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Relativity in Rotating Frames

Fundamental Theories of Physics
An International Book Series on The Fundamental Theories of Physics: Their Clarification, Development and Application Editor: ALWYN VAN DER MERWE, University of Denver, U.S.A.
Editorial Advisory Board: JAMES T. CUSHING, University of Notre Dame, U.S.A. GIANCARLO GHIRARDI, University of Trieste, Italy LAWRENCE P. HORWITZ, Tel-Aviv University, Israel BRIAN D. JOSEPHSON, University of Cambridge, U.K. CLIVE KILMISTER, University of London, U.K. PEKKA J. LAHTI, University of Turku, Finland ASHER PERES, Israel Institute of Technology, Israel EDUARD PRUGOVECKI, University of Toronto, Canada TONY SUDBURY, University of York, U.K. HANS-JÜRGEN TREDER, Zentralinstitut für Astrophysik der Akademie der
Wissenschaften, Germany
Volume 135

Relativity in Rotating Frames
Relativistic Physics in Rotating Reference Frames
Edited by Guido Rizzi
Politecnico di Torino and Istituto Nazionale di Fisica Nucleare, Torino, Italy and
Matteo Luca Ruggiero
Politecnico di Torino and Istituto Nazionale di Fisica Nucleare, Torino, Italy
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-90-481-6514-8

ISBN 978-94-017-0528-8 (eBook)

DOI 10.1007/978-94-017-0528-8

Printed on acid-free paper
All Rights Reserved © 2004 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2004 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

Contents

Preface

xi

Contributing Authors

xv

Introduction

xix

Acknowledgments

xxiii

Part I Historical Papers

1

Uniform Rotation of Rigid Bodies and the Theory of Relativity

3

Paul Ehrenfest

2

The existence of the luminiferous ether demonstrated by means of the

5

effect of a relative ether wind in an uniformly rotating interferometer

M. Georges Sagnac

1 Principles of the Method

5

2 Optical vortex effect

6

Part II Papers

1

The Sagnac Effect in the Global Positioning System

11

Neil Ashby

1 Introduction

11

2 Local Inertial Frames

12

3 The GPS

13

4 Relativity of Simultaneity

15

5 Time Transfer with the GPS

18

6 GPS Navigation Equations and the ECEF Frame

20

7 Sagnac-like effects due to rotation of the ECEF frame

25

8 Summary

27

v

vi

Contents

2

Space, Time and Coordinates in a Rotating World

29

Dennis Dieks

1 Introduction

29

2 The rotating frame of reference

31

3 Rods and clocks

32

4 Space and time without rods and clocks

33

5 Accelerating measuring devices

35

6 Space and time in the rotating frame

37

7 Simultaneity, slow clock transport and conventionality

38

8 The rotating Ehrenfest cylinder

40

3

The Hypothesis of Locality and its Limitations

43

Bahram Mashhoon

1 Introduction

43

2 Background

45

3 Length measurement

47

4 Discussion

53

Appendix: Null acceleration

54

4

Sagnac effect: end of the mystery

57

Franco Selleri

1 History: 1913 - 2003

57

2 The Sagnac Correction on the Earth Surface

60

3 Rotating Platforms

63

4 Absolute simultaneity in inertial systems

67

5 The impossible defense of orthodoxy

69

6 New proofs of absolute simultaneity

71

Appendix: A - The Equivalent Transformations

73

Appendix: B - The Inertial Transformations

74

5

Synchronization and desynchronization on rotating platforms

79

Guido Rizzi and Alessio Seraﬁni

1 Introduction

80

2 The many choices of synchronization in a physical reference frame 84

3 Sagnac effect and its universality

89

4 The time-lag as a “theoretical artefact”

89

5 The time-lag as an observable desynchronization

92

6 Exploiting Selleri gauge freedom

97

7 Conclusions

99

Appendix

100

6

Toward a Consistent Theory of Relativistic Rotation

103

Robert D. Klauber

1 Traditional Analysis Conundrums

104

2 Resolution of the Conundrums: Differential Geometry and Non-

time-orthogonality

114

3 Experiment and Non-time-orthogonal Analysis

122

Appendix: Deriving Sagnac Result from the Lab Frame

129

Contents

vii

7

Elementary Considerations of the Time and Geometry of Rotating

139

Reference Frames

Thomas A. Weber

1 Introduction

139

2 Time synchronization on rotating systems

140

3 Time and space coordinates on a rotating disk

142

4 Paradoxes

146

5 Synchronization and the Brillet and Hall experiment

149

6 Conclusion

151

8

Local and Global Anisotropy in the Speed of Light

155

Francesco Sorge

1 Introduction

155

2 Light Speed, Locality and Lorentz-invariance

157

3 The Byl et al. Experiment

158

4 The Byl et al. Experiment Revisited

159

5 Discussion

162

6 Concluding Remarks

164

9

Isotropy of the velocity of light and the Sagnac effect

167

José-Fernando Pascual-Sánchez Angel San Miguel Francisco Vicente

1 Introduction

167

2 The rotating disk and the Sagnac effect

169

3 Measurement of relative speeds in Minkowski space-time

171

4 Equivalent formulation of the problem

172

5 Reduction to the Minkowskian plane

174

6 Concluding remarks

177

10

The relativistic Sagnac effect: two derivations

179

Guido Rizzi Matteo Luca Ruggiero

1 Introduction

179

2 A little historical review of the Sagnac effect

181

3 Direct derivation: Sagnac effect for material and light particles

185

4 The Sagnac effect from an analogy with the Aharonov-Bohm effect 196

5 Conclusions

204

Appendix: Space-Time Splitting and Cattaneo’s Approach

205

11

Inertial forces: the special relativistic assessment

221

Donato Bini and Robert T. Jantzen

1 Introduction

221

2 Inertial forces in classical mechanics

223

3 Inertial forces geometrized

229

4 Application to rotating observers in Minkowski spacetime

233

5 Conclusions

236

Appendix: Adapted spacetime frames

237

12

Eppur, si muove !

241

viii

Contents

Lluís Bel

1 Galilean frames of reference

242

2 Uniformly rotating frames of reference

247

3 The Wilson and Wilson experiment

254

4 The Michelson-Morley experiment

258

13

Does anything happen on a rotating disk?

261

Angelo Tartaglia

1 Introduction

261

2 Posing and deﬁning the problem

262

3 Local measurements

263

4 Global measurements

266

5 Conclusion

271

14

Proper co-ordinates of non-inertial observers and rotation

275

Hrvoje Nikolic´

1 Proper non-inertial co-ordinates

275

2 Application to rotation

279

Appendix

282

15

Space geometry in rotating reference frames: A historical appraisal

285

Øvynd Grøn

1 Introduction

285

2 The discussion of 1910 and 1911 in Physikalische Zeitschrift

286

3 Einstein’s realization that the geometry on the rotating disk is non

Euclidean

288

4 Spatial geodesics on the rotating disk

297

5 Relativity of simultaneity and coordinates in rotating frames

298

6 What is the effect of the Lorentz contraction upon a disk that is put

into rotation?

301

7 Curved space and discussion of Einstein’s and Eddington’s analysis

of the rotating disk

305

8 Uniform contra rigid rotation

306

9 Relativistically rigid motion and rotation

307

10 The theory of elastic media applied to the rotating disk

309

11 The metric in a rotating frame as solution of Einstein’s ﬁeld equa-

tions

314

12 Kinematical solution of Ehrenfest’s paradox

315

13 Energy associated with tangential stress in a rotating disk

316

14 A rotating disk with angular acceleration

318

15 A rolling disk

320

16 The rotating disk and the Thomas precession

323

17 Contracted rotating disk

324

18 Conclusion

326

16

Quantum Physics in Inertial and Gravitational Fields

335

Giorgio Papini

1 Introduction

335

2 Quantum phases

336

3 Inertial ﬁelds in particle accelerators

344

Contents

ix

4 Maximal acceleration

350

5 Conclusions

355

17

Quantum Mechanics in a Rotating Frame

361

Jeeva Anandan and Jun Suzuki

1 Introduction

361

2 Lorentz and Galilei transformations in Quantum Mechanics

362

3 Non Relativistic Aspects of the Rotating Frame

364

4 Relativistic Aspects of the Rotating Frame

366

5 Conclusion

367

Appendix: Conventions and Notations

368

Jeeva Anandan (1948-2003)

