation law (4.204) follows from the bi-gyroassociative law (4.203) with P1 = P2 , noting that lgyr[P2 , P2 ] is trivial by (4.115), p. 125. The right cancellation law (4.205) follows from the bi-gyroassociative law (4.203) with P3 = P2 , noting that rgyr[P2 , P2 ] is trivial.  The bi-gyroassociative law gives rise to the left and the right bi-gyroassociative laws in the following theorem.

Bi-gyrogroups and Bi-gyrovector Spaces – P

Theorem 4.40. (Left and Right Bi-gyroassociative Law in (Rn×m , ⊕)). The bi-gyroaddition ⊕ in Rn×m possesses the left bi-gyroassociative law P1 ⊕(P2 ⊕P3 ) = (P1 rgyr[P3 , P2 ]⊕P2 )⊕lgyr[P1 rgyr[P3 , P2 ], P2 ]P3

(4.206)

and the right bi-gyroassociative law (P1 ⊕P2 )⊕P3 = P1 rgyr[P2 , lgyr[P2 , P1 ]P3 ]⊕(P2 ⊕lgyr[P2 , P1 ]P3 )

(4.207)

for all P1 , P2 , P3 ∈ Rn×m . Proof. The left bi-gyroassociative law (4.206) is obtained from the bi-gyroassociative law (4.203) by replacing P1 by P1 rgyr[P3 , P2 ] and noting the bi-gyration inversion law (4.197). The right bi-gyroassociative law (4.207) is obtained from the bi-gyroassociative law (4.203) by replacing P3 by lgyr[P2 , P1 ]P3 and by employing the bi-gyration inversion law (4.197). 

4.17. Bi-gyration Reduction Properties in Bi-gyrogroupoids A reduction property of a gyration lgyr[P1 , P2 ] or rgyr[P1 , P2 ] is a property enabling the gyration to be expressed as a gyration that involves P1 ⊕P2 . Several reduction properties are derived in Subsects. 4.17.1 – 4.17.4.

4.17.1. Bi-gyration Reduction Properties I When P3 = P2 , (4.202) specializes to lgyr[P1 ⊕P2 , lgyr[P1 , P2 ]P2 ]lgyr[P1 , P2 ] = In rgyr[P1 , P2 ]rgyr[P1 ⊕P2 , lgyr[P1 , P2 ]P2 ] = Im

(4.208)

or, equivalently by bi-gyration inversion, (4.197), lgyr[P1 , P2 ] = lgyr[lgyr[P1 , P2 ]P2 , P1 ⊕P2 ] rgyr[P1 , P2 ] = rgyr[lgyr[P1 , P2 ]P2 , P1 ⊕P2 ] .

(4.209)

Similarly, when P2 = P1 , (4.202) specializes to In = lgyr[P1 rgyr[P1 , P3 ], P1 ⊕P3 ]lgyr[P1 , P2 ] Im = rgyr[P1 , P3 ]rgyr[P1 rgyr[P1 , P3 ], P1 ⊕P3 ]

(4.210)

or, equivalently by bi-gyration inversion, (4.197), and renaming P3 as P2 , lgyr[P1 , P2 ] = lgyr[P1 ⊕P2 , P1 rgyr[P1 , P2 ]] rgyr[P1 , P2 ] = rgyr[P1 ⊕P2 , P1 rgyr[P1 , P2 ]] .

(4.211)

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Formalizing the results in (4.209) and (4.211) we obtain the following theorem. Theorem 4.41. (Left and Right Gyration Reduction Properties). lgyr[P1 , P2 ] = lgyr[lgyr[P1 , P2 ]P2 , P1 ⊕P2 ] rgyr[P1 , P2 ] = rgyr[lgyr[P1 , P2 ]P2 , P1 ⊕P2 ] .

(4.212)

and lgyr[P1 , P2 ] = lgyr[P1 ⊕P2 , P1 rgyr[P1 , P2 ]] rgyr[P1 , P2 ] = rgyr[P1 ⊕P2 , P1 rgyr[P1 , P2 ]] .

(4.213)

for all P1 , P2 ∈ Rn×m .

4.17.2. Bi-gyration Reduction Properties II In general, the product of bi-boosts in a pseudo-orthogonal group S O(m, n) is a Lorentz transformation which is not a boost. In some special cases, however, the product of bi-boosts is again a bi-boost, as shown below. Let P1 , P2 ∈ Rn×m , and let J(P1 , P2 ) be the bi-boost symmetric product ⎛ ⎞⎛ ⎞⎛ ⎞ ⎜⎜⎜P1 ⎟⎟⎟ ⎜⎜⎜P2 ⎟⎟⎟ ⎜⎜⎜P1 ⎟⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ (4.214) J(P1 , P2 ) = B(P1 )B(P2 )B(P1 ) = ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ , ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ Im Im Im which is symmetric with respect to the central bi-boost factor (P2 , In , Im )t . Then, by the Lorentz product law (4.180), ⎞⎛ ⎞ ⎛ ⎜⎜⎜ P1 ⊕P2 ⎟⎟⎟ ⎜⎜⎜P1 ⎟⎟⎟ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ J(P1 , P2 ) = ⎜⎜⎜⎜⎜lgyr[P1 , P2 ]⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ rgyr[P1 , P2 ] Im (4.215) ⎛ ⎞ ⎛ ⎞ (P1 ⊕P2 )⊕lgyr[P1 , P2 ]P1 ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ P3 ⎟⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ = ⎜⎜⎜⎜⎜ lgyr[P1 ⊕P2 , lgyr[P1 , P2 ]P1 ]lgyr[P1 , P2 ] ⎟⎟⎟⎟⎟ =: ⎜⎜⎜⎜⎜ On ⎟⎟⎟⎟⎟ . ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ Om rgyr[P1 , P2 ]rgyr[P1 ⊕P2 , lgyr[P1 , P2 ]P1 ] By means of (4.97), p. 120, it is clear from (4.214) that J(P1 , P2 )−1 = J(−P1 , −P2 ) .

(4.216)

Hence, by the gyroautomorphic inverse property (4.125), p. 126, and by the bi-gyration

Bi-gyrogroups and Bi-gyrovector Spaces – P

even property, (4.126), p. 126, it is clear from (4.215) that ⎞ ⎛ ⎜⎜⎜−P3 ⎟⎟⎟ ⎟⎟ ⎜⎜ J(P1 , P2 )−1 = J(−P1 , −P2 ) = ⎜⎜⎜⎜⎜ On ⎟⎟⎟⎟⎟ . ⎟⎠ ⎜⎝ Om

(4.217)

But, it follows from the inverse Lorentz transformation (4.99), p. 120, that ⎞ ⎛ −1 ⎜⎜⎜−On P3 O−1 m ⎟ ⎟⎟⎟ ⎜⎜ ⎟⎟⎟ O−1 (4.218) J(P1 , P2 )−1 = ⎜⎜⎜⎜⎜ ⎟⎟⎟ . n ⎟ ⎜⎝ ⎠ O−1 m Comparing the right-hand sides of (4.218) and (4.217), we find that On = O−1 n and Om = O−1 , implying O = I and O = I . Hence, by (4.215), the bi-boost product n n m m m J(P1 , P2 ) is, again, a bi-boost, ⎞ ⎛ ⎜⎜⎜(P1 ⊕P2 )⊕lgyr[P1 , P2 ]P1 ⎟⎟⎟ ⎟⎟⎟ ⎜⎜ ⎟⎟⎟ . In (4.219) J(P1 , P2 ) = ⎜⎜⎜⎜⎜ ⎟⎟⎠ ⎜⎝ Im Following (4.219) and (4.215) we have the bi-gyration identities lgyr[P1 ⊕P2 , lgyr[P1 , P2 ]P1 ]lgyr[P1 , P2 ] = In rgyr[P1 , P2 ]rgyr[P1 ⊕P2 , lgyr[P1 , P2 ]P1 ] = Im ,

(4.220)

implying, by the bi-gyration inversion law (4.197), lgyr[P1 , P2 ] = lgyr[lgyr[P1 , P2 ]P1 , P1 ⊕P2 ] rgyr[P1 , P2 ] = rgyr[lgyr[P1 , P2 ]P1 , P1 ⊕P2 ] ,

(4.221)

for all P1 , P2 ∈ Rn×m . The results in (4.219) – (4.220) can readily be extended to the symmetric product of any number of bi-boosts that appear symmetrically with respect to a central factor. Thus, for instance, the symmetric bi-boost product J, J = B(Pk )B(Pk−1 ) . . . B(P2 )B(P1 )B(P0 )B(P1 )B(P2 ) . . . B(Pk−1 )B(Pk ) ,

(4.222)

is symmetric with respect to the central factor B(P0 ), for any k ∈ N, and all Pi ∈ Rn×m , i = 0, 1, 2, . . . , k. As such, the bi-boost product J in (4.222) is, again, a bi-boost. We now manipulate the first bi-gyration identity in (4.220) into an elegant form that will be elevated to the status of a theorem (Theorem 4.42). Let us consider the

147

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

following chain of equations, which are numbered for subsequent explanation: (1)

 In === lgyr[P1 ⊕P2 , lgyr[P1 , P2 ]P1 ]lgyr[P1 , P2 ] (2)

 === lgyr[P1 , P2 ]lgyr[lgyr[P2 , P1 ](P1 ⊕P2 ), P1 ] (3)

 === lgyr[P1 , P2 ]lgyr[lgyr[P2 , P1 ](P1 ⊕P2 )rgyr[P2 , P1 ], P1 rgyr[P2 , P1 ]]

(4.223)

(4)

 === lgyr[P1 , P2 ]lgyr[P2 ⊕P1 , P1 rgyr[P2 , P1 ]] . Derivation of the numbered equalities in (4.223): (1) This equation is the first equation in (4.220). (2) Follows from the commuting relation (4.169), p. 135, with On = lgyr[P1 , P2 ], noting that lgyr[P2 , P1 ] = lgyr−1 [P1 , P2 ]. (3) Follows from the bi-gyration invariance relation (4.176), p. 137, with Om = rgyr[P2 , P1 ]. (4) Follows from the bi-gyrocommutative law (4.194), p. 142. By (4.223) and the bi-gyration inversion law (4.197), lgyr[P2 , P1 ] = lgyr[P2 ⊕P1 , P1 rgyr[P2 , P1 ]] .

(4.224)

Renaming (P1 , P2 ) in (4.224) as (P2 , P1 ), we obtain the first identity in the following theorem. Theorem 4.42. (Left Gyration Reduction Properties). lgyr[P1 , P2 ] = lgyr[P1 ⊕P2 , P2 rgyr[P1 , P2 ]]

(4.225)

lgyr[P1 , P2 ] = lgyr[P1 rgyr[P2 , P1 ], P2 ⊕P1 ]

(4.226)

and for all P1 , P2 ∈ R

n×m

.

Proof. The bi-gyration identity (4.225) is identical with (4.224). The bi-gyration identity (4.226) is obtained from (4.225) by applying the bi-gyration inversion law (4.197) followed by renaming (P1 , P2 ) as (P2 , P1 ).  When m = 1 right gyrations are trivial, rgyr[P1 , P2 ] = Im=1 = 1. Hence, in the special case when m = 1, the bi-gyration reduction properties (4.225) – (4.226) descend to the gyration properties of gyrogroup theory found, for instance, in [84]. The bi-gyration identity (4.226) involves both left and right gyrations. We manipulate

Bi-gyrogroups and Bi-gyrovector Spaces – P

it into an identity that involves only left gyrations in the following chain of equations, which are numbered for subsequent explanation: (1)

 lgyr[P1 , P2 ] === lgyr[P1 rgyr[P2 , P1 ], P2 ⊕P1 ] (2)

 === lgyr[lgyr[P1 , P2 ]P1 rgyr[P2 , P1 ], lgyr[P1 , P2 ](P2 ⊕P1 )] (3)

 === lgyr[lgyr[P1 , P2 ]P1 rgyr[P2 , P1 ]rgyr[P1 , P2 ], lgyr[P1 , P2 ](P2 ⊕P1 )rgyr[P1 , P2 ]] (4)

 === lgyr[lgyr[P1 , P2 ]P1 , P1 ⊕P2 ] . (4.227) Derivation of the numbered equalities in (4.227): (1) This equation is the bi-gyration identity 4.226. (2) Follows from Result (4.173) of Corollary 4.28, p. 137, noting that, by Item (1), the left gyrations lgyr[P1 rgyr[P2 , P1 ], P2 ⊕P1 ] and lgyr[P1 , P2 ] commute since they are equal. (3) Follows from (4.176), p. 137, with Om = rgyr[P1 , P2 ]. (4) Follows from Item (3) by applying both the bi-gyration inversion law (4.197) and the bi-gyrocommutative law (4.194), p. 142. By means of the bi-gyration inversion law (4.197), the second bi-gyration identity in (4.220) gives rise to the bi-gyration identity rgyr[P1 , P2 ] = rgyr[lgyr[P1 , P2 ]P1 , P1 ⊕P2 ] ,

(4.228)

leading to the following theorem. Theorem 4.43. (Right Gyration Reduction Properties). rgyr[P1 , P2 ] = rgyr[lgyr[P1 , P2 ]P1 , P1 ⊕P2 ]

(4.229)

rgyr[P1 , P2 ] = rgyr[P2 ⊕P1 , lgyr[P2 , P1 ]P2 ]

(4.230)

and for all P1 , P2 ∈ Rn×m . Proof. The bi-gyration identity (4.229) is identical with (4.228). The bi-gyration identity (4.230) is obtained from (4.229) by applying the bi-gyration inversion law (4.197) followed by renaming (P1 , P2 ) as (P2 , P1 ).  The bi-gyration identity (4.230) involves both left and right gyrations. We manipulate

149

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it into an identity that involves only right gyrations in the following chain of equations, which are numbered for subsequent explanation: (1)

 rgyr[P1 , P2 ] === rgyr[P2 ⊕P1 , lgyr[P2 , P1 ]P2 ] (2)

 === rgyr[(P2 ⊕P1 )rgyr[P1 , P2 ], lgyr[P2 , P1 ]P2 rgyr[P1 , P2 ]] (3)

 === rgyr[lgyr[P1 , P2 ](P2 ⊕P1 )rgyr[P1 , P2 ], lgyr[P1 , P2 ]lgyr[P2 , P1 ]P2 rgyr[P1 , P2 ]] (4)

 === rgyr[P1 ⊕P2 , P2 rgyr[P1 , P2 ]] . (4.231) Derivation of the numbered equalities in (4.231): (1) This equation is the bi-gyration identity 4.230. (2) Follows from Result (4.174) of Corollary 4.28, p. 137, noting that, by Item 1, the right gyrations rgyr[P2 ⊕P1 , lgyr[P2 , P1 ]P2 ] and rgyr[P1 , P2 ] commute since they are equal. (3) Follows from (4.177), p. 137. (4) Follows from Item (3) by applying both the bi-gyration inversion law (4.197) and the bi-gyrocommutative law (4.194), p. 142. Formalizing the results in (4.227) and (4.231) we obtain the following theorem. Theorem 4.44. (Bi-gyration Reduction Properties). lgyr[P1 , P2 ] = lgyr[lgyr[P1 , P2 ]P1 , P1 ⊕P2 ]

(4.232)

rgyr[P1 , P2 ] = rgyr[P1 ⊕P2 , P2 rgyr[P1 , P2 ]]

(4.233)

and for all P1 , P2 ∈ Rn×m .

4.17.3. Bi-gyration Reduction Properties III As in Subsect. 4.17.2, let P1 , P2 ∈ Rn×m , and let J(P1 , P2 ) be the bi-boost symmetric product ⎛ ⎞⎛ ⎞⎛ ⎞ ⎜⎜⎜P1 ⎟⎟⎟ ⎜⎜⎜P2 ⎟⎟⎟ ⎜⎜⎜P1 ⎟⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ (4.234) J(P1 , P2 ) = ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ , ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ Im Im Im

Bi-gyrogroups and Bi-gyrovector Spaces – P

which is symmetric with respect to the central bi-boost factor (P2 , In , Im )t . Then, ⎞ ⎛ ⎞⎛ ⎜⎜⎜P1 ⎟⎟⎟ ⎜⎜⎜ P2 ⊕P1 ⎟⎟⎟ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ J(P1 , P2 ) = ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜lgyr[P2 , P1 ]⎟⎟⎟⎟⎟ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ Im rgyr[P2 , P1 ] (4.235) ⎛ ⎞ ⎛ ⎞ P1 rgyr[P2 , P1 ]⊕(P2 ⊕P1 ) ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ P3 ⎟⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ = ⎜⎜⎜⎜⎜ lgyr[P1 rgyr[P2 , P1 ], P2 ⊕P1 ]lgyr[P2 , P1 ] ⎟⎟⎟⎟⎟ =: ⎜⎜⎜⎜⎜ On ⎟⎟⎟⎟⎟ . ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ Om rgyr[P2 , P1 ]rgyr[P1 rgyr[P2 , P1 ], P2 ⊕P1 ] By means of (4.97), p. 120, it is clear from (4.234) that J(P1 , P2 )−1 = J(−P1 , −P2 ) .

(4.236)

Hence, by the gyroautomorphic inverse property (4.125), p. 126, and by the bi-gyration even property, (4.126), p. 126, it is clear from (4.235) that ⎞ ⎛ ⎜⎜⎜−P3 ⎟⎟⎟ ⎟⎟ ⎜⎜ (4.237) J(P1 , P2 )−1 = J(−P1 , −P2 ) = ⎜⎜⎜⎜⎜ On ⎟⎟⎟⎟⎟ . ⎟⎠ ⎜⎝ Om But, it follows from the inverse Lorentz transformation (4.99), p. 120, that ⎞ ⎛ −1 ⎜⎜⎜−On P3 O−1 m ⎟ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ −1 −1 ⎜ O (4.238) J(P1 , P2 ) = ⎜⎜⎜ ⎟⎟⎟ . n ⎟⎠ ⎜⎝ −1 Om Comparing the right-hand sides of (4.238) and (4.237), we find that Om = Im and On = In . Hence, the bi-boost product J(P1 , P2 ) is, again, a bi-boost, so that by (4.235), ⎞ ⎛ ⎜⎜⎜P1 rgyr[P2 , P1 ]⊕(P2 ⊕P1 )⎟⎟⎟ ⎟⎟⎟⎟ ⎜⎜ In (4.239) J(P1 , P2 ) = ⎜⎜⎜⎜⎜ ⎟⎟⎟ . ⎟⎠ ⎜⎝ Im Following (4.239) and (4.235) we have the bi-gyration identities lgyr[P1 rgyr[P2 , P1 ], P2 ⊕P1 ]lgyr[P2 , P1 ] = In rgyr[P2 , P1 ]rgyr[P1 rgyr[P2 , P1 ], P2 ⊕P1 ] = Im ,

(4.240)

implying lgyr[P1 , P2 ] = lgyr[P1 rgyr[P2 , P1 ], P2 ⊕P1 ] rgyr[P1 , P2 ] = rgyr[P1 rgyr[P2 , P1 ], P2 ⊕P1 ] ,

(4.241)

151

152

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

for all P1 , P2 ∈ Rn×m . The first entries of (4.215) and (4.235) imply the interesting identity (P1 ⊕P2 )⊕lgyr[P1 , P2 ]P1 = P1 rgyr[P2 , P1 ]⊕(P2 ⊕P1 ) .

(4.242)

4.17.4. Bi-gyration Reduction Properties IV Let (P1 , In , Im )t and (P2 , In , Im )t be two given bi-boosts in the pseudo-Euclidean space Rm,n , and let the bi-boost (X, On , Om )t be given by the equation ⎛ ⎞ ⎛ ⎞−1 ⎛ ⎞ ⎜⎜⎜ X ⎟⎟⎟ ⎜⎜⎜P1 ⎟⎟⎟ ⎜⎜⎜P2 ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟ ⎜ ⎟ ⎜⎜⎜ On ⎟⎟⎟ := ⎜⎜⎜ In ⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ . (4.243) ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ Om Im Im Then, the following two consequences of (4.243) are equivalent: ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎜⎜⎜ X ⎟⎟⎟ ⎜⎜⎜P1 ⎟⎟⎟ ⎜⎜⎜P2 ⎟⎟⎟ ⎜⎜⎜ P1 ⊕P2 ⎟⎟⎟ ⎟⎟ ⎜⎜⎜⎜ ⎟⎟⎟⎟ ⎜⎜⎜⎜ ⎟⎟⎟⎟ ⎜⎜⎜⎜ ⎟⎟⎟⎟ ⎜⎜⎜⎜ ⎜⎜⎜ On ⎟⎟⎟ = ⎜⎜⎜ In ⎟⎟⎟ ⎜⎜⎜ In ⎟⎟⎟ = ⎜⎜⎜lgyr[P1 , P2 ]⎟⎟⎟⎟⎟ ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ Om Im Im rgyr[P1 , P2 ] and

⎛ ⎞ ⎛ ⎞ ⎞⎛ ⎞ ⎛ P1 Om ⊕X ⎜⎜⎜P2 ⎟⎟⎟ ⎜⎜⎜P1 ⎟⎟⎟ ⎜⎜⎜ X ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎜⎜⎜ In ⎟⎟⎟ = ⎜⎜⎜ In ⎟⎟⎟ ⎜⎜⎜ On ⎟⎟⎟ = ⎜⎜⎜ lgyr[P1 Om , X]On ⎟⎟⎟⎟⎟ . ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ Im Im Om Om rgyr[P1 Om , X]

(4.244)

(4.245)

The matrix equation (4.245) in Rm,n implies On = lgyr[X, P1 Om ] Om = rgyr[X, P1 Om ] ,

(4.246)

so that, by the first entry of the matrix equation (4.244), On = lgyr[P1 ⊕P2 , P1 Om ] Om = rgyr[P1 ⊕P2 , P1 Om ] .

(4.247)

Inserting On and Om from the second and the third entries of the matrix equation (4.244) into (4.247), we obtain the reduction properties lgyr[P1 , P2 ] = lgyr[P1 ⊕P2 , P1 rgyr[P1 , P2 ]] rgyr[P1 , P2 ] = rgyr[P1 ⊕P2 , P1 rgyr[P1 , P2 ]] ,

(4.248)

thus recovering (4.213). As a first example, the first reduction property in (4.248) gives rise to the reduction

Bi-gyrogroups and Bi-gyrovector Spaces – P

property lgyr[P1 , P2 ] = lgyr[(P1 ⊕P2 )rgyr[P2 , P1 ], P1 ]

(4.249)

in the following chain of equations, which are numbered for subsequent explanation: (1)

 lgyr[P1 , P2 ] === lgyr[P1 ⊕P2 , P1 rgyr[P1 , P2 ]] (2)

 === lgyr[(P1 ⊕P2 )rgyr[P2 , P1 ], P1 rgyr[P1 , P2 ]rgyr[P2 , P1 ]] (4.250) (3)

 === lgyr[(P1 ⊕P2 )rgyr[P2 , P1 ], P1 ] . Derivation of the numbered equalities in (4.250): (1) This is the first identity in (4.248). (2) Item (2) is derived from Item (1) by applying Identity (4.176) of Theorem 4.30, p. 137, with Om = rgyr[P2 , P1 ]. (3) Item 3 follows immediately from Item 2 by the bi-gyration inversion law (4.197), p. 143. As a second example, the second reduction property in (4.248) gives rise to the reduction property rgyr[P1 , P2 ] = rgyr[(P1 ⊕P2 )rgyr[P2 , P1 ], P1 ]

(4.251)

in the following chain of equations, which are numbered for subsequent explanation: (1)

 rgyr[P1 , P2 ] === rgyr[P1 ⊕P2 , P1 rgyr[P1 , P2 ]] (2)

 === rgyr[(P1 ⊕P2 )rgyr[P2 , P1 ], P1 rgyr[P1 , P2 ]rgyr[P2 , P1 ]] (4.252) (3)

 === rgyr[(P1 ⊕P2 )rgyr[P2 , P1 ], P1 ] . Derivation of the numbered equalities in (4.252): (1) This is the second identity in (4.248). (2) Being the inverse of rgyr[P1 , P2 ], the right gyrations rgyr[P2 , P1 ] and rgyr[P1 , P2 ] commute. Hence, by Item 1, the right gyrations rgyr[P2 , P1 ]

and

rgyr[P1 ⊕P2 , P1 rgyr[P1 , P2 ]]

commute. The latter commutativity, in turn, implies Item 2 by Corollary 4.28, p. 137, with Om = rgyr[P2 , P1 ]. (3) Item 3 follows immediately from Item 2 by the bi-gyration inversion law (4.197), p. 143.

153

154

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Formalizing the results in (4.249) and (4.251) in terms of a new binary operation in Rn×m , we obtain the elegant gyration reduction property in the following theorem. Theorem 4.45. (The Gyration Reduction Property). For all P1 , P2 ∈ Rn×m , lgyr[P1 , P2 ] = lgyr[P1 ⊕ P2 ,  P1 ] rgyr[P1 , P2 ] = rgyr[P1 ⊕ P2 ,  P1 ] ,

(4.253)

where ⊕ is a binary operation in Rn×m given by P1 ⊕ P2 = (P1 ⊕P2 )rgyr[P2 , P1 ]

(4.254)

 P = P = −P .

(4.255)

and, consequently, for all P ∈ R

n×m

.

4.18. Bi-gyrogroups – P The elegance and simplicity of the bi-gyration reduction property (4.253) in terms of the parameter P ∈ Rn×m along with the binary operation ⊕ in Theorem 4.45 is remarkable. It indicates that it would prove useful to replace the bi-gyrogroupoid binary operation ⊕ in Rn×m by the bi-gyrogroup binary operation ⊕ in Rn×m , which is formally defined in Def. 4.46. Definition 4.46. (Bi-gyrogroup Operation, Bi-gyrogroups). Let (Rn×m , ⊕) be a bigyrogroupoid (Def. 4.16, p. 121; Theorem 4.18, p. 124). The bi-gyrogroup binary operation ⊕ in Rn×m is given by P1 ⊕ P2 = (P1 ⊕P2 )rgyr[P2 , P1 ]

(4.256)

for all P1 , P2 ∈ Rn×m . The resulting groupoid (Rn×m , ⊕ ) is called a bi-gyrogroup. The term bi-gyrogroup, coined in Def. 4.46, will be justified in Sect. 4.23. Following (4.256) we have, by right gyration inversion, (4.197), p. 143, P1 ⊕P2 = (P1 ⊕ P2 )rgyr[P1 , P2 ]

(4.257)

for all P1 , P2 ∈ R . We will find in the sequel that the bi-gyrogroups (Rn×m , ⊕ ), rather than the bigyrogroupoids (Rn×m , ⊕), form the desired elegant algebraic structure that the parametric realization of the Lorentz group S O(m, n) encodes. The point is that we must study bi-gyrogroupoids in order to pave the way to the study of bi-gyrogroups. The bi-gyrogroup operation ⊕ is given in (4.256) in terms of the bi-gyrogroupoid n×m

Bi-gyrogroups and Bi-gyrovector Spaces – P

operation ⊕ and a right gyration. However, the right gyration is not privileged, since the operation ⊕ can be determined equivalently by ⊕ and a left gyration as well. Indeed, it follows from (4.256) and the bi-gyrocommutative law (4.194), p. 142, in (Rn×m , ⊕) that P1 ⊕ P2 = lgyr[P1 , P2 ](P2 ⊕P1 )

(4.258)

P1 ⊕P2 = lgyr[P1 , P2 ](P2 ⊕ P1 )

(4.259)

and hence for all P1 , P2 ∈ Rn×m . Following Def. 4.46 of the bi-gyrogroup binary operation ⊕ in Rn×m , it proves useful to express the bi-gyrations of Rn×m in terms of ⊕ rather than ⊕, resulting in the following theorem. Theorem 4.47. (Bi-gyrogroup Bi-gyrations). The left and right gyrations in a bigyrogroup (Rn×m , ⊕ ) are given by the equations lgyr[P1 , P2 ] =



In + c−2 (P1 ⊕ P2 )(P1 ⊕ P2 )t

−1



c−2 P1 Pt2 +



In + c−2 P1 Pt1



In + c−2 P2 Pt2



rgyr[P1 , P2 ]     −1 −2 t t t −2 −2 Im + c−2 (P2 ⊕ P1 )t (P2 ⊕ P1 ) = c P1 P2 + Im + c P1 P1 Im + c P2 P2 (4.260) for all P1 , P2 ∈ Rn×m . Proof. Noting that rgyr[P1 , P2 ] ∈ S O(m), the first equation in (4.260) follows from (4.257) and the third equation in (4.114), p. 124. Similarly, noting that lgyr[P1 , P2 ] ∈ S O(n), the second equation in (4.260) follows from (4.259) and the fourth equation in (4.114).  Note that the first equation in (4.260) and the third equation in (4.114), p. 124, are identically the same equations with a single exception: the binary operation ⊕ in (4.114) is replaced by the binary operation ⊕ in (4.260). Note also that the order of gyrosummation in the second equation in (4.260) is P2 ⊕ P1 rather than P1 ⊕ P2 . Clearly, the identity element of the groupoid (Rn×m , ⊕ ) is 0n,m , and the inverse  P of P ∈ (Rn×m , ⊕ ) is  P = P = −P, as stated in (4.255), noting that rgyr[P, P] = Im is trivial according to Corollary 4.19, p. 125. Hence, rgyr[ P, P] = Im for all P ∈

155

156

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Rn×m . Theorem 4.48. (Bi-gyrogroup Left and Right Automorphisms). Let (Rn×m , ⊕ ) be a bi-gyrogroup, m, n ∈ N. Then, On (P1 ⊕ P2 ) = On P1 ⊕ On P2 (P1 ⊕ P2 )Om = P1 Om ⊕ P2 Om On (P1 ⊕ P2 )Om = On P1 Om ⊕ On P2 Om

(4.261)

for all P1 , P2 ∈ Rn×m , On ∈ S O(n) and Om ∈ S O(m). Proof. The first identity in (4.261) is proved in the following chain of equations, which are numbered for subsequent explanation. (1)

 On (P1 ⊕ P2 ) === On (P1 ⊕P2 )rgyr[P2 , P1 ] 

(2)

 === (On P1 ⊕On P2 )rgyr[P2 , P1 ] (3)

 === (On P1 ⊕On P2 )rgyr[On P2 , On P1 ]

(4.262)

(4)

 === On P1 ⊕ On P2 . Derivation of the numbered equalities in (4.262): (1) (2) (3) (4)

Follows from Def. 4.46. Follows from the first identity in (4.153), p. 131. Follows from (4.177), p. 137. Follows from Def. 4.46.

The second identity in (4.261) is proved in the following chain of equations, which are numbered for subsequent explanation: (1)

 (P1 ⊕ P2 )Om === (P1 ⊕P2 )rgyr[P2 , P1 ]Om (2)

 === (P1 ⊕P2 )Om rgyr[P2 Om , P1 Om ] (3)

 === (P1 Om ⊕P2 Om )rgyr[P2 Om , P1 Om ] (4)

 === P1 Om ⊕ P2 Om . Derivation of the numbered equalities in (4.263): (1) Follows from Def. 4.46.

(4.263)

Bi-gyrogroups and Bi-gyrovector Spaces – P

(2) Follows from in (4.170), p. 135. (3) Follows from the second identity in (4.153), p. 131. (4) Follows from Def. 4.46. Finally, the third identity in (4.261) follows immediately from the first two identities in (4.261).  The maps On : P → On P, Om : P → POm , and (On , Om ) : P → On POm of Rn×m onto itself are bijective. Hence, by Theorem 4.48, 1. the map On : P → On P is a left automorphism of the bi-gyrogroup (Rn×m , ⊕ ); 2. the map Om : P → POm is a right automorphism of the bi-gyrogroup (Rn×m , ⊕ ); and 3. the map (On , Om ) : P → On POm is a bi-automorphism of the bi-gyrogroup (Rn×m , ⊕ ) (A bi-automorphism being an automorphism consisting of a left and a right automorphism). Theorem 4.49. (Left Cancellation law in (Rn×m , ⊕ )). The bi-gyrogroup (Rn×m , ⊕ ) possesses the left cancellation law  P1 ⊕ (P1 ⊕ P2 ) = P2 ,

(4.264)

for all P1 , P2 ∈ Rn×m . Proof. The proof is provided by the following chain of equations, which are numbered for subsequent explanation: (1)

  P1 ⊕ (P1 ⊕ P2 ) === P1 ⊕ (P1 ⊕P2 )rgyr[P2 , P1 ] (2)

 === (P1 ⊕(P1 ⊕P2 )rgyr[P2 , P1 ])rgyr[(P1 ⊕P2 )rgyr[P2 , P1 ], P1 ] (3)

 === (P1 ⊕(P1 ⊕P2 )rgyr[P2 , P1 ])rgyr[P1 , P2 ] (4)

 === P1 rgyr[P1 , P2 ]⊕(P1 ⊕P2 ) (5)

 === P2 . (4.265) Derivation of the numbered equalities in (4.265): (1) Follows from (4.255) and from Def. 4.46 of ⊕ applied to P1 ⊕ P2 . (2) Follows from Def. 4.46 of ⊕ . (3) Follows from (4.251), p. 153.

157

158

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

(4) Follows from the second identity in (4.153) of Theorem 4.25, p. 131, applied with Om = rgyr[P2 , P1 ], and from the bi-gyration inversion law (4.197), p. 143. (5) Follows from the left cancellation law (4.204), p. 144, in (Rn×m , ⊕).  Clearly, the left cancellation law (4.264) of ⊕ is simpler than the left cancellation law (4.204) of ⊕, so that the passage from ⊕ to ⊕ is rewarding.

4.19. Bi-gyration Decomposition and Polar Decomposition In this section we present manipulations that lead to the bi-gyroassociative law of the binary operation ⊕ in Theorem 4.50, p. 164. The product of two bi-boosts, B(P1 ) and B(P2 ), P1 , P2 ∈ Rn×m , is a Lorentz transformation Λ = B(P1 )B(P2 ) ∈ S O(m, n) that need not be a bi-boost. As such, it possesses the bi-gyration decomposition (4.88), p. 118, as well as the polar decomposition (4.91), p. 119, along with the bi-gyration in (4.106). The bi-gyration decomposition of the bi-boost product gives rise to the binary operation ⊕ in Rn×m as follows. By (4.105) – (4.106), p. 122, the bi-boost product B(P1 )B(P2 ) possesses the unique bi-gyration decomposition B(P1 )B(P2 ) = ρ(rgyr[P1 , P2 ])B(P12 )λ(lgyr[P1 , P2 ])

(4.266)

where, by Def. 4.16, p. 121, P12 =: P1 ⊕P2 .

(4.267)

Similarly, the polar decomposition of the bi-boost product gives rise to the binary operation ⊕ in Rn×m as follows. By (4.91), p. 119, and (4.106), p. 122, the bi-boost product B(P1 )B(P2 ) possesses the unique polar decomposition B(P1 )B(P2 ) = B(P12 )ρ(rgyr[P1 , P2 ])λ(lgyr[P1 , P2 ])

(4.268)

where, by definition, P12 =: P1 ⊕ P2 .

(4.269)

In order to see the relationship between the binary operations ⊕ and ⊕ in Rn×m we employ the second identity in (4.78), p. 115, with Om = rgyr[P1 , P2 ] to manipulate the polar decomposition (4.268) into the equivalent bi-gyration decomposition (4.266), B(P1 )B(P2 ) = B(P12 )ρ(rgyr[P1 , P2 ])λ(lgyr[P1 , P2 ]) = ρ(rgyr[P1 , P2 ])B(P12 rgyr[P1 , P2 ])λ(lgyr[P1 , P2 ]) .

(4.270)

Comparing (4.270) and (4.266), noting that the bi-gyration decomposition is unique by Theorem 4.12, p. 118, we find that P12 rgyr[P1 , P2 ] = P12 , or equivalently by means

Bi-gyrogroups and Bi-gyrovector Spaces – P

of (4.267), (4.269), and the bi-gyration inversion law (4.197), p. 143, P1 ⊕ P2 = (P1 ⊕P2 )rgyr[P2 , P1 ]

(4.271)

P1 ⊕P2 = (P1 ⊕ P2 )rgyr[P1 , P2 ] . Comparing the first equation in (4.271) and (4.256), we have ⊕ = ⊕ .

(4.272) 



It follows from (4.272) that the bi-gyrogroup operation ⊕ = ⊕ in Def. 4.46 stems from the polar decomposition (4.268), just as the bi-gyrogroupoid operation ⊕ stems from the bi-gyration decomposition (4.266). It is convenient here to temporarily use the short notation LP1 ,P2 := lgyr[P1 , P2 ] RP1 ,P2 := rgyr[P1 , P2 ]

(4.273)

in intermediate results, turning back to the full notation in final results. We note −1 that L−1 P1 ,P2 = LP2 ,P1 and RP1 ,P2 = RP2 ,P1 , as we see from the bi-gyration inversion law (4.197), p. 143. Identities (4.268) and (4.271) imply ρ(RP1 ,P2 )λ(LP1 ,P2 ) = B((P1 ⊕P2 )RP2 ,P1 )B(P1 )B(P2 ) .

(4.274)

Identities (4.266) and (4.271) imply, by right gyration inversion, the following chain of equations, which are numbered for subsequent explanation: (1)

 B(P1 ⊕P2 )λ(LP1 ,P2 ) === ρ(RP2 ,P1 )B(P1 )B(P2 ) (2)

 === B(P1 RP1 ,P2 )ρ(RP2 ,P1 )B(P2 )

(4.275)

(3)

 === B(P1 RP1 ,P2 )B(P2 RP1 ,P2 )ρ(RP2 ,P1 ) . Derivation of the numbered equalities in (4.275): (1) This identity is obtained from (4.266) and (4.267) by using the right gyration inversion law in (4.197) according to which ρ(rgyr[P1 , P2 ])−1 = ρ(RP2 ,P1 ). (2) Follows from Item (1) by an application to B(P1 ) of the second identity in (4.78), p. 115, with Om = RP2 ,P1 , noting the right gyration inversion law, RP1 ,P2 RP2 ,P1 = Im . (3) Like Item (2), Item (3) follows from an application to B(P2 ) of the second identity in (4.78), p. 115, with Om = RP2 ,P1 , noting the right gyration inversion law, RP1 ,P2 RP2 ,P1 = Im .

159

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

By means of (4.275) and right gyration inversion we have B(P1 ⊕P2 ) = B(P1 RP1 ,P2 )B(P2 RP1 ,P2 )ρ(RP2 ,P1 )λ(LP2 ,P1 )

(4.276)

so that, by bi-boost inversion, ρ(RP2 ,P1 )λ(LP2 ,P1 ) = B(P2 RP1 ,P2 )B(P1 RP1 ,P2 )B(P1 ⊕P2 ) .

(4.277)

Inverting both sides of (4.277) and noting that the matrices λ(LP1 ,P2 ) and ρ(RP1 ,P2 ) commute, we obtain the identity ρ(RP1 ,P2 )λ(LP1 ,P2 ) = B((P1 ⊕P2 ))B(P1 RP1 ,P2 )B(P2 RP1 ,P2 ) .

(4.278)

The left-hand sides of (4.274) and (4.278) are identical. Hence, we have the chain of equations B((P1 ⊕P2 )RP2 ,P1 )B(P1 )B(P2 ) = B((P1 ⊕P2 ))B(P1 RP1 ,P2 )B(P2 RP1 ,P2 ) = ρ(RP1 ,P2 )λ(LP1 ,P2 ) ,

(4.279)

which, in full notation, takes the form B((P1 ⊕P2 )rgyr[P2 , P1 ])B(P1 )B(P2 ) = B((P1 ⊕P2 ))B(P1 rgyr[P1 , P2 ])B(P2 rgyr[P1 , P2 ])

(4.280)

= ρ(rgyr[P1 , P2 ])λ(lgyr[P1 , P2 ]) . By Def. 4.46, p. 154, and by (4.255), the extreme sides of (4.280) yield the identity ρ(rgyr[P1 , P2 ])λ(lgyr[P1 , P2 ]) = B( (P1 ⊕ P2 ))B(P1 )B(P2 ) ,

(4.281)

so that for all P1 , P2 , X ∈ Rn×m , ρ(rgyr[P1 , P2 ])λ(lgyr[P1 , P2 ])B(X) = B( (P1 ⊕ P2 ))B(P1 )B(P2 )B(X) .

(4.282)

Let J1 (J2 ) denote the left(right)-hand side of (4.282). Using the column notation in (4.89), p. 118, we manipulate the left-hand side, J1 , of (4.282) as follows, where we apply the Lorentz transformation product law (4.180), p. 139, and note Corollary

Bi-gyrogroups and Bi-gyrovector Spaces – P

4.19, p. 125, on trivial bi-gyrations: J1 = ρ(rgyr[P1 , P2 ])λ(lgyr[P1 , P2 ])B(X) ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎞⎛ ⎛ 0n,m 0n,m 0n,m ⎟⎟⎟ ⎜⎜⎜ X ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ X ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎜ ⎟ ⎟⎟⎟ ⎜⎜⎜ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ In ⎟⎟⎟ = ⎜⎜⎜lgyr[P1 , P2 ]⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ In = ⎜⎜⎜⎜lgyr[P1 , P2 ]⎟⎟⎟⎟ ⎜⎜⎜⎜ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ ⎟⎠ ⎜⎝ ⎜⎝ rgyr[P1 , P2 ] Im Im rgyr[P1 , P2 ] Im

(4.283)

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ lgyr[P1 , P2 ]X ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜lgyr[P1 , P2 ]X ⎟⎟⎟ ⎜⎜⎜ A1 ⎟⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ = ⎜⎜⎜⎜⎜ lgyr[0n,m , lgyr[P1 , P2 ]X]lgyr[P1 , P2 ] ⎟⎟⎟⎟⎟ = ⎜⎜⎜⎜⎜ lgyr[P1 , P2 ] ⎟⎟⎟⎟⎟ =: ⎜⎜⎜⎜⎜ B1 ⎟⎟⎟⎟⎟ . ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ C1 rgyr[P1 , P2 ]rgyr[0n,m , lgyr[P1 , P2 ]X] rgyr[P1 , P2 ] Similarly, applying the Lorentz transformation product law (4.180) we manipulate the right-hand side, J2 , of (4.282) as follows: J2 = B( (P1 ⊕ P2 ))B(P1 )B(P2 )B(X) ⎞ ⎞⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎛  ⎜⎜⎜ (P1 ⊕ P2 )⎟⎟⎟ ⎜⎜⎜P1 ⎟⎟⎟ ⎜⎜⎜P2 ⎟⎟⎟ ⎜⎜⎜ X ⎟⎟⎟ ⎜⎜⎜ (P1 ⊕ P2 )⎟⎟⎟ ⎜⎜⎜P1 ⎟⎟⎟ ⎜⎜⎜ P2 ⊕X ⎟⎟⎟ ⎟ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎜⎜ ⎟⎟⎟ ⎜⎜⎜ In ⎟⎟⎟ ⎜⎜⎜ In ⎟⎟⎟ ⎜⎜⎜ In ⎟⎟⎟ = ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ In ⎟⎟⎟ ⎜⎜⎜lgyr[P2 , X]⎟⎟⎟⎟⎟ In I = ⎜⎜⎜⎜⎜ n ⎟⎟⎠ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎜⎝ Im Im Im Im Im Im rgyr[P2 , X]

(4.284)

⎛  ⎞ ⎞⎛ P1 rgyr[P2 , X]⊕(P2 ⊕X) ⎜⎜⎜ (P1 ⊕ P2 )⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ lgyr[P1 rgyr[P2 , X], P2 ⊕X]lgyr[P2 , X] ⎟⎟⎟⎟⎟ . In = ⎜⎜⎜⎜ ⎟⎟⎠ ⎟⎟⎠ ⎜⎜⎝ ⎜⎝ Im rgyr[P2 , X]rgyr[P1 rgyr[P2 , X], P2 ⊕X] In equations (4.285) we adjust each entry of the right column of the extreme righthand side of (4.284) to our needs. By the second equation in (4.153), p. 131, with Om = rgyr[P2 , X], and the right gyration inversion law in (4.197), and by (4.256) – (4.257), we have P1 rgyr[P2 , X]⊕(P2 ⊕X) = {P1 ⊕(P2 ⊕X)rgyr[X, P2 ]}rgyr[P2 , X] = {P1 ⊕(P2 ⊕ X)}rgyr[P2 , X]

(4.285a)

= {P1 ⊕ (P2 ⊕ X)}rgyr[P1 , P2 ⊕ X]rgyr[P2 , X] . By (4.176) with Om = rgyr[X, P2 ], and the right gyration inversion law in (4.197), and by (4.256), we have lgyr[P1 rgyr[P2 , X], P2 ⊕X] = lgyr[P1 , (P2 ⊕X)rgyr[X, P2 ]] = lgyr[P1 , P2 ⊕ X] .

(4.285b)

By (4.170) with Om = rgyr[P2 , X], and the right gyration inversion law in (4.197),

161

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

and by (4.256), we have rgyr[P2 , X]rgyr[P1 rgyr[P2 , X], P2 ⊕X] = rgyr[P1 , (P2 ⊕X)rgyr[X, P2 ]]rgyr[P2 , X] = rgyr[P1 , P2 ⊕ X]rgyr[P2 , X] . (4.285c) By means of the equations in (4.285), the extreme right-hand side of (4.284) can be expressed in terms of ⊕ , rather than ⊕, as ⎞ ⎛ ⎞ ⎞⎛ ⎛  ⎜⎜⎜ (P1 ⊕ P2 )⎟⎟⎟ ⎜⎜⎜{P1 ⊕ (P2 ⊕ X)}rgyr[P1 , P2 ⊕ X]rgyr[P2 , X]⎟⎟⎟ ⎜⎜⎜ A2 ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎜⎜  ⎟⎟⎟ =: ⎜⎜⎜ B2 ⎟⎟⎟ . (4.286) ⎟⎟⎟ ⎜⎜⎜ In lgyr[P J2 = ⎜⎜⎜⎜⎜ 1 , P2 ⊕ X]lgyr[P2 , X] ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ ⎟⎟⎠ ⎜⎜⎝ ⎜⎝ Im C2 rgyr[P1 , P2 ⊕ X]rgyr[P2 , X] We now face the task of calculating A2 , B2 , and C2 by means of the Lorentz product law (4.180). Applying the Lorentz product law to (4.286), we calculate the second entry, B2 , of J2 and simplify it in the following chain of equations, which are numbered for subsequent explanation, and where we use the notation Om := rgyr[P1 , P2 ⊕ X]rgyr[P2 , X] .

(4.287)

(1)

 B2 === lgyr[ (P1 ⊕ P2 )Om , {P1 ⊕ (P2 ⊕ X)}Om ]lgyr[P1 , P2 ⊕ X]lgyr[P2 , X] (2)

 === lgyr[ (P1 ⊕ P2 ), P1 ⊕ (P2 ⊕ X)]lgyr[P1 , P2 ⊕ X]lgyr[P2 , X] . (4.288) Derivation of the numbered equalities in (4.288): (1) This equation is obtained by calculating the Lorentz transformation product in (4.286) by means of (4.180), selecting the resulting second entry, and using the notation in (4.287). (2) Follows from Item (1) by omitting the matrix Om from the two entries of lgyr according to (4.176), p. 137. By (4.282), J1 = J2 and hence, by (4.283) and (4.286), B2 = B1 , that is, by (4.288) and (4.283), lgyr[ (P1 ⊕ P2 ), P1 ⊕ (P2 ⊕ X)]lgyr[P1 , P2 ⊕ X]lgyr[P2 , X] = lgyr[P1 , P2 ]

(4.289)

for all P1 , P2 , X ∈ Rn×m . Similarly, we calculate the third entry, C2 , of J2 and simplify it in the following

Bi-gyrogroups and Bi-gyrovector Spaces – P

chain of equations, which are numbered for subsequent explanation: (1)

 C2 === rgyr[P1 , P2 ⊕ X]rgyr[P2 , X] rgyr[ (P1 ⊕ P2 )rgyr[P1 , P2 ⊕ X]rgyr[P2 , X], {P1 ⊕ (P2 ⊕ X)}rgyr[P1 , P2 ⊕ X]rgyr[P2 , X]] (2)

 === rgyr[P1 , P2 ⊕ X]rgyr[ (P1 ⊕ P2 )rgyr[P1 , P2 ⊕ X], {P1 ⊕ (P2 ⊕ X)}rgyr[P1 , P2 ⊕ X]] rgyr[P2 , X]

(4.290)

(3)

 === rgyr[ (P1 ⊕ P2 ), P1 ⊕ (P2 ⊕ X)]rgyr[P1 , P2 ⊕ X]rgyr[P2 , X] . Derivation of the numbered equalities in (4.290): (1) This equation is obtained by calculating the Lorentz transformation product in (4.286) by means of (4.180), and selecting the resulting third entry. (2) Follows from Item (1) by applying Identity (4.170), p. 135, with Om = rgyr[P2 , X]. (3) Follows from Item (2) by applying Identity (4.170), p. 135, with Om = rgyr[P1 , P2 ⊕ X]. By (4.282), J1 = J2 and hence, by (4.283) and (4.286), C2 = C1 , that is, by (4.290) and (4.283), rgyr[ (P1 ⊕ P2 ), P1 ⊕ (P2 ⊕ X)]rgyr[P1 , P2 ⊕ X]rgyr[P2 , X] = rgyr[P1 , P2 ] (4.291) for all P1 , P2 , X ∈ Rn×m . We are now in a position to calculate the first entry, A2 , of J2 and simplify it in the following chain of equations, which are numbered for subsequent explanation: (1)

 A2 ===  (P1 ⊕ P2 )rgyr[P1 , P2 ⊕ X]rgyr[P2 , X] ⊕{P1 ⊕ (P2 ⊕ X)}rgyr[P1 , P2 ⊕ X]rgyr[P2 , X] (2)

  === { (P1 ⊕ P2 )⊕{P1 ⊕ (P2 ⊕ X)}}rgyr[P1 , P2 ⊕ X]rgyr[P2 , X] (3)

  === { (P1 ⊕ P2 )⊕ {P1 ⊕ (P2 ⊕ X)}}rgyr[ (P1 ⊕ P2 ), P1 ⊕ (P2 ⊕ X)] rgyr[P1 , P2 ⊕ X]rgyr[P2 , X] (4)

  === { (P1 ⊕ P2 )⊕ {P1 ⊕ (P2 ⊕ X)}}rgyr[P1 , P2 ] . Derivation of the numbered equalities in (4.292):

(4.292)

163

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

(1) This equation is obtained by calculating the Lorentz transformation product in (4.286) by means of (4.180), and selecting the resulting first entry. (2) Item (2) is obtained from Item (1) by using the second Identity in (4.153), p. 131, with Om given by Om = rgyr[P1 , P2 ⊕ X]rgyr[P2 , X]. (3) The binary operation ⊕ that appears in Item (2) is expressed here in terms of the binary operation ⊕ by means of (4.257). (4) Item (4) follows from Item (3) by Identity (4.291). By (4.282), J1 = J2 and hence, by (4.283) and (4.286), A2 = A1 , that is, by (4.292) and (4.283), { (P1 ⊕ P2 )⊕ {P1 ⊕ (P2 ⊕ X)}}rgyr[P1 , P2 ] = lgyr[P1 , P2 ]X .

(4.293)

Hence, by right gyration inversion,  (P1 ⊕ P2 )⊕ {P1 ⊕ (P2 ⊕ X)} = lgyr[P1 , P2 ]Xrgyr[P2 , P1 ]

(4.294)

for all P1 , P2 , X ∈ Rn×m . Left gyroadding (P1 ⊕ P2 )⊕ to both sides of (4.294) and applying the left cancellation law (4.264), we obtain the left bi-gyroassociative law, (P1 ⊕ P2 )⊕ lgyr[P1 , P2 ]Xrgyr[P2 , P1 ] = (P1 ⊕ P2 )⊕ { (P1 ⊕ P2 )⊕ {P1 ⊕ (P2 ⊕ X)}}

(4.295)

= P1 ⊕ (P2 ⊕ X) .

4.20. The Bi-gyroassociative Law in Bi-gyrogroups Theorem 4.50. (Bi-gyrogroup Left and Right Bi-gyroassociative Law). The binary operation ⊕ in Rn×m possesses the left bi-gyroassociative law P1 ⊕ (P2 ⊕ X) = (P1 ⊕ P2 )⊕ lgyr[P1 , P2 ]Xrgyr[P2 , P1 ]

(4.296)

and the right bi-gyroassociative law (P1 ⊕ P2 )⊕ X = P1 ⊕ (P2 ⊕ lgyr[P2 , P1 ]Xrgyr[P1 , P2 ])

(4.297)

for all P1 , P2 , X ∈ Rn×m . Proof. The left bi-gyroassociative law (4.296) is proved in (4.295). The right bi-gyroassociative law (4.297) results from an application of the left bi-gyroassociative law to the right-hand side of (4.297), by means of bi-gyration

Bi-gyrogroups and Bi-gyrovector Spaces – P

inversion, P1 ⊕ (P2 ⊕ lgyr[P2 , P1 ]Xrgyr[P1 , P2 ]) = (P1 ⊕ P2 )⊕ lgyr[P1 , P2 ]lgyr[P2 , P1 ]Xrgyr[P1 , P2 ]rgyr[P2 , P1 ]

(4.298)

= (P1 ⊕ P2 )⊕ X . 

4.21. The Bi-gyrocommutative Law in Bi-gyrogroups The bi-gyroassociative law in bi-gyrogroups (Rn×m , ⊕ ) is obtained in Sects. 4.19 – 4.20 by comparing the bi-gyration decomposition and the polar decomposition of the biboost product Λ = B(P1 )B(P2 ). In this section we derive the bi-gyrocommutative law in bi-gyrogroups (Rn×m , ⊕ ) from its counterpart (4.194), p. 142, in bi-gyrogroupoids (Rn×m , ⊕). Theorem 4.51. (Bi-gyrocommutative Law in (Rn×m , ⊕ )). The binary operation ⊕ in Rn×m possesses the bi-gyrocommutative law P1 ⊕ P2 = lgyr[P1 , P2 ](P2 ⊕ P1 )rgyr[P2 , P1 ]

(4.299)

for all P1 , P2 ∈ Rn×m . Proof. By means of (4.257), p. 154, and right gyration inversion (4.197), p. 143, the bi-gyrocommutative law (4.194), p. 142, in (Rn×m , ⊕) can be expressed in terms of ⊕ rather than ⊕, obtaining (P1 ⊕ P2 )rgyr[P1 , P2 ] = lgyr[P1 , P2 ](P2 ⊕ P1 )rgyr[P2 , P1 ]rgyr[P1 , P2 ] = lgyr[P1 , P2 ](P2 ⊕ P1 ) .

(4.300)

Identity (4.299) of the theorem follows immediately from (4.300) by right gyration inversion.  The bi-gyrocommutative law (4.299) in the bi-gyrogroup (Rn×m , ⊕ ) is similar to the bi-gyrocommutative law (4.194), p. 142, in the bi-gyrogroupoid (Rn×m , ⊕). We may, therefore, note that the former involves the right gyration rgyr[P2 , P1 ], while the latter involves the right gyration rgyr[P1 , P2 ]. The following theorem is similar to Theorem 4.20, p. 127, in which the binary operation ⊕ is replaced by the binary operation ⊕ in Rn×m .

165

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Theorem 4.52. Let P1 , P2 , P3 ∈ Rn×m . Then, P3 = P1 ⊕ P2

(4.301)

and lgyr[P1 , P2 ] = In rgyr[P1 , P2 ] = Im

(4.302)

if and only if B(P1 )B(P2 ) = B(P3 ) .

(4.303)

Proof. If P1 , P2 , and P3 satisfy (4.303), then, by Theorem 4.20, p. 127, rgyr[P1 , P2 ] = In , lgyr[P1 , P2 ] = Im , and P3 = P1 ⊕P2 . Hence, by (4.256), p. 154, P3 = P1 ⊕P2 = P1 ⊕ P2 . Conversely, if P1 , P2 , and P3 satisfy (4.301) and (4.302), then, by (4.256), P3 = P1 ⊕ P2 = P1 ⊕P2 . Hence, by Theorem 4.20, B(P1 )B(P2 ) = B(P3 ). 

4.22. Bi-gyrogroup Gyrations The bi-gyroassociative laws (4.296) – (4.297) and the bi-gyrocommutative law (4.299) suggest the definition of gyrations in terms of bi-gyrations, that is, in terms of left and right gyrations. Left gyrations, lgyr[P1 , P2 ] ∈ SO(n), and right gyrations, rgyr[P2 , P1 ] ∈ SO(m), of Rn×m are given by (4.114), p. 124, in terms of ⊕ and by (4.260), p. 155, in terms of ⊕ . They give rise to gyrations of Rn×m according to the following definition. Definition 4.53. (Gyrogroup Gyrations). The gyrator gyr : Rn×m × Rn×m → Aut(Rn×m , ⊕ ) generates automorphisms called gyrations, gyr[P1 , P2 ] ∈ Aut(Rn×m , ⊕ ), given by the equation gyr[P1 , P2 ]X = lgyr[P1 , P2 ]Xrgyr[P2 , P1 ]

(4.304)

for all P1 , P2 , X ∈ Rn×m , m, n ∈ N. The gyration gyr[P1 , P2 ] is said to be the gyration generated by P1 , P2 ∈ Rn×m . Being automorphisms of the bi-gyrogroup (Rn×m , ⊕ ), gyrations are also called gyroautomorphisms. Def. 4.53 is rewarding, leading to the result that any bi-gyrogroup (Rn×m , ⊕ ) is a gyrocommutative gyrogroup, as we will see in Sect. 4.23. Theorem 4.54. (Bi-gyrogroup Gyroassociative and Gyrocommutative Laws). Any

Bi-gyrogroups and Bi-gyrovector Spaces – P

bi-gyrogroup (Rn×m , ⊕ ), m, n ∈ N, possesses the left and the right gyroassociative law P1 ⊕ (P2 ⊕ X) = (P1 ⊕ P2 )⊕ gyr[P1 , P2 ]X

(4.305)

(P1 ⊕ P2 )⊕ X = P1 ⊕ (P2 ⊕ gyr[P2 , P1 ]X)

(4.306)

and

and the gyrocommutative law P1 ⊕ P2 = gyr[P1 , P2 ](P2 ⊕ P1 ) .

(4.307)

Proof. Identities (4.305) – (4.306) follow immediately from Def. 4.53 and the left and right bi-gyroassociative law (4.296) – (4.297). Similarly, (4.307) follows immediately from Def. 4.53 and the bi-gyrocommutative law (4.299).  It is anticipated in Def. 4.53 that gyrations are automorphisms. The following theorem asserts that this is indeed the case. Theorem 4.55. (Gyroautomorphisms). Gyrations gyr[P1 , P2 ] of a bi-gyrogroup n×m  (R , ⊕ ) are automorphisms of the bi-gyrogroup. Proof. It follows from the bi-gyration inversion law in Theorem 4.37, p. 142, and from (4.304) that gyr[P1 , P2 ] is invertible, gyr−1 [P1 , P2 ] = gyr[P2 , P1 ]

(4.308)

for all P1 , P2 ∈ Rn×m . Noting that lgyr[P1 , P2 ] ∈ S O(n) and rgyr[P1 , P2 ] ∈ S O(m) it follows from (4.304) and the third identity in (4.261), p. 156, that gyr[P1 , P2 ](P⊕ Q) = gyr[P1 , P2 ]P⊕ gyr[P1 , P2 ]Q

(4.309)

for all P1 , P2 , P, Q ∈ Rn×m . Hence, by (4.308) and (4.309), gyrations of (Rn×m , ⊕ ) are automorphisms of (Rn×m , ⊕ ), and the proof is complete.  Theorem 4.56. (Left Gyration Reduction Properties). Left gyrations of a bigyrogroup (Rn×m , ⊕ ) possess the left gyration left reduction property lgyr[P1 , P2 ] = lgyr[P1 ⊕ P2 , P2 ]

(4.310)

and the left gyration right reduction property lgyr[P1 , P2 ] = lgyr[P1 , P2 ⊕ P1 ] .

(4.311)

Proof. By (4.225), p. 148, (4.176), p. 137, with Om = rgyr[P2 , P1 ], gyration inversion,

167

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

and (4.256), p. 154, we have the following chain of equations: lgyr[P1 , P2 ] = lgyr[P1 ⊕P2 , P2 rgyr[P1 , P2 ]] = lgyr[(P1 ⊕P2 )rgyr[P2 , P1 ], P2 rgyr[P1 , P2 ]rgyr[P2 , P1 ]] = lgyr[(P1 ⊕P2 )rgyr[P2 , P1 ], P2 ]

(4.312)

= lgyr[P1 ⊕ P2 , P2 ] , thus proving (4.310). By (4.226), p. 148, (4.177), p. 137, with Om = rgyr[P1 , P2 ], gyration inversion, and (4.256), p. 154, we have the following chain of equations: lgyr[P1 , P2 ] = lgyr[P1 rgyr[P2 , P1 ], P2 ⊕P1 ] = lgyr[P1 rgyr[P2 , P1 ]rgyr[P1 , P2 ], (P2 ⊕P1 )rgyr[P1 , P2 ]] = lgyr[P1 , (P2 ⊕P1 )rgyr[P1 , P2 ]]

(4.313)

= lgyr[P1 , P2 ⊕ P1 ] , 

thus proving (4.311).

Theorem 4.57. (Right Gyration Reduction Properties). Right gyrations of a bigyrogroup (Rn×m , ⊕ ) possess the right gyration left reduction property rgyr[P1 , P2 ] = rgyr[P1 ⊕ P2 , P2 ]

(4.314)

and the right gyration right reduction property rgyr[P1 , P2 ] = rgyr[P1 , P2 ⊕ P1 ] .

(4.315)

Proof. By (4.230), p. 149, (4.177), p. 137, with On = lgyr[P1 , P2 ], gyration inversion, and (4.258), p. 155, we have the following chain of equations: rgyr[P1 , P2 ] = rgyr[P2 ⊕P1 , lgyr[P2 , P1 ]P2 ] = rgyr[lgyr[P1 , P2 ](P2 ⊕P1 ), lgyr[P1 , P2 ]lgyr[P2 , P1 ]P2 ] = rgyr[lgyr[P1 , P2 ](P2 ⊕P1 ), P2 ]

(4.316)

= rgyr[P1 ⊕ P2 , P2 ] , thus proving (4.314). By (4.229), p. 149, (4.177), p. 137, with On = lgyr[P2 , P1 ], gyration inversion, and

Bi-gyrogroups and Bi-gyrovector Spaces – P

(4.258), p. 155, we have the following chain of equations: rgyr[P1 , P2 ] = rgyr[lgyr[P1 , P2 ]P1 , P1 ⊕P2 ] = rgyr[lgyr[P2 , P1 ]lgyr[P1 , P2 ]P1 , lgyr[P2 , P1 ](P1 ⊕P2 )] = rgyr[P1 , lgyr[P2 , P1 ](P1 ⊕P2 )]

(4.317)

= rgyr[P1 , P2 ⊕ P1 ] , 

thus proving (4.315).

Theorem 4.58. (Gyration Reduction Properties). The gyrations of any bi-gyrogroup (Rn×m , ⊕ ), m, n ∈ N, possess the left and right reduction property gyr[P1 , P2 ] = gyr[P1 ⊕ P2 , P2 ]

(4.318)

gyr[P1 , P2 ] = gyr[P1 , P2 ⊕ P1 ] .

(4.319)

and

Proof. Identities (4.318) and (4.319) follow from Def. 4.53 of gyr in terms of lgyr and rgyr, and from Theorems 4.56 and 4.57. 

4.23. Bi-gyrogroups are Gyrocommutative Gyrogroups We are now in a position to see that bi-gyrogroups (Rn×m , ⊕ ) are gyrocommutative gyrogroups. Following Defs. 2.13 – 2.14, Theorem 4.59 asserts that any bi-gyrogroup (Rn×m , ⊕ ) is a concrete example of the abstract gyrocommutative gyrogroup. Theorem 4.59. (Bi-gyrogroups Are Gyrocommutative Gyrogroups). Any bi-gyrogroup (Rn×m , ⊕ ), n, m ∈ N, is a gyrocommutative gyrogroup. Proof. We will validate each of the six gyrocommutative gyrogroup axioms (G1)–(G6) in Defs. 2.13 and 2.14. 1. The bi-gyrogroup (Rn×m , ⊕ ) possesses the left identity 0n,m , thus validating Axiom (G1). 2. Every element P ∈ Rn×m possesses the left inverse  P := −P ∈ Rn×m , thus validating Axiom (G2). 3. The binary operation ⊕ obeys the left gyroassociative law (4.305) by Theorem 4.54, thus validating Axiom (G3). 4. The map gyr[P1 , P2 ] is an automorphism of (Rn×m , ⊕ ) by Theorem 4.55, that is, gyr[P1 , P2 ] ∈ Aut(Rn×m , ⊕ ), thus validating Axiom (G4).

169

170

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

5. The binary operation ⊕ in Rn×m possesses the left reduction property (4.318) by Theorem 4.58, thus validating Axiom (G5). 6. The binary operation ⊕ in Rn×m possesses the gyrocommutative law (4.307) by Theorem 4.54, thus validating Axiom (G6). 

4.24. Bi-gyrovector Spaces We introduce scalar multiplication ⊗ into the bi-gyrogroup (Rn×m , ⊕ ), m, n ∈ N, turning it into a bi-gyrovector space (Rn×m , ⊕ , ⊗). Definition 4.60. (Scalar Multiplication, P). Let ⎞ ⎛ 1 t ⎟⎟⎟ ⎜⎜⎜ Im + c−2 Pt P 2P c ⎟⎟⎟ ∈ SO(m, n) B(P) := ⎜⎜⎜⎜⎝  ⎟⎠ −2 t P In + c PP

(4.320)

be a bi-boost of signature (m, n), parametrized by P ∈ Rn×m , m, n ∈ N. The scalar multiplication r⊗P = P⊗r between r ∈ R and P ∈ Rn×m is the unique element r⊗P ∈ Rn×m that satisfies the equation c B(r⊗P) = (B(P))r .

(4.321)

It is anticipated in Def. 4.60 of the scalar multiplication r⊗P in Rn×m that r⊗P is uniquely determined by (4.321). By employing the Singular Value Decomposition (SVD) of P ∈ Rn×m , Theorem 4.61 demonstrates that this is indeed the case and presents two equivalent formulas for r⊗P. The singular value decomposition of P ∈ Rn×m is given by the equation [39, Sect. 7.3]

 Σk 0k,m−k Ot , (4.322) P = On 0n−k,k 0n−k,m−k m where 1. k = rank(P) is the rank of P, k ≤ min{m, n}; 2. Σk is a k × k diagonal matrix of which the diagonal entries are positive; 3. On ∈ O(n) and Om ∈ O(m), where O(n) is the group of all n × n real orthogonal matrices. The diagonal matrix Σk = diag(σ21 , σ22 , . . . , σ2k ) ,

(4.323)

σ21 ≥ σ22 ≥ . . . ≥ σ2k > 0, where σ2i , i = 1, . . . , k are the positive eigenvalues of PPt (or, equivalently, of Pt P), is determined uniquely by P in (4.322). The square roots

Bi-gyrogroups and Bi-gyrovector Spaces – P

σ1 ≥ σ2 ≥ . . . ≥ σk > 0 are the positive singular values of P [39]. Contrasting Σk , the orthogonal matrices On and Om in (4.322) are not determined uniquely by P. Theorem 4.61. (Scalar Multiplication, P). The unique scalar multiplication r⊗P in the space Rn×m , m, n ∈ N, that results from (4.321) is given by each of the following two equivalent equations: √ √ r   r  1   In + c−2 PPt + c−2 PPt − In + c−2 PPt − c−2 PPt r⊗P = 2

 Ik 0k,m−k × On Ot 0n−k,k 0n−k,m−k m (4.324)

 Ik 0k,m−k = On Ot × 0n−k,k 0n−k,m−k m √ √ r   r  1   Im + c−2 Pt P + c−2 Pt P − Im + c−2 Pt P − c−2 Pt P , 2 for any r ∈ R and P ∈ Rn×m , where On ∈ O(n) and Om ∈ O(m) are given by the SVD (4.322) of P. Proof. For the sake of simplicity we assume c = 1. The proof for any c > 0 is similar. Let P be represented by its SVD (4.322),

 0k,m−k Σk Ot , P = On (4.325) 0n−k,k 0n−k,m−k m where On ∈ O(n) and Om ∈ O(m), so that

 0k,n−k Σk t Ot . P = Om 0m−k,k 0m−k,n−k n Then,

PP = On

Σ2k

t

and

0n−k,k

Pt P = Om

Σ2k 0m−k,k

implying √

PPt

= On

Σk 0n−k,k

(4.326)

 0k,n−k Ot 0n−k,n−k n

(4.327)

 0k,m−k Ot , 0m−k,m−k m

(4.328)

 0k,n−k Ot 0n−k,n−k n

(4.329)

171

172

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

and √

Pt P

= Om

 0k,m−k Ot . 0m−k,m−k m

Σk 0m−k,k

(4.330)

By means of (4.327) we have the chain of equations ⎞ ⎫2 ⎛ ⎞2 ⎧ ⎛ ⎪ ⎪ ⎜ ⎟ ⎜ 2 2 ⎪ ⎪ ⎜ ⎟ ⎜ ⎨ ⎜⎜ Ik + Σk 0k,n−k ⎟⎟ t ⎬ ⎜⎜⎜ Ik + Σk 0k,n−k ⎟⎟⎟⎟⎟ t ⎜ ⎟ O O = O ⎪ ⎪ ⎟⎠ n ⎪ ⎟⎠ On n⎜ ⎪ ⎝ ⎩ n ⎜⎝ 0 ⎭ In−k 0n−k,k In−k n−k,k

 Ik + Σ2k 0k,n−k t On = On 0n−k,k In−k

= In + On

Σ2k 0n−k,k

(4.331)

 0k,n−k Ot 0n−k,n−k n

= In + PPt , implying 

⎛ ⎞ ⎜⎜⎜ I + Σ2 0 ⎟⎟⎟ k k,n−k ⎟⎟⎟ Ot . k In + PPt = On ⎜⎜⎜⎝ ⎠ n 0n−k,k In−k

(4.332)

Similarly,

⎛ ⎞ ⎜⎜⎜ I + Σ2 0 ⎟⎟ k,m−k ⎟ ⎟⎟⎟ Ot . k Im + Pt P = Om ⎜⎜⎜⎝ k (4.333) ⎠ m 0m−k,k Im−k √ √ Substituting P and Pt from (4.325) – (4.326) and In + PPt and Im + Pt P from (4.332) – (4.333) into the bi-boost B(P) in (4.320) yields  t 

Om 0m,n Om 0m,n M , (4.334) B(P) = 0n,m On 0n,m Otn 

where

⎛ ⎜⎜⎜ I + Σ2 0 k,m−k ⎜⎜⎜ k k ⎜⎜⎜ 0 I m−k M = ⎜⎜⎜⎜⎜ m−k,k ⎜⎜⎜ Σk 0k,m−k ⎜⎜⎝ 0n−k,k 0n−k,m−k

Σk 0m−k,k Ik + Σ2k 0n−k,k

⎞ 0k,n−k ⎟⎟⎟⎟ ⎟⎟ 0m−k,n−k ⎟⎟⎟⎟⎟ ⎟⎟⎟ . 0k,n−k ⎟⎟⎟⎟ ⎟⎠ In−k

(4.335)

Bi-gyrogroups and Bi-gyrovector Spaces – P

The matrix M, in turn, possesses the decomposition ⎞ ⎛ 0k,k 0k,m−k 0k,n−k ⎟⎟ ⎜⎜⎜ Ik ⎟ ⎜⎜⎜⎜0m−k,k 0m−k,k Im−k 0m−k,n−k ⎟⎟⎟⎟⎟ ⎜ M = ⎜⎜⎜ ⎟M Ik 0k,m−k 0k,n−k ⎟⎟⎟⎟ o ⎜⎜⎝ 0k,k ⎠ 0n−k,k 0n−k,k 0n−k,m−k In−k ⎞ ⎛ 0k,m−k 0k,k 0k,n−k ⎟⎟ ⎜⎜⎜ Ik ⎟ ⎜⎜⎜ 0 0k,m−k Ik 0k,n−k ⎟⎟⎟⎟⎟ × ⎜⎜⎜⎜⎜ k,k ⎟, Im−k 0m−k,k 0m−k,n−k ⎟⎟⎟⎟ ⎜⎜⎝0m−k,k ⎠ 0n−k,k 0n−k,m−k 0n−k,k In−k where

⎛ ⎜⎜⎜ I + Σ2 ⎜⎜⎜ k k ⎜⎜⎜ Mo = ⎜⎜⎜⎜⎜ Σk ⎜⎜⎜ ⎜⎜⎝ 0m−k,k 0n−k,k



Σk

0k,m−k

Ik + Σ2k 0k,m−k 0m−k,k Im−k 0n−k,k 0n−k,m−k

⎞ 0k,n−k ⎟⎟⎟⎟ ⎟⎟⎟ ⎟ 0k,n−k ⎟⎟⎟⎟⎟ . ⎟⎟ 0m−k,n−k ⎟⎟⎟⎟ ⎠ In−k

(4.336)

(4.337)

Substituting (4.335) – (4.337) into (4.334), and omitting obvious indices for clarity, we obtain the equation ⎞ ⎞⎛√ ⎛ 2 0 0⎟⎟⎟ ⎛ ⎞ ⎜⎜⎜⎜ I 0 0 0⎟⎟⎟⎟ ⎜⎜⎜⎜ I + Σ √ Σ ⎟⎟ ⎜⎜Om 0 ⎟⎟⎟⎟ ⎜⎜⎜⎜0 0 I 0⎟⎟⎟⎟ ⎜⎜⎜⎜ Σ I + Σ2 0 0⎟⎟⎟⎟ B(P) = ⎜⎜⎜⎝ ⎟⎟⎟ ⎜⎜⎜ ⎟⎠ ⎜⎜⎜ ⎟ 0 I 0⎟⎟⎟⎟ 0 On ⎜⎜⎝0 I 0 0⎟⎟⎠ ⎜⎜⎜ 0 ⎝ ⎠ 0 0 0 I 0 0 0 I (4.338) ⎞ ⎛ ⎜⎜⎜ I 0 0 0⎟⎟⎟ ⎛ ⎞ ⎜⎜⎜0 0 I 0⎟⎟⎟ ⎜⎜Ot 0 ⎟⎟⎟ m ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟⎟ . × ⎜⎜⎜ ⎟⎜ ⎜⎝⎜0 I 0 0⎟⎟⎟⎠⎟ ⎝ 0 Otn ⎠ 0 0 0 I It is clear from (4.338) that for any r ∈ R, ⎞r ⎞ ⎞ ⎛⎛ √ ⎛ I 0 0 0 0 0 ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜⎜⎜⎜ I + Σ2 √ Σ ⎟⎟⎟ ⎜ ⎞ ⎜⎜⎜ ⎛ ⎜ ⎟ ⎟ ⎠ ⎜⎜Om 0 ⎟⎟⎟ ⎜⎜⎜0 0 I 0⎟⎟⎟⎟ ⎜⎜⎜⎜⎝ Σ 0 0 ⎟⎟⎟⎟ I + Σ2 ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎠ ⎜⎜⎜ ⎟⎟  

(B(P))r = ⎜⎜⎜⎝ I 0 ⎟⎟⎟⎟ 0 0 0 On ⎜⎜⎜⎝0 I 0 0⎟⎟⎟⎠ ⎜⎜⎜⎜ ⎝ ⎠ 0 0 0 I 0 I 0 0 ⎛ ⎜⎜⎜ I ⎜⎜⎜0 × ⎜⎜⎜⎜⎜ ⎜⎜⎝0 0

⎞ 0 0 0⎟⎟ ⎛ ⎞ ⎟ t 0 I 0⎟⎟⎟⎟⎟ ⎜⎜⎜⎜Om 0 ⎟⎟⎟⎟ ⎟⎜ ⎟. I 0 0⎟⎟⎟⎟ ⎝ 0 Otn ⎠ ⎠ 0 0 I

(4.339)

173

174

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Let Λ+ = Λ− = Then,

and, hence,

⎛√ ⎜⎜⎜ I + Σ2 ⎝⎜ Σ ⎛√ ⎜⎜⎜ I + Σ2 ⎜⎝ Σ

√ √

I + Σ2 + Σ I+

Σ2

(4.340)

− Σ.