369

18

On rotating spacetimes

371

Fernando de Felice

1 Introduction

371

2 A rotating spacetime

372

3 Basic measurements

373

4 The radial motion

374

5 The time-like geodesic motion

376

6 The behaviour of light

377

7 Conclusions

379

Part III Round Table

I

Dialogue on the velocity of light in a rotating frame

383

II Dialogue on synchronization and Sagnac effect

396

III Dialogue on the measurement of lengths in a rotating frame

411

IV Dialogue on the Brillet-Hall experiment

432

V Dialogue on quantum effects in rotating systems

438

VI Dialogue on non uniform motions and other details about Klauber’s

and Selleri’s challenges

443

Index

449

Preface
For today’s physicist, steeped in the Newtonian and post-Newtonian traditions, it takes a real effort of the imagination to realize that, for by far the longest stretch of time, the prevailing scientiﬁc efforts to understand the universe (to say nothing of pre- or non-scientiﬁc views, past or contemporary) were based on the concept that uniform circular motion is higher, nobler, and more natural than any other form of motion. Michel Blay has well described the situation:
Since Antiquity and more precisely since the elaboration of the Aristotelian conceptual outlook, circular motion was conceived as both the prime motion and the natural motion. This motion, for example that of the stars, proceeded from an internal principle and, contrary to violent motion, such as that of a stone that is thrown, did not presuppose the action of some exterior motor in order to continue. In the Aristotelian Cosmos, divided into two worlds, the movement of the stars belonged to the celestial sphere of perfect motions and incorruptible bodies, the motion of a thrown stone belonged to the sublunar sphere (the earth and its surrounding space) of more or less chaotic motions and bodies subject to decay.
Of course, in the ancient world the atomists opposed this viewpoint, their atoms falling straight downward perpetually through the void, with an occasion clinamen, or random deviation from their downward fall, to explain the formation of worlds such as ours. But, for whatever scientiﬁc or sociological reasons, the Aristotelian world view triumphed and atomism languished for almost two millennia.
Even Galileo, a determined opponent of the peripatetics, as the followers of Aristotle were called, succumbed to the appeal of circular motion in his ﬁrst attempts to formulate the concept of inertial motion. It was only with Descartes and then Newton that the law of inertia in its modern form triumphed, and uniform motion in a straight line came to be regarded as the natural standard, deviations from it being attributed to forces, notably gravitation as understood by Newton. Now it was uniform circular motion that required an external
xi

xii
explanation. And, as Newton’s famous bucket experiment demonstrated, there was a great dynamical difference between non-rotating and rotating frames of reference. Since we live on such a rotating frame of reference, this difference is of great practical as well as theoretical signiﬁcance. But as long as the Newtonian kinematical framework stood intact, both rotating and non-rotating observers shared a common, absolute time.
With the advent of the special theory of relativity and its new kinematics, the absolute time concept shattered, falling apart into two distinct concepts.
One is the proper time, associated with every time-like world-line and directly measurable by a clock travelling along that world-line. The proper time is an invariant but one that is no longer independent of the world-line. The time elapsed between two events depends on the history of the clock travelling between the two. However, the concept of proper time survives the transition to general relativity more-or-less unscathed.
In special relativity, one can also deﬁne a family of global times, one associated with each inertial frame of reference, but each depending on a conventional stipulation (such as the constancy of the one-way velocity of light in that frame) and only measurable indirectly, in terms of the appropriate proper time and proper length measurements in the frame in question. Once one ventures beyond the class of inertial frames, the concept of a global time becomes even more problematic, and the problem only becomes more acute when one ventures into the general theory of relativity. Solutions exist, such as the Gödel universe, for which it is impossible to formulate such a concept.
What are the implications of the new kinematics for rotating frame of reference? This question early became the subject of an intense discussion among relativists (to say nothing of their opponents!), a discussion that continues down to our times. This book constitutes an important and fascinating contribution to this ongoing discussion. One can hardly agree with everything found in it, since many of the authors disagree among themselves; and it is not the function of a preface to pass judgement on merit of the individual contributions. That must be left to each reader in the short run, and to the scientiﬁc community in the long run. But what is certain is that each reader will spend many absorbing hours reading the papers in this book, and perhaps even more trying to form and justify his or her opinions on the questions raised in them.
JOHN STACHEL

This book is in memory of Jeeva Anandan

Contributing Authors
Neil Ashby (1934) is Professor at the University of Colorado in Boulder, and Guest Researcher at National Institute of Standards and Technology, Boulder. His Ph.D. is from Harvard University (1961). He is interested in electrons in solids, solar system tests of relativity, relativistic effects on clocks in satellites.
Dennis Dieks (1949) is Professor of Philosophy and Foundations of Science at Utrecht University, The Netherlands. His Ph.D. is from Utrecht University (1981). He is interested in the foundations of physics: quantum theory and its interpretation, questions about space and time, and problems relating to causality and determinism.
Bahram Mashhoon (1947) is Professor of Physics at the University of Missouri - Columbia. His Ph.D. in Physics is from Princeton (1972). He is interested in theoretical physics, relativity and gravitation.
Franco Selleri (1946) is Professor of Physics at the University of Bari, Italy. He is interested in elementary particles, quantum physics, relativistic physics.
Guido Rizzi (1939) is Professor of Physics at Politecnico di Torino, Italy. He is interested in special and general relativity.
Alessio Seraﬁni (1977) is Ph.D. student at the University of Salerno, Italy. His laurea is from the University of Pisa, Italy (2001). He is interested in relativity, quantum mechanics and quantum optics.
Robert D. Klauber (1943) is retired (from industry) and part time teacher. His Ph.D. is from Virginia Polytechnic Institute and State University (1982). He is interested in relativity, cosmology and quantum ﬁeld theory.
xv

xvi
Thomas A. Weber (1934) is Professor of Physics at Iowa State University. His Ph.D is from the University of Notre Dame (1961). He is interested in mathematical physics, relativity, formal scattering theory, inverse problems, isospectral and phase equivalent hamiltonians.
Francesco Sorge (1961) is member of INFN in Padua, Italy, and teaches Mathematics and Physics at high schools. His Ph.D. in Physics is from the University of Padua (2002). He is interested in relativity and theory of measure, relativistic astrophysics and gravitational waves.
José-Fernando Pascual-Sánchez (1954) is Associate Professor of Applied Mathematics at the University of Valladolid, Spain. His Ph.D. is from the University of Valladolid (1983). He is interested in gravitomagnetism, GNSS and relativity, inhomogeneous cosmology.
Angel San Miguel (1956) is Associate Professor of Applied Mathematics at the University of Valladolid, Spain. His Ph.D. is from the University of Valladolid (1985). He is interested in relativity, geometrical mechanics.
Francisco Vicente (1956) is Associate Professor of Applied Mathematics at the University of Valladolid, Spain. His Ph.D. is from the University of Valladolid (1985). He is interested in relativistic mechanics, general relativity.
Matteo Luca Ruggiero (1975) is Ph.D. student at Politecnico di Torino, Italy. His laurea is from the University of Pisa, Italy (2000). He is interested in relativity, Einstein-Cartan theory, foundations of relativity, theoretical physics.
Donato Bini (1963) is Researcher at Istituto per le Applicazioni del Calcolo "M. Picone" CNR, Rome, Italy. His Ph.D. in Physics is from the University of Rome "La Sapienza" (1995). He is interested in differential geometry with applications to relativity, gravitational waves, spacetime splitting techniques, black hole perturbations, spinning test particles.
Robert T. Janzten (1952) is Professor at Villanova University, USA. His Ph.D. in Physics is from the University of California at Berkeley (1978). He is interested in differential geometry and Lie group theory with applications to relativity, spatially homogeneous cosmology, higher dimensional spacetimes with symmetry, spacetime splitting techniques.

Contributing Authors

xvii

Lluís Bel (1933) is Visiting Professor at the University of the Basc Country, Spain. His Ph.D. in Mathematical Sciences his from the University of Paris (1960). He is interested in dynamical systems, frames of reference, cosmology.

Angelo Tartaglia (1943) is Professor of Physics at Politecnico di Torino, Italy. He is interested in special and general relativity, gravitomagnetism.

Hrvoje Nikolic´ (1970) is Post-Doc at the University of Zagreb, Croatia. His Ph.D. is from the University of Zagreb (2001). He is interested in the theory of relativity, gravitation, foundations of quantum mechanics, quantum ﬁeld theory.

Øyvind Grøn (1944) is Professor of Physics at the University of Oslo, Norway. He is interested in the theory of relativity, cosmology, history of physics.

Giorgio Papini (1934) is Professor Emeritus at the University of Regina, Canada. His Ph.D. is from the University of Saskatechewan, Canada. He is interested in the interaction of inertia and gravitation with quantum systems, and in conformal invariant theories of Weyl-Dirac type.

Jeeva Anandan (1948) is Professor of Physics at the University of South Carolina. His Ph.D. in Physics is from the University of Pittsburgh (1978), his Ph.D. in Philosophy is from Oxford (1997). He is interested in the foundations of quantum mechanics, general relativity and gauge ﬁelds.

Jun Suzuki (1976) is Ph.D. student at the University of South Carolina. His B.A. is from the University of Tokio (2001). He is interested in the foundations of quantum theory.

Fernando de Felice (1943) is Professor of Relativity at the University of Padua, Italy. He is interested in relativity and theoretical astrophysics.

Introduction
When Alwyn Van der Merwe asked us to edit a book on relativistic physics in rotating reference frames, about a year ago, we had just published a paper on the space geometry of a rotating disk. Our interest in this ﬁeld dates back to some years ago, when one of us came across the Sagnac effect, and thoroughly studied this intriguing issue. Since then, we have been studying this ﬁeld with increasing interest and now we are well acquainted with the pertaining literature, which encompasses a lot of seminal discussions and debates that have stimulated the development of relativistic physics for a century.
We must admit that it was not easy to ﬁnd our way through the great number of papers: a real mare magnum without any guiding light, where the same topics were treated using very different approaches, and often different notations, which made it difﬁcult to compare them. Indeed, we think that many physicists believe that nothing new or interesting can be found in this ﬁeld, and the only open questions are a matter of philosophy rather than of some interest in the current development of theoretical and experimental physics. On the contrary, carefully digging in the literature, we discovered problems still controversial, viewpoints in contention, and open issues pertaining to the very foundations of relativity.
Therefore a monograph on these subjects seemed to us really necessary, not only to tidy up a little the various papers published here and there over the years, but also to critically re-examine and discuss the most controversial issues. For these reasons we accepted Van der Merwe’s kind invitation with enthusiasm.
We started our work by collecting contributions from scientists who, though having different and at times opposite viewpoints on these subjects, share a common physical-mathematical background. We asked them to write their papers with a clear and plain style and, also, with a somewhat pedagogical aim: according to our purposes, this book should be a guide and a reference for all those who, now and in the future, have an interest in this ﬁeld. As you
xix

xx
well know, the higher the ambitions, the greater the efforts; as to the results, only the reader can judge.
We deliberately decided to conﬁne ourselves to special relativity, and not to consider rotating frames in curved space-times. This would have enormously increased the material at our disposal: on the other hand we found it expedient (and easier, of course) to understand the crucial problems and concepts in ﬂat space-time, avoiding the formal complications deriving from gravitation.
The ﬁrst historical contributions to the study of rotating frames in relativity date back to ﬁrst decades of the last century, when the papers by Ehrenfest and Sagnac shattered the foundations of the brand-new theory of relativity: these papers, translated into English, are published in our book, as an ideal beginning to this long-standing debate.
A glance at the contributing papers shows that, even now, a century later, interest in these problems is not purely academic or philosophical. In his paper, Ashby explains the relevance of the relativistic study of rotation for a modern technical device, such as the Global Positioning System. After all, we must remember that we are living in a rotating frame, the Earth, and every experimental expectation, based on a relativistic approach, should take this into account as an obvious fact. Another fundamental issue, developed by Mashhoon, is the so-called hypothesis of locality and its limitations in accelerating systems. However, this hypothesis, together with other fundamental ones, are questioned by those who ﬁnd, in the relativistic approach to rotation, arguments against the self-consistency of the theory. For instance, this is the case of Klauber and Selleri who, though adopting different approaches, claim that the special theory of relativity is not valid when it is applied to rotating frames, and to this purpose they raise several stimulating arguments. The standard formulation of the theory is defended against these ”attacks”, by Dieks, Grøn, Weber, Rizzi and Seraﬁni, and an interesting debate develops about the fundamental problems of measurements of space, time, synchronization, that also involves Nikolic´, Bel and Tartaglia. Other topics related to these fundamental issues, such as the isotropy of the velocity of light and the universality of the Sagnac effect for matter and light beams, are dealt with in the papers by Sorge, Pascual-Sánchez, San Miguel and Vicente, Rizzi and Ruggiero. The relativistic approach to inertial forces and the role of rotating observers are examined by Bini and Jantzen, while the mathematical properties of rotating space-times are studied by De Felice. Finally, halfway between the classical theory of relativity and the quantum world, the quantum-inertial effects are thoroughly described in the papers by Papini, Anandan and Suzuki.
In order to give the reader a deeper insight into the most controversial issues, we organized a sort of virtual round table. After the publication of the drafts of the contributing papers at our web site, we asked the authors to comment on the papers, and confront the various viewpoints. Then, we collected their

INTRODUCTION

xxi

comments, which resulted in a lively on-line discussion. The dialogues that you can read at the end of this book are based on this discussion. However the raw contributions of the authors to the on-line debate, which could not be published as they were, have been supplemented by fragments that we borrowed from their papers. During this editing job, we aimed at composing the material at our disposal to obtain something like a real discussion about the main topics of the book. We are well aware of the arbitrariness of this job; however we did it with the greatest care in order to accurately quote the ideas and the opinions of the authors. The ﬁnal result should not be considered just as an appendix, but as an integral and fundamental part of the book: we have attempted to offer the reader a vademecum to ﬁnd her/his way through the papers. Indeed, the round table can also be read independently of the papers, to get a bird’s eye view of the main subjects, or it can be read after the papers, for a direct comparison of the viewpoints on the most controversial and interesting topics. The underlying aim, which is actually the aim of the whole book, is to offer the reader the possibility of understanding these opinions, and the subtleties on which their differences are based: to this end, we have tried to write as clearly as possible, having in mind this pedagogical purpose. We hope that, in this way, the reader can shape his own ideas, which is our ultimate goal.
We do not know whether we have succeeded or not: what we know for sure is that, at the end of this work we feel richer in knowledge and also in doubts, and ready and willing to go on in this stimulating ﬁeld. If you share these feelings, there is a chance that we did not fail in our purposes; however, if we bored you, be assured, we did not do it on purpose.

Last, but deﬁnitely not least, a thought for Prof. Jeeva Anandan, who passed away during the making of this book, and whose last paper we are honoured to publish here: a great loss for all physicists, and one which we feel deeply, ﬁrst of all because he was one of our collaborators, then because he wrote, during his long scientiﬁc activity, many important and fundamental papers about the main matter of this book, such as the Sagnac effect and relativistic rotation. This book is dedicated to him.

Guido Rizzi and Matteo Luca Ruggiero Torino, September 15, 2003

Acknowledgments
We are grateful to Alwyn Van der Merwe, for giving us the opportunity of editing this book, and to all contributing authors, for their willingness, patience and clever collaboration during all phases of the project. During our work, friendly relationships were established among all collaborators: it is all thanks to them that we have succeeded in carrying out this project. In addition, we also thank those who could not participate in our book, because of previous commitments, but who strongly encouraged our work.
A special word of thanks to John Stachel, who wrote the Preface with excellent insight.
We would also like to thank the staff at Kluwer Academic Publisher, in particular publishing editor Sabine Freisem and her assistant Kirsten Theunissen, and Berendina Schermers-van Straalen who helped us for the copyright permissions.
We thank Alessio Seraﬁni and Maria Blandin Savoia, for carefully reading the virtual round table, and Ugo Montrucchio for translating the historical papers. G Rizzi and ML Ruggiero
Of my indebtedness to my mother, my father and my brother Marco, for their encouragement, and to Dara, for her inﬁnite patience, I can make no adequate acknowledgments. ML Ruggiero
xxiii

I
HISTORICAL PAPERS

Chapter 1
UNIFORM ROTATION OF RIGID BODIES AND THE THEORY OF RELATIVITY
Paul Ehrenfest∗
St. Petersburg, September 1909
In order to generalize the relativistic kinematics of rigid bodies in rectilinear uniform motion to whatever kind of motion, following Minkowski’s ideas we obtain the following statement:
To say that a body remains relativistically rigid means: it deforms continuously by arbitrary motion so that each of its inﬁnitesimal elements Lorentz contract (relative to its rest length) all the time in accordance with the instantaneous velocity of each of its elements, as observed by an observer at rest.
According to me, as I am going to explain, the consequences of this statement, applied to a very simply motion, result in a contradiction.
Recently, Born2 has given a deﬁnition of relativistic rigidity, which applies to all motions. Born’s deﬁnition - which is in agreement with the relativistic principles - is not based on the system of measurements performed by an observer at rest; rather, its deﬁnition is based on the (Minkowskian) measurements performed by a continuum of inertial observers: as measured by these observers, co-moving with each point of an arbitrarily moving body, each element of the body remains undeformed.
Both deﬁnitions of relativistic rigidity, as far as I can understand, are equivalent.
∗Translated from Physikalische Zeitschrift, 10, 918 (1909), Courtesy of S. Hirzel Verlag. 2M. Born, Die Theorie des starren Electrons in der Kinematik des Relativitäts Prinzipes, Ann. d. Phys, 30, 1 (1909), Phys. Zeits., 9, 844 (1908).
3 G. Rizzi and M.L. Ruggiero (eds.), Relativity in Rotating Frames, 3-4. © 2004 Kluwer Academic Publishers.