⎞   

⎟⎟⎟ 1 I I Λ+ 0 I I Σ √ ⎟⎠ = 2 I −I 0 Λ− I −I I + Σ2

(4.341)

⎞r r 

 ⎟⎟⎟ 1 I I Λ+ 0 I I Σ √ ⎟⎠ = 2 I −I 0 Λ− I −I I + Σ2   

1 I I Λr+ 0 I I 2 I −I 0 Λr− I −I ⎛1 r ⎞ ⎜⎜⎜ 2 (Λ+ + Λr− ) 12 (Λr+ − Λr− )⎟⎟⎟ ⎟⎟ = ⎜⎜⎝ 1 r 1 r r r ⎠ (Λ − Λ ) (Λ + Λ ) + − + − 2 2 =

for all r ∈ R. Substituting (4.342) into (4.339) yields ⎛1 r ⎞ ⎛ ⎛1 r ⎜⎜ (Λ+ − Λr− ) ⎜⎜⎜ ⎜⎜⎜ 2 (Λ+ + Λr− ) 0⎟⎟⎟ t ⎜ ⎟ ⎜⎜⎜Om ⎜⎝ ⎟⎠ Om Om ⎜⎜⎜⎝ 2 ⎜⎜ 0 I 0 ⎛1 r ⎞ (B(P))r = ⎜⎜⎜⎜⎜ ⎛ 1 r r ⎜ (Λ + Λr− ) ⎜⎜⎜ ⎜⎜⎜ 2 (Λ+ − Λ− ) 0⎟⎟⎟ t ⎟⎟⎠ Om On ⎜⎜⎜⎜⎝ 2 + ⎜⎝ On ⎜⎜⎝ 0 0 0

(4.342)

⎞ ⎞ 0⎟⎟⎟ t ⎟⎟ ⎟⎟⎠ On ⎟⎟⎟  ⎟⎟⎟ t 0 A Q ⎟ ⎞ ⎟⎟ =: . Q C 0⎟⎟⎟ t ⎟⎟⎟⎟⎟ ⎟⎟⎠ On ⎟⎠ I (4.343)

Clearly Λ+ Λ− = I

(4.344)

( 12 (Λ+ + Λ− ))2 − ( 12 (Λ+ − Λ− ))2 = I .

(4.345)

and, hence,

Following (4.345) and the definition of A, C, and Q in (4.343), we have A2 = I + Qt Q C 2 = I + QQt

(4.346)

Bi-gyrogroups and Bi-gyrovector Spaces – P

Hence, by means of (4.343) and (4.346), ⎛√ ⎜⎜ I + Qt Q r (B(P)) = ⎜⎜⎜⎝ Q

√

Qt I + QQt

⎞ ⎟⎟⎟ ⎟⎟⎠ .

(4.347)

Comparing (4.347) and (4.320) – (4.321) (c=1 for simplicity), we see that r⊗P is given uniquely by r⊗P = Q .

(4.348)

We, therefore, wish to express Q in terms of P. By the definition of Q in (4.343) we have ⎛1 r ⎞ ⎜⎜⎜ 2 (Λ+ − Λr− ) 0⎟⎟⎟ t ⎟⎟⎠ Om . Q = On ⎜⎜⎝ 0 0

(4.349)

Following (4.329), (4.332), and (4.340) we have 

√

 √ √ Σ 0 t I + Σ2 0 t t t O + On O I + PP + PP = On 0 0 n 0 I n (4.350)

 Λ+ 0 t O = On 0 I n and, similarly, √

I+

PPt

−

√

PPt

 Λ− 0 t O . = On 0 I n

Hence, √

( I+ √

PPt

+

( I + PPt − so that

√ √

(4.351)

 Λr+ 0 t O = On 0 I n

PPt )r

 Λr− 0 t r t O PP ) = On 0 I n

(4.352)

⎛1 r ⎞ r  √  ⎜ ⎟⎟⎟ √ √ √ (Λ − Λ ) 0 ⎜ + − 2 1 t+ t )r − ( I + PPt − t )r = O ⎜ ⎜⎜⎝ ⎟⎟⎠ Otn . I + PP PP PP ( n 2 0 I (4.353)

175

176

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Hence, by (4.353) and (4.349),   √  √ √ √ 1 t+ t )r − ( I + PPt − t )r O I 0 Ot I + PP PP PP ( n 2 0 0 m ⎛1 r ⎞  ⎜⎜⎜ 2 (Λ+ − Λr− ) 0⎟⎟⎟ I 0 t ⎜ ⎟ Om = On ⎜⎝ ⎟⎠ 0 I 0 0

(4.354)

⎛1 r ⎞ ⎜⎜⎜ 2 (Λ+ − Λr− ) 0⎟⎟⎟ t ⎟⎟⎠ Om = Q . = On ⎜⎜⎝ 0 0 Following (4.354) we have, by means of (4.348),  √  I 0 √ √ √ 1 r r t t t t Ot , (4.355) r⊗P = Q = 2 ( I + PP + PP ) − ( I + PP − PP ) On 0 0 m thus verifying the first equation for r⊗P in (4.324). It remains to prove the second equation for r⊗P in (4.324). Following (4.330), (4.333), and (4.340) we have 

√

 √ √ Σ 0 t I + Σ2 0 t t t O + Om O I + P P + P P = Om 0 0 m 0 I m

= Om

(4.356)



Λ+ 0 t O 0 I m

and, similarly, √

I+

Pt P

−

√

Pt P

= Om

 Λ− 0 t O . 0 I m

Hence, √

( I+ √

Pt P

+

( I + Pt P −

√ √

(4.357)

 Λr+ 0 t O 0 I m

Pt P)r

= Om

Pt P) = Om r

 Λr− 0 t O 0 I m

(4.358)

so that ⎛1 r ⎞  √  ⎜⎜⎜ 2 (Λ+ − Λr− ) 0⎟⎟⎟ √ √ √ 1 r r t t t t ⎟⎟⎠ Otm . (4.359) P P) − I + P P − P P) = Om ⎜⎜⎝ 2 ( I+PP+ 0 I

Bi-gyrogroups and Bi-gyrovector Spaces – P

Hence, by (4.359) and (4.349), 

 √ √ √ I 0 t  √ 1 t O Pt P)r − ( I + Pt P − Pt P)r 2 n 0 0 Om ( I + P P + ⎞  ⎛1 r r I 0 ⎜⎜⎜⎜ 2 (Λ+ − Λ− ) 0⎟⎟⎟⎟ t = On ⎜ ⎟⎠ Om 0 0 ⎝ 0 I

(4.360)

⎛1 r ⎞ ⎜⎜⎜ 2 (Λ+ − Λr− ) 0⎟⎟⎟ t ⎟⎟⎠ Om = Q . = On ⎜⎜⎝ 0 0 Following (4.360) we have, by means of (4.348),

  √ √ √ I 0 t  √ 1 Om ( I + Pt P + Pt P)r − ( I + Pt P − Pt P)r , (4.361) r⊗P = Q = 2 On 0 0 thus verifying the second equation for r⊗P in (4.324). The proof of the theorem is thus complete. 

4.25. On the Pseudo-inverse of a Matrix In this section we explore the presence of a Moore-Penrose pseudo-inverse matrix in each of the two r⊗P formulas (4.324). √ It√follows from (4.329) that the Moore-Penrose pseudo-inverse ( PPt )# of the matrix PPt is given by ⎛ −1 ⎞ √ ⎜⎜⎜ Σk 0k,n−k ⎟⎟⎟ t # t ⎜ ⎟⎟⎠ On , (4.362) ( PP ) = On ⎜⎝ 0n−k,k 0n−k,n−k where, following (4.323), −2 −2 −2 Σ−1 k = diag(σ1 , σ2 , . . . , σk ) .

(4.363)

Hence, by means of (4.362) and (4.322), ⎛ −1 ⎞⎛ ⎞ √ ⎜⎜⎜ Σk 0k,n−k ⎟⎟⎟ ⎜⎜⎜ Σk 0k,m−k ⎟⎟⎟ t # ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ Om ( PPt ) P = On ⎜⎜⎝ 0n−k,k 0n−k,n−k 0n−k,k 0n−k,m−k ⎛ ⎞ ⎜⎜⎜ Ik 0k,m−k ⎟⎟⎟ t ⎟⎟⎠ Om . = On ⎜⎜⎝ 0n−k,k 0n−k,m−k Similarly, P(

√

Pt P)#

⎛ ⎞ ⎜⎜⎜ Ik 0k,m−k ⎟⎟⎟ t ⎟⎟⎠ Om , = On ⎜⎜⎝ 0n−k,k 0n−k,m−k

(4.364)

(4.365)

177

178

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

so that

√ √ ( PPt )# P = P( Pt P)# .

(4.366)

The right-hand side of each of (4.364) and (4.365) is a factor that appears in each of the two r⊗P formulas (4.324). Hence, by means of (4.364) and (4.365), the two r⊗P formulas (4.324) can be written as (where c > 0 reappears) √ √ r   r  1   −2 t −2 t −2 t −2 t In + c PP + c PP − In + c PP − c PP r⊗P = 2 √ × ( PPt )# P (4.367) √ = P( Pt P)# × √ √ r   r  1   Im + c−2 Pt P + c−2 Pt P − Im + c−2 Pt P − c−2 Pt P . 2 If the matrix PPt is invertible then √ √ In ( PPt )# P = ( PPt )−1 P = √ P PPt

(4.368)

and, similarly, if the matrix Pt P is invertible then √ √ Im P( Pt P)# = P( Pt P)−1 = P √ . (4.369) Pt P Hence, when matrix inversion is justified, the two r⊗P formulas (4.324) can be written as r   r  √ √ −2 t −2 t In + c−2 PPt − c−2 PPt 1 In + c PP + c PP − r⊗P := P √ 2 c−2 PPt r   r (4.370)  √ √ −2 t −2 t −2 t −2 t Im + c P P + c P P − Im + c P P − c P P 1 = P √ 2 c−2 Pt P for all r ∈ R and P ∈ Rn×m . Interestingly, we have the limit lim r⊗P = rP

c→∞

(4.371)

for all r ∈ R and P ∈ Rn×m , as one can check by applying L’Hospital’s rule. Example 4.62. In the special case when m = 1, P ∈ Rn×1 = Rn is a column vector, and

Bi-gyrogroups and Bi-gyrovector Spaces – P

Pt P = P 2 , so that (4.370) yields the scalar multiplication in Rn , ⎞r ⎛  ⎞r ⎫ ⎧⎛  ⎪ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎪ ⎪ 2 2 ⎪ ⎪ P P ⎟⎟ ⎜⎜ P P ⎟⎟⎟⎟ ⎪ c ⎨⎜⎜ ⎬ P ⎜ ⎟ ⎟ ⎜ 1 + + − 1 + − r⊗P = ⎪ ⎪ ⎜ ⎟ ⎟ ⎜ ⎪ ⎜ ⎟ ⎟ ⎜ 2 2 ⎪ ⎪ ⎝ ⎠ ⎠ ⎝ 2⎩ c c ⎪ c c ⎭ P

(4.372)

for all r ∈ R and P ∈ Rn , P  0. Clearly, r⊗0 = 0. The gyrovector space scalar multiplication (4.372) is studied, for instance, in [84, Sect. 6.18].

4.26. Properties of Bi-gyrovector Space Scalar Multiplication The triple (Rn×m , ⊕ , ⊗), where ⊕ is the binary operation in Rn×m given by (4.256), p. 154, and where ⊗ is the scalar multiplication in Rn×m given by (4.324), p. 171, is the bi-gyrovector space of the space Rn×m of the bi-boost parameter P. Theorem 4.63. (Bi-gyrovector Space Properties). Let (Rn×m , ⊕ , ⊗) be the gyrovector space of a space Rn×m , m, n ∈ N. Then, its scalar multiplication ⊗ obeys the scalar distributive law (r1 + r2 )⊗P = r1 ⊗P⊕ r2 ⊗P ,

(4.373)

(r1 r2 )⊗P = r1 ⊗(r2 ⊗P) ,

(4.374)

r⊗(r1 ⊗P⊕ r2 ⊗P) = r⊗(r1 ⊗P)⊕ r⊗(r2 ⊗P),

(4.375)

the scalar associative law

the monodistributive law

and the scalar-matrix transpose law (r⊗P)t = r⊗Pt

(4.376)

for all r, r1 , r2 ∈ R and P ∈ Rn×m . Proof. By means of (4.321) and Theorem 4.20, p. 127, B((r1 + r2 )⊗P) = (B(P))r1 +r2 = (B(P))r1 (B(P))r2 = B(r1 ⊗P)B(r2 ⊗P) = B(r1 ⊗P⊕r2 ⊗P)

(4.377)

179

180

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

and rgyr[r1 ⊗P, r2 ⊗P] = In .

(4.378)

It follows from (4.378) and (4.257), p. 154, that r1 ⊗P⊕r2 ⊗P = r1 ⊗P⊕ r2 ⊗P ,

(4.379)

so that (4.377) and (4.379) imply B((r1 + r2 )⊗P) = B(r1 ⊗P⊕ r2 ⊗P) .

(4.380)

Owing to the bijective correspondence between bi-boosts B(P) ∈ SO(m, n) and their parameter P ∈ Rn×m , (4.373) follows from (4.380). By means of (4.321), B((r1 r2 )⊗P) = (B(P))r1 r2 = ((B(P))r2 )r1 = (B(r2 ⊗P))r1

(4.381)

= B(r1 ⊗(r2 ⊗P)) . Owing to the bijective correspondence between bi-boosts B(P) ∈ SO(m, n) and their parameter P ∈ Rn×m , (4.374) follows from (4.381). The proof of (4.375) follows from the scalar distributive law (4.373) and the scalar associative law (4.374), as shown in the chain of equations below. r⊗(r1 ⊗P⊕ r2 ⊗P) = r⊗((r1 + r2 )⊗P) = (r(r1 + r2 ))⊗P = (rr1 + rr2 )⊗P

(4.382)

= (rr1 )⊗P⊕ (rr2 )⊗P = r⊗(r1 ⊗P)⊕ r⊗(r2 ⊗P) . Finally, the scalar-matrix transpose law (4.376) follows immediately from the two equivalent formulas for r⊗P in (4.324), p. 171. 

Bi-gyrogroups and Bi-gyrovector Spaces – P

Theorem 4.64. (A Trivial Bi-gyration). Let (Rn×m , ⊕ , ⊗) be the gyrovector space of a space Rn×m , m, n ∈ N. Then, lgyr[r1 ⊗P, r2 ⊗P] = In rgyr[r1 ⊗P, r2 ⊗P] = Im

(4.383)

for all r1 , r2 ∈ R and P ∈ Rn×m . Proof. By (4.377) we have B(r1 ⊗P)B(r2 ⊗P) = B((r1 + r2 )⊗P) , from which (4.383) follows by means of Theorem 4.20, p. 127.

(4.384) 

4.27. A Commuting Relation by SVD Having seen the usefulness of matrix singular value decomposition (SVD) in Theorem 4.61, p. 171, it is instructive to prove the commuting relation (4.54), p. 111, by means of SVD in the following lemma. Lemma 4.65.

  P Im + c−2 Pt P = In + c−2 PPt P

(4.385)

for all P ∈ Rn×m , m, n ∈ N. Proof. Without loss of generality we assume c = 1. Let P be represented by its SVD (4.322) – (4.323), p. 170. Then, by (4.332) and (4.322), ⎛ ⎞

 ⎜⎜⎜ I + Σ2 0 ⎟⎟  Σk 0k,m−k t k k,n−k ⎟ ⎜ ⎟ t k ⎜ ⎟ Otm In + PP P = On ⎜⎝ ⎟⎠ On On 0 0 n−k,k n−k,m−k 0n−k,k In−k ⎛ ⎞  ⎜⎜⎜ I + Σ2 0 ⎟⎟⎟ Σ 0 k k,m−k k k,n−k ⎜ ⎟ k ⎟⎟⎠ Otm = On ⎜⎝⎜ 0 0 n−k,k n−k,m−k 0n−k,k In−k ⎛ ⎞ ⎜⎜⎜ I + Σ2 Σ ⎟⎟ 0 k,m−k ⎟ ⎟⎟⎟ Ot k k = On ⎜⎜⎜⎝ k ⎠ m 0n−k,k 0n−k,m−k

(4.386)

181

182

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

and



P Im + Pt P = On

0n−k,k

= On

Σk

Σk 0n−k,k

⎛ ⎞  ⎜⎜⎜ I + Σ2 0 ⎟⎟ 0k,m−k k,m−k ⎟ ⎟⎟⎟ Ot k Otm Om ⎜⎜⎜⎝ k ⎠ m 0n−k,m−k 0m−k,k Im−k 0k,m−k 0n−k,m−k

⎞  ⎛⎜  ⎜⎜⎜ Ik + Σ2 0k,m−k ⎟⎟⎟⎟ t k ⎟⎟⎠ Om ⎜⎜⎝ 0m−k,k Im−k

(4.387)

⎛  ⎞ ⎜⎜⎜Σ I + Σ2 0 ⎟⎟ k,m−k ⎟ ⎟⎟⎟ Ot . k = On ⎜⎜⎜⎝ k k ⎠ m 0n−k,k 0n−k,m−k The matrix Σk is diagonal, so that   Σk Ik + Σ2k = Ik + Σ2k Σk . Finally, (4.386) – (4.388) imply (4.385), as desired.

(4.388) 

4.28. Einstein Bi-gyrogroups and Bi-gyrovector Spaces – P The main goal of this section is to summarize the introduction of two algebraic objects, the bi-gyrogroup and the bi-gyrovector space. These objects form a natural generalization of the concepts of the gyrogroups and the gyrovector spaces studied in Chaps. 2 and 3, to which they descend when m = 1. In the following two subsections we summarize the bi-gyrogroup and the bi-gyrovector space that underlie the space Rn×m of all n × m real matrices P ∈ Rn×m , m, n ∈ N.

4.28.1. Einstein Bi-gyrogroups Let Rn×m be the set of all n × m real matrices, m, n ∈ N, and let ⊕E := ⊕ be the Einstein addition of signature (m, n) in Rn×m , given by (4.256), p. 154. The resulting pair (Rn×m , ⊕E ) is the Einstein bi-gyrogroup of signature (m, n) that underlies the space Rn×m . Einstein bi-gyrogroups BE = (Rn×m , ⊕E ) are regulated by gyrations, possessing the following properties: 1. The binary operation ⊕E := ⊕ in Rn×m is Einstein addition of signature (m, n), given by (4.256), p. 154. It descends to the common Einstein addition of proper velocities in special relativity theory when m = 1 (one temporal dimension) and n = 3 (three spatial dimensions). 2. BE possesses the unique identity element 0n,m . 3. Every element P ∈ Rn×m possesses a unique inverse, E P = −P. 4. Any two elements P1 , P2 ∈ Rn×m determine in (4.135), p. 128,

Bi-gyrogroups and Bi-gyrovector Spaces – P

a. a left gyration lgyr[P1 , P2 ] ∈ SO(n) and b. a right gyration rgyr[P1 , P2 ] ∈ SO(m). A left and a right gyration, in turn, determine a gyration, gyr[P1 , P2 ] : Rn×m → Rn×m , according to (4.304), p. 166, gyr[P1 , P2 ]P3 = lgyr[P1 , P2 ]P3 rgyr[P2 , P1 ]

(4.389)

for all P1 , P2 , P3 ∈ R . 5. Left and right gyrations are automorphisms of BE . 6. Left and right gyrations obey the gyration inversion law (4.197), p. 143, n×m

lgyr−1 [P1 , P2 ] = lgyr[P2 , P1 ] rgyr−1 [P1 , P2 ] = rgyr[P2 , P1 ] ,

(4.390)

for all P1 , P2 ∈ Rn×m . 7. Left and right gyrations possess the reduction properties in Theorem 4.56, p. 167, lgyr[P1 , P2 ] = lgyr[P1 ⊕E P2 , P2 ] lgyr[P1 , P2 ] = lgyr[P1 , P2 ⊕E P1 ]

(4.391)

and in Theorem 4.57, p. 168, rgyr[P1 , P2 ] = rgyr[P1 ⊕E P2 , P2 ] rgyr[P1 , P2 ] = rgyr[P1 , P2 ⊕E P1 ] .

(4.392)

8. Einstein addition ⊕E in Rn×m obeys the left and the right bi-gyroassociative laws (4.305)–(4.306), p. 167, P1 ⊕E (P2 ⊕E P3 ) = (P1 ⊕E P2 )⊕E gyr[P1 , P2 ]P3 (P1 ⊕E P2 )⊕E P3 = P1 ⊕E (P2 ⊕E gyr[P2 , P1 ]P3 )

(4.393)

and the bi-gyrocommutative law (4.307), p. 167, P1 ⊕E P2 = gyr[P1 , P2 ](P2 ⊕E P1 )

(4.394)

for all P1 , P2 , P3 ∈ Rn×m . By Theorem 4.59, p. 169, Einstein bi-gyrogroups are gyrocommutative gyrogroups.

4.28.2. Einstein Bi-gyrovector Spaces Introducing a scalar multiplication, ⊗, into Einstein bi-gyrogroups BE = (Rn×m , ⊕E ), m, n ∈ N, yields the Einstein gyrovector spaces VE = (Rn×m , ⊕E , ⊗), which possess the following properties:

183

184

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

1. The scalar multiplication ⊗ in an Einstein bi-gyrovector space VE = (Rn×m , ⊕E , ⊗) is a multiplication r⊗P ∈ Rn×m between real numbers r ∈ R and elements P ∈ (Rn×m , ⊕E ) of the Einstein bi-gyrogroup, given by Theorem 4.61, p. 171. 2. The scalar multiplication ⊗ obeys the scalar distributive law (r1 + r2 )⊗P = r1 ⊗P⊕E r2 ⊗P ,

(4.395)

(r1 r2 )⊗P = r1 ⊗(r2 ⊗P) ,

(4.396)

r⊗(r1 ⊗P⊕E r2 ⊕P) = r⊗(r1 ⊗P)⊕E r⊗(r2 ⊗P),

(4.397)

the scalar associative law

the monodistributive law

and the scalar matrix transpose law (r⊗P)t = r⊗Pt

(4.398)

for all r1 , r2 , r ∈ R and P ∈ Rn×m . 3. Left and right trivial gyrations with scalar multiplications, (4.383), p. 181, are lgyr[r1 ⊗P, r2 ⊗P] = In rgyr[r1 ⊗P, r2 ⊗P] = Im

(4.399)

for all r1 , r2 ∈ R and P ∈ Rn×m . 4. Scalar multiplication respects orthogonal transformations r⊗(On POm ) = On (r⊗P)Om for all P ∈ R

n×m

, Om ∈ SO(m), On ∈ SO(n), and r ∈ R.

(4.400)

CHAPTER 5

Bi-gyrogroups and Bi-gyrovector Spaces – V

5.1. Introduction This chapter results from Chap. 4 by a change of parameter, changing the parameter of the ambiP in the space Rn×m into a new parameter V in the spectral c-ball Rn×m c ent space Rn×m . In Chap. 4 we have realized parametrically the Lorentz transformations Λ ∈ SO(m, n) in pseudo-Euclidean spaces Rm,n of signature (m, n), m, n ∈ N. Every Lorentz transformation Λ ∈ SO(m, n) possesses the parametric realization (4.89), p. 118, ⎛ ⎞ ⎜⎜⎜ P ⎟⎟⎟ ⎜ ⎟ (5.1) Λ = Λ(Om , P, On ) = ρ(Om )B(P)λ(On) = ⎜⎜⎜⎜ On ⎟⎟⎟⎟ , ⎝ ⎠ Om in terms of the main parameter P ∈ Rn×m and the two orientation parameters Om ∈ SO(m) and On ∈ SO(n). We have seen in Chap. 4 that the group structure of SO(m, n) gives rise to the bigyrogroup structure (Rn×m , ⊕ ) of the parameter space Rn×m . The binary operation ⊕ in Rn×m turns out to be both bi-gyrocommutative and bi-gyroassociative, justifying calling it a bi-gyroaddition. Finally, in Chap. 4 we have introduced a scalar multiplication ⊗ into bi-gyrogroups, turning them into bi-gyrovector spaces (Rn×m , ⊕ , ⊗). Accordingly, Chap. 4 is about the P-parametric realization of Lorentz transformations in pseudo-Euclidean spaces of signature (m, n), m, n ∈ N. Furthermore, in the special case when m = 1, the binary operation ⊕ in Rn×1 = Rn descends to the special relativistic addition of proper velocities, as indicated in Example 5.33. In this chapter we change the parameter P whose domain is the space Rn×m into a of the ambient space Rn×m . Accordnew parameter, V, whose domain is the ball Rn×m c ingly, this chapter is about the V-parametric realization of Lorentz transformations in pseudo-Euclidean spaces of any signature (m, n). In the special case when m = 1, the = Rnc of the ambient space Rn descends to the binary operation ⊕ in the ball Rn×1 c Beyond Pseudo-rotations in Pseudo-Euclidean Spaces http://dx.doi.org/10.1016/B978-0-12-811773-6.50005-0 Copyright © 2018 Elsevier Inc. All rights reserved.

185

186

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

common special relativistic Einstein addition of coordinate velocities, as indicated in Example 5.32 and explained in Sect. 5.17. The goal of this chapter is to reveal the bi-gyrogroup and the bi-gyrovector space structures that underlie the ball Rn×m c , which are finally presented in Sect. 5.35.

5.2. Bi-boost Parameter Change, P → V Definition 5.1. For any m, n ∈ N let φ : Rn×m → Rn×m be the map given by  −1 φ : P → V = In + c−2 PPt P,

(5.2)

where c > 0 is an arbitrarily fixed positive constant. The image Bn×m := φ(Rn×m ) ⊂ Rn×m

(5.3)

of φ in Rn×m is the spectral c-ball (or c-ball, in short) of Rn×m . The term “spectral c-ball” will be justified by Theorem 5.10 following the introduction of the spectral norm in (5.69). Furthermore, we will see in Example 5.26 that of Rn×1 descends to the Euclidean in the special case when m = 1, the c-ball Rn×m c n n c-ball Rc of the Euclidean n-space R . Clearly, φ maps the zero element of Rn×m into the zero element of Rn×m c , and it maps −P into −φ(P), φ(0n,m ) = 0n,m φ(−P) = −φ(P) ,

(5.4)

for all P ∈ Rn×m . Lemma 5.2. The map φ in Def. 5.1 can be written equivalently as  −1 φ : P → V = P Im + c−2 Pt P .

(5.5)

Proof. The proof follows immediately from (5.2) and from the commuting relation   P Im + c−2 Pt P = In + c−2 PPt P (5.6) for all P ∈ Rn×m , proved in (4.54), p. 111.



Lemma 5.3. φ(Pt ) = φ(P)t for all P ∈ Rn×m .

(5.7)

Bi-gyrogroups and Bi-gyrovector Spaces – V

Proof. By (5.2), φ(Pt ) = for any Pt ∈ Rm×n , and by (5.5) φ(P)t =





Im + c−2 Pt P

Im + c−2 Pt P

−1

−1

Pt

(5.8)

Pt

(5.9) 

for any P ∈ Rn×m , implying (5.7) as desired. Theorem 5.4. (Parameter Change). For any P ∈ Rn×m ,   −1 −1 V = In + c−2 PPt P = P Im + c−2 Pt P

(5.10)

if and only if P=



In − c−2 VV t

−1

 V = V Im − c−2 V t V

−1

.

(5.11)

Proof. The proof is divided into four parts. In Parts IA and IB we prove that (5.10) implies (5.11), and in Parts IIA and IIB we prove that (5.11) implies (5.10). Part I A : Assuming (5.10), we have  P = In + c−2 PPt V (5.12) and the commuting relation  Pt In + c−2 PPt

−1

=



Im + c−2 Pt P

−1

Pt

so that by (5.13) and (5.10)    −1 −1 PPt In + c−2 PPt = P Im + c−2 Pt P Pt = In + c−2 PPt

(5.13) −1

PPt .

(5.14)

Then, by (5.10) and (5.14), VV t =



In + c−2 PPt

−1

 PPt In + c−2 PPt

= PPt (In + c−2 PPt )−1

−1

(5.15)

= (In + c−2 PPt )−1 PPt . Hence, PPt = (In + c−2 PPt )VV t

(5.16)

PPt = VV t (In + c−2 PPt ) .

(5.17)

and

187

188

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

A rearrangement of (5.16) yields VV t = PPt (In − c−2 VV t ),

(5.18)

PPt = VV t (In − c−2 VV t )−1 .

(5.19)

implying

Similarly, a rearrangement of (5.17) yields VV t = (In − c−2 VV t )PPt ,

(5.20)

PPt = (In − c−2 VV t )−1 VV t .

(5.21)

implying

Following (5.19) we have In + c−2 PPt = In + c−2 VV t (In − c−2 VV t )−1 = (In − c−2 VV t )(In − c−2 VV t )−1 + c−2 VV t (In − c−2 VV t )−1 = (In − c−2 VV t + c−2 VV t )(In − c−2 VV t )−1

(5.22)

= (In − c−2 VV t )−1 so that



In + c−2 PPt =



In − c−2 VV t

−1

Hence, by (5.12) and (5.23),   P = In + c−2 PPt V = In − c−2 VV t

. −1

(5.23)

V,

(5.24)

thus validating the first equation in (5.11). In Part IB of the proof we validate the second equation in (5.11), Part I B : Assuming (5.10), we have  P = V Im + c−2 Pt P (5.25) and the commuting relation, as in (5.13),   −1 −1 Pt In + c−2 PPt = Im + c−2 Pt P Pt so that by (5.10) and (5.26),   −1 Pt P Im + c−2 Pt P = Pt In + c−2 PPt

−1

P=



Im + c−2 Pt P

(5.26) −1

Pt P .

(5.27)

Bi-gyrogroups and Bi-gyrovector Spaces – V

Then, by (5.10) and (5.27),   −1 −1 V t V = Im + c−2 Pt P Pt P Im + c−2 Pt P = Pt P(Im + c−2 Pt P)−1

(5.28)

= (Im + c−2 Pt P)−1 Pt P . Hence, Pt P = (Im + c−2 Pt P)V t V

(5.29)

Pt P = V t V(Im + c−2 Pt P).

(5.30)

and

A rearrangement of (5.29) yields V t V = Pt P(Im − c−2 V t V),

(5.31)

Pt P = V t V(Im − c−2 V t V)−1 .

(5.32)

implying

Similarly, a rearrangement of (5.30) yields V t V = (Im − c−2 V t V)Pt P,

(5.33)

Pt P = (Im − c−2 V t V)−1 V t V .

(5.34)

implying

Following (5.32) we have Im + c−2 Pt P = Im + c−2 V t V(Im − c−2 V t V)−1 = (Im − c−2 V t V)(Im − c−2 V t V)−1 + c−2 V t V(Im − c−2 V t V)−1 = (Im − c−2 V t V + c−2 V t V)(Im − c−2 V t V)−1

(5.35)

= (Im − c−2 V t V)−1 so that



Im + c−2 Pt P =



Im − c−2 V t V

−1

.

Hence, by (5.25) and (5.36),   P = V Im + c−2 Pt P = V Im − c−2 V t V

(5.36) −1

.

Equations (5.24) and (5.37) validate the two equations in (5.11). Conversely, in Parts IIA and IIB we show that (5.11) implies (5.10).

(5.37)

189

190

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Part II A: Assuming (5.11), we have V=



In − c−2 VV t P

and the commuting relation  V t In − c−2 VV t

−1

so that, by (5.39) and (5.11)  VV t In − c−2 VV t

−1

Then, by (5.11) and (5.40),  PPt = In − c−2 VV t

=



Im − c−2 V t V

(5.38) −1

Vt

 −1 = V Im − c−2 V t V V t  −1 = In − c−2 VV t VV t . −1

 VV t In − c−2 VV t

= VV t (In − c−2 VV t )−1

(5.39)

(5.40)

−1

(5.41)

= (In − c−2 VV t )−1 VV t . Hence, VV t = (In − c−2 VV t )PPt

(5.42)

VV t = PPt (In − c−2 VV t ) .

(5.43)

and

A rearrangement of (5.42) yields PPt = VV t (In + c−2 PPt ),

(5.44)

VV t = PPt (In + c−2 PPt )−1 .

(5.45)

implying

Similarly, a rearrangement of (5.43) yields PPt = (In + c−2 PPt )VV t ,

(5.46)

VV t = (In + c−2 PPt )−1 PPt .

(5.47)

implying

Bi-gyrogroups and Bi-gyrovector Spaces – V

Following (5.45) we have In − c−2 VV t = In − c−2 PPt (In + c−2 PPt )−1 = (In + c−2 PPt )(In + c−2 PPt )−1 − c−2 PPt (In + c−2 PPt )−1 = (In + c−2 PPt − c−2 PPt )(In + c−2 PPt )−1

(5.48)

= (In + c−2 PPt )−1 so that



In − c−2 VV t =



In + c−2 PPt

−1

Hence, by (5.38) and (5.49),   V = In − c−2 VV t P = In + c−2 PPt

.

(5.49)

−1

P,

(5.50)

thus validating the first equation in (5.10). In Part IIB of the proof we validate the second equation in (5.10). Part II B: Assuming (5.11), we have  V = P Im − c−2 V t V (5.51) and the commuting relation, as in (5.39),   −1 V t In − c−2 VV t = Im − c−2 V t V so that by (5.11) and (5.52),   −1 V t V Im − c−2 V t V = V t In − c−2 VV t Then, by (5.11) and (5.53),  Pt P = Im − c−2 V t V

−1

−1

V=



−1

Vt

Im − c−2 V t V

 V t V Im − c−2 V t V

= V t V(Im − c−2 V t V)−1

(5.52) −1

V tV .

(5.53)

−1

(5.54)

= (Im − c−2 V t V)−1 V t V . Hence, V t V = (Im − c−2 V t V)Pt P

(5.55)

V t V = Pt P(Im − c−2 V t V) .

(5.56)

and

A rearrangement of (5.55) yields Pt P = V t V(Im + c−2 Pt P),

(5.57)

191

192

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

implying V t V = Pt P(Im + c−2 Pt P)−1 .

(5.58)

Similarly, a rearrangement of (5.56) yields Pt P = (Im + c−2 Pt P)V t V,

(5.59)

V t V = (Im + c−2 Pt P)−1 Pt P .

(5.60)

implying

Following (5.58) we have Im − c−2 V t V = Im − c−2 Pt P(Im + c−2 Pt P)−1 = (Im + c−2 Pt P)(Im + c−2 Pt P)−1 − c−2 Pt P(Im + c−2 Pt P)−1 = (Im + c−2 Pt P − c−2 Pt P)(Im + c−2 Pt P)−1

(5.61)

= (Im + c−2 Pt P)−1 so that



Im − c−2 V t V =



Im + c−2 Pt P

−1

.

Hence, by (5.51) and (5.62),   −1 V = P Im − c−2 V t V = P Im + c−2 Pt P . Equations (5.50) and (5.63) validate the two equations in (5.10), as desired.

(5.62)

(5.63) 

Theorem 5.5. Let φ : Rn×m → Bn×m , m, n ∈ N, be the map given by each of the two mutually equivalent equations  −1 φ : P → V = In + c−2 PPt P (5.64)  −1 φ : P → V = P Im + c−2 Pt P where Bn×m = φ(Rn×m ) is the image of Rn×m under φ. Then, φ is bijective, and the inverse φ−1 : Bn×m → Rn×m of φ is given by each of the two mutually equivalent equations  −1 φ−1 : V → P = In − c−2 VV t V (5.65)  −1 −1 −2 t φ : V → P = V Im − c V V . Proof. The proof follows immediately from Theorem 5.4.



Bi-gyrogroups and Bi-gyrovector Spaces – V

It is clear from (5.64) – (5.65) that φ(0n,m ) = 0n,m φ(−P) = −φ(P) φ−1 (0n,m ) = 0n,m

(5.66)

φ−1 (−V) = −φ−1 (V) , for all V ∈ Rn×m c .

5.3. Matrix Balls of the Parameter V Definition 5.6. (Spectrum, Spectral Radius [39, p. 35]). The set of all complex numbers that are eigenvalues of a square matrix A ∈ Rn×n , n ∈ N, is called the spectrum of A and is denoted by σ(A). The spectral radius of A is the nonnegative number ρ(A) = max{|λ| : λ ∈ σ(A)}.

(5.67)

The spectral radius ρ(A) is thus the radius of the smallest disc centered at the origin of the complex plane that includes all the eigenvalues of A. For Def. 5.7, we note that for any V ∈ Rn×m the set of nonzero eigenvalues of VV t equals the set of nonzero eigenvalues of V t V. Definition 5.7. (Matrix Ball, Matrix Spectral Norm). For any m, n ∈ N and c > 0, of the set Rn×m of all n × m real matrices is the set the c-ball Rn×m c √ Rn×m = {V ∈ Rn×m : ∀λ ∈ σ(VV t ), λ < c} c (5.68) √ = {V ∈ Rn×m : ∀λ ∈ σ(V t V), λ < c} . The matrix spectral norm V of V ∈ Rn×m , or norm in short, is defined by √ V = max{ λ : λ ∈ σ(VV t )} √ = max{ λ : λ ∈ σ(V t V)} .

(5.69)

It is clear from Def. 5.7 that Rn×m = {V ∈ Rn×m : V < c} c

(5.70)

− V = V

(5.71)

and that for all V ∈ Rn×m c .

193

194

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Well-known properties of the matrix spectral norm are [39, p. 290]: A ≥ 0 A = 0 if and only if A = 0 rA = |r| A A + B ≤ A + B AB ≤ A B

Nonnegative Positive Homogeneity Property Triangle Inequality Submultiplicity (5.72)

for any r ∈ R and square matrices A, B ∈ Rk×k , k ∈ N. Furthermore, the matrix spectral norm is invariant under bi-rotations (orthogonally invariant [39, p. 296]), as shown in the following theorem. Theorem 5.8. (Orthogonally Invariant Norm). Let m, n ∈ N and V ∈ Rn×m . Then, On VOm = V

(5.73)

for any On ∈ SO(n) and Om ∈ SO(m). Proof. Let U = On VOm .

(5.74)

Then, UU t = On VV t Otn U t U = Otm V t VOm .

(5.75)

Hence, the eigenvalues of UU t are independent of Om and the eigenvalues of U t U are independent of On . But, the nonzero eigenvalues of UU t are equal to the nonzero eigenvalues of U t U. Hence, the nonzero eigenvalues of U t U are independent of both On and Om , implying U = On VOm = In VIm = V , as desired.

(5.76) 

The only matrix norm that appears in the book is the spectral norm. Hence, the word spectral may be omitted. In order to characterize the image Bn×m = φ(Rn×m ) of Rn×m under φ in terms of eigenvalues, we present the following well-known theorem.

Bi-gyrogroups and Bi-gyrovector Spaces – V

Theorem 5.9. ([5, p. 56]). If a square matrix A has the eigenvalue λ and the corresponding eigenvector x, then any rational function R(A) of A has the eigenvalue R(λ) and the eigenvector x. Theorem 5.9 enables us to prove the following theorem, which characterizes Bn×m in terms of eigenvalues. Theorem 5.10. For any m, n ∈ N, let Bn×m = φ(Rn×m )

(5.77)

be the image under the bijection φ in (5.64) – (5.65). Then, Bn×m = Rn×m . c

(5.78)

Proof. Let V ∈ Bn×m = φ(Rn×m ). Then, there exists P ∈ Rn×m such that  −1 V = φ(P) = In + c−2 PPt P

(5.79)

and, hence, by (5.47), VV t = (In + c−2 PPt )−1 PPt .

(5.80)

Let λi , i = 1, . . . , n, be the eigenvalues of PPt . Then, λi ≥ 0 and, by (5.80) and Theorem 5.9, the eigenvalues μi of VV t are ⎧ ⎪ ⎪ 0, if λi = 0 , ⎪ ⎪ ⎪ λi ⎨ = (5.81) μi = ⎪ c2 ⎪ 1 + c−2 λi ⎪ ⎪ ⎪ , if λ > 0 , i ⎩ 1 + c2 /λi n×m so that 0 ≤ μi < c2 . Hence V ∈ Rn×m ⊆ Rn×m c , implying the inclusion B c . n×m To prove the reverse inclusion, let V ∈ Rc and let μi , i = 1, . . . , n be the eigenvalues of VV t . Then, 0 ≤ μi < c2 , so that we can define P ∈ Rn×m by the equation  −1 P = In − c−2 VV t V . (5.82)

By means of Theorem 5.4, (5.82) implies  V = In + c−2 PPt

−1

P

(5.83)

so that V = φ(P) ∈ Bn×m , implying the reverse inclusion Rn×m ⊆ Bn×m . Hence, Bn×m = c  Rn×m c , as desired. We will see in Example 5.26, p. 209, that in the special case when m = 1, the ball of spectral radius c and the Euclidean ball Rnc of radius c coincide.

Rn×1 c

195

196

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Following Theorems 5.5 and 5.10, for any m, n ∈ N the map φ is a bijection φ : Rn×m → Rn×m c

(5.84)

from the space Rn×m onto its c-ball Rn×m c , given by each of the two equations  −1 φ : P → V = In + c−2 PPt P  −1 φ : P → V = P Im + c−2 Pt P

(5.85)

satisfying each of the two equations φ−1 : V → P = φ−1



−1

In − c−2 VV t V  −1 : V → P = V Im − c−2 V t V .

(5.86)

possesses the following commuting relations Lemma 5.11. The map φ : Rn×m → Rn×m c with On and Om : On φ(P) = φ(On P) (5.87)

φ(P)Om = φ(POm ) and On φ−1 (V) = φ−1 (On V)

(5.88)

φ−1 (V)Om = φ−1 (VOm ) for any P ∈ Rn×m , V ∈ Rn×m c , On ∈ SO(n), and Om ∈ SO(m), m, n ∈ N. Proof. By means of (5.85),  −1 φ(On P) = On P Im + c−2 (On P)t (On P)  −1 = On P Im + c−2 Pt P

(5.89)

= On φ(P) and φ(POm ) = =

 

In + c−2 (POm )(POm )t In + c−2 PPt

= φ(P)Om .

−1

POm

−1

POm (5.90)

Bi-gyrogroups and Bi-gyrovector Spaces – V

Similarly, by means of (5.86),  −1 φ−1 (On V) = On V Im − c−2 (On V)t (On V)  = On V Im − c−2 V t V

−1

(5.91)

= On φ−1 (V) and φ−1 (VOm ) = =

 

In − c−2 (VOm )(VOm)t In − c−2 VV t

−1

−1

VOm

VOm

(5.92)

= φ−1 (V)Om .  Let V ∈ Rn×m , m, n ∈ N. The square matrices VV t and V t V are called Gram matrices [39]. The eigenvalues of VV t and V t V are nonnegative. The positive eigenvalues of both VV t and V t V are the same, denoted by σ21 , σ22 , . . . , σ2k , k ≤ min(m, n). Their positive square roots, σ1 , σ2 , . . . , σk , are called the singular values of V [39]. Let vi ∈ Rn , i = 1, . . . , m, be the columns of V, that is, V = (v1 v2 . . . vm ) ∈ Rn×m .

(5.93)

Then, the following properties of Gram matrices are well known: tr(VV t ) = tr(V t V) = v21 + v22 + . . . + v2m

(5.94)

= σ21 + σ22 + . . . + σ2k , where k = rank(V), k ≤ min(m, n). Here, tr(A) denotes the trace of a square matrix A, and σ2i , i = 1, . . . , k, are all the positive eigenvalues of VV t and, hence, all the positive eigenvalues of V t V. Theorem 5.12. For any m, n ∈ N and c > 0, let V = (v1 v2 . . . vm ) ∈ Rn×m , √ . where vi ∈ Rnc , i = 1, 2, . . . , m, are the columns of V. Then, V ∈ Rn×m mc

(5.95)

197

198

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Proof. By assumption v2k < c2 , k = 1, 2, . . . , m. Hence, by (5.94), tr(VV t ) = tr(V t V) = σ21 + σ22 + . . . + σ2k

(5.96)

= v21 + v22 + . . . + v2m < mc2 . Hence, σ2i < mc2

(5.97)

√ , as desired. for i = 1, . . . , k. Hence, by the c-ball definition in (5.68), V ∈ Rn×m mc



The following two lemmas lead to the proof of Theorem 5.15. Lemma 5.13. Let A ∈ Rn×n be a symmetric matrix. Then, λmax (A) = max n

x∈R ,x0

xt Ax , xt x

(5.98)

where λmax (A) is the largest eigenvalue of A. Proof. Since A is symmetric, there exists an orthogonal matrix O ∈ O(n) and a diagonal matrix D = diag(d1 , d2 , . . . , dn ) ∈ Rn×n such that the diagonal elements of D are the eigenvalues of A. Then, max n

x∈R ,x0

xt Ax xt Ot DOx (Ox)t D(Ox) = max = max x∈Rn ,x0 xt Ot Ox x∈Rn ,x0 (Ox)t (Ox) xt x d1 y21 + d2 y22 + . . . dn y2n yt Dy = max , = max y∈Rn ,y0 yt y y∈Rn ,y0 y21 + y22 + . . . y2n

(5.99)

where y = (y1 , y2 , . . . , yn ) ∈ Rn . Clearly, max n

y∈R ,y0

d1 y21 + d2 y22 + . . . dn y2n y21 + y22 + . . . y2n

≤ max{d1 , d2 , . . . , dn } = d j

(5.100)

for some 1 ≤ j ≤ n. Equality is reached in (5.100) when y j = 1 and yi j = 0. Hence, max n

y∈R ,y0

d1 y21 + d2 y22 + . . . dn y2n y21 + y22 + . . . y2n

= max{d1 , d2 , . . . , dn } = λmax (A) .

Result (5.98) of the Lemma follows immediately from (5.99) and (5.101).

(5.101) 

Bi-gyrogroups and Bi-gyrovector Spaces – V

Lemma 5.14. Let A = (ai j ) ∈ Rn×n be a symmetric matrix. Then, λmax (A) ≥ aii

(5.102)

for all i = 1, 2, . . . , n. Proof. Let ei ∈ Rn be the unit column vector whose ith component is 1 and whose remaining components are 0, i = 1, 2, . . . , n. Then, by Lemma 5.13, λmax (A) ≥

eti Aei = aii , eti ei

(5.103) 

as desired. Theorem 5.15. For any m, n ∈ N and c > 0, let V = (v1 v2 . . . vm ) ∈ Rn×m ,

(5.104)

n where vi ∈ Rn , i = 1, 2, . . . , m, are the columns of V. If V ∈ Rn×m c , then vi ∈ Rc .

Proof. Following (5.104), V t V ∈ Rm×m is a square matrix given by (V t V)i j = vti v j and, by definition, V = σmax (V) =



λmax (V t V),

(5.105)

(5.106)

where σmax (V) is the largest singular value of V. Hence, by (5.106) and Lemma 5.14, V 2 = (σmax (V))2 = λmax (V t V) ≥ vti vi = v 2 for all i = 1, 2, . . . , n. then vi ∈ Rnc , i = 1, 2, . . . , n, as desired. Hence, if V ∈ Rn×m c

(5.107) 

Theorem 5.16. For any m, n ∈ N and c > 0, let V ∈ Rn×m . Then, V ∈ Rn×m if and only c if the matrix  −1 L Γn,V,c := In − c−2 VV t ∈ Rn×n (5.108) exists. Proof. By Theorem 5.10, Rn×m = φ(Rn×m ). Hence, V ∈ Rn×m if and only if there exists c c n×m such that V = φ(P). The equation V = φ(P), in turn, is equivalent to the P∈R

199

200

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

equation 

In + c−2 PPt =



In − c−2 VV t

−1

,

(5.109)

as we see from (5.49). The matrix on the left-hand side of (5.109) exists for any P ∈ Rn×m . Hence, if V = φ(P) then the matrix on the right-hand side of (5.109) exists.  −1 L Conversely, if the matrix Γn,V,c = In − c−2 VV t ∈ Rn×n on the right-hand side of (5.109) exists for some V ∈ Rn×m , then we define P by the equation  −1 P := In − c−2 VV t V ∈ Rn×m . (5.110) Then, by (5.65), P satisfies the equation φ(P) = V. L exists if and only if V = φ(P) for some P ∈ Rn×m . The Hence, the matrix Γn,V,c equation V = φ(P), in turn, holds, by definition, if and only if V ∈ Rn×m c . Hence, the L exists if and only if V ∈ Rn×m .  matrix Γn,V,c c Theorem 5.16 possesses the following dual theorem with a similar proof. Instructively, we present the dual theorem as well, along with its dual proof. Theorem 5.17. For any m, n ∈ N and c > 0, let V ∈ Rn×m . Then, V ∈ Rn×m if and only c if the matrix  −1 (5.111) ΓRm,V,c := Im − c−2 V t V ∈ Rm×m exists. Proof. By Theorem 5.10, Rn×m = φ(Rn×m ). Hence, V ∈ Rn×m if and only if there exist c c n×m such that V = φ(P). The equation V = φ(P), in turn, is equivalent to the P∈R equation   −1 Im + c−2 Pt P = Im − c−2 V t V , (5.112) as we see from (5.62). The matrix on the left-hand side of (5.112) exists for any P ∈ Rn×m . Hence, if V = φ(P) then the matrix on the right-hand side of (5.112) exists.  −1 Conversely, if the matrix ΓRm,V,c = Im − c−2 V t V ∈ Rm×m on the right-hand side of (5.112) exists for some V ∈ Rn×m , then we define P by the equation  −1 P := V Im − c−2 V t V ∈ Rn×m . (5.113) Then, by (5.65), P satisfies the equation φ(P) = V. Hence, the matrix ΓRm,V,c exists if and only if V = φ(P) for some P ∈ Rn×m . The

Bi-gyrogroups and Bi-gyrovector Spaces – V

equation V = φ(P), in turn, holds, by definition, if and only if V ∈ Rn×m c . Hence, the R n×m  matrix Γm,V,c exists if and only if V ∈ Rc . L The left gamma factor Γn,V,c ∈ Rn×n and the the right gamma factor ΓRm,V,c ∈ Rm×m n×m of V ∈ Rc , defined in (5.108) and in (5.111), play an important role in the book. The values of the indices m and n are clear from the signature (m, n) of V ∈ Rn×m c , and the n×m value of the index c is clear from the spectral radius of the c-ball Rc of V. Hence, for simplicity, in general the indices m, n, and c are omitted when their values are clear from the context, as shown in (5.115) – (5.116).

Remark 5.18. The complex counterpart SU(p, q), p, q ∈ N, of SO(m, n), m, n ∈ N, is studied in complex pseudo-Euclidean geometry. According to Y.A. Neretin [57, Chap. 2], this geometry was a subject of some work by L.S. Pontrjagin, M.G. Krein, M.S. Livshits, and V.P. Potapov in the 1940s-1950s. It should be emphasized that our study of SO(m, n) can straightforwardly be extended to SU(p, q).

5.4. Reparametrizing the Bi-boost We now wish to change the bi-boost parameter P ∈ Rn×m , m, n ∈ N, the domain of which is the space Rn×m of all n × m real matrices, to the new parameter V = φ(P) ∈ n×m of the ambient space Rn×m . We, therefore, Rn×m c , the domain of which is the c-ball Rc recall the following relations between P and V, which are taken from (5.11), (5.36), and (5.23):   −1 −1 P = In − c−2 VV t V = V Im − c−2 V t V   −1 (5.114) Im + c−2 Pt P = Im − c−2 V t V   −1 In + c−2 PPt = In − c−2 VV t , where, following Theorem 5.5, V = φ(P) and P = φ−1 (V). in the c-ball Rn×m is constructed by constructing A generic parameter V ∈ Rn×m c c n×m a generic parameter P ∈ R , which is any n × m real matrix, and then constructing V = φ(P) by means of (5.64). The equations in (5.114) along with analogies with the gamma factor of special relativity theory suggest the definition of a left gamma factor ΓVL and a right gamma factor ΓRV by the equations  −1 ΓVL := In − c−2 VV t ∈ Rn×n (5.115)  −1 ΓRV := Im − c−2 V t V ∈ Rm×m ,

201

202

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

V ∈ Rn×m c . It clearly follows from (5.115) that  ΓVL t = Im − c−2 V t V  ΓRV t = In − c−2 VV t

−1 −1

= ΓRV =

(5.116)

ΓVL

Rn×m c .

Along with (5.116) we have interesting identities as, for instance, for all V ∈ the commuting relations (ΓVL V)t = V t ΓRV t (VΓRV )t = ΓVL t V t

(5.117)

for any V ∈ Rn×m c . The matrix identities (5.117) are analogous to their square-matrix counterpart, (GH)t = H t Gt for any n × n matrices G and H. The matrices ΓVL and ΓRV are symmetric. Furthermore, they are even in the parameter V ∈ Rn×m c , that is, L Γ−V = ΓVL

ΓR−V = ΓRV .

(5.118)

Naturally, the pair (ΓVL , ΓRV ) of a left and a right gamma factor is called a bi-gamma factor. Following the first equation in (5.114), and (5.115), the left and right gamma factors are related by the first commuting relation in (5.119). The remaining commuting relations in (5.119) follow immediately from the first one, noting that left and right gamma factors are symmetric matrices. ΓVL V = VΓRV ΓRV V t = V t ΓVL ΓVL VV t = VV t ΓVL

(5.119)

ΓRV V t V = V t VΓRV . Moreover, by Theorem 5.4 with E = P, we have  −1 E = ΓVL V = VΓRV ⇐⇒ V = In + c−2 EE t E  −1 = E Im + c−2 E t E .

(5.120)

The result in (5.120) is presented here for later reference. It will prove useful in (5.204), p. 220.

Bi-gyrogroups and Bi-gyrovector Spaces – V

In the bi-gamma notation (5.115), the equations in (5.114) take the form  

P = ΓVL V = VΓRV ∈ Rn×m In + c−2 PPt = ΓVL ∈ Rn×n

(5.121)

Im + c−2 Pt P = ΓRV ∈ Rm×m .

Following the change of parameter from the parameter P ∈ Rn×m of the bi-boost of the bi-boost Bv (V), we have by means of B p (P) := B(P) to the parameter V ∈ Rn×m c (4.75), p. 115, and (5.121) ⎛ ⎞ 1 t ⎜⎜⎜ Im + c−2 Pt P ⎟⎟⎟ 2P c ⎟⎟⎟ B p (P) = ⎜⎜⎜⎜⎝  ⎟⎠ −2 t P In + c PP (5.122) ⎛ 1 R t 1 t L⎞ R ⎜⎜⎜ ⎟ ΓV Γ V = c2 V ΓV ⎟⎟ c2 V ⎟⎟⎟ =: Bv (V) , = ⎜⎜⎜⎝ ⎠ L R L ΓV V = VΓV ΓV where V = φ(P). When m = 1 the bi-boost Bv (V) specializes to the standard Lorentz boost in one time dimension and n space dimensions parametrized by V, where V represents relativistically admissible velocities [74]. Furthermore, when m = 1 the bi-boost B p (P) specializes to the Lorentz boost in one time dimension and n space dimensions, where its parameter P represents the relativistic proper velocity [84, Sect. 3.8]. It, therefore, proves useful to study the bi-boost B p (P) parametrized by P ∈ Rn×m along with its equivalent bi-boost Bv (V) parametrized by V ∈ Rn×m c . Following (5.122) we, thus, have B p (P) = Bv (V) = Bv (φ(P)) , where Bv (V) is the “velocity” bi-boost ⎛ R ⎜⎜⎜ ΓV Bv (V) = ⎜⎜⎜⎝ ΓVL V

1 R t⎞ Γ V ⎟⎟ c2 V ⎟ ⎟

ΓVL

⎟⎟⎠ ∈ SO(m, n) ,

V ∈ Rn×m c , and where B p (P) is the “proper velocity” bi-boost ⎛ ⎞ 1 t ⎜⎜⎜ Im + c−2 Pt P ⎟⎟⎟ P 2 c ⎟⎟⎟ ∈ SO(m, n) , B p (P) = ⎜⎜⎜⎜⎝  ⎟⎠ P In + c−2 PPt

(5.123)

(5.124)

(5.125)

P ∈ Rn×m . By means of (5.122) – (5.125) and Theorem 5.5 we clearly have the following Lemma.

203

204

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Lemma 5.19. Let B p (P) and Bv (V) be a proper velocity bi-boost and a velocity biboost, given by (5.124) – (5.125). Then, B p (P) = Bv (V)

(5.126)

if and only if V = φ(P) or, equivalently, if and only if P = φ−1 (V). Accordingly, the generic Lorentz transformation Λ(P, On , Om ), with main parameter P ∈ Rn×m , of signature (m, n), m, n ∈ N, in (4.88) – (4.89), p. 118, becomes Λ = Λ(V, On , Om ), with main parameter V ∈ Rn×m c , given by the unique bi-gyration decomposition in theorem 5.20.

5.5. Lorentz Transformation Decomposition Following (5.124), the Lorentz bi-boost Bv (V) of signature (m, n), parametrized by V ∈ Rn×m c , is given by ⎛ R ⎞ ⎜⎜⎜ ΓV c12 ΓRV V t ⎟⎟⎟ ⎜ ⎟⎟⎟ ∈ SO(m, n) . (5.127) Bv (V) = ⎜⎜⎝ L L ⎠ ΓV V ΓV Owing to the bijective correspondence φ in Theorem 5.5, p. 192, between the parameters P ∈ Rn×m and V ∈ Rn×m c , Theorem 4.12, p. 118, can be translated from a formalism that involves P into a formalism that involves V in the following theorem. Theorem 5.20. (Lorentz Transformation Bi-gyration Decomposition, V). A matrix Λ ∈ R(m+n)×(m+n) is the matrix representation of a Lorentz transformation of signature (m, n), Λ ∈ S O(m, n), if and only if it possesses the bi-gyration decomposition ⎞⎛ ⎞⎛ ⎞ ⎛ ⎜⎜⎜ Om 0m,n ⎟⎟⎟ ⎜⎜⎜ ΓRV c12 ΓRV V t ⎟⎟⎟ ⎜⎜⎜ Im 0m,n ⎟⎟⎟ ⎜ ⎜ ⎟ ⎟ ⎟⎟⎟ ∈ SO(m, n) ⎜ ⎟⎠⎟ ⎜⎝⎜ ⎟⎟ ⎜⎜ (5.128) Λ = ⎜⎝⎜ ⎠ L L ⎠⎝ 0n,m In ΓV V ΓV 0n,m On for any V ∈ Rn×m c , Om ∈ SO(m), and On ∈ SO(n). Proof. Equation (5.128), expressed in terms of the parameter V ∈ Rn×m c , is equivalent to (4.88), p. 118, expressed in terms of the parameter P ∈ Rn×m , as we see from the relations between the parameters P and V in (5.121).  Similarly, Theorem 4.13, p. 119, is translated from its P-formalism into a corresponding V-formalism in the following theorem.

Bi-gyrogroups and Bi-gyrovector Spaces – V

Theorem 5.21. (Lorentz Transformation Polar Decomposition, V). A matrix Λ ∈ R(m+n)×(m+n) is the matrix representation of a Lorentz transformation of signature (m, n), Λ ∈ S O(m, n), if and only if it possesses the polar decomposition ⎞⎛ ⎞⎛ ⎞ ⎛ R ⎜⎜⎜ ΓV c12 ΓRV V t ⎟⎟⎟ ⎜⎜⎜ Om 0m,n ⎟⎟⎟ ⎜⎜⎜ Im 0m,n ⎟⎟⎟ ⎜ ⎜ ⎟ ⎟ ⎟⎟⎟ ⎜ ⎟⎟ ⎜⎜ ⎟⎠⎟ ⎜⎝⎜ Λ = ⎜⎝⎜ ⎠ L L ⎠⎝ 0n,m In 0n,m On ΓV V ΓV (5.129) ⎛ R ⎞⎛ ⎞ ⎜⎜⎜ ΓV c12 ΓRV V t ⎟⎟⎟ ⎜⎜⎜ Om 0m,n ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ∈ SO(m, n) , = ⎜⎜⎜⎝ ⎠ L L ⎠⎝ ΓV V ΓV 0n,m On for any V ∈ Rn×m c , Om ∈ SO(m), and On ∈ SO(n). Proof. Equation (5.129), expressed in terms of the parameter V ∈ Rn×m c , is equivalent n×m to (4.91), p. 119, expressed in terms of the parameter P ∈ R , as we see from the relations between the parameters P and V in (5.121).  Initially, we pay attention to the bi-gyration decomposition (5.128). Results obtained by means of this decomposition will enable us to pay further attention to the polar decomposition (5.129). It is convenient to write (5.128) parametrically, as ⎛ ⎞ ⎜⎜⎜ V ⎟⎟⎟ ⎜ ⎟ (5.130) Λ = Λ(Om , V, On ) = ρ(Om )Bv(V)λ(On ) = ⎜⎜⎜⎜ On ⎟⎟⎟⎟ , ⎝ ⎠ Om where ρ(Om) and λ(On ) are given by (4.76) – (4.77), p. 115, and Bv (V) is given by (5.127). Lemma 5.22. For all V ∈ Rn×m c , Om ∈ S O(m), and On ∈ S O(n), we have the commuting relations ΓRV Om = Om ΓRVOm On ΓVL = ΓOL n V On

(5.131)

and the relations ΓROn V = ΓRV L ΓVO = ΓVL . m

(5.132)

Proof. The proof is similar to the proof of Lemma 4.6, p. 111. Manipulating the second

205

206

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

equation in (5.115) we have the following chain of equations: (ΓRV )−2 = Im − c−2 V t V = Om Im Otm − c−2 Om Otm V t VOm Otm = Om (Im − c−2 Otm V t VOm )Otm   = Om Im − c−2 Otm V t VOm Im − c−2 Otm V t VOm Otm   = Om Im − c−2 Otm V t VOm Otm Om Im − c−2 Otm V t VOm Otm  = (Om Im − c−2 Otm V t VOm Otm )2 . Hence

 (ΓRV )−1 = Om Im − c−2 Otm V t VOm Otm  = Om Im − c−2 (VOm )t VOm Otm

(5.133)

(5.134)

= Om (ΓRVOm )−1 Otm , so that (ΓRV )−1 Om = Om (ΓRVOm )−1 ,

(5.135)

thus implying the first equation in (5.131). Similarly, manipulating the first equation in (5.115) we have the following chain of equations: (ΓVL )−2 = In − c−2 VV t = Otn In On − c−2 Otn On VV t Otn On = Otn (In − c−2 On VV t Otn )On   = Otn In − c−2 On VV t Otn In − c−2 On VV t Otn On   = Otn In − c−2 On VV t Otn On Otn In − c−2 On VV t Otn On  = (Otn In − c−2 On VV t Otn On )2 . Hence

 (ΓVL )−1 = Otn In − c−2 On VV t Otn On  = Otn In − c−2 On V(On V)t On = Otn (ΓOL n V )−1 On ,

(5.136)

(5.137)

Bi-gyrogroups and Bi-gyrovector Spaces – V

so that On (ΓVL )−1 = (ΓOL n V )−1 On ,

(5.138)

thus implying the second equation in (5.131). Finally, the proof of (5.132) follows immediately from (5.115).



Lemma 5.23. The commuting relation Bv (V)ρ(Om) = ρ(Om )Bv(VOm )

(5.139)

holds for all V ∈ Rn×m and Om ∈ SO(m). c Proof. Let J1 and J2 denote, respectively, the left-hand side and the right-hand side of (5.139), so that we have to prove that J1 = J2 . Then, ⎛ R ⎞⎛ ⎞ ⎜⎜⎜ ΓV c12 ΓRV V t ⎟⎟⎟ ⎜⎜⎜ Om 0m,n ⎟⎟⎟ ⎟⎟⎟⎠ ⎜⎜⎜⎝ ⎟⎟⎟⎠ J1 := Bv (V)ρ(Om) = ⎜⎜⎜⎝ ΓVL V ΓVL 0n,m In (5.140) ⎛ R 1 R t⎞ ⎜⎜⎜ ΓV Om c2 ΓV V ⎟⎟⎟ ⎟⎟⎟ . = ⎜⎜⎜⎝ L L ⎠ ΓV VOm ΓV Hence, by Lemma 5.22,

⎛ ⎞⎛ ⎜⎜⎜ Om 0m,n ⎟⎟⎟ ⎜⎜⎜ ΓRVOm ⎜ ⎟⎟⎟ ⎜⎜⎜ J2 := ρ(Om )Bv(VOm ) = ⎜⎜⎝ ⎠⎝ L VOm 0n,m In ΓVO m ⎛ ⎞⎛ ⎜⎜⎜ Om 0m,n ⎟⎟⎟ ⎜⎜⎜ ΓRVOm ⎟⎟⎟ ⎜⎜⎜ = ⎜⎜⎜⎝ ⎠⎝ 0n,m In ΓVL VOm

⎞ 1 R Γ Ot V t ⎟⎟ c2 VOm m ⎟ ⎟ ⎞ 1 R Γ Ot V t ⎟⎟ c2 VOm m ⎟ ⎟ ΓVL

⎛ ⎜⎜⎜Om ΓRVOm = ⎜⎜⎜⎝ ΓVL VOm

⎞ 1 O ΓR Ot V t ⎟⎟ c2 m VOm m ⎟ ⎟

⎛ R ⎜⎜⎜ ΓV Om = ⎜⎜⎜⎝ ΓVL VOm

⎞ 1 R Γ O Ot V t ⎟⎟ c2 V m m ⎟ ⎟

⎛ R ⎜⎜⎜ ΓV Om = ⎜⎜⎜⎝ ΓVL VOm

1 R t⎞ Γ V ⎟⎟ c2 V ⎟ ⎟

⎟⎟⎠

ΓVL

ΓVL

ΓVL

⎟⎟⎠

L ΓVO m

⎟⎟⎠

(5.141)

⎟⎟⎠

⎟⎟⎠ .

Following (5.140) and (5.141) we have J1 = J2 , as desired.



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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Lemma 5.24. The commuting relation λ(On )Bv (V) = Bv (On V)λ(On )

(5.142)

and On ∈ SO(n). holds for all V ∈ Rn×m c Proof. Let J3 and J4 denote, respectively, the left-hand side and the right-hand side of (5.142), so that we have to prove that J3 = J4 . Then, ⎛ ⎞⎛ ⎞ ⎜⎜⎜ Im 0m,n ⎟⎟⎟ ⎜⎜⎜ ΓRV c12 ΓRV V t ⎟⎟⎟ ⎟⎟⎟⎠ ⎜⎜⎜⎝ ⎟⎟⎟⎠ J3 := λ(On )Bv(V) = ⎜⎜⎜⎝ 0n,m On ΓVL V ΓVL (5.143) ⎛ R 1 R t⎞ ⎜⎜⎜ ΓV Γ V ⎟⎟ c2 V ⎟ ⎟⎟⎟ . = ⎜⎜⎜⎝ L L ⎠ On ΓV V On ΓV Hence, by Lemma 5.22,

⎛ R ⎜⎜⎜ ΓOn V J4 := Bv (On V)λ(On ) = ⎜⎜⎜⎝ ΓOL n V On V ⎛ ⎜⎜⎜ ΓRV = ⎜⎜⎜⎝ ΓOL n V On V

⎞⎛ 1 R Γ V t Otn ⎟⎟⎟ ⎜⎜⎜ Im c2 On V ⎟⎜ ΓOL n V

⎟⎟⎠ ⎜⎜⎝

0n,m

1 R t t⎞⎛ Γ V On ⎟⎟⎟ ⎜⎜⎜ Im c2 V ⎟⎜

ΓOL n V

⎟⎟⎠ ⎜⎜⎝ 0n,m

⎛ 1 R t⎞ ⎜⎜⎜ ΓRV 2 ΓV V ⎟ ⎟⎟⎟ c ⎟⎟⎠ = ⎜⎜⎜⎝ ΓOL n V On V ΓOL n V On

⎞ 0m,n ⎟⎟⎟ ⎟⎟⎟ ⎠ On ⎞ 0m,n ⎟⎟⎟ ⎟⎟⎟ ⎠ On (5.144)

⎛ R 1 R t⎞ ⎜⎜⎜ ΓV 2 ΓV V ⎟ ⎟⎟⎟ c ⎟⎟ . = ⎜⎜⎜⎝ L L ⎠ On ΓV V On ΓV V Following (5.143) and (5.144) we have J3 = J4 , as desired.



Combining Lemmas 5.23 and 5.24, we obtain the following lemma, which is analogous to Lemma 4.10, p. 115.

Bi-gyrogroups and Bi-gyrovector Spaces – V

Lemma 5.25. The commuting relations λ(On )Bv(V) = Bv (On V)λ(On ) Bv (V)ρ(Om) = ρ(Om)Bv (VOm )

(5.145)

λ(On )Bv(V)ρ(Om) = ρ(Om)Bv (On VOm )λ(On ) and Bv (V)ρ(Om)ρ(On ) = ρ(Om )ρ(On )Bv (O−1 n VOm )

(5.146)

hold for any V ∈ Rn×m c , Om ∈ S O(m), and On ∈ S O(n).

5.6. Examples Example 5.26. In this example we show that in the special case when m = 1 the c-ball specializes to the Euclidean c-ball Rnc of the Euclidean n-space Rn×1 = Rn . Rn×m c For m = 1, V ∈ Rn×m = Rn is a column vector in the Euclidean n-space Rn , and t V V = V 2 is a 1 × 1 matrix the eigenvalue of which is λ = V 2 . Hence, following (5.68) we have √ = {V ∈ Rn×1 : λ = V 2 , λ = V < c} Rn×1 c (5.147) = {V ∈ Rn : V < c} = Rnc . Indeed, in special relativity, the relativistically admissible velocities are elements of the Euclidean c-ball R3c , where c represents the vacuum speed of light. Example 5.27. Let the matrix V ∈ Rn×m be equi-column, that is, V is an n × m matrix the m columns of which are equal to each other, V = (v v . . . v) ,

(5.148)

v ∈ Rn . Then, each of the square matrices VV t and V t V possesses a single nonzero eigenvalue λ, λ = mv2 .

(5.149)

V ∈ Rn×m c

(5.150)

Hence, by (5.68), p. 193, √ if and only if mv < c , that is, if and only if v < c/ m. Hence, 2

2

√ V := (v v . . . v) ∈ Rn×m mc

⇐⇒

v ∈ Rnc .

(5.151)

Since c is an arbitrarily fixed positive constant, it seems that it makes no difference √ whether we select the constant c or the constant mc. This is, however, not the case

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

as shown in Sects. 6.6 and 6.11, where we encounter a situation in which two different √ values of the time dimension m are involved and the use of mc rather than c is necessary for the existence of a related identity. The equivalence in (5.151) suggests that Einstein addition in Rnc between elements √ √ . between equi-column elements of Rn×m of Rnc coincides with Einstein addition in Rn×m mc mc It will be indicated by Fig. 7.2, p. 366, and Fig. 7.4, p. 377, that this is indeed the case. Example 5.28. In this example we show that when m = 1 the right gamma factor equals the gamma factor of special relativity theory. When m = 1, P ∈ Rn×1 = Rn is a column vector so that Pt P = P 2 . Then, by (5.64),  −1 P (5.152) V = φ(P) = P Im + c−2 Pt P =  1 + c−2 P 2 so that V ∈ Rn is a column vector and V 2 = V t V =

P 2 . 1 + c−2 P 2

(5.153)

Hence, 0 ≤ V < c and, by (5.115), 1 = γV ΓRV =  1 − c−2 V 2

(m = 1)

(5.154)

= φ(Rn×1 ). Here γV is the gamma factor (2.3), p. 11, that plays an for all V ∈ Rn×1 c important role in special relativity and in its underlying hyperbolic geometry [81, 84, 93, 95, 96, 98]. Example 5.29. (Relations Between Left and Right Gamma Factors). Extending (5.154) to m ≥ 1, the left and the right gamma factor in (5.115),   −1 ΓVL := In − c−2 VV t = In + c−2 PPt (5.155) and ΓRV :=



Im − c−2 V t V

−1

=



Im + c−2 Pt P ,

(5.156)

are related by the equations −In + ΓVL =

1 P(Im + ΓRV )−1 Pt c2

(5.157)

−Im + ΓRV =

1 t P (In + ΓVL )−1 P , c2

(5.158)

and

Bi-gyrogroups and Bi-gyrovector Spaces – V

where P and V are related by Theorem 5.4, p. 187. Indeed, by means of (5.155) – (5.156), Identities (5.157) – (5.158) are the elegant matrix identities (5.159) – (5.160) presented and proved in the following lemma. Lemma 5.30. The matrix identities   −1 1 −In + In + c−2 PPt = 2 P Im + Im + c−2 Pt P Pt c and   −1 1 −Im + Im + c−2 Pt P = 2 Pt In + In + c−2 PPt P c hold for all P ∈ Rn×m , m, n ∈ N. Proof. Clearly,  

2

 Im + Im + c−2 Pt P = 2 Im + Im + c−2 Pt P + c−2 Pt P . Let

 −1

R := Im + Im + c−2 Pt P

(5.159)

(5.160)

(5.161)

(5.162)

so that (5.161) can be written as 2R−1 + c−2 Pt P − (R−1 )2 = 0m,m .

(5.163)

Left multiplying and right multiplying (5.163) by R yields 2R + c−2 RPt PR − Im = 0m,m .

(5.164)

Left multiplying by P and right multiplying by Pt , (5.164) yields P(2R + c−2 RPt PR − Im )Pt = 0n,n

(5.165)

2PRPt + c−2 PRPt PRPt = PPt

(5.166)

In + c−2 (2PRPt + c−2 PRPt PRPt ) = In + c−2 PPt .

(5.167)

so that

and hence

Identity (5.167) can be written as (In + c−2 PRPt )2 = In + c−2 PPt , implying In + c−2 PRPt =



In + c−2 PPt .

(5.168)

(5.169)

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Finally, by means of (5.162), (5.169) yields (5.159), as desired. The proof of (5.160) is similar to that of (5.159).