4

P. Ehrenfest

Let me show a very simple kind of motion, for which the previous deﬁnition bring about a contradiction: namely, the motion I am going to refer to is the uniform rotation around a ﬁxed axis.

Consider a relativistically rigid cylinder with radius R and height H. It is given a rotating motion around its axis, which ﬁnally becomes constant. As measured by an observer at rest, the radius of the rotating cylinder is R . Then R has to fulﬁll the following two contradictory requirements:
1 The circumference of the cylinder must obtain a contraction
2πR < 2πR

relative to its rest length, since each of its elements moves with an instantaneous velocity R ω.
2 If one considers each element along a radius, then the instantaneous velocity of each element is directed perpendicular to the radius. Hence, the elements of a radius cannot show any contraction relative to their rest length. This means that: R =R

Chapter 2
THE EXISTENCE OF THE LUMINIFEROUS ETHER DEMONSTRATED BY MEANS OF THE EFFECT OF A RELATIVE ETHER WIND IN AN UNIFORMLY ROTATING INTERFEROMETER
M. Georges Sagnac∗
introduced by M.E. Bouty
1. Principles of the Method
I let a horizontal platform rotate uniformly, at one or two revolutions per second, around a vertical axis; on the platform I have ﬁrmly ﬁxed the pieces of an interferometer equal to that I used in my previous experiments, described in 1910 (Comptes rendus, 150, 1676). The two interfering beams, which are reﬂected by four mirrors placed on the rim of the rotating platform, are superimposed after the propagation in opposite directions along the same circular horizontal circuit which encloses an area S. The rotating system contains also the light source, a little electric lamp, the receiver, a ﬁne grain photographic plate which records the interference fringe. On the photographs d and s, obtained respectively by a dextrorsum and a sinistrorsum rotation with the same frequency, the centre of the central fringe has different positions. I measure this displacement with respect to the centre of interference.
∗Translated from Comptes rendus de l’Académie des sciences,157, 708 (1913), Courtesy of Elsevier-France. 5
G. Rizzi and M.L. Ruggiero (eds.), Relativity in Rotating Frames, 5-7. © 2004 Kluwer Academic Publishers.

6

M. G. Sagnac

1.1 First Method
On d and then on s, I look for the position of the central fringe with respect to the images of the micrometrical vertical traits in the focal plane of the collimator.

1.2 Second Method
I measure directly the distance between the vertical central fringe of a d photograph and the central fringe of a s photograph exactly contiguous to the ﬁrst, below a sharp horizontal separation line. I get these photographs in a direct way without touching the photographic chassis, giving, before each of the images d and s, the contiguous positions that correspond to the illuminating slit with cutting horizontal edges, in the focal plane of the collimator.

2. Optical vortex effect
The fringe shift z of the interference centre that I measured by the method just outlined, turns out to be a particular case of the optical vortex effect deﬁned in my previous works (Congrés de Bruxelles de septembre, 1910 I, 217; Comptes Rendus, 152 1911, 310; Le Radium, VIII, 1911, 1) and which, according to the current ideas, should be thought of as a direct manifestation of the luminiferous ether.
In a system, moving as a whole with respect to the ether, the time of propagation between any couple of points should be modiﬁed as though the system were at rest under the action of an ether wind, whose relative velocity turns out to be equal and opposite to the one of those points, and dragging the light just like the atmosphere wind drags the sound waves.
The observation of the optical effect of the wind relative to the ether will constitute a proof of the ether, in the same way as the observation of the inﬂuence of the wind relative to atmosphere - in a moving system - on the speed of sound allows to deduce the existence of the atmosphere around the moving system, if no other effects are present.
The necessity of getting from the same point-like source the light waves that are recombined in another point to obtain interference, cancels the ﬁrst order interference effect due to the motion of the whole optical system, unless the matter that drags the ether provokes a circulation C of ether in the light path, spanning an area S, that is an ether vortex bS (Comptes rendus, 141, 1905, 1220; 1910 and 1911, loc.cit.). I showed by interferometrical techniques (1910 and 1911 loc.cit.), using a circuit having a vertical projection of 20 m2 that the ether dragging near the ground does not produce a vortex density greater than 1/1000 radians per second.

Luminiferous ether demonstrated in a uniformly rotating interferometer

7

In a horizontal optical circuit, at a given latitude α, the daily terrestrial rotation, if the ether is at rest, should produce a ether-relative vortex whose density is 4π sin α/T , where T is the period of the sidereal day, or 4π sin α/86164, which is noticeably smaller than the upper limit 1/1000 that I established using a vertical circuit. I hope to be able to ﬁnd out whether the corresponding small optical vortex effect exists or not.
At ﬁrst, it was easier to ﬁnd out the proof of the existence of the ether using a small rotating optical circuit. A frequency N of two revolutions per second produces a vortex density b = 4πN , that is 25 radians per second. A dextrorsum uniform rotation of the interferometer produces a sinistrorsum ether wind; the rotation causes a backward shift x of the phase of the beam T, whose propagation around the area S is dextrorsum, and an equal forward shift of the phase of the counter-propagating beam R, so that the relative displacement of the fringes turns out to be 2x. The displacement z that I notice between a photograph s and a photograph d should be twice the previous one. Hence, using the value of x that I gave before (loc.cit. 1910 and 1911), I obtained:

bS 16πN S

z = 4x = 4 =

λV0

λV0

where V0 is the velocity of light in vacuum and λ is the wavelength of light. Using a frequency N = 2Hz and an area S = 860 cm2, the displacement z (for the indigo light) turns out to be 0.07, and it can be easily seen; the interfringe is from 0, 5 mm to 1 mm.
The interference displacement z, which is a constant fraction of the interfringe for a given frequency of rotation N , is not visible on the photographs when the fringes are too close; this shows that the observed effect depends on a phase difference due to the rotational motion of the system, and the displacement of the interference centre, observed by a comparison between a photograph d and a photograph s, does not depend on the random or elastic displacement of the optical pieces during rotation (thanks to the counter-screws that block the regulation screws of the optical system).
An air vortex, produced above the interferometer by a ventilator with vertical axis (blowing down the air towards the interferometer) does not displace the interference centre, thanks to the careful regulated superimposition of the counter-propagating beams. The air vortex, similar but less strong, produced by the rotation of the interferometer, does not cause any noticeable effect.
Hence, the observed interference effect turns out to be the optical vortex effect due to the motion of the system with respect to the ether and it is a direct proof of the existence of the ether, which is a necessary support for the light waves of Huygens and Fresnel.

II
PAPERS

Chapter 1
THE SAGNAC EFFECT IN THE GLOBAL POSITIONING SYSTEM

Neil Ashby
Department of Physics, UCB 390. University of Colorado. Boulder, CO 80309-0390 USA. n ashby@mobek.colorado.edu

Abstract

In the Global Positioning System (GPS) the reference frame used for navigation is an earth-centered, earth-ﬁxed rotating frame, the WGS-84 frame. The time reference is deﬁned in an underlying earth-centered locally inertial frame, freely falling with the earth but non-rotating, with a time unit determined by atomic clocks at rest on earth’s rotating geoid. Therefore GPS receivers must apply signiﬁcant Sagnac or Sagnac-like corrections, depending on how information is processed by the receiver. These corrections can be described either from the point of view of the local inertial frame, in which light travels with uniform speed c in all directions, or from the point of view of an earth-centered rotating frame, in which the Sagnac effect is described by terms in the fundamental scalar invariant that couple space and time. Such corrections are very important for comparing time standards world-wide.