Example 5.31. In the special case when m = 1, P ∈ Rn×1 = Rn is a column vector, Pt P = P 2 , and PPt is an n × n matrix, so that (5.159) specializes to  1 1 In + c−2 PPt = In + 2 PPt . (5.170)  c 1 + 1 + c−2 P 2 We now manipulate (5.170) in the following chain of equations, which are num= Rnc , bered for subsequent explanation. For all V ∈ Rn×1 c 

In − c−2 VV t

−1

(1)

1 1  === In + 2 c 1+ √

1 1−c−2 V 2

ΓVL

2

VV t

2 1 1  === In + 2 V ΓRV V t c 1 + γV

(m = 1)

(2)

(5.171)

(3) 1 γV2  === In + 2 VV t . c 1 + γV

Derivation of the numbered equalities in (5.171): (1) This equation is equivalent to (5.170) since (i) the left-hand sides of the two equations are equal by (5.23); and (ii) their right-hand sides are equal by (5.36) with m = 1, and by (5.41) along with (5.115). (2) Follows from Item (1) by (5.154) and by the first commuting relation in (5.119). (3) Follows from Item (2) by (5.154), noting that m = 1. Noting (5.115), the chain of equation (5.171) yields the important equation ΓVL = In +

1 γV2 VV t , c2 1 + γV

(m = 1) ,

(5.172)

V ∈ Rn×1 = Rnc , which holds for m = 1 and all n ∈ N. c The importance of (5.172) is revealed in Example 5.32, enabling us to show straightforwardly that the bi-boost Bv (V), V ∈ Rn×m c , m, n ∈ N, specializes to the Lorentz boost n = R , of special relativity in the special case when m = 1. Bv(V), V ∈ Rn×1 c c Example 5.32. When m = 1 the bi-boost Bv (V) ∈ SO(m, n) in (5.127) can be manipulated by means of (5.119) and by means of (5.154) and (5.172), obtaining the

Bi-gyrogroups and Bi-gyrovector Spaces – V

following chain of equations: ⎛ ⎞ 1 R ⎜⎜⎜ΓR t⎟ ⎟⎟⎟ Γ V ⎜ Bv (V) = ⎜⎜⎜⎜ m=1,V c2 m=1,V ⎟⎟⎟⎟ ⎝ L ⎠ ΓV V ΓVL ⎛ ⎜⎜⎜ ΓR ⎜ = ⎜⎜⎜⎜ m=1,V ⎝ R VΓm=1,V ⎛ ⎜⎜⎜ γ ⎜⎜⎜ V ⎜ = ⎜⎜⎜⎜ ⎜⎜⎜ ⎝γV V

⎞ 1 R t⎟ ⎟⎟ Γ V c2 m=1,V ⎟⎟⎟⎟⎟ ⎠ ΓVL

(5.173)

⎞ 1 ⎟⎟⎟ t γ V ⎟⎟⎟ c2 V ⎟⎟⎟ ⎟⎟ ∈ SO(1, n) , 1 γV2 t⎟ In + 2 VV ⎟⎟⎠ c 1 + γV

(m = 1)

where V ∈ Rn×1 ⊂ Rn×1 = Rn is a column vector in the ball Rn×1 = Rnc of Rn , c c Rnc = {V ∈ Rn : V < c} ,

(5.174)

and where γV is given by (5.154). The extreme right-hand side of (5.173) turns out to be the standard special relativistic (n + 1) × (n + 1) matrix representation of the Lorentz group in one time dimension and n space dimensions [74] [81, p. 254] [93, p. 447]. Accordingly, it follows from (5.173) that in the special case when m = 1 the Lorentz group of signature (m, n) specializes to the Lorentz group of special relativity theory. Example 5.33. In the special case when m = 1, P ∈ Rn×1 = Rn is a column vector so that Pt P = P 2 . Accordingly, when m = 1 Identity (5.159) specializes to the identity  PPt 1 In + c−2 PPt = In + 2 . (5.175)  c 1 + 1 + c−2 P 2 Hence, when m = 1, the bi-boost B p (P) ∈ SO(m, n) in (5.122) specializes to the proper velocity (PV) boost ⎛ ⎞ 1 t ⎜⎜⎜ 1 + c−2 P 2 ⎟⎟⎟ P 2 c ⎟⎟⎟ ∈ SO(1, n) t (5.176) B p (P) = ⎜⎜⎜⎜⎝ ⎟⎠ P In + 12 √ PP c 1+

1+c−2 P 2

in one proper-time dimension and n space dimensions, where P ∈ Rn is the proper velocity of special relativity (in physical applications n = 3). The PV-boost (5.176) leaves invariant the relativistic inner product in (5.186), with m = 1. The PV-boost B p (P) ∈ SO(1, n) involves the proper-velocity parameter P ∈ Rn ,

213

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

which is measured by means of proper-time. The need for a search for a proper-time boost, like the one in (5.176), arises in several papers as, for instance, [28, 29, 30, 31, 32, 33, 34] and [84, 86, 90, 99]. The application B p (P)(t, x)t of the PV-boost B p (P) ∈ SO(1, n) to time space coordinates (t, x)t is linear, and it keeps the relativistic norm  τ = t2 − x2 /c2 (5.177) invariant.  Similarly, the application B p (P)( τ2 + x2 /c2 , x)t of the PV-boost B p (P) ∈ SO(1, n) to proper-time space coordinates (τ, x)t is nonlinear, and it keeps the proper-time τ invariant.

5.7. Inverse Lorentz Transformation Theorem 5.34. (The Inverse Bi-boost). The inverse of the bi-boost Bv (V), V ∈ Rn×m c , is Bv (−V), Bv (V)−1 = Bv(−V) . Proof. By (5.127), (5.119), and (5.115) we have ⎛ R ⎞⎛ ⎞ ⎜⎜⎜ ΓV c12 ΓRV V t ⎟⎟⎟ ⎜⎜⎜ ΓRV − c12 ΓRV V t ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟⎠ ⎜⎜⎜⎝ Bv (V)Bv(−V) = ⎜⎜⎜⎝ ⎠ ΓVL V ΓVL −ΓVL V ΓVL ⎛ R 2 1 R t L ⎞   ⎜⎜⎜ ΓV − c2 ΓV V ΓV V − c12 ΓRV 2 V t + c12 ΓRV V t ΓVL ⎟⎟⎟ ⎟⎟ = ⎜⎜⎜⎜⎝  L 2  L 2 ⎟⎟⎠ 1 L L R R t ΓV VΓV − ΓV V − c2 ΓV VΓV V + ΓV ⎛ R 2 1  R 2 t ⎞     ⎜⎜⎜ ΓV − c2 ΓV V V c12 {− ΓRV 2 V t + ΓRV 2 V t }⎟⎟⎟ ⎟⎟ = ⎜⎜⎜⎜⎝    L 2  L 2 1  L 2 t ⎟⎟⎠ L 2 ΓV − c2 ΓV VV ΓV V − ΓV V ⎛ R 2  ⎞  ⎜⎜⎜ ΓV Im − c12 V t V ⎟⎟⎟ 0m,n ⎟⎟⎟⎟ = ⎜⎜⎜⎜⎝  L 2  1 t ⎠ ΓV In − c2 VV 0n,m ⎛ ⎞ ⎜⎜⎜ Im 0m,n ⎟⎟⎟ ⎜ ⎟⎟⎟ = Bv (0n,m ) = Im+n , = ⎜⎝⎜ ⎠ 0n,m In implying (5.178).

(5.178)

(5.179)



Bi-gyrogroups and Bi-gyrovector Spaces – V

Using the notation in (5.130), p. 205, Result (5.179) takes the form ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎜⎜⎜ V ⎟⎟⎟ ⎜⎜⎜−V ⎟⎟⎟ ⎜⎜⎜0n,m ⎟⎟⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ Bv (V)Bv(−V) = ⎜⎜⎜⎜ In ⎟⎟⎟⎟ ⎜⎜⎜⎜ In ⎟⎟⎟⎟ = ⎜⎜⎜⎜ In ⎟⎟⎟⎟ = Bv (0n,m ) = Im+n , ⎝ ⎠⎝ ⎠ ⎝ ⎠ Im Im Im

(5.180)

(0n,m, In , Im )t = Bv (0n,m ) being the identity Lorentz transformation of signature (m, n). Let Λ ∈ SO(m, n), m, n ∈ N, be a Lorentz transformation of signature (m, n). Then, by (5.130), p. 205, Λ = Λ(Om , V, On ) = ρ(Om )Bv (V)λ(On) .

(5.181)

Hence, by (5.181) and by Lemma 5.25, and by noting that ρ(Om ) and λ(On ) commute, we obtain the inverse Lorentz transformation in the following chain of equations: Λ−1 = λ(Otn )Bv(−V)ρ(Otm) = λ(Otn )ρ(Otm)Bv(−VOtm ) = ρ(Otm )λ(Otn )Bv(−VOtm )

(5.182)

= ρ(Otm )Bv(−Otn VOtm )λ(Otn ) −1 −1 −1 = ρ(O−1 m )Bv (−On VOm )λ(On ) .

In the column notation (5.130), p. 205, Result (5.182) takes the form shown in (5.183) in the following theorem, which is associated with the parameter V ∈ Rn×m c , and is similar to Theorem 4.15, p. 120, which is associated with the parameter P ∈ Rn×m . Theorem 5.35. (Inverse Lorentz Transformation). The inverse of a Lorentz transformation Λ = (V, On , Om )t is given by the equation ⎛ ⎞−1 ⎛ −1 −1 ⎞ ⎜⎜⎜−On VOm ⎟⎟⎟ ⎜⎜⎜ V ⎟⎟⎟ ⎟⎟⎟ ⎜ ⎜⎜⎜ ⎟⎟⎟ ⎟⎟⎟ . ⎜⎜⎜ On ⎟⎟⎟ = ⎜⎜⎜⎜⎜ O−1 (5.183) n ⎟⎟⎠ ⎜⎜⎝ ⎜⎜⎝ ⎟⎟⎠ −1 Om Om

5.8. Bi-boosts We know by construction that the bi-boost Bv (V), V ∈ Rn×m c , of signature (m, n), m, n ∈ N, preserves the inner product of signature (m, n) in the pseudo-Euclidean space Rm,n . However, solely owing to the commuting relations in (5.119), a direct proof is straightforward and simple and, hence, instructive. Accordingly, the aim of this section is to prove directly that the bi-boost Bv(V) preserves the pseudo-Euclidean inner product of

215

216

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

signature (m, n), m, n ∈ N, in (5.186). Let ⎛ ⎞ ⎜⎜⎜ t1 ⎟⎟⎟ ⎜⎜ ⎟⎟ t = ⎜⎜⎜⎜ ... ⎟⎟⎟⎟ ∈ Rm , ⎜⎝ ⎟⎠ tm so that

⎛ ⎞ ⎜⎜⎜ x1 ⎟⎟⎟ ⎜⎜ ⎟⎟ x = ⎜⎜⎜⎜ ... ⎟⎟⎟⎟ ∈ Rn , ⎜⎝ ⎟⎠ xn

  t = (t1 , . . . , tm , x1 , . . . , xn )t ∈ Rm,n x

(5.184)

(5.185)

is a generic point of the pseudo-Euclidean space Rm,n . The inner product of signature (m, n) in Rm,n involves the constant c > 0 according to the equation      t    t1 t2 t1 Im t2 0m,n · := = t1 ·t2 − c−2 x1 ·x2 (5.186) x1 x2 x1 0n,m −c−2 In x2 for all (t1 , x1 )t , (t2 , x2 )t ∈ Rm,n , where t1 ·t2 = tt1 t2 and x1 ·x2 = xt1 x2 are the standard inner product in Rm and Rn , respectively. The bi-boost Bv (V) is given by its (m + n)×)(m + n) matrix representation (5.127), ⎛ R ⎞ ⎜⎜⎜ ΓV c−2 ΓRV V t ⎟⎟⎟ ⎟⎟⎠⎟ (5.187) Bv(V) = ⎜⎜⎝⎜ ΓVL V ΓVL m, n ∈ N, where the left and the right gamma factors are symmetric matrices given by (5.115),  −1 ΓVL = In − c−2 VV t ∈ Rn×n (5.188)  −1 ΓRV = Im − c−2 V t V ∈ Rm×m . of the ambient The space of the parameter V in (5.187) – (5.188) is the c-ball Rn×m c n×m space R , which is given by (5.68), p. 193, or equivalently, by (5.70). in the c-ball Rn×m of Rn×m is constructed by conThe generic parameter V ∈ Rn×m c c n×m and employing (5.64), structing a generic parameter P ∈ R   −1 −1 V = φ(P) = In + c−2 PPt P = P Im + c−2 Pt P , (5.189) as shown in Theorem 5.10. Theorem 5.36. The bi-boost

⎛ R ⎞ ⎜⎜⎜ ΓV c−2 ΓRV V t ⎟⎟⎟ ⎟⎟⎟ Bv(V) = ⎜⎜⎜⎝ ⎠ ΓVL V ΓVL

(5.190)

Bi-gyrogroups and Bi-gyrovector Spaces – V

V ∈ Rn×m c , m, n ∈ N, leaves the pseudo-Euclidean inner product (5.186) invariant, that is,         t t t1 t ·B (V) 2 = 1 · 2 (5.191) Bv (V) x1 v x2 x1 x2 for any t1 , t2 ∈ Rm and x1 , x2 ∈ Rn . Proof. Following (5.190) we have ⎞ ⎞  ⎛ R   ⎛ R ⎜⎜⎜ΓV t + c−2 ΓRV V t x⎟⎟⎟ ⎜⎜⎜ ΓV c−2 ΓRV V t ⎟⎟⎟ t t ⎟⎟⎠ . ⎟⎟⎠ = ⎜⎜⎝ Bv (V) = ⎜⎜⎝ x x ΓVL Vt + ΓVL x ΓVL V ΓVL

(5.192)

Hence, by (5.184) – (5.186), (5.119), and (5.121), we have the following chain of equations:     t t Bv (V) 1 ·Bv(V) 2 x1 x2 ⎞t  ⎞ ⎛ R ⎛ R ⎜⎜⎜ΓV t2 + c−2 ΓRV V t x2 ⎟⎟⎟ ⎜⎜⎜ΓV t1 + c−2 ΓRV V t x1 ⎟⎟⎟ Im 0 m,n = ⎜⎜⎝ ⎜⎜⎝ L ⎟⎟⎠ ⎟⎟⎠ −2 ΓVL Vt1 + c−2 ΓVL x1 0n,m −c In ΓV Vt2 + ΓVL x2 ⎛ R ⎞ ⎜⎜ΓV t2 + c−2 ΓRV V t x2 ⎟⎟⎟ t R −2 t R −2 t t L −2 t L ⎜ ⎜ ⎟⎟⎠ = (t1 ΓV + c x1 VΓV , − c t1 V ΓV − c x1 ΓV ) ⎜⎝ ΓVL Vt2 + ΓVL x2 = tt1 (ΓRV )2 t2 + c−2 tt1 (ΓRV )2 V t x2 + c−2 xt1 V(ΓRV )2 t2 + c−4 xt1 V(ΓRV )2 V t x2 − c−2 (tt1 V t (ΓVL )2 Vt2 + tt1 V t (ΓVL )2 x2 + xt1 (ΓVL )2 Vt2 + xt1 (ΓVL )2 x2 ) =

tt1 (ΓRV )2 t2

+

c−2 tt1 (ΓRV )2 V t x2

+

c−2 xt1 V(ΓRV )2 t2

+

(5.193)

c−4 xt1 (ΓVL )2 VV t x2

− (c−2 tt1 V t V(ΓRV )2 t2 + c−2 tt1 (ΓRV )2 V t x2 + c−2 xt1 V(ΓRV )2 t2 + c−2 xt1 (ΓVL )2 ) = tt1 (Im − c−2 V t V)(ΓRV )2 − c−2 xt1 (ΓVL )2 (In − c−2 VV t )x2 = tt1 t2 − c−2 xt1 x2 = t1 ·t2 − c−2 x1 ·x2     t t = 1 · 2 . x1 x2 It follows from (5.193) that the bi-boost Bv (V) preserves the inner product (5.186) of signature (m, n), as desired. 

217

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Example 5.37. Following (5.188) – (5.68) we have the obvious limits of large c lim ΓRV = Im

c→∞

lim ΓL c→∞ V lim Rn×m c→∞ c

= In =R

n×m

(5.194) .

Hence, in that limit we have

⎛ R ⎞ ⎛ ⎞ ⎜⎜⎜ ΓV c−2 ΓRV V t ⎟⎟⎟ ⎜⎜Im 0m,n ⎟⎟ ⎜ ⎟⎟⎟ , ⎜ ⎟ ⎟⎟⎠ = ⎜⎜⎝ B∞ (V) := lim Bv (V) = lim ⎜⎜⎝ ⎠ c→∞ c→∞ L L V In ΓV V ΓV

(5.195)

where V ∈ Rn×m under the limit and V ∈ Rn×m otherwise. c The limit of (5.192) as c approaches infinity yields an obvious generalization of the familiar Galilei transformation in a pseudo-Euclidean space of signature (m, n),        t t Im 0m,n t = , (5.196) B∞ (V) = x + Vt V In x x t ∈ Rm , x ∈ Rn , V ∈ Rn×m . Being the limit as c → ∞ of the Lorentz bi-boost Bv (V), V ∈ Rn×m c , of signature (m, n), we naturally call B∞ (V), V ∈ Rn×m , a Galilei bi-boost of signature (m, n). Low signature Galilei bi-boosts are studied in Sects. 6.1, p. 297, and 6.2, p. 298. It is clear from (5.195) that the product of two Galilei bi-boosts of signature (m, n) is a Galilei bi-boost of signature (m, n) again, given by parameter addition, that is, B∞ (V1 )B∞(V2 ) = B∞ (V1 + V2 )

(5.197)

for all V1 , V2 ∈ Rn×m . Unlike the simplicity of the Galilei bi-boost product in (5.197), its Lorentz counterpart gives rise in Sect. 5.9 to a rich, interesting algebraic structure.

5.9. Bi-boost Product The Lorentz transformation product law is expressed in terms of the old parameter P ∈ Rn×m in Theorem 4.31, p. 139. The objective of this section is to derive the Lorentz transformation product law expressed in terms of the new parameter V ∈ Rn×m c . , Let Bv (Vk ), k = 1, 2, be two bi-boosts parametrized by Vk ∈ Rn×m c ⎛ R ⎞ 1 R t ⎜⎜ ΓV Γ V ⎟⎟ c2 Vk k ⎟ ⎟⎟ . (5.198) Bv (Vk ) = ⎜⎜⎜⎝ L k ΓVk Vk ΓVLk ⎠

Bi-gyrogroups and Bi-gyrovector Spaces – V

By matrix multiplication and the commuting relations (5.119), ⎛ R ⎞⎛ R ⎞ 1 R t⎟ ⎜ 1 R t⎟ ⎜⎜⎜ ΓV1 ΓV2 2 ΓV V1 ⎟ 2 ΓV V2 ⎟ ⎜ c c ⎜ ⎟ ⎟⎟⎟ 1 2 Bv (V1 )Bv (V2 ) = ⎜⎜⎝ L ⎟⎟ ⎜⎜ ΓV1 V1 ΓVL1 ⎠ ⎝ΓVL2 V2 ΓVL2 ⎠ ⎛ R R ⎜⎜ΓV ΓV + 12 ΓRV V t ΓVL V2 = ⎜⎜⎜⎝ 1L 2 R c 1L 1 L 2 ΓV1 V1 ΓV2 + ΓV1 ΓV2 V2

⎞

1 (ΓRV1 ΓRV2 V2t + ΓRV1 V1t ΓVL2 )⎟⎟⎟ c2 ⎟⎟ 1 L R t L L ⎠ Γ V Γ V + Γ Γ 1 V2 2 V1 V2 c2 V1

⎛ R ⎞ ⎛ ⎜⎜⎜ΓV1 (Im + c12 V1t V2 )ΓRV2 c12 ΓRV1 (V1 + V2 )t ΓVL2 ⎟⎟⎟ ⎜⎜⎜E 11 ⎜ ⎟⎟ =: ⎜⎜ = ⎜⎝ L ΓV1 (V1 + V2 )ΓRV2 ΓVL1 (In + c12 V1 V2t )ΓVL2 ⎠ ⎝E 21

(5.199) ⎞

1 E ⎟⎟ c2 12 ⎟ ⎟

E 22

⎟⎠ .

As we see from (5.199), the product of two bi-boosts need not be a bi-boost. However, it is a Lorentz transformation and, as such, it uniquely possesses the bi-gyration decomposition (5.128). Hence, by (5.128) – (5.130), p. 204, we can express the biboost product Bv (V1 )Bv(V2 ) by the following unique decomposition. Λ = Bv (V1 )Bv (V2 ) = ρ(rgyr[V1 , V2 ])Bv(V1 ⊕V2 )λ(lgyr[V1 , V2 ]) ⎞⎛ ⎛ ⎜⎜⎜rgyr[V1 , V2 ] 0m,n ⎟⎟⎟ ⎜⎜⎜ ΓRV12 ⎟⎟⎠ ⎜⎜⎝ = ⎜⎜⎝ 0n,m In ΓVL12 V12 ⎛ ⎜⎜rgyr[V1 , V2 ]ΓRV12 = ⎜⎜⎜⎝ ΓVL12 V12 ⎛ ⎜⎜E 11 = ⎜⎜⎜⎝ E 21

⎞⎛

1 R Γ V t ⎟⎟ ⎜⎜⎜ Im c2 V12 12 ⎟ ⎟⎟ ⎜⎜ ΓVL12 ⎠ ⎝0n,m

0m,n lgyr[V1 , V2 ]

⎞ 1 t rgyr[V1 , V2 ]ΓRV12 V12 lgyr[V1 , V2 ]⎟⎟⎟ c2 ⎟⎟⎠ ΓVL12 lgyr[V1 , V2 ]

⎞ ⎟⎟⎟ ⎟⎟⎠ (5.200)

⎞

1 E ⎟⎟ c2 12 ⎟ ⎟

E 22

⎟⎠ ,

where the composite parameter V12 ∈ Rn×m c , V12 =: V1 ⊕V2 ,

(5.201)

and the bi-gyration (lgyr[V1 , V2 ], rgyr[V1 , V2 ]) ∈ SO(n) × SO(m) are to be determined in terms of V1 and V2 in (5.204) –(5.205) and (5.217). The uniqueness of the Lorentz transformation bi-gyration decomposition, insured by the Bi-gyration Decomposition Theorem 5.20, implies that the matrix entries E i j , i, j = 1, 2, defined in (5.199) and the matrix entries E i j defined in (5.200) are identically equal.

219

220

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Hence, the expressions V12 =: V1 ⊕V2 ∈ Rn×m c lgyr[V1 , V2 ] ∈ SO(n)

(5.202)

rgyr[V1 , V2 ] ∈ SO(m) , V1 , V2 ∈ Rn×m c , that appear in (5.200) are uniquely determined by the Bi-gyration Decomposition Theorem 5.20. Employing (5.199) – (5.200), in the following Subsections 5.9.1 –5.9.3, we determine each of the expressions in (5.202) in terms of V1 and V2 . Remark 5.38. Thus, the Bi-gyration Decomposition Theorem 5.20, p. 204, gives rise to a binary operation, ⊕, and a bi-gyrator, gyr = (lgyr, rgyr), in the ball Rn×m of the c parameter V. (Compare with Remark 4.17, p. 123).

5.9.1. E 21 In this subsection we study the consequences of the equality between E 21 in (5.199) and E 21 in (5.200). With V12 = V1 ⊕V2 , we see from (5.200) that E 21 = ΓVL1 ⊕V2 (V1 ⊕V2 ) . Hence, by (5.120), the binary operation ⊕ in Rn×m is given by c   −1 −1 t t V1 ⊕V2 = In + c−2 E 21 E 21 E 21 = E 21 Im + c−2 E 21 E 21 ,

(5.203)

(5.204)

where, by (5.199), E 21 = ΓVL1 (V1 + V2 )ΓRV2 ,

(5.205)

V1 , V2 ∈ Rn×m c . Thus, the bi-gyrosum V1 ⊕V2 is expressed in (5.204) – (5.205) in terms of V1 and V2 . It follows immediately from (5.118) and (5.204) – (5.205) that the binary operation ⊕ possesses the automorphic inverse property in Rn×m c (−V1 )⊕(−V2 ) = −(V1 ⊕V2 ) .

(5.206)

It is interesting to note that following (5.68), (5.204) – (5.205), and (5.194), we have the limits lim Rn×m = Rn×m c

c→∞

lim (V1 ⊕V2 ) = V1 + V2 .

(5.207)

c→∞

Thus, as expected, in the limit of large c, the binary operation ⊕ in the c-ball Rn×m c

Bi-gyrogroups and Bi-gyrovector Spaces – V

tends to the common matrix addition, +, in the ambient space Rn×m . specializes In the special case when m = 1, the binary operation ⊕ in the ball Rn×m c to Einstein velocity addition of special relativity in the Euclidean ball Rnc , as indicated in Example 5.32, p. 212, and explained in Sect. 5.17, p. 248. Einstein velocity addition (2.2), p. 10, in the Euclidean ball Rnc is studied in Chaps. 2 and 3 and, for instance, in [81, 84, 93, 94, 95, 96, 98]. Additionally, the equality between E 21 in (5.200) and in (5.199), along with the first commuting relation in (5.119), yields the elegant equations ΓVL1 ⊕V2 (V1 ⊕V2 ) = ΓVL1 (V1 + V2 )ΓRV2 (V1 ⊕V2 )ΓRV1 ⊕V2 = ΓVL1 (V1 + V2 )ΓRV2

(5.208)

which show how closely the binary operations ⊕ and + are related to each other in terms of the bi-gamma factor. Lemma 5.39. The expression E 21 in (5.203) possesses the commuting relations   −1 −1 t t t t E 21 E 21 In + c−2 E 21 E 21 = In + c−2 E 21 E 21 E 21 E 21 (5.209)   −1 −1 t t t t E 21 E 21 Im + c−2 E 21 E 21 = Im + c−2 E 21 E 21 E 21 E 21 and the identities ΓVL1 ⊕V2 ΓRV1 ⊕V2

:= :=

 

In −

c−2 (V

1 ⊕V2 )(V1 ⊕V2

)t

Im − c−2 (V1 ⊕V2 )t (V1 ⊕V2 )

−1

−1

= =



t In + c−2 E 21 E 21



(5.210) Im +

t c−2 E 21 E 21 .

Proof. The commuting relations in (5.209) follow immediately from the commuting relation in (5.204). By (5.204) and (5.209) we have   −1 −1 t t t t −2 (V1 ⊕V2 )(V1 ⊕V2 ) = In + c E 21 E 21 E 21 E 21 In + c−2 E 21 E 21 (5.211) −2 t −1 t = (In + c E 21 E 21 ) E 21 E 21 .

221

222

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Hence, t −1 t In − c−2 (V1 ⊕V2 )(V1 ⊕V2 )t = In − c−2 (In + c−2 E 21 E 21 ) E 21 E 21 t −1 t t −1 t = (In + c−2 E 21 E 21 ) (In + c−2 E 21 E 21 ) − c−2 (In + c−2 E 21 E 21 ) E 21 E 21 t −1 t t = (In + c−2 E 21 E 21 ) (In + c−2 E 21 E 21 − c−2 E 21 E 21 )

(5.212)

t −1 ) , = (In + c−2 E 21 E 21

thus proving the first identity in (5.210). The proof of the second identity in (5.210) is similar. 

5.9.2. E 11 and E 22 In this subsection we study the consequences of the equality between E 11 (E 22 ) in (5.199) and E 11 (E 22 ) in (5.200). With V12 = V1 ⊕V2 , we see from (5.200) that E 11 = rgyr[V1 , V2 ]ΓRV1 ⊕V2 E 22 = ΓVL1 ⊕V2 lgyr[V1 , V2 ] ,

(5.213)

so that for all V1 , V2 ∈ Rn×m c , lgyr[V1 , V2 ] = (ΓVL1 ⊕V2 )−1 E 22 rgyr[V1 , V2 ] = E 11 (ΓRV1 ⊕V2 )−1 ,

(5.214)

where, by (5.199), 1 t V V2 )ΓRV2 c2 1 1 E 22 = ΓVL1 (In + 2 V1 V2t )ΓVL2 c and where, by (5.115), p. 201, and Lemma 5.39,   −1 1 1 t ΓVL1 ⊕V2 = In − 2 (V1 ⊕V2 )(V1 ⊕V2 )t = In + 2 E 21 E 21 c c   −1 1 1 t ΓRV1 ⊕V2 = Im − 2 (V1 ⊕V2 )t (V1 ⊕V2 ) = Im + 2 E 21 E 21 . c c E 11 = ΓRV1 (Im +

(5.215)

(5.216)

Bi-gyrogroups and Bi-gyrovector Spaces – V

Following (5.214) - (5.216) and (5.115) we have lgyr[V1 , V2 ] =   −1  −1 −1 1 1 1 1 t t t t In + 2 E 21 E 21 In − 2 V1 V1 (In + 2 V1 V2 ) In − 2 V2 V2 c c c c 1 = (ΓVL1 ⊕V2 )−1 ΓVL1 (In + 2 V1 V2t )ΓVL2 c rgyr[V1 , V2 ] =   −1 −1  −1 1 t 1 t 1 t 1 t Im − 2 V1 V1 (Im + 2 V1 V2 ) Im − 2 V2 V2 Im + 2 E 21 E 21 c c c c 1 = ΓRV1 (Im + 2 V1t V2 )ΓRV2 (ΓRV1 ⊕V2 )−1 , c where E 21 is given by (5.205). It follows immediately from (5.217), (5.205), and (5.118) that

(5.217)

lgyr[−V1 , −V2 ] =lgyr[V1 , V2 ] rgyr[−V1 , −V2 ] =rgyr[V1 , V2 ] .

(5.218)

Equations (5.213) and (5.215) yield the right and the left bi-gamma identity 1 t V V2 )ΓRV2 c2 1 1 ΓVL1 ⊕V2 lgyr[V1 , V2 ] = ΓVL1 (In + 2 V1 V2t )ΓVL2 . c

rgyr[V1 , V2 ]ΓRV1 ⊕V2 = ΓRV1 (Im +

(5.219)

For V ∈ Rn×m , the left (right) gamma factor ΓVL (ΓRV ) is real if and only if V ∈ Rn×m c , as we see from (5.68) and (5.115), p. 201. Hence, each of the two equations in (5.219) ⇒ V1 ⊕V2 ∈ Rn×m yields the following implication: V1 , V2 ∈ Rn×m c c , so that ⊕ is a binary n×m operation in Rc as expected. Example 5.40. In the special case when m = 1, rgyr[V1 , V2 ] ∈ SO(1) = {1}, so that rgyr[V1 , V2 ] = 1. Moreover, when m = 1 we have ΓRm=1,V = γV by (5.154), p. 210. Hence, the first identity in (5.219) descends to the gamma identity, 1 V1 ·V2 ), (m = 1) , (5.220) c2 V1 ·V2 = V1t V2 , which plays an important role in special relativity and its underlying hyperbolic geometry [81, 93, 96]. In fact, the gamma identity (5.220) signaled the emergence of hyperbolic geometry in special relativity when it was first studied by Sommerfeld [65] and Variˇcak [102, γV1 ⊕V2 = γV1 γV2 (1 +

223

224

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

103] in terms of rapidities [93, p. 90]. Formalizing the results in (5.204) and (5.217), we obtain the following theorem. Theorem 5.41. (Bi-gyroaddition and Bi-gyration, V). The bi-gyroaddition and the bi-gyration in the parameter bi-gyrogroupoid (Rn×m c , ⊕) are given by the equations   −1 −1 t t −2 V1 ⊕V2 = In + c E 21 E 21 E 21 = E 21 Im + c−2 E 21 E 21 (5.221) and lgyr[V1 , V2 ] =   −1  −1 −1 1 1 1 1 t t t t In + 2 E 21 E 21 In − 2 V1 V1 (In + 2 V1 V2 ) In − 2 V2 V2 c c c c 1 = (ΓVL1 ⊕V2 )−1 ΓVL1 (In + 2 V1 V2t )ΓVL2 c rgyr[V1 , V2 ] =   −1 −1  −1 1 t 1 t 1 t 1 t Im − 2 V1 V1 (Im + 2 V1 V2 ) Im − 2 V2 V2 Im + 2 E 21 E 21 c c c c 1 = ΓRV1 (Im + 2 V1t V2 )ΓRV2 (ΓRV1 ⊕V2 )−1 , c where E 21 = ΓVL1 (V1 + V2 )ΓRV2 ∈ Rn×m  −1 ΓVL := In − c−2 VV t ∈ Rn×n  −1 ΓRV := Im − c−2 V t V ∈ Rm×m ,

(5.222)

(5.223)

for all V, V1 , V2 ∈ Rn×m c .

5.9.3. E 12 In this subsection we study the consequences of the equality between E 12 in (5.199) and E 12 in (5.200). The equality between E 12 in (5.200) and in (5.199) yields the equation rgyr[V1 , V2 ]ΓRV1 ⊕V2 (V1 ⊕V2 )t lgyr[V1 , V2 ] = ΓRV1 (V1 + V2 )t ΓVL2 for all V1 , V2 ∈ Rn×m c , m, n ∈ N.

(5.224)

Bi-gyrogroups and Bi-gyrovector Spaces – V

5.10. Product of Bi-boosts Theorem 5.42. (The Bi-boost Product Law). Let m, n ∈ N be any two positive integers, and let Λ = Bv (V1 )Bv(V2 ) be the product of two bi-boosts parametrized by V1 , V2 ∈ Rn×m c . Then, ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎜⎜⎜V1 ⎟⎟⎟ ⎜⎜⎜V2 ⎟⎟⎟ ⎜⎜⎜ V1 ⊕V2 ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ Λ = Bv (V1 )Bv (V2 ) = ⎜⎜⎜⎜ In ⎟⎟⎟⎟ ⎜⎜⎜⎜ In ⎟⎟⎟⎟ = ⎜⎜⎜⎜lgyr[V1 , V2 ]⎟⎟⎟⎟ , ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎠ ⎝ ⎠⎝ ⎠ ⎝ Im Im rgyr[V1 , V2 ]

(5.225)

(5.226)

where the gyrosum V1 ⊕V2 ∈ Rn×m and the bi-gyration (lgyr[V1 , V2 ], rgyr[V1 , V2 ]) ∈ c SO(n) × SO(m) are given by Theorem 5.41. Proof. The equations in (5.226) are the first equations in (5.200), p. 219, written in the column notation (5.130), p. 205. Hence, the proof follows immediately from Theorem 5.41.  In the special case when V1 = V and V2 = 0n,m , (5.226) specializes to ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎜⎜⎜ V ⎟⎟⎟ ⎜⎜⎜0n,m ⎟⎟⎟ ⎜⎜⎜ V⊕0n,m ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ Bv (V)Bv(0n,m ) = ⎜⎜ In ⎟⎟ ⎜⎜ In ⎟⎟ = ⎜⎜lgyr[V, 0n,m ]⎟⎟⎟⎟ . ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎠ ⎝ ⎠ ⎝ ⎠⎝ Im Im rgyr[V, 0n,m ]

(5.227)

But Bv (0n,m ) = Im+n , so that

⎛ ⎞ ⎜⎜⎜ V ⎟⎟⎟ ⎜⎜⎜⎜ ⎟⎟⎟⎟ Bv (V)Bv(0n,m ) = Bv (V) = ⎜⎜⎜⎜ In ⎟⎟⎟⎟ . ⎜⎜⎜⎝ ⎟⎟⎟⎠ Im

(5.228)

Comparing (5.227) and (5.228), we obtain the identities V⊕0n,m = V lgyr[V, 0n,m ] = In rgyr[V, 0n,m ] = Im for all V ∈ Rn×m c .

(5.229)

225

226

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Similarly to (5.227) – (5.229), by considering Bv(0n,m )Bv (V) one can show that 0n,m ⊕V = V lgyr[0n,m , V] = In rgyr[0n,m , V] = Im

(5.230)

n×m for all V ∈ Rn×m is the identity element of the gyrogroupoid c . Hence, 0n,m ∈ Rc n×m (Rc , ⊕), and the gyrations in (5.229) – (5.230) are trivial. In the special case when V1 = V and V2 = −V, (5.226) specializes to

⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎜⎜⎜ V ⎟⎟⎟ ⎜⎜⎜−V ⎟⎟⎟ ⎜⎜⎜ V⊕(−V) ⎟⎟⎟ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ Bv (V)Bv(−V) = ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ = ⎜⎜⎜⎜⎜lgyr[V, −V]⎟⎟⎟⎟⎟ . ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ rgyr[V, −V] Im Im

(5.231)

Comparing (5.231) and (5.180), p. 215, we have V⊕(−V) = 0n,m lgyr[V, −V] = In rgyr[V, −V] = Im

(5.232)

for any V ∈ Rn×m c . Similarly to (5.231) – (5.232), by considering Bv(−V)Bv (V) one can show that (−V)⊕V = 0n,m lgyr[−V, V] = In rgyr[−V, V] = Im

(5.233)

V = −V .

(5.234)

for any V ∈ Rn×m c . Hence, the inverse V of V is

Accordingly, we use the natural notation V1 V2 = V1 ⊕(−V2 )

(5.235)

and write VV = 0n,m lgyr[V, V] = lgyr[V, V] = In rgyr[V, V] = rgyr[V, V] = Im for any V ∈ Rn×m c , etc.

(5.236)

Bi-gyrogroups and Bi-gyrovector Spaces – V

Let Λ = Bv (V1 )Bv (V2 ),

(5.237)

where V1 , V2 ∈ Rn×m and, hence, Λ ∈ SO(m, n), for any m, n ∈ N. Bi-boost products c are given by matrix products. Hence, the inverse, Λ−1 , of Λ can be calculated in two different ways, Λ−1 = (Bv (V1 )Bv (V2 ))−1 = Bv (V2 )−1 Bv (V1 )−1 .

(5.238)

By (5.226) and by the Inverse Lorentz Transformation Theorem 5.35, p. 215, ⎞ ⎞−1 ⎛ ⎛ ⎜⎜⎜ V1 ⊕V2 ⎟⎟⎟ ⎜⎜⎜−lgyr−1 [V1 , V2 ](V1 ⊕V2 )rgyr−1 [V1 , V2 ]⎟⎟⎟ ⎟⎟⎟⎟ ⎟⎟ ⎜⎜ ⎜⎜ (5.239) Λ−1 = ⎜⎜⎜⎜⎜lgyr[V1 , V2 ]⎟⎟⎟⎟⎟ = ⎜⎜⎜⎜⎜ lgyr−1 [V1 , V2 ] ⎟⎟⎟ , ⎟⎠ ⎟⎠ ⎜⎝ ⎜⎝ −1 rgyr[V1 , V2 ] rgyr [V1 , V2 ] where we use the notation lgyr−1 [V1 , V2 ] = (lgyr[V1 , V2 ])−1

and

rgyr−1 [V1 , V2 ] = (rgyr[V1 , V2 ])−1 .

Calculating Λ−1 in a different way, as shown in (5.238), we have by (5.178), p. 214, (5.226), (5.206), and (5.218), ⎞⎛ ⎞ ⎛ ⎜⎜⎜−V2 ⎟⎟⎟ ⎜⎜⎜−V1 ⎟⎟⎟ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ Λ−1 = ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ Im Im ⎛ ⎞ ⎜⎜⎜ (−V2 )⊕(−V1 ) ⎟⎟⎟ ⎜⎜ ⎟⎟ = ⎜⎜⎜⎜⎜lgyr[−V2 , −V1 ]⎟⎟⎟⎟⎟ (5.240) ⎜⎝ ⎟⎠ rgyr[−V2 , −V1 ] ⎛ ⎞ ⎜⎜⎜ −(V2 ⊕V1 ) ⎟⎟⎟ ⎜⎜ ⎟⎟ = ⎜⎜⎜⎜⎜lgyr[V2 , V1 ]⎟⎟⎟⎟⎟ . ⎜⎝ ⎟⎠ rgyr[V2 , V1 ] Comparing (5.239) and (5.240) we obtain the identities lgyr−1 [V1 , V2 ] = lgyr[V2 , V1 ] rgyr−1 [V1 , V2 ] = rgyr[V2 , V1 ] lgyr[V2 , V1 ](V1 ⊕V2 )rgyr[V2 , V1 ] = V2 ⊕V1 . Formalizing the results in (5.241), we obtain the following two theorems.

(5.241)

227

228

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

n×m Theorem 5.43. (Bi-gyration Inversion Law in (Rn×m c , ⊕)). Let (Rc , ⊕), m, n ∈ N, be a bi-gyrogroupoid. Then,

lgyr−1 [V1 , V2 ] = lgyr[V2 , V1 ] rgyr−1 [V1 , V2 ] = rgyr[V2 , V1 ]

(5.242)

for all V1 , V2 ∈ Rn×m c . n×m Theorem 5.44. (Bi-gyrocommutative Law in (Rn×m c , ⊕)). Let (Rc , ⊕), m, n ∈ N, be a bi-gyrogroupoid. Then,

V1 ⊕V2 = lgyr[V1 , V2 ](V2 ⊕V1 )rgyr[V1 , V2 ] .

(5.243)

for all V1 , V2 ∈ Rn×m c . Example 5.45. Transposing (5.224), noting that (lgyr[V1 , V2 ])t = (lgyr[V1 , V2 ])−1 = lgyr[V2 , V1 ] (rgyr[V1 , V2 ])t = (rgyr[V1 , V2 ])−1 = rgyr[V2 , V1 ]

(5.244)

we obtain the equation lgyr[V2 , V1 ](V1 ⊕V2 )ΓRV1 ⊕V2 rgyr[V2 , V1 ] = ΓVL2 (V1 + V2 )ΓRV1 .

(5.245)

Manipulating the left-hand side of (5.245) by means of the first commuting relation in (5.119), p. 202, and manipulating the right-hand side of (5.245) by means of (5.208) we obtain the equation lgyr[V2 , V1 ]ΓVL1 ⊕V2 (V1 ⊕V2 )rgyr[V2 , V1 ] = ΓVL2 ⊕V1 (V2 ⊕V1 )

(5.246)

for all V1 , V2 ∈ Rn×m c . The resulting elegant equation demonstrates that the application of the bi-gyration (lgyr[V2 , V1 ], rgyr[V2 , V1 ]) takes ΓVL1 ⊕V2 (V1 ⊕V2 ) into ΓVL2 ⊕V1 (V2 ⊕V1 ). Equation (5.246) thus gives rise to a nice bi-gyrocommutative-like law.

5.11. Product of Lorentz Transformations Let Λ1 and Λ2 be two Lorentz transformations of signature (m, n), m, n ∈ N, so that, according to (5.130), Λ1 = Λ(On,1 , V1 , Om,1 ) = ρ(Om,1 )Bv(V1 )λ(On,1 ) = (V1 , On,1 , Om,1 )t Λ2 = Λ(On,2 , V2 , Om,2 ) = ρ(Om,2 )Bv(V2 )λ(On,2 ) = (V2 , On,2 , Om,2 )t .

(5.247)

The product Λ1 Λ2 of Λ1 and Λ2 is obtained in the following chain of equations,

Bi-gyrogroups and Bi-gyrovector Spaces – V

which are numbered for subsequent explanation: (1)

 Λ1 Λ2 === ρ(Om,1 )Bv (V1 )λ(On,1 )ρ(Om,2 )Bv(V2 )λ(On,2 ) (2)

 === ρ(Om,1 )Bv (V1 )ρ(Om,2 )λ(On,1 )Bv(V2 )λ(On,2 ) (3)

 === ρ(Om,1 )ρ(Om,2 )Bv(V1 Om,2 )Bv(On,1 V2 )λ(On,1 )λ(On,2 ) (4)

 === ρ(Om,1 Om,2 )Bv(V1 Om,2 )Bv(On,1 V2 )λ(On,1 On,2 ) (5)

 === ρ(Om,1 Om,2 ) × ρ(rgyr[V1 Om,2 , On,1 V2 ])Bv(V1 Om,2 ⊕On,1 V2 )λ(lgyr[V1 Om,2 , On,1 V2 ]) × λ(On,1 On,2 ) (6)

 === ρ(Om,1 Om,2 rgyr[V1 Om,2 , On,1 V2 ]) × Bv (V1 Om,2 ⊕On,1 V2 ) × λ(lgyr[V1 Om,2 , On,1 V2 ]On,1 On,2 ) . (5.248) Derivation of the numbered equalities in (5.248): (1) (2) (3) (4)

This equation follows from (5.247). Follows from (1) since λ(On,1 ) and ρ(Om,2 ) commute. Follows from (2) by Lemma 5.25, p. 208. Follows from (3) by the obvious matrix identities ρ(Om,1 )ρ(Om,2 ) = ρ(Om,1 Om,2 ) and λ(On,1 )λ(On,2 ) = λ(On,1 On,2 ). (5) Follows from (4) by the bi-boost product law (5.226), p. 225. (6) Obvious (Similar to the argument in Item (4)). In the column notation (5.130), p. 205, the result of (5.248) gives the product law of Lorentz transformations in the following theorem. Theorem 5.46. (Lorentz Transformation Product Law, V). The product of two generic Lorentz transformations Λ1 = (V1 , On,1 , Om,1 )t Λ2 = (V2 , On,2 , Om,2 )t

(5.249)

229

230

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

of signature (m, n), m, n ∈ N, in terms of parameter composition is given by ⎞⎛ ⎞ ⎛ ⎜⎜⎜ V1 ⎟⎟⎟ ⎜⎜⎜ V2 ⎟⎟⎟ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ Λ1 Λ2 = ⎜⎜⎜⎜⎜ On,1 ⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ On,2 ⎟⎟⎟⎟⎟ ⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ Om,1 Om,2 ⎛ ⎞ V1 Om,2 ⊕On,1 V2 ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟ = ⎜⎜⎜⎜ lgyr[V1 Om,2 , On,1 V2 ]On,1 On,2 ⎟⎟⎟⎟⎟ , ⎜⎝ ⎟⎠ Om,1 Om,2 rgyr[V1 Om,2 , On,1 V2 ]

(5.250)

where ⊕, lgyr and rgyr are given by Theorem 5.41, p. 224, in terms of the parameters V1 , V2 ∈ Rn×m c . Remarkably, the Lorentz transformation product laws in (5.250) for the parameter V and in (4.180) for the parameter P of Theorem 5.46 and of Theorem 4.31, p. 139, respectively, have the same form, while the definitions of ⊕, lgyr, and rgyr in Theorems 5.46 and 4.31 do not share the same form.

5.12. Bi-gyroassociative Law in Bi-gyrogroupoids Matrix multiplication is associative. Hence (Λ1 Λ2 )Λ3 = Λ1 (Λ2 Λ3 ) .

(5.251)

Let V1 , V2 , V3 ∈ Rn×m . On the one hand, by (5.226) and (5.250), ⎧⎛ ⎞ ⎛ ⎞⎫ ⎛ ⎞ ⎪ ⎜⎜⎜V3 ⎟⎟⎟ ⎜⎜⎜V1 ⎟⎟⎟ ⎜⎜⎜V2 ⎟⎟⎟⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎜⎜⎜⎜ ⎟⎟⎟⎟ ⎨⎜⎜⎜⎜ ⎟⎟⎟⎟ ⎜⎜⎜⎜ ⎟⎟⎟⎟⎪ (B(V1 )B(V2))B(V3) = ⎪ ⎜⎜⎜ In ⎟⎟⎟ ⎜⎜⎜ In ⎟⎟⎟ ⎜⎜⎜ In ⎟⎟⎟⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎪ ⎪ ⎪ ⎪ ⎭ ⎝ Im ⎠ ⎩⎝ Im ⎠ ⎝ Im ⎠⎪ ⎛ ⎞⎛ ⎞ ⎜⎜⎜ V1 ⊕V2 ⎟⎟⎟ ⎜⎜⎜V3 ⎟⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ = ⎜⎜⎜⎜⎜lgyr[V1 , V2 ]⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ rgyr[V1 , V2 ] Im ⎛ ⎞ (V1 ⊕V2 )⊕lgyr[V1 , V2 ]V3 ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟ = ⎜⎜⎜⎜ lgyr[V1 ⊕V2 , lgyr[V1 , V2 ]V3 ]lgyr[V1 , V2 ] ⎟⎟⎟⎟⎟ . ⎜⎝ ⎟⎠ rgyr[V1 , V2 ]rgyr[V1 ⊕V2 , lgyr[V1 , V2 ]V3 ]

(5.252)

Bi-gyrogroups and Bi-gyrovector Spaces – V

On the other hand, similarly, by (5.226) and (5.250), ⎞ ⎛ ⎞ ⎧⎛ ⎞ ⎛ ⎞⎫ ⎛ ⎞ ⎛ ⎜⎜⎜V2 ⎟⎟⎟ ⎜⎜⎜V3 ⎟⎟⎟⎪ ⎜⎜⎜V1 ⎟⎟⎟ ⎜⎜⎜ V2 ⊕V3 ⎟⎟⎟ ⎜⎜⎜V1 ⎟⎟⎟ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎟⎟ ⎜⎜ ⎟⎟ ⎪ ⎨⎜⎜⎜⎜ ⎟⎟⎟⎟ ⎜⎜⎜⎜ ⎟⎟⎟⎟⎪ ⎬ ⎜⎜⎜⎜ ⎟⎟⎟⎟ ⎜⎜⎜⎜ = ⎜⎜⎜ In ⎟⎟⎟ ⎜⎜⎜lgyr[V2 , V3 ]⎟⎟⎟⎟⎟ B(V1 )(B(V2)B(V3 )) = ⎜⎜⎜⎜⎜ In ⎟⎟⎟⎟⎟ ⎪ ⎜⎜⎜ In ⎟⎟⎟ ⎜⎜⎜ In ⎟⎟⎟⎪ ⎪ ⎪ ⎪ ⎪ ⎟⎠ ⎜⎝ ⎟⎠ ⎪ ⎪⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠⎪ ⎪ ⎜⎝ ⎟⎠ ⎜⎝ Im ⎩ Im Im ⎭ Im rgyr[V2 , V3 ] ⎛ ⎞ V1 rgyr[V2 , V3 ]⊕(V2 ⊕V3 ) ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟ = ⎜⎜⎜⎜ lgyr[V1 rgyr[V2 , V3 ], V2 ⊕V3 ]lgyr[V2 , V3 ] ⎟⎟⎟⎟⎟ . ⎜⎝ ⎟⎠ rgyr[V2 , V3 ]rgyr[V1 rgyr[V2 , V3 ], V2 ⊕V3 ]

(5.253)

Hence, by (5.251) – (5.253), corresponding entries of the extreme right-hand sides of (5.252) and (5.253) are equal, giving rise to the bi-gyroassociative law (V1 ⊕V2 )⊕lgyr[V1 , V2 ]V3 = V1 rgyr[V2 , V3 ]⊕(V2 ⊕V3 )

(5.254)

and to the bi-gyration identities lgyr[V1 ⊕V2 , lgyr[V1 , V2 ]V3 ]lgyr[V1 , V2 ] = lgyr[V1 rgyr[V2 , V3 ], V2 ⊕V3 ]lgyr[V2 , V3 ] rgyr[V1 , V2 ]rgyr[V1 ⊕V2 , lgyr[V1 , V2 ]V3 ] = rgyr[V2 , V3 ]rgyr[V1 rgyr[V2 , V3 ], V2 ⊕V3 ] (5.255) for all V1 , V2 , V3 ∈ Rn×m c . Interestingly, (5.254) – (5.255) and their P-counterparts, (4.201) – (4.202), p. 143, share the same form. Formalizing the result in (5.254) we obtain the following theorem. Theorem 5.47. (Bi-gyroassociative Law in (Rn×m c , ⊕)). The bi-gyroaddition ⊕ in the possesses the bi-gyroassociative law c-ball Rn×m c (V1 ⊕V2 )⊕lgyr[V1 , V2 ]V3 = V1 rgyr[V2 , V3 ]⊕(V2 ⊕V3 )

(5.256)

for all V1 , V2 , V3 ∈ Rn×m c . Note that in the bi-gyroassociative law (5.256), V1 and V2 are grouped together on the left-hand side, while V2 and V3 are grouped together on the right-hand side. When m = 1 right gyrations are trivial, rgyr[V1 , V2 ] = Im=1 = 1. Hence, in the special case when m = 1, the bi-gyroassociative law (5.256) descends to the gyroassociative law of gyrogroup theory, presented in Def. 2.13, p. 22. The bi-gyroassociative law gives rise to the left and the right cancellation laws in the following theorem.

231

232

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Theorem 5.48. (Left and Right Cancellation Laws in (Rn×m c , ⊕)). n×m The ball bi-gyrogroupoid (Rc , ⊕) possesses the left and the right cancellation laws V2 = V1 rgyr[V1 , V2 ]⊕(V1 ⊕V2 )

(5.257)

V1 = (V1 ⊕V2 )lgyr[V1 , V2 ]V2

(5.258)

and for all V1 , V2 ∈ Rn×m c . Proof. The left cancellation law (5.257) follows from the bi-gyroassociative law (5.256) with V1 = V2 , noting that lgyr[V2 , V2 ] is trivial by (5.236). The right cancellation law (5.258) follows from the bi-gyroassociative law (5.256) with V3 = V2 , noting  that rgyr[V2 , V2 ] is trivial by (5.236). The bi-gyroassociative law gives rise to the left and the right bi-gyroassociative laws in the following theorem. Theorem 5.49. (Left and Right Bi-gyroassociative Law in (Rn×m c , ⊕)). possesses the left bi-gyroassociative law The bi-gyroaddition ⊕ in Rn×m c V1 ⊕(V2 ⊕V3 ) = (V1 rgyr[V3 , V2 ]⊕V2 )⊕lgyr[V1 rgyr[V3 , V2 ], V2 ]V3

(5.259)

and the right bi-gyroassociative law (V1 ⊕V2 )⊕V3 = V1 rgyr[V2 , lgyr[V2 , V1 ]V3 ]⊕(V2 ⊕lgyr[V2 , V1 ]V3 )

(5.260)

for all V1 , V2 , V3 ∈ Rn×m c . Proof. The left bi-gyroassociative law (5.259) is obtained from the bi-gyroassociative law (5.256) by replacing V1 by V1 rgyr[V3 , V2 ] and employing the bi-gyration inversion law (5.242). The right bi-gyroassociative law (5.260) is obtained from the bi-gyroassociative law (5.256) by replacing V3 by lgyr[V2 , V1 ]V3 and employing the bi-gyration inversion law (5.242).  Remark 5.50. Remarkably, this Sect. 5.12 is identical with Sect. 4.16, p. 143, with one exception: the parameter P of the space Rn×m in Sect. 4.16 is replaced by the of the ambient space Rn×m in the present Sect. 5.12. The parameter V of the ball Rn×m c reason for this remarkable coincidence is clear: 1. The theorems in Sect. 4.16, p. 143, are derived from the Lorentz transformation product law (4.180), p. 139, for the parameter P;

Bi-gyrogroups and Bi-gyrovector Spaces – V

2. The theorems in the present Sect. 5.12 are derived from the Lorentz transformation product law (5.250), p. 230, for the parameter V; and 3. The Lorentz transformation product laws in (4.180) and in (5.250) are identically the same with one exception: a. The Lorentz transformation product law in (4.180) involves the space parameter P ∈ Rn×m , while b. The Lorentz transformation product law in (5.250) involves the ball parameter V ∈ Rn×m c . Indeed, the present Sect. 5.12 indicates that results derived from the Lorentz transformation product law in (4.180) for P give rise to results derived from the Lorentz transformation product law in (5.250) for V just by replacing P ∈ Rn×m by V ∈ Rn×m c . To understand this coincidence, relationships between P and V are studied in Sect. 5.13.

5.13. Relationships Between the Bi-boost Parameters P and V Let V = φ(P). Then, by (5.122), p. 203, B p (P) = Bv (V), where

⎛ ⎜⎜⎜ Im + c−2 Pt P B p (P) = ⎜⎜⎜⎜⎝ P ⎛ ⎜⎜⎜ Bv (V) = ⎜⎜⎜⎝

ΓRV

(5.261)

⎞ ⎟⎟⎟ ⎟⎟⎟  ⎟⎠ −2 t In + c PP 1 t P c2

1 R t Γ V c2 V

ΓVL V = VΓRV

= ΓVL

1 t L⎞ V ΓV ⎟⎟⎟ c2 ⎟

(5.262)

⎟⎟⎠ ,

P ∈ Rn×m , V ∈ Rn×m c . Accordingly, let Vk = φ(Pk ) ,

(5.263)

Λ := B p (P1 )B p (P1 ) = Bv (V1 )Bv (V1 ) ∈ SO(m, n) ,

(5.264)

Pk ∈ Rn×m , Vk ∈ Rn×m c , k = 1, 2. Then, the element Λ ∈ SO(m, n),

has two decompositions in (5.265) and (5.267).

233

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

By (4.108), p. 123, Λ possesses the decomposition Λ = B p (P1 )B p (P2 ) = ρ(rgyr p [P1 , P2 ])B p (P1 ⊕ p P2 )λ(lgyr p [P1 , P2 ]) ⎞⎛ ⎛ ⎜⎜⎜rgyr p [P1 , P2 ] 0m,n ⎟⎟⎟ ⎜⎜⎜ BRm,P1 ⊕p P2 ⎟⎟⎟ ⎜⎜⎜ = ⎜⎜⎜⎝ ⎠⎜ 0n,m In ⎝ P1 ⊕ p P2

⎞⎛ 1 (P1 ⊕ p P2 )t ⎟⎟⎟ ⎜⎜ Im c2 ⎟⎜ L Bn,P 1 ⊕ p P2

⎟⎟⎟ ⎜⎜⎜ ⎠⎝

0n,m

⎞ (5.265) ⎟⎟⎟ ⎟⎟⎟ , ⎠ lgyr p [P1 , P2 ] 0m,n

where for clarity we use the notation B p := B,

⊕ p := ⊕,

lgyr p := lgyr and rgyr p := rgyr

(5.266)

in order to emphasize that the associated bi-boost parameter is P ∈ Rn×m as opposed to V ∈ Rn×m c . By (5.200), p. 219, Λ possesses the decomposition Λ = Bv (V1 )Bv (V2 ) = ρ(rgyrv [V1 , V2 ])Bv(V1 ⊕V2 )λ(lgyrv [V1 , V2 ]) ⎞⎛ ⎛ ⎜⎜⎜rgyrv [V1 , V2 ] 0m,n ⎟⎟⎟ ⎜⎜⎜ ΓRV1 ⊕v V2 ⎜ ⎟ ⎜ = ⎜⎝ ⎟⎠ ⎜⎝ 0n,m In ΓVL1 ⊕v V2 (V1 ⊕v V2 )

⎞⎛

1 R Γ (V ⊕ V )t ⎟⎟ ⎜⎜⎜ Im c2 V1 ⊕v V2 1 v 2 ⎟ ⎟⎟⎠ ⎜⎜⎝ ΓVL1 ⊕v V2 0n,m

0m,n

⎞ ⎟⎟⎟ ⎟⎟⎠

lgyrv [V1 , V2 ] (5.267)

where for clarity we use the notation ⊕v := ⊕,

lgyrv := lgyr and rgyrv := rgyr

(5.268)

in order to emphasize that the associated bi-boost parameter is V ∈ Rn×m as opposed c n×m to P ∈ R . Theorem 5.51. Let m, n ∈ N. For any Pk ∈ Rn×m let Vk = φ(Pk ) ∈ Rn×m c , k = 1, 2. Then, lgyr p [P1 , P2 ] = lgyrv [V1 , V2 ] = lgyrv [φ(P1 ), φ(P2 )] rgyr p [P1 , P2 ] = rgyrv [V1 , V2 ] = rgyrv [φ(P1 ), φ(P2 )]

(5.269)

L P1 ⊕ p P2 = ΓVL1 ⊕v V2 (V1 ⊕v V2 ) = Γφ(P (φ(P1 )⊕v φ(P2 ) 1 )⊕v φ(P2 )

(P1 ⊕ p P2 )t = ΓRV1 ⊕v V2 (V1 ⊕v V2 )t = ΓRφ(P1 )⊕v φ(P2 ) (φ(P1 )⊕v φ(P2 )t

(5.270)

L L = ΓVL1 ⊕v V2 = Γφ(P Bn,P 1 ⊕ p P2 1 )⊕v φ(P2 )

BRm,P1 ⊕ p P2 = ΓRV1 ⊕v V2 = ΓRφ(P1 )⊕v φ(P2 )

(5.271)

and B p (P1 ⊕ p P2 ) = Bv (V1 ⊕v V2 ) = Bv (φ(P1 )⊕v φ(P2 )) .

(5.272)

Bi-gyrogroups and Bi-gyrovector Spaces – V

Proof. The decomposition of Λ in the form that appears in (5.265) and in (5.267) is unique. Hence, the two decompositions of Λ in (5.265) and in (5.267) coincide. Noting (5.263), the equality of the two decompositions of Λ in (5.265) and in (5.267) implies the identities in (5.269) – (5.272), which are identities between corresponding blocks of corresponding matrices of the two decompositions.  In Theorems 5.52 and 5.53, 1. the binary operation ⊕ p and the bi-gyration (lgyr p , rgyr p ) are the binary operation ⊕ in Rn×m and the bi-gyration (lgyr, rgyr) given by Theorem 4.18, p. 124 and 2. the binary operation ⊕v and the bi-gyration (lgyrv , rgyrv ) are the binary operation and the bi-gyration (lgyr, rgyr) given by Theorem 5.41, p. 224. ⊕ in Rn×m c Theorem 5.52. Let m, n ∈ N. Then, for any P1 , P2 ∈ Rn×m and any V1 , V2 ∈ Rn×m c , φ(P1 ⊕ p P2 ) = φ(P1 )⊕v φ(P2 ) φ−1 (V1 ⊕v V2 ) = φ−1 (V1 )⊕ p φ−1 (V2 ). Proof. The proof follows immediately from Lemma 5.19, p. 203, and (5.272).

(5.273) 

Theorem 5.53. φ(lgyr p [P1 , P2 ]P3 ) = lgyrv [φ(P1 ), φ(P2 )]φ(P3 ) φ(P3 rgyr p [P1 , P2 ]) = φ(P3 )rgyrv [φ(P1 ), φ(P2 )] ,

(5.274)

m, n ∈ N, for all P1 , P2 , P3 ∈ Rn×m . Proof. Noting that lgyr[P1 , P2 ] ∈ SO(n) and rgyr[P1 , P2 ] ∈ SO(m), we have by (5.87), p. 196, and by (5.269) φ(lgyr p [P1 , P2 ]P3 ) = lgyr p [P1 , P2 ]φ(P3 ) = lgyrv [φ(P1 ), φ(P2 )]φ(P3 )

(5.275)

and φ(P3 rgyr p [P1 , P2 ]) = φ(P3 )rgyr p [P1 , P2 ] = φ(P3 )rgyrv [φ(P1 ), φ(P2 )] .

(5.276) 

235

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Remark 5.54. When the distinction between the binary operations ⊕ p and ⊕v in Rn×m and Rn×m c , respectively, is obvious from the context and need not be emphasized, we may omit the indices p and v. Thus, for instance, writing (5.273) without these indices, φ(P1 ⊕P2 ) = φ(P1 )⊕φ(P2 ) φ−1 (V1 ⊕V2 ) = φ−1 (V1 )⊕φ−1 (V2 ) ,

(5.277)

raises no confusion. In the same way we may omit these indices from bi-gyrations (lgyr p , rgyr p ) and bi-gyrations (lgyrv , rgyrv ) when no confusion arises. We can now employ the bijection φ between Rn×m and Rn×m to show that the inverse c V of V ∈ (Rn×m , ⊕) is given by V = −V

(5.278)

for all V ∈ Rn×m c . Indeed, by (5.277) and (5.66), p. 193, we have φ−1 (V⊕(−V)) = φ−1 (V)⊕φ−1 (−V) = φ−1 (V)⊕(−φ−1 (V) (5.279) = P⊕(−P) = 0n,m , for any V ∈ Rn×m and P = φ−1 (V). Applying φ to the two extreme sides of (5.279) c yields the equation V⊕(−V) = 0n,m , which implies (5.278). Let V = φ(P). Since P = −P we have by means of (5.4), p. 186, and (5.278) φ(P) = φ(−P) = −φ(P) = −V = V = φ(P) .

(5.280)

Similarly, let P = φ−1 (V). Since V = −V we have by means of (5.66), p. 193, and (5.278) φ−1 (V) = φ−1 (−V) = −φ−1 (V) = −P = P = φ−1 (V) .

(5.281)

Hence, by (5.280) and (5.281), φ(P) = φ(P) φ−1 (V) = φ−1 (V) , for all P ∈ Rn×m and V ∈ Rn×m c .

(5.282)

Bi-gyrogroups and Bi-gyrovector Spaces – V

5.14. Properties of Bi-gyrations in the Ball Owing to the bijection φ between the space Rn×m and its c-ball Rn×m c , n, m ∈ N, properties of bi-gyrations in Rn×m induce the same properties for bi-gyrations in Rn×m c , resulting in theorems 5.55 and 5.61. n×m Theorem 5.55. For any V, V1 , V2 ∈ Rn×m c , m, n ∈ N, bi-gyrations in the c-ball Rc possess the following bi-gyration identities:

lgyrv [0n,m , V] = lgyrv [V, 0n,m ] = In rgyrv [0n,m , V] = rgyrv [V, 0n,m ] = Im lgyrv [V, V] = lgyrv [V, V] = In rgyrv [V, V] = rgyrv [V, V] = Im lgyrv [V, V] = In rgyrv [V, V] = Im lgyrv [V1 , V2 ] = lgyrv [V1 , V2 ] rgyrv [V1 , V2 ] = rgyrv [V1 , V2 ] lgyr−1 v [V1 , V2 ] = lgyrv [V2 , V1 ] rgyr−1 v [V1 , V2 ] = rgyrv [V2 , V1 ] .

(5.283)

(5.284)

(5.285)

(5.286)

(5.287)

and the bi-gyration exclusion property lgyrv [V1 , V2 ]  −In rgyrv [V1 , V2 ]  −Im .

(5.288)

Proof. Proof of (5.283): To prove the left gyration identity lgyrv [0n,m , V] = In in (5.283) we have to show that lgyrv [0n,m , V]V1 = V1

(5.289)

for all V, V1 ∈ Rn×m c . Identity (5.289), in turn, follows from its P-counterpart in (4.115), p. 125, which is equivalent to the identity lgyr p [0n,m , P]P1 = P1 ,

(5.290)

for all P, P1 ∈ Rn×m . Indeed, the application of φ to (5.290) yields (5.289) as shown in the chain of equations (5.291) for V = φ(P) and V1 = φ(P1 ).

237

238

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

By means of the first equation in (5.274) and the first equation in (5.4), p. 186, V1 = φ(P1 ) = φ(lgyr p [0n,m , P]P1 ) = lgyrv [φ(0n,m ), φ(P)]φ(P1 )

(5.291)

= lgyrv [0n,m , V]V1 , for all P, P1 ∈ Rn×m and all V, V1 ∈ Rn×m c , thus verifying one of the bi-gyration identities in (5.283). The proof of the remaining bi-gyration identities in (5.283) is similar. Proof of (5.284): To prove the left gyration identity lgyrv [V, V] = In in (5.284) we have to show that lgyrv [V, V]V1 = V1

(5.292)

for all V, V1 ∈ Rn×m c . Identity (5.292), in turn, follows from its P-counterpart in (4.115), p. 125, which is equivalent to the identity lgyr p [P, P]P1 = P1

(5.293)

for all P, P1 ∈ Rn×m . Indeed, the application of φ to (5.293) yields (5.292) as shown in the chain of equations (5.294) below for V = φ(P) and V1 = φ(P1 ). By means of the first equation in (5.274) and the first equation in (5.282), V1 = φ(P1 ) = φ(lgyr p [P, P]P1 ) = lgyrv [φ(P), φ(P)]φ(P1 ) = lgyrv [φ(P), φ(P)]φ(P1 )

(5.294)

= lgyrv [V, V]V1 , for all P, P1 ∈ Rn×m and all V, V1 ∈ Rn×m c , thus verifying one of the bi-gyration identities in (5.284). The proof of the remaining bi-gyration identities in (5.284) is similar. Proof of (5.285): To prove the first bi-gyration identity in (5.285) we have to show that lgyrv [V, V]V1 = V1

(5.295)

for all V, V1 ∈ Rn×m c . Identity (5.295), in turn, follows from its P-counterpart in (4.168), p. 135, which is equivalent to the identity lgyr p [P, P]P1 = P1

(5.296)

for all P, P1 ∈ Rn×m . Indeed, the application of φ to (5.296) yields (5.295) as shown in the chain of equations (5.297) for V = φ(P) and V1 = φ(P1 ).

Bi-gyrogroups and Bi-gyrovector Spaces – V

By means of the first equation in (5.274), V1 = φ(P1 ) = φ(lgyr p [P, P]P1 ) = lgyrv [φ(P), φ(P)]φ(P1 )

(5.297)

= lgyrv [V, V]V1 for all P, P1 ∈ Rn×m and all V, V1 ∈ Rn×m c , thus verifying the first bi-gyration identity in (5.285). The proof of the second bi-gyration identity in (5.285) is similar. Proof of (5.286): By the first identity in (5.269), by the first bi-gyration identity in (4.127), p. 126, and by (5.280), we have for V1 = φ(P1 ) and V2 = φ(P2 ) the following chain of equations that verifies the first bi-gyration identity in (5.286): lgyrv [V1 , V2 ] = lgyrv [φ(P1 ), φ(P2 )] = lgyr p [P1 , P2 ] = lgyr p [P1 , P2 ] (5.298) = lgyrv [φ(P1 ), φ(P2 )] = lgyrv [φ(P1 ), φ(P2 )] = lgyrv [V1 , V2 ] Rn×m c .

for all V1 , V2 ∈ The proof of the second bi-gyration identity in (5.286) is similar. Proof of (5.287): To prove the first bi-gyration identity in (5.287) we have to show that lgyrv [V2 , V1 ]lgyrv [V1 , V2 ]V3 = V3

(5.299)

for all V1 , V2 , V3 ∈ Rn×m c . The P-counterpart of (5.299) follows from the bi-gyration inversion law in (4.197), p. 143, lgyr p [P2 , P1 ]lgyr p [P1 , P2 ]P3 = P3

(5.300)

for all P1 , P2 , P3 ∈ Rn×m . Hence, for Vk = φ(Pk ), k = 1, 2, 3, we have by means of the

239

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

first equation in (5.274) the following chain of equations: V3 = φ(P3 ) = φ(lgyr p [P2 , P1 ]lgyr p [P1 , P2 ]P3 ) = lgyrv [φ(P2 ), φ(P1 )]φ(lgyr p [P1 , P2 ]P3 ) (5.301) = lgyrv [φ(P2 ), φ(P1 )]lgyrv [φ(P1 ), φ(P2 )]φ(P3 ) = lgyrv [V2 , V1 ]lgyrv [V1 , V2 ]V3 for all V1 , V2 , V3 ∈ Rn×m c , thus verifying the first bi-gyration identity in (5.287). The proof of the second bi-gyration identity in (5.287) is similar. Proof of (5.288): To prove the first bi-gyration exclusion property in (5.288) we have to show that lgyrv [V1 , V2 ]V3  −V3

(5.302)

Rn×m c .

for all V1 , V2 , V3 ∈ The bi-gyration exclusion property (5.302) follows from its P-counterpart in (4.145), p. 129, which is equivalent to the bi-gyration exclusion property lgyr p [P1 , P2 ]P3  −P3

(5.303)

for all P1 , P2 , P3 ∈ Rn×m . Indeed, the application of φ to (5.303) yields (5.302) as shown in (5.304) for Vk = φ(Pk ), k = 1, 2, 3. −V3 = −φ(P3 )  φ(lgyr p [P1 , P2 ]P3 ) = lgyrv [φ(P1 ), φ(P2 )]φ(P3 )

(5.304)

= lgyrv [V1 , V2 ]V3 Rn×m c ,

for all V1 , V2 , V3 ∈ thus verifying the first bi-gyration exclusion property in (5.288). The proof of the second bi-gyration exclusion property in (5.288) is similar. Finally, by proving (5.283) – (5.288), the proof of the theorem is complete.  Theorem 5.56. Let V1 , V2 , V3 ∈ Rn×m . Then, V3 = V1 ⊕V2

(5.305)

and lgyr[V1 , V2 ] = In rgyr[V1 , V2 ] = Im

(5.306)

Bi-gyrogroups and Bi-gyrovector Spaces – V

if and only if B(V1 )B(V2) = B(V3 ) .

(5.307)

Proof. The proof is similar to that of Theorem 4.20. Following (5.200), p. 219, the bi-boost product B(V1 )B(V2 ) possesses the unique decomposition B(V1 )B(V2 ) = ρ(rgyr[V1 , V2 ])B(V1⊕V2 )λ(lgyr[V1 , V2 ]) .

(5.308)

If V1 , V2 , and V3 satisfy (5.307) then the uniqueness of the decomposition of the bi-boost product B(V1 )B(V2) in (5.308) implies (5.305) and (5.306). Conversely, if  V1 , V2 , and V3 satisfy (5.305) and (5.306) then (5.307) follows from (5.308).

5.15. Bi-gyrogroups – V As in Def. 4.46, p. 154, with the parameter P ∈ Rn×m , it proves useful with the paramn×m by eter V ∈ Rn×m c , as well, to replace the bi-gyrogroupoid binary operation ⊕ in Rc  n×m the bi-gyrogroup binary operation ⊕ in Rc , which is defined below. Definition 5.57. (Bi-gyrogroup Operation, Bi-gyrogroups). Let (Rn×m c , ⊕) be a biis given by gyrogroupoid, m, n ∈ N. The bi-gyrogroup binary operation ⊕ in Rn×m c V1 ⊕ V2 = (V1 ⊕V2 )rgyr[V2 , V1 ]

(5.309)

for all V1 , V2 ∈ Rn×m c , where ⊕ and rgyr[·, ·] are given by (5.221) – (5.223). The result ing groupoid (Rn×m c , ⊕ ) is called a bi-gyrogroup. Remark 5.58. In the special case when m = 1, the binary operations ⊕ and ⊕ in Rn×m c coincide since rgyr[V2 , V1 ] = 1, as noted in Example 5.40, p. 223. Accordingly, when = Rnc coincide with Einstein velocity m = 1, the two binary operations ⊕ and ⊕ in Rn×1 c addition of special relativity. Indeed, the primed binary operation ⊕ is destined to be the Einstein addition of signature (m, n) in Def. 6.6, p. 316. As in (5.266) and in (5.268), when we wish to emphasize the distinction between of the parameter V we ⊕ in the space Rn×m of the parameter P and in the ball Rn×m c   n×m   n×m and ⊕v := ⊕ in Rc . use the notation ⊕ p := ⊕ in R Each of the two binary operations ⊕ and ⊕ descends to the common special relativistic Einstein (velocity) addition (2.2), p. 10, in the special case when m = 1, as shown in Sect. 5.17, p. 248. However, we will find in the sequel that the bi-gyrogroups  n×m (Rn×m c , ⊕ ), rather than the bi-gyrogroupoids (Rc , ⊕), form the desired elegant algebraic structure that the parametrization of the Lorentz group S O(m, n) encodes. The point is that we must study bi-gyrogroupoids in order to pave the way to the study of bi-gyrogroups.

241

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Theorem 5.59. φ(P1 ⊕p P2 ) = φ(P1 )⊕v φ(P2 ) φ−1 (V1 ⊕v V2 ) = φ−1 (V1 )⊕p φ−1 (V2 ) ,

(5.310)

m, n ∈ N, for all P1 , P2 ∈ Rn×m and V1 , V2 ∈ Rn×m c . Proof. Let φ(Pk ) = Vk , k = 1, 2. Then, by means of (4.256), p. 154, (5.274), and (5.309), we have φ(P1 ⊕p P2 ) = φ((P1 ⊕ p P2 )rgyr p [P2 , P1 ]) = φ(P1 ⊕ p P2 )rgyrv [φ(P2 ), φ(P1 )] = (φ(P1 )⊕v φ(P2 ))rgyrv [φ(P2 ), φ(P1 )]

(5.311)

= φ(P1 )⊕v φ(P2 ) , thus verifying the first identity in (5.310). The second identity in (5.310) follows from the first one. Indeed, with Pk = −1 φ (Vk ), k = 1, 2, the first identity in (5.310) yields φ(φ−1 (V1 )⊕p φ−1 (V2 )) = φ(φ−1 (V1 ))⊕v φ(φ−1 (V2 )) = V1 ⊕v V2 , implying the second identity in (5.310).