1. Introduction
The purpose of the Global Positioning System (GPS) is accurate navigation on or near earth’s surface. GPS also provides an accurate world-wide clock synchronization and timing system. Most GPS users are interested in knowing their position on earth; the developers of GPS have therefore adopted an Earth-Centered, Earth-Fixed (ECEF) rotating reference frame as the basis for navigation. Speciﬁcally, in the WGS-84(873) frame, the model earth rotates about a ﬁxed axis with a deﬁned rotation rate, ωE = 7.2921151247 × 10−5 rad s−1.[1],[2]
In an inertial frame, a network of self-consistently synchronized clocks can be established either by transmission of electromagnetic signals that propagate

11
G. Rizzi and M.L. Ruggiero (eds.), Relativity in Rotating Frames, 11-28. © 2004 Kluwer Academic Publishers.

12

N. Ashby

with the universal constant speed c (this is called Einstein synchronization), or by slow transport of portable atomic clocks. On the other hand it is wellknown[3] that in a rotating reference frame, the Sagnac effect prevents a network of self-consistently synchronized clocks from being established by such processes. This is a signiﬁcant issue in using timing signals to determine position in the GPS. The Sagnac effect can amount to hundreds of nanoseconds; a timing error of one nanosecond can lead to a navigational error of 30 cm.
To account for the Sagnac effect, a hypothetical non-rotating reference frame is introduced. Time in this so-called Earth-Centered Inertial (ECI) Frame is adopted as the basis for GPS time; this is discussed in Section 2. Of course the earth’s mass encompasses the origin of the ECI frame and has signiﬁcant gravitational effects. To an extremely good approximation in the GPS, however, gravitational effects can be simply added to other effects arising from special relativity. In this article gravitational effects will not be considered. Even time dilation, which is an effect of second order in the small parameter v/c, where v is the velocity of some clock, will be neglected. I shall conﬁne this discussion to effects which are of ﬁrst order (linear) in velocities. The Sagnac effect is such an effect.
A description of the GPS system, of the signal structure, and the navigation message, needed to understand how navigation calculations are performed, is given in Section 3. In comparing synchronization processes in the ECI frame with those in the ECEF frame, taking into account relativity principles, it becomes evident that the Sagnac effect is a manifestation of the relativity of simultaneity. Observers in the rotating ECEF frame using Einstein synchronization will not agree that clocks in the ECI frame are synchronized, due to the relative motion. In fact observers in the rotating frame cannot even globally synchronize their own clocks, due to the rotation. This is discussed in Section 4. Section 5 discusses Sagnac corrections that are necessary when comparing remote clocks on earth by observations of GPS satellites in common-view. Section 6 introduces the GPS navigation equations and discusses synchronization processes from the point of view of the rotating ECEF frame. Section 7 develops implications of the fact that GPS navigation messages provide satellite ephemerides in the ECEF frame.

2. Local Inertial Frames
Einstein’s Principle of Equivalence allows one to discuss frames of reference which are freely falling in the gravitational ﬁelds of external bodies. Sufﬁciently near the origin of such a freely falling frame, the laws of physics are the same as they are in an inertial frame; in particular electromagnetic waves propagate with uniform speed c in all directions when measured with standard rods and atomic clocks. Such freely falling frames are called locally inertial

The Sagnac effect in the GPS

13

frames. For the GPS, it is very useful to introduce such a frame that is nonrotating, with its origin ﬁxed at earth’s center, and which falls freely along with the earth in the gravitational ﬁelds of the other solar system bodies. This is called an Earth-Centered Inertial (ECI) frame.
Clocks in the GPS are synchronized in the ECI frame, in which self - consistency can be achieved. Thus imagine the underlying ECI frame, unattached to the spin of the earth, but with its origin at the center of the earth. In this non-rotating frame a ﬁctitious set of standard clocks is introduced, available anywhere, all of them synchronized by the Einstein synchronization procedure and running at agreed rates such that synchronization is maintained. These clocks read the coordinate time t. Next one introduces the rotating earth with a set of standard clocks distributed around upon it, possibly roving around. One applies to each of the standard clocks a set of corrections based on the known positions and motions of the clocks. This generates a “coordinate clock time" in the earth-ﬁxed, rotating system. This time is such that at each instant the coordinate clock agrees with a ﬁctitious atomic clock at rest in the local inertial frame, whose position coincides with the earth-based standard clock at that instant. Thus coordinate time is equivalent to time which would be measured by standard clocks at rest in the local inertial frame. [4]
In the ECEF frame used in the GPS, the unit of time is the SI second as realized by the clock ensemble of the U. S. Naval Observatory, and the unit of length is the SI meter. In summary, the reference frame for navigation is the rotating WGS-84 frame, but clocks are synchronized in the underlying hypothetical ECI frame with a unit of time deﬁned by clocks (essentially on the geoid) and a unit of length determined by the deﬁned value of the speed of light, c = 299792458 m/s.

3. The GPS
The Global Positioning System can be described in terms of three principal “segments:" a Space Segment, a Control Segment, and a User Segment. The Space Segment consists essentially of 24 satellites carrying atomic clocks. (Spare satellites and spare clocks in satellites exist.) There are four satellites in each of six orbital planes inclined at 55◦ with respect to earth’s equatorial plane, distributed so that from any point on the earth, four or more satellites are almost always above the local horizon. Tied to the clocks are navigation and timing signals that will be discussed below.
The Control Segment is comprised of a number of ground-based monitoring stations which continually gather information from the satellites. These data are sent to a Master Control Station in Colorado Springs, CO, which analyzes the constellation and projects the satellite ephemerides and clock behav-

14

N. Ashby

ior forward for the next few hours. This information is then uploaded into the satellites for retransmission to users.
The User Segment consists of all users who, by receiving signals transmitted from the satellites, are able to determine their position, velocity, and the time on their local clocks.
The timing signals transmitted from each satellite are right circularly polarized. A carrier signal of frequency 1.542 MHz is modulated with a series of phase reversals; these phase reversals carry information bits from the transmitter to the receiver. Such phase reversals are conceptually important because the phase of an electromagnetic wave is a relativistic scalar. The phase reversals correspond to physical points in spacetime at which - for all observers - the electric and magnetic ﬁelds vanish.
The navigation message contained in these bit streams include values of parameters from which the receiver can compute the satellite’s position in the rotating ECEF frame, as a function of time of transmission. Also the GPS time on the satellite clock is indicated by a particular phase reversal in the sequence. A receiver distinguishes the signal from a particular satellite by comparing the bit streams, that are unique to each satellite, with bit streams generated by electronic circuitry within the receiver.
Additional information contained in the messages includes an almanac for the entire satellite constellation, information about satellite vehicle health, and information from which Universal Coordinated Time as maintained by the U. S. Naval Observatory–UTC(USNO)–can be determined.
The GPS is a navigation and timing system that is operated by the United States Department of Defense (DoD), and therefore has a number of aspects to it which are classiﬁed. Several organizations monitor GPS signals independently and provide services from which satellite ephemerides and clock behavior can be obtained. Accuracies in the neighborhood of 5-10 cm are not unusual. Carrier phase measurements of the transmitted signals are commonly done to better than a millimeter.
For purposes of the remainder of this article, I shall think of a signal from a GPS satellite as containing within itself information about the position and time of a transmission "event". The position is speciﬁed in the rotating ECEF frame. GPS time is time in an underlying local inertial frame. The signal propagates with speed c in a straight line in the ECI frame to the receiver, where it is decoded and its arrival time tR is compared to the time of transmission tT . The receiver can then form the so-called pseudoranges

ρ = c(tR − tT ).

(1.1)

A receiver continually forms such pseudoranges for each satellite being observed. A signal can be imagined abstractly as propagating with speed c from transmitter to receiver in a straight line in the ECI frame, with position and

The Sagnac effect in the GPS

15

u’

v

ux’

dx’

Figure 1.1. Synchronization by transmission of a signal

time of the transmission event “known" by the receiver. Possible clock biases in the receiver prevent the GPS time of the reception event from being known a priori.

4. Relativity of Simultaneity

To establish the connection between the Sagnac effect and the relativity of simultaneity, consider an observer moving with velocity v in the x direction relative to an inertial frame such as the ECI frame. To be speciﬁc, one can imagine measurements of unprimed quantities such as v and signal velocity u to be performed in the ECI frame, while primed quantities such as u are measured in the rest frame of the moving observer. Referring to Figure 1.1, let a signal be travelling with speed components (ux, uy) (measured in the moving observer’s frame). The vertical lines represent planes at x and x +dx . The signal travels a distance dx in the x direction and the moving observer desires to use this signal to transfer time from clocks in the plane at x to clocks in the plane at x + dx . Here I am neglecting higher-order terms in the velocity so dx = dx , there being no appreciable Lorentz contraction. Let the components of signal speed in the ECI frame be (ux, uy). The well-known Lorentz transformations for speed include the expression

ux

=

ux + v

1

+

uxv c2

.

(1.2)

16

N. Ashby

The terms in the denominator of this expression arise from the time-component of the ordinary Lorentz transformation. In particular the second term in the denominator arises from the relativity of simultaneity, a consequence of the constancy of the speed of light. We wish to compare the propagation time of this signal, measured by the moving observer, with the propagation time measured in the ECI frame. The analysis is performed in the ECI frame.
If the moving observer moves a distance vdt in time dt, then the total distance travelled by the signal in the x-direction is uxdt, which is comprised of two contributions: the distance dx, plus the distance vdt required to catch up to the plane at x + dx . Thus

uxdt = dx + vdt,

(1.3)

and therefore the time required is

dx

dt =

.