(5.312) 

It is interesting to note that following (5.310) and (5.273) the bijective map φ preserves both the primed and the unprimed binary operations, that is, φ(P1 ⊕p P2 ) = φ(P1 )⊕v φ(P2 ) φ(P1 ⊕ p P2 ) = φ(P1 )⊕v φ(P2 )

(5.313)

for any P1 , P2 ∈ Rn×m . n×m can Employing the bijection φ : Rn×m → Rn×m c , properties of bi-gyrogroups R n×m be translated into properties of bi-gyrogroups Rc . Thus, for instance, Theorem 5.60 below is the V-counterpart of Theorem 4.48, p. 156, obtained by employing the bijection φ.  Theorem 5.60. (Bi-gyrogroup Left and Right Automorphisms). Let (Rn×m c , ⊕v ), m, n ∈ N, be a bi-gyrogroup. Then, On ∈ SO(n) and Om ∈ SO(m) are, respectively, a

Bi-gyrogroups and Bi-gyrovector Spaces – V

 left automorphism and a right automorphism of the bi-gyrogroup (Rn×m c , ⊕ ), that is,

On (V1 ⊕v V2 )Om = On V1 Om ⊕v On V2 Om ,

(5.314)

for all V1 , V2 ∈ Rn×m c , Om ∈ SO(m), and On ∈ SO(n). n×m Proof. Given V1 , V2 ∈ Rn×m be given by c , let P1 , P2 ∈ R

V1 = φ(P1 ) V2 = φ(P2 ) ,

(5.315)

and let Om ∈ SO(m) and On ∈ SO(n). Then, we have the following chain of equations, which gives the desired result. The equations are numbered for subsequent explanation. (1)   On (V1 ⊕v V2 )Om ===

On (φ(P1 )⊕v φ(P2 ))Om

(2)

 === On φ(P1 ⊕p P2 )Om (3)

 === φ(On (P1 ⊕p P2 )Om ) (4)

 === φ(On P1 Om ⊕p On P2 Om )

(5.316)

(5)

 === φ(On P1 Om )⊕v φ(On P2 Om ) (6)

 === On φ(P1 )Om ⊕v On φ(P2 )Om (7)

 === On V1 Om ⊕v On V2 Om . Derivation of the numbered equalities in (5.316): (1) (2) (3) (4) (5) (6) (7)

Follows from (5.315). Follows from (5.310). Follows from Lemma 5.11, p.196. Follows from Theorem 4.48, p. 156. Follows from the first equation in (5.310). Follows from Theorem 4.48, p. 156. Follows from (5.315). 

243

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Theorem 5.61. (Bi-gyration Reduction Properties). Bi-gyrations in the c-ball Rn×m c , m, n ∈ N, possess the bi-gyration reduction properties lgyrv [V1 , V2 ] = lgyrv [V1 ⊕v V2 , V2 ] = lgyrv [V1 , V2 ⊕v V1 ] rgyrv [V1 , V2 ] = rgyrv [V1 ⊕v V2 , V2 ] = rgyrv [V1 , V2 ⊕v V1 ] .

(5.317)

Proof. The bi-gyration reduction properties in (5.317) are the image under the bijection φ of their P-counterparts (4.310) – (4.311) and (4.314) – (4.315), p. 168. By the first equation in (5.269), p. 234, the bi-gyration reduction property (4.314) in Rn×m , which is rgyr p [P1 , P2 ] = rgyr p [P1 ⊕p P2 , P2 ] ,

(5.318)

P1 , P2 ∈ Rn×m , implies by means of (5.310), rgyrv [φ(P1 ), φ(P2 )] = rgyrv [φ(P1 ⊕p P2 ), φ(P2 )] = rgyrv [φ(P1 )⊕v φ(P2 ), φ(P2 )] .

(5.319)

Hence, with Vk = φ(Pk ), k = 1, 2, (5.319) yields rgyrv [V1 , V2 ] = rgyrv [V1 ⊕v V2 , V2 ] ,

(5.320)

for all V1 , V2 ∈ Rn×m c , thus verifying one of the bi-gyration reduction properties in (5.317). The proof of the remaining bi-gyration reduction properties in (5.317) follows similarly from their respective P-counterparts in Theorems 4.56 and 4.57, p. 167.  The definition of ⊕ in (5.309) leads to the following lemma of four identities that exhibit the symmetry between the binary operations ⊕ and ⊕ in Rn×m c . Lemma 5.62. V1 ⊕ V2 V1 ⊕V2 V1 ⊕ V2 V1 ⊕V2

= (V1 ⊕V2 )rgyr[V2 , V1 ] = (V1 ⊕ V2 )rgyr[V1 , V2 ] = lgyr[V1 , V2 ](V2 ⊕V1 ) = lgyr[V1 , V2 ](V2 ⊕ V1 )

(5.321)

for all V1 , V2 ∈ Rn×m c , m, n ∈ N. Proof. The four equations in (5.321) are validated by demonstrating that each of these equations is the image under the bijection φ of its P-counterpart in (4.256) – (4.259), p. 154. The first equation in (5.321) is valid by Def. 5.57. The second equation in (5.321) follows from the first one by the bi-gyration

Bi-gyrogroups and Bi-gyrovector Spaces – V

inversion law (5.287), p. 237. The third equation in (5.321) is established by demonstrating that it is the image under the bijection φ of its P-counterpart (4.258) with Vk = φ(Pk ), k = 1, 2, Indeed, following (4.258), p. 155, we have V1 ⊕v V2 = φ(P1 )⊕v φ(P2 ) = φ(P1 ⊕p P2 ) = φ(lgyr p [P1 , P2 ](P2 ⊕ p P1 )) = lgyrv [φ(P1 ), φ(P2 )]φ(P2 ⊕ p P1 )

(5.322)

= lgyrv [φ(P1 ), φ(P2 )](φ(P2 )⊕v φ(P1 )) = lgyrv [V1 , V2 ](V2 ⊕v V1 ) , thus validating the third equation in (5.321). Finally, the fourth equation in (5.321) follows from the third one by the bi-gyration inversion law (5.287), p. 237.   Bi-gyrogroups (Rn×m c , ⊕ ) possess a commutative-like and an associative-like law, which are more elegant than those possessed by bi-gyrogroupoids (Rn×m c , ⊕). By Theorem 4.51, p. 165, and Theorem 4.50, p. 164, with P replaced by V we have the following two theorems.   Theorem 5.63. (Bi-gyrocommutative Law in (Rn×m c , ⊕ )). The binary operation ⊕ in Rn×m possesses the bi-gyrocommutative law c

V1 ⊕ V2 = lgyr[V1 , V2 ](V2 ⊕ V1 )rgyr[V2 , V1 ]

(5.323)

for all V1 , V2 ∈ Rn×m c . Proof. The bi-gyrocommutative law (5.323) is established by demonstrating that it is the image under the bijection φ of its P-counterpart (4.299), p. 165, with Vk = φ(Pk ), k = 1, 2.

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Indeed, following (4.299) with the application of φ we have V1 ⊕v V2 = φ(P1 ⊕p P2 ) = φ(lgyr p [P1 , P2 ](P2 ⊕p P1 )rgyr p [P2 , P1 ]) = lgyrv [φ(P1 ), φ(P2 )]φ((P2 ⊕p P1 )rgyr p [P2 , P1 ]) = lgyrv [φ(P1 ), φ(P2 )](φ(P2 )⊕v φ(P1 ))rgyrv [φ(P2 ), φ(P1 )] = lgyrv [V1 , V2 ](V2 ⊕v V1 )rgyrv [V2 , V1 ] , (5.324) 

thus verifying (5.323).

Theorem 5.64. (Bi-gyrogroup Left and Right Bi-gyroassociative Law of ⊕ ). The binary operation ⊕ in Rn×m c , m, n ∈ N, possesses the left bi-gyroassociative law V1 ⊕ (V2 ⊕ X) = (V1 ⊕ V2 )⊕ lgyr[V1 , V2 ]Xrgyr[V2 , V1 ]

(5.325)

and the right bi-gyroassociative law (V1 ⊕ V2 )⊕ X = V1 ⊕ (V2 ⊕ lgyr[V2 , V1 ]Xrgyr[V1 , V2 ])

(5.326)

for all V1 , V2 , X ∈ Rn×m c . Proof. The left bi-gyroassociative law (5.325) is established by demonstrating that it is the image under the bijection φ of its P-counterpart (4.296), p. 164, with Vk = φ(Pk ), k = 1, 2, 3. Indeed, following (4.296) with X = P3 , along with the application of φ, we have (V1 ⊕v (V2 ⊕v V3 ) = φ(P1 ⊕p (P2 ⊕p P3 )) = φ{(P1 ⊕p P2 )⊕p lgyr p [P1 , P2 ]P3 rgyr p [P2 , P1 ]} = φ(P1 ⊕p P2 )⊕v φ(lgyr p [P1 , P2 ]P3 rgyr p [P2 , P2 ])

(5.327)

= (V1 ⊕v V2 )⊕v lgyrv [V1 , V2 ]V3 rgyrv [V2 , V1 ] , for all V1 , V2 , V3 ∈ Rn×m c , thus verifying (5.325). The proof of the right bi-gyroassociative law(5.326) follows similarly by the application of φ to its P-counterpart (4.297), p. 164. 

5.16. Einstein Bi-gyroaddition and Bi-gyration We are now in the position to present a theorem similar to Theorem 5.41, p. 224, related to the with ⊕ replaced by ⊕v = ⊕ , where ⊕ is the binary operation in Rn×m c n×m  binary operation ⊕ in Rc by (5.321). The binary operation ⊕ is called Einstein

Bi-gyrogroups and Bi-gyrovector Spaces – V

bi-gyroaddition of signature (m, n), a definition that will be formalized in Def. 6.6, p. 316. Theorem 5.65. (Einstein Bi-gyroaddition and Bi-gyration). Einstein bi-gyroaddition  ⊕ and its associated bi-gyration in the Einstein bi-gyrogroup (Rn×m c , ⊕ ) are given by n×m the following equations. For any V1 , V2 ∈ Rc ,   −1  −2 t −1 t −2 V1 ⊕ V2 = In − c V1 V1 (In + c V2 V1 ) (V1 + V2 ) Im − c−2 V1t V1 (5.328)   −1 V1 ⊕ V2 = In − c−2 V1 V1t (V1 + V2 )(Im + c−2 V1t V2 )−1 Im − c−2 V1t V1 and lgyr[V2 , V1 ] = rgyr[V2 , V1 ] =

 

In − c−2 V2 V2t (In + c−2 V1 V2t )−1 t Im + c−2 E 21 E 21





In − c−2 V1 V1t



t In + c−2 E 21 E 21

Im − c−2 V2t V2 (Im + c−2 V1t V2 )−1



Im − c−2 V1t V1 , (5.329)

where E 21 =



In − c−2 V1 V1t

−1

(V1 + V2 )



Im − c−2 V2t V2

−1

.

(5.330)

Proof. The two equations in (5.329) follow immediately from (5.242), p. 228, and (5.222), p. 224, where E 21 is given by (5.330). It remains to prove the two equations in (5.328). By means of (5.321) and (5.309), we have V2 ⊕ V1 = lgyr[V2 , V1 ](V1 ⊕V2 )

(5.331)

V1 ⊕ V2 = (V1 ⊕V2 )rgyr[V2 , V1 ] , where, by (5.221), p. 224, V1 ⊕V2 =



t In + c−2 E 21 E 21

= E 21



Im +

−1

E 21

t c−2 E 21 E 21

−1

(5.332) .

Substituting from (5.329), (5.332), and (5.330) into (5.331) yields (5.328), and the proof is complete.  The notation, ⊕ , for Einstein bi-gyroaddition will be revised in (6.54), where ⊕ is denoted by ⊕E , following Def. 6.6, p. 316.

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5.17. On the Special Relativistic Einstein Gyroaddition In order to justify calling the binary operation ⊕ in (5.328) an Einstein bi-gyroaddition of signature (m, n), m, n ∈ N, we show in this section that Einstein bi-gyroaddition of signature (1, n), n ≥ 1, also called Einstein gyroaddition or Einstein addition, is the common Einstein velocity addition (2.2), p. 10, of special relativity theory (n = 3 in physical applications). Thus, what is known in the language of special relativity as Einstein addition is called Einstein gyroaddition in gyrolanguage, and Einstein bigyroaddition in bi-gyrolanguage. Accordingly, we find in the chain of equations (5.334) that the binary operation ⊕ in (5.328) descends to the special relativistic Einstein addition ⊕ in (2.2), p. 10, when m = 1. with m = 1 and n ∈ N. Then, by (5.171), p. 212, Let V1 , V2 ∈ Rn×m c 

In − c−2 V1 V1t

−1

= In +

2 1 γV1 V1 V1t . c2 1 + γV1

(m = 1)

(5.333)

By means of (5.328) and (5.333) we have the following chain of equations, some = Rnc , of which are numbered for subsequent explanation. For any V1 , V2 ∈ Rn×1 c ⎧ ⎫ (1) ⎪ ⎪ γV2 ⎪ ⎪ 1 1 1  ⎨ 1 t⎬ I V1 ⊕ V2 === ⎪ (V1 + V2 ) + V V n 1 1⎪ ⎪ ⎪ 2 −2 ⎩ ⎭ c 1 + γV1 1 + c V1 ·V2 γV1 ⎧ ⎫ 2 2 (2) ⎪ ⎪ ⎪ 1 1 1 γV1 1 γV1  ⎪ ⎨ ⎬ 2 === ⎪ V + V + V V + V (V ·V ) ⎪ 1 2 1 1 1 1 2 ⎪ ⎪ 2 2 −2 ⎩ ⎭ 1 + c V1 ·V2 γV c 1 + γV1 c 1 + γV1 1 ⎧ ⎫ 2 2 2 (3) ⎪ γV γV − 1 ⎪ ⎪ 1 1 1 γV1  ⎪ ⎨ ⎬ 1 1 === ⎪ V1 + V2 + V + V (V ·V ) ⎪ 1 1 1 2 ⎪ ⎪ ⎩ ⎭ 1 + c−2 V1 ·V2 γV 1 + γV1 γV2 c2 1 + γV1 1 1 ⎧ ⎫ γV − 1 ⎪ ⎪ 1 1 1 γV1 ⎨ 1 ⎬ === V1 + V2 + 1 V1 + 2 (V1 ·V2 )V1 ⎪ ⎪ ⎩ ⎭ −2 γV1 γV1 1 + c V1 ·V2 γV1 c 1 + γV1 ⎧ ⎫ ⎪ ⎪ 1 1 1 γV1 ⎨ ⎬ === V1 + V2 + 2 (V1 ·V2 )V1 ⎪ ⎪ ⎩ ⎭ −2 γ 1 + c V ·V c 1+γ 1

2

V1

V1

(4)

 === V1 ⊕V2 .

(m = 1) (5.334)

Derivation of the numbered equalities in (5.334): (1) This equation is obtained by an obvious substitution from (5.333)  into the second equation in (5.328) and noting that V1t V2 = V1 ·V2 and that 1 − c−2 V1t V1 =  1 − c−2 V1 2 = γV−1 . 1

Bi-gyrogroups and Bi-gyrovector Spaces – V

(2) Follows from Item (1) by noting that V1 V1t V1 = V1 V1 2 and V1 V1t V2 = V1 (V1 ·V2 ). (3) Follows from Item (2) by noting that V1 2 /c2 = (γV2 − 1)/γV2 according to (2.12), 1 1 p. 13. (4) Here ⊕ is the Einstein special relativistic addition, given by (2.2), p. 10. Result (5.334) is compatible with Result (5.173), p. 213, according to which the bi-boost Bv (V) ∈ SO(m, n) descends to the common Lorentz boost of special relativity theory when m = 1.

5.18. Bi-gamma Identities for the Primed Binary Operation The passage in Def. 5.57 from the binary operation ⊕ to the primed binary operation introduces further elegance into the bi-gyrocommutative and the ⊕ in the ball Rn×m c bi-gyroassociative laws as Theorems 5.63 and 5.64 indicate. This further elegance will be exploited in Sect. 5.19. Accordingly, in this section we present a theorem in which we express the bi-gamma identities (5.219), p. 223, in terms of the primed binary operation ⊕ rather than ⊕. Theorem 5.66. (The Bi-gamma Identities). Let m, n ∈ N. Then, we have the right and the left bi-gamma identity ΓRV1 ⊕ V2 rgyr[V1 , V2 ] = ΓRV1 (Im + ΓVL1 ⊕ V2 lgyr[V1 , V2 ]

=

ΓVL1 (In

1 t V V2 )ΓRV2 c2 1

(5.335)

1 + 2 V1 V2t )ΓVL2 c

for all V1 , V2 ∈ Rn×m c . Proof. We note the relations ΓRVOm = Otm ΓRV Om L ΓVO = ΓVL m

(5.336)

and Om ∈ SO(m), in Lemma 5.22, p. 205. for all V ∈ Rn×m c Following Def. 5.57 of the primed binary operation, ⊕ , and the first relation in (5.336), noting that rgyr[V1 , V2 ] ∈ SO(m) and that (rgyr[V1 , V2 ])t = rgyr[V2 , V1 ], we have ΓRV1 ⊕V2 = ΓR(V1 ⊕ V2 )rgyr[V1 ,V2 ]

(5.337)

= rgyr[V2 , V1 ]ΓRV1 ⊕ V2 rgyr[V1 , V2 ] . Substituting ΓRV1 ⊕V2 from (5.337) into the first bi-gamma identity in (5.219), p. 223,

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

and noting that rgyr[V1 , V2 ]rgyr[V2 , V1 ] = Im , yields the first bi-gamma identity in (5.335). Similarly, following Def. 5.57 of the primed binary operation, ⊕ , and the second relation in (5.336), noting that rgyr[V1 , V2 ] ∈ SO(m), we have ΓVL1 ⊕V2 = ΓVL1 ⊕ V2 rgyr[V1 ,V2 ] = ΓVL1 ⊕ V2 .

(5.338)

Substituting ΓVL1 ⊕V2 from (5.338) into the second bi-gamma identity in (5.219), p. 223, yields the second bi-gamma identity in (5.335). The proof is thus complete. 

5.19. Bi-gyrogroup Gyrations The bi-gyroassociative laws (5.325) – (5.326) and the bi-gyrocommutative law (5.323) exhibit a pattern that suggests the following definition of gyrations in terms of left and right gyrations. Definition 5.67. (Bi-gyrogroup Gyrations) ([99, Definition 43]). The gyrator, gyr, of  a bi-gyrogroup (Rn×m c , ⊕ ),  gyr : Rn×m × Rn×m → Aut(Rn×m c c c ,⊕ ),

(5.339)

 generates automorphisms called gyrations, gyr[V1 , V2 ] ∈ Aut(Rn×m c , ⊕ ), given by the equation

gyr[V1 , V2 ]X = lgyr[V1 , V2 ]Xrgyr[V2 , V1 ]

(5.340)

or, equivalently, by the equation gyr[V1 , V2 ]X = lgyr[V1 , V2 ]X(rgyr[V1 , V2 ])−1

(5.341)

for all V1 , V2 , X ∈ Rn×m c , where left and right gyrations, lgyr[V1 , V2 ] and rgyr[V2 , V1 ], are given in (5.217), p. 223. The gyration gyr[V1 , V2 ] is said to be the gyration generated by V1 , V2 ∈ Rn×m c . Being represented by left and right gyrations, bi-gyrogroup  gyrations are called bi-gyrations. Being automorphisms of (Rn×m c , ⊕ ), bi-gyrations are also called bi-gyroautomorphisms. Def. 5.67 is rewarding, leading to the elegant result of Theorem 5.75, p. 255, ac cording to which any bi-gyrogroup (Rn×m c , ⊕ ), m, n ∈ N, is a gyrocommutative gyrogroup. Theorem 5.68. (Gyrogroup Gyroassociative and gyrocommutative Laws). obeys the left and the right gyroassociative law binary operation ⊕ in Rn×m c V1 ⊕ (V2 ⊕ X) = (V1 ⊕ V2 )⊕ gyr[V1 , V2 ]X

The

(5.342)

Bi-gyrogroups and Bi-gyrovector Spaces – V

and (V1 ⊕ V2 )⊕ X = V1 ⊕ (V2 ⊕ gyr[V2 , V1 ]X)

(5.343)

and the gyrocommutative law V1 ⊕ V2 = gyr[V1 , V2 ](V2 ⊕ V1 ) .

(5.344)

Proof. Identities (5.342) – (5.343) follow immediately from Def. 5.67 and the left and right bi-gyroassociative law (5.325) – (5.326). Similarly, (5.344) follows immediately from Def. 5.67 and the bi-gyrocommutative law (5.323).  It is anticipated in Def. 5.67 that gyrations are automorphisms. The following theorem asserts that this is indeed the case. Theorem 5.69. (Gyroautomorphisms) For all V1 , V2 ∈ Rn×m c , gyrations gyr[V1 , V2 ] n×m  of a bi-gyrogroup (Rc , ⊕ ) are automorphisms of the bi-gyrogroup. Proof. The result of this theorem is established by applying the bijection φ to its Pcounterpart result of Theorem 4.55, p. 167.  Theorem 5.70. (Left Gyration Reduction Properties) Left gyrations of a  bi-gyrogroup (Rn×m c , ⊕ ) possess the left gyration left reduction property lgyr[V1 , V2 ] = lgyr[V1 ⊕ V2 , V2 ]

(5.345)

and the left gyration right reduction property lgyr[V1 , V2 ] = lgyr[V1 , V2 ⊕ V1 ] .

(5.346)

Proof. The results (5.345) – (5.346) of this theorem are established by applying the bijection φ to their P-counterpart results (4.310) – (4.311) of Theorem 4.56, p. 167, as shown in the proof of Theorem 5.61.  Theorem 5.71. (Right Gyration Reduction Properties) Right gyrations of a bi gyrogroup (Rn×m c , ⊕ ) possess the right gyration left reduction property rgyr[V1 , V2 ] = rgyr[V1 ⊕ V2 , V2 ]

(5.347)

and the right gyration right reduction property rgyr[V1 , V2 ] = rgyr[V1 , V2 ⊕ V1 ] .

(5.348)

Proof. The results (5.347) – (5.348) of this theorem are established by applying the

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

bijection φ to their P-counterpart results (4.314) – (4.315) of Theorem 4.57, p. 168, as shown in the proof of Theorem 5.61.  Theorem 5.72. (Gyration Reduction Properties) The gyrations of any bi-gyrogroup  (Rn×m c , ⊕ ), m, n ∈ N, possess the left and right reduction property gyr[V1 , V2 ] = gyr[V1 ⊕ V2 , V2 ]

(5.349)

gyr[V1 , V2 ] = gyr[V1 , V2 ⊕ V1 ] .

(5.350)

and

Proof. Identities (5.349) and (5.350) follow from Def. 5.67 of gyr in terms of lgyr and rgyr, and from Theorems 5.70 and 5.71. Indeed, for any V1 , V2 , X ∈ Rn×m we have c gyr[V1 , V2 ]X = lgyr[V1 , V2 ]Xrgyr[V1 , V2 ] = lgyr[V1 ⊕ V2 , V2 ]Xrgyr[V1 ⊕ V2 , V2 ]

(5.351)

= gyr[V1 ⊕ V2 , V2 ]X , 

thus proving (5.349). The proof of (5.350) is similar.

5.20. Uniqueness of Left and Right Gyrations A bi-gyration gyr = (lgyr, rgyr) determines its left and right constituents lgyr and rgyr uniquely, as we see from the following two lemmas. Lemma 5.73. Let On ∈ S O(n) and Om ∈ S O(m), n, m ∈ N. Then, On POm = P

(5.352)

for all P ∈ Rn×m if and only if Otn = λIn and Om = λIm , where λ ∈ R satisfies the equation λn = λm = 1. Thus, λ = 1 when n or m is odd, and λ = ±1 when both n and m are even. Proof. If On = ±In and Om = ±Im , then obviously (5.352) is true for all P ∈ Rn×m . Conversely, assuming On POm = P or equivalently Otn P = POm ,

(5.353)

On ∈ S O(n), Om ∈ S O(m), for all P ∈ Rn×m , we will show that On = λIn and Om = λIm where λ = 1 when m or n is odd, and λ = ±1 when both m and n are even. (We interject here an important remark. Owing to the bi-gyration exclusion property 4.145, p. 129, one may note that if On or Om is a gyration then the ambiguous sign

Bi-gyrogroups and Bi-gyrovector Spaces – V

of λ descends to the positive sign, implying On = In and Om = Im ). Let ⎞ ⎛ ... a1n ⎟⎟ ⎜⎜⎜a11 ⎟⎟⎟ ⎜⎜⎜⎜ . ⎟⎟⎟ t ⎜ . On = ⎜⎜⎜ . ⎟⎟⎟ ∈ S O(n) ⎟⎟⎠ ⎜⎜⎝ ... ann an1 and

⎛ ⎜⎜⎜ b11 ⎜⎜⎜ Om = ⎜⎜⎜⎜⎜ ... ⎜⎜⎝ bm1

Furthermore, let Pi j ∈ R

... ...

⎞ b1m ⎟⎟ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟⎟ ∈ S O(m) . ⎟⎠ bmm

(5.354)

(5.355)

n×m

be the matrix ⎛ ⎜⎜⎜0 . . . 0 . . . ⎜⎜⎜ ⎜⎜⎜ .. ⎜⎜⎜ . ⎜⎜ Pi j = ⎜⎜⎜⎜⎜0 . . . 1 . . . ⎜⎜⎜ ⎜⎜⎜ .. ⎜⎜⎜ . ⎜⎝ 0 ... 0 ...

⎞ 0⎟⎟ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟ 0⎟⎟⎟⎟ ∈ Rn×m ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟⎟ ⎟⎠ 0

with one at the i j-entry and zeros elsewhere, i = 1, . . . , n, j = 1, . . . , m. Then, the matrix product Otn Pi j , ⎛ ⎞ ⎜⎜⎜0 . . . a1i . . . 0⎟⎟⎟ ⎜⎜⎜⎜ . ⎟⎟⎟⎟ ⎜⎜⎜ .. ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟ t On Pi j = ⎜⎜⎜0 . . . aii . . . 0⎟⎟⎟⎟⎟ ∈ Rn×m , ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ .. ⎟⎟⎟ ⎜⎜⎜ . ⎟⎟⎟ ⎜⎝ ⎟⎠ 0 . . . ani . . . 0

(5.356)

(5.357)

is a matrix with jth column (a1i , . . . , aii , . . . , ani )t and zeros elsewhere. Shown explicitly in (5.357) are the first column, the jth column, and the mth column of the matrix Otn Pi j , along with its first row, ith row and nth row.

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Similarly, the matrix product Pi j Om , ⎛ ⎞ ⎜⎜⎜ 0 . . . 0 . . . 0 ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ .. ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜⎜ . ⎟⎟ ⎜⎜⎜ Pi j Om = ⎜⎜⎜b1 j . . . b j j . . . b jm ⎟⎟⎟⎟⎟ ∈ Rn×m , ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ .. ⎟⎟⎟ ⎜⎜⎜ . ⎟⎟⎟ ⎜⎝ ⎟⎠ 0 ... 0 ... 0

(5.358)

is a matrix with ith row (b1 j , . . . , b j j , . . . , b jm ) and zeros elsewhere. Shown explicitly in (5.358) are the first column, the jth column, and the mth column of the matrix Pi j Om , along with its first row, ith row and nth row. It follows from (5.353) that (5.357) and (5.358) are equal. Hence, by comparing entries of the matrices in (5.357) – (5.358) we have aii = b j j

(5.359)

and aii1 = 0 b j j1 = 0

(5.360)

for all i, i1 = 1, . . . , n and all j, j1 = 1, . . . , m, i1  i and j1  j. By (5.359) – (5.360) and (5.354) – (5.355) we have Otn = λIn Om = λIm .

(5.361)

By assumption On ∈ SO(n) and Om ∈ SO(m), so that their determinants are det(On ) = det(Om ) = 1. Hence, following (5.361), λn = λm = 1. Hence λ = 1 when either n or m is odd, and λ = ±1 when both n and m are even.  The following Lemma 5.74 is an immediate consequence of Lemma 5.73. Lemma 5.74. Let On,k ∈ S O(n) and Om,k ∈ S O(m), n, m ∈ N, k = 1, 2. Then, On,1 POm,1 = On,2 POm,2

(5.362)

for all P ∈ Rn×m (1) if and only if On,1 = On,2 and Om,1 = Om,2 , when n or m is odd and (2) if and only if On,1 = ±On,2 and Om,1 = ±Om,2 , when both n and m are even. Owing to the bi-gyration exclusion property 4.145, p. 129, the ambiguous sign in

Bi-gyrogroups and Bi-gyrovector Spaces – V

Lemma 5.74(2) descends to the positive sign in the special case when the bi-rotation (On , Om ) is a bi-gyration, (On , Om ) = (lgyr[V1 , V2 ], rgyr[V1 , V2 ]).

5.21. Bi-gyrogroups Are Gyrocommutative Gyrogroups This section is the V-counterpart of Sect. 4.23. We are now in a position to see that  the bi-gyrogroups (Rn×m c , ⊕ ), called Einstein bi-gyrogroups (see Def. 6.6, p. 316), are gyrocommutative gyrogroups. Following Defs. 2.13 – 2.14, p. 22, Theorem 5.75  asserts that any Einstein bi-gyrogroup (Rn×m c , ⊕ ), m, n ∈ N, is a concrete example of the abstract gyrocommutative gyrogroup. Theorem 5.75. (Einstein Bi-gyrogroups Are Gyrocommutative Gyrogroups). Any  bi-gyrogroup (Rn×m c , ⊕ ), n, m ∈ N, is a gyrocommutative gyrogroup. Proof. We validate below each of the six gyrocommutative gyrogroup axioms (G1)– (G6) in Defs. 2.13 and 2.14.  1. The bi-gyrogroup (Rn×m c , ⊕ ) possesses the left identity 0n,m , thus validating Axiom (G1). possesses the left inverse  V := −V ∈ Rn×m 2. Every element V ∈ Rn×m c c , thus validating Axiom (G2). 3. The binary operation ⊕ obeys the left gyroassociative law (5.342) in Theorem 5.68, thus validating Axiom (G3).  4. The map gyr[V1 , V2 ] is an automorphism of (Rn×m c , ⊕ ) by Theorem 5.69, that is, n×m  gyr[V1 , V2 ] ∈ Aut(Rc , ⊕ ), thus validating Axiom (G4). possesses the left reduction property (5.349) in 5. The binary operation ⊕ in Rn×m c Theorem 5.72, thus validating Axiom (G5). possesses the gyrocommutative law (5.344) in 6. The binary operation ⊕ in Rn×m c Theorem 5.68, thus validating Axiom (G6).

 Being gyrocommutative gyrogroups, Einstein bi-gyrogroups enjoy all the results in gyrocommutative gyrogroup theory that have been developed in [81, 84, 93, 94, 95, 96, 98]. As important examples of results that Einstein bi-gyrogroups inherit from the abstract gyrocommutative gyrogroup, we consider the following definition and theorems. As in Sect. 2.7, Einstein addition ⊕E , ⊕E := ⊕ , comes with Einstein coaddition, E , the definition of which follows.

(5.363)

255

256

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Definition 5.76. (Einstein Coaddition). Let (Rn×m c , ⊕E ) be an Einstein bi-gyrogroup of signature (m, n), m, n ∈ N. The bi-gyrogroup cooperation is a second binary operation in Rn×m c , related to the bi-gyrogroup operation, ⊕E , by the equation A E B = A⊕E gyr[A, E B]B ,

(5.364)

for any A, B ∈ Rn×m c . Naturally, A E B = A E (E B) .

(5.365)

When it is necessary to emphasize that Einstein coaddition is a binary operation in a bi-gyrogroup (Rn×m c , ⊕E ) of signature (m, n) and spectral radius c, we use the notation E = E,(m,n),c . By Theorem 5.75, any Einstein bi-gyrogroup (Rn×m c , ⊕E ) is a gyrocommutative gyrogroup, the gyrator, gyr, of which is given by Def. 5.67, p. 250. As such, the binary cooperation E of Einstein bi-gyrogroup (Rn×m c , ⊕E ) is commutative according to Theorem 2.44, p. 50. Furthermore, by Theorem 3.10, p. 71, we have A⊕E (E A⊕E B)⊗ 12 = 12 ⊗(A E B) .

(5.366)

Theorem 5.77. (Left and Right Cancellation Laws in (Rn×m c , ⊕E )). For any m, n ∈ N . Then, we have the left cancellation law and c > 0, let A, B ∈ Rn×m c A⊕E (E A⊕E B) = B ,

(5.367)

(B E A)⊕E A = B

(5.368)

the first right cancellation law

and the second right cancellation law (BE A) E A = B .

(5.369)

Proof. The theorem follows from corresponding results in (2.96) – (2.98), p. 33, and from the fact that bi-gyrogroups are gyrocommutative gyrogroups.  Theorem 5.78. (Left Bi-gyrotranslation Theorem). For any m, n ∈ N and c > 0, let A, B, C ∈ (Rn×m c , ⊕E ). Then, E (E A⊕E B)⊕E (E A⊕E C) = gyr[E A, B](E B⊕E C) (A⊕E B)E (A⊕E C) = gyr[A, B](BE C) .

(5.370)

Proof. The theorem follows from Theorem 2.52, p. 54, and from the result of Theorem 5.75 according to which bi-gyrogroups (Rn×m c , ⊕E ) are gyrocommutative gyrogroups.

Bi-gyrogroups and Bi-gyrovector Spaces – V

 Theorem 5.79. For any m, n ∈ N and c > 0, let A, B, C ∈ (Rn×m c , ⊕E ). Then, gyr[E A⊕E B, AE C] = gyr[A, E B]gyr[B, E C]gyr[C, E A]

(5.371)

and, accordingly, lgyr[E A⊕E B, AE C] = lgyr[A, E B]lgyr[B, E C]lgyr[C, E A] rgyr[E A⊕E B, AE C] = rgyr[A, E B]rgyr[B, E C]rgyr[C, E A] .

(5.372)

Proof. By [84, Theorem 3.14, p. 55], the gyration identity (5.371) is valid in any gyrocommutative gyrogroup. By Theorem 5.75, in turn, bi-gyrogroups (Rn×m c , ⊕E ) are gyrocommutative gyrogroup. Hence, (5.371) is valid in any bi-gyrogroup (Rn×m c , ⊕E ). By the bi-gyrogroup gyration Def. 5.67, p. 250, gyr[E A⊕E B, AE C]X = lgyr[E A⊕E B, AE C]X(rgyr[E A⊕E B, AE C])−1

(5.373)

and gyr[A, E B]gyr[B, E C]gyr[C, E A]X = gyr[A, E B]gyr[B, E C]lgyr[C, E A]X(rgyr[C, E A]−1 = gyr[A, E B]lgyr[B, E C]lgyr[C, E A]X(rgyr[C, E A]−1 (rgyr[B, E C]−1 = lgyr[A, E B]lgyr[B, E C]lgyr[C, E A]X(rgyr[C, E A]−1 (rgyr[B, E C]−1 (rgyr[A, E B]−1 = lgyr[A, E B]lgyr[B, E C]lgyr[C, E A]X(rgyr[A, E B]rgyr[B, E C]rgyr[C, E A])−1 (5.374) for all A, B, C ∈ Rn×m and X ∈ Rn×m . c Hence, by (5.371), (5.373), and (5.374), along with Lemma 5.74, we have the following weaker form of (5.372): lgyr[E A⊕E B, AE C] = ±lgyr[A, E B]lgyr[B, E C]lgyr[C, E A] rgyr[E A⊕E B, AE C] = ±rgyr[A, E B]rgyr[B, E C]rgyr[C, E A]

(5.375)

for all A, B, C ∈ Rn×m c . It remains to show that the ambiguous sign in (5.375) descends to the positive sign. Indeed, in the special case when A = 0n,m , the two identities in (5.375) descend to the

257

258

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

identities lgyr[B, E C] = ±lgyr[B, E C] rgyr[B, E C] = ±rgyr[B, E C]

(5.376)

for all B, C ∈ Rn×m c , demonstrating that the negative sign of the ambiguous sign in (5.375) is not valid, thus verifying (5.371). 

5.22. Matrix Division Notation Let M1 and M2 be two matrices such that the inverse, M2−1 , of M2 exists. If the two matrices satisfy the commuting relation M1 M2−1 = M2−1 M1 ,

(5.377)

then we may adopt the matrix division notation M1 := M1 M2−1 = M2−1 M1 . M2

(5.378)

In order to establish the commutativity relation (5.377) between matrix functions M1 = f (A) and M2 = g(B) for square matrices A and B and their functions f (A) and g(B), the following theorem is useful. Theorem 5.80. ([27, p. 113]). If the function f (λ) can be expanded in a power series in the circle |λ − λ0 | < r, ∞  αk (λ − λ0 )k , (5.379) f (λ) = k=0

then this expansion remains valid when the scalar argument λ is replaced by a matrix A whose eigenvalues lie within the circle of convergence, ∞  αk (A − λ0 I)k . (5.380) f (A) = k=0

Example 5.81. Following the matrix division notation (5.377) – (5.378) we have ∞  1 In t −1 = (In − 2 VV ) = (c−2 VV t )k , (5.381) In − c−2 VV t c k=0 V ∈ Rn×m , if λi < c2 , where λi , i = 1, 2, . . . , n, are the n eigenvalues of VV t , that is, if

Bi-gyrogroups and Bi-gyrovector Spaces – V

V ∈ Rn×m lies in the c-ball Rn×m of the matrix space Rn×m . Similarly, c c Im 1 t −1  −2 t k − V V) = (c V V) , = (I m Im − c−2 V t V c2 k=0 ∞

(5.382)

V ∈ Rn×m , if λ j < c2 , where λ j , j = 1, 2, . . . , m, are the m eigenvalues of V t V, that is, lies in the c-ball Rn×m if V ∈ Rn×m c c . If two matrices A and B commute then the matrix functions f (A) and g(B) commute. Hence, in particular, f (A) and g(A) and g(A)−1 commute with each other. Hence, for instance, by means of the matrix division notation, the inverse matrix notation in (5.383) – (5.384) and in (5.453) and (5.454), p. 273, and in (5.493) – (5.495), p. 281, is justified.

5.23. Additive Decomposition of the Lorentz Bi-boost In the limit of large c, c → ∞, the Lorentz bi-boost B(V) of signature (m, n) descends to its corresponding Galilei bi-boost B∞ (V) of signature (m, n), m, n ∈ N, as shown in (5.195), p. 218. Interestingly, the relation between Lorentz and Galilei bi-boosts is deeper. The Lorentz bi-boost B(V), V ∈ Rn×m c , of signature (m, n) turns out to be the sum of its corresponding Galilei bi-boost B∞ (V) and an elegant (m + n) × (m + n) matrix multiplied by c−2 . This sum gives rise to the additive relation between the Lorentz bi-boost B(V) and its Galilei counterpart B∞ (V) that we establish in Theorem 5.83 below. In the following Lemma we use the matrix division notation (5.377) – (5.378). Lemma 5.82. ΓVL = In +

1 (ΓVL )2 VV t c2 In + ΓVL

(5.383)

1 (ΓRV )2 t ΓRV = Im + 2 VV c Im + ΓRV and (ΓVL )2 − In 1 t VV = c2 (ΓVL )2 1 t VV= c2

(ΓRV )2 − Im (ΓRV )2

(5.384)

259

260

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

for all V ∈ Rn×m c , m, n ∈ N. Proof. By means of the first commuting relation in (5.119), p. 202, we have the commuting relation (In + ΓVL )V = V(Im + ΓRV )

(5.385)

and hence the commuting relation V(Im + ΓRV )−1 = (In + ΓVL )−1 V

(5.386)

for all V ∈ Rn×m c . Following (5.157), p. 210, we have 1 P(Im + ΓRV )−1 Pt c2 and following the first equation in (5.121), p. 203, we have ΓVL = In +

P = ΓVL V = VΓRV Pt = V t ΓVL = ΓRV V t .

(5.387)

(5.388)

Substituting from (5.388) into (5.387) yields the first identity in (5.389). ΓVL = In +

1 VΓRV (Im + ΓRV )−1 ΓRV V t 2 c

= In +

1 L Γ (In + ΓVL )−1 ΓVL VV t c2 V

(5.389)

1 VV t ΓVL (In + ΓVL )−1 ΓVL . c2 Each of the second and third identities in (5.389) follows from the first one by means of the commuting relations (5.388) and (5.386). The first identity in (5.383) follows from (5.389) where we use the matrix division notation. The proof of the second identity in (5.383) is similar. Finally (5.384) follows immediately from (5.383).  = In +

Theorem 5.83. (Additive Decomposition of the Lorentz Bi-boost). Let ⎛ R ⎞ ⎜⎜⎜ ΓV c12 ΓRV V t ⎟⎟⎟ ⎟⎟⎟ ∈ SO(m, n) , Bv (V) = ⎜⎜⎜⎝ ⎠ ΓVL V ΓVL

(5.390)

V ∈ Rn×m ⊂ Rn×m , in (5.127), p. 204, be the Lorentz bi-boost of signature (m, n), m, n ∈ c

Bi-gyrogroups and Bi-gyrovector Spaces – V

N, and let

 Im 0m,n , B∞ (V) = V In 

(5.391)

V ∈ Rn×m , in (5.195), p. 218, be the Galilei bi-boost of signature (m, n), m, n ∈ N. Then, ⎛ (ΓR )2 ⎞ R t ⎟ ⎜⎜⎜ V R V t V Γ V ⎟⎟⎟ V I +Γ 1 ⎜ m ⎟⎟⎟ (5.392) Bv (V) = B∞ (V) + 2 ⎜⎜⎜⎜⎜ L 2 V c ⎝ (ΓV ) VV t V (ΓVL )2 VV t ⎟⎟⎠ In +ΓVL

In +ΓVL

for all V ∈ Rn×m c . 

Proof. The proof follows immediately from (5.390) – (5.391) and (5.383).

Definition 5.84. (Additive Decomposition of the Lorentz Bi-boost). Let V ∈ Rn×m c , m, n > 1, and let ⎛ R ⎞ ⎜⎜⎜ ΓV c12 ΓRV V t ⎟⎟⎟ ⎟⎟⎟ (5.393) Bv (V) = ⎜⎜⎜⎝ L L ⎠ ΓV V ΓV and

⎛ ⎞ ⎜⎜⎜Im 0m,n ⎟⎟⎟ ⎜ ⎟⎟⎠ B∞ (V) = ⎜⎝ V In

(5.394)

be the Lorentz bi-boost of signature (m, n) and its corresponding Galilei bi-boost of signature (m, n). Furthermore, let ⎛ (ΓR )2 ⎞ R t ⎟ ⎜⎜⎜ V R V t V Γ V ⎟⎟⎟ V ⎜ Im +Γ ⎟⎟⎟ , (5.395) E(V) = ⎜⎜⎜⎜⎜ L 2 V ⎝ (ΓV ) VV t V (ΓVL )2 VV t ⎟⎟⎠ L L I +Γ I +Γ n

V

n

V

so that, by Theorem 5.83, we have the Lorentz bi-boost additive decomposition 1 E(V) . (5.396) c2 We say that B∞ (V) is the Galilean part and E(V) is the entanglement part of the Lorentz bi-boost Bv (V) of signature (m, n). Bv (V) = B∞ (V) +

Calling E(V) the entanglement part of a Lorentz bi-boost Bv (V) will be justified in Chap. 6, where relativistic entanglements generated by Lorentz bi-boosts of signature (m, n), m, n > 1, emerge.

261

262

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

The commuting relation (ΓVL )2

(ΓRV )2

I n + ΓV

Im + ΓRV

V=V L

,

(5.397)

V ∈ Rn×m c , follows immediately from the commuting relation (5.386) and the first commuting relation in (5.119), p. 202. Hence, by (5.395) and (5.397), the entanglement part E(V), V ∈ Rn×m c , possesses the following representations: ⎞ ⎛ (ΓR )2 ⎞ ⎛ (ΓR )2 R t ⎟ R t ⎟ ⎜⎜⎜ V R V t V ⎜⎜⎜ V R V t V Γ V Γ V ⎟ ⎟⎟⎟ ⎟ V V I +Γ I +Γ ⎜ m ⎟⎟⎟⎟ = ⎜⎜⎜⎜ m V ⎟⎟⎟ E(V) = ⎜⎜⎜⎜⎜ L 2 V ⎝ (ΓV ) VV t V (ΓVL )2 VV t ⎟⎟⎠ ⎜⎜⎝V (ΓRV )2 V t V V (ΓRV )2 V t ⎟⎟⎠ In +ΓVL

In +ΓVL

Im +ΓRV

⎛ ⎞⎛ ⎞ ⎛ ΓRV t ⎜⎜⎜ Im 0m,m ⎟⎟⎟ ⎜⎜⎜ ΓRV 0m,m ⎟⎟⎟ ⎜⎜⎜⎜ Im +Γ RV V ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ = ⎜⎜⎜⎝ ⎜ R ⎠⎝ Γ R ⎠⎜ 0n,m V 0m,m ΓV ⎝ V R V t Im +ΓV

Im +ΓRV

⎞⎛ ⎞ V ⎟⎟⎟⎟ ⎜⎜ V 0n,n ⎟⎟ ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎟⎠ . ⎟⎟⎠ ⎜⎝ ΓRV t 0n,m In RV

(5.398)

t

Im +ΓV

We may note that the expression (ΓRV /(Im + ΓRV ))V t that appears three times on the extreme right-hand side of (5.398) is the bi-gyrohalf, 12 ⊗V t , of V t as shown in (5.496), p. 281. Remarkably, the first representation of E(V) in (5.398) involves both the left and right gamma factors, ΓVL and ΓRV , while the second one involves only the right gamma factor ΓRV . The advantage of the second representation of E(V) in (5.398) over the first one emerges when we study the special case when m = 1, which is of interest in special relativity theory, that is, when the dimension of time is 1. Clearly, the right gamma L since ΓRm=1,V is a scalar while factor ΓRm=1,V is simpler than the left gamma factor Γn>1,V L is an n × n matrix. Hence, indeed, the second representation of E(V) in (5.398) Γn>1,V is simpler than the first one when m = 1. This simplification proves useful in our study in Sect. 6.3 of the Lorentz bi-boost of signature (1, n), n > 1, when the bi-boost descends to a boost.

5.24. Bi-gyrovector Spaces  We introduce a scalar multiplication ⊗ into the bi-gyrogroup (Rn×m c , ⊕ ), m, n ∈ N, turnn×m  ing it into a bi-gyrovector space (Rc , ⊕ , ⊗). When no confusion arises between Bv (V) and B p (P) = B(P) we may write B(V) instead of Bv (V).

Definition 5.85. (Scalar Multiplication, V). Noting the bi-gamma factor (5.115),

Bi-gyrogroups and Bi-gyrovector Spaces – V

p. 201, and its commuting relations (5.119), let ⎞ ⎛  −1 −1 ⎟ 1  ⎜⎜⎜ t⎟ −2 V t V −2 V t V ⎟⎟⎟ I − c I − c V m m ⎜ ⎟⎟⎟ c2  B(V) = ⎜⎜⎜⎜  −1 −1 ⎠ ⎝ In − c−2 VV t V In − c−2 VV t ⎛  −1 ⎜⎜⎜ ⎜⎜⎜ Im − c−2 V t V = ⎜⎜  −1 ⎝ V Im − c−2 V t V ⎛ ⎜⎜⎜ = ⎜⎜⎜⎝

ΓRV ΓVL V = VΓRV

⎞ −1 ⎟ 1 t −2 VV t ⎟⎟⎟ V I − c n ⎟⎟ c2  ⎟⎠ −1 ⎟ In − c−2 VV t

(5.399)

⎞ c−2 ΓRV V t = c−2 V t ΓVL ⎟⎟⎟ ⎟⎟⎟ ∈ SO(m, n) ⊂ R(m+n)×(m+n) ⎠ L ΓV

be a bi-boost of signature (m, n), parametrized by V ∈ Rn×m c . The scalar multiplicais the unique element r⊗V ∈ Rn×m that tion r⊗V = V⊗r between r ∈ R and V ∈ Rn×m c c satisfies the equation B(r⊗V) = (B(V))r .

(5.400)

It is anticipated in Def. 5.85 of the scalar multiplication r⊗V in the ball Rn×m that c r⊗V is uniquely determined by (5.400). By employing the Singular Value Decompo⊂ Rn×m , Theorem 5.86 demonstrates that this is indeed the sition (SVD) of V ∈ Rn×m c case. The singular value decomposition of V ∈ Rn×m is given by the equation   Σk 0k,m−k Ot (5.401) V = On 0n−k,k 0n−k,m−k m [39, Sect. 7.3], where 1. k = rank(V) is the rank of V, k ≤ min{m, n}; 2. Σk is a k × k diagonal matrix of which the diagonal entries are positive; and 3. On ∈ O(n) and Om ∈ O(m), where O(n) is the group of all n × n real orthogonal matrices. The diagonal matrix Σ2k = diag(σ21 , σ22 , . . . , σ2k ) ,

(5.402)

σ21 ≥ σ22 ≥ . . . ≥ σ2k > 0, where σ2i , i = 1, . . . , k, are the positive eigenvalues of VV t (or, equivalently, of V t V), is determined uniquely by V in (5.401). The square roots σ1 ≥ σ2 ≥ . . . ≥ σk > 0 are the positive singular values of V. Contrasting Σk , the orthogonal matrices On and Om in (5.401) are not determined uniquely by V.

263

264

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Theorem 5.86. (Scalar Multiplication, V). The unique scalar multiplication r⊗V in the ball Rn×m c , m, n ∈ N, that results from (5.400) is given by each of the following two equivalent equations:

r r √ √   In + c−2 VV t − In − c−2 VV t Ik 0k,m−k r⊗V = c Otm r r On √ √ 0 0 −2 t −2 t n−k,k n−k,m−k In + c VV + In − c VV (5.403)

r r √ √   −2 V t V −2 V t V I + c − I − c m m Ik 0k,m−k Ot = c On   , √ √ 0n−k,k 0n−k,m−k m I + c−2 V t V r + I − c−2 V t V r m m for any r ∈ R and V ∈ Rn×m c , where On ∈ O(n) and Om ∈ O(m) are derived from V by the SVD (5.401) of V. Proof. For the sake of simplicity we assume c = 1. The proof for any c > 0 is similar. We divide the proof into three parts. Part I: (Background). Let V be represented by its SVD (5.401),   Σk 0k,m−k Ot , (5.404) V = On 0n−k,k 0n−k,m−k m where On ∈ O(n) and Om ∈ O(m), so that   Σk 0k,n−k t Ot . V = Om 0m−k,k 0m−k,n−k n Then,

 VV = On

Σ2k

t

and

0n−k,k

 V V = Om

Σ2k

t

0m−k,k

implying



√ VV t = On and √

0n−k,k

 V tV

= Om

Σk

Σk

0m−k,k

(5.405)

 0k,n−k Ot 0n−k,n−k n

(5.406)

 0k,m−k Ot , 0m−k,m−k m

(5.407)

 0k,n−k Ot 0n−k,n−k n

(5.408)

 0k,m−k Ot . 0m−k,m−k m

(5.409)

Bi-gyrogroups and Bi-gyrovector Spaces – V

By means of (5.406) we have the chain of equations ⎞ ⎫2 ⎛ ⎞2 ⎧ ⎛ ⎪ ⎜⎜⎜ I − Σ2 0 ⎟⎟ ⎪ ⎪ ⎬ ⎨ ⎜⎜⎜⎜ Ik − Σ2k 0k,n−k ⎟⎟⎟⎟ t ⎪ k,n−k ⎟ ⎜⎜⎜ k ⎟⎟⎟ Ot k ⎜ ⎟ O O = O ⎪ ⎪ ⎜ ⎟ n n ⎪ ⎠ n⎪ ⎝ ⎠ n ⎭ ⎩ ⎝ 0 I 0 I n−k,k n−k n−k,k n−k 

 Ik − Σ2k 0k,n−k t On = On 0n−k,k In−k  = In − On

Σ2k

(5.410) 

0k,n−k Ot 0n−k,k 0n−k,n−k n

= In − VV t , implying

⎛ ⎞ ⎜⎜⎜ I − Σ2 0 ⎟⎟⎟ k k,n−k ⎜ ⎟⎟⎟ Ot . k In − VV t = On ⎜⎝⎜ ⎠ n 0n−k,k In−k

(5.411)

⎛ ⎞ ⎜⎜⎜ I − Σ2 0 ⎟⎟ k,m−k ⎟ ⎟⎟⎟ Ot . k Im − V t V = Om ⎜⎜⎜⎝ k ⎠ m 0m−k,k Im−k

(5.412)

 Similarly,  Hence 



In − VV t

Im − V t V

−1

⎛ ⎞ −1 ⎜⎜⎜ ⎟ 2 0k,n−k ⎟⎟⎟⎟ Ot = On ⎜⎜⎜⎝ Ik − Σk ⎟⎠ n 0n−k,k In−k

−1

⎛ ⎞ −1 ⎜⎜⎜ ⎟ 2 0k,m−k ⎟⎟⎟⎟ Ot . = Om ⎜⎜⎝⎜ Ik − Σk ⎟⎠ m 0m−k,k Im−k

(5.413)

By means of (5.413) and (5.404) we have ⎛ ⎞ −1  ⎜⎜⎜ ⎟⎟⎟ Σ  2 −1 0k,m−k k I − Σ 0 ⎜ ⎟ t k k,n−k k ⎟⎟⎠ Otm In − VV V = On ⎜⎜⎝ 0 0 n−k,k n−k,m−k 0n−k,k In−k ⎛ ⎞ −1 ⎜⎜⎜ ⎟⎟ 2 I − Σ Σ 0 ⎜ ⎟⎟⎟ Ot . k k,m−k ⎟ k = On ⎜⎝⎜ k ⎠ m 0n−k,k 0n−k,m−k

(5.414)

265

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Similarly, by means of (5.413) and (5.405) we have ⎛ ⎞ −1  ⎜⎜⎜ ⎟⎟⎟ Σ  2 −1 t 0k,n−k k I − Σ 0 ⎜ ⎟ t k k,m−k k ⎜ ⎟ Otn Im − V V V = Om ⎜⎝ ⎟⎠ 0m−k,k Im−k 0m−k,k 0m−k,n−k (5.415) ⎛ ⎞ −1 ⎜⎜⎜ ⎟ 2 ⎟⎟ = Om ⎜⎜⎜⎝ Ik − Σk Σk 0k,n−k ⎟⎟⎟⎠ Otn . 0m−k,k 0m−k,n−k √ √ √ √ −1 −1 −1 −1 Substituting Im − V t V , In − VV t , Im − V t V V t , and In − VV t V from (5.413) – (5.415) into the first equation in (5.399) yields the bi-boost SVD (with c = 1), as shown below: ⎛ ⎞ ⎛ ⎞ ⎞ ⎛ −1 −1 ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟ ⎟⎟⎟ ⎜⎜⎜ 2 2 I − Σ 0 I − Σ Σ 0 ⎟⎟ Ot ⎜⎜ k k,m−k ⎟ k k,n−k ⎟ k k ⎜ ⎟⎟⎟⎠ Otn ⎟⎟⎟⎟ ⎜⎜⎜⎜ Om ⎜⎜⎜⎝ k O m m ⎠ ⎝ ⎟⎟⎟ ⎜⎜⎜ 0 I 0 0 m−k,k m−k m−k,k m−k,n−k ⎟⎟⎟ ⎜⎜⎜ B(V) = ⎜⎜ ⎛  ⎟⎟⎟ ⎞ ⎛ ⎞ −1 −1 ⎜⎜⎜ ⎜⎜ ⎟ ⎜⎜⎜ ⎟⎟⎟ 2 ⎜⎜⎜ ⎜⎜⎜ Ik − Σ2k Σk 0k,m−k ⎟⎟⎟⎟⎟ t 0k,n−k ⎟⎟ Ot ⎟⎟⎟⎟ On ⎜⎜⎜⎝ Ik − Σk ⎜⎝On ⎜⎝ ⎟⎠ Om ⎟⎠ n ⎟⎠ 0n−k,k 0n−k,m−k 0n−k,k In−k

⎛ ⎜⎜ Om = ⎜⎜⎜⎝ 0n,m

⎞ ⎛ ⎛ −1 ⎟ ⎜⎜⎜ ⎜⎜⎜ 2 0k,m−k ⎟⎟⎟⎟ ⎜⎜⎜ ⎜⎜⎜ Ik − Σk ⎟⎠ ⎞⎜ ⎝ 0m,n ⎟⎟⎟ ⎜⎜⎜⎜ 0m−k,k Im−k ⎟⎟⎠ ⎜⎜⎜⎛  ⎞ −1 ⎟ On ⎜⎜⎜⎜⎜⎜⎜⎜ 2 ⎜⎜⎜⎜⎜⎜ Ik − Σk Σk 0k,m−k ⎟⎟⎟⎟⎟ ⎝⎝ ⎠ 0n−k,k 0n−k,m−k

⎛ ⎞⎞ −1 ⎜⎜⎜ ⎟⎟ 2 ⎜⎜⎜ Ik − Σk Σk 0k,n−k ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟ ⎞ ⎝ ⎠⎟ ⎛ 0m−k,k 0m−k,n−k ⎟⎟⎟⎟ ⎜⎜⎜ Otm 0m,n ⎟⎟⎟ ⎟ ⎜ ⎟⎟ ⎛ ⎞ ⎟⎟⎟⎟ ⎜⎝ −1 t ⎠ ⎟ ⎜⎜⎜ ⎟ 0 O n 0k,n−k ⎟⎟⎟⎟ ⎟⎟⎟⎟ n,m ⎜⎜⎜ Ik − Σ2k ⎟⎠ ⎟⎠ ⎝ 0n−k,k In−k (5.416)

For clarity, we continue the matrix manipulations with obvious indices omitted. Following (5.416) we have the bi-boost SVD ⎛ √ ⎞ √ −1 −1 ⎜⎜ 2 0 I − Σ2 Σ 0⎟⎟⎟⎟   ⎜⎜⎜⎜ I − Σ   ⎟ Om 0 ⎜⎜⎜ 0 I 0 0⎟⎟⎟⎟ Otm 0 ⎜√ ⎟⎟⎟ . (5.417) B(V) = √ −1 0 On ⎜⎜⎜⎜⎜ I − Σ2 −1 Σ 0 ⎟⎟⎟ 0 Otn 2 I − Σ 0 ⎜⎝ ⎟⎠ 0 0 0 I Let

 Om 0 ∈ O(m + n) R= 0 On 

(5.418)

Bi-gyrogroups and Bi-gyrovector Spaces – V

and

⎛ 1 ⎞ ⎜⎜⎜ √2 Ik 0 √12 Ik 0⎟⎟⎟ ⎜⎜⎜ ⎟ ⎜⎜⎜ 0 I 0 0⎟⎟⎟⎟⎟ Q = ⎜⎜⎜ 1 ⎟ ∈ O(m + n) . 1 ⎜⎜⎜ √2 Ik 0 − √2 Ik 0⎟⎟⎟⎟⎟ ⎝ ⎠ 0 0 0 I

Then, by means of (5.417) – (5.419), ⎛√ ⎜⎜⎜ I − Σ2 −1 (I + Σ) ⎜⎜⎜⎜ 0 ⎜ B(V) = RQ ⎜⎜⎜⎜ ⎜⎜⎜ 0 ⎜⎝ 0

⎞ 0⎟⎟⎟⎟ ⎟ 0⎟⎟⎟⎟ t t ⎟⎟⎟ Q R . √ −1 I − Σ2 (I − Σ) 0⎟⎟⎟⎟ ⎠ 0 I

0 I

0 0

0 0

Let √

so that

I−

−1

(5.420)



I+Σ , I−Σ so that the matrix T is diagonal positive definite, satisfying the equations  −1 I−Σ √ = I − Σ2 (I − Σ) . T −1 = I+Σ Then, by means of (5.421) – (5.422), we have ⎛ ⎞ ⎜⎜⎜T 0 0 0⎟⎟⎟ ⎜⎜⎜ 0 I 0 0⎟⎟⎟⎟⎟ t t B(V) = RQ ⎜⎜⎜⎜⎜ ⎟Q R , ⎜⎜⎝ 0 0 T −1 0⎟⎟⎟⎟⎠ 0 0 0 I T :=

Σ2

(5.419)

(I + Σ) =

(5.421)

(5.422)

(5.423)

⎛ r ⎞ ⎜⎜⎜T 0 0 0⎟⎟⎟ ⎜⎜⎜ 0 I 0 0⎟⎟⎟ ⎟⎟⎟ Qt Rt (B(V))r = RQ ⎜⎜⎜⎜⎜ ⎜⎜⎝ 0 0 T −r 0⎟⎟⎟⎟⎠ 0 0 0 I ⎛ r −r ⎜⎜⎜ T + T ⎜⎜⎜⎜ 2 ⎜⎜⎜ 0 ⎜⎜⎜ = R ⎜⎜⎜⎜ r −r ⎜⎜⎜⎜ T − T ⎜⎜⎜ 2 ⎜⎝ 0

0 I 0 0

T r − T −r 2 0 T r + T −r 2 0

⎞ ⎟ 0⎟⎟⎟⎟ ⎟⎟⎟ 0⎟⎟⎟⎟⎟ t ⎟⎟⎟ R . ⎟⎟ 0⎟⎟⎟⎟⎟ ⎟⎟⎠ I

(5.424)

267

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

By means of (5.418) and (5.424), we have ⎛ r ⎞ ⎛ ⎛ T r + T −r −r ⎜⎜⎜ ⎜⎜⎜ ⎜⎜⎜ T − T ⎟⎟⎟ 0 ⎜⎜⎜⎜Om ⎜⎜⎜⎝ ⎟⎟⎟⎠ Otm Om ⎜⎜⎜⎝ 2 2 ⎜⎜⎜ 0 I 0 ⎜ (B(V))r = ⎜⎜⎜⎜⎜ ⎛ r ⎛ ⎞ −r r −r ⎜⎜⎜ ⎜⎜ T − T ⎜⎜⎜ T + T ⎜⎜⎜ ⎜⎜⎜ 0⎟⎟⎟⎟⎟ t ⎜ 2 2 ⎝⎜ On ⎝⎜ ⎠⎟ Om On ⎜⎝⎜ 0 0 0

⎞ ⎞ 0⎟⎟⎟⎟⎟ t ⎟⎟⎟⎟⎟ ⎟⎠ On ⎟⎟⎟ ⎟⎟⎟ 0 ⎟ ⎞ ⎟⎟⎟⎟⎟ . ⎟ 0⎟⎟⎟⎟⎟ t ⎟⎟⎟⎟ ⎟⎠ On ⎟⎠ I

(5.425)

Part II: (Proof of the first equation for r⊗V in (5.403)). Let Z ∈ Rn×m be given by c the equation ⎛ √ √ −1 −1 t ⎞ tZ tZ ⎜ ⎜ I − Z I − Z Z ⎟⎟⎟⎟ (5.426) (B(V))r = B(Z) = ⎜⎜⎜⎝ √ √ −1 −1 ⎟ ⎠. I − ZZ t Z I − ZZ t If Z ∈ Rn×m that satisfies (5.426) exists, then, by definition 5.85, r⊗V = Z. c Furthermore, if Z exists, then by means of (5.425) – (5.426), ⎛ r ⎞ T + T −r ⎜ ⎟ ⎜ √ ⎜⎜⎜ −1 0⎟⎟⎟⎟ t t ⎜ ⎟⎟⎟ On 2 I − ZZ = On ⎜⎜ ⎝ ⎠ 0 I ⎛ r −r ⎜⎜⎜ T − T √ −1 ⎜ 2 I − ZZ t Z = On ⎜⎜⎜⎜ ⎝ 0

(5.427)

⎞ ⎟ 0⎟⎟⎟⎟ t ⎟⎟⎟ Om . ⎠ 0

Hence, by (5.427), ⎧ ⎛ r −r ⎪ ⎜T + T ⎪ ⎪ ⎨ ⎜⎜⎜⎜ Z=⎪ On ⎜⎜ 2 ⎪ ⎪ ⎩ ⎝ 0

⎞ ⎫−1 ⎛ r −r ⎪ ⎜⎜⎜ T − T ⎪ 0⎟⎟⎟⎟⎟ t ⎪ ⎬ ⎜ On ⎜⎜⎜⎝ ⎟⎟⎠ On ⎪ 2 ⎪ ⎪ ⎭ I 0

⎛ r −r ⎜⎜⎜ T + T = On ⎜⎜⎜⎜⎝ 2 0

⎞−1 ⎛ r T − T −r 0⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ ⎟⎟⎠ ⎜⎜⎝ 2 I 0

⎛ r −r ⎜⎜⎜ T − T ⎜ = On ⎜⎜⎜⎝ T r + T −r 0

⎞ 0⎟⎟⎟⎟⎟ t ⎟⎟⎠ Om . 0

⎞ 0⎟⎟⎟⎟⎟ t ⎟⎟⎠ Om 0

⎞ 0⎟⎟⎟⎟⎟ t ⎟⎟⎠ Om 0

(5.428)

Conversely, if Z is given by (5.428) then (B(V))r = B(Z) ,

(5.429)

Bi-gyrogroups and Bi-gyrovector Spaces – V

where (B(V))r is given by (5.425), as one can check. Hence, by Def. 5.85, r⊗V = Z. We therefore wish to express Z in terms of V. Following (5.421) we have r  I+Σ 2r , (5.430) T = I−Σ so that T r − T −r T 2r − I (I + Σ)r − (I − Σ)r = . = T r + T −r T 2r + I (I + Σ)r + (I − Σ)r Following (5.428), Z can be written as ⎛ r −r ⎜⎜⎜ T − T ⎜ Z = On ⎜⎜⎜⎝ T r + T −r 0

(5.431)

⎞   I 0 t 0⎟⎟⎟⎟⎟ t O , ⎟⎟⎠ On On 0 0 m 0

(5.432)

Following (5.408) we have

  √ I+Σ 0 t t O I + VV = On 0 I n (5.433)

  √ I−Σ 0 t t O , I − VV = On 0 I n implying (I +

  √ (I + Σ)r 0 t VV t )r = On O 0 I n (5.434)

  √ (I − Σ)r 0 t r t (I − VV ) = On O . 0 I n Hence, by means of (5.434) and (5.431) we have ⎛ (I + Σ)r − (I − Σ)r √ √ ⎜⎜⎜ r r t t (I + VV ) − (I − VV ) ⎜⎜ (I + Σ)r + (I − Σ)r = O √ √ n⎜ ⎜⎝ t r t r (I + VV ) + (I − VV ) 0 ⎛ r −r ⎜⎜⎜ T − T ⎜ r = On ⎜⎜⎜⎝ T + T −r 0

⎞ 0⎟⎟⎟⎟⎟ t ⎟⎟⎠ On . 0

⎞ 0⎟⎟⎟⎟⎟ t ⎟⎟⎠ On 0 (5.435)

Substituting the extreme right-hand side of (5.435) into (5.432), and noting that

269

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Z = r⊗V, yields

√

√

 I 0 t r⊗V = O , On √ √ 0 0 m (I + VV t )r + (I − VV t )r (I +

VV t )r − (I −

VV t )r



(5.436)

thus verifying the first equation for r⊗V in (5.403). that satisPart III: (Proof of the second equation for r⊗V in (5.403)). If Z ∈ Rn×m c fies (5.426) exists, then, by definition 5.85, r⊗V = Z. Furthermore, if Z exists, then by means of (5.425) – (5.426), ⎛ r ⎞ −r ⎜⎜⎜ T + T ⎟⎟⎟ √ −1 0 ⎜ ⎟⎟⎟ t ⎜ t 2 I − Z Z = Om ⎜⎜⎜ ⎟⎟⎠ Om ⎝ 0 I (5.437) ⎛ r ⎞ −r T − T ⎜⎜⎜ ⎟ √ −1 0⎟⎟⎟⎟ t ⎜ ⎟⎟⎟ On . 2 I − Z t Z Z t = Om ⎜⎜⎜⎜ ⎝ ⎠ 0 0 Hence, by (5.437)

⎧⎛ ⎛ r −r ⎪ ⎪ ⎪ ⎪⎜⎜⎜⎜ ⎜⎜⎜⎜ T + T ⎨ ⎜⎜⎜Om ⎜⎜⎜ Z=⎪ 2 ⎪ ⎝ ⎝ ⎪ ⎪ ⎩ 0 ⎧ ⎛ r −r ⎪ ⎪ ⎜T + T ⎪ ⎪ ⎨ ⎜⎜⎜⎜ Om ⎜⎜⎝ =⎪ 2 ⎪ ⎪ ⎪ ⎩ 0 ⎛ r −r ⎜⎜⎜ T − T = On ⎜⎜⎜⎜⎝ T r + T −r 0

⎛ r ⎞ ⎞−1 −r ⎜⎜⎜ T − T 0⎟⎟⎟⎟⎟ t ⎟⎟⎟⎟⎟ ⎟⎟⎠ Om ⎟⎟⎠ Om ⎜⎜⎜⎜⎝ 2 I 0 ⎞−1 ⎛ r T − T −r 0⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ ⎟⎟⎠ ⎜⎜⎝ 2 I 0

t ⎞ ⎫ ⎪ ⎪ ⎪ 0⎟⎟⎟⎟⎟ t ⎪ ⎬ ⎟⎟⎠ On ⎪ ⎪ ⎪ ⎪ ⎭ 0

t ⎞ ⎫ ⎪ ⎪ ⎪ 0⎟⎟⎟⎟⎟ t ⎪ ⎬ ⎟⎟⎠ On ⎪ ⎪ ⎪ ⎪ ⎭ 0

(5.438)

⎞ 0⎟⎟⎟⎟⎟ t ⎟⎟⎠ Om . 0

Conversely, if Z is given by (5.438) then (B(V))r = B(Z) ,

(5.439)

where (B(V))r is given by (5.425), as one can check. Hence, by Def. 5.85, r⊗V = Z. We therefore wish to express Z in terms of V. Following (5.438), Z can be written as ⎛ r ⎞   T − T −r ⎜ ⎜ ⎜ I 0 t 0⎟⎟⎟⎟⎟ t Om Om ⎜⎜⎜⎜⎝ T r + T −r (5.440) Z = On ⎟⎟⎠ Om . 0 0 0 0

Bi-gyrogroups and Bi-gyrovector Spaces – V

Following (5.409) we have

  √ I+Σ 0 t t O I + V V = Om 0 I m (5.441)

  √ I−Σ 0 t t O , I − V V = Om 0 I m implying

  √ (I + Σ)r 0 t r t O (I + V V) = Om 0 I m (5.442)

  √ (I − Σ)r 0 t r t (I − V V) = Om O . 0 I m Hence, by means of (5.442) and (5.431) we have ⎛ (I + Σ)r − (I − Σ)r √ √ ⎜⎜⎜ (I + V t V)r − (I − V t V)r = Om ⎜⎜⎜⎜⎝ (I + Σ)r + (I − Σ)r √ √ (I + V t V)r + (I − V t V)r 0 ⎛ r −r ⎜⎜⎜ T − T ⎜ = Om ⎜⎜⎜⎝ T r + T −r 0

⎞ 0⎟⎟⎟⎟⎟ t ⎟⎟⎠ Om . 0

⎞ 0⎟⎟⎟⎟⎟ t ⎟⎟⎠ Om 0 (5.443)

Substituting the extreme right-hand side of (5.443) into (5.440), and noting that Z = r⊗V, yields √ √   I 0 t (I + V t V)r − (I − V t V)r r⊗V = On O , (5.444) √ √ 0 0 m (I + V t V)r + (I − V t V)r thus verifying the second equation for r⊗V in (5.403), so that the proof of the theorem is complete. 

5.25. On the Pseudo-inverse of a Matrix In this section we explore the presence of a Moore-Penrose pseudo-inverse matrix in each of the two r⊗V formulas (5.403). √ It follows from (5.408) that the Moore-Penrose pseudo-inverse ( VV t )# of the √ matrix VV t is given by ⎛ −1 ⎞ √ ⎜⎜⎜ Σk 0k,n−k ⎟⎟⎟ t # t ( VV ) = On ⎜⎜⎝ (5.445) ⎟⎟⎠ On , 0n−k,k 0n−k,n−k

271

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

where, following (5.402), k = rank(V) and −1 −1 −1 Σ−1 k = diag(σ1 , σ2 , . . . , σk ) .

(5.446)

Hence, by means of (5.445) and (5.404), ⎛ −1 ⎞⎛ ⎞ √ ⎜⎜⎜ Σk 0k,n−k ⎟⎟⎟ ⎜⎜⎜ Σk 0k,m−k ⎟⎟⎟ t # t ⎜ ⎜ ⎟ ⎟⎟⎠ Om ( VV ) V = On ⎜⎝ ⎟⎠ ⎜⎝ 0n−k,k 0n−k,n−k 0n−k,k 0n−k,m−k ⎛ ⎞ ⎜⎜⎜ Ik 0k,m−k ⎟⎟⎟ t ⎟⎟⎠ Om . = On ⎜⎜⎝ 0n−k,k 0n−k,m−k Similarly,

so that

⎛ ⎞ √ ⎜⎜⎜ Ik 0k,m−k ⎟⎟⎟ t # ⎟⎟⎠ Om , V( V t V) = On ⎜⎜⎝ 0n−k,k 0n−k,m−k √ √ ( VV t )# V = V( V t V)# .

(5.447)

(5.448)

(5.449)

The right-hand side of each of (5.447) and (5.448) is a factor that appears in each of the two r⊗V formulas (5.403). Hence, by means of (5.447) and (5.448), the two r⊗V formulas (5.403) can be written as

r r √ √ In + VV t − In − VV t √ r⊗V = r r ( VV t )# V √ √ In + VV t + In − VV t (5.450)

r r √ √ Im + V t V − Im − V t V √ = V( V t V)# r r . √ √ Im + V t V + Im − V t V If the matrix VV t is invertible then √ √ In ( VV t )# V = ( VV t )−1 V = √ V VV t

(5.451)

and, similarly, if the matrix V t V is invertible then √ √ Im . (5.452) V( V t V)# = V( V t V)−1 = V √ V tV Hence, when matrix inversion is justified, the two r⊗V formulas (5.403) can be

Bi-gyrogroups and Bi-gyrovector Spaces – V

written as (where c > 0 reappears)

r r √ √ In + c−2 VV t − In − c−2 VV t In r⊗V := V r r √ √ √ c−2 VV t In + c−2 VV t + In − c−2 VV t

r r √ √ Im + c−2 V t V − Im − c−2 V t V Im =V√

r r √ √ c−2 V t V Im + c−2 V t V + Im − c−2 V t V

(5.453)

for all r ∈ R and V ∈ Rn×m c . Noting the definition of the bi-gamma factor in (5.115), p. 201, the scalar multiplication (5.453) can be expressed in terms of the bi-gamma factor as follows: 2r   In − ΓVL − (ΓVL )2 − In ΓVL V r⊗V :=  2r   L 2 L L 2 (ΓV ) − In In + Γ − (Γ ) − In V

V





2r

(5.454)

Im − ΓRV − (ΓRV )2 − Im =V 2r   (ΓRV )2 − Im Im + ΓR − (ΓR )2 − Im V V ΓRV

for all r ∈ R and V ∈ Rn×m c , when matrix inversion is justified. In the special case when m = 1, the scalar multiplication in (5.454) specializes to the ones in [84, Eqs. (6.267), p. 195, and (6.266), p. 194] and, hence, in (3.19), p. 66. The r⊗V formulas (5.454) prove useful in the study of the bi-gyrohalf in Sect. 5.30.