(1.4)

ux − v

But from the expression for the Lorentz transformation of speed, keeping only terms of linear order in v,

ux

−v

≈

ux

1

+

uxv c2

.

(1.5)

and therefore

dx vdx

dt = + ux

c2

.

(1.6)

The ﬁrst term in this result is just the time required, in the moving frame, for

the signal to travel from the x plane to the x + dx plane. If the moving

observer ignores the motion relative to the ECI frame, this would be the time

used to synchronize clocks in the x + dx plane to clocks in the x plane. The

second term is the additional time required to synchronize the clocks in the ECI

frame. Note that in this second term, the value of ux has cancelled out, so that the value of the signal speed is irrelevant. The signal could be a light signal

travelling in a ﬁber of index of refraction n, or it could even be an acoustic

signal. The signal speed could even be variable, the last term would not be

affected.

Consider for example an optical ﬁber loop of length L and index of refrac-

tion n which by means of a system of pulleys is made to move with speed v

around in a closed circuit, relative to an inertial frame. The circuit itself could

be of any shape, such as a ﬁgure 8 or an oval. In such a case it is not useful

to speak of rotation, although Eq. 1.6 applies to the rotational case as well.

Eq. 1.6 applies to each inﬁnitesimal segment of the moving loop, since one

can imagine a sequence of moving reference frames each of which is instan-

taneously at rest with respect to the moving ﬁber loop and in which Eq. 1.6 is

The Sagnac effect in the GPS

17

valid. If a signal travels around the loop in a direction parallel to the velocity, then from Eq. 1.6, the total time required for the signal to make one circuit is

dx vL Δt+ = dt = ux + c2 ,

(1.7)

and the time required for the signal to make one circuit in the direction opposite

to the velocity is

dx vL Δt− = ux − c2 ,

(1.8)

The difference is

2vL

Δt = Δt+ − Δt− = c2 ,

(1.9)

and for two counterpropagating monochromatic beams this can be converted

into an observable interference fringe shift. If the beams are recombined in the

ECI frame where they have angular frequency ω, then the phase difference will

be

Δφ = ωΔt .

(1.10)

The Sagnac effect in a moving ﬁber loop is independent of the ﬁber’s index of refraction or of the shape of the loop. This has been conﬁrmed in recent experiments.[5]
For example for electrons of energy E = ω, the phase difference will be

2EvL Δφ = c2 .

(1.11)

Interference experiments with electrons have been reported in reference [6],

which also has a comprehensive discussion of the many different points of

view of the Sagnac effect that can be taken.

In the GPS, a decision was made to synchronize GPS clocks in the ECI

reference frame. The above discussion demonstrates that observers on earth, in

the ECEF frame, must apply a “Sagnac" correction (the second term in Eq. 1.6)

to their synchronization processes in order to synchronize their clocks to GPS

time.

The correction can be generalized slightly by noting that the distance dx is

in the same direction as the relative velocity v. If dr is the vector increment

of path in the direction of signal propagation, then the Sagnac correction term

can be written

v · dr dtSagnac = c2 .

(1.12)

For applications in the GPS, it is useful to describe this correction term another way, in terms of accounting for motion of the receiver during propagation of signals from transmitters to receivers. Henceforth only signals propagating

18

N. Ashby

with speed c will be considered. This assumption also applies to measurements made locally by the moving observer in the ECEF frame, since at each instant the measurements of distance and time intervals are the same as they would be in an inertial frame which instantaneously coincides with the observer in the ECEF frame and which moves with the instantaneous velocity v of the ECEF observer. In Eq. 1.3, the velocity v is present to account for the fact that the signal must catch up to the position at x + dx which is moving with velocity v, and to ﬁrst order in the small quantity v/c leads directly to the Sagnac correction term in Eqs. 1.3 and 1.12. The Sagnac correction can thus be interpreted as an effect which arises in the ECEF frame when one accounts for motion of the receiver during propagation of the electromagnetic signal with speed c.

5. Time Transfer with the GPS

In the GPS navigation is accomplished by means of signals from four or
more satellites, whose arrival times are measured at the location of the receiver. I now consider one such signal in space, transmitted from satellite position rT at GPS time tT . Let the receiver position at GPS time tT be rR, and let the receiver have velocity v in the ECI frame. Let the signal (considered abstractly as a pulse) arrive at the receiver at time tR. During the time interval Δt = tR − tT , the displacement of the receiver is vΔt. Since the signal travels with speed c, the constancy of the speed of light c implies that

c2(Δt)2 = (rR + vΔt − rT )2 .

(1.13)

To simplify the equation, I deﬁne

R = rR − rT . Then to leading order in v,
c2(Δt)2 = (R + vΔt)2 ≈ R2 + 2v · RΔt .

(1.14) (1.15)

Taking the square root of both sides of Eq. (1.15) and again expanding to leading order in v gives

cΔt = R + v · RΔt . R

(1.16)

This equation can be solved approximately for Δt to give

R v·R

Δt = + c

c2

.

(1.17)

The second term in Eq. 1.17 is the Sagnac correction term, which arises when one accounts for motion of the receiver while the signal propagates from transmitter to receiver. This is illustrated in Figure 1.2.

The Sagnac effect in the GPS

19

ω

v Δt

c Δt R

tR = tT +

_R_ c

+

_R_._v_ c2

t*

T

(Sagnac correction)

Figure 1.2. Sagnac correction arising from motion of the ECEF observer.

Suppose that the receiver is ﬁxed to the surface of the earth, at a well-

surveyed location so that the receiver position rR is well known at all times.

The velocity of the receiver will be just that due to rotation of the earth with

angular velocity ωE, so

v = ωE × rR ,

(1.18)

We take rR to be the vector from earth’s center to the receiver position. Then the Sagnac correction term can be rewritten as

ΔtSagnac

=

ωE

× rR c2

·R

=

2ωE c2

·

1 2 rR × R

.

(1.19)

The quantity 2ωE/c2 has the value

2ωE c2

=

1.6227 × 10−21

s/m2

=

1.6227 × 10−6

ns/km2 .

The last factor in Eq. 1.19 can be interpreted as a vector area A:

(1.20)

1 A = 2 rR × R .

(1.21)

The only component of A which contributes to the Sagnac correction is along earth’s angular velocity vector ωE, because of the dot product that appears in the expression. This component is the projection of the area onto a plane normal to earth’s angular velocity vector. This leads to a simple description

20

N. Ashby

of the Sagnac correction: ΔtSagnac is 2ωE/c2 time the area swept out by the electromagnetic pulse as it travels from the GPS transmitter to the receiver, projected onto earth’s equatorial plane. This is depicted in Figure 1.3, in which the receiver is on earth’s surface at the tip of the path vector R.
In the early 1980s clocks in remotely situated timing laboratories were being compared by using GPS satellites in "common view", that is when one GPS satellite is observed at the same time by more than one timing laboratory. In one experiment[7] signals from GPS satellites were utilized in simultaneous common view between three pairs of earth timing centers to accomplish a circumnavigation of the globe. The centers were the National Bureau of Standards (now the National Institute of Standards and Technology) in Boulder, Colorado; Physikalisch-Technische Bundesanstalt in Braunschweig, West Germany; and Tokyo Astronomical Observatory. A typical geometrical conﬁguration of ground stations and satellites, with the corresponding projected areas, is illustrated in Figure 1.4. The size of the Sagnac effect calculated varies from about 240 ns to 350 ns depending on the location of the satellites at a particular moment. Sufﬁcient data were collected to perform 90 independent circumnavigations. As Figure 1.4 shows, when a satellite is eastward of one timing center and westward of another, one of the Sagnac corrections is positive and the other is negative, so when computing the difference of times between the two terrestrial clocks, the Sagnac corrections actually add up in a positive sense.
The mean value of the residuals over 90 days of observation was 5 ns, less than 2 percent of the magnitude of the calculated total Sagnac correction. A signiﬁcant part of these residuals can be attributed to random noise processes in the clocks.
Sagnac corrections of the form of Eq. 1.19 are routinely used in comparisons between distant time standards laboratories on earth.

6. GPS Navigation Equations and the ECEF Frame
The navigation problem in GPS is to determine the position of the receiver in the ECEF reference frame. A by-product of this process is the accurate determination of GPS time at the receiver. In general neither the position nor the time is known, so the assumptions used in previous sections regarding the Sagnac effect are of little use. The principles of position determination and time transfer in the GPS can be very simply stated. Let there be four synchronized atomic clocks which transmit sharply deﬁned pulses from the positions rj at times tj, with j = 1, 2, 3, 4 an index labelling the different transmission events. Suppose that these four signals are received at position r at one and the same instant t. This is called "time-tagging at the receiver", meaning that observations of the various signals are made simultaneously at the receiver at

The Sagnac effect in the GPS
R

21
ω
Receiver

Figure 1.3. Sagnac correction arising from motion of the ECEF observer.

time t. Then from the principle of the constancy of the speed of light,

c2(t − tj)2 = |r − rj|2 , j = 1, 2, 3, 4.

(1.22)

These four equations can be solved for the unknown space-time coordinates of the reception event, (t, r). The solution will provide the position of the receiver at the time of the simultaneous reception events, t. No knowledge of the receiver velocity is needed. The Sagnac effect becomes irrelevant. At most one can say that because the solution gives the ﬁnal position and time of the reception event, the Sagnac effect has been automatically accounted for.
However there are complications from the fact that the navigation equations, Eqs. 1.22, are valid in the ECI frame, whereas users almost always want to know their position in the ECEF frame. For discussions of relativity, the particular choice of ECEF frame is immaterial. Also, the fact the the earth truly rotates about an axis slightly different from the WGS-84 axis, with a variable rotation rate, has little consequence for relativity and I shall not go into this here.
It should be emphasized strongly that the transmitted navigation messages provide the user only with a function from which the satellite position can be calculated in the ECEF as a function of the transmission time. Usually the satellite transmission times tj are unequal, so the coordinate system in which the satellite positions are speciﬁed changes orientation from one measurement

22

N. Ashby

ω
P N T

Figure 1.4. Common-view signals from three satellites provide an Around-the-World Sagnac experiment.

to the next. Therefore to implement Eqs. (1.22), the receiver must generally perform a different rotation for each measurement made, into some common inertial frame, so that Eqs. (1.22) apply. After solving the propagation delay equations, a ﬁnal rotation must then be performed into the ECEF to determine the receiver’s position. This can become exceedingly complicated and confusing. I shall discuss this in a later section.
The purpose of the present discussion is to examine ﬁrst-order relativistic effects from the point of view of the ECEF frame. Consider the simplest instance of a transformation from an inertial frame, in which the space-time is Minkowskian, to a rotating frame of reference. Thus ignoring gravitational potentials, the metric in an inertial frame in cylindrical coordinates is

−ds2 = −(c dt)2 + dr2 + r2dφ2 + dz2 ,

(1.23)

and the transformation to a coordinate system {t , r , φ , z } rotating at the uniform angular rate ωE is

t = t , r = r , φ = φ + ωEt , z = z .

(1.24)

This results in the following well-known metric (Langevin metric) in the rotating frame:

−ds2 = −

1

−

ωE2 r c2

2

(cdt )2 + 2ωEr 2dφ dt + (dσ )2 ,

(1.25)

The Sagnac effect in the GPS

23

where the abbreviated expression (dσ )2 = (dr )2 + (r dφ )2 + (dz )2 for the

square of the coordinate distance has been used.

The time transformation t = t in Eqs. (1.24) is a result of the convention to

determine time t in the rotating frame in terms of time in the underlying ECI

frame.

Now consider a process in which observers in the rotating frame attempt to

use Einstein synchronization (that is, the principle of the constancy of the speed

of light) to establish a network of synchronized clocks. Light travels along a null worldline so I may set ds2 = 0 in Eq. (1.25). Also, it is sufﬁcient for

this discussion to keep only terms of ﬁrst order in the small parameter ωEr /c.

Then

(cdt )2 − 2ωEr 2dφ (cdt ) − (dσ )2 = 0 , c

(1.26)

and solving for (cdt ),

cdt = dσ + ωEr 2dφ . c

(1.27)

The quantity r 2dφ /2 is just the inﬁnitesimal area dAz in the rotating coordinate system swept out by a vector from the rotation axis to the light pulse, and projected onto a plane parallel to the equatorial plane. Thus the total time required for light to traverse some path is

dt =
path

dσ path c

+

2ωE c2

dAz .
path

[ light ]

(1.28)

Observers ﬁxed on the earth, who were unaware of earth rotation, would use just dσ /c for synchronizing their clock network. Observers at rest in the underlying inertial frame would say that this leads to signiﬁcant path-dependent inconsistencies, which are proportional to the projected area encompassed by the path. Consider for example a synchronization process which follows earth’s equator in the eastwards direction. For earth, 2ωE/c2 = 1.6227 × 10−21 s/m2 and the equatorial radius is a1 = 6, 378, 137 m, so the area is πa21 = 1.27802 × 1014 m2 . Thus the last term in Eq. (1.28) is

2ωE c2

dAz = 207.4 ns.
path

(1.29)

From the underlying inertial frame, this can be regarded as the additional travel time required by light to catch up to the moving reference point. Simpleminded use of Einstein synchronization in the rotating frame gives only dσ /c, and thus leads to a signiﬁcant error. Traversing the equator once eastward, the last clock in the synchronization path would lag the ﬁrst clock by 207.4 ns.

24

N. Ashby

Traversing the equator once westward, the last clock in the synchronization path would lead the ﬁrst clock by 207.4 ns.
In an inertial frame a portable clock can be used to disseminate time. The clock must be moved so slowly that changes in the moving clock’s rate due to time dilation, relative to a reference clock at rest on earth’s surface, are extremely small. On the other hand, observers in a rotating frame who attempt this ﬁnd that the proper time elapsed on the portable clock is affected by earth’s rotation rate. Factoring (dt )2 out of the right side of Eq. (1.25), the proper time increment dτ on the moving clock is given by

(dτ )2 = (ds/c)2 = dt 2 1 −

ωE r c

2

−

2ωEr 2dφ c2dt

−

dσ cdt

2

.

(1.30) For a slowly moving clock (dσ /cdt )2 << 1 so the last term in brackets in

Eq. (1.30) can be neglected. Also, keeping only ﬁrst order terms in the small

quantity ωEr /c,

dτ

=

dt

−

ωEr 2dφ c2

(1.31)

which leads to

dt =
path

path

dτ

+

2ωe c2

dAz .
path

[ portable clock ]

(1.32)

This should be compared with Eq. (1.28). Path-dependent discrepancies in the rotating frame are thus inescapable whether one uses light or portable clocks to disseminate time, while synchronization in the underlying inertial frame using either process is self-consistent.
Eqs. 1.28 and 1.32 can be reinterpreted as a means of realizing coordinate time t = t in the rotating frame, if after performing a synchronization process appropriate corrections of the form +2ωE path dAz/c2 are applied. It is remarkable how many different ways this can be viewed. The different ways discussed so far in this article include the fact that from the inertial frame it appears that the reference clock from which the synchronization process starts is moving, requiring light to traverse a different path than it appears to traverse in the rotating frame. The Sagnac effect can also be regarded as arising from the relativity of simultaneity in a Lorentz transformation to a sequence of local inertial frames co-moving with points on the rotating earth, or as the difference between proper times of a slowly moving portable clock and a Master reference clock ﬁxed on earth’s surface.
This was recognized in the early 1980s by the Consultative Committee for the Deﬁnition of the Second and the International Radio Consultative Committee who formally adopted procedures incorporating such corrections for the

The Sagnac effect in the GPS

25

comparison of time standards located far apart on earth’s surface. For the GPS it means that synchronization of the entire system of ground-based and orbiting atomic clocks is performed in the local inertial frame, or ECI coordinate system.

7. Sagnac-like effects due to rotation of the ECEF frame

By design, the ephemerides (positions) of the GPS satellites are broadcast in such a way that the receiver can compute their positions at the instant of transmission in the rotating WGS-84 reference frame. For time-tagging at the receiver, the propagation delays from different satellites can vary from about 67 ms to 86 ms. During this approximately 19 ms transmission time variation, the ECEF reference frame can rotate more than a microradian and the positions of the satellites due to this rotation alone can vary by over 30 meters while the satellites move in inertial space by as much as 60 meters. If this is not carefully accounted for, unacceptable navigation errors can occur.
It would lead to serious error to assert Eqs. 1.22 were valid in the ECEF frame. What the receiver must do is rotate the positions of each of the satellites, that have been computed in the rotating frame, into some chosen ECI frame. Then Eqs. 1.22 are valid and can be solved in the ECI frame. The resulting position found in the ECI frame is ﬁnally rotated into the WGS-84 frame and used for navigation.
To illustrate that these rotations give rise to Sagnac-like effects, suppose the chosen ECI frame instantaneously coincides with the WGS-84 frame at the instant of arrival of the earliest of the four signals. I denote the GPS time of arrival of this particular signal by t1, and the position of this particular satellite at this time as r1. Let the time intervals between the arrival of this signal and the other three signals be denoted by

Δti = ti − t1 , i = 1, 2, 3, 4

(1.33)

where for simplicity I have taken Δt1 = 0. During the time interval Δti the ECEF frame has rotated the amount ωEΔti. An active rotation of the satellite position ri(ECEF ) by the amount +ωEΔti is necessary in order to express the position of satellite i in the inertial frame in which the position r1 is expressed. This rotation operation can be expressed as

ri(ECI) = ri(ECEF ) + ωE × ri(ECEF )Δti .