5.26. Scalar Multiplication: The SVD Formula The two r⊗V formulas in (5.403), p. 264, are partially expressed in terms of the SVD of V ∈ Rn×m c . When fully expressed in terms of the SVD of V, the distinct forms of these two formulas coincide, giving rise to the following theorem. Theorem 5.87. (Scalar Multiplication, SVD). For any m, n ∈ N and c > 0, let V ∈ be represented by its SVD (5.401), Rn×m c   Σk 0k,m−k Ot , V = On (5.455) 0n−k,k 0n−k,m−k m

273

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

where Σk ∈ Rk×k is diagonal, On ∈ O(n), Om ∈ O(m), and k = rank(V). Then, for any r ∈ R, ⎛ ⎞ ⎜⎜⎜ (Ik + c−1 Σk )r − (Ik − c−1 Σk )r ⎟⎟ 0 ⎜⎜⎜ ⎟⎟⎟ t k,m−k ⎟ (5.456) r⊗V = c On ⎜⎜ (Ik + c−1 Σk )r + (Ik − c−1 Σk )r ⎟⎟⎠ Om . ⎝ 0n−k,k 0n−k,m−k Proof. By means of (5.455) and (5.408), p. 264, we have ⎛ ⎞ √ ⎜⎜⎜Ik ± c−1 Σk 0k,n−k ⎟⎟⎟ t −1 t ⎟⎟⎠ On . In ± c VV = On ⎜⎜⎝ 0n−k,k In−k

(5.457)

Substituting (5.457) into the first r⊗V formula in (5.403), and using the notation T ± := Ik ± c−1 Σk

(5.458)

yields the following chain of equations: ⎧⎛ r ⎞ ⎛ r ⎞⎫ ⎪ ⎪ ⎪ ⎬ t ⎨⎜⎜⎜⎜ T + 0k,n−k ⎟⎟⎟⎟ ⎜⎜⎜⎜ T − 0k,n−k ⎟⎟⎟⎟⎪ O On ⎪ ⎜⎝ ⎟⎠ − ⎜⎝ ⎟⎠⎪ ⎛ ⎞ ⎪ ⎭ n ⎩ 0n−k,k In−k ⎜⎜⎜ Ik 0n−k,k In−k ⎪ 0k,m−k ⎟⎟⎟ t ⎜ ⎟⎟⎠ Om ⎧⎛ r On ⎜⎝ r⊗V = c ⎞ ⎛ r ⎞⎫ ⎪ ⎪ ⎜ ⎜ ⎟ ⎟ 0 0 ⎪ ⎪ T T 0 0 n−k,k n−k,m−k ⎟⎟ ⎜⎜⎜ − k,n−k ⎟ k,n−k ⎟ ⎨⎜⎜⎜ + ⎟⎟⎟⎬ Ot On ⎪ ⎜ ⎟+⎜ ⎪ ⎪ ⎭ n ⎩⎝0n−k,k In−k ⎠ ⎝0n−k,k In−k ⎠⎪ ⎛ r ⎞ ⎜⎜⎜T + − T −r 0k,n−k ⎟⎟⎟ ⎟⎟⎠ Otn On ⎜⎜⎝ 0n−k,k 0n−k,n−k =c On ⎛ r ⎞ ⎜⎜⎜T + + T −r 0k,n−k ⎟⎟⎟ t ⎟⎟⎠ On On ⎜⎜⎝ 0n−k,k 0n−k,n−k ⎛ T r −T r ⎞ ⎜⎜⎜ T+r +T−r 0k,n−k ⎟⎟⎟ + − ⎜ ⎟⎟⎠ = c On ⎜⎝ 0n−k,k 0n−k,n−k

⎛ ⎞ ⎜⎜⎜ Ik 0k,m−k ⎟⎟⎟ t ⎜⎜⎝ ⎟⎟⎠ Om 0n−k,k 0n−k,m−k

⎛ ⎞ ⎜⎜⎜ Ik 0k,m−k ⎟⎟⎟ t ⎜⎜⎝ ⎟⎟⎠ Om 0n−k,k 0n−k,m−k

⎛ T r −T r ⎞ ⎜⎜⎜ T+r +T−r 0k,m−k ⎟⎟⎟ t + − ⎟⎟⎠ Om , = c On ⎜⎜⎝ 0n−k,k 0n−k,m−k (5.459) as desired.



Bi-gyrogroups and Bi-gyrovector Spaces – V

By L’Hospital’s rule ⎛ ⎜⎜⎜ (Ik + c−1 Σk )r − (Ik − c−1 Σk )r ⎜ lim r⊗V = lim c On ⎜⎜⎜⎜ (Ik + c−1 Σk )r + (Ik − c−1 Σk )r ⎝ c→∞ c→∞ 0n−k,k ⎛ (Ik + c−1 Σk )r − (Ik − c−1 Σk )r ⎜⎜⎜ lim c ⎜⎜⎜c→∞ = On ⎜⎜ (Ik + c−1 Σk )r + (Ik − c−1 Σk )r ⎝ 0n−k,k

0k,m−k 0n−k,m−k

⎞ ⎟⎟⎟ ⎟⎟⎟ t ⎟⎟⎠ Om

⎞ ⎟ 0k,m−k ⎟⎟⎟⎟ t ⎟⎟⎟ Om ⎠ 0n−k,m−k

(5.460)

⎛ ⎞ ⎛ ⎞ 0k,m−k ⎟⎟⎟ t 0k,m−k ⎟⎟⎟ t ⎜⎜⎜ rΣk ⎜⎜⎜ Σk ⎟⎟⎠ Om = r On ⎜⎜⎝ ⎟⎟⎠ Om = rV . = On ⎜⎜⎝ 0n−k,k 0n−k,m−k 0n−k,k 0n−k,m−k Hence, as expected, in the Euclidean limit, c → ∞, of large c, Einstein scalar multiplitends to the common matrix multiplication by scalars in Rn×m . cation in Rn×m c

5.27. Properties of Bi-gyrovector Space Scalar Multiplication   n×m The triple (Rn×m given by Def. 5.57, c , ⊕ , ⊗), where ⊕ is the binary operation in Rc n×m p. 241 and ⊗ is the scalar multiplication in Rc given by (5.403), p. 264, is the biof the bi-boost parameter V. gyrovector space of the ball Rn×m c  Theorem 5.88. (Bi-gyrovector Space Properties). Let (Rn×m c , ⊕ , ⊗) be the bigyrovector space of a ball Rn×m c , m, n ∈ N. Then, its scalar multiplication ⊗ obeys the scalar distributive law

(r1 + r2 )⊗V = r1 ⊗V⊕ r2 ⊗V ,

(5.461)

(r1 r2 )⊗V = r1 ⊗(r2 ⊗V) ,

(5.462)

r⊗(r1 ⊗V⊕ r2 ⊗V) = r⊗(r1 ⊗V)⊕ r⊗(r2 ⊗V),

(5.463)

the scalar associative law

the monodistributive law

and the scalar-matrix transpose law (r⊗V)t = r⊗V t for all r, r1 , r2 ∈ R and V ∈ Rn×m c .

(5.464)

275

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Proof. By means of (5.400) and Theorem 5.56, p. 240, B((r1 + r2 )⊗V) = (B(V))r1+r2 = (B(V))r1 (B(V))r2 = B(r1 ⊗V)B(r2⊗V)

(5.465)

= B(r1 ⊗V⊕r2 ⊗V) and rgyr[r1 ⊗V, r2 ⊗V] = In .

(5.466)

It follows from (5.466) and (5.309), p. 241, that r1 ⊗V⊕r2 ⊗V = r1 ⊗V⊕ r2 ⊗V ,

(5.467)

so that (5.465) and (5.467) imply B((r1 + r2 )⊗V) = B(r1 ⊗V⊕ r2 ⊗V) .

(5.468)

Owing to the bijective correspondence between bi-boosts B(V) and their parameter V, (5.461) follows from (5.468). By means of (5.400), B((r1r2 )⊗V) = (B(V))r1r2 = ((B(V))r2 )r1 = (B(r2⊗V))r1

(5.469)

= B(r1 ⊗(r2 ⊗V)) . Owing to the bijective correspondence between bi-boosts B(V) and their parameter V, (5.462) follows from (5.469). The proof of the monodistributive law (5.463) follows from the scalar distributive law (5.461) and the scalar associative law (5.462), as shown in the chain of equations

Bi-gyrogroups and Bi-gyrovector Spaces – V

below: r⊗(r1 ⊗V⊕ r2 ⊗V) = r⊗((r1 + r2 )⊗V) = (r(r1 + r2 ))⊗V = (rr1 + rr2 )⊗V

(5.470)

= (rr1 )⊗V⊕ (rr2 )⊗V = r⊗(r1 ⊗V)⊕ r⊗(r2 ⊗V) . Finally, the scalar-matrix transpose law (5.464) follows immediately from the two equivalent formulas for r⊗V in (5.403), p. 264.  Theorem 5.89. (A Trivial Bi-gyration). Let (Rn×m , ⊕ , ⊗) be the gyrovector space of a space Rn×m , m, n ∈ N. Then, lgyr[r1 ⊗V, r2 ⊗V] = In rgyr[r1 ⊗V, r2 ⊗V] = Im

(5.471)

for all r1 , r2 ∈ R and V ∈ Rn×m . Proof. By (5.465) we have B(r1 ⊗V)B(r2⊗V) = B((r1 + r2 )⊗V) , from which (5.471) follows by means of Theorem 5.56, p. 240.

(5.472) 

Example 5.90. In this example we show that in the special case when m = 1 the n×m ⊂ Rn×m , scalar multiplication (5.454) in the ball gyrovector space (Rn×m c , ⊕, ⊗), Rc descends to Einstein scalar multiplication in the Euclidean ball gyrovector space (Rnc , ⊕, ⊗), Rnc ⊂ Rn , studied for instance in [84, Eq. (6.267)]. We may note that for m = 1 right gyrations are trivial so that ⊕ = ⊕ when m = 1. descends when m = 1 to the Euclidean c-ball By (5.147), p. 209, the c-ball Rn×m c n×1 n Rc = Rc of column vectors v = V, so that V t V = v 2 . By (5.154), p. 210, the right gamma factor ΓRV descends when m = 1 to the gamma factor, ΓRm=1,V = γV . Hence, by (5.454) and (5.453), when m = 1 the scalar multiplication r⊗V in the gyrovector space

277

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

n×m (Rn×m descends to the scalar multiplication c , ⊕, ⊗) of the c-ball Rc  1 − (γv − γv2 − 1)2r γv r⊗v = v   1 + (γv − γv2 − 1)2r γv2 − 1 r  r  v v − 1− 1+ c c v v v = c = c tanh(r tanh−1 ) , r  r v c v v v 1+ + 1− c c

(5.473)

v  0, r⊗0 = 0. We thus recover in (5.473) the well-known Einstein scalar multiplication in the Euclidean ball Rnc , studied for instance in [81, 84, 93, 94, 95, 96, 98] in the context of analytic hyperbolic geometry, and presented in (3.19), p. 66.

5.28. Scaling Property Theorem 5.91. (Scaling Property). For any m, n ∈ N and c > 0, let V ∈ Rn×m c . If the involved matrix inversions are justified, then In In |r|⊗V = √ V √ t (r⊗V)(r⊗V) VV t

(5.474)

Im Im |r|⊗V √ =V√ (r⊗V)t (r⊗V) VtV

(5.475)

and

for all r ∈ R, r  0. Proof. By means of (5.453), noting that functions of the matrix VV t ∈ Rn×n are symmetric, we have the matrix equation ⎧ ⎫2 √ √ ⎪ ⎪ ⎪ ⎨ (In + c−2 VV t )r − (In − c−2 VV t )r ⎪ ⎬ t c (5.476) (r⊗V)(r⊗V) = ⎪ √ √ ⎪ ⎪ ⎩ (I + c−2 VV t )r + (I − c−2 VV t )r ⎪ ⎭ n

n

Rn×m c .

for all r ∈ R and V ∈ Hence, √ √  (In + c−2 VV t )r − (In − c−2 VV t )r c = ± (r⊗V)(r⊗V)t . √ √ (In + c−2 VV t )r + (In − c−2 VV t )r Substituting (5.477) into (5.453) yields  In V r⊗V = ± (r⊗V)(r⊗V)t √ VV t

(5.477)

(5.478)

Bi-gyrogroups and Bi-gyrovector Spaces – V

so that

⎧ √ In ⎪ ⎪ (r⊗V)(r⊗V)t √ V, i f r ≥ 0 ⎪ ⎪ ⎪ ⎪ VV t ⎨ r⊗V = ⎪ ⎪ ⎪ √ In ⎪ ⎪ ⎪ ⎩− (r⊗V)(r⊗V)t √ t V, i f r ≤ 0 . VV Hence, we have the equation  In V, |r|⊗V = (r⊗V)(r⊗V)t √ VV t from which (5.474) follows immediately. The proof of (5.475) is similar.

(5.479)

(5.480) 

In the special case when m = 1, the scaling property (5.475) of signature (1, n) descends to the common scaling property in vector spaces, |r|⊗V V = , r⊗V V

(m = 1)

(5.481)

for all r ∈ R and V ∈ Rnc , V  0, where V is the Euclidean norm of V.

5.29. Homogeneity Property Theorem 5.92. (Homogeneity Property). Scalar multiplication in a bi-gyrovector  space (Rn×m c , ⊕ , ⊗) possesses the homogeneity property r⊗V = |r|⊗ V

(5.482)

for any r ∈ R and V ∈ Rn×m c . Proof. For simplicity we assume c = 1. The proof for any c > 0 is similar. By means of (5.456) we have ⎛ 2 ⎞ ⎜⎜⎜ S k 0k,n−k ⎟⎟⎟ t t (r⊗V)(r⊗V) = On ⎜⎜⎝ ⎟⎟⎠ On , 0n−k,k 0n−k,n−k

(5.483)

k = rank(V), where S k ∈ Rk×k is diagonal, Sk =

T +r − T −r (Ik + Σk )r − (Ik − Σk )r = =: diag(s1 , s2 , . . . , sk ) T +r + T −r (Ik + Σk )r + (Ik − Σk )r

(5.484)

and where Σk ∈ Rk×k is diagonal, Σk =: diag(σ1 , σ2 , . . . , σk ) .

(5.485)

Consequently, the diagonal elements of S k are related to the diagonal elements of

279

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Σk by the equations si =

(1 + σi )r − (1 − σi )r , (1 + σi )r + (1 − σi )r

(5.486)

i = 1, 2, . . . , k. The diagonal elements σi , i = 1, 2, . . . , k, of Σk are the singular values of V ∈ Rn×m c=1 , so that 0 < σi < 1. Similarly, the diagonal elements si , i = 1, 2, . . . , k, of S k are the singular values of r⊗V ∈ Rn×m c=1 , so that 0 < si < 1. Relation (5.486) is a discrete realization of the function f (x) =

(1 + x)r − (1 − x)r ∈ R, (1 + x)r + (1 − x)r

(5.487)

x ∈ (0, 1), r > 0. The function f (x) is strictly monotonically increasing. Hence, if σ j = max(σi )i=1,2,...,k

(5.488)

for some j ∈ N, 1 ≤ j ≤ k, then correspondingly, s j = max(si )i=1,2,...,k .

(5.489)

V = σ j ,

(5.490)

Hence, the norm of V is σ j ,

and, correspondingly, the norm of r⊗V is s j , r⊗V = s j =

(1 + σ j )r − (1 − σ j )r . (1 + σ j )r + (1 − σ j )r

(5.491)

In calculating the scalar multiplication r⊗ V , we note that V ∈ R1×1 c=1 is a 1 × 1 matrix with the single eigenvalue σ j , as we see from (5.490). Hence, by means of (5.456) with c = 1, where necessarily On=1 = Om=1 = 1, we have (1 + σ j )r − (1 − σ j )r r⊗ V = . (1 + σ j )r + (1 − σ j )r It follows from (5.491) and (5.492) that r⊗V = r⊗ V , as desired.

(5.492) 

Remark 5.93. One should note that the scalar multiplication, ⊗, on the left-hand side of (5.482) represents a scalar multiplication between a real number r ∈ R and a bi-gyropoint V ∈ Rn×m c , while the scalar multiplication, ⊗, on the right-hand side of (5.482) represents a scalar multiplication between a nonnegative number |r| ∈ [0, ∞) and a nonnegative number V ∈ [0, c). These two interpretations of the scalar multiplication are embedded in Theorem 5.86, p. 263, where ⊗ in r⊗V corresponds to a signature (m, n), m, n ∈ N, and ⊗ in |r|⊗ V corresponds to a signature (1, 1).

Bi-gyrogroups and Bi-gyrovector Spaces – V

5.30. Bi-gyrohalf With r = 1/2, the scalar multiplication (5.454) yields the bi-gyrohalf formula in the following theorem. Theorem 5.94. (Bi-gyrohalf). The bi-gyrohalf, 12 ⊗V, of any V ∈ Rn×m is given by c 1 2 ⊗V

=

ΓVL

V =V L

I n + ΓV

ΓRV Im + ΓRV

,

(5.493)

and the bi-gyrohalf, 12 ⊗V t , of V t is given by 1 t 2 ⊗V

=

ΓVL t Im + ΓVL t

Vt = Vt

ΓRV t In + ΓRV t

.

(5.494)

As expected from the bi-gyrohalf of V, we have the following identity: ΓVL

V⊕ L

I n + ΓV

ΓVL In + ΓVL

V =V,

(5.495)

for all V ∈ Rn×m c . Following (5.464), p. 275, and (5.493) we have 1 t 2 ⊗V

ΓL

= ( 12 ⊗V)t = V t I +ΓV L = n

V

ΓRV Vt Im +ΓRV

.

(5.496)

Interestingly, the extreme right-hand side of (5.496) appears several times in (5.398), p. 262. By means of bi-gyrohalf formulas in (5.493) and (5.496), Result (5.383) of Lemma 5.82 gives rise to the following lemma, the proof of which is immediate. Lemma 5.95. ΓVL = In +

1 L 1 Γ ( ⊗V)V t ∈ Rn×n c2 V 2

(5.497)

1 ΓRV = Im + 2 ΓRV ( 12 ⊗V t )V ∈ Rm×m , c for all V ∈ Rn×m c . Lemma 5.95, in turn, gives rise to the following lemma, the proof of which is immediate.

281

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Lemma 5.96. ΓVL =

ΓRV =

In ∈ Rn×n 1 1 In − 2 ( 2 ⊗V)V t c Im 1 Im − 2 ( 12 ⊗V t )V c

(5.498)

∈ Rm×m ,

for all V ∈ Rn×m c . By means of the definition of the bi-gamma factor in (5.115), p. 201, Lemma 5.96 gives rise to the following lemma that presents elegant dual matrix identities. Lemma 5.97.

 

In − c−2 VV t = In − c−2 ( 12 ⊗V)V t

Im −

c−2 V t V

= Im −

c−2 ( 12 ⊗V t )V

(5.499) .

for all V ∈ Rn×m c . Lemme 5.97 demonstrates the simplicity and elegance that the bi-gyrohalf offers. Further demonstration is provided by the following chain of equations (5.500) that leads to the elegant additive-multiplicative decomposition of the Lorentz transformation of signature (m, n). Employing (5.390) - (5.391), p. 260, along with (5.497), in the following chain of equations we manipulate the Lorentz transformation of signature (m, n), m, n ∈ N, obtaining its additive-multiplicative decomposition. ⎞ ⎛ 1 R t ⎟⎟ ⎜⎜⎜ ΓR Γ V ⎟⎟ ⎜ B(V) = ⎜⎜⎜⎜ V c2 V ⎟⎟⎟⎟ ⎠ ⎝ L ΓV V ΓVL ⎛ ⎞ ⎛ ⎞ ⎜⎜⎜Im 0m,n ⎟⎟⎟ 1 ⎜⎜⎜ ΓRV ( 12 ⊗V t )V ΓRV V t ⎟⎟⎟ ⎟⎟⎟ + ⎜⎜⎜ ⎟⎟⎟ = ⎜⎜⎜⎝ ⎠ c2 ⎝ L 1 t L 1 t⎠ V In ΓV ( 2 ⊗V)V V ΓV ( 2 ⊗V)V ⎛ ⎞ ⎛ ⎞⎛ ⎜⎜⎜Im 0m,n ⎟⎟⎟ 1 ⎜⎜⎜ ΓRV 0m,n ⎟⎟⎟ ⎜⎜⎜ 12 ⊗V t ⎟⎟⎟ + ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ = ⎜⎜⎜⎝ ⎠ c2 ⎝ ⎠⎝ V In 0n,m ΓVL ( 12 ⊗V)V t

(5.500)

⎞⎛ ⎞ 0n,n ⎟⎟⎟ Im ⎟⎟⎟ ⎜⎜⎜ V ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ . ⎠⎝ 1 t ⎠ 0 V ⊗V m,m 2

Evidently, in terms of bi-gyrohalf V, the Lorentz additive-multiplicative decomposition (5.500) takes on unexpected grace and elegance.

Bi-gyrogroups and Bi-gyrovector Spaces – V

5.31. A Commuting Relation for Scalar Multiplication and Rotations Theorem 5.98. (A Commuting Relation, Scalar Multiplication and Rotations). be any element of an Einstein bi-gyrovector space (Rn×m Let V ∈ Rn×m c c , ⊕E , ⊗) of signature (m, n), m, n ∈ N. Then, r⊗(On VOm ) = On (r⊗V)Om

(5.501)

for all On ∈ SO(n), Om ∈ SO(m) and r ∈ R. Proof. Let B(V) be a bi-boost of signature (m, n), m, n ∈ N. By means of the third identity in (5.145), p. 209, and the commutativity of ρ(Otm ) and λ(On ) we have B(On VOm ) = ρ(Otm )λ(On )B(V)ρ(Om)λ(Otn ) = λ(On )ρ(Otm)B(V)ρ(Om)λ(Otn )

(5.502)

for all On ∈ SO(n), Om ∈ SO(m) and V ∈ Rn×m c . Hence, by (5.400), p. 263, and by (5.502), B(r⊗(On VOm )) = (B(On VOm ))r = λ(On )ρ(Otm)(B(V))rρ(Om )λ(Otn ) = λ(On )ρ(Otm)B(r⊗V)ρ(Om )λ(Otn )

(5.503)

= B(On (r⊗V)Om ) , so that B(r⊗(On VOm )) = B(On (r⊗V)Om ) .

(5.504)

Finally, (5.504) yields (5.501) since a bi-boost B(V) determines its parameter V ∈ Rn×m uniquely.  c The following theorem is an immediate result of Theorem (5.98). Theorem 5.99. (A Commuting Relation, Scalar Multiplication and Gyrations). Let (Rn×m c , ⊕E , ⊗) be an Einstein gyrovector space of signature (m, n), m, n ∈ N. Then, r⊗(gyr[A, B]V) = gyr[A, B](r⊗V)

(5.505)

for all r ∈ R and A, B, V ∈ Rn×m c . Proof. By Def. (5.67), p. 250, and by (5.501), noting that lgyr[A, B] ∈ SO(n) and

283

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

rgyr[A, B] ∈ SO(m), we have r⊗(gyr[A, B]V) = r⊗(lgyr[A, B]Vrgyr[A, B]) = lgyr[A, B](r⊗V)rgyr[A, B]

(5.506)

= gyr[A, B](r⊗V) , for all r ∈ R, and A, B, V ∈ Rn×m c , as desired.



5.32. P-V Scalar Multiplication Relationship The scalar multiplication for the parameter V and for the parameter P are related by Theorem 5.101, following a remark about notation. Remark 5.100. (See also Remark 5.58, p. 241). We use the notation ⊗v := ⊗ for the and ⊗ p := ⊗ for the scalar multiplication scalar multiplication (5.403), p. 264, in Rn×m c (4.324), p. 171, in Rn×m when we wish to emphasize the distinction between the scalar  multiplication ⊗v in the gyrovector space (Rn×m c , ⊕v , ⊗v ) of the parameter V and the scalar multiplication ⊗ p in the gyrovector space (Rn×m , ⊕p , ⊗ p ) of the parameter P. This explicit distinction is necessary, for instance, in Theorem 5.101. Theorem 5.101. Let φ : Rn×m → Rn×m be the bijection (5.84) – (5.86), p. 196, from the c , m, n ∈ N. Then, space Rn×m onto its c-ball Rn×m c φ(r⊗ p P) = r⊗v φ(P)

(5.507)

φ−1 (r⊗v V) = r⊗ p φ−1 (V)

(5.508)

and for all r ∈ R, P ∈ Rn×m and V ∈ Rn×m c . Proof. Let V = φ(P). Then, by means of (5.400), (5.261), (4.321), and (5.261) again: we obtain the following chain of equations: Bv (r⊗v φ(P)) = (Bv(φ(P)))r = (B p (P))r = B p (r⊗ p P)

(5.509)

= Bv (φ(r⊗ p P)) . The relationship between bi-boosts Bv (V) and their parameter V is bijective. Hence, (5.509) implies (5.507). The proof of (5.508) is similar. 

Bi-gyrogroups and Bi-gyrovector Spaces – V

5.33. Bi-gyrovector Space Isomorphism Guided by key features of the bijection φ we are led to the following definition of bigyrovector space isomorphism as a map between bi-gyrovector Spaces that respects the bi-gyrovector Space structure. Definition 5.102. (Bi-gyrovector Space Isomorphism). Let (S k , ⊕k , ⊗k ), k = 1, 2, be two bi-gyrovector spaces with respective bi-gyrations (lgyrk [s1,k , s2,k ], rgyrk [s1,k , s2,k ]), where s1,k , s2,k ∈ S k . A bi-gyrovector space isomorphism is a bijective map f : S 1 → S 2 satisfying f (a⊕1 b) = f (a)⊕2 f (b) lgyr1 [a, b] = lgyr2 [ f (a), f (b)] rgyr1 [a, b] = rgyr2 [ f (a), f (b)] f (lgyr1 [a, b]c) = lgyr2 [ f (a), f (b)] f (c) f (c rgyr1 [a, b]) = f (c)rgyr2 [ f (a), f (b)]

(5.510)

f (r⊗1 a) = r⊗2 f (a)

(5.511)

and for all a, b, c ∈ S 1 and r ∈ R. Two bi-gyrovector spaces are isomorphic if a bi-gyrovector space isomorphism exists between them. Theorem 5.103. (Isomorphic Bi-gyrovector Spaces). For any m, n ∈ N, the P-bi gyrovector space (Rn×m , ⊕p , ⊗ p ) and the V-bi-gyrovector space (Rn×m c , ⊕v , ⊗v ) are isomorphic under the bijection φ. Proof. Let φ : Rm×n → Rn×m be the bijection given by (5.84) – (5.86), p. 196. Then, by c (5.310), p. 242, φ(P1 ⊕p P2 ) = φ(P1 )⊕v φ(P2 )

(5.512)

φ(r⊗ p P) = r⊗v φ(P)

(5.513)

and by (5.507), for all P1 , P2 , P ∈ Rn×m and r ∈ R. Furthermore, by (5.269), p. 234, we have lgyr p [P1 , P2 ] = lgyrv [φ(P1 ), φ(P2 )] rgyr p [P1 , P2 ] = rgyrv [φ(P1 ), φ(P2 )]

(5.514)

285

286

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

and by (5.274), p. 235, φ(lgyr p [P1 , P2 ]P3 ) = lgyrv [φ(P1 ), φ(P2 )]φ(P3 ) φ(P3 rgyr p [P1 , P2 ]) = φ(P3 )rgyrv [φ(P1 ), φ(P2 )]

(5.515)

for all P1 , P2 , P3 ∈ Rn×m . Hence, by Def. 5.102, φ is a bi-gyrovector space isomorphism and, accordingly,  the P-bi-gyrovector space (Rn×m , ⊕p , ⊗ p ) and the V-bi-gyrovector space (Rn×m c , ⊕v , ⊗v ) are isomorphic under φ. 

5.34. Group Isomorphism Between Lorentz Groups Let SOp (m, n) be the group SO(m, n), m, n ∈ N, of all Lorentz transformations of signature (m, n) parametrized by the main parameter P ∈ Rn×m and the two orientation parameters On ∈ SO(n) and Om ∈ SO(m). Similarly, let SOv (m, n) be the group SO(m, n) of all Lorentz transformations of signature (m, n) parametrized by the main parameand the two orientation parameters On ∈ SO(n) and Om ∈ SO(m). We ter V ∈ Rn×m c show in this section that the bi-gyrovector space isomorphism φ gives rise to a group isomorphism ψ between the Lorentz groups SOp (m, n) and SOv (m, n). Definition 5.104. (Lorentz Transformation Group Isomorphism). Let SOp (m, n), m, n ∈ N, be the Lorentz group SO(m, n) of all Lorentz transformations Λ of signature (m, n), ⎛ ⎞ ⎜⎜⎜ P ⎟⎟⎟ ⎜ ⎟ (5.516) Λ = Λ(Om , P, On ) = ⎜⎜⎜⎜ On ⎟⎟⎟⎟ , ⎝ ⎠ Om parametrized by the main parameter P ∈ Rn×m and by the left and right rotation parameters On ∈ SO(n) and Om ∈ SO(m), given by (4.88) – (4.89), p. 118, with group operation given in terms of parameter composition by (4.180), p. 139. Similarly, let SOv (m, n) be the Lorentz group SO(m, n) of all Lorentz transformations Λ of signature (m, n), ⎛ ⎞ ⎜⎜⎜ V ⎟⎟⎟ ⎜ ⎟ (5.517) Λ = Λ(Om , V, On ) = ⎜⎜⎜⎜ On ⎟⎟⎟⎟ , ⎝ ⎠ Om parametrized by the main parameter V ∈ Rn×m and by the left and right rotation pac rameters On ∈ SO(n) and Om ∈ SO(m), given by (5.128) – (5.130), p. 204, with group operation given in terms of parameter composition by (5.250), p. 230.

Bi-gyrogroups and Bi-gyrovector Spaces – V

Then, the map ψ : SOp (m, n) → SOv (m, n)

(5.518)

is defined in terms of the bijection φ, φ : Rn×m → Rn×m c ,

(5.519)

⎛ ⎞ ⎛ ⎞ ⎜⎜⎜φ(P)⎟⎟⎟ ⎜⎜⎜ P ⎟⎟⎟ ⎜⎜⎜ ⎟ ⎜⎜⎜ ⎟⎟⎟ ψ ⎜⎜ On ⎟⎟ = ⎜⎜ On ⎟⎟⎟⎟ . ⎝ ⎠ ⎝ ⎠ Om Om

(5.520)

in (5.85), p. 196, by the equation

Theorem 5.105. (Isomorphic Lorentz Groups). The Lorentz groups SOp (m, n) and SOv (m, n) are isomorphic under the group isomorphism ψ. Proof. We have to show that the map ψ is a group isomorphism, that is, it preserves the group operation bijectively. In order to show that ψ preserves (or, respects) the group operation, let us consider

287

288

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

the following chain of equations, which are numbered for subsequent explanation: ⎛ ⎞ ⎞⎛ ⎞⎫ ⎧⎛ P1 Om,2 ⊕ p On,1 P2 ⎜⎜⎜ ⎟⎟⎟ ⎪ ⎪ (1) P P ⎜ ⎜ ⎟ ⎟ 1 2 ⎪ ⎪ ⎜ ⎜ ⎟ ⎟ ⎟⎟ ⎪ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟⎪ ⎬  ⎜⎜⎜⎜ ⎨⎜⎜⎜ ⎜⎜⎜ On,1 ⎟⎟⎟ ⎜⎜⎜ On,2 ⎟⎟⎟⎪ === ψ ⎜⎜⎜ lgyr p [P1 Om,2 , On,1 P2 ]On,1 On,2 ⎟⎟⎟⎟⎟ ψ⎪ ⎪ ⎪ ⎪ ⎜⎝ ⎟⎠ ⎭ ⎩⎝O ⎠ ⎝O ⎠⎪ m,1 m,2 Om,1 Om,2 rgyr p [P1 Om,2 , On,1 P2 ] ⎛ ⎞ φ(P1 Om,2 ⊕ p On,1 P2 ) ⎜⎜⎜ ⎟⎟⎟ ⎟⎟  ⎜⎜⎜⎜ === ⎜⎜⎜ lgyr p [P1 Om,2 , On,1 P2 ]On,1 On,2 ⎟⎟⎟⎟⎟ ⎜⎝ ⎟⎠ Om,1 Om,2 rgyr p [P1 Om,2 , On,1 P2 ] (2)

⎛ ⎞ φ(P1 Om,2 )⊕v φ(On,1 P2 ) ⎜⎜⎜ ⎟⎟⎟ ⎟⎟  ⎜⎜⎜⎜ === ⎜⎜⎜ lgyrv [φ(P1 Om,2 ), φ(On,1 P2 )]On,1 On,2 ⎟⎟⎟⎟⎟ ⎜⎝ ⎟⎠ Om,1 Om,2 rgyrv [φ(P1 Om,2 ), φ(On,1 P2 )] (3)

⎛ ⎞ φ(P1 )Om,2 ⊕v On,1 φ(P2 ) ⎜⎜⎜ ⎟⎟⎟ (4) ⎟⎟  ⎜⎜⎜⎜ === ⎜⎜⎜ lgyrv [φ(P1 )Om,2 , On,1 φ(P2 )]On,1 On,2 ⎟⎟⎟⎟⎟ ⎜⎝ ⎟⎠ Om,1 Om,2 rgyrv [φ(P1 )Om,2 , On,1 φ(P2 )]

(5.521)

⎛ ⎞ V1 Om,2 ⊕v On,1 V2 ⎜⎜⎜ ⎟⎟⎟ ⎟⎟  ⎜⎜⎜⎜ === ⎜⎜⎜ lgyrv [V1 Om,2 , On,1 V2 ]On,1 On,2 ⎟⎟⎟⎟⎟ ⎜⎝ ⎟⎠ Om,1 Om,2 rgyrv [V1 Om,2 , On,1 V2 ] (5)

⎛ ⎞⎛ ⎞ ⎜⎜⎜ V1 ⎟⎟⎟ ⎜⎜⎜ V2 ⎟⎟⎟ (7) ⎟⎟ ⎜⎜ ⎟⎟   ⎜⎜⎜⎜ === ⎜⎜⎜ On,1 ⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ On,2 ⎟⎟⎟⎟⎟ === ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ Om,1 Om,2 (6)

⎛ ⎛ ⎞⎛ ⎞ ⎞ ⎛ ⎞ ⎜⎜⎜φ(P1 )⎟⎟⎟ ⎜⎜⎜φ(P2 )⎟⎟⎟ (8) ⎜⎜⎜ P1 ⎟⎟⎟ ⎜⎜⎜ P2 ⎟⎟⎟ ⎜⎜⎜ ⎜⎜⎜ ⎟⎜ ⎟ ⎟ ⎜ ⎟ ⎜⎜⎜ On,1 ⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ On,2 ⎟⎟⎟⎟⎟  ⎜⎜⎜ On,1 ⎟⎟⎟⎟⎟ ψ ⎜⎜⎜⎜⎜ On,2 ⎟⎟⎟⎟⎟ . = = = ψ ⎜⎜⎝ ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ Om,1 Om,2 Om,1 Om,2

Derivation of the numbered equalities in (5.521): (1) (2) (3) (4) (5) (6) (7) (8)

This equation follows from Result (4.180) of Theorem 4.31, p. 139. Follows from the definition of ψ in (5.520). Follows from (5.273) and from (5.269), p. 234. Follows from (5.87), p. 196. Follows by defining V1 = φ(P1 ) and V2 = φ(P2 ) Follows from (5.250), p. 230. Follows from the definition of V1 and V2 in Item 5. Follows from the definition of the map ψ in (5.520). It follows from (5.521) that the map ψ preserves the group operation. Moreover,

Bi-gyrogroups and Bi-gyrovector Spaces – V

the map ψ is bijective, possessing the inverse map

given by

ψ−1 : SOv (m, n) → SOp (m, n)

(5.522)

⎛ −1 ⎞ ⎛ ⎞ ⎜⎜⎜φ (V)⎟⎟⎟ ⎜⎜⎜ V ⎟⎟⎟ ⎜ ⎜ ⎟ ⎟ ψ−1 ⎜⎜⎜⎜ On ⎟⎟⎟⎟ = ⎜⎜⎜⎜ On ⎟⎟⎟⎟ , ⎝ ⎝ ⎠ ⎠ Om Om

(5.523)

where φ−1 is the inverse map of φ, given by (5.86), p. 196. Hence, the map ψ is a group isomorphism so that the Lorentz groups SOp (m, n)  and SOv (m, n) are isomorphic. It should be noted that we employ in (5.521) the unprimed binary operations ⊕ p and ⊕v rather than their primed counterparts.

5.35. Einstein Bi-gyrogroups and Bi-gyrovector Spaces – V The main goal of this section is to summarize the introduction of two algebraic objects, the bi-gyrogroup and the bi-gyrovector space, which are isomorphic to those presented given by (5.2), p. 186, in Sect. 4.28 via the isomorphism φ : Rn×m → Rn×m c  −1 φ : P → V = In + c−2 PPt P . (5.524) These objects form a natural generalization of the concepts of the gyrogroups and the gyrovector spaces studied in Chaps. 2 and 3, to which they descend when m = 1. In the following two subsections we summarize properties of the bi-gyrogroup and of the ambient space Rn×m of all the bi-gyrovector space that underlie the c-ball Rn×m c n × m real matrices, m, n ∈ N.

5.35.1. Einstein Bi-gyrogroups Let Rn×m be the set of all n × m real matrices, m, n ∈ N, and let ⊕E := ⊕ be the Einstein addition of signature (m, n) in Rn×m c , given by (5.309), p. 241 and by Theorem 5.65, n×m p. 247. The resulting pair (Rc , ⊕E ) is the Einstein bi-gyrogroup of signature (m, n) that underlies the ball Rn×m c . = (Rn×m Einstein bi-gyrogroups Rn×m c c , ⊕E ) are regulated by gyrations, possessing the following properties: 1. The binary operation ⊕E := ⊕ in Rn×m is Einstein addition of signature (m, n), c given by (5.309), p. 241, and by Theorem 5.65, p. 247. It descends to the common Einstein addition of coordinate velocities in special relativity theory when m = 1 (one temporal dimension) and n = 3 (three spatial dimensions), as explained in Sect. 5.17.

289

290

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

2. Einstein addition, ⊕E , comes with an associated coaddition, E , defined in Def. 5.76, p. 256, obeying the left and the right cancellation laws in Theorem 5.77, p. 256, V1 ⊕E (E V1 ⊕E V2 ) = V2 (V2 E V1 )⊕E V1 = V2

(5.525)

(V2 E V1 ) E V1 = V2 , for all V1 , V2 ∈ Rn×m c , m, n ∈ N. possesses the unique identity element 0n,m . 3. Rn×m c possesses a unique inverse, E V = −V. 4. Every element V ∈ Rn×m c determine in (4.135), p. 128, and in Theorem 5. Any two elements V1 , V2 ∈ Rn×m c 5.65, p. 247, both a. a left gyration lgyr[V1 , V2 ] ∈ SO(n) and b. a right gyration rgyr[V1 , V2 ] ∈ SO(m). → A left and a right gyration, in turn, determine a gyration, gyr[V1 , V2 ] : Rn×m c , according to (4.304), p. 166 and (5.340), p. 250, Rn×m c gyr[V1 , V2 ]V3 = lgyr[V1 , V2 ]V3 rgyr[V2 , V1 ] = lgyr[V1 , V2 ]V3 (rgyr[V1 , V2 ])t

(5.526)

for all V1 , V2 , V3 ∈ Rn×m c . 6. Left and right gyrations are automorphisms of Rn×m c . 7. Left and right gyrations are even, that is, by (5.286), p. 237, lgyr[E V1 , E V2 ] = lgyr[V1 , V2 ] rgyr[E V1 , E V2 ] = rgyr[V1 , V2 ]

(5.527)

gyr[E V1 , E V2 ] = gyr[V1 , V2 ] , for all V1 , V2 ∈ Rn×m c . 8. Left and right gyrations obey the gyration inversion law in (4.197), p. 143, and in (5.287), p. 237, lgyr−1 [V1 , V2 ] = lgyr[V2 , V1 ] rgyr−1 [V1 , V2 ] = rgyr[V2 , V1 ] gyr−1 [V1 , V2 ] = gyr[V2 , V1 ] , for all V1 , V2 ∈ Rn×m c .

(5.528)

Bi-gyrogroups and Bi-gyrovector Spaces – V

9. Left and right gyrations possess the reduction properties in Theorem 4.56, p. 167, and in Theorem 5.70, p. 251, lgyr[V1 , V2 ] = lgyr[V1 ⊕E V2 , V2 ] = lgyr[V1 , V2 ⊕E V1 ]

(5.529)

and in Theorem 4.57, p. 168, and in Theorem 5.71, p. 251, rgyr[V1 , V2 ] = rgyr[V1 ⊕E V2 , V2 ] = rgyr[V1 , V2 ⊕E V1 ] ,

(5.530)

so that gyr[V1 , V2 ] = gyr[V1 ⊕E V2 , V2 ] = gyr[V1 , V2 ⊕E V1 ] .

(5.531)

A useful gyration identity that follows immediately from the reduction properties along with a left cancellation is gyr[V2 , E V2 ⊕E V1 ] = gyr[V1 , E V2 ]

(5.532)

for all V1 , V2 ∈ Rn×m c . obeys the left and the right bi-gyroassociative laws 10. Einstein addition ⊕E in Rn×m c in (4.305) – (4.306), p. 167, in (5.325) – (5.326), p. 246, and in (5.342) – (5.343), p. 250, V1 ⊕E (V2 ⊕E V3 ) = (V1 ⊕E V2 )⊕E gyr[V1 , V2 ]V3 (V1 ⊕E V2 )⊕E V3 = V1 ⊕E (V2 ⊕E gyr[V2 , V1 ]V3 )

(5.533)

and the bi-gyrocommutative law in (4.307), p. 167, in (5.323), p. 245, and in (5.344), p. 251, V1 ⊕E V2 = gyr[V1 , V2 ](V2 ⊕E V1 ) for all V1 , V2 , V3 ∈

(5.534)

Rn×m c .

By Theorem 5.75, p. 255, Einstein bi-gyrogroups are gyrocommutative gyrogroups.

5.35.2. Einstein Bi-gyrovector Spaces Introducing a scalar multiplication, ⊗, into Einstein bi-gyrogroups Rn×m = (Rn×m c c , ⊕E ), n×m m, n ∈ N, yields the Einstein gyrovector spaces VE = (Rc , ⊕E , ⊗), which possess the following properties: 1. The scalar multiplication ⊗ in an Einstein bi-gyrovector space VE = (Rn×m c , ⊕E , ⊗) n×m is a multiplication r⊗V ∈ Rc between real numbers r ∈ R and elements V ∈ (Rn×m c , ⊕E ) of the Einstein bi-gyrogroup, given by Theorem 4.61, p. 171. 2. The scalar multiplication ⊗ obeys the scalar distributive law (5.461), p. 275, (r1 + r2 )⊗V = r1 ⊗V⊕E r2 ⊗V ,

(5.535)

291

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

the scalar associative law (5.462), (r1 r2 )⊗V = r1 ⊗(r2 ⊗V) ,

(5.536)

the monodistributive law (5.463), r⊗(r1 ⊗V⊕E r2 ⊕V) = r⊗(r1 ⊗V)⊕E r⊗(r2 ⊗V),

(5.537)

and the scalar matrix transpose law (5.464), (r⊗V)t = r⊗V t

(5.538)

for all r1 , r2 , r ∈ R and V ∈ Rn×m c . 3. Scalar multiplication enjoys the homogeneity property (5.482), p. 279, r⊗V = |r|⊗ V ,

(5.539)

for all r ∈ R and V ∈ Rn×m c . 4. Left and right trivial gyrations with scalar multiplications, (4.383), p. 181, are lgyr[r1 ⊗V, r2 ⊗V] = In rgyr[r1 ⊗V, r2 ⊗V] = Im

(5.540)

for all r1 , r2 ∈ R and V ∈ Rn×m c . Hence, gyr[r1⊗V, r2 ⊗V] is trivial, that is, gyr[r1 ⊗V, r2 ⊗V]U = U

(5.541)

for all r1 , r2 ∈ R and U, V ∈ Rn×m c . 5. Scalar multiplication respects orthogonal transformations, (5.501), p. 283, r⊗(On VOm ) = On (r⊗V)Om

(5.542)

for all V ∈ Rn×m c , Om ∈ SO(m), On ∈ SO(n), and r ∈ R. Hence, in particular, lgyr[V1 , V2 ](V⊗r) = {lgyr[V1 , V2 ]V}⊗r (r⊗V)rgyr[V1 , V2 ] = r⊗{Vrgyr[V1 , V2 ]}

(5.543)

gyr[V1 , V2 ](V⊗r) = {gyr[V1 , V2 ]V}⊗r

5.36. The Bi-gamma Norm The scaling property, r⊗V = |r|⊗ V , expressed in terms of the matrix spectral norm, proves useful in the geometry of bi-gyrovector spaces. As such, it demonstrates that the matrix norm that suits the geometry of bi-gyrovector spaces is the matrix spectral norm. The following theorem presents results that indicate, as well, that the matrix norm that suits the geometry of bi-gyrovector spaces is the matrix spectral norm.

Bi-gyrogroups and Bi-gyrovector Spaces – V

Theorem 5.106. (Bi-gamma Norm). For any m, n ∈ N and c > 0, let V ∈ Rn×m c . Then, L Γ V = ΓVL

ΓR V

=

(5.544)

ΓRV .

Proof. By means of (5.115), p. 201, and (5.411), p. 265, we have ⎛ ⎞ −1 ⎜⎜⎜ ⎟ 2 −2  −1 0k,n−k ⎟⎟⎟⎟ t ⎜⎜⎜ Ik − c Σk L −2 t ⎟⎟⎟ On , = On ⎜⎜ ΓV = In − c VV ⎝ ⎠ 0n−k,k In−k

(5.545)

k = rank(V), where Σk is diagonal, as in (5.402), Σk = diag(σ1 , σ2 , . . . , σk ) .

(5.546)

The diagonal elements of Σk are positive, said to be the singular values of V. They satisfy the condition 0 < σi < c , i = 1, 2, . . . , k, since V ∈ Rn×m c . Following (5.545) we have ΓVL (ΓVL )t

(5.547)

⎛ ⎞ ⎜⎜⎜(Ik − c−2 Σ2k )−1 0k,n−k ⎟⎟⎟ ⎟⎟⎟ Ot = On ⎜⎜⎜⎝ ⎠ n 0n−k,k In−k  =: On

S k2

0n−k,k



(5.548)

0k,n−k t On , In−k

where S k2 ∈ Rk×k is diagonal, S k2 = (Ik − c−2 Σ2k )−1 =: diag(s21 , s22 , . . . , s2k ) .

(5.549)

The diagonal elements si , i = 1, 2, . . . , k, of S k are k singular values of ΓVL ∈ Rn×m . Each of the remaining n − k singular values of ΓVL is 1, as we see from (5.548). Consequently, the diagonal elements of S k are related to the diagonal elements of Σk by the equations si = 

1 1 − c−2 σ2i

> 1,

(5.550)

0 < σi < c. Let 1 ≤ j ≤ k be the integer for which σ j = max(σi )i=1,2,...,k .

(5.551)

293

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Then, it is clear from (5.550) that s j = max(si )i=1,2,...,k .

(5.552)

is It follows from (5.551) that the norm of V ∈ Rn×m c V = σ j .

(5.553)

Similarly, it follows from (5.552) that the norm of ΓVL ∈ Rn×n is ΓVL = s j .

(5.554)

Hence, by (5.550), (5.553), and (5.554), and by the definition of ΓVL in (5.115), p. 201, we have the following chain of equations: ΓVL = max(si )i=1,2,...,k = s j = 

1 1 − c−2 σ2j

= 

1 1−

c−2 V 2

L = Γ V .

(5.555)

The chain of equations (5.555) completes the proof of the first identity in (5.544). The proof of the second identity in (5.544) is similar. 

5.37. The Bi-gyrotriangle Inequality The following lemma proves useful in the derivation of the bi-gyrotriangle inequality in Theorem 5.108. Lemma 5.107. (Bi-gamma Inequality). For any m, n ∈ N and c > 0, let V1 , V2 ∈ Rn×m c . Then, ΓVL1 (In +

1 1 V1 V2t )ΓVL2 ≤ ΓVL1 (1 + 2 V1 V2 ) ΓVL2 . 2 c c

(5.556)

Proof. As stated in (5.72), p. 194, the norm of a product is not bigger than the product of the norms, and the norm of a sum is not bigger than the sum of the norms. Hence, ΓVL1 (In + V1 V2t )ΓVL2 ≤ ΓVL1 In + V1 V2t ΓVL2 ≤ ΓVL1 (1 + V1 V2 ) ΓVL2 ,

(5.557) 

as desired.

Theorem 5.108. (Bi-gyrotriangle Inequality, I). For any m, n ∈ N and c > 0, let V1 , V2 ∈ (Rn×m c , ⊕E ). Then, V1 ⊕E V2 ≤ V1 ⊕E V2 .

(5.558)

Bi-gyrogroups and Bi-gyrovector Spaces – V

Proof. Evaluating the norms of both sides of the second bi-gamma identity in (5.335), p. 249, yields the equation ΓVL1 (In ⊕E

1 V1 V2t )ΓVL2 = ΓVL1 ⊕ V2 E c2

(5.559)

for any V1 , V2 ∈ Rn×m c . By (5.154), p. 210, L Γ V = γ V ,

(5.560)

where V ∈ Rn×m and hence V ∈ R1×1 c c . But, γ V is the special relativistic gamma factor, satisfying the gamma identity (2.10), p. 12, γ V1 ⊕ V2 = γ V1 γ V2 (1 + E

1 V1 V2 ) . c2

(5.561)

Hence, by (5.560) and (5.561), 1 L V1 V2 )Γ V , 2 E c2 so that, by the Bi-gamma Norm Theorem 5.106, L L = Γ V (1 + Γ V 1 ⊕ V2 1

1 V1 V2 ) ΓVL2 . c2 Hence, by (5.559), (5.563), and Lemma 5.107, L = ΓVL1 (1 + Γ V 1 ⊕ V2 E

L ΓVL1 ⊕ V1 ≤ Γ V 1 ⊕ V2 E

E

(5.562)

(5.563)

(5.564)

and, finally, by the Bi-gamma Norm Theorem 5.106, L L Γ V ≤ Γ V . 1 ⊕ V2 1 ⊕ V2 E

E

(5.565)

The left gamma factor ΓrL , −c < r < c, is a strictly monotonically increasing function of r. Hence, by means of (5.565) we have the bi-gyrotriangle inequality (5.558).  Theorem 5.109. (Bi-gyrotriangle Inequality, II). For any m, n ∈ N and c > 0, let A, B, C ∈ (Rn×m c , ⊕E ). Then, E A⊕E C ≤ E A⊕E B ⊕E E B⊕E C .

(5.566)

Proof. By (5.370), p. 256, E (E A⊕E B)⊕E (E A⊕E C) = gyr[E A, B](E B⊕E C) .

(5.567)

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Hence, by the left cancellation law (5.367), p. 256, E A⊕E C = (E A⊕E B)⊕E gyr[E A, B](E B⊕E C) .

(5.568)

Hence, by the bi-gyrotriangle inequality (5.558), and, since the norm is orthogonally invariant, by (5.73), p. 194, E A⊕E C = (E A⊕E B)⊕E gyr[E A, B](E B⊕E C) ≤ E A⊕E B ⊕E gyr[E A, B](E B⊕E C)

(5.569)

= E A⊕E B ⊕E E B⊕E C , as desired.



CHAPTER 6

Applications to Time-Space of Signature (m,n)

6.1. Application of the Galilei Bi-boost of Signature (1,n) In this section and in Sect. 6.2 about the Galilei bi-boost of signature (m, n), m, n ∈ N, we set the stage for the presentation of the Lorentz bi-boost of signature (m, n) in Sects. 6.3 and 6.4. A Galilei bi-boost of signature (1, n), n ∈ N, is a Galilei boost. Let B∞ (V) = B∞ (v) be the Galilei bi-boost of signature (m, n) = (1, 3), parametrized by the velocity parameter V = v, ⎛ ⎞ ⎜⎜⎜v1 ⎟⎟⎟ ⎜ ⎟ V = v = ⎜⎜⎜⎜v2 ⎟⎟⎟⎟ ∈ R3 = R3×1 , (6.1) ⎝ ⎠ v3 presented in (5.195) – (5.196), p. 218. In order to conform to the formalism of biboosts of signature (m, n), m, n > 1, in Sect. 6.2 we view V as a 3 × 1 matrix in R3×1 , and v as a vector in R3 , noting that R3×1 = R3 . Furthermore, let ⎛ ⎞  ⎜⎜⎜⎜ t ⎟⎟⎟⎟ ⎜⎜ x ⎟⎟ t = ⎜⎜⎜⎜⎜ 1 ⎟⎟⎟⎟⎟ ∈ R4×1 (6.2) x ⎜⎜⎝ x2 ⎟⎟⎠ x3 be a 4 × 1 matrix that represents the time-space coordinates of a particle with position x = (x1 , x2 , x3 ) ∈ R3 at time t ∈ R. The point (t, x) is said to be a particle of signature (m, n)=(1, 3) with position x ∈ R3 at time t ∈ R. A particle of signature (m, n) is also called an (m, n)-particle, in short. Following (5.195) – (5.196), p. 218, the application of the Galilei bi-boost B∞ (V) of signature (m, n) = (1, 3) to a (1, 3)-particle (t, x) in m + n = 1 + 3 time-space

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces http://dx.doi.org/10.1016/B978-0-12-811773-6.50006-2 Copyright © 2018 Elsevier Inc. All rights reserved.

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

dimensions yields

⎛    ⎜⎜⎜⎜ 1 ⎜⎜⎜v1 t t ⎜⎜⎜ := B (V) = ∞ ⎜⎜⎜v2 x x ⎝ v3

0 1 0 0

0 0 1 0

⎞⎛ ⎞ ⎛ ⎞ 0⎟⎟ ⎜⎜ t ⎟⎟ ⎜⎜ t ⎟⎟  ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟ t 0⎟⎟⎟ ⎜⎜⎜ x1 ⎟⎟⎟ ⎜⎜⎜v1 t + x1 ⎟⎟⎟⎟⎟ . ⎟⎜ ⎟ = ⎜ ⎟= x + vt 0⎟⎟⎟⎟ ⎜⎜⎜⎜ x2 ⎟⎟⎟⎟ ⎜⎜⎜⎜v2 t + x2 ⎟⎟⎟⎟ ⎠⎝ ⎠ ⎝ ⎠ 1 x3 v3 t + x 3

(6.3)

Accordingly, the Galilei bi-boost B∞ (V) of signature (1, 3) is a Galilei boost that keeps the time invariant, t = t, and boosts the position x ∈ R3 of the particle into the position x = x + vt ∈ R3 , v ∈ R3 , of the boosted particle, at time t. The extension of (6.1) – (6.3) from signature (1, 3) to signature (1, n) is obvious. The Galilei bi-boost B∞ (V) = B∞ (v) of signature (1, n), n ∈ N, is a Galilei boost that keeps the time of a (1, n)-particle invariant, t = t, and boosts the position x ∈ Rn of the (1, n)-particle into the position x = x + vt ∈ Rn , v ∈ Rn , of the boosted (1, n)-particle, at time t.

6.2. Application of the Galilei Bi-boost of Signature (m,n) Let B∞ (V) = B∞ (v1 , v2 ) be the Galilei bi-boost of signature (2, 3), parametrized by the velocity matrix V = (v1 v2 ), ⎛ ⎞ ⎜⎜⎜v11 v12 ⎟⎟⎟ ⎜ ⎟ V = (v1 v2 ) = ⎜⎜⎜⎜v21 v22 ⎟⎟⎟⎟ ∈ R3×2 , (6.4) ⎝ ⎠ v31 v32 of two velocities vk = (v1k , v2k , v3k )t columns of the matrix V. Furthermore, let ⎛ ⎜⎜⎜ t1  ⎜⎜⎜⎜ 0 ⎜⎜ T = ⎜⎜⎜⎜ x11 X ⎜⎜⎜ ⎜⎜⎝ x21 x31

∈ R3 , k = 1, 2. The two velocities vk are the two ⎞ 0 ⎟⎟ ⎞ ⎟ ⎛ t2 ⎟⎟⎟⎟⎟ ⎜⎜⎜ t1 0 ⎟⎟⎟ ⎟ ⎜ ⎟ x12 ⎟⎟⎟⎟ = ⎜⎜⎜⎜ 0 t2 ⎟⎟⎟⎟ ∈ R5×2 ⎟⎟⎟ ⎝ ⎠ x22 ⎟⎟ x1 x2 ⎠ x32

(6.5)

be a 5 × 2 matrix that represents a (2, 3)-particle consisting of the time-space coordinates of two subparticles, (tk , xk ), k = 1, 2, with positions xk = (x1k , x2k , x3k )t ∈ R3 , at time tk ∈ R, respectively. Here  t1 0 T= , (6.6) 0 t2 t1 , t2 > 0, is a 2 × 2 diagonal, positive definite matrix that represents the times t1 and

Applications to Time-space of Signature (m,n)

t2 when two subparticles are observed at positions x1 and x2 in R3 , respectively, and ⎛ ⎞ ⎜⎜⎜ x11 x12 ⎟⎟⎟ ⎜ ⎟ X = ⎜⎜⎜⎜ x21 x22 ⎟⎟⎟⎟ = (x1 x2 ) ∈ R3×2 (6.7) ⎝ ⎠ x31 x32 is a 3 × 2 matrix the columns of which represent the positions x1 , x2 ∈ R3 of two subparticles at times t1 , t2 ∈R, respectively. T Accordingly, the point = (T, X) ∈ R5×2 represents a (2, 3)-particle consisting X of a system of two subparticles (t1 , x1 ) and (t2 , x2 ) with positions x1 and x 2 in R3 at T , and the times t1 and t2 , respectively, t1 , t2 ∈ R. We use the displayed notation, X inline notation, (T, X), interchangeably. Following (5.195) – (5.196), p. 218, the collective application of the Galilei biboost B∞ (V) of signature (m, n) = (2, 3) to the pair of subparticles (T, X) in m + n = 2 + 3 time-space dimensions yields ⎛ ⎞ ⎞⎛ 1 0 0 0 0⎟⎟ ⎜⎜ t1 0 ⎟⎟ ⎜ ⎜ ⎛ ⎛  ⎞ ⎞ ⎜ ⎟ ⎟⎜ 1 0 0 0⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ 0 t2 ⎟⎟⎟⎟⎟ ⎜⎜⎜ t1 0 ⎟⎟⎟ ⎜⎜⎜⎜⎜ 0 ⎜⎜⎜ t1 0 ⎟⎟⎟ ⎜ ⎜ ⎟ ⎜ ⎜⎜⎜ ⎟ ⎟ ⎟  ⎜⎜⎝ 0 t2 ⎟⎟⎟⎟⎠ :=B∞ (V) ⎜⎜⎜⎜⎝ 0 t2 ⎟⎟⎟⎟⎠ = ⎜⎜⎜⎜⎜v11 v12 1 0 0⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ x11 x12 ⎟⎟⎟⎟⎟ ⎜⎜⎜v ⎟ ⎟⎜ x1 x2 x1 x2 ⎜⎝ 21 v22 0 1 0⎟⎟⎟⎠ ⎜⎜⎜⎝ x21 x22 ⎟⎟⎟⎠ v31 v32 0 0 1 x31 x32 (6.8) ⎛ ⎞ t1 0 ⎜⎜⎜ ⎟⎟⎟ ⎛ ⎞ ⎜⎜⎜ ⎟⎟⎟ ⎜ t 0 t2 0 ⎟⎟⎟ 1 ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ ⎟ 0 t2 ⎟⎟⎟⎟ . = ⎜⎜v11 t1 + x11 v12 t2 + x12 ⎟⎟ = ⎜⎜ ⎜⎜⎜ ⎟⎟⎟ ⎝ ⎠ x1 + v1 t1 x2 + v2 t2 ⎜⎜⎝v21 t1 + x21 v22 t2 + x22 ⎟⎟⎠ v31 t1 + x31 v32 t2 + x32 The chain of equations (6.8) describes the application of a Galilei bi-boost B∞ (V) of signature (2, 3) to collectively bi-boost two subparticles, (t1 , x1 ) and (t2 , x2 ), into the two bi-boosted subparticles, (t1 , x1 + v1 t1 ) and (t2 , x2 + v2 t2 ), by three-dimensional velocities v1 = (v11 , v21 , v31 )t and v2 = (v12 , v22 , v32 )t in R3 . It is important to note that the two collectively bi-boosted subparticles are not entangled in the sense that the boost of each boosted subparticle is independent of the boost of the other boosted subparticle. Interestingly, this observation fails when we replace Galilei bi-boosts of signature (m, n), m, n ≥ 2, by corresponding Lorentz bi-boosts of signature (m, n), as shown in Sect. 6.4. The extension of (6.4) – (6.8) from signature (2, 3) to signature (2, n), n ∈ N, is obvious. The Galilei bi-boost B∞ (V) of signature (2, n) is parametrized by a velocity matrix V ∈ Rn×2 of order n × 2 that consists of two columns, V = (v1 v2 ), which represent the two velocities v1 , v2 ∈ Rn .

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It is clear from (6.8) that 1. the Galilei bi-boost B∞ (V) of signature (2, 3) keeps invariant each of the times t1 and t2 of the two particles (t1 , x1 ) and (t2 , x2 ), that is, tk = tk , k = 1, 2, and 2. the Galilei bi-boost B∞ (V) of signature (2, 3) bi-boosts the particle positions x1 , x2 ∈ Rn into the bi-boosted positions x1 + v1 t1 , x2 + v2 t2 ∈ Rn at times t1 , t2 ∈ R, respectively. Each of the two particles (t1 , x1 ) and (t2 , x2 ) possesses a one-dimensional time, t1 ∈ R and t2 ∈ R, respectively. Accordingly, the system consisting of the two particles possesses the two-dimensional time, (t1 , t2 ) ∈ R2 . Each of the two particles possesses its own clock, so that the two-dimensional time of the system is measured by two clocks. In general, a system consisting of m particles possesses an m-dimensional time, measured by m clocks, m ∈ N. Further extension of (6.4) – (6.8) from signature (2, n) to signature (m, n), m, n ∈ N, is now obvious. The Galilei bi-boost B∞ (V) of signature (m, n) is parametrized by a velocity matrix V ∈ Rn×m of order n × m that consists of m columns, V = (v1 v2 . . . vm ), which represent the m velocities v1 , v2 , . . . , vm ∈ Rn . Furthermore, when B∞ (V) is applied to collectively bi-boost m particles in Rn it keeps invariant each of the times tk , k = 1, . . . , m of the m particles (tk , xk ), that is, tk = tk , and it bi-boosts their positions xk ∈ Rn into the bi-boosted positions xk + vk tk ∈ Rn at times tk , respectively. The m collectively bi-boosted particles are not entangled in the sense that (i) the boost of each boosted particle is independent of the boosts of the other boosted particles and (ii) the time of each boosted particle is independent of the times of the other boosted particles. A Galilei bi-boost of signature (m, n), applied collectively to the m subparticles of an (m, n)-particle, is thus equivalent to m Galilei boosts applied individually to each subparticle, yielding no entanglement. In contrast, we will see in Sect. 6.4 that the Lorentz bi-boost of signature (m, n), applied collectively to the m subparticles of an (m, n)-particle, yields subparticle entanglement.

6.3. Application of the Lorentz Bi-boost of Signature (1,n) A Lorentz bi-boost of signature (1, n), n ∈ N, is a Lorentz boost. In particular, the Lorentz boost of signature (1, 3) is the Lorentz transformation, without space rotation, of Einstein’s special theory of relativity. Let B(V) = B(v) be the Lorentz bi-boost of signature (m, n) = (1, 3), parametrized

Applications to Time-space of Signature (m,n)

by the velocity parameter V = v,

⎛ ⎞ ⎜⎜⎜v1 ⎟⎟⎟ ⎜ ⎟ V = v = ⎜⎜⎜⎜v2 ⎟⎟⎟⎟ ∈ R3c = R3×1 c . ⎝ ⎠ v3

(6.9)

In order to conform to the formalism of bi-boosts of signature (m, n), m, n > 1, in Sect. 6.4, we view V as a 3 × 1 matrix in the ball R3×1 c , and v as a vector in the Euclidean ball R3c of all relativistically admissible velocities, noting that R3×1 = R3c . c Furthermore, let ⎛ ⎞  ⎜⎜⎜⎜ t ⎟⎟⎟⎟ ⎜⎜ x ⎟⎟ t = ⎜⎜⎜⎜⎜ 1 ⎟⎟⎟⎟⎟ ∈ R1,3 (6.10) x ⎜⎜⎝ x2 ⎟⎟⎠ x3 be the time-space coordinates of a particle with position x = (x1 , x2 , x3 ) ∈ R3c at time t ∈ R. The point (t, x) ∈ R1,3 of the pseudo-Euclidean space R1,3 of signature (1, 3) is thus said to be a particle of signature (1, 3), or a (1, 3)-particle in short, with position x ∈ R3 at time t ∈ R. This (1, 3)-particle has the squared relativistic norm  



2

t

1 t t =



= t2 − 2 x2 · (6.11)

x

x x c of signature (1, 3) that results from the inner product (4.4), p. 103. Following (5.173), p. 213, the Lorentz boost B(v) of signature (1, 3) is a 4 × 4 block matrix, and its application to the (1, 3)-particle (t, x) in m + n = 1 + 3 time-space

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dimensions takes the form

⎛ ⎞ 1 ⎜⎜⎜ γ ⎟⎟⎟ ⎛ ⎞ γ (v v v ) 1 2 3 v v ⎜ ⎟⎟⎟ ⎜⎜⎜ t ⎟⎟⎟ 2  ⎜⎜⎜   c ⎟⎟⎟ ⎜⎜⎜ x ⎟⎟⎟ ⎜⎜⎜ ⎛ ⎞ ⎛ ⎞ t t ⎟⎟⎟ ⎜⎜⎜ 1 ⎟⎟⎟ ⎜ v v ⎜ ⎟ ⎜ ⎟ := B(v) = 1 1 ⎜ 2 ⎜⎜⎜ ⎜⎜⎜ ⎟⎟⎟ ⎟⎟ ⎜⎜ x ⎟⎟ 1 γv ⎜⎜⎜⎜ ⎟⎟⎟⎟ x x ⎜ ⎟ ⎜⎜⎜γv ⎜⎜v2 ⎟⎟ I3 + ⎜⎜⎜v2 ⎟⎟⎟ (v1 v2 v3 )⎟⎟⎟⎟ ⎜⎜⎝ 2 ⎟⎟⎠ ⎜ ⎟ 2 ⎝ ⎝ ⎠ ⎠ x3 c 1 + γv ⎝ ⎠ v3 v3 ⎛ ⎞ 1 ⎜⎜⎜ γ ⎟⎟⎟  t γ v ⎜⎜⎜ v ⎟⎟⎟ t 2 v c ⎟⎟⎟ = ⎜⎜⎜⎜⎜ ⎟⎟⎟ x 1 γv2 ⎜⎜⎝ γv v I3 + 2 vvt ⎟⎠ c 1 + γv

(6.12)

⎛ ⎞ 1 ⎜⎜⎜ ⎟⎟⎟ t  γ t + γ v x v ⎜⎜⎜ ⎟⎟⎟ 2 v c t ⎜ ⎟ −→ ⎜ ⎟ = ⎜⎜⎜ , ⎟⎟⎟ c→∞ 1 γv2 x + vt ⎜⎜⎝ ⎟ γv vt + x + 2 vvt x⎟⎠ c 1 + γv where γv is the gamma factor of special relativity theory, γv = ΓRm=1,v = 

1

. (6.13) v2 1− 2 c The Lorentz bi-boost B(V) of signature (1, 3) in (6.12) keeps invariant the squared relativistic norm (6.11) of the particle (t, x) ∈ R1,3 in the pseudo-Euclidean space R1,3 , that is, 1 1 (6.14) (t )2 − 2 (x )2 = t2 − 2 x2 . c c Furthermore, in the limit of large c, when c → ∞, the application of the Lorentz biboost of signature (1, 3) descends to the application of its Galilei counterpart, B∞ (v), in (6.3), as expected. Finally, the extension of the Lorentz bi-boost of signature (1, 3) to the Lorentz bi-boost of signature (1, n), n ∈ N, is obvious. Its application to time-space coordinates takes the form (6.12) where we replace v, x ∈ R3 by v, x ∈ Rn , n ∈ N. Example 6.1. If v ∈ R3c ⊂ R3 and x ∈ R3 are parallel, we select space coordinates with respect to which v = (v, 0, 0) and x = (x, 0, 0). Then, the Lorentz boost application in

Applications to Time-space of Signature (m,n)

(6.12) specializes to

⎞ ⎛ 1 ⎟⎟⎟  ⎜⎜⎜ γ   γ v ⎟⎟⎟ t ⎜⎜⎜ v 2 v t c ⎜⎜ ⎟⎟⎟ 2 2⎟  :=⎜ γ v ⎜⎜⎝ x v ⎟⎠ x γv v 1 + 1 + γ v c2 ⎛ ⎜⎜⎜ γ = ⎜⎜⎜⎝ v γv v

⎞ 1 ⎟⎟ γ v v ⎟ t c2 ⎟⎟⎟⎠ x γv

(6.15)

⎛ ⎞ ⎜⎜⎜γ (t + 1 vx)⎟⎟⎟ = ⎜⎜⎜⎝ v c2 ⎟⎟⎟⎠ . γv (vt + x) This simple form of the application of the Lorentz transformation is found invariably in every book on special relativity theory (sometimes with v replaced by −v).

6.3.1. Additive Decomposition, Signature (1,n) This subsection sets the road to Subsect. 6.4.1. Let us express the boost B(v) in (6.12) in terms of its additive decomposition (5.396), p. 261, of the special relativistic signature (1, 3), with E(v), v = (v1 , v2 , v3 )t ∈ R3×1 = R3c , given by (5.398) with m = 1, obtaining c ⎛ γ2 ⎞ ⎞ ⎛ γv vt ⎟⎟⎟⎟ ⎜⎜⎜1 01,3 ⎟⎟⎟ 1 ⎜⎜⎜⎜ 1+γv v2 ⎟⎟ ⎟⎠⎟ + ⎜⎜⎜ 2 v B(V) = ⎜⎝⎜ 2 c2 ⎝ γv v2 v γv vvt ⎟⎠ v I3 1+γ 1+γ v

⎛ ⎜⎜⎜ 1 ⎜⎜⎜ ⎜⎜⎜v1 = ⎜⎜⎜⎜ ⎜⎜⎜v2 ⎜⎜⎝ v3

0 0 1 0 0 1 0 0

v

⎞ ⎛ γ 0⎟⎟⎟ v ⎜⎜⎜ v2 ⎟⎟⎟ 1+γv ⎜ ⎜ ⎛ ⎞ 0⎟⎟⎟⎟ 1 ⎜⎜⎜⎜ ⎜⎜v1 ⎟⎟ ⎟⎟⎟ + 2 γv ⎜⎜⎜ γ ⎜⎜⎜ v v2 ⎜⎜⎜⎜⎜v2 ⎟⎟⎟⎟⎟ 0⎟⎟⎟ c ⎟⎟⎠ ⎜⎝ 1+γv ⎜⎝ ⎟⎠ v3 1

γv 1+γv

⎞ ⎟⎟⎟ (v1 v2 v3 ) ⎟⎟⎟ ⎛ ⎞ ⎟⎟⎟ ⎜⎜⎜v1 ⎟⎟⎟ ⎟. ⎜⎜⎜⎜v2 ⎟⎟⎟⎟ (v1 v2 v3 )⎟⎟⎟⎟⎟ ⎟⎠ ⎜⎝ ⎟⎠ v3

(6.16)

Similarly to (6.12), the application of the Lorentz boost B(v) of signature (1, 3) in (6.16) to a particle (t, x) results in a particle (t , x ) boosted by the velocity v ∈ R3×1 = c 3 Rc , as shown in (6.17).    t t := B(v) x x ⎛ 1 ⎜⎜⎜ t + 2 γv ⎜⎜⎜⎜⎝ = x + vt c 



γv v2 t 1+γv

γv v2 vt 1+γv

+

⎞ ⎟⎟⎟ ⎟⎟⎟ . ⎟ v(v·x)⎠

+ v·x γv 1+γv

(6.17)

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The application of the Lorentz boost B(v) of signature (1, 3) to a (1, 3)-particle (t, x) in (6.17) boosts the particle into the boosted (1, 3)-particle (t , x ). This application is additively decomposed by (6.17) into two parts called the Galilean part (GP) and the entanglement part (or, Einsteinian part) (EP) as follows:    t t (6.18) GP of = x + vt x and

⎛   ⎜⎜⎜ 1 t ⎜⎜⎜ = EP of γ x c2 v ⎜⎝

γv v2 t 1+γv

γv v2 vt 1+γv

+

⎞ ⎟⎟⎟ ⎟⎟⎟ =: 1 E(v) . ⎟ c2 v(v·x)⎠

+ v·x γv 1+γv

(6.19)

The impact of each of the two parts, (6.18) and (6.19), of (6.17) is important in our understanding of special relativity theory, where we are guided by analogies with Galilean-Newtonian physics, as we explain below: (1) Galilean Part (GP). This part, (6.18), of the boost application to a particle in (6.17) provides an intuitively clear physical interpretation. It enables us to interpret and measure with respect to time-space coordinates the time t ∈ R and the position x ∈ R3 of particles (t, x) as well as the velocity v ∈ R3 of boosted particles (t, x + vt). (2) Entanglement Part (EP). This part, (6.19), of the boost application to a particle in (6.17) reveals the counterintuitive relativistic effects that result from Lorentzboosting particles (t, x) into boosted particles (t , x ) by velocities v ∈ R3×1 = R3c . c Owing to the presence of the coefficient c−2 of the EP in the additive decomposition (6.17), non-Galilean, relativistic effects are directly noticeable only at very high speeds. Unlike the GP, where time and space are separated, time and space are entangled in the EP. We will see that this entanglement is richer in the collective application of a Lorentz bi-boost of signature (m, n), m, n > 1, to m particles. The non-Galilean, relativistic effects that the EP reveals include (1) length contraction; (2) time dilation; (3) relativity of simultaneity; (4) Thomas precession; and (5) energy levels in quantum mechanics, some details of which are presented below: (1) Length Contraction was introduced by George Francis FitzGerald in 1889 [6] and Hendrix Antoon Lorentz in 1897 [54], before the introduction of special relativity by Einstein in 1905. (2) Time Dilation was predicted by Joseph Larmor in 1897, at least for orbiting electrons [50], prior to Lorentz and Einstein [59]. It helps to explain delayed disintegration of muons from cosmic showers [14]. (3) Relativity of Simultaneity is the concept that distant simultaneity, whether two spatially separated events occur at the same time, is not absolute, but depends on

Applications to Time-space of Signature (m,n)

the observer’s reference frame. The relativity of simultaneity results in Thomas precession; see, for instance, [72, p. 170]. (4) Thomas Precession is a relativistic space rotation of time-space coordinates. It is an effect predicted by Einstein’s special theory of relativity, as Llewellyn Hilleth Thomas discovered in 1926 [73],[91],[98, Chap. 13]. The mathematical extension by abstraction of Thomas precession gives rise to gyrations and to bi-gyrations of signature (m, n), which regulate Einstein addition of special relativity and Einstein addition, ⊕E := ⊕ , of signature (m, n). (5) Energy Levels in Quantum Mechanics. One of the cornerstones of quantum mechanics was the prediction of the energy levels of the hydrogen atom. When attempts were made to explain the fine structure of the hydrogen spectral lines, it was found that the splitting of the lines was in error by a factor of 2. L.H. Thomas realized that relativistic time dilation must be used in calculating the frequencies, and calculations showed that this relativistic correction, which results in Thomas precession, was the factor of two which was needed for agreement with experiment. We thus see that the passage from the Galilei bi-boost of signature (1, 3) (which is the Galilei boost of classical mechanics) to the Lorentz bi-boost of signature (1, 3) (which is the Lorentz boost of special relativity theory) introduces relativistic phenomena, some of which were discovered before the 1905 appearance of special relativity. In a similar way we will see that the passage from the Galilei bi-boost of signature (m, n) to the Lorentz bi-boost of signature (m, n), m, n ∈ N, introduces new phenomena, including m-dimensional time, temporal, and spatial precession, and entanglement between particles.

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6.4. Application of the Lorentz Bi-boost of Signature (m,n) Following (5.127), p. 204, the collective application of the Lorentz bi-boost B(V) of signature (2, 3) to the pair of subparticles of a (2, 3)-particle is given by ⎛ ⎛  ⎞  ⎞ 0 ⎟⎟ ⎜⎜⎜ t1 ⎜⎜⎜ t11 t12 ⎟⎟⎟ ⎟ ⎜ ⎜⎜⎜ t ⎟  ⎜⎜⎜ 0 t2 ⎟⎟⎟⎟⎟ ⎜⎜⎜ 21 t22 ⎟⎟⎟⎟⎟ ⎜ ⎜ ⎜⎜⎜ x x ⎟⎟⎟ = B(V) ⎜⎜⎜ x11 x12 ⎟⎟⎟⎟ 12 ⎟ ⎜⎜⎜ ⎜⎜⎜ 11 ⎟ ⎟⎟⎟   ⎟ ⎜⎜⎜ x21 x22 ⎟⎟⎟⎟⎟ ⎜⎜⎜ x21 x22 ⎟ ⎝ ⎝  ⎠  ⎠ x31 x32 x31 x32 (6.20)  ⎞ ⎛ ⎛ ⎞ 1 v v v ⎜⎜⎜ ⎟ t 0 ⎜ ⎟ 11 21 31 ⎟⎟⎟ ⎜⎜ 1 ⎟⎟ ΓRm=2,V ΓR ⎜⎜⎜ t2 ⎟⎟⎟⎟⎟ c2 m=2,V v12 v22 v32 ⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ 0 ⎜⎜⎜ ⎟⎟⎟ ⎜⎜ ⎛ ⎞ ⎟ = ⎜⎜⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ x11 x12 ⎟⎟⎟⎟ . ⎜⎜⎜v11 v12 ⎟⎟⎟ ⎜⎜⎜ L ⎟⎟⎟ ⎜⎜⎜ x ⎜ ⎟ ⎟⎟ L Γn=3,V ⎜⎜⎝Γn=3,V ⎜⎜⎜⎜⎝v21 v22 ⎟⎟⎟⎟⎠ ⎟⎟⎠ ⎜⎜⎝ 21 x22 ⎟⎟⎟⎠ x31 x32 v31 v32 Here V ∈ R3×2 is the velocity matrix c ⎞ ⎛ ⎜⎜⎜v11 v12 ⎟⎟⎟ ⎟ ⎜ V = ⎜⎜⎜⎜v21 v22 ⎟⎟⎟⎟ ∈ R3×2 c , ⎠ ⎝ v31 v32

(6.21)

L , ΓRm=2,V ) is given in terms of V by (5.115), p. 201. and the bi-gamma factor (Γn=3,V Noting the limits in (5.194), p. 218, the application of the Lorentz bi-boost B(V) of signature (2, 3) in (6.20) descends in the limit when c → ∞ to the application of the Galilei bi-boost B∞ (V) of signature (2, 3) in (6.8), as expected. In (6.20) the Lorentz bi-boost B(V) bi-boosts the pair of particles p1 and p2 ,

p1 = (t1 , 0, x11 , x21 , x31 )t ∈ R2,3 p2 = (0, t2 , x12 , x22 , x32 )t ∈ R2,3 ,

(6.22)

in the pseudo-Euclidean space R2,3 into the pair of the bi-boosted particles p1 and p2 ,      t p1 = (t11 , t21 , x11 , x21 , x31 ) ∈ R2,3      t p2 = (t12 , t22 , x12 , x22 , x32 ) ∈ R2,3 .

(6.23)

Here the two particles p1 and p2 are not entangled, but their Lorentz-bi-boosted particles p1 and p2 are entangled, each having a two-dimensional time. In the pseudo-Euclidean space R2,3 the two particles p1 and p2 have squared relativistic norms that remain equal to those of their bi-boosted particles p1 and p2 ,

Applications to Time-space of Signature (m,n)

respectively, that is,  2  2 ) + (t21 ) − (t11

1  2 1 2  2  2 2 2 ((x11 ) + (x21 ) + (x31 ) ) = t12 − 2 (x11 + x21 + x31 ) 2 c c

(6.24) 1 1  2  2  2  2  2 2 2 2 (t12 ) + (t22 ) − 2 ((x12 ) + (x22 ) + (x32 ) ) = t22 − 2 (x12 + x22 + x32 ). c c In the application of a Galilei bi-boost of signature (2, 3) in (6.8) to a pair of particles, the bi-boosts of the two bi-boosted particles are not coupled. In this sense we say that the two Galilei bi-boosted particles are not entangled. In contrast, in the application of a Lorentz bi-boost of signature (2, 3) in (6.20) to a pair of particles p1 and p2 , the bi-boosts of the two bi-boosted particles p1 and p2 are coupled. In this sense we say that the two Lorentz-bi-boosted particles are entangled. Moreover, it follows from (6.20) that each of the two entangled particles p1 and p2 in (6.23) necessitates a two-dimensional time, as opposed to the common one-dimensional time that its Galilei counterpart (6.8) necessitates. Thus, by analogy, we expect that each of m entangled particles necessitates an m-dimensional time, as shown in (6.25) – (6.26). The obvious extension of the application of a Lorentz bi-boost of signature (2, 3) to collectively bi-boost two particles in the pseudo-Euclidean space R2,3 gives rise to the application of a Lorentz bi-boost of signature (m, n) to collectively bi-boost m particles in the pseudo-Euclidean space Rm,n . This extension of (6.20) from signature (2, 3) to signature (m, n) is shown in (6.25). By means of (5.127), p. 204, the application of a Lorentz bi-boost B(V), V ∈ Rn×m c , of signature (m, n), m, n ∈ N, to bi-boost collectively m particles (tk , xk ), k = 1 . . . , m, in a pseudo-Euclidean space Rm,n takes the following form, which is the obvious extension of (6.20) from signature (2, 3) to signature (m, n), m, n ∈ N: ⎛ ⎞ ⎛ ⎞ 1 R t ⎟⎟ 1 R t ⎟⎟  ⎜⎜ R    ⎜⎜ R ⎜ ⎟⎟⎟ ⎜ ⎟ Γ Γ Γ V T + Γ V X T T ⎜ ⎜ ⎟ T ⎟⎟⎟ ∈ R(m+n)×(m+n) , c2 V = ⎜⎜⎜⎜ V = ⎜⎜⎜⎜ V c2 V ⎟⎟⎟⎟  := B(V) X X X ⎝ ⎠ ⎝ L ⎠ ΓVL VT + ΓVL X ΓV V ΓVL (6.25) where ⎛ ⎞ ⎜⎜⎜t1 0 . . . 0 ⎟⎟⎟ ⎜⎜⎜ 0 t 0 ⎟⎟⎟⎟⎟ 2 ⎜ ⎟⎟⎟ ∈ Rm×m (6.26) T = (t1 t2 . . . tm ) = ⎜⎜⎜⎜⎜ .. ⎜⎜⎜ . ⎟⎟⎟⎟ ⎝ ⎠ 0 0 tm is a diagonal m × m time matrix the diagonal elements tk > 0, k = 1, . . . , m, of which

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represent, respectively, the times of the m particles (tk , xk ); and where ⎛ ⎞ ⎜⎜⎜ x11 x12 . . . x1m ⎟⎟⎟ ⎜⎜⎜⎜ x21 x22 x2m ⎟⎟⎟⎟⎟ ⎜ ⎜ ⎟⎟⎟ ∈ Rn×m X = (x1 x2 . . . xm ) = ⎜⎜⎜ .. ⎟⎟⎟ ⎜⎜⎜ . ⎟⎠ ⎝ xn1 xn2 xnm

(6.27)

is an n × m matrix the m columns of which represent, respectively, the positions of the m particles (tk , xk ) at respective times tk . Following the application of the Lorentz bi-boost B(V) in (6.25) the m bi-boosted particles become entangled, each particle of which possesses an m-dimensional time tk . The m × m matrix T  is, then, a positive definite “time” matrix times an orthogonal matrix, the m columns of which are tk , k = 1, . . . , m, ⎛   ⎞ . . . t1m ⎜⎜⎜ t11 t12 ⎟⎟ ⎜⎜⎜ t ⎟⎟   ⎟ t t ⎜⎜⎜ 21 22 ⎟⎟⎟ 2m ⎟     m×m T = (t1 t2 . . . tm ) = ⎜⎜⎜ .. . (6.28) . ⎟⎟⎟⎟ ∈ R . ⎜⎜⎜ . . ⎟ ⎟ ⎝   ⎠ tm1 tm2 tmm Similarly, the n × m matrix X  is a space k = 1, . . . , m, ⎛  ⎜⎜⎜ x11 ⎜⎜⎜ x ⎜ 21     X = (x1 x2 . . . xm ) = ⎜⎜⎜⎜⎜ .. ⎜⎜⎜ . ⎝ xn1

matrix the m columns of which are xk ,  x12 ...  x22  xn2

 ⎞ x1m ⎟⎟ ⎟⎟⎟  ⎟ x2m ⎟⎟⎟ ⎟⎟⎟ ∈ Rn×m . ⎟⎟⎟  ⎠ xnm

(6.29)

The kth column xk of X  is the position of the kth particle (tk , xk ). The point (T, X) ∈ R(m+n)×m is said to be a particle (point) of signature (m, n), or an (m, n)-particle (point) for short. It consists of a system of m subparticles (subpoints) (tk , xk ) ∈ Rm,n in the pseudo-Euclidean space Rm,n , each of which has m temporal dimensions and n spatial dimensions. The m-dimensional time tk of the kth subparticle (tk , xk ) consists of the (i) primary t time tkk and the (ii) secondary, (m − 1)-dimensional time (tk1 , . . . , t kk , . . . , tkm ) from which the component tkk is removed. The secondary time is noticeable only at very high speeds. At low speeds the primary time is approximately equal to the classical, Galilean one-dimensional time. In general, an (m, n)-particle, m ≥ 2, consists of m n-dimensional subparticles that have different positions at the same time. As such the (m, n)-particle is, in general, disintegrated while its constituent subparticles are entangled. If, however, the constituent subparticles of an (m, n)-particle possess equal positions at equal times, the (m, n)-particle is said to be integrated.

Applications to Time-space of Signature (m,n)

Illustrative examples are presented in Subsects. 6.5.1 – 6.5.3, of Sect. 6.5.

6.4.1. Additive Decomposition, Signature (m,n) In this subsection we extend the study in subsection (6.3.1) from signature (1, n) to signature (m, n), m, n ∈ N. Let us express the bi-boost B(v) in (6.25) in terms of the additive decomposition (5.396), p. 261, of signature (m, n), m, n ∈ N, with E(V), V = (v1 v2 . . . vm ) ∈ Rn×m c , given by (5.398). We then obtain 1 B(V) = B∞ (V) + 2 E(V) c ⎛ (ΓR )2 ⎞ (6.30) t  ⎜ V ΓRV V t ⎟⎟⎟⎟ 1 ⎜⎜⎜⎜⎜ Im +ΓRV V V Im 0m,n ⎟ + 2 ⎜⎜⎜ L 2 = ⎟⎟ . V In c ⎝ (ΓV ) VV t V (ΓVL )2 VV t ⎟⎟⎠ In +ΓVL

In +ΓVL

Similarly to (6.25), the collective application of the Lorentz bi-boost B(V) of signature (m, n) in (6.30) to the m subparticles of the (m, n)-particle (T, X) results in the bi-boosted m subparticles of the bi-boosted (m, n)-particle (T  , X  ). The m bi-boosting velocities of the m subparticles are v1 , v2 , . . . , vm , which are the m columns of the n × m velocity matrix V = (v1 v2 . . . vm ) ∈ Rn×m c . Accordingly, the application of a bi-boost B(V) to the m subparticles of an (m, n)particle (T, X) in (6.25) takes the following form, where T ∈ Rm×m and X ∈ Rn×m are given by (6.26) and (6.27):    T T = B(V) X X  =

Im V

⎛ (ΓR )2 t  ⎜ V 1 ⎜⎜⎜⎜⎜ Im +ΓRV V V 0m,n T + 2 ⎜⎜⎜ L 2 In X c ⎝ (ΓV ) VV t V In +ΓVL

⎞ ΓRV V t ⎟⎟⎟⎟ T ⎟⎟⎟ ⎟ X (ΓVL )2 t⎟ VV ⎠

In +ΓVL

(6.31)

⎛ (ΓR )2 ⎞ t R t V ⎜⎜⎜ ⎟⎟⎟ V VT + Γ V X R V 1 ⎜⎜⎜ Im +ΓV ⎟⎟⎟ T = + 2 ⎜⎜⎜ L 2 ⎟. L 2 (Γ ) (Γ ) X + VT c ⎝ V VV t VT + V VV t X ⎟⎟⎠ 



In +ΓVL

In +ΓVL

The collective application of the Lorentz bi-boost B(V) of signature (m, n) to the m subparticles represented by the columns of (T, X) in (6.31) bi-boosts the subparticles into the m bi-boosted subparticles represented by the columns of (T  , X  ) in (6.31). This application is additively decomposed in (6.31) into the following GP and EP.

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By means of (6.26) and (6.27), it follows from (6.31) that     t1 T T t2 ... tm = GP of = X + VT X x1 + v1 t1 x2 + v2 t2 xm + vm tm and

⎛ (ΓR )2 ⎞ t R t V   ⎜⎜⎜ ⎟⎟⎟ V VT + Γ V X R V 1 1 ⎜⎜⎜ Im +ΓV ⎟⎟⎟ T = 2 ⎜⎜⎜ L 2 EP of ⎟⎟⎟ =: 2 E(V) ,  L 2 X c ⎝ (ΓV ) VV t VT + (ΓV ) VV t X ⎠ c In +ΓVL

(6.32)

(6.33)

In +ΓVL

in full analogy with (6.18) – (6.19). The GP (6.32) of (6.31) involves m subparticles, which are in no entanglement. Furthermore, the m subparticles have collectively time dimension m since each of the m subparticles has its own one-dimensional time. The m-dimensional time is measured by m clocks, one clock for each subparticle. Like the GP, the EP (6.33) of (6.31) involves m subparticles with collective time dimension m. Contrasting GP, however, the EP involves m subparticles that are in entanglement. The impact of the two parts, (6.32) and (6.33) of (6.31), to our study of the resulting geometric entanglement is important. It motivates viewing V ∈ Rn×m as an c n × m matrix of m columns, each of which is interpreted as a velocity that represents an n-dimensional point in a velocity space Rnc . Accordingly, we consider each point  V of the bi-gyrovector space (Rn×m c , ⊕ , ⊗) as a system of m n-dimensional subpoints. n×m The point V = (v1 v2 . . . vm ) ∈ Rc is called a bi-gyropoint of order (m, n), or an (m, n)-bi-gyropoint, and its subpoints vk ∈ Rnc , k = 1, . . . , m, are gyropoints. In general, the subpoints of a bi-gyropoint are entangled. The entanglement of subpoints and subsegments that join subpoints is explored graphically in Chap. 7.

6.5. Particles that are Systems of Subparticles Definition 6.2. (Integrated Particles). Let m, n ∈ N. (1) A matrix in Rn×m is a one-number matrix if all its entries are equal. (2) A square matrix in Rn×n is one-number diagonal if the main diagonal entries of the matrix are equal, and all off main diagonal entries are equal. (For instance, the matrix (6.36) is one-number diagonal). (3) A matrix in Rn×m is equi-column if all its columns are equal. (For instance, the matrix (6.37) is equi-column). The velocity matrix V = (v1 v2 . . . vm ) ∈ Rn×m has m n-dimensional columns vk , c each of which is a subvelocity vk ∈ Rn of V that represents the velocity of a subparticle pk = (tk , xk ) ∈ Rm,n , k = 1, . . . , m. The position matrix X = (x1 x2 . . . xm ) ∈ Rn×m has m n-dimensional columns

Applications to Time-space of Signature (m,n)

xk , each of which is a subposition xk ∈ Rn of X that represents the position of the subparticle pk = (tk , xk ) ∈ Rm,n , k = 1, . . . , m, as in (6.27) and (6.29). The time matrix T = (t1 t2 . . . tm ) ∈ Rm×m has m m-dimensional columns tk , each of which is a subtime tk ∈ Rm of T that represents the m-dimensional time tk ∈ Rm of a subparticle pk = (tk , xk ) ∈ Rm,n , k = 1, . . . , m, as in (6.26) and (6.28). The main diagonal entries tkk of T represent primary times, tkk ∈ R being the primary time of subparticle pk . Accordingly, the block matrix   T t1 t2 . . . tm = ∈ R(m+n)×m (6.34) X x1 x2 tm has m columns, each of which represents a subparticle pk = (tk , xk ) ∈ Rm,n , k = 1, . . . , m. Thus, the block matrix (T, X) in (6.34) represents an (m, n)-particle consisting of m subparticles pk = (tk , xk ) ∈ Rm,n in the pseudo-Euclidean space Rm,n . Accordingly, an (m, n)-particle is a system of m subparticles in m temporal dimensions and n spatial dimensions. Definition 6.3. (Integrated and Disintegrated Particles). The (m, n)-particle (6.34) of m subparticles pk = (tk , xk ) ∈ Rm,n , k = 1, . . . , m, m ≥ 2, is integrated if its subparticles possess the same position at the same primary and secondary time. Thus formally, the (m, n)-particle (6.34) is integrated if (1) the time matrix T ∈ Rm×m is a one-number diagonal matrix and (2) the space matrix X ∈ Rn×m is an equi-column matrix. Otherwise, the (m, n)-particle (6.34) is disintegrated. An integrated (m, n)-particle is viewed as a single particle capable of being disintegrated by a bi-boost. Illustrative examples follow.

6.5.1. Integrated (m,n)-Particles: Example For (m, n) = (3, 2) the time-space block matrix ⎞ ⎛ ⎜⎜⎜3 2 2⎟⎟⎟  ⎜⎜⎜⎜2 3 2⎟⎟⎟⎟  ⎟⎟⎟ ⎜⎜⎜ t1 t2 t3 T = ⎜⎜⎜2 2 3⎟⎟⎟ = ∈ R5×3 , X x1 x2 x3 ⎟⎟⎟ ⎜⎜⎜ ⎜⎜⎝4 4 4⎟⎟⎠ 5 5 5

(6.35)

consisting of the two blocks T and X, represents an integrated (3, 2)-particle made of three 2-dimensional subparticles, p1 , p2 , and p3 , as explained below.

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The block T in (6.35) is the square m × m=3 × 3 submatrix of (6.35), ⎞ ⎛ ⎜⎜⎜3 2 2⎟⎟⎟ ⎟ ⎜ T = ⎜⎜⎜⎜2 3 2⎟⎟⎟⎟ = (t1 t2 t3 ) , ⎠ ⎝ 2 2 3 t1 , t2 , t3 ∈ R3 , which is a one-number diagonal time matrix. The block X in (6.35) is the n × m=2 × 3 submatrix of (6.35),  4 4 4 = (x1 x2 x3 ) , X= 5 5 5

(6.36)

(6.37)

x1 = x2 = x3 ∈ R2 , which is an equi-column space matrix. The three constituent subparticles pk , k = 1, 2, 3, of the (3, 2)-particle (6.35) are described below: (1) p1 = (t1 , x1 ) = (3, 2, 2, 4, 5)t ∈ R3,2 . This subparticle is represented by the first column of (T, X). Accordingly, it has a primary time t11 = 3, and a secondary, twot t dimensional time (t 11 , t21 , t31 ) =(t21 , t31 )=(2, 2) ; and a position x1 = (4, 5) . t 3,2 (2) p2 = (t2 , x2 ) = ((2, 3, 2, 4, 5) ∈ R . This subparticle is represented by the second column of (T, X). Accordingly, it has a primary time t22 = 3, and a secondary, t t two-dimensional time (t12 , t 22 , t32 ) =(t12 , t32 )=(2, 2) ; and a position x2 = (4, 5) . t 3,2 (3) p3 = (t3 , x3 ) = ((2, 2, 3, 4, 5) ∈ R . This subparticle is represented by the third column of (T, X). Accordingly, it has a primary time t33 = 3, and a secondary, t t two-dimensional time (t13 , t23 , t 33 ) =(t13 , t23 )=(2, 2) ; and a position x3 = (4, 5) . The three subparticles p1 , p2 , and p3 have equal primary times, 3, equal secondary, two-dimensional times, (2, 2)t , and equal positions, (4, 5)t . Hence the (3, 2)-particle that consists of these three subparticles can be viewed as a single particle with a primary time 3, a secondary time (2, 2)t , and a position (4, 5)t . This (3, 2)-particle, viewed as a subparticle system, is therefore integrated, consisting of three subparticles that form a single integrated particle.

6.5.2. Bi-boosts that Preserve Integrated Particles: Example Let V ∈ R2×3 c=1 be an equi-column 2×3 velocity matrix,  0.4 0.4 0.4 V= ∈ R2×3 c=1 , 0.3 0.3 0.3

(6.38)

consisting of the three equal two-dimensional subvelocities v1 = v2 = v3 = (0.4, 0.3)t .

(6.39)

2×3 The velocity matrix V lies in the spectral disc R2×3 since the two c=1 of the space R t eigenvalues of the 2 × 2 matrix VV , λ1 = 0 and λ2 = 0.75, are smaller than c2 = 1.

Applications to Time-space of Signature (m,n)

The bi-boost Bc=1 (V) ∈ SOc=1 (3, 2) parametrized by V, given by (5.190), p. 216, turns out to be approximately, rounded to four decimal places, ⎞ ⎛ ⎜⎜⎜1.3333 0.3333 0.3333 0.8000 0.6000⎟⎟⎟ ⎜⎜⎜0.3333 1.3333 0.3333 0.8000 0.6000⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ Bc=1 (V) ≈ ⎜⎜⎜⎜0.3333 0.3333 1.3333 0.8000 0.6000⎟⎟⎟⎟ ∈ SOc=1 (3, 2) . (6.40) ⎟ ⎜⎜⎜ ⎜⎜⎝0.8000 0.8000 0.8000 1.6400 0.4800⎟⎟⎟⎟⎠ 0.6000 0.6000 0.6000 0.4800 1.3600 Applying the bi-boost Bc=1 (V) to bi-boost the integrated (3, 2)-particle (T, X) in (6.35) yields the bi-boosted (3, 2)-particle (T  , X  ), ⎞ ⎛ ⎜⎜⎜11.5333 10.5333 10.5333⎟⎟⎟    ⎜⎜⎜⎜10.5333 11.5333 10.5333⎟⎟⎟⎟ ⎟⎟ ⎜⎜ T T (6.41) Bc=1 (V) =:  ≈ ⎜⎜⎜⎜10.5333 10.5333 11.5333⎟⎟⎟⎟ . ⎟⎟⎟ X X ⎜⎜⎜ ⎜⎜⎝14.5600 14.5600 14.5600⎟⎟⎠ 12.9200 12.9200 12.9200 Remarkably, as one may expect by analogies with Galilean mechanics, the biboosted (3, 2)-particle (T  , X  ) remains integrated. Indeed, it is integrated since (i) its time matrix T  is one-number diagonal and (ii) its position matrix X  is equi-column. Thus, bi-boosting the three subparticles of the integrated (3, 2)-particle (T, X) by equal subvelocities does not disintegrate the integrated particle. Rather, it results in the biboosted integrated (3, 2)-particle (T  , X  ). Let us now repeat the numerical example in (6.38)-(6.41) with c = 1 replaced by, say, c = 6. Accordingly, let V be given by (6.38). Then, the bi-boost parametrized by V, given by (5.190), p. 216, turns out to be approximately, rounded to four decimal places, ⎞ ⎛ ⎜⎜⎜1.0035 0.0035 0.0035 0.0112 0.0084⎟⎟⎟ ⎜⎜⎜⎜0.0035 1.0035 0.0035 0.0112 0.0084⎟⎟⎟⎟ ⎟⎟ ⎜⎜ Bc=6 (V) ≈ ⎜⎜⎜⎜0.0035 0.0035 1.0035 0.0112 0.0084⎟⎟⎟⎟ ∈ SOc=6 (3, 2) . (6.42) ⎟⎟⎟ ⎜⎜⎜ ⎜⎝⎜0.4042 0.4042 0.4042 1.0068 0.0051⎟⎠⎟ 0.3032 0.3032 0.3032 0.0051 1.0038 We may note that the bi-boost matrix Bc=1 (V) in (6.40) is symmetric owing to the choice c = 1. In contrast, Bc=6 (V) in (6.42) is not symmetric. Evidently, Bc=6 (V) is closer than Bc=1 (V) to their Galilean counterpart Bc=∞ (V)

313

314

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

which, according to (5.195), p. 218, is given by ⎛ 0 ⎜⎜⎜ 1 ⎜⎜⎜ 0 1 ⎜⎜⎜ ⎜ ⎜ 0 0 Bc=∞ (V) = ⎜⎜ ⎜⎜⎜ ⎜⎜⎝0.4 0.4 0.3 0.3

0 0 1 0.4 0.3

0 0 0 1 0

⎞ 0⎟⎟ ⎟ 0⎟⎟⎟⎟⎟ ⎟ 0⎟⎟⎟⎟ . ⎟ 0⎟⎟⎟⎟ ⎠ 1

(6.43)

Applying the bi-boost Bc=6 (V) to bi-boost the integrated (3, 2)-particle (T, X) in (6.35) yields the bi-boosted (3, 2)-particle (T  , X  ), ⎞ ⎞ ⎛ ⎛ ⎜⎜⎜3 2 2⎟⎟⎟ ⎜⎜⎜3.1117 2.1117 2.1117⎟⎟⎟ ⎜⎜⎜2 3 2⎟⎟⎟  ⎜⎜⎜2.1117 3.1117 2.1117⎟⎟⎟  ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎜⎜⎜ T T = Bc=6 (V) ⎜⎜⎜⎜2 2 3⎟⎟⎟⎟ =:  ≈ ⎜⎜⎜⎜2.1117 2.1117 3.1117⎟⎟⎟⎟ . (6.44) Bc=6 (V) X X ⎟ ⎟ ⎜⎜⎜ ⎜⎜⎜ ⎜⎜⎝4 4 4⎟⎟⎟⎟⎠ ⎜⎜⎝6.8821 6.8821 6.8821⎟⎟⎟⎟⎠ 5 5 5 7.1616 7.1616 7.1616 Two more numerical demonstrations similar to (6.44) are instructive: ⎞ ⎞ ⎛ ⎛ ⎜⎜⎜4.1152 2.1152 2.1152⎟⎟⎟ ⎜⎜⎜4 2 2⎟⎟⎟ ⎜⎜⎜2 4 2⎟⎟⎟  ⎜⎜⎜2.1152 4.1152 2.1152⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎜⎜⎜ T Bc=6 (V) ⎜⎜⎜⎜2 2 4⎟⎟⎟⎟ =:  ≈ ⎜⎜⎜⎜2.1152 2.1152 4.1152⎟⎟⎟⎟ . X ⎟ ⎟ ⎜⎜⎜ ⎜⎜⎜ ⎜⎝⎜7.2864 7.2864 7.2864⎟⎟⎟⎠⎟ ⎜⎝⎜4 4 4⎟⎟⎟⎠⎟ 7.4648 7.4648 7.4648 5 5 5 and

⎛ ⎜⎜⎜3 ⎜⎜⎜0 ⎜⎜⎜ Bc=6 (V) ⎜⎜⎜⎜0 ⎜⎜⎜ ⎜⎜⎝4 5

0 3 0 4 5

⎞ ⎛ 0⎟⎟ ⎜⎜⎜3.0976 ⎟⎟⎟ ⎜⎜⎜0.0976  0⎟⎟⎟ ⎜⎜⎜ T ⎟⎟⎟ 3⎟⎟ =:  ≈ ⎜⎜⎜⎜0.0976 X ⎟ ⎜⎜⎜ 4⎟⎟⎟⎟ ⎜⎜⎝5.2652 ⎠ 5.9489 5

0.0976 3.0976 0.0976 5.2652 5.9489

⎞ 0.0976⎟⎟ ⎟ 0.0976⎟⎟⎟⎟⎟ ⎟ 3.0976⎟⎟⎟⎟ . ⎟ 5.2652⎟⎟⎟⎟ ⎠ 5.9489

(6.45)

(6.46)

As in (6.41), in (6.44) – (6.46) a bi-boost parametrized by an equi-column velocity matrix takes an integrated particle into a bi-boosted integrated particle. Interestingly, the bi-boosts in (6.44) – (6.46) leave invariant the difference between the primary time and the common value of the secondary time (which is, respectively, 1, 2, 3 in (6.44) – (6.46)) invariant, as expected from (6.31). In order to see in Lemma 6.5 why any bi-boost parametrized by an equi-column velocity matrix takes an integrated particle into a bi-boosted integrated particle we need the following lemma. Lemma 6.4. Let V ∈ Rn×m ⊂ Rn×m and X ∈ Rn×m be two equi-column matrices, and c m×m let T ∈ R be a one-number-diagonal matrix. Furthermore, let f (V t V) ∈ Rm×m be

Applications to Time-space of Signature (m,n)

an arbitrary analytic matrix function of the matrix V t V ∈ Rm×m and let g(VV t ) ∈ Rn×n be an arbitrary analytic matrix function of the matrix VV t ∈ Rn×n . Then, (1) the two matrices f (V t V)V t X and f (V t V)V t VT are one-number matrices in Rm×m and (2) the two matrices g(VV t )VV t X and g(VV t )VT are equi-column matrices in Rn×m . be an equi-column matrix, and let Bc (V) ∈ SOc (m, n) be Lemma 6.5. Let V ∈ Rn×m c the bi-boost parametrized by V. Furthermore, let T ∈ Rm×m be a one-number diagonal matrix and let X ∈ Rn×m be an equi-column matrix, so that the (m, n)-particle (T, X) is integrated. Finally, let    T T = B (V) . (6.47) c X X Then, the bi-boosted (m, n)-particle (T  , X  ) that results from bi-boosting the integrated (m, n)-particle (T, X) by the bi-boost Bc (V) is integrated as well. Proof. The proof follows immediately from Lemma 6.4, noting (1) that the bi-boost application in (6.47) is given by (6.31), p. 309; (2) that ΓRV in (6.31) is an analytic matrix function of the matrix V t V for all V ∈ Rn×m c ; and (3) that ΓVL in (6.31) is an analytic matrix function of the matrix VV t for all V ∈ Rn×m c . 

6.5.3. Bi-boosts that Disintegrate Integrated Particles: Example Let V ∈ R2×3 c=1 be a non-equi-column 2×3 velocity matrix,  0.4 0.3 0.2 (6.48) V= ∈ R2×3 c=1 , 0.3 0.4 0.5 consisting of the three distinct two-dimensional subvelocities v1 = (0.4, 0.3)t v2 = (0.3, 0.4)t v3 = (0.2, 0.5)t .

(6.49)

2×3 since the two The velocity matrix V lies in the spectral disc R2×3 c=1 of the space R t eigenvalues of the 2 × 2 matrix VV , λ1 ≈ 0.0140 and λ2 ≈ 0.6160, are smaller than c2 = 1.

315

316

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

The bi-boost Bc=1 (V) ∈ SOc=1 (3, 2) parametrized by V, given by (5.190), p. 216, turns out to be approximately, rounded to four decimal places, ⎞ ⎛ ⎜⎜⎜1.2450 0.2414 0.1719 0.6048 0.5216⎟⎟⎟ ⎜⎜⎜0.2414 1.2479 0.1780 0.5065 0.6250⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ Bc=1 (V) ≈ ⎜⎜⎜⎜0.1719 0.1780 1.1280 0.3478 0.4612⎟⎟⎟⎟ ∈ SOc=1 (3, 2) . (6.50) ⎟ ⎜⎜⎜ ⎜⎜⎝0.6048 0.5065 0.3478 1.2852 0.3023⎟⎟⎟⎟⎠ 0.5216 0.6250 0.4612 0.3023 1.3356 Applying the bi-boost Bc=1 (V) to bi-boost the integrated (3, 2)-particle (T, X) in (6.35) yields the bi-boosted (3, 2)-particle (T  , X  ), ⎞ ⎛ ⎜⎜⎜ 9.5889 8.5853 8.5158 ⎟⎟⎟    ⎜⎜⎜⎜ 8.7267 9.7331 8.6632 ⎟⎟⎟⎟ ⎟⎟ ⎜⎜ T T (6.51) Bc=1 (V) =:  ≈ ⎜⎜⎜⎜ 6.8244 6.8305 7.7805 ⎟⎟⎟⎟ . ⎟⎟⎟ X X ⎜⎜⎜ ⎜⎜⎝10.1756 10.0773 9.9185 ⎟⎟⎠ 11.6246 11.7280 11.5642 We see in (6.51) that the bi-boost Bc=1 (V), parametrized by a non-equi-column velocity matrix V, bi-boosts an integrated (3, 2)-particle, (T, X), into a disintegrated (3, 2)-particle, (T  , X  ). Accordingly, we say that the bi-boost Bc=1 (V) disintegrates an integrated particle. The constituents of the integrated (3, 2)-particle, (T, X), are three subparticles that share their times and positions and, hence, may be considered as a single particle. In contrast, the constituents of the bi-boosted, disintegrated (3, 2)-particle, (T  , X  ), are three subparticles that possess distinct times and distinct positions and, in general, do not share the same position at the same time.

6.6. Einstein Addition of Signature (m,n) The binary operation ⊕ in the ball Rn×m c , m, n ∈ N, is defined in Def. 5.57, p. 241. The n×m  resulting bi-gyrogroup (Rc , ⊕ ) turns out in Theorem 5.75, p. 255, to be a gyrocommutative gyrogroup that generalizes the Einstein gyrogroups studied in Chap. 2. In the special case when m = 1 we have Rn×1 = Rnc , as explained in Example 5.26, c  n  p. 209, and the bi-gyrogroup (Rn×m c , ⊕ ) descends to the gyrogroup (Rc , ⊕ ), as implied from Example 5.32, p. 212. The binary operation ⊕ in Rnc turns out in Sect. 5.17, p. 248, to be the common Einstein addition ⊕ of relativistically admissible velocities, given by (2.2), p. 10. We therefore introduce the following formal definition in which we call ⊕ Einstein addition of signature (m, n). Definition 6.6. (Einstein Addition of Signature (m,n)). The binary operation ⊕ in the ball Rn×m c , defined in Def. 5.57, p. 241, is presented in (5.328), p. 247. This binary operation is called Einstein addition of signature (m, n), denoted ⊕E := ⊕ , m, n ∈ N.

Applications to Time-space of Signature (m,n)

When it is necessary to emphasize that the binary operation ⊕E in Rn×m depends on the c constant c and on the signature (m, n), we use the full notation, ⊕E =: ⊕E,(m,n),c . The common Einstein addition of special relativity theory is the binary operation ⊕ defined in (2.2), p. 10. Following Def. 6.6 and the result of Sect. 5.17, the relativistic Einstein addition ⊕, studied in Chaps. 2 and 3, takes in full notation the form ⊕ = ⊕E = ⊕E,(1,n),c .

(6.52)

According to Def. 6.6 and (5.328), p. 247, Einstein addition, ⊕E , of signature (m, n), ⊕E = ⊕E,(m,n),c ,

(6.53)

is given by each of the following two mutually equivalent equations,   −1 V1 ⊕E,(m,n),c V2 = In − c−2 V1 V1t (In + c−2 V2 V1t )−1 (V1 + V2 ) Im − c−2 V1t V1 V1 ⊕E,(m,n),c V2 =



In −

c−2 V1 V1t

−1

−2

(V1 + V2 )(Im + c

V1t V2 )−1



(6.54) Im −

c−2 V1t V1

,

V1 , V2 ∈ Rn×m c . Einstein addition of signature (m, n) is a binary operation between elements V ∈ n×m Rn×m c , which are n × m velocity matrices that lie in the ball Rc . In the special case when m = 1 it descends to Einstein velocity addition ⊕E,(1,n),c in the Euclidean ball Rnc . The latter, in turn, is identical with Einstein addition ⊕ of special relativity theory (n = 3 in physical applications), given by (2.2), p. 10, as shown in Sect. 5.17. It is clear from (6.54) and a commuting relation in (5.117), p. 202, that the neutral element of (Rn×m c , ⊕E ) is 0n,m . Furthermore, it is clear from (6.54) that E V = −V

(6.55)

for all V ∈ Rn×m and that Einstein addition possesses the automorphic inverse property, c E (V1 ⊕E V2 ) = E V1 E V2 ,

(6.56)

for all V1 , V2 ∈ Rn×m c .

6.7. On the Bi-boost Product The bi-boost product B(V1 )B(V2 ), V1 , V2 ∈ Rn×m c , is expressed in (5.199), p. 219, in terms of the bi-gyration (lgyr[V1 , V2 ], rgyr[V1 , V2 ]) generated by V1 and V2 along with

317

318

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

V1 ⊕V2 ,

⎛ ⎜⎜rgyr[V1 , V2 ]ΓRV1 ⊕V2 B(V1 )B(V2 ) = ⎜⎜⎜⎝ L ΓV1 ⊕V2 (V1 ⊕V2 )  E11 E12 =: . E21 E22

⎞

1 rgyr[V1 , V2 ]ΓRV1 ⊕V2 (V1 ⊕V2 )t lgyr[V1 , V2 ]⎟⎟⎟ c2 ⎟⎟⎠ ΓVL1 ⊕V2 lgyr[V1 , V2 ]

(6.57) Einstein bi-gyroaddition, ⊕E , in Def. 6.6 and in Def. 5.57, p. 241, via (5.309), V1 ⊕E V2 = V1 ⊕ V2 = (V1 ⊕V2 )rgyr[V2 , V1 ] ,

(6.58)

is presented in (5.328), p. 247. It introduces the elegance observed in both the bigyrocommutative and the bi-gyroassociative laws, giving rise to its bi-gyrogroup structure. We, therefore, wish to express (6.57) in terms of V1 ⊕E V2 rather than V1 ⊕V2 . Following (6.58) we have V1 ⊕V2 = (V1 ⊕E V2 )rgyr[V1 , V2 ] .

(6.59)

6.7.1. Manipulating E11 In this subsection we manipulate the entry E11 of the bi-boost product in (6.57). By means of (6.59) and a commuting relation in (5.131), p. 205, ΓRV1 ⊕V2 = ΓR(V1 ⊕ V2 )rgyr[V1 ,V2 ] = rgyr[V2 , V1 ]ΓRV1 ⊕ V2 rgyr[V1 , V2 ] . E

E

(6.60)

Hence, by (6.57) and (6.60), E11 = rgyr[V1 , V2 ]ΓRV1 ⊕V2 = ΓRV1 ⊕E V2 rgyr[V1 , V2 ] .

(6.61)

6.7.2. Manipulating E21 In this subsection we manipulate the entry E12 of the bi-boost product in (6.57). By means of (6.57), (6.59), a relation in (5.132), p. 205, and a commuting relation in (5.119), p. 202, we have E21 = ΓVL1 ⊕V2 (V1 ⊕V2 ) L = Γ(V (V1 ⊕E V2 )rgyr[V1 , V2 ] 1 ⊕ V2 )rgyr[V1 ,V2 ] E

= ΓVL1 ⊕E V2 (V1 ⊕E V2 )rgyr[V1 , V2 ] = (V1 ⊕E V2 )ΓRV1 ⊕ V2 rgyr[V1 , V2 ] . E

(6.62)

Applications to Time-space of Signature (m,n)

6.7.3. Manipulating E12 In this subsection we manipulate the entry E21 of the bi-boost product in (6.57). By mans of (6.59), a commuting relation in (5.131), p. 205, and a commuting relation in (5.119), p. 202, we have ΓRV1 ⊕V2 (V1 ⊕V2 )t lgyr[V1 , V2 ] = ΓR(V1 ⊕ V2 )rgyr[V1 ,V2 ] rgyr[V2 , V1 ](V1 ⊕V2 )t lgyr[V1 , V2 ] E

= rgyr[V2 , V1 ]ΓRV1 ⊕E V2 lgyr[V1 , V2 ]lgyr[V2 , V1 ](V1 ⊕V2 )t lgyr[V1 , V2 ] = rgyr[V2 , V1 ]ΓRV1 ⊕ V2 (V1 ⊕E V2 )t lgyr[V1 , V2 ] E

= rgyr[V2 , V1 ](V1 ⊕E V2 )t ΓVL1 ⊕ V2 lgyr[V1 , V2 ] . E

(6.63) Hence, by (6.57) and (6.63), c2 E12 = rgyr[V1 , V2 ]ΓRV1 ⊕V2 (V1 ⊕V2 )t lgyr[V1 , V2 ] = (V1 ⊕E V2 )t ΓVL1 ⊕ V2 lgyr[V1 , V2 ] .

(6.64)

E

6.7.4. Manipulating E22 In this subsection we manipulate the entry E22 of the bi-boost product in (6.57). By means of (6.57), (6.59), and a relation in (5.132), p. 205, we have E22 = ΓVL1 ⊕V2 lgyr[V1 , V2 ] L = Γ(V lgyr[V1 , V2 ] 1 ⊕ V2 )rgyr[V1 ,V2 ] E

(6.65)

= ΓVL1 ⊕ V2 lgyr[V1 , V2 ] . E

6.7.5. The Bi-boost Product Representation Finally, by means of the results of Subsects. 6.7.1 – 6.7.4, the bi-boost product expressed in terms of Einstein bi-gyroaddition is given by the following theorem. Theorem 6.7. (Bi-boost Product Representation). For any m, n ∈ N and c > 0, let V1 , V2 , V ∈ (Rn×m c , ⊕E ) and let B(V) be the bi-boost of signature (m, n), ⎞ ⎛ 1 R t⎟ ⎜⎜⎜ ΓRV 2 ΓV V ⎟ c ⎟⎟⎟ ∈ R(m+n)×(m+n) . (6.66) B(V) = ⎜⎜⎝ L ⎠ ΓV V = VΓRV ΓVL

319

320

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Then, we have the bi-boost product representation ⎞ ⎛ ⎟⎟⎟ ⎜⎜⎜rgyr[V1 , V2 ] 0m,n ⎟⎟⎠ . ⎜ B(V1 )B(V2 ) = B(V1 ⊕E V2 ) ⎜⎝ 0n,m lgyr[V1 , V2 ]

(6.67)

Proof. By means of (6.57) along with the results of Subsects. 6.7.1 – 6.7.3, we have the following chain of equations, which completes the proof: ⎞ ⎛ 1 t L ⎜⎜⎜ ΓRV1 ⊕ V2 rgyr[V1 , V2 ] 2 (V1 ⊕E V2 ) ΓV ⊕ V lgyr[V1 , V2 ]⎟ ⎟⎟⎟ c 1 2 E E ⎟⎟⎠ B(V1 )B(V2 ) = ⎜⎜⎜⎝ (V1 ⊕E V2 )ΓRV1 ⊕ V2 rgyr[V1 , V2 ] ΓVL1 ⊕ V2 lgyr[V1 , V2 ] E

E

⎛ ⎜⎜⎜ ΓRV1 ⊕ V2 rgyr[V1 , V2 ] E = ⎜⎜⎜⎝ L ΓV1 ⊕ V2 (V1 ⊕E V2 )rgyr[V1 , V2 ] E

⎛ ⎜⎜⎜ ΓRV1 ⊕ V2 E ⎜ = ⎜⎜⎝ L ΓV1 ⊕ V2 (V1 ⊕E V2 ) E

⎞ 1 R Γ (V ⊕ V )t lgyr[V1 , V2 ]⎟⎟⎟ c2 V1 ⊕E V2 1 E 2 ⎟⎟⎟ ⎠ ΓVL1 ⊕ V2 lgyr[V1 , V2 ] E

⎞⎛ 1 R Γ (V ⊕ V )t ⎟⎟ ⎜⎜rgyr[V1 , V2 ] c2 V1 ⊕E V2 1 E 2 ⎟ ⎟⎟⎟ ⎜⎜⎜ ⎠⎝ 0n,m ΓVL1 ⊕ V2 E

⎞ ⎟⎟⎟ ⎟⎟⎠

0m,n lgyr[V1 , V2 ]

⎞ ⎛ ⎟⎟⎟ ⎜⎜⎜rgyr[V1 , V2 ] 0m,n ⎟⎟⎠ . = B(V1 ⊕E V2 ) ⎜⎜⎝ 0n,m lgyr[V1 , V2 ] (6.68)  Identity (6.67) of Theorem 6.7 suggests the following notation for the bi-gyration: ⎞ ⎛ ⎟⎟⎟ ⎜⎜⎜rgyr[V1 , V2 ] 0m,n Gyr[V1 , V2 ] = ⎜⎜⎝ (6.69) ⎟⎟⎠ ∈ R(m+n)×(m+n) , 0n,m lgyr[V1 , V2 ] so that (6.67) is written as B(V1 )B(V2 ) = B(V1 ⊕E V2 )Gyr[V1 , V2 ]

(6.70)

B(V1 ⊕E V2 ) = B(V1 )B(V2 )Gyr[V2 , V1 ] ,

(6.71)

and hence, V1 , V2 ∈ Rn×m c . Comparing block entries of the bi-boost product B(V1 )B(V2 ) in (6.68) and in (5.199),

Applications to Time-space of Signature (m,n)

p. 219, we obtain the following four matrix identities: ΓRV1 ⊕E V2 rgyr[V1 , V2 ] = ΓRV1 (Im +

1 t V V2 )ΓRV2 c2 1

(6.72)

1 ΓVL1 ⊕ V2 lgyr[V1 , V2 ] = ΓVL1 (In + 2 V1 V2t )ΓVL2 E c and ΓVL1 ⊕ V2 (V1 ⊕E V2 )rgyr[V1 , V2 ] = ΓVL1 (V1 + V2 )ΓRV2 E

ΓRV1 ⊕E V2 (V1 ⊕E V2 )t lgyr[V1 , V2 ] = ΓRV1 (V1 + V2 )t ΓVL2 .

(6.73)

The identities in (6.72) turn out to be the bi-gamma identities (5.335), p. 249.

6.7.6. Applications Being matrix multiplication, bi-boost multiplication is associative, {B(V1 )B(V2 )}B(V3 ) = B(V1 ){B(V2 )B(V3 )} ,

(6.74)

V1 , V2 , V3 ∈ Rn×m c . Let us calculate each side of (6.74). By means of (6.70) we have {B(V1 )B(V2 )}B(V3 ) = B(V1 ⊕E V2 )Gyr[V1 , V2 ]B(V3 )

(6.75)

and by means of (6.70), the left bi-gyroassociative law in Theorem 5.64, p. 246, and (6.71), we have B(V1 ){B(V2 )B(V3 )} = B(V1 )B(V2 ⊕E V3 )Gyr[V2 , V3 ] = B(V1 ⊕E (V2 ⊕E V3 ))Gyr[V1 , V2 ⊕E V3 ]Gyr[V2 , V3 ] = B((V1 ⊕E V2 )⊕E lgyr[V1 , V2 ]V3 rgyr[V2 , V1 ]) × Gyr[V1 , V2 ⊕E V3 ]Gyr[V2 V3 ]

(6.76)

= B(V1 ⊕E V2 )B(lgyr[V1 , V2 ]V3 rgyr[V2 , V1 ]) × Gyr[lgyr[V1 , V2 ]V3 rgyr[V2 , V1 ], V1 ⊕E V2 ] × Gyr[V1 , V2 ⊕E V3 ]Gyr[V2 , V3 ] . Hence, by means of (6.74) – (6.76), we have the identity Gyr[V1 , V2 ]B(V3 ) = B(lgyr[V1 , V2 ]V3 rgyr[V2 , V1 ]) × Gyr[lgyr[V1 , V2 ]V3 rgyr[V2 , V1 ], V1 ⊕E V2 ] × Gyr[V1 , V2 ⊕E V3 ]Gyr[V2 , V3 ] ,

(6.77)

321

322

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

for all V1 , V2 , V3 ∈ Rn×m c . The special case of (6.77) when V1 = r⊗V2 , r ∈ R, is elegant and interesting. Substituting V1 = r⊗V2 into (6.77) and noting that by Theorem 5.89, p. 277, lgyr[r⊗V2 , V2 ] = In

and

rgyr[r⊗V2 , V2 ] = Im ,

yields B(V3 ) = B(V3 )Gyr[V3 , r⊗V2 ⊕E V2 ]Gyr[r⊗V2 , V2 ⊕E V3 ]Gyr[V2 , V3 ] ,

(6.78)

implying, by gyration inversion, Gyr[(r + 1)⊗V2 , V3 ] = Gyr[r⊗V2 , V2 ⊕E V3 ]Gyr[V2 , V3 ] .

(6.79)

Renaming (V2 , V3 ) as (V1 , V2 ), (6.79) yields the elegant identity Gyr[(r + 1)⊗V1 , V2 ] = Gyr[r⊗V1 , V1 ⊕E V2 ]Gyr[V1 , V2 ] ,

(6.80)

for any V1 , V2 ∈ Rn×m and r ∈ R. c Similarly, substituting V2 = r⊗V1 into (6.77) yields the identity B(V3 ) = B(V3 )Gyr[V3 , V1 ⊕E r⊗V1 ]Gyr[V1 , r⊗V1 ⊕E V3 ]Gyr[r⊗V1 , V3 ] ,

(6.81)

implying Gyr[(r + 1)⊗V1 , V3 ] = Gyr[V1 , r⊗V1 ⊕E V3 ]Gyr[r⊗V1 , V3 ] .

(6.82)

Renaming V3 and V2 , (6.82) yields the identity Gyr[(r + 1)⊗V1 , V2 ] = Gyr[V1 , r⊗V1 ⊕E V2 ]Gyr[r⊗V1 , V2 ] ,

(6.83)

for any V1 , V2 ∈ Rn×m and r ∈ R. c Finally, comparing (6.80) and (6.83) yields the identity Gyr[r⊗V1 , V1 ⊕E V2 ]Gyr[V1 , V2 ] = Gyr[V1 , r⊗V1 ⊕E V2 ]Gyr[r⊗V1 , V2 ] .

(6.84)

6.7.7. Bi-boost Application to Time-Space Events Taking the first m columns of the (m + n) × (m + n) matrix identity (6.68) yields the following bi-boost application to a time-space event of signature (m, n), ⎞  ⎛ R ⎞ ⎛ ΓRV1 ⊕ V2 rgyr[V1 , V2 ] ⎜⎜⎜ ΓV2 ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟⎟ T E ⎟⎠⎟ = ⎜⎜⎜ ⎟⎟⎠ =: ∈ R(n+m)×m , B(V1 ) ⎜⎝⎜ (6.85) ⎝ R R S (V1 ⊕E V2 )ΓV1 ⊕ V2 rgyr[V1 , V2 ] V2 Γ V2 E

Rn×m c .

V1 , V2 ∈ The time part, T , of the time-space event (T, S ) in (6.85) is nonsingular since it possesses uniquely the polar decomposition T = ΓRV1 ⊕E V2 rgyr[V1 , V2 ] ∈ Rm×m × SO(m)

(6.86)

Applications to Time-space of Signature (m,n)

as the product of a positive definite matrix, ΓRV1 ⊕ V2 ∈ Rm×m , and a special orthogonal E matrix, rgyr[V1 , V2 ] ∈ SO(m). In the special case when m = 1, the following results hold: (1) Right gyrations are trivial, being elements of SO(1) = {1}. (2) The right gamma factor ΓRV , V ∈ Rn×m c , descends to the special relativistic Lorentz n gamma factor γv , v ∈ Rc , as shown in (5.154), p. 210. (3) The bi-boost B(V) of signature (m, n), V ∈ Rn×m c , descends to the special relativistic boost, as shown in Example 5.32, p. 212. Hence, when m = 1, (6.85) descends to the well-known special relativistic identity, ⎛ ⎞ ⎛ ⎞ ⎜⎜⎜ γv2 ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ γv1 ⊕E v2 ⎟⎟⎠ = ⎜⎜⎝ ⎟⎟ ∈ Rn+1 , B(v1 ) ⎜⎜⎝ (6.87) γv1 ⊕E v2 (v1 ⊕E v2 )⎠ γv2 v2 v1 , v2 ∈ Rnc , where γv is the Lorentz gamma factor (2.3), p. 11. Interestingly, the (n + 1)-vectors (γv , γv v) ∈ Rn+1 , v ∈ Rnc , in (6.87), where n = 3 in physical applications, have relativistic norm 1. Indeed, the squared relativistic norm of (γv , γv v) is ⎛ ⎞2 ⎜⎜⎜ γv ⎟⎟⎟ 1 1 ⎜⎜⎝ ⎟⎟⎠ = γv2 − 2 γv2 v2 = γv2 (1 − 2 v2 ) = 1 . (6.88) c c  γv v  Guided by analogies between (6.85) and (6.87) we define in Sect. 6.8 the relativistic bi-norm of signature (m, n), obtaining in Theorem 6.12, p. 326, a result analogous to (6.88).

6.7.8. Bi-boost Application to Space-Time Events Taking the last n columns of the (m + n) × (m + n) matrix identity (6.68) yields the following bi-boost application to a space-time event of signature (m, n), ⎞ ⎛ ⎛1 ⎞ ⎛ ⎞ ⎟⎟⎟ ⎜⎜ 1 ⎟⎟ ⎜⎜⎜ V t ΓL ⎟⎟⎟ ⎜⎜⎜ 1 (V ⊕ V )t ΓL lgyr[V , V ] ⎜ S 1 2 1 2 ⎜ ⎟ E ⎜ ⎜ ⎟ ⊕ V V 1 E 2 ⎟⎟⎟ =: ⎜⎜ c2 ⎟⎟⎟⎟ ∈ R(m+n)×n , (6.89) B(V1 ) ⎜⎜⎜⎝ c2 2 V2 ⎟⎟⎟⎠ = ⎜⎜⎜⎜ c2 ⎟⎠ ⎝ ⎠⎟ ⎜⎝ L ΓV1 ⊕ V2 lgyr[V1 , V2 ] ΓVL2 T E V1 , V2 ∈ Rn×m c . The time part, T , of the space-time event (S , T ) in (6.89) is nonsingular since it possesses uniquely the polar decomposition T = ΓVL1 ⊕E V2 lgyr[V1 , V2 ] ∈ Rn×n × SO(n)

(6.90)

as the product of a positive definite matrix, ΓVL1 ⊕ V2 ∈ Rn×n , and a special orthogonal E matrix, lgyr[V1 , V2 ] ∈ SO(n).

323

324

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

In order to emphasize the distinction between ⎞ ⎛ ⎞ ⎛ ⎟⎟⎟ ΓRV1 ⊕ V2 rgyr[V1 , V2 ] ⎜⎜⎜T ⎟⎟⎟ ⎜⎜⎜ E ⎟⎟⎟ ∈ R(n+m)×m ⎜⎝⎜ ⎟⎠⎟ := ⎜⎜⎜ ⎝ R (V1 ⊕E V2 )ΓV1 ⊕ V2 rgyr[V1 , V2 ]⎠ S

(6.91)

E

in (6.85), which involves the right gamma factor, and ⎞ ⎛ ⎞ ⎛ ⎜⎜⎜S ⎟⎟⎟ ⎜⎜⎜(V1 ⊕E V2 )t ΓVL1 ⊕E V2 lgyr[V1 , V2 ]⎟⎟⎟ ⎜ ⎟⎟⎟ ∈ R(m+n)×n ⎜⎜⎝ ⎟⎟⎠ := ⎜⎝⎜ ⎠ L lgyr[V , V ] Γ T 1 2 V1 ⊕E V2

(6.92)

in (6.89), which involves the left gamma factor, we call (T, S ) a time-space event, and (S , T ) a space-time event.

6.8. Relativistic Time-Space Bi-norm of Signature (m,n) Definition 6.8. (Time-Space Events of Signature (m,n)). For any m, n ∈ N, let T ∈ Rm×m be a nonsingular matrix. It possesses uniquely the polar decomposition T = tOm ∈ Rm×m ,

(6.93)

where t ∈ Rm×m is either positive definite or negative definite, and Om ∈ SO(m). Furthermore, let S ∈ Rn×m and let V be related to S by the equation V = S T −1 ∈ Rn×m .

(6.94)

Then, the matrix (T, S ) ∈ R(n+m)×m is said to be a time-space event of signature (m, n) with a time part T = tOm ∈ Rm×m and a space part S = VT = VtOm ∈ Rn×m . By means of (6.93) and (6.94) we have   T tOm ∈ R(n+m)×m , = VtOm S

(6.95)

where the positive definite m × m matrix t is associated with m-dimensional time in the sense that each column of t is an m-dimensional time. Definition 6.9. (Relativistic Time-Space Bi-norm of Signature (m,n)). For any m, n ∈ N, the time-space event (T, S ) ∈ R(n+m)×m is relativistically admissible if the matrix T t T − c−2 S t S ∈ Rm×m is positive definite. The relativistic time-space bi-norm

(T, S ) of signature (m, n) of a relativistically admissible time-space event (T, S ) ∈ R(n+m)×m is given by     T  1 m×m . (6.96)  S  = T t T − 2 S t S ∈ R c

Applications to Time-space of Signature (m,n)

Noting that by the definition of V in (6.94), S = VT ,

(6.97)

where T is nonsingular, we seek a condition on V that insures that the matrix under the square root in (6.96) is positive definite and, hence, relativistically admissible. The following lemma proves useful in the search for the desired condition. m×m be nonsingular. Then, the Lemma 6.10. Let m, n ∈ N, V ∈ Rn×m c , and let T ∈ R t −2 t matrix T (Im − c V V)T is positive definite if and only if the matrix Im − c−2 V t V is positive definite.

Proof. A real, positive definite matrix A ∈ Rm×m is a nonsingular symmetric matrix such that xt Ax > 0 for all nonzero x ∈ Rn . Clearly, xt (Im − c−2 V t V)x > 0 holds for all nonzero x ∈ Rn if and only if xt T t (Im − c−2 V t V)T x=(T x)t (Im − c−2 V t V)T x holds for all x ∈ Rn , x  0, since T is invertible.  Theorem 6.11. (Relativistically Admissible Time-Space Events). Let m, n ∈ N, S ∈ Rn×m and let T ∈ Rm×m be nonsingular, and V = S T −1 ∈ Rn×m . Then, the matrix T t T − c−2 S t S is positive definite if and only if V ∈ Rn×m c . Proof. By means of (6.97) we have T tT −

1 t 1 S S = T t T − 2 (VT )t (VT ) 2 c c = T tT −

1 t t T V VT c2

= T t (Im −

(6.98)

1 t V V)T ∈ Rm×m , c2

where T ∈ Rm×m is nonsingular. Hence, by Lemma 6.10, the matrix T t T − c−2 S t S is positive definite if and only if the matrix Im − c−2 V t V is positive definite. But, the latter is positive definite if and only if V ∈ Rn×m (by [39, Theorem 7.2.1, p. 402], according to which a symmetric c matrix is positive definite if and only if all its eigenvalues are positive).  It follows from Theorem (6.11) that (i) a time-space event (T, S = VT ) is relativistically admissible if and only if V ∈ Rn×m and that (ii) the bi-norm of this time-space c event is positive definite if and only if V ∈ Rn×m c . For arbitrary square matrices M and N we write M > N if M − N is positive definite. Accordingly, the bi-norm (T, S ) ∈ Rm×m is positive definite, that is, (T, S ) >

325

326

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

0m,m , if and only if S t S < c2 T t T or, equivalently, if and only if V t V < c2 Im , where V = S T −1 . Theorem 6.12. (Unit Bi-norm, Time-Space). ⎛ ⎞ ⎜⎜⎜ ΓRV Om ⎟⎟⎟ ⎜⎝⎜ R ⎟⎠⎟ = Im  VΓV Om 

(6.99)

for any m, n ∈ N, V ∈ Rn×m c , and Om ∈ SO(m). Proof. By means of the fourth commuting relation in (5.119), p. 202, and the definition of ΓRV in (5.115), p. 201, the time-space bi-norm definition in (6.96) yields ⎛ ⎞2 ⎜⎜⎜ ΓRV Om ⎟⎟⎟ 1 ⎜⎜⎝ R ⎟⎟⎠ = (ΓRV Om )t (ΓRV Om ) − 2 (VΓRV Om )t (VΓRV Om ) c  VΓV Om  = Otm (ΓRV )2 Om −

1 t R t R O Γ V VΓV Om c2 m V

= Otm (ΓRV )2 Om −

1 t t O V V(ΓRV )2 Om c2 m

= Otm (Im −

(6.100)

1 t V V)(ΓRV )2 Om c2

= Otm Im Om = Im , as desired.



In order to enhance an analogy with special relativity we impose the spectral norm on both sides of (6.99), obtaining the result  ⎛ ⎞   ⎜⎜⎜ ΓRV Om ⎟⎟⎟   ⎜⎝⎜ R ⎟⎠⎟  = Im = 1 , (6.101)   VΓV Om   and Om ∈ SO(m), which is analogous to the special relativistic result for any V ∈ Rn×m c in (6.88). We may note that the inside norm on the left-hand side of (6.101) is the relativistic bi-norm of signature (m, n), while the outside norm is the spectral norm. The left-hand side of (6.101) is said to be the relativistic spectral norm of the timespace event (T, S ) := (ΓRV Om , VΓRV Om ). By analogy with special relativity theory, we call V ∈ Rn×m a velocity of signature (m, n) or, simply, an (m, n)-velocity. We say that an (m, n)-velocity V ∈ Rn×m (n+m)×m is relativistically admissible if V ∈ Rn×m a c . Similarly, we call (T, S ) ∈ R

Applications to Time-space of Signature (m,n)

time-space event of signature (m, n) or, simply, an (m, n)-time-space event. We say that an (m, n)-time-space event is relativistically admissible if T ∈ Rm×m is nonsingular and if S t S < c2 T t T . A time-space event of signature (m, n) whose bi-norm is Im is said to be a unit time-space event. Accordingly, by Theorem 6.12, (T, S ) = (ΓRV Om , VΓRV Om ) are unit time-space events for any V ∈ Rn×m and Om ∈ SO(m). c

6.9. Relativistic Time-Space Bi-inner Product of Signature (m,n) Definition 6.13. (Relativistic Time-Space Bi-inner Product). The relativistic biinner product (T 1 , S 1 )·(T 2 , S 2 ) of signature (m, n) of two time-space events (T k , S k ) ∈ R(n+m)×m , k = 1, 2, is given by   1 T1 T2 (6.102) · = T 1t T 2 − 2 S 1t S 2 ∈ Rm×m . S1 S2 c Accordingly, the relativistic spectral bi-inner product of these time-space events is given by         T 1 T1 T T    2 =  1 · 2  = T 1t T 2 − 2 S 1t S 2  ∈ R . (6.103)  S1 S2  S1 S2 c Clearly,

and

   2 T  T T =   ∈ Rm×m · S  S S     2     2  T    T  T T =     =     ∈ R .  S S  S    S 

(6.104)

(6.105)

Theorem 6.14. (Invariance Under Bi-boosts). The relativistic bi-inner product of time-space events is invariant under bi-boosts, that is,     T T T T (6.106) B(V) 1 ·B(V) 2 = 1 · 2 , S1 S2 S1 S2 n×m for any m, n ∈ N, V ∈ Rn×m , and T 1 , T 2 ∈ Rm×m . c , S 1, S 2 ∈ R In particular, the relativistic bi-norm of a time-space event is invariant under biboosts, that is,       T  T  (6.107)  B(V) S  =  S  ,

for any S ∈ Rn×m and T ∈ Rm×m .

327

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Proof. Let (T 3 , S 3 ) and (T 4 , S 4 ) be the time-space events (T 1 , S 1 ) and (T 2 , S 2 ) biboosted by B(V). Then, ⎛ ⎞ ⎛ ⎛ ⎞ 1 R t ⎞⎟⎟ 1 R t ⎞⎟⎟ ⎛⎜T ⎞⎟ ⎛⎜⎜ R ⎜⎜⎜T 1 ⎟⎟⎟ ⎜⎜⎜ ΓR ⎜⎜⎜T 3 ⎟⎟⎟ ⎜ 1 ⎟ Γ Γ V T + ⎜⎜⎝ ⎟⎟⎠ := B(V) ⎜⎜⎝ ⎟⎟⎠ = ⎜⎜⎜ V c2 V ⎟⎟⎟⎟ ⎜⎜⎜⎝ ⎟⎟⎟⎠ = ⎜⎜⎜⎜ V 1 c2 ΓV V S 1 ⎟⎟⎟⎟ (6.108) ⎝ R ⎝ ⎠ S ⎠ S3 S1 1 VΓV ΓVL VΓRV T 1 + ΓVL S 1 and, similarly, ⎛ ⎞ ⎛ ⎛ ⎞ ⎜⎜T 2 ⎟⎟ ⎜⎜⎜ ΓR ⎜⎜⎜T 4 ⎟⎟⎟ ⎜⎜⎝ ⎟⎟⎠ := B(V) ⎜⎜⎜⎝ ⎟⎟⎟⎠ = ⎜⎜⎜⎝ V S4 S2 VΓR V

1 R t ⎞⎟⎟ ⎛⎜T ⎞⎟ ⎛⎜⎜ R 1 R t ⎞⎟⎟ ⎜ 2⎟ Γ Γ V T + Γ V S 2 ⎟⎟⎟ ⎜ ⎟ ⎜ ⎟ 2 ⎜ ⎟ c2 V ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ = ⎜⎜⎝ V c2 V ⎟⎠ . L R S 2 ΓV VΓV T 2 + ΓVL S 2

(6.109)

Hence, by straightforward matrix algebra and commuting relations in (5.119), p. 202, and the definition of ΓRV and ΓVL in (5.115), we have the following chain of equations:   1 T3 T4 · = T 3t T 4 − 2 S 3t S 4 S3 S4 c = (T 1t ΓRV +

1 t R R 1 1 S 1 VΓV )(ΓV T 2 + 2 ΓRV V t S 2 )− 2 (T 1t ΓRV V t + S 1t ΓVL )(V1 ΓRV T 2 + ΓVL S 2 ) 2 c c c

= T 1t (ΓRV )2 T 2 +

1 t 1 1 S 1 V(ΓRV )2 V t S 2 − 2 T 1t ΓRV V t VΓRV T 2 − 2 S 1t (ΓVL )2 S 2 4 c c c

= T 1t (ΓRV )2 T 2 +

1 t 1 1 S VV t (ΓVL )2 S 2 − 2 T 1t V t V(ΓRV )2 T 2 − 2 S 1t (ΓVL )2 S 2 c4 1 c c

1 t 1 1 V V)(ΓRV )2 T 2 − 2 S 1t (In − 2 VV t )(ΓVL )2 S 2 2 c c c   1 T T = T 1t T 2 − 2 S 1t S 2 = 1 · 2 , S1 S2 c = T 1t (Im −

(6.110) as desired.



Theorem 6.15. (Invariance Under Lorentz Transformations). The relativistic biinner product is invariant under Lorentz transformations, that is,     T1 T T T ·Λ(Om , V, On ) 2 = 1 · 2 (6.111) Λ(Om , V, On ) S1 S2 S1 S2 n×m , and T 1 , T 2 ∈ for any m, n ∈ N, V ∈ Rn×m c , Om ∈ SO(m), On ∈ SO(n), S 1 , S 2 ∈ R m×m R , where Λ(Om , V, On ) ∈ SO(m, n) is the Lorentz transformation of signature (m, n). In particular, the relativistic bi-norm is invariant under Lorentz transformations,

Applications to Time-space of Signature (m,n)

that is,

      T  T  Λ(Om , V, On ) S  =  S 

(6.112)

for any S ∈ Rn×m and T ∈ Rm×m . Proof. The Lorentz transformation of signature (m, n) possesses the polar decomposition (5.129), p. 205, ⎞ ⎛ ⎜⎜⎜ Om 0m,n ⎟⎟⎟ ⎟⎟⎠ , ⎜ (6.113) Λ(Om , V, On ) = B(V) ⎜⎝ 0n,m On according to which the Lorentz transformation is a bi-boost preceded by a bi-rotation. Owing to Theorem 6.14, it is enough to prove that the relativistic bi-inner product is invariant under bi-rotations, that is, ⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜⎜⎜ Om 0m,n ⎟⎟⎟ ⎜⎜⎜T 1 ⎟⎟⎟ ⎜⎜⎜ Om 0m,n ⎟⎟⎟ ⎜⎜⎜T 2 ⎟⎟⎟ ⎜⎜⎜T 1 ⎟⎟⎟ ⎜⎜⎜T 2 ⎟⎟⎟ (6.114) ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ · ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ = ⎜⎜⎝ ⎟⎟⎠ · ⎜⎜⎝ ⎟⎟⎠ . 0n,m On S 1 0n,m On S 2 S1 S2 Indeed,

⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜⎜⎜ Om 0m,n ⎟⎟⎟ ⎜⎜⎜T 1 ⎟⎟⎟ ⎜⎜⎜ Om 0m,n ⎟⎟⎟ ⎜⎜⎜T 2 ⎟⎟⎟ ⎜⎜⎜Om T 1 ⎟⎟⎟ ⎜⎜⎜Om T 2 ⎟⎟⎟ ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠· ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ = ⎜⎜⎝ ⎟⎟⎠ · ⎜⎜⎝ ⎟⎟⎠ 0n,m On S 1 0n,m On S 2 On S 1 On S 2 1 (On S 1 )t On S 2 c2 ⎛ ⎞ ⎛ ⎞ ⎜⎜T 1 ⎟⎟ ⎜⎜T 2 ⎟⎟ 1 t t = T 1 T 2 − 2 S 1 S 2 = ⎜⎜⎜⎝ ⎟⎟⎟⎠ · ⎜⎜⎜⎝ ⎟⎟⎟⎠ , c S1 S2

= (Om T 1 )t Om T 2 −

as desired.

(6.115)



6.10. A Bi-gyrotriangle Bi-gamma Identity For any m, n ∈ N and c > 0, let A, B, C ∈ (Rn×m c , ⊕E , ⊗). The bi-gyrotriangle with bigyrovertices A, B, and C is denoted by ABC. Its bi-gyrosides opposite to A, B, and C are, respectively, BC = E B⊕E C, AC = E A⊕E C, and AB = E A⊕E B, the bi-gyrolengths of which are E B⊕E C , E A⊕E C , and E A⊕E B . As an example, a bi-gyrotriangle in (R2×3 c=1 , ⊕E , ⊗) is a system of three 2-dimensional entangled gyrotriangles, as shown in Fig. 7.3, p. 376. The right bi-gamma identity in (5.335), p. 249, with V1 replaced by E V1 = −V1

329

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

takes the form ΓR E V1 ⊕E V2 rgyr[ E V1 , V2 ] = ΓRV1 (Im −

1 t V V2 )ΓRV2 c2 1

(6.116)

 for any V1 , V2 ∈ Rn×m c , noting that ⊕ = ⊕E . In this section we manipulate (6.116) into a form that gives a bi-gyrotriangle bigamma identity. Accordingly, let

V1 = E A⊕E B ∈ Rn×m c V2 = E A⊕E C ∈ Rn×m . c

(6.117)

Then, by (5.370), p. 256, E V1 ⊕E V2 = E ( E A⊕E B)⊕E ( E A⊕E C) = gyr[ E A, B]( E B⊕E C) .

(6.118)

Substituting V1 and V2 from (6.117) – (6.118) into the left-hand side of (6.116) yields the following chain of equations, which are numbered for subsequent explanation: (1)

 LHS === ΓRgyr[

E

A,B]( E B⊕E C) rgyr[ E ( E A⊕E B), E A⊕E C]

(2)

 R === Γlgyr[

E

A,B]( E B⊕E C)rgyr[B, E A] rgyr[ E A⊕E B, A E C]

(3)

 R === Γ(

E

B⊕E C)rgyr[B, E A] rgyr[ E A⊕E B, A E C]

(4)

 === rgyr[ E A, B]ΓR E B⊕E C rgyr[B, E A]rgyr[ E A⊕E B, A E C] (5)

 === rgyr[ E A, B]ΓR E B⊕E C rgyr[B, E A]rgyr[A, E B]rgyr[B, E C]rgyr[C, E A] (6)

 === rgyr[ E A, B]ΓR E B⊕E C rgyr[B, E C]rgyr[C, E A] . (6.119) Derivation of the numbered equalities in (6.119): (1) This equation results from the above-mentioned substitution into the left-hand side of (6.116). (2) This equation results from the bi-gyration definition in Def. 5.67, p. 250, from (5.218), p. 223, and by the automorphic inverse property (6.56), p. 317, of Einstein addition. (3) Follows from (5.132), p. 205, noting that lgyr[ E A, B] ∈ SO(n). (4) Follows from Item (3) by a commuting relation in (5.131), p. 205.

Applications to Time-space of Signature (m,n)

(5) Follows from Item (4) by the right gyration identity in (5.372), p. 257. (6) Follows from Item (5) by the gyration inversion law (5.242), p. 228. Substituting V1 and V2 from (6.117) – (6.118) into the right-hand side of (6.116) yields 1 ( A⊕ B)t )( E A⊕E C))ΓR E A⊕E C . c2 E E Hence, (6.116), (6.119), and (6.120) imply the identity RHS = ΓR E A⊕E B (Im −

ΓR E A⊕E B (Im −

1 ( A⊕ B)t )( E A⊕E C))ΓR E A⊕E C c2 E E

(6.120)

(6.121)

= rgyr[ E A, B]ΓR E B⊕E C rgyr[B, E C]rgyr[C, E A] . Rearranging (6.121), we obtain the elegant bi-gyrotriangle bi-gamma identity ΓR

E

B⊕E C rgyr[ E B, C]

(6.122) 1 t R ( A⊕ B) ( A⊕ C))Γ rgyr[ A, C] . E A⊕E C E E E c2 E E We may note here that the left counterpart of the right version (6.122) of the bigyrotriangle bi-gamma identity is = (ΓR E A⊕E B rgyr[ E A, B])t (Im −

Γ L

E

B⊕E C lgyr[ E B, C]

(6.123) 1 t L t ( A⊕ B)( A⊕ C) )(Γ lgyr[ A, C]) . E A⊕E C E E E c2 E E The bi-gyrotriangle bi-gamma identity (6.122) and its left counterpart (6.123) provide a relationship between = Γ LE A⊕E B lgyr[ E A, B](In −

(1) the two bi-gyrosides AB = E A⊕B and AC = E A⊕C of bi-gyrotriangle ABC, which emanate from bi-gyrovertex A, and (2) the bi-gyroside BC = E B⊕C opposite to bi-gyrovertex A. In order to discover further bi-gyrogeometric elegance hidden in (6.122) we temporarily use the notation T 1 = ΓR A⊕ B rgyr[ E A, B] ∈ Rm×m E E T 2 = ΓR E A⊕E C rgyr[ E A, C] ∈ Rm×m

(6.124)

T 3 = ΓR E B⊕E C rgyr[ E B, C] ∈ Rm×m along with the notation in (6.117). With the notation in (6.124) and (6.117), along with the relativistic bi-inner product of signature (m, n), studied in Sect. 6.9, the bi-gyrotriangle bi-gamma identity (6.122)

331

332

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

takes the following graceful form: T 3 = T 1t (Im − = T 1t T 2 −

1 t V V2 )T 2 c2 1 1 (V1 T 1 )t V2 T 2 c2

1 t S S2 c2 1   T T = 1 · 2 ∈ Rm×m , S1 S2

(6.125)

= T 1t T 2 −

where S k = Vk T k , k = 1, 2. Returning to the original notation, the bi-gyrotriangle bi-gamma identity (6.125) reads ΓR

E

B⊕E C rgyr[ E B, C]

⎞ ⎛ ⎞ ⎛ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ΓR A⊕ C rgyr[ E A, C] ΓR A⊕ B rgyr[ E A, B] E E E E ⎟⎟⎟ · ⎜⎜⎜ ⎟⎟⎟ ∈ Rm×m . = ⎜⎜⎜⎜⎝ ⎟ ⎜ ⎟ ( E A⊕E B)ΓR A⊕ B rgyr[ E A, B]⎠ ⎝( E A⊕E C)ΓR A⊕ C rgyr[ E A, C]⎠ E E E E (6.126) The bi-gyrotriangle bi-gamma identity (6.126) of signature (m, n) presents the right bi-gamma-gyration expression ΓR B⊕ C rgyr[ E B, C], associated with bi-gyroside BC of E E bi-gyrotriangle ABC, as the bi-inner product of two unit time-space events, associated with the bi-gyrosides AB and AC of the bi-gyrotriangle. Noting Items (1) – (3), p. 323, we see that in the special case when m = 1, (6.126) descends to its special relativistic counterpart, ⎛ ⎞ ⎛ ⎞ γ A⊕ C γ A⊕ B ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ E E E E ⎟⎠ · ⎜⎜⎝ γ B⊕ C = ⎜⎜⎝ ⎟ ⎟⎟ E E ( E A⊕E B)γ A⊕ B ( E A⊕E C)γ A⊕ C ⎠ E E E E (6.127) 1 = γ A⊕ B γ A⊕ C (1 − 2 ( E A⊕E B)t ( E A⊕E C)) ∈ R . E E E E c The gyrotriangle gamma identity (6.127) is well known in special relativity, giving rise to the law of gyrocosines of gyrotrigonometry, as shown in [93, Chap. 12] and [98, Chap. 7]. We may note here that the left counterpart of the right version (6.126) of the

Applications to Time-space of Signature (m,n)

bi-gyrotriangle bi-gamma identity is Γ L

E

B⊕E C lgyr[ E B, C]

⎞ ⎛ ⎞ ⎛ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ Γ L A⊕ C lgyr[ E A, C] Γ L A⊕ B lgyr[ E A, B] E E E E ⎜ ⎟ ⎟⎟⎟ ∈ Rn×n . ⎜ ⎜ ⎟ ⎜ · = ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎟ t L t L ( E A⊕E B) Γ A⊕ B lgyr[ E A, B] ( E A⊕E C) Γ A⊕ C lgyr[ E A, C]⎠ E E E E (6.128)

6.11. (m,n)-Velocity Structures that Einstein Addition Preserves √ , k ∈ N. The Definition 6.16. (Equi-block and Equi-column Matrices). Let v ∈ Rn×k kc block matrix √ V = (v v . . . v) ∈ Rn×(km)

(6.129)

kmc

generated by m copies of v is said to be an equi-block matrix, possessing the equi-block √ . √ with the block v ∈ Rn×k structure of matrices in Rn×(km) kc kmc In the special case when k = 1, the matrix (6.129) is said to be an equi-column matrix. Example 6.17. Let v1 , v2 ∈ Rn×1 = Rnc . The matrix c √ , V = (v1 v2 v1 v2 v1 v2 ) ∈ Rn×6 6c

(6.130)

generated by the columns v1 and v2 , is an equi-block matrix with the block √ . v = (v1 v2 ) ∈ Rn×2 2c

(6.131)

1m = (Ik Ik . . . Ik ) ∈ Rk×km

(6.132)

Let be the equi-block matrix generated by m copies of the identity matrix Ik , k, m ∈ N, so that 1m 1tm = mIk .

(6.133)

Furthermore, let U and V be the equi-block matrices √ U = (u u . . . u) = u1m ∈ Rn×km kmc

(6.134)

√ V = (v v . . . v) = v1m ∈ Rn×km kmc √ . generated, respectively, by m copies of u and m copies of v, u, v ∈ Rn×k kc

333

334

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Then, UU t = u1m 1tm ut = muut VU t = v1m 1tm ut = mvut

(6.135)

U t U = 1tm ut u1m . Following the first equation in (6.134) and the third equation in (6.135), we have   −1 −1 √ √ −2 t Ikm − ( kmc) U U = Ikm − ( kmc)−2 1tm ut u1m . (6.136) The symmetric matrix on the left-hand side of (6.136) is positive definite since √ . Hence, the matrix on the right-hand side of (6.136) is positive definite as U ∈ Rn×km kmc well. The main goal of this section is to prove Theorem 6.23. Lemma 6.18 is used in the proof of Lemma 6.19, and Lemma 6.19 is used in the proof of Lemma 6.20, which in turn, is used in the proof of Theorem 6.23. Lemma 6.18. Let α ∈ Rk×k , let c1 =

√

kmc ,

(6.137)

and let t km×km J1 = Ikm − c−2 1 1m α1m ∈ R

(6.138)

k×k , J2 = Ik − c−2 1 mα ∈ R

(6.139)

and k, m ∈ N. Then, the k eigenvalues of J2 are equal to k eigenvalues of J1 , and each of the remaining (m − 1)k eigenvalues of J1 is 1. Accordingly, J2 is positive definite if and only if J1 is positive definite. Proof. Sylvester’s determinant identity states that det(In + AB) = det(Im + BA)

(6.140)

for any m, n ∈ N, A ∈ Rn×m , and B ∈ Rm×n . By means of (6.133), (6.138) – (6.139), and straightforward algebra, along with Sylvester’s determinant identity (6.140), we establish the following chain of equations.

Applications to Time-space of Signature (m,n)

For any λ ∈ R, λ  1, t det(J1 − λIkm ) = det(Ikm − c−2 1 1m α1m − λIkm ) t = det((1 − λ)Ikm − c−2 1 1m α1m )

α 1m ) 1−λ α t ) = (1 − λ)km det(Ik − c−2 1 1m 1m 1−λ mα = (1 − λ)km det(Ik − c−2 ) 1 1−λ t = (1 − λ)km det(Ikm − c−2 1 1m

(by (6.140))

(6.141)

(by (6.133))

= (1 − λ)(m−1)k det((1 − λ)Ik − c−2 1 mα) = (1 − λ)(m−1)k det(J2 − λIk ) . The condition λ  1 in (6.141) can be removed from the extreme sides of (6.141) by continuity considerations, obtaining det(J1 − λIkm ) = (1 − λ)(m−1)k det(J2 − λIk )

(6.142)

for all λ ∈ R. We see from (6.142) that the k eigenvalues of J2 are equal to k eigenvalues of J1 and that each of the remaining (m − 1)k eigenvalues of J1 is 1. Hence, J2 is positive definite if and only if J1 is positive definite.  √ , let α ∈ Rk×k be given by the equation Lemma 6.19. For any u ∈ Rn×k



kc

t t Ikm − c−2 1 1m u u1m

−1

t = Ikm − c−2 1 1m α1m ,

where c1 is given by (6.137). Then,   −1 c21 −2 t α= Ik − Ik − c1 mu u m

(6.143)

(6.144)

so that, equivalently, Ik −

c−2 1 mα

=



t Ik − c−2 1 mu u

−1

.

(6.145)

Proof. The definition of α in (6.143) implies t 2 −2 t t Ikm = (Ikm − c−2 1 1m α1m ) (Ikm − c1 1m u u1m )

(6.146)

335

336

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

and, hence, t t −1 t t 2 = (Ikm − c−2 (Ikm − c−2 1 1m u u1m ) 1 1m u u1m ) t −4 t 2 = Ikm − c−2 1 1m (2α)1m + c1 1m (mα )1m .

(6.147)

Hence, we have the following chain of equations, which are numbered for subsequent explanation: (1)

t c−2 1 1m (2α)1m

−

t 2 c−4 1 1m (mα )1m

 t t −1 === Ikm − (Ikm − c−2 1 1m u u1m ) (2)

 t t −2 t t −1 === −c−2 1 1m u u1m (Ikm − c1 1m u u1m )

(6.148)

(3)

 t t −2 t −1 === −c−2 1 1m u u(Ik − c1 mu u) 1m . Derivation of the numbered equalities in (6.148): (1) This equation follows immediately from (6.147) (2) This equation is verified by multiplying each side of the equation by the factor t t (Ikm − c−2 1 1m u u1m ). (3) This equation is verified by showing that a. this equation is equivalent to the equation t t −1 t −1 1m (Ikm − c−2 = (Ik − c−2 1 1m u u1m ) 1 mu u) 1m

which, in turn, b. is equivalent to the equation t −2 t t (Ik − c−2 1 mu u)1m = 1m (Ikm − c1 1m u u1m )

which is obvious, noting the commuting relation Ik 1m = 1m Ikm and (6.133). The extreme sides of (6.148) yield the equation −4 2 −2 t −2 t −1 1tm {−2c−2 1 α + c1 mα − c1 u u(Ik − c1 mu u) }1m = 0km,km ,

(6.149)

implying −4 2 −2 t −2 t −1 = 0k,k . −2c−2 1 α + c1 mα − c1 u u(Ik − c1 mu u)

(6.150)

t −2 t By means of the obvious equation −c−2 1 mu u = Ik − c1 mu u − Ik we have the obvious equation t −2 t −1 t −1 = Ik − (Ik − c−2 −c−2 1 mu u(Ik − c1 mu u) 1 mu u) .

(6.151)

Applications to Time-space of Signature (m,n)

By means of (6.150) we have −4 2 2 −2 t −2 t −1 −2c−2 1 mα + c1 m α = c1 mu u(Ik − c1 mu u) .

(6.152)

Hence, by means of (6.152) and (6.151), we have the chain of equations 2 −2 t −2 t −1 (Ik − c−2 1 mα) = Ik + c1 mu u(Ik − c1 mu u) t −1 = Ik − {Ik − (Ik − c−2 1 mu u) }

(6.153)

t −1 = (Ik − c−2 1 mu u) ,

so that Ik − c−2 1 mα = ± and, hence,



t Ik − c−2 1 mu u

−1

  −1 c21 −2 t α= Ik ∓ Ik − c1 mu u . m

(6.154)

(6.155)

It remains to determine the value of the ambiguous sign in (6.154) – (6.155). Following (6.143) an the third equation in (6.135) we have the equation  −1 t −2 t 1 α1 = I − c U U , (6.156) Ikm − c−2 m km 1 m 1 . Hence, the where the right-hand side is a positive definite matrix, since U ∈ Rn×km c1 left-hand side of (6.156) is a positive definite matrix as well. Hence, the matrix t J1 = Ikm − c−2 1 1m α1m

(6.157)

in (6.138) is positive definite. Hence, by Lemma 6.18, the matrix J2 = Ik − c−2 1 mα

(6.158)

in (6.139) is positive definite. The matrix

√ −2 t t J3 = Ik − c−2 1 mu u = Ik − ( kc) u u

(6.159) √ −1 √ . Equation (6.154) can be written as J2 = ± J3 , is positive definite since u ∈ Rn×k kc √ −1 where both J2 and J3 are positive definite matrices. This result determines the value of the ambiguous sign in (6.154) – (6.155) to be the upper sign, thus obtaining (6.144) – (6.145), as desired. 

337

338

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

√ , Lemma 6.20. For any u ∈ Rn×k kc

 1m

Ikm −

t t c−2 1 1m u u1m

−1

 =

Ik − (

√

−1

kc)−2 ut u

1m ,

(6.160)

where c1 is given by (6.137). Proof. The proof is given by the following chain of equations, which are numbered for subsequent explanation:  −1 (1)  t t t 1m Ikm − c−2 === 1m (Ikm − c−2 1 u u1 m 1 1m α1m ) 1 m (2)

 === (Ik − c−2 1 mα)1m (3)  −1  tu Ik − c−2 mu 1m === 1  (4) −1 √  −2 t Ik − ( kc) u u 1m . ===

(6.161)

Derivation of the numbered equalities in (6.161): (1) (2) (3) (4)

Follows from the definition of α in (6.143). Follows from Item 1 by the commuting relation 1m Ikm = Ik 1m , and by (6.133). Follows from Item 2 by Result (6.145) of Lemma 6.19. Follows from Item 3 by (6.137). 

Theorem 6.21. Let k, m, n ∈ N, and let U = (u u . . . u) ∈ Rn×(km)

(6.162)

be an equi-block matrix generated by m copies of u ∈ Rn×k . Then, √ U ∈ Rn×(km) kmc

Proof. Let

 ΓRkm,U, √kmc

:=

⇐⇒

√ . u ∈ Rn×k kc

√ Ikm − ( kmc)−2 U t U

if the right-hand side of (6.164) exists, and let  −1 √ R √ Γk,u, kc := Ik − ( kc)−2 ut u if the right-hand side of (6.165) exists.

(6.163)

−1

(6.164)

(6.165)

Applications to Time-space of Signature (m,n)

√ √ exists if and only if U ∈ Rn×(km) . km,U, kmc kmc R √ n×k matrix Γ exists if and only if u ∈ R √ . k,u, kc kc

By Theorem 5.17, p. 200, the matrix ΓR

Similarly, by Theorem 5.17, the But, by means (6.164) – (6.165), (6.135), and (6.160), we have the chain of equations  −1 √ R −2 t √ 1m Γkm,U, kmc = 1m Ikm − ( kmc) U U  = 1m  =

−1 √ Ikm − ( kmc)−2 1tm ut u1m

√ Ik − ( kc)−2 ut u

(6.166)

−1

1m

= ΓRk,u, √kc 1m . The extreme sides of (6.166) demonstrate that ΓR

√ km,U, kmc

ΓR

√ k,u, kc

exists if and only if

√ . √ exists. Hence, as desired, U ∈ Rn×(km) if and only if u ∈ Rn×k kc

kmc



Theorem 6.22 is a special case of Theorem 6.21 in which equi-block matrices are specialized to equi-column matrices. As such, this theorem provides a link between matrix balls Rn×m of Rn×m , m, n ∈ N, and Euclidean balls Rn×1 = Rnc of Rn . Matrix balls c c are the home of bi-velocities along with their Einstein bi-gyroaddition of signature (m, n), while Euclidean balls are the home of n-dimensional relativistically admissible velocities along with their relativistic Einstein velocity addition (2.2). p. 10. Theorem 6.22. Let U = (u u . . . u) ∈ Rn×m

(6.167)

= Rnc . Then, be an equi-column matrix generated by m copies of u ∈ Rn×1 c √ U ∈ Rn×m mc

⇐⇒

u ∈ Rn×1 = Rnc . c

(6.168)

Proof. This theorem is the special case of Theorem 6.21, obtained by selecting k = of Rn×1 coincides with the Euclidean ball Rnc of 1, noting that the spectral ball Rn×1 c n R. 

339

340

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Theorem 6.23. (Einstein Addition Preserves Equi-block Structures). Let √ U = ((u1 u2 . . . uk ) (u1 u2 . . . uk ) . . . (u1 u2 . . . uk )) ∈ Rn×(km) kmc

(6.169) √ V = ((v1 v2 . . . vk ) (v1 v2 . . . vk ) . . . (v1 v2 . . . vk )) ∈ Rn×(km) kmc

be two equi-block matrices in copies of v,

√ Rn×(km) kmc

generated, respectively, by m copies of u and m

√ u = (u1 u2 . . . uk ) ∈ Rn×k kc √ , v = (v1 v2 . . . vk ) ∈ Rn×k kc

(6.170)

= Rnc , i = 1, . . . , k. k, m, n ∈ N, where ui and vi are column vectors, ui , vi ∈ Rn×1 c Furthermore, let √ W = U⊕E,(km,n), √kmc V ∈ Rn×(km)

(6.171)

√ . w = u⊕E,(k,n), √kc v ∈ Rn×k kc

(6.172)

kmc

and

Then, W is the equi-block matrix generated by m copies of w, √ W = (w w . . . w) = w1m ∈ Rn×(km) ,

(6.173)

U⊕E,(km,n), √kmc V = (u⊕E,(k,n), √kc v)1m ,

(6.174)

1m = (Ik Ik . . . Ik ) ∈ Rk×(km) .

(6.175)

kmc

that is, equivalently,

where

Proof. Let us consider the following chain of equations, which are numbered for

Applications to Time-space of Signature (m,n)

subsequent explanation: (1)

 U⊕E,(km,n), √kmc V === U⊕E,(km,n),c1 V (2)    −2 t −1 −2 t t In − c1 UU (In + c1 VU ) (U + V) Ikm − c−2 === 1 U U (3)

 ===



−2 t −1 t In − c−2 1 muu (In + c1 mvu ) (u + v)1m



−1

−1

t t Ikm − c−2 1 1m u u1m   (4) −1 √ √ −2 t −1 √  −2 t −2 t In − ( kc) uu (In + ( kc) vu ) (u + v) Ik − ( kc) u u 1m === (5)

 === (u⊕E,(k,n), √kc v)1m . (6.176) Derivation of the numbered equalities in (6.176): (1) Follows immediately from the definition of c1 in (6.137). (2) Follows from the first formula in (6.54) of the binary operation ⊕E,(m,n),c with m √ replaced by km and c replaced by c1 = kmc. (3) Follows from (6.134) – (6.135). √ −2 (4) Follows from Item 3 by noting that c−2 kc) and by (6.160). m = ( 1 (5) Follows from the first formula in (6.54) of the binary operation ⊕E,(m,n),c with m √ replaced by k and c replaced by kc. Finally, the extreme sides of (6.176) establish the result, (6.174), of the theorem.  Theorem 6.23 presents an interplay between Einstein addition ⊕E,(km,n), √kmc of signa√ , and Einstein addition ⊕E,(k,n), √kc of signature (k, n) in the ture (km, n) in the ball Rn×(km) √ . ball Rn×k

kmc

kc

Example 6.24. In this example we present the special case of Theorem 6.23 that corresponds to k = 3, m = 3, and n = 2. In Identity (6.177a), Einstein addition ⊕E,(9,2), √9c √ is expressed in terms of of signature (m, n)=(9, 2) of two equi-block matrices in R2×9 9c

341

342

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

√ . Einstein addition ⊕E,(3,2), √3c of signature (m, n)=(3, 2) of two matrices in R2×3 3c            a1 b1 c1 a1 b1 c1 a1 b1 c1 a b c a b c a b c ⊕E,(9,2), √9c 1 1 1 1 1 1 1 1 1 a2 b2 c2 a2 b2 c2 a2 b2 c2 a2 b2 c2 a2 b2 c2 a2 b2 c2

     a1 b1 c1 a1 b1 c1 √ = ⊕ a2 b2 c2 E,(3,2), 3c a2 b2 c2

     a1 b1 c1 a1 b1 c1 √ ⊕ a2 b2 c2 E,(3,2), 3c a2 b2 c2 

    a1 b1 c1 a b c ⊕E,(3,2), √3c 1 1 1 . a2 b2 c2 a2 b2 c2 (6.177a)

Each of the two matrices on the left-hand side of (6.177a) is an equi-block matrix of order 2 × 9 with a block of order 2 × 3, given, respectively, by      a1 b1 c1 a1 b1 c1 and . (6.177b) a2 b2 c2 a2 b2 c2 The matrix on the right-hand side of (6.177a) is an equi-block matrix of order 2 × 9 with a block of order 2 × 3, given by      a1 b1 c1 a1 b1 c1 √ ⊕ . (6.177c) a2 b2 c2 E,(3,2), 3c a2 b2 c2 Identity (6.177a) asserts that one can (1) either first assemble each of two blocks into two equi-block matrices and then add the resulting two matrices by an appropriate Einstein addition (2) or, equivalently, first add the two blocks into a single block by an appropriate Einstein addition and then assemble the resulting block into an appropriate equiblock matrix.

6.12. (m,n)-Velocity Structures that Einstein Scalar Multiplication Preserves Theorem 6.25 is the scalar multiplication counterpart of Theorem 6.23. Theorem 6.25. (Einstein Scalar Multiplication Preserves Equi-block Structures). Let √ , V = (v v . . . v) = v1m ∈ Rn×(km)

(6.178)

1m = (Ik Ik . . . Ik ) ∈ Rk×(km) ,

(6.179)

kmc

where

Applications to Time-space of Signature (m,n)

√ be an equi-block matrix in Rn×(km) generated by m copies of the block v, kmc

√ , v = (v1 v2 . . . vk ) ∈ Rn×k kc

(6.180)

k, m, n ∈ N, where vi , i = 1, . . . , k, are column vectors, vi ∈ Rn×1 = Rnc . c Furthermore, let √ W = r⊗V ∈ Rn×(km)

(6.181)

√ w = r⊗v ∈ Rn×k kc

(6.182)

kmc

and

for any r ∈ R. Then, W is the equi-block matrix generated by m copies of w, √ , W = (w w . . . w) = w1m ∈ Rn×(km)

(6.183)

r⊗(v1m ) = r⊗V = (r⊗v)1m .

(6.184)

kmc

that is, equivalently,

Proof. Following (6.178) we have VV t = v1m 1tm vt = mvvt .

(6.185)

By means of the first equation in (5.453), p. 273, when matrix inversion is justified, r  r   √  √ −2 t −2 t In + ( kmc) VV − In − ( kmc) VV In V. r⊗V =  r  r  √  √  √ −2 VV t −2 t −2 t ( kmc) In + ( kmc) VV + In − ( kmc) VV (6.186) Substituting (6.178) and (6.185) into the right-hand side of (6.186) yields  r  r  √  √ In + ( kc)−2 vvt − In − ( kc)−2 vvt In r⊗V =  v1m r  r  √  √  √ −2 t (6.187) −2 t −2 t ( kc) vv In + ( kc) vv + In − ( kc) vv = (r⊗v)1m , √ , as desired. v ∈ Rn×k kc

√ regardless of whether the inverse of In order to validate (6.187) for any v ∈ Rn×k kc

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 √ the matrix ( kc)−2 vvt exists we employ the quasi inverse matrix by means of the equation  √  √ −1 ( kc)−2 vvt v = ( kc)−2 vvt # v , (6.188) as explained in Sect. 5.25. 

6.13. Partially Integrated Particles: Example In this section we extend the example of Subsect. 6.5.1 from an integrated particle to a partially integrated particle. Similar to the integrated (3, 2)-particle (T, X) in (6.35), p. 311, let us consider the partially integrated ((m, n) = (6, 3))-particle (T, X), ⎞ ⎛ ⎜⎜⎜3.0000 2.0000 2.0000 1.2000 1.2000 1.2000⎟⎟⎟ ⎟ ⎜⎜⎜ ⎜⎜⎜2.0000 3.0000 2.0000 1.2000 1.2000 1.2000⎟⎟⎟⎟⎟ ⎜⎜⎜2.0000 2.0000 3.0000 1.2000 1.2000 1.2000⎟⎟⎟ ⎟⎟ ⎜⎜  ⎜⎜⎜⎜2.4000 2.4000 2.4000 5.0000 1.0000 1.0000⎟⎟⎟⎟ ⎟⎟ ⎜⎜ T (6.189) = ⎜⎜⎜⎜2.4000 2.4000 2.4000 1.0000 5.0000 1.0000⎟⎟⎟⎟ . ⎟⎟⎟ X ⎜⎜⎜ 2.4000 2.4000 2.4000 1.0000 1.0000 5.0000 ⎟ ⎜⎜⎜ ⎜⎜⎜4.0000 4.0000 4.0000 2.0000 2.0000 2.0000⎟⎟⎟⎟⎟ ⎟ ⎜⎜⎜ ⎜⎜⎜5.0000 5.0000 5.0000 3.0000 3.0000 3.0000⎟⎟⎟⎟⎟ ⎟⎠ ⎜⎝ 6.0000 6.0000 6.0000 4.0000 4.0000 4.0000 of (T, X) in (6.189) is the m × m = 6 × 6 time matrix in R6×6 ⎞ 2.0000 2.0000 1.2000 1.2000 1.2000⎟⎟⎟ ⎟ 3.0000 2.0000 1.2000 1.2000 1.2000⎟⎟⎟⎟ ⎟  2.0000 3.0000 1.2000 1.2000 1.2000⎟⎟⎟⎟  ⎟⎟⎟ = t1 t2 t3 t4 t5 t6 , 2.4000 2.4000 5.0000 1.0000 1.0000⎟⎟⎟ ⎟ 2.4000 2.4000 1.0000 5.0000 1.0000⎟⎟⎟⎟ ⎠ 2.4000 2.4000 1.0000 1.0000 5.0000 (6.190) tk ∈ R6 , k = 1, . . . , 6. We say that T is a two-number diagonal time matrix, just as T in (6.36) is a one-number diagonal time matrix. The block X of (T, X) in (6.189) is the n × m=3 × 6 space matrix in R3×6 , ⎞ ⎛ ⎜⎜⎜4.0000 4.0000 4.0000 2.0000 2.0000 2.0000⎟⎟⎟   ⎟ ⎜ X = ⎜⎜⎜⎜5.0000 5.0000 5.0000 3.0000 3.0000 3.0000⎟⎟⎟⎟ = x1 x2 x3 x4 x5 x6 , ⎠ ⎝ 6.0000 6.0000 6.0000 4.0000 4.0000 4.0000 (6.191) The block T ⎛ ⎜⎜⎜3.0000 ⎜⎜⎜ ⎜⎜⎜2.0000 ⎜⎜⎜2.0000 T = ⎜⎜⎜⎜ ⎜⎜⎜2.4000 ⎜⎜⎜ ⎜⎜⎝2.4000 2.4000

Applications to Time-space of Signature (m,n)

xk ∈ R6 , k = 1, . . . , 6. We say that the space matrix X is equi-block since (x1 x2 x3 ) = (x4 x5 x6 ) ,

(6.192)

just as the time matrix X in (6.37) is equi-column. The (m + n) × m=9 × 6 matrix (T, X) in (6.189) represents an (m, n)=(6, 3)-particle consisting of (m = 6) three-dimensional subparticles pk ∈ R3 , k = 1, . . . , m = 6. The kth column of (T, X) represents the kth subparticle. The six subparticles are described below: (1) The subparticles p1 , p2 , and p3 possess a. the same primary time, 3; b. the same (m − 1) = 5-dimensional secondary time, (2, 2, 2.4, 2.4, 2.4)t ; and c. the same position (4, 5, 6)t ∈ R3 . (2) The subparticles p4 , p5 , and p6 possess a. the same primary time, 5; b. the same (m − 1) = 5-dimensional secondary time, (1.2, 1.2, 1.2, 1, 1)t ; and c. the same position (2, 3, 4)t ∈ R3 . Hence, the three subparticles p1 , p2 , and p3 (resp., p4 , p5 , and p6 ) can be viewed as a single subparticle p123 (resp., p456 ). Hence, the (m, n)=(6, 3)-particle (T, X) of m = 6 subparticles in (6.189) can be viewed as a system of two subparticles p123 and p456 , which are described as follows: (1) The subparticle p123 possesses a primary time 3; a secondary (m-1)=5-dimensional time, (2, 2, 2.4, 2.4, 2.4)t ; and a position (4, 5, 6)t ∈ R3 . (2) The subparticle p456 possesses a primary time 5; a secondary (m-1)=5-dimensional time, (1.2, 1.2, 1.2, 1, 1)t ; and a position (2, 3, 4)t ∈ R3 . In the sense of Items (1) and (2), the (6,3)-particle (T, X) is partially integrated. It can be viewed as a system of two particles, each of which is capable of being disintegrated into three subparticles.

6.13.1. Bi-boosts that Preserve Partially Integrated Particles: Example This subsection is similar in structure to Subsect. 6.5.2. √ √ , Keeping in mind the equi-block structure of (T, X) in (6.189), let V ∈ Rn×m = R3×6 mc 6c with, say, c = 0.75, be the equi-block velocity matrix ⎞ ⎛ ⎜3 3 3 6 6 6⎟⎟⎟ 1 ⎜⎜⎜⎜ √ . ⎜⎜2 2 2 4 4 4⎟⎟⎟⎟⎟ ∈ R3×6 (6.193) V= 6c ⎠ 10 ⎜⎝ 5 5 5 1 1 1

345

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

The equi-block velocity matrix V is generated by the two subvelocities v1 = (0.3, 0.2, 0.5)t ∈ R3 and v2 = (0.6, 0.4, 0.1)t ∈ R3 with respective norms (approximately) 0.6164 and 0.7280. Hence, the two ((m, n) = 3 (1, 3))-velocities v1 and v2 lie in the ball R3×1 c=0.75 =Rc=0.75 . Hence, the ((m, n) = (6, 3))√ velocity V lies in the ball R3×6 . 60.75 Let Bc= √60.75 (V) be the bi-boost of signature (m, n)=(6, 3) parametrized by V. Then, approximately, rounded to four decimal places, we have by means of (5.127), p. 204, Bc= √60.75 (V) ≈ ⎛ ⎜⎜⎜1.1144 ⎜⎜⎜⎜0.1144 ⎜⎜⎜ ⎜⎜⎜0.1144 ⎜⎜⎜ ⎜⎜⎜0.1172 ⎜⎜⎜ ⎜⎜⎜0.1172 ⎜⎜⎜0.1172 ⎜⎜⎜ ⎜⎜⎜0.6140 ⎜⎜⎜ ⎜⎜⎜0.4093 ⎝ 0.7068

0.1144 1.1144 0.1144 0.1172 0.1172 0.1172 0.6140 0.4093 0.7068

0.1144 0.1144 1.1144 0.1172 0.1172 0.1172 0.6140 0.4093 0.7068

0.1172 0.1172 0.1172 1.1711 0.1711 0.1711 1.0136 0.6757 0.3272

0.1172 0.1172 0.1172 0.1711 1.1711 0.1711 1.0136 0.6757 0.3272

0.1172 0.1172 0.1172 0.1711 0.1711 1.1711 1.0136 0.6757 0.3272

0.1819 0.1819 0.1819 0.3003 0.3003 0.3003 1.4497 0.2998 0.2382

0.1213 0.1213 0.1213 0.2002 0.2002 0.2002 0.2998 1.1999 0.1588

⎞ 0.2094⎟⎟⎟ ⎟ 0.2094⎟⎟⎟⎟ ⎟ 0.2094⎟⎟⎟⎟ ⎟ 0.0969⎟⎟⎟⎟⎟ ⎟ 0.0969⎟⎟⎟⎟ ⎟ 0.0969⎟⎟⎟⎟ ⎟ 0.2382⎟⎟⎟⎟⎟ ⎟ 0.1588⎟⎟⎟⎟ ⎠ 1.2072 (6.194)

in SOc= √60.75 (6, 3). Applying the bi-boost Bc= √60.75 (V) to bi-boost the partially integrated (6, 3)-particle (T, X) in (6.189) yields the bi-boosted (6, 3)-particle (T  , X  ),    T T √ =:  ≈ Bc= 60.75 (V) X X ⎞ ⎛ ⎜⎜⎜ 7.2355 6.2355 6.2355 3.9978 3.9978 3.9978 ⎟⎟⎟ ⎟ ⎜⎜⎜ ⎜⎜⎜ 6.2355 7.2355 6.2355 3.9978 3.9978 3.9978 ⎟⎟⎟⎟⎟ ⎜⎜⎜ 6.2355 6.2355 7.2355 3.9978 3.9978 3.9978 ⎟⎟⎟ ⎟ ⎜⎜⎜ ⎜⎜⎜ 7.2366 7.2366 7.2366 8.2090 4.2090 4.2090 ⎟⎟⎟⎟⎟ ⎟ ⎜⎜⎜ ⎜⎜⎜ 7.2366 7.2366 7.2366 4.2090 8.2090 4.2090 ⎟⎟⎟⎟⎟ . ⎟ ⎜⎜⎜ ⎜⎜⎜ 7.2366 7.2366 7.2366 4.2090 4.2090 8.2090 ⎟⎟⎟⎟⎟ ⎜⎜⎜20.3223 20.3223 20.3223 14.0567 14.0567 14.0567⎟⎟⎟ ⎟ ⎜⎜⎜ ⎜⎜⎜15.8816 15.8816 15.8816 11.0378 11.0378 11.0378⎟⎟⎟⎟⎟ ⎜⎝ ⎠⎟ 16.2931 16.2931 16.2931 10.6161 10.6161 10.6161

(6.195)

Interestingly, as one may expect by analogies with Galilean mechanics, the biboosted (6, 3)-particle (T  , X  ) remains partially integrated, having the same partial integrated structure as that of the original (6, 3)-particle (T, X) in (6.189). Indeed, (1) like the block T in (6.189) – (6.190), the block T  in (6.195) is a two-number diagonal time matrix and

Applications to Time-space of Signature (m,n)

(2) like the block X in (6.189) – (6.191), the block X  in (6.195) is an equi-block space matrix having the same equi-block structure as that of the original space matrixX. Analogous examples can be constructed for any partially integrated (m, n)-particle (by a numerically corroborated conjecture). The resulting conjecture is important. It demonstrates that any partially integrated structure of any (m, n)-particle (which represents a system of m n-dimensional subparticles) is preserved by bi-boosts of signature (m, n) parametrized by related equi-block velocity matrices of signature (m, n). The analogous conjecture for Galilean bi-boosts is obvious since these involve no entanglements. In the special case when m = 1 there is no entanglement, and the resulting geometry turns out to be the analytic hyperbolic geometry studied in [81, 84, 93, 94, 95, 96, 98], which is of interest in non-Euclidean geometry. It is, therefore, expected that the resulting extended analytic hyperbolic geometry that corresponds to m > 1 and, hence, involves entanglement will prove interesting as well. Accordingly, in Chap. 7 we will pay special attention to the study of analytic hyperbolic geometry of signature (3, 2).

6.14. Linking Einstein Addition of Signature (m,n) with the Special Relativistic Einstein Addition Theorem 6.26 is a special case of Theorem 6.23 in which equi-block matrices are specialized to equi-column matrices. As such, this theorem provides a link between Einstein addition of signature (m,n), m, n ∈ N, and the special relativistic Einstein addition (2.2), p. 10, which is of signature (1, n). This link suggests that Einstein addition of signature (m, n) and its associated Lorentz transformation of signature (m, n) be considered as an integral part of Einstein’s special relativity theory for all m, n ∈ N. The common, special relativistic Lorentz transformation of signature (1, n) individually boosts a single, n-dimensional particle, entangling the space-time coordinates of the particle. Similarly, the Lorentz transformation of signature (m, n), m, n ∈ N, collectively boosts a system of m ndimensional particles, entangling the space-time coordinates of the particle system. As a result, the boosted constituent subparticles of the particle system are in entanglement. Theorem 6.26. Let √ U = (u u . . . u) ∈ Rn×m mc √ V = (v v . . . v) ∈ Rn×m mc

(6.196)

√ generated, respectively, by m copies of the be two equi-column matrices in Rn×m mc

347

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

columns u, v ∈ Rn×1 = Rnc . Furthermore, let c √ W = U⊕E,(m,n), √mc V ∈ Rn×m mc

(6.197)

be the bi-gyrosum of U and V obtained by Einstein bi-gyroaddition of signature (m, n) √ , and let in Rn×m mc w = u⊕E,(1,n),c v ∈ Rnc

(6.198)

be the gyrosum of u and v obtained by the special relativistic Einstein gyroaddition of signature (1, n) in Rnc . Then, W is the equi-column matrix generated by m copies of the column w, √ . W = (w w . . . w) ∈ Rn×m mc

(6.199)

Proof. This theorem is a special case of Theorem 6.23, obtained by selecting k = 1.  Being of signature (1, n), Einstein gyroaddition ⊕E,(1,n),c/ √m in (6.198) coincides with the special relativistic Einstein addition in the ball Rnc/ √m , m, n ∈ N, as explained in Sect. 5.17, p. 248. In contrast, Einstein bi-gyroaddition ⊕E,(m,n),c in (6.197) is a generalization of the special relativistic Einstein addition from the special relativistic signature (1, n) to signature (m, n), for all m, n ∈ N. Result (6.199) of Theorem 6.26 asserts that Einstein bi-gyrosum (6.197) of equicolumn elements of Rn×m is given by the special relativistic Einstein gyrosum (6.198) c of elements of Rnc/ √m . Accordingly, Theorem 6.26 provides a link between Einstein bi-gyroaddition ⊕E,(m,n),c in (6.197) of any signature (m, n), m, n ≥ 1, and the special relativistic Einstein gyroaddition ⊕E,(1,n),c/ √m of signature (1, n), n ≥ 1. To see clearly the link that Theorem 6.26 provides we note that in this theorem (1) U ∈ Rn×m is considered as m copies of u ∈ Rnc/ √m ; c is considered as m copies of v ∈ Rnc/ √m ; and, consequently, (2) V ∈ Rn×m c (3) the Einstein bi-gyrosum of signature (m, n), U⊕E,(m,n),c V ∈ Rn×m c , can be considered as m copies of the special relativistic Einstein gyrosum of signature (1, n), u⊕E,(1,n),c/ √m v ∈ Rnc/ √m . Graphically, a presentation of a point coincides with the presentation of m > 1 copies of the point. Accordingly, the presentation of m copies of the gyropoint v ∈ Rnc/ √m , that is, the presentation of the bi-gyropoint V = (v v . . . v) ∈ Rn×m c , coincides with the presentation of the gyropoint v. Hence, in Theorem 6.26 (1) the graphical presentation of the bi-gyropoints U, V, W ∈ Rn×m c coincides with

Applications to Time-space of Signature (m,n)

(2) the graphical presentation of the gyropoints u, v, w ∈ Rnc/ √m . A related graphical presentation with n = 2 (that is, plane geometry) and m = 3 is shown in Fig. 7.1, p. 364. The link between Einstein addition of signature (m, n), m, n ∈ N, and the special relativistic Einstein addition, which is of signature (1, n), is thus clear. Example 6.27. In this example, a special case of Theorem 6.26, suitable for a two-dimensional graphical illustration, is presented. Let ⎛ ⎞ ⎜⎜⎜a1 a1 a1 ⎟⎟⎟ √ A = (a a a) = ⎜⎜⎝ ⎟⎟⎠ ∈ R2×3 3c a2 a2 a2 (6.200) ⎛ ⎞ ⎜⎜⎜b1 b1 b1 ⎟⎟⎟ ⎟⎟⎠ ∈ R2×3 √ B = (b b b) = ⎜⎜⎝ 3c b2 b2 b2 be two equi-column matrices generated, respectively, by the columns ⎛ ⎞ ⎜⎜a1 ⎟⎟ = R2c a = ⎜⎜⎜⎝ ⎟⎟⎟⎠ ∈ R2×1 c a2 (6.201)

⎛ ⎞ ⎜⎜b1 ⎟⎟ b = ⎜⎜⎜⎝ ⎟⎟⎟⎠ ∈ R2×1 = R2c . c b2 Then, by Theorem 6.26 with n = 2 and m = 3, A⊕E,(3,2), √3c B = (a⊕E,(1,2),c b a⊕E,(1,2),c b a⊕E,(1,2),c b) .

(6.202)

It is important to note that ⊕E,(1,2),c is the special relativistic Einstein addition in the disc R2c , presented in (2.2), p. 10, with n = 2. Clearly, no confusion may arise if we write (6.202) in the short notation A⊕E B = (a⊕E b a⊕E b a⊕E b) ,

(6.203)

√ , and for gywhere we use the same notation, ⊕E , for bi-gyroadding matrices in R2×3 3c

= R2c . roadding matrices in R2×1 c The use of the short notation in (6.203) is fully analogous to the commonly used notation in (6.204). In the limit where c → ∞ (6.203) tends to its Euclidean counterpart A + B = (a + b a + b a + b)

(6.204)

where A, B ∈ R2×3 , a, b ∈ R2 . As in (6.203), we commonly use the same notation, +, for adding matrices in R2×3 and for adding matrices in R2 = R2×1 , as shown in (6.204).

349

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

6.15. Linking Einstein Scalar Multiplication of Signature (m,n) with the Special Relativistic Einstein Scalar Multiplication Theorem 6.28 is a special case of Theorem 6.25 in which equi-block matrices are specialized to equi-column matrices. As such, this theorem provides a link between Einstein scalar multiplication of signature (m,n), m, n ∈ N, and the special relativistic Einstein scalar multiplication, which is of signature (1, n). Theorem 6.28. Let √ V = (v v . . . v) ∈ Rn×m mc

(6.205)

√ generated by m copies of the column v ∈ Rn×1 = Rnc . be an equi-column matrix in Rn×m c mc Furthermore, let √ W = r⊗V ∈ Rn×m mc

(6.206)

√ , and let be the scalar bi-gyromultiplication of r ∈ R and V of signature (m, n) in Rn×m mc

= Rnc w = r⊗v ∈ Rn×1 c

(6.207)

be the scalar gyromultiplication of r ∈ R and v obtained by the special relativistic Einstein scalar gyromultiplication of signature (1, n) in Rnc . Then, W is the equi-column matrix generated by m copies of the column w, √ . W = (w w . . . w) ∈ Rn×m mc

(6.208)

Proof. This theorem is a special case of Theorem 6.25, obtained by selecting k = 1.  Being of signature (1, n), Einstein scalar gyromultiplication ⊗ in (6.207) coincides with the special relativistic Einstein scalar gyromultiplication in the ball Rnc/ √m , m, n ∈ N, as explained in Example 5.90, p. 277. In contrast, Einstein scalar bi-gyromultiplication ⊗ in (6.206) is a generalization of the special relativistic Einstein scalar gyromultiplication from the special relativistic signature (1, n) to signature (m, n), for all m, n ∈ N. Result (6.208) of Theorem 6.28 asserts that Einstein scalar bi-gyromultiplication (6.206) of equi-column elements of Rn×m is given by the special relativistic Einstein c scalar gyromultiplication (6.207) of elements of Rnc/ √m . Accordingly, Theorem 6.28 provides a link between Einstein scalar bi-gyromultiplication ⊗ in (6.206) of any signature (m, n), m, n ≥ 1, and the special relativistic Einstein scalar gyromultiplication ⊗ of signature (1, n), n ≥ 1. To see clearly the link that Theorem 6.28 provides we note that in this theorem

Applications to Time-space of Signature (m,n)

(1) V ∈ Rn×m is considered as m copies of v ∈ Rnc/ √m and, consequently, c (2) the Einstein scalar bi-gyromultiplication of signature (m, n), r⊗V ∈ Rn×m c , can be considered as m copies of the special relativistic Einstein scalar gyromultiplication of signature (1, n), r⊗v ∈ Rnc/ √m . Graphically, a presentation of a point coincides with the presentation of m > 1 copies of the point. Accordingly, the presentation of m copies of the gyropoint v ∈ Rnc/ √m , that is, the presentation of the bi-gyropoint V = (v v . . . v) ∈ Rn×m c , coincides with the presentation of the gyropoint v. Hence, (1) the graphical presentation of the bi-gyropoints V, W ∈ Rn×m in Theorem 6.26 coc incides with (2) the graphical presentation of the gyropoints v, w ∈ Rnc/ √m . A related graphical presentation with n = 2 (that is, plane geometry) and m = 3 is shown in Fig. 7.1, p. 364. The link between Einstein scalar multiplication of signature (m, n), m, n ∈ N, and the special relativistic Einstein scalar multiplication, which is of signature (1, n), is thus clear. Example 6.29. In this example, a special case of Theorem 6.28, suitable for a two-dimensional graphical illustration, is presented. Let ⎛ ⎞ ⎜⎜⎜v1 v1 v1 ⎟⎟⎟ ⎜ ⎟⎟⎠ ∈ R2×3 √ V = (v v v) = ⎜⎝ (6.209) 3c v2 v2 v2 be an equi-column matrix generated by the column ⎛ ⎞ ⎜⎜v1 ⎟⎟ v = ⎜⎜⎜⎝ ⎟⎟⎟⎠ ∈ R2c . v2

(6.210)

Then, by Theorem 6.28 with n = 2 and m = 3, r⊗V = (r⊗v r⊗v r⊗v) .

(6.211)

Example 6.30. Combining Result (6.203) of Example 6.27 and Result (6.211) of Example 6.29 yields the following identity, A⊕E ( E A⊕E B)⊗t = (a⊕E ( E a⊕E b)⊗t a⊕E ( E a⊕E b)⊗t a⊕E ( E a⊕E b)⊗t) ,

(6.212)

t ∈ R, where A and B are equi-column, given by (6.200). It is implicit in (6.212) that √ on the left-hand side of (6.212), and that they are ⊕E and ⊗ are operations in R2×3 3c

operations in R2c on the right-hand side of (6.212).

351

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6.16. Natural Sub-bi-gyrovector Spaces n×m For any m, n, p, q ∈ N and c > 0, let V ∈ Rn×m → R(n+p)×(m+q) be c c , and let ψ : Rc given by the equation ⎛ ⎞ ⎜⎜⎜ V 0n,q ⎟⎟⎟ ⎟⎟⎠ ∈ R(n+p)×(m+q) . (6.213) ψ(V) = ⎜⎜⎝ 0 p,m 0 p,q

Lemma 6.31. For any m, n, p, q ∈ N and c > 0, V ∈ Rn×m c

⇐⇒

ψ(V) ∈ R(n+p)×(m+q) . c

(6.214)

Proof. The matrices V and ψ(V) have the same nonzero singular values, and hence the result.  Lemma 6.32. For any m, n, p, q ∈ N and c > 0, let V1 , V2 , V ∈ Rn×m c , and let r ∈ R. Then, ψ(V1 ⊕E,(m,n),c V2 ) = ψ(V1 )⊕E,(m+q,n+p),c ψ(V2 ) ⎛ ⎞ ⎜⎜⎜lgyr[V1 , V2 ] 0n,p ⎟⎟⎟ ⎜ ⎟⎟⎠ ∈ SO(n + p) lgyr[ψ(V1 ), ψ(V2 )] = ⎜⎝ 0 p,n Ip ⎞ ⎛ ⎜⎜⎜rgyr[V1 , V2 ] 0m,q ⎟⎟⎟ rgyr[ψ(V1 ), ψ(V2 )] = ⎜⎜⎝ ⎟⎟⎠ ∈ SO(m + q) 0q,m Iq

(6.215)

ψ(r⊗V) = r⊗ψ(V)

(6.216)

ψ(V1 E,(m,n),c V2 ) = ψ(V1 ) E,(m+q,n+p),c ψ(V2 ) .

(6.217)

and

Proof. The proof of (6.215) follows straightforwardly from (6.54), p. 317, and the proof of (6.216) follows straightforwardly from (5.403), p. 264. The proof of (6.217) follows straightforwardly from the definition of E in Def. 5.76 in terms of ⊕E and gyr = (lgyr, rgyr) and from (6.215).  with ψ(V) ∈ R(n+p)×(m+q) the set Theorem 6.33. Identifying V ∈ Rn×m c c (n+p)×(m+q) n×m {ψ(V) : V ∈ Rc } ⊂ Rc is a sub-bi-gyrovector space (Rn×m c , ⊕E , ⊗) of signature (m, n) of the bi-gyrovector space (Rm+q,n+p , ⊕ , ⊗) of signature (m + q, n + p). c E Proof. The proof follows immediately from Lemma 6.31 and Lemma 6.32.



Applications to Time-space of Signature (m,n)

6.17. Bi-gyrohyperbolic Matrix Functions In this section we show that our study of the group SO(m, n), m, n ∈ N, enables the hyperbolic functions cosh and sinh to be extended to what we call bi-gyrohyperbolic matrix functions. The latter are related to bi-boosts in the same way that the former are related to boosts. The right bi-gamma identity (6.126) and the left bi-gamma identity (6.128) suggest the definition of the four bi-gyrohyperbolic matrix functions of signature (m, n), m, n ∈ N, in the following definition. Definition 6.34. (Bi-gyrohyperbolic Functions of Signature (m,n)). Let m, n ∈ N. For all A, B ∈ Rn×m c , 1. the left bi-gyrocosh function of signature (m, n), lcosh(A, B) : Rn×m × Rn×m → c c n×n R , is given by lcosh(A, B) = ΓL⊖ A⊕ B lgyr[⊖E A, B] ∈ Rn×n , E

E

(6.218)

2. the left bi-gyrosinh function of signature (m, n), lsinh(A, B) : Rn×m × Rn×m → c c m×n R , is given by lsinh(A, B) = c−1 (⊖E A⊕E B)t ΓL⊖ A⊕ B lgyr[⊖E A, B] ∈ Rm×n , E

E

(6.219)

3. the right bi-gyrocosh function of signature (m, n), rcosh(A, B) : Rn×m × Rn×m → c c m×m R , is given by rcosh(A, B) = ΓR⊖ A⊕ B rgyr[⊖E A, B] ∈ Rm×m , E

E

(6.220)

and 4. the right bi-gyrosinh function of signature (m, n), rsinh(A, B) : Rn×m × Rn×m → c c n×m R , is given by rsinh(A, B) = c−1 (⊖E A⊕E B)ΓR⊖ A⊕ B rgyr[⊖E A, B] ∈ Rn×m . E

E

(6.221)

In terms of the bi-gyrohyperbolic functions of signature (m, n) in Def. 6.34, the left bi-gamma identity (6.128) can be written as lcosh(A, B)t lcosh(A, C) − lsinh(A, B)t lsinh(A, C) = lcosh(B, C) ,

(6.222)

for all A, B, C ∈ Rn×m c , and the right bi-gamma identity (6.126) can be written as rcosh(A, B)trcosh(A, C) − rsinh(A, B)t rsinh(A, C) = rcosh(B, C) ,

(6.223)

for all A, B, C ∈ Rn×m c . Clearly, (6.223) is the right counterpart of (6.222). Furthermore, Identity (6.99), p. 326, with V = ⊖E A⊕E B and Om = rgyr[⊖E A, B], can be written

353

354

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

as rcosh(A, B)trcosh(A, B) − rsinh(A, B)t rsinh(A, B) = Im ,

(6.224)

for all A, B ∈ Rn×m c . Similarly, one can establish the left counterpart of (6.224), lcosh(A, B)t lcosh(A, B) − lsinh(A, B)t lsinh(A, B) = In ,

(6.225)

for all A, B ∈ Rn×m c . Obviously, lcosh(⊖E A, ⊖E B) = lcosh(A, B) rcosh(⊖E A, ⊖E B) = rcosh(A, B) (6.226) lsinh(⊖E A, ⊖E B) = −lsinh(A, B) rsinh(⊖E A, ⊖E B) = −rsinh(A, B) , for all A, B ∈ Rn×m c , noting that ⊖E A = −A. Example 6.35. In this example we show that left and right bi-gyrohyperbolic functions of signature (m, n), m, n ∈ N, descend to the common hyperbolic functions cosh and sinh when m = n = 1. In the special case when the signature is (m, n) = (1, 1), we have A, B ∈ Rc := (−c, c), and all bi-gyrations are trivial, being elements of SO(1) = {1}, lgyr[⊖E A, B] = rgyr[⊖E A, B] = 1 .

(6.227)

Here, Einstein addition ⊕E in Rc is the Einstein addition of signature (1, 1), given by A⊕E B =

A+B , 1 + c−2 AB

(6.228)

for all A, B ∈ Rc . Also, in this special case, bi-gamma factors descend to gamma factors, Γ⊖LE A⊕E B = ΓR⊖E A⊕E B = γ⊖E A⊕E B ,

(6.229)

m = n = 1, as shown in Example 5.28, p. 210. Hence, it follows from Def. 6.34 and from (6.222) – (6.225) that in this special case of signature (1, 1), lcosh(A, B) = rcosh(A, B) = γ⊖ A⊕ B = cosh φ⊖E A⊕E B E

E

lsinh(A, B) = rsinh(A, B) = c−1 (⊖E A⊕E B)γ⊖ A⊕ B = sinh φ⊖E A⊕E B E

E

cosh2 φ⊖E A⊕E B − sinh2 φ⊖E A⊕E B = 1 ,

(6.230)

Applications to Time-space of Signature (m,n)

where ⊖E A⊕E B (6.231) c is known in special relativity theory as the rapidity associated with the one-dimensional velocity ⊖E A⊕E B ∈ Rc . Accordingly, the Lorentz transformation Λ ∈ SO(1, 1) ⊂ R2×2 of signature (1, 1) is what is known as a hyperbolic rotation, ! cosh φ⊖E A⊕E B sinh φ⊖E A⊕E B Λ= ∈ SO(1, 1) , (6.232) sinh φ⊖E A⊕E B cosh φ⊖E A⊕E B φ⊖E A⊕E B = tanh−1

A, B ∈ Rc . Rapidities are commonly used in special relativity theory as a measure for relativistic velocities for one-dimensional motion. Our study thus enables hyperbolic rotations in SO(1, 1) to be extended to hyperbolic rotations in SO(m, n) for all m, n > 1. Identities (6.224) – (6.225) can be recast as the matrix identity, ! ! rcosh(A, B)t −rsinh(A, B)t rcosh(A, B) lsinh(A, B) = Im+n , −lsinh(A, B)t lcosh(A, B)t rsinh(A, B) lcosh(A, B)

(6.233)

so that ! !−1 rcosh(A, B)t −rsinh(A, B)t rcosh(A, B) lsinh(A, B) = −lsinh(A, B)t lcosh(A, B)t rsinh(A, B) lcosh(A, B)

(6.234)

in R(m+n)×(m+n) , for all A, B ∈ Rn×m c . Identities (6.222) – (6.223) suggest the matrix identity ! ! rcosh(A, B)t −rsinh(A, B)t rcosh(A, C) lsinh(A, C) −lsinh(A, B)t lcosh(A, B)t rsinh(A, C) lcosh(A, C) (6.235) ! rcosh(B, C) lsinh(B, C) = ∈ R(m+n)×(m+n) , rsinh(B, C) lcosh(B, C) for all A, B, C ∈ Rn×m c . Finally, (6.234) – (6.235) yield the elegant product rule ! ! rcosh(A, B) lsinh(A, B) rcosh(B, C) lsinh(B, C) rsinh(A, B) lcosh(A, B) rsinh(B, C) lcosh(B, C) (6.236) ! rcosh(A, C) lsinh(A, C) = ∈ R(m+n)×(m+n) , rsinh(A, C) lcosh(A, C) for all A, B, C ∈ Rn×m c .

355

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

We call the matrix ! rcosh(A, B) lsinh(A, B) H(A, B) := ∈ R(m+n)×(m+n) , rsinh(A, B) lcosh(A, B)

(6.237)

A, B ∈ Rn×m c , the bi-gyrohyperbolic matrix of signature (m, n), m, n ∈ N, generated by A and B. Following (6.236), bi-gyrohyperbolic matrices of signature (m, n) possess the product rule H(A, B)H(B, C) = H(A, C)

(6.238)

for all A, B, C ∈ Rn×m c . Moreover, bi-gyrohyperbolic matrices have determinant 1, possessing the following properties, H(A, B)−1 = H(B, A) H(A, B)−1 = H(⊖E A, ⊖E B)t H(A, B) = H(A, 0n,m )H(0n,m , B) H(A, A) = Im+n det(H(A, B)) = 1 ,

(6.239)

for all A, B ∈ Rn×m c .

6.18. Bi-gyrohyperbolic Bi-boosts We now explore the impact of the bi-gyrohyperbolic matrix product rule (6.238) on the bi-boost. Following (6.237), (6.218) – (6.221), and (5.127), p. 204, and noting the first two commuting relations in (5.119), p. 202, we have the following chain of equations. ! ! ! c−1 Im 0m,n cIm 0m,n rcosh(A, B) c−1 lsinh(A, B) H(A, B) = 0n,m In 0n,m In c rsinh(A, B) lcosh(A, B)    ΓR⊖ A⊕ B rgyr[⊖E A, B] c−2 (⊖E A⊕E B)t Γ⊖L A⊕ B lgyr[⊖E A, B] E E E E   =   R L (⊖E A⊕E B)Γ⊖ A⊕ B rgyr[⊖E A, B] Γ⊖ A⊕ B lgyr[⊖E A, B] E

E

  ΓR⊖ A⊕ B E E   =  (⊖E A⊕E B)ΓR⊖ A⊕ E

E

E

E

(6.240)  !   A⊕E B  0m,n E  rgyr[⊖E A, B]  0n,m lgyr[⊖E A, B]

c−2 (⊖E A⊕E B)t Γ⊖L

B

Γ⊖L

E

A⊕E B

! rgyr[⊖E A, B] 0m,n = Bc (⊖E A⊕E B) ∈ SO(m, n) , 0n,m lgyr[⊖E A, B]

Applications to Time-space of Signature (m,n)

for all A, B ∈ Rn×m c , m, n ∈ N. Here, Bc (⊖E A⊕E B) is the familiar bi-boost (5.127) of signature (m, n) generated by ⊖E A⊕E B ∈ Rn×m c ,    ΓR⊖ A⊕ B c−2 (⊖E A⊕E B)t Γ⊖L A⊕ B  E E E E   . (6.241) Bc (⊖E A⊕E B) =   R L (⊖E A⊕E B)Γ⊖ A⊕ B Γ⊖ A⊕ B E

E

E

E

The matrix product on the extreme right-hand side of (6.240) is called the bigyrohyperbolic bi-boost generated by A, B ∈ Rn×m c . It follows immediately from (6.240) and from the product rule (6.238) of bi-hyperbolic matrices that the bi-gyrohyperbolic boost enjoys the elegant product rule ! rgyr[⊖E A, B] 0m,n Bc (⊖E A⊕E B) 0n,m lgyr[⊖E A, B] ! rgyr[⊖E B, C] 0m,n ×Bc (⊖E B⊕E C) 0n,m lgyr[⊖E B, C] ! rgyr[⊖E A, C] 0m,n =Bc (⊖E A⊕E C) , 0n,m lgyr[⊖E A, C] for all A, B, C ∈ Rn×m c .

(6.242)

357

CHAPTER 7

Analytic Bi-hyperbolic Geometry: The Geometry of Bi-gyrovector Spaces

7.1. Introduction We have seen in Chaps. 2 and 3 that the analytic geometry that regulates Einstein gyrovector spaces is analytic hyperbolic geometry, that is, the hyperbolic geometry of Lobachevsky and Bolyai, studied analytically. In this chapter we will see that, in full analogy, the analytic geometry that regulates Einstein bi-gyrovector spaces is a natural generalization of analytic hyperbolic geometry that, naturally, we call analytic bi-hyperbolic geometry. The concepts of Einstein gyrogroups and gyrovector spaces are presented in Chaps. 2 and 3, along with the resulting analytic hyperbolic geometry. Historically, these concepts emerged from the 1988 study of the parametric realization of the Lorentz group SO(1, n), n ∈ N, in [74]. We therefore expect that a generalization of these concepts, along with a resulting generalized analytic hyperbolic geometry, would emerge from the analogous parametric realization of the Lorentz group SO(m, n) for all m, n ∈ N. We, therefore, studied the Lorentz groups SO(m, n) and their parametric realizations in Chaps. 4 and 5. In Chap. 4 we have studied the parametric realization Λ = Λ(P, On , Om ) ∈ SO(m, n) of each Lorentz group SO(m, n), m, n ∈ N, leading to bi-gyrogroups in Sect. 4.23, and to bi-gyrovector spaces in Sect. 4.24, for the parameter P ∈ Rn×m . The parameter P represents multi-velocities that descend to the proper velocity [90] of special relativity theory when (m, n) = (1, 3). Proper velocities as opposed to coordinate velocities in special relativity are also known as traveler’s velocities, as opposed to observer’s velocities. Observer’s speeds are bounded by the vacuum speed of light c, while traveler’s speeds are unbounded. Special relativity is commonly studied in terms of observer’s velocities, the magnitudes of which are bounded by c. Hence, in Chap. 5 we have changed the parameter P ∈ Rn×m into the parameter V ∈ Rn×m in the c-ball Rn×m of the ambient space Rn×m . c c Following the change of parameter from P ∈ Rn×m to V ∈ Rn×m c , bi-gyrogroups and bigyrovector spaces of c-balls Rn×m , m, n ∈ N, are presented in Sects. 5.15 – 5.21 and in c Beyond Pseudo-rotations in Pseudo-Euclidean Spaces http://dx.doi.org/10.1016/B978-0-12-811773-6.50007-4 Copyright © 2018 Elsevier Inc. All rights reserved.

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Sect. 5.24. We are now in the position to study in this chapter the ball bi-gyrovector spaces of signature (m, n), m, n ∈ N, and their underlying analytic bi-hyperbolic geometry of signature (m, n). For graphical illustrations we select (1) n = 2, resulting in plane analytic bi-hyperbolic geometry and (2) m=3, resulting in bi-gyropoints in R2×3 c , each of which is a system of three subpoints. Each subpoint, in turn, is a gyropoint in R2×1 = R2c . Here the spectral disc c R2×1 and the Euclidean disc R2c coincide, as shown in (5.147), p. 209. c

7.2. Bi-gyropoints in Bi-gyrovector Spaces Any Einstein bi-gyrogroup = (Rn×m Rn×m c c , ⊕E )

(7.1)

of signature (m, n), m, n ∈ N, is a bi-gyrocommutative bi-gyrogroup that admits a scalar multiplication, turning itself into a bi-gyrovector space, Rn×m = (Rn×m c c , ⊕E , ⊗),

(7.2)

of signature (m, n). thus possesses two algebraic structures: (i) the bi-gyrocommutative The ball Rn×m c bi-gyrogroup structure, according to Theorem 5.75, p. 255, and (ii) the bi-gyrovector space structure. For any V ∈ Rn×m , the square matrices VV t ∈ Rn×n and V t V ∈ Rm×m have no negative eigenvalues, and have the same positive eigenvalues. Consequently, they need not have the same number of zero eigenvalues. By the ball definition in (5.68), p. 193, the matrix V ∈ Rn×m is an element of the ball Rn×m if and only if each eigenvalue of VV t (or, equivalently, of V t V) is smaller c 2 than c . Einstein addition ⊕E of signature (m, n), ⊕E = ⊕E,(m,n),c ,

(7.3)

is a binary operation in the ball Rn×m c , given by Def. 6.6, p. 316, and by each of the two equations in (6.54). We use the short notation, ⊕E , when the signature (m, n) and the constant c are obvious from the context. Einstein scalar multiplication ⊗ is the scalar multiplication in Rn×m given by each c of the two equations in (5.453), p. 273. The elements of the ball Rn×m are called bi-gyropoints of signature (m,n) or, for c convenience, (m, n)-bi-gyropoints. In the special case when m = 1 a bi-gyropoint of signature (m = 1, n) descends to a gyropoint in the Euclidean ball Rn×1 = Rnc , studied c in Chap. 3.

Analytic Bi-hyperbolic Geometry: The Geometry of Bi-gyrovector Spaces

Let V ∈ Rn×m be a bi-gyropoint of signature (m, n). c m columns, ⎛ ⎜⎜⎜v11 v12 . . . ⎜⎜⎜v ⎜ 21 v22 . . . V = (v1 v2 . . . vm ) = ⎜⎜⎜⎜⎜ .. ⎜⎜⎜ . ⎝ vn1 vn2 . . .

Then, V is an n × m matrix of ⎞ v1m ⎟⎟ ⎟ v2m ⎟⎟⎟⎟⎟ ⎟⎟⎟ ∈ Rn×m . c ⎟⎟⎟ ⎟⎠ vnm

(7.4)

= Rnc , said to be a Each column vk , k = 1, . . . , m, of V is an n × 1 matrix in Rn×1 c subpoint of the bi-gyropoint V. A bi-gyropoint of order (m, n), lying in the ball Rn×m c , is thus a system of m n-dimensional subpoints, each of which is a gyropoint lying in the Euclidean ball Rn×1 = Rnc according to Theorem 5.15, p. 199. c Graphically, a bi-gyropoint V of order (m, n) = (3, 2) is represented by its m = 3 subpoints v1 , v2 , v3 ∈ R2c ⊂ R2 in the disc R2c of the ambient Euclidean plane R2 . An illustrative example follows. Example 7.1. Let  0.25 −0.12 −0.22 A= 0.35 −0.20 −0.35

 and

B=

−0.20 −0.42 0.40 , 0.25 0.48 −0.15

(7.5)

2×3 A, B ∈ R2×3 c=1 , be two bi-gyropoints of signature (3, 2) in the unit spectral disc Rc=1 . Indeed, the eigenvalues of AAt and At A (of BBt and Bt B) are nonnegative and smaller than c2 = 1. The bi-gyropoint A is represented graphically in Fig. 7.1, p. 364, by its three subpoints a1 , a2 , a3 ∈ R2c=1 , where

a1 = (0.25, 0.35),

a2 = (−0.12, −0.20),

and

a3 = (−0.22, −0.35)

(7.6)

are the three columns of A. Similarly, the bi-gyropoint B is represented graphically in Fig. 7.1 by its three subpoints b1 , b2 , b3 ∈ R2c , where b1 = (−0.20, 0.25),

b2 = (−0.42, 0.48),

and

b3 = (0.40, −0.15)

(7.7)

are the three columns of B. As shown in Fig. 7.1, and as expected from Theorem 5.15, p. 199, the subpoints ak , bk , k = 1, 2, 3, are gyropoints that lie in the unit disc R2c=1 . The three subpoints ak (bk ), k = 1, 2, 3, in Fig. 7.1 represent the single bi-gyropoint A (B) of signature (m, n) = (3, 2). The subpoints are presented in Fig. 7.1 with respect to Cartesian coordinates. In general, however, the Cartesian coordinates that underlie graphical presentations are not shown.

361

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

7.3. Bi-gyrodistance Definition 7.2. (Bi-gyrodistance). For any m, n ∈ N and c > 0, let A, B ∈ Rn×m be c two bi-gyropoints in an Einstein bi-gyrogroup (Rn×m , ⊕ ). The bi-gyrodistance d(A, B) c E between A and B is given by the matrix spectral norm (see Def. 5.7, p. 193), d(A, B) = E A⊕E B .

(7.8)

0 ≤ d(A, B) < c ,

(7.9)

Clearly, since E A⊕E B ∈ Rn×m c ; and d(A, B) = 0 if and only if A = B. Theorem 7.3. (Bi-gyrodistance Invariance Under Left Bi-gyrotranslations). Bigyrodistance in Einstein bi-gyrogroups (Rn×m c , ⊕E ), m, n ∈ N, is invariant under left bi-gyrotranslations, that is, d(A, B) = d(X⊕E A, X⊕E B)

(7.10)

for all A, B, X ∈ (Rn×m c , ⊕E ). = (Rn×m Proof. Let Rn×m c c , ⊕E ) be any Einstein bi-gyrogroup of signature (m, n), m, n ∈ N, and let A, B, X ∈ Rn×m c . Then, by the Left Bi-gyrotranslation Theorem 5.78, p. 256, along with the definition of the bi-gyrator gyr = (lgyr, rgyr) in Def. 5.67, p. 250, we have E (X⊕E A)⊕E (X⊕E B) = gyr[X, A](E A⊕E B) = lgyr[X, A](E A⊕E B)rgyr[A, X]

(7.11)

= On (E A⊕E B)Om where On = lgyr[X, A] ∈ SO(n) Om = rgyr[A, X] ∈ SO(m) .

(7.12)

Hence, by means of Def. 7.2, (7.11), and Theorem 5.8, p. 194, asserting that the norm is orthogonally invariant, we have d(X⊕E A, X⊕E B) = E (X⊕E A)⊕E (X⊕E B) = On (E A⊕E B)Om  = E A⊕E B = d(A, B) , as desired.

(7.13) 

Analytic Bi-hyperbolic Geometry: The Geometry of Bi-gyrovector Spaces

Theorem 7.4. (Bi-gyrodistance Invariance Under Bi-rotations). Bi-gyrodistance in Einstein bi-gyrogroups (Rn×m c , ⊕E ), m, n ∈ N, is invariant under bi-rotations, that is, d(A, B) = d(On AOm , On BOm )

(7.14)

for any A, B ∈ (Rn×m c , ⊕E ), and any On ∈ SO(n) and Om ∈ SO(m). Proof. By means of Def. 7.2 and Theorem 5.60, p. 242, noting that ⊕ = ⊕E by Def. 6.6, p. 316, and by means of Theorem 5.8, p. 194, asserting that the norm is orthogonally invariant, we have d(On AOm , On BOm ) = E (On AOm )⊕E (On BOm ) = On (E A⊕E B)Om  = E A⊕E B

(7.15)

= d(A, B) , 

as desired.

Theorem 7.5. (Bi-gyrodistance Symmetry). Bi-gyrodistance in Einstein bi-gyrogroups (Rn×m , ⊕ ), m, n ∈ N, is symmetric, that is, c E d(A, B) = d(B, A)

(7.16)

for all A, B ∈ (Rn×m c , ⊕E ). Proof. By means of Def. 7.2, the bi-gyrocommutative law (5.534), p. 291, the norm invariance (5.73), p. 194, and (5.71), p. 193, we have d(A, B) = E A⊕E B = lgyr[E A, B](BE A)rgyr[B, E A] = BE A

(7.17)

= E B⊕E A = d(B, A) , as desired.



7.4. Bi-gyrolines in Bi-gyrovector Spaces Gyrolines in gyrovector spaces, studied in Sect. 3.6, are extended here to bi-gyrolines in bi-gyrovector spaces.

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1

0.8

0.6

b2 0.4

x1 a1

b1 x2

0.2

0

b3

a2

−0.2

x3

a3

−0.4

−0.6

E A⊕E X⊕E E X⊕E B = E A⊕E B

−0.8

d(A, X)⊕E d(X, B) = d(A, B)

−1

−1

−0.5

0

0.5

1

Figure 7.1 The (3, 2)-bi-gyroline LAB that passes through the bi-gyropoints A and B. Two bi-gyropoints of signature (3,2), A, B ∈ (R2×3 c=1 , ⊕E , ⊗), given by (7.5), are represented graphically by their subpoints, given by (7.6) and (7.7). By Theorem 5.15, p. 199, each of the subpoints is a gyropoint in the disc R2c=1 . The bi-gyropoint A is the system of the m = 3 subpoints a1 , a2 , a3 and, similarly, the bi-gyropoint B is the system of the m = 3 subpoints b1 , b2 , b3 . The bi-gyrosegment AB that joins A and B is shown as well. In the same way that a bi-gyropoint of signature (m, n) = (3, 2) is represented graphically by its m = 3 subpoints, the bi-gyrosegment of signature (m, n) = (3, 2) that joins A and B is represented graphically by its m = 3 subsegments (which are in geometric entanglement) that join respective subpoints, as shown here. Also an arbitrary bi-gyropoint X ∈ R2×3 c=1 that lies between A and B is shown, being represented by it subpoints x1 , x2 , and x3 .

Definition 7.6. (Bi-gyrolines and bi-gyrosegments). For any m, n ∈ N and c > 0, let A, B ∈ Rn×m be two bi-gyropoints in a bi-gyrovector space Rn×m = (Rn×m c c c , ⊕E , ⊗). The bi-gyroline LAB = LAB (t) of signature (m, n), m, n ∈ N, which passes through A and B is the set of bi-gyropoints , LAB (t) = A⊕E (E A⊕E B)⊗t ⊂ Rn×m c

(7.18)

parametrized by t ∈ R. The subset of bi-gyroline LAB parametrized by 0 ≤ t ≤ 1 is a bi-gyrosegment of signature (m, n), denoted by AB. A bi-gyrosegment AB of signature (3, 2) is shown in Fig. 7.1. The bi-gyroline LAB (t) passes through the bi-gyropoint A when t = 0 and, owing to the left cancellation law (5.367), p. 256, it passes through the bi-gyropoint B when t = 1. Furthermore, we will see in Sect. 7.7 that it passes through the bi-gyromidpoint MAB of A and B when t = 1/2.

Analytic Bi-hyperbolic Geometry: The Geometry of Bi-gyrovector Spaces

For graphical presentations we select the signature (m, n) = (3, 2). Thus, for instance, A = (a1 , a2 , a3 ) ∈ (R2×3 c=1 , ⊕E , ⊗)

(7.19)

is a (3, 2)-bi-gyropoint represented by its three constituents a1 , a2 , a3 ∈ R2c=1 . Each 2 of the constituents of A ∈ R2×3 c=1 is a gyropoint in the disc Rc=1 . Indeed, according to 2×3 Theorem 5.15, p. 199, the constituents of A ∈ Rc=1 lie in the disc R2c=1 , as shown in Fig. 7.1. Similarly, the three constituents of the (3, 2)-bi-gyropoint B, B = (b1 , b2 , b3 ) ∈ (R2×3 c=1 , ⊕E , ⊗) ,

(7.20)

are the three gyropoints b1 , b2 , b3 ∈ R2c=1 , also shown in Fig. 7.1. Let X ∈ LAB ⊂ R2×3 c=1 be a bi-gyropoint on a bi-gyroline LAB . Then, X is represented by its three constituents x1 , x2 , x3 ∈ R2c=1 , X = (x1 , x2 , x3 ) = A⊕E (E A⊕E B)⊗t0 ∈ (R2×3 c=1 , ⊕E , ⊗) ,

(7.21)

for some t0 ∈ R, as shown in Fig. 7.1. If 0 < t0 < 1, then X lies between A and B, as shown in Fig. 7.1, giving rise to the bi-gyrotriangle equality. Betweenness and the bi-gyrotriangle equality will be studied in Sects. 7.6 and 7.7. Accordingly, a bi-gyroline LAB ⊂ R2×3 c=1 is represented graphically by its three constituents, which are the three gyrolines in the disc R2c=1 shown in Fig. 7.1. The three constituent sub-bi-gyrolines of the (3, 2)-bi-gyroline LAB in Fig. 7.1 are curved since they are in entanglement. The amount of curving indicates the strength of the entanglement. Accordingly, the three coincident constituent sub-bi-gyrolines of the (3, 2)-bi-gyroline LAB in Fig. 7.2 are straight, since they are not in entanglement.

7.5. Reduction of Bi-gyrolines to Gyrolines Gyrolines in special relativity, studied and shown graphically in Chap. 3, are Euclidean segments in the ball. In contrast, in general, bi-gyrolines of signature (m, n), m, n > 1, are curved owing to geometric entanglement, as shown in Fig. 7.1. However, bi-gyrolines that are determined by equi-column bi-gyropoints are not in geometric entanglement and, as such, they are not curved. Rather, they descend to special relativistic gyrolines, as explained below and illustrated in Fig. 7.2. Let A and B be two equi-column bi-gyropoints, √ A = (a a a) ∈ R2×3 3c √ , B = (b b b) ∈ R2×3 3c

(7.22)

= R2c . Here A (B) is a repeated bi-gyropoint with multiplicity 3. where a, b ∈ R2×1 c

365

366

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

1

0.8

0.6

0.4

0.2

a

0

b

−0.2

−0.4

−0.6

−0.8

−1

−1

−0.5

0

0.5

1

√ , ⊕ , ⊗) are equi-column, given by (7.24). Bi-gyropoint Figure 7.2 The (3, 2)-bi-gyropoints A, B ∈ (R2×3 E 3c A (resp. B) is represented by a repeated gyropoint a (resp. b) of multiplicity 3, given by (7.25). Accordingly, the bi-gyroline that passes through A and B is a repeated gyroline of multiplicity 3. The resulting gyroline is shown. As expected, the resulting gyroline is a gyroline in the special relativistic 2 model of hyperbolic geometry, that is, a Euclidean segment in the disc R2×1 c=1 = Rc=1 .

Then, by (6.212), p. 351, the bi-gyroline LAB that passes through A and B is repeated with multiplicity m = 3, that is, LAB = A⊕E (E A⊕E B)⊗t √ , = (a⊕E (E a⊕E b)⊗t a⊕E (E a⊕E b)⊗t a⊕E (E a⊕E b)⊗t) ∈ R2×3 3c

(7.23)

t ∈ R. √ . In Einstein addition, ⊕E , on the left equation in (7.23) is a binary operation in R2×3 3c contrast, Einstein addition, ⊕E , on the extreme right-hand side of (7.23) is the special relativistic Einstein addition (2.2), p. 10, in R2c . Similarly, the scalar multiplication, ⊗, on the left equation in (7.23) is a scalar √ . In contrast, the scalar multiplication, ⊗, on the extreme rightmultiplication in R2×3 3c hand side of (7.23) is the special relativistic Einstein scalar multiplication, studied in Sect. 3.2. The three constituents of the bi-gyroline LAB in (7.23) coincide with each other since A and B are equi-column. Being equi-column, each of A and B is a repeated gyropoint with multiplicity 3. Hence, while the bi-gyroline in Fig. 7.1 is represented graphically by three distinct curved gyrolines, the bi-gyroline LAB in (7.23) is

Analytic Bi-hyperbolic Geometry: The Geometry of Bi-gyrovector Spaces

represented graphically by a single, repeated straight gyroline of multiplicity 3, as shown in Fig. 7.2. Moreover, this single, repeated gyroline of multiplicity 3 is generated by the special relativistic Einstein addition and scalar multiplication. Hence, as shown in Fig. 7.2, the repeated bi-gyroline LAB in (7.23) coincides with a gyroline in the special relativistic model of hyperbolic geometry, that is, it is a Euclidean segment in the disc. A numerical example illustrated by Fig. 7.2 follows. Example 7.7. Let √ A = (a a a) ∈ R2×3 3c √ B = (b b b) ∈ R2×3 3c

(7.24)

(c = 1) be equi-column (3,2)-bi-gyropoints, where 2 a = (0.70 0.15)t ∈ R2×1 c=1 = Rc=1 2 b = (0.30 0.20)t ∈ R2×1 c=1 = Rc=1 .

(7.25)

Then, the bi-gyropoint A (resp. B) is the repeated gyropoint a (resp. b) of multiplicity 3. Accordingly, the bi-gyroline that passes through A and B, given by (7.23), is a repeated gyroline of multiplicity 3. The resulting gyroline is shown in Fig. 7.2. As expected from Example 6.30, p. 351, it turns out to be the special relativistic gyroline that passes through a and b, studied in Chap. 3 and shown in the figures of Chap. 3. Accordingly, the bi-hyperbolic geometry that underlies the Einstein bi-gyrovector space (Rn×m c , ⊕E , ⊗) of signature (m, n), m, n > 1, descends to the hyperbolic geometry that underlies the Einstein gyrovector space (Rnc , ⊕E , ⊗) in the special case when the involved bi-gyropoints of Rn×m are equi-column. Thus, the hyperbolic geometry of c Lobachevsky and Bolyai is a special case of bi-hyperbolic geometry.

7.6. Betweenness Definition 7.8. (Betweenness). A bi-gyropoint A2 lies between bi-gyropoints A1 and n×m A3 in a bi-gyrovector space (Rn×m are c , ⊕E , ⊗) (i) if the bi-gyropoints A1 , A2 , A3 ∈ Rc bi-gyrocollinear (that is, they lie on the same bi-gyroline), that is, they are related by the equations Ak = A⊕E (E A⊕E B)⊗tk ,

(7.26)

k = 1, 2, 3, for some A, B ∈ Rn×m c , A  B, and some tk ∈ R and (ii) if, in addition, either t1 < t2 < t3 or t3 < t2 < t1 .

367

368

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Lemma 7.9. In a bi-gyrovector space (Rn×m c , ⊕E , ⊗), three distinct bi-gyropoints A1 ,A2 , and A3 are bi-gyrocollinear if and only if any one of these bi-gyropoints, say A2 , can be expressed in terms of the two other bi-gyropoints by the equation A2 = A1 ⊕E (E A1 ⊕E A3 )⊗t

(7.27)

for some t ∈ R. Proof. If bi-gyropoints A1 , A2 , A3 are bi-gyrocollinear, then there exist bi-gyropoints A, B ∈ Rn×m c , A  B, and distinct real numbers tk , k = 1, 2, 3, such that Ak = A⊕E (E A⊕E B)⊗tk .

(7.28)

Let t2 − t1 (7.29) t3 − t 1 and let us consider the following chain of equations, which are numbered for subsequent explanation. Temporarily, in the following chain of equations we use the notation ⊕ = ⊕E and  = E . t=

(1)

 A1 ⊕(A1 ⊕A3 )⊗t === [A⊕(A⊕B)⊗t1 ]⊕{[A⊕(A⊕B)⊗t1 ]⊕[A⊕(A⊕B)⊗t3 ]}⊗t (2)

 === [A⊕(A⊕B)⊗t1 ]⊕gyr[A, (A⊕B)⊗t1 ]{(A⊕B)⊗t1 ⊕(A⊕B)⊗t3 }⊗t (3)

 === [A⊕(A⊕B)⊗t1 ]⊕gyr[A, (A⊕B)⊗t1 ](A⊕B)⊗((−t1 + t3 )t) (4)

 === A⊕{(A⊕B)⊗t1 ⊕(A⊕B)⊗((−t1 + t3 )t)} (5)

 === A⊕(A⊕B)⊗(t1 + (−t1 + t3 )t) (6)

 === A⊕(A⊕B)⊗t2 (7)

 === A2 . (7.30) Derivation of the numbered equalities in (7.30): (1) Follows from (7.28) (2) Follows from Item (1) by the Left Bi-gyrotranslation Theorem 5.78, p. 256. (3) Follows from Item (2) by the monodistributive law (5.537), p. 292, the scalar associative law (5.536), and the scalar distributive law (5.535). (4) Item (4) is equivalent to Item (3) by the left bi-gyroassociative law (5.533), p. 291.

Analytic Bi-hyperbolic Geometry: The Geometry of Bi-gyrovector Spaces

(5) Follows from Item (4) by the scalar distributive law. (6) Follows from (7.29). (7) Follows from (7.28). The chain of equations (7.30) verifies (7.27). Conversely, if A2 is given by (7.27), then the three bi-gyropoints A1 , A2 , A3 are bi-gyrocollinear, since A2 lies on the bi-gyroline that passes through the other two bi-gyropoints A1 and A3 .  Lemma 7.10. A bi-gyropoint A2 lies between bi-gyropoints A1 and A3 in a bi-gyrovector space (Rn×m c , ⊕E , ⊗) if and only if A2 = A1 ⊕E (E A1 ⊕E A3 )⊗t

(7.31)

for some 0 < t < 1. Proof. If A2 lies between A1 and A3 , then bi-gyropoints A1 , A2 , A3 are bi-gyrocollinear, by Def. 7.8. Hence, there exist distinct bi-gyropoints A, B ∈ Rn×m and real numbers tk , c k = 1, 2, 3, such that Ak = A⊕E (E A⊕E B)⊗tk ,

(7.32)

and either t1 < t2 < t3 or t3 < t2 < t1 . Let t2 − t1 . (7.33) t3 − t 1 Then, 0 < t < 1, and by means of the result of the chain of equations (7.30) we have t=

A1 ⊕E (E A1 ⊕E A2 )⊗t = A2 ,

(7.34)

thus verifying (7.31) for 0 < t < 1. Conversely, if (7.31) holds then, by Def. 7.8 with t1 = 0, t2 = t, and t3 = 1, A2 lies between A1 and A3 . 

7.7. The Bi-gyrotriangle Equality The following lemma proves useful in establishing the bi-gyrotriangle equality in theorem 7.12. Lemma 7.11. A⊕E (E A⊕E B)⊗t = B⊕E (E B⊕E A)⊗(1 − t) for any t ∈ R and A, B ∈ (Rn×m c , ⊕E , ⊗), m, n ∈ N, c > 0.

(7.35)

369

370

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Proof. The proof is given by the following chain of equations, which are numbered for subsequent explanation. Temporarily, in the following chain of equations we use the notation ⊕ = ⊕E and  = E . (1)

 B⊕(B⊕A)⊗(1 − t) === B⊕{(B⊕A)(B⊕A)⊗t} (2)

 === {B⊕(B⊕A)}gyr[B, B⊕A]{(B⊕A)⊗t} (3)

 === Agyr[A, B]{(B⊕A)⊗t}

(7.36)

(4)

 === A{gyr[A, B](B⊕A)}⊗t (5)

 === A{AB}⊗t . Derivation of the numbered equalities in (7.36): (1) This equation results from the scalar distributive law (5.535), p. 291. (2) Follows from Item (1) by the left bi-gyroassociative law (5.533), p. 291. (3) Follows from Item (2) by a left cancellation, and by the gyration identity (5.532), p. 291. (4) Follows from Item (3) by the third identity in (5.543), p. 292. (5) Follows from Item (4) by the bi-gyrocommutative law (5.534), p. 291.  The bi-gyrotriangle inequality, studied in Sect. 5.37, is augmented in the following theorem, giving rise to the bi-gyrotriangle equality, illustrated in Fig. 7.1. Theorem 7.12. (Bi-gyrotriangle Equality). Let X be a bi-gyropoint between bigyropoints A and B in a bi-gyrovector space (Rn×m c , ⊕E , ⊗). Then, d(A, X)⊕E d(X, B) = d(A, B) .

(7.37)

Proof. Bi-gyropoint X lies between A and B. Hence, by Lemma 7.10, X = A⊕E (E A⊕E B)⊗t

(7.38)

for some 0 < t < 1, and, by Lemma 7.11, X = B⊕E (E B⊕E A)⊗(1 − t) .

(7.39)

Analytic Bi-hyperbolic Geometry: The Geometry of Bi-gyrovector Spaces

Hence, by left cancellations, E A⊕E X = (E A⊕E B)⊗t E B⊕E X = (E B⊕E A)⊗(1 − t) .

(7.40)

Taking norms, noting the homogeneity property (5.539), p. 292, (7.40) yields E A⊕E X = E A⊕E B⊗t E X⊕E B = E B⊕E X = E B⊕E A⊗(1 − t) .

(7.41)

Hence, by the scalar distributive law (5.535), p. 291, we have E A⊕E X⊕E E X⊕E B = E A⊕E B⊗t ⊕E E A⊕E B⊗(1 − t) = E A⊕E B⊗{t + (1 − t)}

(7.42)

= E A⊕E B , 

thus verifying (7.37).

7.8. A Bi-gyroruler for Bi-gyrolines A ruler for lines in Euclidean geometry is studied, for instance, in [55], and a gyroruler for hyperbolic gyrolines is presented in [66]. Extending this concept to bigyrogeometry, in this section we present a bi-gyroruler for bi-gyrolines. In a bi-gyrovector space (Rn×m c , ⊕E , ⊗), let P1 = A⊕E (E A⊕E B)⊗t1 P2 = A⊕E (E A⊕E B)⊗t2 ,

(7.43)

t1 , t2 ∈ R, be two bi-gyropoints on a bi-gyroline LAB . The bi-gyrodistance between P1 and P2 is manipulated in the following chain of equations, which are numbered for

371

372

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

subsequent explanation: (1)

 d(P1 , P2 ) === E P1 ⊕E P2  (2)

 === E {A⊕E (E A⊕E B)⊗t1 }⊕E {A⊕E (E A⊕E B)⊗t2 } (3)

 === gyr[A, (E A⊕E B)⊗t1 ]{E (E A⊕E B)⊗t1 ⊕E (E A⊕E B)⊗t2 } (4)

 === E (E A⊕E B)⊗t1 ⊕E (E A⊕E B)⊗t2 

(7.44)

(5)

 === (E A⊕E B)⊗(−t1 + t2 ) (6)

 === E A⊕E B ⊗ | − t1 + t2 | . Derivation of the numbered equalities in (7.44): (1) (2) (3) (4) (5) (6)

Follows from the bi-gyrodistance definition. Follows from (7.43). Follows from Item (2) by the Left Bi-gyrotranslation Theorem 5.78, p. 256. Follows from Item (3) by the norm invariance stated in Theorem 5.8, p. 194. Follows from Item (4) by the scalar distributive law (5.535), p. 291. Follows from Item (5) by the homogeneity property (5.539), p. 292. Formalizing the main result in (7.43) – (7.44), we have the following theorem.

Theorem 7.13. In a bi-gyrovector space (Rn×m c , ⊕E , ⊗), let P1 = A⊕E (E A⊕E B)⊗t1 P2 = A⊕E (E A⊕E B)⊗t2

(7.45)

be two bi-gyropoints lying on the bi-gyroline LAB = A⊕E (E A⊕E B)⊗t ,

(7.46)

t ∈ R, corresponding to t = t1 and t = t2 , respectively. Then, d(P1 , P2 ) = d(A, B)⊗|t2 − t1 | .

(7.47)

Definition 7.14. (Bi-gyroruler). Let L be a bi-gyroline in a bi-gyrovector space (Rn×m c , ⊕E , ⊗). A function f : L → R is a bi-gyroruler for L if (i) f is bijective and (ii) for each pair of bi-gyropoints P1 , P2 ∈ L, f satisfies the bi-gyroruler equation | f (P1 )E f (P2 )| = d(P1 , P2 ) .

(7.48)

Analytic Bi-hyperbolic Geometry: The Geometry of Bi-gyrovector Spaces

Theorem 7.15. (Bi-gyroruler Theorem). Let A, B be two distinct bi-gyropoints in a bi-gyrovector space (Rn×m c , ⊕E , ⊗), and let LAB = {P : P = A⊕E (E A⊕E B)⊗t, t ∈ R}

(7.49)

be the bi-gyroline that passes through A and B. Then, the function f : LAB → R given by f (P) = f (A⊕E (E A⊕E B)⊗t) = d(A, B)⊗t

(7.50)

is a bi-gyroruler for LAB . Proof. The proof is obtained by the chain of equations (7.52), which are numbered for subsequent explanation. Let P1 = A⊕E (E A⊕E B)⊗t1 P2 = A⊕E (E A⊕E B)⊗t2

(7.51)

be any two bi-gyropoints on a bi-gyroline LAB in a bi-gyrovector space (Rn×m c , ⊕E , ⊗). Then, (1)

 d(P1 , P2 === P2 E P1  (2)

 === {A⊕E (E A⊕E B)⊗t2 }E {A⊕E (E A⊕E B)⊗t1 } (3)

 === gyr[A, (E A⊕E B)⊗t2 ]{(E A⊕E B)⊗t2 }E {(E A⊕E B)⊗t1 } (4)

 === (E A⊕E B)⊗t2 E (E A⊕E B)⊗t1  (5)

 === E A⊕E B)⊗(t2 − t1 ) (6)

 === |t2 − t1 |⊗E A⊕E B (7)

 === |t2 − t1 |⊗d(A, B) (8)

 === |(t2 − t1 )⊗d(A, B)| (9)

 === |t2 ⊗d(A, B)E t1 ⊗d(A, B)| (10)

 === | f (P2 )E f (P1 )| .

(7.52)

373

374

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

Derivation of the numbered equalities in (7.52): (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

This equation follows from the bi-gyrodistance definition. Follows from (7.51). Follows from Item (2) by the Left Bi-gyrotranslation Theorem 5.78, p. 256. Follows from Item (3) by the norm invariance under orthogonal transformations. Follows from Item (4) by the scalar distributive law (5.535), p. 291. Follows from Item (5) by the homogeneity property (5.539), p. 292. Follows from the bi-gyrodistance definition. Is obvious. Follows from Item (8) by the scalar distributive law (5.535), p. 291. Follows from Item (9) by the definition of f in (7.50).

Clearly, the function f : LAB → R is bijective. Hence, it follows from (7.52) that f is a bi-gyroruler for the bi-gyroline LAB . 

7.9. Bi-gyromidpoints Let A, B ∈ Rn×m = (Rn×m c c , ⊕E , ⊗), m, n ∈ N, c > 0, be two distinct bi-gyropoints, and let . MAB = A⊕E (E A⊕E B)⊗ 12 ∈ Rn×m c

(7.53)

Then, MAB is a bi-gyropoint between A and B, lying on the bi-gyroline LAB , LAB = A⊕E (E A⊕E B)⊗t ,

(7.54)

t ∈ R, that passes through A and B. In the special case when t1 = 0 and t2 = 12 , Theorem 7.13 asserts that for P1 = A⊕E (E A⊕E B)⊗0 = A P2 = A⊕E (E A⊕E B)⊗ 12 = MAB

(7.55)

we have d(A, MAB ) = d(A, B)⊗ 12 .

(7.56)

Similarly, in the special case when t1 = 1 and t2 = 12 , Theorem 7.13 asserts that for P1 = A⊕E (E A⊕E B)⊗1 = B P2 = A⊕E (E A⊕E B)⊗ 12 = MAB

(7.57)

we have d(B, MAB ) = d(A, B)⊗ 12 .

(7.58)

Hence, by (7.56) and (7.58), d(A, MAB ) = d(B, MAB ) = 12 ⊗d(A, B) ,

(7.59)

Analytic Bi-hyperbolic Geometry: The Geometry of Bi-gyrovector Spaces

so that MAB lies between A and B and is equi-bi-gyrodistant from A and B. As such, MAB is said to be the bi-gyromidpoint of A and B. The bi-gyromidpoint (7.53) of A and B in Rn×m is fully analogous to its Euclidean c E counterpart MAB ∈ Rn×m , E = A + (−A + B) 12 ∈ Rn×m , MAB

(7.60)

A, B ∈ Rn×m . A second bi-gyromidpoint formula in Rn×m follows from the first bi-gyromidpoint c formula (7.53) along with (5.366), p. 256, , MAB = 12 ⊗(A E B) ∈ Rn×m c

(7.61)

n×m , A, B ∈ Rn×m c . The latter is fully analogous to its Euclidean counterpart in R E MAB = 12 (A + B) ∈ Rn×m ,

(7.62)

A, B ∈ Rn×m . As shown in Fig. 7.1, a bi-gyroline in R2×3 is a system of m = 3 entangled gyrolines c in the disc, which are in geometric entanglement. Hence, similarly, a bi-gyrotriangle in R2×3 is a system of m = 3 entangled gyrotriangles in the disc, which are in geometric c entanglement, as shown in Fig. 7.3. Fig. 7.3 presents a bi-gyrotriangle ABC in R2×3 whose bi-gyrosides are AB, AC, c and BC, where A = (a1 a2 a3 ) ∈ R2×3 c B = (b1 b2 b3 ) ∈ R2×3 c

(7.63)

C = (c1 c2 c3 ) ∈ R2×3 c . According to Theorem 5.15, p. 199, A, B, C ∈ R2×3 implies ak , bk , ck ∈ R2c , k = c 1, 2, 3, as shown in Fig. 7.3. Bi-gyroside AB is the bi-gyrosegment AB shown in Fig. 7.1. Also shown in Fig. 7.3 are the bi-gyromidpoints X = MBC , Y = MAC , and Z = MAB of the bi-gyrosides, where X, Y, Z are given by their constituents, X = (x1 x2 x3 ) ∈ R2×3 c Y = (y1 y2 y3 ) ∈ R2×3 c

(7.64)

Z = (z1 z2 z3 ) ∈ R2×3 c . The bi-gyrosegment that joins a bi-gyrovertex of a bi-gyrotriangle and the bigyromidpoint of the opposing bi-gyroside is called a bi-gyromedian. In general, the three bi-gyromedians of a bi-gyrotriangle are not concurrent. However, when the

375

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

b2

y3

a1

z1

b1 z2

c3 x2

x1

x3 b3

a2 z3

y2 c2

c1 y1

a3

Figure 7.3 A bi-gyrotriangle ABC in (R2×3 c , ⊕E , ⊗) is shown here in terms of its three constituents in R2×1 = R2c . The three constituents are the entangled gyrotriangles ak bk ck , k = 1, 2, 3, in R2×1 = R2c . c c The bi-gyrosegment AB in Fig. 7.1 is one of the three bi-gyrosides of bi-gyrotriangle ABC shown here. Also the bi-gyromidpoint of each bi-gyroside of the bi-gyrotriangle is shown. In general, the three bi-gyromedians of a bi-gyrotriangle are not concurrent. However, in the special case when the bi-gyrotriangle bi-gyrovertices A, B, C are equi-column, the bi-gyromedians of bi-gyrotriangle ABC are concurrent, as they should in hyperbolic geometry.

bi-gyrovertices of a bi-gyrotriangle are equi-column, the bi-gyrotriangle descends to a gyrotriangle in the special relativistic model of hyperbolic geometry, as demonstrated in Fig. 7.2 for a bi-gyroline. Moreover, in this case the three bi-gyromedians of a bi-gyrotriangle descend to the gyromedians of a gyrotriangle, which are concurrent, as they should in the special relativistic model of hyperbolic geometry. This case is illustrated in Fig. 7.4.

7.10. Reduction of Bi-gyrotriangles to Gyrotriangles The geometry that regulates a gyrovector space (Rnc , ⊕E , ⊗) is n-dimensional hyperbolic geometry, as demonstrated in Chaps. 2 and 3, and studied in [98]. Hence, the geometry that regulates a bi-gyrovector space (Rn×m c , ⊕E , ⊗) is called bi-hyperbolic geometry. Remarkably, the binary operation ⊕E in a gyrovector space (Rnc , ⊕E , ⊗) and in

Analytic Bi-hyperbolic Geometry: The Geometry of Bi-gyrovector Spaces

mbc b

c mac

mab a

Figure 7.4 This figure is generated by the same formulas that generate Fig. 7.3 in (R2×3 c , ⊕E , ⊗). Unlike the bi-gyrovertices of the bi-gyrotriangle in Fig. 7.3, however, in this figure these are equi-column. As a result, the (3,2)-bi-gyrotriangle in Fig. 7.3 descends to the gyrotriangle of multiplicity 3 in this figure. Being a gyrotriangle, its three gyromedians are concurrent, as they should in hyperbolic geometry. The transition from Fig. 7.3 to this figure is performed by moving from (m, n)-bi-gyropoints to equi-column (m, n)-bi-gyropoints, which are n-dimensional gyropoints of multiplicity m. For graphical presentation, (m, n) = (3, 2) in Fig. 7.3 and in this figure. This transition illustrates the way bihyperbolic geometry descends to hyperbolic geometry.

its hyperbolic geometry is the special relativistic Einstein addition (n = 3 in physical applications), while the binary operation ⊕E in a bi-gyrovector space (Rn×m c , ⊕E , ⊗) and in its bi-hyperbolic geometry is Einstein addition of signature (m, n). It is important to note that Einstein addition of signature (m, n) descends to m copies of the special relativistic Einstein addition in the special case when the bi-gyropoints that are involved are equi-column, as explained in Sect. 6.14. A result in a bi-gyrovector space (Rn×m c , ⊕E , ⊗) based on bi-gyropoints descends to a corresponding result in the gyrovector space (Rnc , ⊕E , ⊗) when the involved bigyropoints are equi-column, as explained in Sect. 7.5 and illustrated in the transition from Fig. 7.1 to Fig. 7.2. Similarly, any result in bi-hyperbolic geometry descends to a corresponding result in hyperbolic geometry in the special case when all the involved bi-gyropoints are equi-column.

377

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

This reduction of bi-hyperbolic geometry to hyperbolic geometry is now demonstrated in the transition from Fig. 7.3 to Fig. 7.4. A (3,2)-bi-gyrotriangle ABC is graphically represented in Fig. 7.3 by its three constituents, which are the three entangled gyrotriangles ak bk ck , k = 1, 2, 3. In contrast, the gyrotriangle in Fig. 7.4 is represented by its single, repeated gyrotriangle abc of multiplicity 3. The repeated gyrotriangle abc in Fig. 7.4 is regulated by hyperbolic geometry so that, for instance, its repeated gyromedians of multiplicity 3 are concurrent, as shown in Fig. 7.4. Thus, in the transition from Fig. 7.3 to Fig. 7.4, where bi-gyroangle bi-gyrovertices in Rn×m ((m,n)=(3,2) in the figures) descend to equi-column bi-gyrovertices, c (1) bi-gyropoints in the bi-hyperbolic geometry of (Rn×m c , ⊕E , ⊗) descend to gyropoints of multiplicity m in the hyperbolic geometry of (Rnc , ⊕E , ⊗); (2) bi-gyromidpoints in the bi-hyperbolic geometry of (Rn×m c , ⊕E , ⊗) descend to gyromidpoints of multiplicity m in the hyperbolic geometry of (Rnc , ⊕E , ⊗); and (3) bi-gyromedians in the bi-hyperbolic geometry of (Rn×m c , ⊕E , ⊗) descend to gyromedians of multiplicity m in the hyperbolic geometry of (Rnc , ⊕E , ⊗). We thus realize that bi-hyperbolic geometry is a natural generalization of the hyperbolic geometry of Lobachevsky and Bolyai, in which geometric entanglement takes place.

7.11. Bi-gyroparallelograms Definition 7.16. (Bi-gyroparallelograms). In a bi-gyrovector space (Rn×m c , ⊕E , ⊗), m, n ∈ N, let A,B, and C be any three non-bi-gyrocollinear bi-gyropoints, and let D ∈ Rn×m be given by the bi-gyroparallelogram condition c D = (B E C)E A .

(7.65)

Then, the four bi-gyropoints A, B, C, and D are said to be the bi-gyrovertices of the bi-gyroparallelogram ABDC, where (1) bi-gyrovertices A and D (resp. B and C) are said to be opposite to each other and (2) bi-gyrosegments AD and BC are the bi-gyrodiagonals of the bi-gyroparallelogram ABDC. A bi-gyroparallelogram ABDC in R2×3 is shown in Fig. 7.5 in terms of its three c constituents, where A, B, C are given by (7.63) and where D = (d1 d2 d3 ) ∈ R2×3 c

(7.66)

is given by the bi-gyroparallelogram condition (7.65). The three constituents of bigyroparallelogram ABDC are the three entangled gyroparallelograms ak bk dk ck , k = 1, 2, 3, in R2×1 = R2c , as shown in Fig. 7.5. c

Analytic Bi-hyperbolic Geometry: The Geometry of Bi-gyrovector Spaces

d2

b2 m2

a1

a2

c2

b3

b1 m1 m3

a3

d3 c1

c3

d1 (E A⊕E B) E (E A⊕E C) = E A⊕E D MAD = MBC = MABDC

Figure 7.5 A bi-gyroparallelogram ABDC in (R2×3 c , ⊕E , ⊗) is shown in terms of its three constituents = R2c . The three constituents are the entangled gyroparallelograms ak bk dk ck , k = 1, 2, 3, in in R2×1 c R2×1 = R2c . The bi-gyromidpoints MAD and MBC of its bi-gyrodiagonals AD and BC coincide, giving rise c to the bi-gyroparallelogram bi-gyrocentroid MABDC := MAD = MBC . The bi-gyrocentroid is presented in = R2c . terms of its three constituents, which are m1 , m2 , m3 ∈ R2×1 c

Theorem 7.17. (The Bi-gyrodiagonals of a Bi-gyroparallelogram Bisect Each Other). Let MAD and MBC be, respectively, the bi-gyromidpoints of the two bi-gyrodiagonals AD and BC of a bi-gyroparallelogram ABDC in a bi-gyrovector space (Rn×m c , ⊕E , ⊗). Then, MAD = MBC .

(7.67)

Proof. By means of the bi-gyroparallelogram condition (7.65) and the right cancellation law (5.369), p. 256, we have D E A = B E C .

(7.68)

Hence, by means of (7.61) and the commutativity of E , MAD = 12 ⊗(A E D) = 12 ⊗(B E C) = MBC , as desired.

(7.69) 

379

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

The gyroparallelogram in Einstein gyrovector spaces (Rnc , ⊕, ⊗) is studied in Chap. 3. Some of its properties are derived in Sects. 3.9 and 3.10 by means of the gyrocommutative gyrogroup structure of (Rnc , ⊕). The gyroparallelogram properties that are presented in Sects. 3.9 and 3.10 remain valid for bi-gyroparallelograms in Einstein n×m bi-gyrovector spaces (Rn×m c , ⊕E , ⊗) as well, since any bi-gyrogroup (Rc , ⊕E ) is a gyrocommutative gyrogroup, according to Theorem 5.75, p. 255. In particular, the gyroparallelogram addition law in Theorem 3.15, p. 83, remains valid, leading to the following theorem. Theorem 7.18. (The Bi-gyroparallelogram (Addition) Law). Let ABDC be a bigyroparallelogram in an Einstein bi-gyrovector space (Rn×m c , ⊕E , ⊗), m, n ∈ N. Then, (E A⊕E B) E (E A⊕E C) = E A⊕E D .

(7.70)

Proof. The proof of this theorem is identical with the proof of Theorem 3.15, p. 83, noting that bi-gyrogroups (Rn×m c , ⊕E ) are gyrocommutative gyrogroups according to Theorem 5.75, p. 255.  A bi-gyroparallelogram ABDC of signature (m, n) = (3, 2) is shown in Fig. 7.5, along with its bi-gyroparallelogram addition law.

7.12. Bi-gyroisometries: The Bi-hyperbolic Isometries In the study of bi-gyroisometries we are guided by analogies with the isometries studied in Sect. 3.11, and with the gyroisometries studied in Sect. 3.13. The bi-gyrodistance function d(A, B) in an Einstein bi-gyrovector space n×m (Rc , ⊕E , ⊗) is given by Def. 7.2, p. 362, d(A, B) = E A⊕E B ,

(7.71)

A, B ∈ Rn×m c . Like the distance function in Sect. 3.11 and the gyrodistance function in Sect. 3.13, the bi-gyrodistance function possesses the following properties: For all A, B, C ∈ Rn×m c , 1. 2. 3. 4. 5.

d(A, B) = d(B, A) d(A, B) ≥ 0 d(A, B) = 0 if and only if A = B. d(A, B) ≤ d(A, C)⊕E d(C, B) (the bi-gyrotriangle inequality). d(A, B) = d(A, C)⊕E d(C, B) (the bi-gyrotriangle equality, where C lies between A and B).

A proof of the bi-gyrotriangle inequality in Item (4) is presented in Theorem 5.108, p. 294, and the proof of the bi-gyrotriangle equality in Item (5) is presented in Theorem 7.12, p. 370.

Analytic Bi-hyperbolic Geometry: The Geometry of Bi-gyrovector Spaces

Definition 7.19. (Bi-gyroisometries). A map φ : Rn×m → Rn×m is a bi-gyroisometry c c n×m of Rc if it preserves the bi-gyrodistance between any two bi-gyropoints of Rn×m c , that is, if d(φA, φB) = d(A, B)

(7.72)

for all A, B ∈ Rn×m c . are two A bi-gyroisometry is injective (one-to-one into). Indeed, if A, B ∈ Rn×m c distinct bi-gyropoints, A  B, then 0  E A⊕E B = E φA⊕E φB ,

(7.73)

so that φA  φB. n×m by X is the map λX : Rn×m → For any X ∈ Rn×m c , a left bi-gyrotranslation of Rc c n×m Rc given by λX A = X⊕E A

(7.74)

for all A ∈ Rn×m c . Theorem 7.20. (Left Bi-gyrotranslational Bi-gyroisometries). Left bi, ⊕ , ⊗) are bi-gyroisometries. Gyrotranslations of an Einstein gyrovector space (Rn×m c E Proof. The proof follows immediately from Theorem 7.3, p. 362.



For any On ∈ SO(n) and Om ∈ SO(m), m, n ∈ N, a bi-rotation of an Einstein bin×m gyrovector space (Rn×m c , ⊕E , ⊗) by (On , Om ) is the map of Rc , given by (On , Om )A = On AOm ,

(7.75)

for all A ∈ Rn×m c . Theorem 7.21. (Bi-rotational Bi-gyroisometries). Bi-rotations of an Einstein gyrovector space (Rn×m c , ⊕E , ⊗) are bi-gyroisometries. Proof. The proof follows immediately from Theorem 7.4, p. 363.



7.13. Bi-gyromotions: The Motions of Bi-hyperbolic Geometry The group of motions of Euclidean geometry is studied in Sect. 3.12. The group of motions of the relativistic model of hyperbolic geometry, called gyromotions, is

381

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

studied in Sect. 3.14. We are now in the position to study the group of motions of bi-hyperbolic geometry, called bi-gyromotions. The group of bi-gyromotions of an Einstein gyrovector space (Rn×m c , ⊕E , ⊗) is a group of triples n×m × SO(n) × SO(m) {(X, On , Om ) : X ∈ Rn×m c , On ∈ SO(n), Om ∈ SO(m)} = Rc (7.76) n×m acting bi-gyroisometrically on Rc according to the equation

(X, On , Om )A = X⊕E On AOtm

(7.77)

n×m is a bi-gyroisometry consisting for all A ∈ Rn×m c . Accordingly, a bi-gyromotion of Rc of a bi-rotation followed by a left bi-gyrotranslation of Rn×m c . Two successive bi-gyromotions are equivalent to a single bi-gyromotion as we see from the following chain of equations, in which we employ the left bi-gyroassociative law (5.533), p. 291, along with the bi-gyration formula (5.526). For all A, Xk ∈ Rn×m c , On,k ∈ SO(n), and Om,k ∈ SO(m), k = 1, 2, we have

(X1 , On,1 , Om,1 )(X2 , On,2 , Om,2 )A = (X1 , On,1 , Om,1 ){X2 ⊕E On,2 AOtm,2 } = X1 ⊕E On,1 {X2 ⊕E On,2 AOtm,2 }Otm,1 = X1 ⊕E {On,1 X2 Otm,1 ⊕E On,1 On,2 A(Om,1 Om,2 )t } = {X1 ⊕E On,1 X2 Otm,1 }⊕E gyr[X1 , On,1 X2 Otm,1 ]On,1 On,2 A(Om,1 Om,2 )t = {X1 ⊕E On,1 X2 Otm,1 }⊕E lgyr[X1 , On,1 X2 Otm,1 ]On,1 On,2 A(rgyr[X1 , On,1 X2 Otm,1 ]Om,1 Om,2 )t   = X1 ⊕E On,1 X2 Otm,1 , lgyr[X1 , On,1 X2 Otm,1 ]On,1 On,2 , (rgyr[X1 , On,1 X2 Otm,1 ]Om,1 Om,2 )t A (7.78) In column notation, the bi-gyromotion composition law (7.78) takes the form ⎞ ⎛ ⎞⎛ ⎞ ⎛ X1 ⊕E On,1 X2 Otm,1 ⎟⎟⎟ ⎜⎜⎜ X1 ⎟⎟⎟ ⎜⎜⎜ X2 ⎟⎟⎟ ⎜⎜⎜ ⎜ ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜ ⎜⎜⎜⎜ ⎜ t ⎜⎜⎜ On,1 ⎟⎟⎟⎟ ⎜⎜⎜⎜ On,2 ⎟⎟⎟⎟ = ⎜⎜⎜ lgyr[X1 , On,1 X2 Om,1 ]On,1 On,2 ⎟⎟⎟⎟ ∈ Rn×m × SO(n) × SO(m) (7.79) c ⎟⎟ ⎜⎝⎜ ⎟⎠⎟ ⎜⎝⎜ ⎟⎠⎟ ⎜⎜⎜ ⎟ ⎝ ⎠ rgyr[X1 , On,1 X2 Otm,1 ]Om,1 Om,2 Om,1 Om,2 for all (Xk , On,k , Om,k ) ∈ Rn×m × SO(n) × SO(m), k = 1, 2. c The product (7.79), called a bi-gyrosemidirect product, is analogous to the gyrosemidirect product studied in Sect. 2.13. The latter, in turn, is analogous to the well-known semidirect product in group theory. The identity element of the bi-gyrosemidirect product (7.79) is (0n,m , In , Im ).

Analytic Bi-hyperbolic Geometry: The Geometry of Bi-gyrovector Spaces

Indeed, following (7.79) we have the bi-gyrosemidirect product ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ X⊕E 0n,m ⎜⎜⎜ X ⎟⎟⎟ ⎜⎜⎜0n,m ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ X ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟ ⎜ ⎟ ⎜⎜⎜ On ⎟⎟⎟ ⎜⎜⎜ In ⎟⎟⎟ = ⎜⎜⎜ lgyr[X, 0n,m ]On In ⎟⎟⎟⎟⎟ = ⎜⎜⎜⎜⎜ On ⎟⎟⎟⎟⎟ , ⎜⎜⎜⎝ ⎟⎟⎟⎠ ⎜⎜⎜⎝ ⎟⎟⎟⎠ ⎜⎜⎜⎝ ⎟⎟⎟⎠ ⎜⎜⎜⎝ ⎟⎟⎟⎠ Om Im rgyr[X, 0n,m ]Om Im Om noting (5.283), p. 237. The inverse of (X, On , Om ) is ⎛ ⎞ −1 ⎛ ⎞ ⎜⎜⎜ X ⎟⎟⎟ ⎜⎜⎜E Otn XOm ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜ ⎟⎟⎟ ⎜⎜⎜ On ⎟⎟⎟ = ⎜⎜⎜⎜⎜ Ot ⎟⎟⎟ . n ⎟⎟⎟ ⎜⎜⎜⎝ ⎟⎟⎟⎠ ⎜⎜⎜⎝ ⎠ Om Otm Indeed, by (7.79) and (7.81) we have the bi-gyrosemidirect product ⎛ ⎞ ⎛ ⎞ −1 ⎛ ⎞ ⎛ ⎞ ⎜⎜⎜ X ⎟⎟⎟ ⎜⎜⎜ X ⎟⎟⎟ ⎜⎜⎜ X ⎟⎟⎟ ⎜⎜⎜E Otn XOm ⎟⎟⎟ ⎜⎜⎜⎜ ⎟⎟⎟⎟ ⎜⎜⎜⎜ ⎟⎟⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜⎜ On ⎟⎟⎟ ⎜⎜⎜ On ⎟⎟⎟ = ⎜⎜⎜⎜⎜ On ⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ Otn ⎟⎟⎟⎟⎟ ⎜⎝⎜ ⎟⎠⎟ ⎜⎝⎜ ⎟⎠⎟ ⎜⎝⎜ ⎟⎠⎟ ⎜⎝⎜ ⎟⎠⎟ Om Om Om Otm ⎛ ⎞ ⎛ ⎞ ⎜⎜⎜X⊕E On (E Otn XOm )Otm = X⊕E (E X)⎟⎟⎟ ⎜⎜⎜0n,m ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎟ ⎜⎜⎜ ⎟⎟⎟ = ⎜⎜⎜ In ⎟⎟⎟⎟⎟ , lgyr[X, E X]On Otn = ⎜⎜⎜⎜ ⎜ ⎟ ⎟⎟⎟ ⎜⎜⎝ ⎟⎟⎠ ⎜⎜⎝ ⎠ t rgyr[X, E X]Om Om Im

(7.80)

(7.81)

(7.82)

noting (5.284), p. 237. In order to apply the subgroup criterion in Theorem 2.12, p. 22, we calculate the following bi-gyrosemidirect product in an Einstein bi-gyrovector space (Rn×m c , ⊕E , ⊗): ⎛ ⎞ ⎛ ⎞⎛ ⎞ −1 ⎛ ⎞ ⎜⎜⎜ X1 ⎟⎟⎟ ⎜⎜⎜E Otn,2 X2 Om,2 ⎟⎟⎟ ⎜⎜⎜ X1 ⎟⎟⎟ ⎜⎜⎜ X2 ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎟ ⎜⎜ ⎟⎟⎟ t ⎜⎜⎜ On,1 ⎟⎟⎟ ⎜⎜⎜ On,2 ⎟⎟⎟ = ⎜⎜⎜ On,1 ⎟⎟⎟⎟⎟ ⎜⎜⎜⎜ O ⎟⎟⎟ n,2 ⎜⎜⎜ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ⎝ ⎝ ⎠⎝ ⎠ ⎠ ⎝⎜ ⎠⎟ t Om,2 Om,1 Om,2 Om,1 ⎛ ⎞ ⎜⎜⎜X1 ⊕E On,1 (E Otn,2 X2 Om,2 )Otm,1 = X1 E On,1 Otn,2 X2 Om,2 Otm,1 ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎟⎟⎟ lgyr[X1 , E On,1 Otn,2 X2 Om,2 Otm,1 ]On,1 Otn,2 = ⎜⎜⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎝ ⎟⎠ t t t rgyr[X1 , E On,1 On,2 X2 Om,2 Om,1 ]Om,1 Om,2

(7.83)

∈ Rn×m × SO(n) × SO(m) , c for any (Xk , On,k , Om,k ) ∈ Rn×m × SO(n) × SO(m), k = 1, 2. c Employing the subgroup criterion, we will find in Sect. 7.14 that the set of all the

383

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Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

bi-gyromotions of an Einstein bi-gyrovector space (Rn×m c , ⊕E , ⊗) forms a group under the bi-gyrosemidirect product.

7.14. Bi-gyrosemidirect Product Groups The study in this section is analogous to the study in Sect. 2.13. = (Rn×m Definition 7.22. (Bi-gyrosemidirect Product Groups). Let Rn×m c c , ⊕E ) be an Einstein bi-gyrogroup. The bi-gyrosemidirect product group G, × SO(n) × SO(m) , G = Rn×m c

(7.84)

is a group of triples (X, On , Om ) ∈ G with group operation given by the bi-gyrosemidirect product (7.79), ⎞ ⎛ ⎞⎛ ⎞ ⎛ X1 ⊕E On,1 X2 Otm,1 ⎟⎟⎟ ⎜⎜⎜ X1 ⎟⎟⎟ ⎜⎜⎜ X2 ⎟⎟⎟ ⎜⎜⎜ ⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎜⎜⎜ On,1 ⎟⎟⎟ ⎜⎜⎜ On,2 ⎟⎟⎟ = ⎜⎜⎜ lgyr[X1 , On,1 X2 Ot ]On,1 On,2 ⎟⎟⎟⎟ ∈ Rn×m × SO(n) × SO(m) , (7.85) c ⎜ ⎟⎟⎟ m,1 ⎜⎜⎜⎝ ⎟⎟⎟⎠ ⎜⎜⎜⎝ ⎟⎟⎟⎠ ⎜⎜⎜ ⎟⎠ ⎝ t rgyr[X1 , On,1 X2 Om,1 ]Om,1 Om,2 Om,1 Om,2 for all (Xk , On,k , Om,k ) ∈ Rn×m × SO(n) × SO(m), k = 1, 2. c It is anticipated in Def. 7.22 that the bi-gyrosemidirect product (7.85) is a group operation. The following theorem asserts that this is indeed the case. = (Rn×m Theorem 7.23. (Bi-gyrosemidirect Product Group). Let Rn×m c c , ⊕E ) be an n×m Einstein bi-gyrogroup. The set G = Rc × SO(n) × SO(m) forms a group under the bi-gyrosemidirect product (7.85). Proof. The proof is similar to the second proof of Theorem 2.28, p. 37. Let S be the group of all bijections of Rn×m onto itself under bijection composition. Let each c element (X, On , Om ) ∈ G act bijectively on the Einstein gyrogroup Rn×m = (Rn×m c c , ⊕E ) according to (7.77). The unique inverse of (X, On , Om ) ∈ G is given by (7.81). Being a set of special bijections of Rn×m onto itself, given by (7.77), G is a subset c of S , G ⊂ S . Let (X1 , On,1 , Om,1 ) and (X2 , On,2 , Om,2 ) be any two elements of G. Then, the product (X1 , On,1 , Om,1 )(X2 , On,2 , Om,2 )−1 is, again, an element of G, as shown in (7.83). Hence, by the subgroup criterion in Theorem 2.12, p. 22, G is a subgroup of S . Hence, in particular, G is a group under bijection composition, where bijection composition is given by the bi-gyrosemidirect product (7.85).  The bijections (X, On , Om ) ∈ G of Rn×m are bi-gyroisometries of the Einstein bic n×m gyrovector space (Rc , ⊕E , ⊗), as we see from Theorem 7.20 and Theorem 7.21. We

Analytic Bi-hyperbolic Geometry: The Geometry of Bi-gyrovector Spaces

call these bi-gyroisometries the bi-gyromotions of the Einstein bi-gyrovector space (Rn×m c , ⊕E , ⊗). Accordingly, we adopt the following formal definition. Definition 7.24. (Bi-gyromotions). The group G of the bi-gyromotions of an Einstein bi-gyrovector space (Rn×m c , ⊕E , ⊗) is the bi-gyrosemidirect product group × SO(n) × SO(m) G = Rn×m c

(7.86)

with group operation given by the bi-gyrosemidirect product (7.85). The importance of Felix Klein’s Erlangen Program in geometry is emphasized in Sect. 3.12. Following the Erlangen Program, a property of an Einstein bi-gyrovector space (Rn×m c , ⊕E , ⊗), m, n > 1, has geometric significance if it is invariant or covariant under the bi-gyromotions of the space. The resulting geometry that regulates the Einstein bi-gyrovector space (Rn×m c , ⊕E , ⊗) is the bi-hyperbolic geometry of signature (m, n). Example 7.25. The bi-gyrodistance function in a bi-gyrovector space (Rn×m c , ⊕E , ⊗) is invariant under the bi-gyromotions of the space, as we see from Theorems 7.3 and 7.4. Hence, the bi-gyrodistance function has geometric significance. Example 7.26. The bi-gyromidpoint MAB , MAB = 12 ⊗(A E B) ,

(7.87)

in an Einstein bi-gyrovector space (Rn×m c , ⊕E , ⊗) is bi-gyrocovariant, that is, it is covariant under the bi-gyromotions of the space. Indeed, MAB (i) is covariant under left bi-gyrotranslations, that is, X⊕E MAB = 12 ⊗((X⊕E A) E (X⊕E B)) ,

(7.88)

for any X ∈ Rn×m c , and (ii) is covariant under bi-rotations, that is, On MAB Om = 12 ⊗(On AOm E On BOm ) ,

(7.89)

for any On ∈ SO(n) and Om ∈ SO(m). As such, the bi-gyromidpoint in an Einstein bi-gyrovector space has geometric significance. For details, see [84, Sect. 6.6]. Example 7.27. The bi-gyroparallelogram condition (7.65) D = (B E C)E A

(7.90)

in an Einstein bi-gyrovector space (Rn×m c , ⊕E , ⊗) is bi-gyrocovariant, that is, it is covariant under the bi-gyromotions of the space. Indeed, (i) D is covariant under

385

386

Beyond Pseudo-rotations in Pseudo-Euclidean Spaces

a1 b3 b1 d2

b2 m

a3

c2

d3 a2 c1

c3 d1

(E A⊕E B) E (E A⊕E C) = E A⊕E D MAD = MBC = MABDC = 02,3 Figure 7.6 A left bi-gyrotranslation by −M of the bi-gyroparallelogram ABDC , with bi-gyrocentroid M in Fig. 7.5, generates in this figure the bi-gyroparallelogram (−M⊕E A)(−M⊕E B)(−M⊕E D)(−M⊕E C) with bi-gyrocentroid −M⊕E M = 02,3 in (R2×3 Accordingly, the bi-gyrocentroid of the bic , ⊕E , ⊗). gyrotranslated bi-gyroparallelogram ABDC in this figure is a repeated two-dimensional zero gyrovector of multiplicity 3.

left bi-gyrotranslations, that is, X⊕E D = ((X⊕E B) E (X⊕E C))E (X⊕E A)

(7.91)

for any X ∈ Rn×m c , and (ii) D is covariant under bi-rotations, that is, On DOm = (On BOm E OnCOm )E On AOm ,

(7.92)

for any On ∈ SO(n) and Om ∈ SO(m). As such, the bi-gyroparallelogram condition in an Einstein bi-gyrovector space has geometric significance. For details, see [98, Theorem 2.58, p. 69]. The covariance of the bi-gyroparallelogram under left bi-gyrations is employed in Fig. 7.6. In this figure, the bi-gyroparallelogram of Fig. 7.5 with bigyrocentroid M = (m1 m2 m3 ) ∈ R2×3 is left bi-gyrotranslated by −M = E M to genc 2 erate a bi-gyroparallelogram with bi-gyrocentroid 02,3 = (0 0 0) ∈ R2×3 c , 0 ∈ Rc .

Notation and Special Symbols ⊕ Einstein addition, (2.2), p. 10, and a gyrogroup operation, in Chaps. 1 and 2; Bi-gyrogroupoid operation in Rn×m , (4.114), p. 124; and in Rn×m c , (5.221), p. 224.  Inverse of ⊕. ⊕ Bi-gyrogroup operation in Rn×m , (4.256), p. 154; and in Rn×m c , (5.328), p. 247. The latter is renamed as Einstein addition, ⊕E , of signature (m, n) in Def. 6.6, p. 316, so that ⊕ = ⊕E .   Inverse of ⊕ .  ⊕E Einstein addition of signature (m, n) in Rn×m c , ⊕E = ⊕ . Def. 6.6, p. 316; (5.363), p. 255. E Inverse of ⊕E .  The cooperation associated with the binary operation ⊕, (2.47), p. 23.  Inverse of . E The cooperation associated with the binary operation ⊕E , Def. 5.76, p. 256. E Inverse of E . ⊗ Einstein special relativistic scalar multiplication, (3.19), p. 66, in Rnc ; Scalar multiplication (scalar bi-gyromultiplication) in Rn×m , Theorem 4.61, p. 171; and in Rn×m c , Theorem 5.86, p. 263. := Equality, where the left-hand side is defined by the right-hand side. =: Equality, where the right-hand side is defined by the left-hand side. · Norm; Matrix spectral norm, Def. 5.7, p. 193. (. . .)t Transpose of (. . .). γ Lorentz gamma factor of special relativity, (2.3), p. 11. ΓRV Right gamma factor parametrized by V ∈ Rn×m c , (5.115), p. 201. ΓVL Left gamma factor parametrized by V ∈ Rn×m c , (5.115), p. 201. λ(On ) Left rotation, (4.77), p. 115. ρ(Om ) Right rotation, (4.76), p. 115. φ A bijective map, φ : P → V, (5.2), p. 186; φ(Rn×m ) = Rn×m c , Theorem 5.10, p. 195. Λ A Lorentz transformation of order (m, n), Λ ∈ SO(m, n). B(P) Bi-boost, parametrized by P ∈ Rn×m , (4.75), p. 115; B(P) ≡ B p (P). B(V) Bi-boost, parametrized by V ∈ Rn×m c , (5.127), p. 204; B(V) ≡ Bv (V). B∞ (V) Galilean boost. B∞ (V) = limc→∞ B(V). d(A, B) Bi-gyrodistance between A and B in (Rn×m c , ⊕E , ⊗); Def. 7.2, p. 362. EP Entanglement Part, (6.19), p. 304. GP Galilean Part, (6.18), p. 304. gyr Gyrator, gyr = (lgyr, rgyr), Def. 4.53, p. 166, for P; and Def. 5.67, p. 250, for V. lgyr Left Gyrator. 1. lgyr[P1 , P2 ] ∈ SO(n) is the left gyration (left gyroautomorphism) generated by P1 , P2 ∈ Rn×m , (4.114), p. 124, and 387

388

Notation and Special Symbols

2. lgyr[V1 , V2 ] ∈ SO(n) is the left gyration (left gyroautomorphism) generated by V1 , V2 ∈ Rn×m by V1 , V2 ∈ Rn×m c c , (5.329), p. 247. rgyr Right Gyrator. 1. rgyr[P1 , P2 ] ∈ SO(m) is the right gyration (right gyroautomorphism) generated by P1 , P2 ∈ Rn×m , (4.114), p. 124, and 2. rgyr[V1 , V2 ] ∈ SO(m) is the right gyration (right gyroautomorphism) generated by V1 , V2 ∈ Rn×m by V1 , V2 ∈ Rn×m c c , (5.329), p. 247. (m, n) N 0n,m In On P R Rn Rm,n Rn×m Rn×m c (Rn×m , ⊕) (Rn×m , ⊕ ) (Rn×m c , ⊕) n×m (Rc , ⊕ ) (Rn×m c , ⊕E ) SO(n) SO(m, n) SVD tr(M) V

Signature (m, n), m, n ∈ N. Set of all Natural numbers. The n × m zero matrix. The n × n identity matrix. An n × n orthogonal matrix with determinant 1, On ∈ SO(n). Generic element of Rn×m . Real line. Euclidean n-space. Pseudo-Euclidean Space of signature (m, n). Set of all real Matrices of order n × m. Spectral ball of spectral radius c, Spectral c-ball, of Rn×m , (5.68), p. 193. Bi-gyrogroupoid. Def. 4.16, p. 121. Bi-gyrogroup. Def. 4.46, p. 154. Bi-gyrogroupoid. Theorem 5.41, p. 224. Bi-gyrogroup. Def. 5.57, p. 241. Einstein bi-gyrogroup. ⊕E = ⊕ . Def. 6.6, p. 316. Special orthogonal group of matrices of order n × n. Pseudo-orthogonal group. (Proper) Lorentz Group of signature (m, n). Singular Value Decomposition. Sect. 4.27. Trace of a square matrix M. Generic element of Rn×m c .

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78. Abraham A. Ungar. Weakly associative groups. Resultate Math., 17(1-2):149–168, 1990. 79. Abraham A. Ungar. Thomas precession and its associated grouplike structure. Amer. J. Phys., 59(9):824–834, 1991. 80. Abraham A. Ungar. Thomas precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics. Found. Phys., 27(6):881–951, 1997. 81. Abraham A. Ungar. Beyond the Einstein addition law and its gyroscopic Thomas precession: The theory of gyrogroups and gyrovector spaces, volume 117 of Fundamental Theories of Physics. Kluwer Academic Publishers Group, Dordrecht, 2001. 82. Abraham A. Ungar. The longest waiting time for print in modern mathematics. Notices Amer. Math. Soc., 48(9):966, 2001. 83. Abraham A. Ungar. On the unification of hyperbolic and Euclidean geometry. Complex Var. Theory Appl., 49(3):197–213, 2004. 84. Abraham A. Ungar. Analytic hyperbolic geometry: Mathematical foundations and applications. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. 85. Abraham A. Ungar. Gyrovector spaces and their differential geometry. Nonlinear Funct. Anal. Appl., 10(5):791–834, 2005. 86. Abraham A. Ungar. The proper-time Lorentz group demystified. J. Geom. Symmetry Phys., 4:69–95, 2005. 87. Abraham A. Ungar. Newtonian and relativistic kinetic energy: analogous consequences of their conservation during elastic collisions. European J. Phys., 27(5):1205–1212, 2006. 88. Abraham A. Ungar. Placing the hyperbolic geometry of Bolyai and Lobachevsky centrally in special relativity theory: an idea whose time has returned. In Non-Euclidean geometries, volume 581 of Math. Appl. (N. Y.), pages 487–506. Springer, New York, 2006. 89. Abraham A. Ungar. The relativistic hyperbolic parallelogram law. In Geometry, integrability and quantization, pages 249–264. Softex, Sofia, 2006. 90. Abraham A. Ungar. The relativistic proper-velocity transformation group. Progress In Electromagnetics Research, 60:85–94, 2006. 91. Abraham A. Ungar. Thomas precession: a kinematic effect of the algebra of Einstein’s velocity addition law. Comments on: “Deriving relativistic momentum and energy. II. Three-dimensional case” [European J. Phys. 26 (2005), no. 5, 851–856; mr2227176] by S. Sonego and M. Pin. European J. Phys., 27(3):L17–L20, 2006. 92. Abraham A. Ungar. Einstein’s velocity addition law and its hyperbolic geometry. Comput. Math. Appl., 53(8):1228–1250, 2007. 93. Abraham A. Ungar. Analytic hyperbolic geometry and Albert Einstein’s special theory of relativity. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. 94. Abraham A. Ungar. A gyrovector space approach to hyperbolic geometry. Morgan & Claypool Pub., San Rafael, California, 2009. 95. Abraham A. Ungar. Barycentric calculus in Euclidean and hyperbolic geometry: A comparative introduction. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. 96. Abraham A. Ungar. Hyperbolic triangle centers: The special relativistic approach. Springer-Verlag, New York, 2010. 97. Abraham A. Ungar. Hyperbolic geometry. J. Geom. Symmetry Phys., 32:61–86, 2013. 98. Abraham A. Ungar. Analytic hyperbolic geometry in n dimensions: An introduction. CRC Press, Boca Raton, FL, 2015. 99. Abraham A. Ungar. Parametric realization of the Lorentz transformation group in pseudo-euclidean spaces. J. Geom. Symmetry Phys., 38:39–108, 2015. 100. Abraham A. Ungar. The intrinsic beauty, harmony and interdisciplinarity in Einstein velocity addition law: gyrogroups and gyrovector space. Math. Interdisciplinary Res., 1(1):5–51, 2016. 101. Abraham A. Ungar. Novel tools to determine hyperbolic triangle centers. In Essays in Mathematics and its Applications: in Honor of Vladimir Arnold, Springer. Springer, New York, 2016. eds: Themistocles M. Rassias and Panos M. Pardalos. 102. Vladimir Variˇcak. Beitr¨age zur nichteuklidischen geometrie [contributions to non-euclidean geometry]. Jber. dtsch. Mat. Ver., 17:70–83, 1908. 103. Vladimir Variˇcak. Anwendung der Lobatschefskjschen Geometrie in der Relativtheorie. Physikalische Zeitschrift, 11:93–96, 1910.

Index addition, Einstein, signature (m,n), 182, 289, 316 addition, Einstein, special relativistic, 10, 289, 347, 349, 366 additive decomposition, 303, 309 additive relation, 259 ambient space, 61, 63, 69 associative law, 17 automorphic inverse property, 317, 330 automorphism, 18, 21, 34 automorphism, bi-gyrogroup, P, 156 automorphism, bi-gyrogroup, V, 242 automorphism, bi-gyrogroupoid, P, 130 automorphism, commuting, 34 automorphism, left, right, 130 ball, c-ball, 186 ball, Euclidean, 195 ball, matrix, 193 ball, spectral radius, 195 betweenness, 367 bi-bi-gamma identity, bi-gyrotriangle, 329 bi-boost, 115 bi-boost product representation, 319 bi-boost product, applications, 321 bi-boost SVD, 266 bi-boost, additive decomposition, 259, 260 bi-boost, application, 322, 323 bi-boost, B(P), 116 bi-boost, bi-gyrohyperbolic, 356 bi-boost, Galilei, 218, 297, 298 bi-boost, inverse, theorem, V, 214 bi-boost, Lorentz, 300 bi-boost, Lorentz, additive decomposition, Definition, 261 bi-boost, Lorentz, additive decomposition, Theorem, 260 bi-boost, Lorentz, application, 306

bi-boost, P and V, 233 bi-boost, parameter change, 186, 201 bi-boost, parameter composition, P, 121 bi-boost, product, 317 bi-boost, Product representation, 320 bi-boost, product, V, 218, 225 bi-boost, square, 132 bi-boost, theorem, V, 216 bi-boost, V, 209, 215 bi-gamma factor, commuting relations, 202, 205 bi-gamma factor, definition, 202 bi-gamma identities, 223, 249, 321 bi-gamma identities, theorem, 249 bi-gamma identity, 295 bi-gamma identity, bi-gyrotriangle, 332 bi-gamma, inequality, 294 bi-gamma, norm, 292 bi-gamma-gyration, right, 332 bi-gyration, 40, 119 bi-gyration decomposition, 114 bi-gyration decomposition, theorem, 123 bi-gyration decomposition, V, 204 bi-gyration inversion, P, 142 bi-gyration inversion, V, 228 bi-gyration, invariance property, 137 bi-gyration, reduction property, 145, 146, 150, 152, 154 bi-gyration, reduction property, left, 148 bi-gyration, reduction property, right, 149 bi-gyration, reduction, V, 244 bi-gyration, theorem, P, 155 bi-gyration, theorem, V, 237 bi-gyration, trivial, 125, 135, 181, 277 bi-gyration, uniqueness, 252 bi-gyrator, 123, 220 bi-gyroaddition, definition, 121 bi-gyroaddition, Einstein, 246 393

394

Index

bi-gyroaddition, Einstein, theorem, 247 bi-gyroaddition, theorem, P, 124 bi-gyroassociative law, bi-gyrogroup, 164 bi-gyroassociative law, bi-gyrogroupoid, 143 bi-gyroassociative law, left and right, bi-gyrogroup, 291 bi-gyroassociative law, left and right, theorem, 145 bi-gyroassociative law, left and right, V, 232 bi-gyroassociative law, theorem, 144 bi-gyroassociative law, V, 231 bi-gyroassociative, left and right, V, 246 bi-gyrocollinear, 367 bi-gyrocommutative law, bi-gyrogroup, 165, 291 bi-gyrocommutative Law, bi-gyrogroupoid, 140 bi-gyrocommutative law, theorem, 142 bi-gyrocommutative law, V, 228 bi-gyrocommutative, V, 245 bi-gyrodistance, 362, 380 bi-gyrodistance, invariance, 362, 363 bi-gyrodistance, symmetric, 363 bi-gyrogroup, 7, 169 bi-gyrogroup, Einstein, 182, 255, 289 bi-gyrogroup, Einstein, P, 182 bi-gyrogroup, Einstein, V, 289 bi-gyrogroup, gyration, definition, P, 166 bi-gyrogroup, gyration, definition, V, 250 bi-gyrogroup, gyrogroup, P, 169 bi-gyrogroup, gyrogroup, V, 255 bi-gyrogroup, operation, definition, 154 bi-gyrogroup, operation, definition, V, 241 bi-gyrogroup, operation, theorem, V, 242 bi-gyrogroup, P, 154 bi-gyrogroup, V, 241 bi-gyrogroupoid, definition, 121 bi-gyrohalf, 262, 281 bi-gyrohalf, Theorem, 281

bi-gyrohyperbolic bi-boost, 356 bi-gyrohyperbolic functions, 353 bi-gyroisometry, 380, 385 bi-gyroisometry, bi-rotation, 381 bi-gyroisometry, def., 381 bi-gyroisometry, left bi-gyrotranslation, 381 bi-gyroline, 363 bi-gyroline, definition, 364 bi-gyroline, reduction to gyrolines, 365 bi-gyromedian, 376 bi-gyromidpoint, 374 bi-gyromotion, 381, 382, 385 bi-gyromotion composition law, 382 bi-gyromotion, definition, 385 bi-gyroparallelogram, 378 bi-gyroparallelogram condition, 378, 379, 385 bi-gyroparallelogram law, 380 bi-gyropoint, 310 bi-gyroruler, 371, 372 bi-gyroruler equation, 372 bi-gyroruler Theorem, 373 bi-gyrosegments, definition, 364 bi-gyrosemidirect product group, 384, 385 bi-gyrosemidirect product group, theorem, 384 bi-gyrotranslation theorem, left, 256 bi-gyrotriangle, 376 bi-gyrotriangle bi-gamma identity, 329, 332 bi-gyrotriangle equality, 365, 369, 370 bi-gyrotriangle inequality, I, 294 bi-gyrotriangle inequality, II, 295 bi-gyrovector space, Einstein, 183, 291 bi-gyrovector space, Einstein, P, 182 bi-gyrovector space, Einstein, V, 289 bi-gyrovector space, P, 179 bi-gyrovector space, V, 275 bi-gyrovector Spaces, 170 bi-inner product, 327, 328, 331 bi-norm, relativistic, 324, 326

Index

bi-norm, relativistic, 324 bi-norm, relativistic spectral, 326 bi-norm, unit, time-space, 326 bi-rotation, 101, 111, 135, 194 binary operation, 21 cancellation law, left, 157 cancellation law, left and right, 144 cancellation law, left and right, V, 231 cancellation law, left-right, 53, 256 cancellation laws, basic, 33 cancellation, left, 25, 34 cancellation, left, general, 24 Cartesian-Beltrami-Klein model, 9, 15, 74, 75, 82 Cauchy-Schwarz inequality, 13, 61 coaddition, Einstein, 255, 290 coaddition, gyrogroup, 23 cogyroautomorphic inverse property, 45 commutative law, 17 commuting relation, 111–113, 115, 202, 205, 207, 208, 260 commuting relation, bi-gyrations and bi-rotations, 135 commuting relation, bi-gyrovector and bi-gamma, 190, 191, 202 commuting relation, SVD, 181 conformal, 78 cooperation, def., 23 cooperation, gyrogroup, 23 coreduction property, 44 covariance, def., 92 criterion, subgroup, 22, 38, 39 decomposition, additive, 4, 6, 259, 260 decomposition, bi-boost product, 123 decomposition, bi-gyration, 114, 158 decomposition, polar, 158 decomposition, polar, P, 119 decomposition, polar, V, 204

decomposition, singular value, 170 duality symmetry, 29, 36 eigenvalue, theorem, 194 Einstein addition, 11 Einstein addition domain extension, 14 Einstein addition, coordinate free, 10 Einstein addition, coordinate, example, 16 Einstein addition, coordinates, 14, 16 Einstein addition, restricted, 11 Einstein addition, signature, 14 Einstein addition, signature (m,n), 182, 289, 305, 316 Einstein addition, Velocity structure, 333 Einstein coaddition, 24, 83 Einstein coaddition, signature (m,n), 255, 290 Einstein groupoid, 11 Einstein half, 66 Einstein scalar multiplication,Velocity structure, 342 entangled, 299, 306, 307 entanglement part, 4, 261, 262 entanglement part, EP, 304 entanglement, geometric, 2, 8 entanglement, physics, 8 entanglement, quantum, 2 EP, entanglement part, 304 equi-block, 345 equi-block matrix, 333 equi-block structure, 333 equi-block structure, theorem, 340, 342 equi-column, 345 equi-column matrix, 333 equi-column structure, theorem, 347, 350 Erlangen program, 92, 99, 385 Euclidean c-ball, 186 Euclidean ball, 339 exclusion property, 128–130, 237, 240, 252, 254

395

396

Index

Galilean part, 261 Galilean part, GP, 304 Galilei transformation, 5 Galilei, transformation, 218 gamma factor, 11 gamma factor, left, right, 201 gamma factor, left, right, relation, 210 gamma identity, 12–14, 96 gamma identity, gyrotriangle, 332 geometry, bi-hyperbolic, 2, 7, 376, 381 geometry, bi-hyperbolic, signature, 385 geometry, Euclidean, 7, 381 geometry, hyperbolic, 2, 7, 381 GP, Galilean part, 304 Gram matrix, 197 group, 22 groupoid, 21 groups, commutative, def., 22 groups, def., 21 gyration, 18, 19, 40 gyration domain extension, 19 gyration even property, 42 gyration inversion, 20, 42 gyration properties, basic, 40 gyration reduction, 252 gyration reduction, left, 251 gyration reduction, property, left, theorem, 167 gyration reduction, property, right, theorem, 168 gyration reduction, property, theorem, 169 gyration reduction, right, 251 gyration respect gyroaddition, 20 gyration, even, 290 gyration, gyrogroup, definition, 166 gyration, inner product invariance, 20 gyration, inversion, 290 gyration, left, right, 119, 290 gyration, nested identity, 18 gyration, reduction property, 291

gyrator, 23, 166, 250, 256 gyrator axioms, 23 gyrator identity, 25, 26 gyroaddition, Einstein, 248 gyroassociative, 7 gyroassociative law, left, 18, 25, 27, 43 gyroassociative law, mixed, 44 gyroassociative law, right, 18, 43 gyroassociative law, theorem, 166 gyroassociative, theorem, 250 gyroautomorphic inverse property, 26, 49, 50, 54 gyroautomorphic inverse property, theorem, 49 gyroautomorphism, 18, 34, 251 gyroautomorphism group, 36 gyroautomorphism inversion law, 40 gyroautomorphism property, 61 gyroautomorphism, theorem, 167 gyrocommutative, 7 gyrocommutative law, 18 gyrocommutative law, theorem, 166 gyrocommutative, theorem, 250 gyrocovariance, 57 gyrocovariance, def., 98 gyrogroup, 7, 255 gyrogroup cooperation, 29, 44 gyrogroup cooperation, def., 23 gyrogroup equations, basic, 31 gyrogroup properties, first, 23, 24 gyrogroup, def., 22 gyrogroup, first properties, 24 gyrogroup, gyrocommutative, 49, 169 gyrogroup, gyrocommutative, def., 23 gyroinvariant, 79 gyroisometry, 93 gyroisometry, characterization, 94 gyroisometry, def., 93 gyroisometry, left gyrotranslation, 94 gyroisometry, unique decomposition, 96

Index

gyrolanguage, 12 gyrolines, 75 gyromidpoint, 71, 83 gyromotion, 97, 381 gyroparallelogram, 81, 85 gyroparallelogram addition law, 57 gyroparallelogram condition, 51, 52, 57, 82, 86 gyroparallelogram gyrocentroid, 81–83 gyroparallelogram gyrodiagonals, 81, 82 gyroparallelogram gyromidpoint condition, 83, 86 gyroparallelogram law, 83 gyroparallelogram symmetries, 82 gyroparallelogram, def., 81 gyroparallelogram, Einstein, 82 gyroparallelogram, opposite gyrosides, 85 gyropoint, 75 gyrosegment, 76 gyrosemidirect product, 36, 39–41, 97, 98 gyrosum inversion, 26, 40, 46 gyrotranslation, left, 33 gyrotranslation, right, 33 gyrotriangle, 376 gyrotriangle equality, 76, 93, 380 gyrotriangle inequality, 13, 62, 93, 380 gyrotrigonometry, 77, 78 gyrovector space, 7, 66 gyrovector space, abstract, 61 gyrovector space, def., 66 gyrovector space, Einstein, 67 gyrovector space, real inner product, 61 half-velocity, classical, 67 half-velocity, relativistic, 67 homogeneity property, 62, 279, 292 hyperbolic rotation, 355 identity map, 25 invariance, under boosts, 327

invariance, under Lorentz transformations, 328 isometry, characterization, 89 isometry, def., 88 isometry, direct, 91 isometry, Euclidean, 88 isometry, hyperbolic, 93 isometry, opposite, 91 isometry, translation, 89 isometry, unique decomposition, 90 isomorphism, bi-gyrovector space, 285 isomorphism, Lorentz group, 286 kinetic energy, 67 left gyrotranslation theorem, 54, 58 left gyrotranslation theorem, I, 30 left gyrotranslation theorem, II, 54 length contraction, 304 Lorentz transformation, 5 Lorentz Transformation, bi-gyration decomposition, V, 204 Lorentz transformation, decomposition, P, 118 Lorentz transformation, decomposition, V, 204 Lorentz transformation, inverse, P, 119, 120 Lorentz transformation, inverse, theorem, V, 215 Lorentz transformation, inverse, V, 214 Lorentz transformation, polar decomposition, P, 119 Lorentz Transformation, polar decomposition, V, 204 Lorentz transformation, product law, 139 Lorentz transformation, product, P, 138 Lorentz transformation, product, V, 228 matrix ball, 193 matrix, equi-block, 347 matrix, equi-column, 209, 310

397

398

Index

matrix, one-number, 310 matrix, one-number diagonal, 310 matrix, positive definite, 105, 106, 128, 130 matrix, pseudo-inverse, 177, 271 matrix, two-number diagonal, 347 monodistributive law, 65, 179, 275, 292 motions, Euclidean, 91 motions, hyperbolic, 97

positive definiteness, 61 product rule, bi-gyrohyperbolic matrix, 356 pseudo, Euclidean, 103 pseudo, rotation, 103

parallelogram, 80 parallelogram addition law, 86 parallelogram center, 80 parallelogram centroid, 80 parallelogram condition, 80, 86 parallelogram diagonals, 80 parallelogram law, 79 parallelogram midpoint condition, 86 parallelogram, def., 80 parameter change, theorem, 187 particle system, 310 particle, (m,n)-, 308 particle, disintegrated, 308, 311 particle, disintegrated, example, 315 particle, integrated, 308, 311 particle, integrated, example, 311, 312 particle, partially integrated, 344 particle, partially integrated, example, 345

scalar multiplication, Einstein, def., 66 Scalar Multiplication, Einstein, special relativistic, 366 scalar multiplication, orthogonal transformations, 292 scalar multiplication, P, 170 scalar multiplication, P, definition, 170 scalar multiplication, P, theorem, 171 scalar multiplication, P-V relationship, 284 scalar multiplication, properties, 179, 275 scalar multiplication, rotations, 283 scalar multiplication, special relativistic, 66 scalar multiplication, SVD, 273 scalar multiplication, trivial gyrations, 292 scalar multiplication, V, 262 scalar multiplication, V, definition, 262 scalar multiplication, V, theorem, 264

quadrangle, 80 quadrilateral, 80

rapidity, 355 reduction property, left, 18 reduction property, right, 18, 43 nested gyroautomorphism identity, 27 relativistically admissible, time-space, 324, norm, Euclidean, 279 325, 327 norm, inside-outside, 326 relativistically admissible, velocity, 9, 11, norm, orthogonally invariant, 194, 362, 17, 23, 203, 209, 301, 316, 326, 363 339 norm, relativistic, 214, 301, 302, 306 relativity of simultaneity, 304 norm, spectral, 186, 326 notation, column, 118, 120, 125, 138, 160, rotation, left, right, 130 215, 225, 229 scalar associative law, 61, 179, 275, 292 notation, displayed, 299 scalar distributive law, 61, 179, 275, 291 notation, inline, 299 scalar multiplication, 291 notation, matrix division, 258–260 scalar multiplication, Einstein, 66

Index

scalar-matrix transpose law, 179, 275, 292 scaling property, 61, 278, 292 semidirect product, 91, 92 singular value decomposition, 170, 263 singular values, 171, 197, 263 SO(m,n), definition, 104 SO(m,n), matrix representation, 105, 107, 110 SO(m,n), parametric representation, P, 111 space-part, 324 spectral c-ball, 185 spectral ball, 339 spectral norm, 186, 193 spectral norm, matrix, 193, 362 spectral norm, properties, 194 spectral radius, 193 spectrum, 193 stellar aberration, 86 sub-bi-gyrovector Spaces, 352 subgroup criterion, 22, 383 subparticle, 308 subpoint, 310, 360

subposition, 311 subtime, 311 subvelocity, 346 Sylvester’s identity, 334 telescopic gyration identity, 51 Thomas precession, 304 time dilation, 304 time-part, 324 time-space event, 324 time-space event, unit, 327 triangle equality, 88 triangle inequality, 88 trivial gyration, 18 trivial map, 25 two-sum identity, 70, 72 vector space, real inner product, 61 velocity, coordinate, 359 velocity, observer’s, 359 velocity, proper, 7, 359 velocity, relativistically admissible, 7 velocity, traveler’s, 359

399