(1.34)

The navigation equations then become c2(t − ti)2 = |r − ri(ECEF ) − ωE × ri(ECEF )Δti|2

(1.35)

and if I put Δt = t − t1 (no subscript on t) and Ri = r − ri(ECEF ) I obtain

c2(Δt − Δti)2 = |Ri − ωE × ri(ECEF )Δti|2

(1.36)

26

N. Ashby

Eqs. 1.36 have within them the four unknowns (Δt, r). The position solution for r will be in the ECI frame chosen for computation. After ﬁnding this position, the result must then be rotated into the ECEF frame for navigation. Since the ECEF frame rotates an amount ωEΔt during the time interval Δt, the ﬁnal solution for the position in the ECEF frame will be

r(ECEF ) = r − ωE × rΔt .

(1.37)

The size of the correction term in this last equation can easily be estimated, since Δt ≈ .015 s and r ≈ 6.4 × 106 m. A typical value will be about 9 meters. Eq. 1.36 can be solved approximately for Δt by expanding the square
on the right side, keeping only linear terms in ωE, and then taking a square root, similar to the approximations made in deriving Eq. 1.17. The result is

Δt

=

Δti

+

Ri c

+

ωE

×

ri(ECEF ) cRi

·

Ri

Δti

.

(1.38)

The last term in the above equation is a Sagnac-like correction. I can estimate its magnitude by substituting in an approximate expression for Δti:

Δti

≈

Ri c

−

R1 c

(1.39)

So the correction term becomes, after interchanging dot and cross products,

ωE

·

ri (E C E F c2

)

×

Ri

(1

−

R1/Ri)

.

(1.40)

Is this really a Sagnac correction? It is linear in the rotational velocity, the

coefﬁcient can be interpreted in terms of an area, and it is relativistic (there is a factor 1/c2).

In the case of time-tagging at the transmitters, signals are chosen for pro-

cessing which leave the transmitters at some chosen time tT . Then the broad-

cast ephemerides will all be calculated by the receiver in one and the same

ECEF frame. It would then be natural to choose for application of the navi-

gation equations (Eqs. 1.22) an inertial frame which coincides with this ECEF

frame at the instant tT of GPS time. But then the signals do not arrive simul-

taneously at the receiver, and the receiver motion during the interval between

arrival of the ﬁrst and last signals must be accounted for.

To illustrate the size of the Sagnac-like effects that occur in this situation,

let r denote the receiver position at transmission time tT , and let ri denote the transmitter position at time tT . Imagine these positions to be expressed in an

inertial frame which coincides instantaneously with the ECEF frame at time

tT . Let ti denote the arrival time at the receiver, of the signal from the ith satellite. The receiver position at time ti will be modiﬁed by earth rotation and

will be

r + ωE × r(ti − tT ) .

(1.41)

REFERENCES

27

The navigation equations in this inertial reference frame will be

c2(ti − tT )2 = |r + ωE × r(ti − tT ) − ri|2

(1.42)

Because of the similarity of this equation to Eq. 1.13 it is clear that Sagnac-

like corrections will enter solution of the equations. The times ti are however

known only to within an added constant, because of a possible error or system-

atic bias in the receiver’s clock. If the arrival times actually measured in the

receiver are ti, then

ti = ti + b .

(1.43)

where b is the receiver clock bias then the navigation equations become

c2(ti + b − tT )2 = |r + ωE × r(ti + b − tT ) − ri|2

(1.44)

and the unknowns are (b, r). Obviously there are many other ways of formulating the problem of accounting for receiver motion. A technical note[8] discusses these issues in more detail, with numerical examples.

8. Summary
In the GPS, the Sagnac effect arises because the primary reference frame of interest for navigation is the rotating Earth-Centered, Earth-Fixed frame, whereas the speed of light is constant in a locally inertial frame, the EarthCentered Inertial frame. Additional Sagnac-like effects arise because the satellite ephemerides are broadcast in a form allowing the receiver to compute satellite positions in the ECEF frame. In the case of time-tagging of observations at the receiver, it is necessary to rotate the satellite positions into a common ECI reference frame in order apply the principle of the constancy of c. In the rotating frame of reference the effect appears to arise from a Coriolis-like term in the fundamental scalar invariant. Whether synchronization procedures are performed by using electromagnetic signals or slowly moving portable clocks, to leading order the same Sagnac effect arises. The effect is of signiﬁcant magnitude and must be taken into account for accurate navigation. It is also necessary to apply Sagnac corrections when comparing remote clocks on earth’s surface.

References

[1] World Geodetic System 1984, (National Imagery and Mapping Agency, 1994) Report No. 8350.2, Third Edition, Amendment 1. NIMA Stock No. DMATR83502WGS84, MSN 7643-01-402-0347.
[2] Malys, S., and Slater, “Maintenance and Enhancement of the World Geodetic System 1984", in Proceedings of the 7th International Technical Meeting of the Satellite Division of the Institute of Navigation, Sept.

28

N. Ashby

20-23, Salt Lake City, UT (ION GPS-94), 17-24. The Institute of Navigation (1994).
[3] Landau, L., and E. Lifshitz, The Classical Theory of Fields, 4th ed., Pergamon, New York p. 254 (1997).
[4] Ashby, N., and Allan, D.W., "Practical Implications of Relativity for a Global Coordinate Time Scale", Radio Science 14, 649-669 (1979).
[5] Wang, R., Y. Zheng, A. Yao, and D. Langley, "Modiﬁed Sagnac experiment for measuring travel-time difference between counter-propagating light beams in a uniformly moving ﬁber", Phys. Letts. A 312, pp 7-10, (2003).
[6] Hasselbach, F., and M. Nicklaus, Phys. Rev. A48, pp 143-151, (1993).
[7] Allan, D. W., M. Weiss, and N. Ashby, "Around-the-World Relativistic Sagnac Experiment", Science 228, pp 69-70 (5 April 1985).
[8] Ashby, N., and M. Weiss, "Global Positioning Receivers and Relativity", NIST Technical Note 1385, U. S. Government Printing Ofﬁce, Washington, D.C., March (1999).

Chapter 2
SPACE, TIME AND COORDINATES IN A ROTATING WORLD

Dennis Dieks
Institute for the History and Foundations of Science Utrecht University, P.O.Box 80.000 3508 TA Utrecht, The Netherlands d.g.b.j.dieks@phys.uu.nl

Abstract

The peculiarities of rotating frames of reference played an important role in the genesis of general relativity. Considering them, Einstein became convinced that coordinates have a different status in the general theory of relativity than in the special theory. This line of thinking was confused, however. To clarify the situation we investigate the relation between coordinates and the results of spacetime measurements in rotating reference frames. We argue that the difference between rotating systems (or accelerating systems in general) and inertial systems does not lie in a different status of the coordinates (which are conventional in all cases), but rather in different global chronogeometric properties of the various reference frames. In the course of our discussion we comment on a number of related issues, such as the question of whether a consideration of the behavior of rods and clocks is indispensable for the foundation of kinematics, the inﬂuence of acceleration on the behavior of measuring devices, the conventionality of simultaneity, and the Ehrenfest paradox.

1. Introduction
In his Autobiographical Notes [1], Einstein relates how important Machian empiricist ideas were for his discovery of a theory that could reconcile the idea that all inertial frames are equivalent with the principle that the velocity of light has a ﬁxed value that is independent of the velocity of the emitting source. It was essential, he states, to realize what the meaning of coordinates in physics is: they are nothing but the outcomes of length and time measurements by means of rods, clocks and light signals. This idea led Einstein to his famous critique of the classical notion of simultaneity, one of the cornerstones of the special theory of relativity.
29
G. Rizzi and M.L. Ruggiero (eds.), Relativity in Rotating Frames, 29-42. © 2004 Kluwer Academic Publishers.

30

D. Dieks

It soon turned out, however, that the special theory of relativity was not able to accommodate gravitation, and the principle of equivalence, in a natural way. Einstein fully recognized this problem in 1908, but it took him another seven years before he succeeded in constructing the general theory. As he explains in his Autobiographical Notes, the main reason for the slowness of his progress in this period was the difﬁculty of abandoning again, in the context of the general theory, the idea that coordinates should possess immediate metrical meaning.
From a systematical (as opposed to a historical or psychological) point of view this emphasis on the different meaning of coordinates, in the context of the two theories, is very odd. For the practice of physics before, during and after Einstein’s days, even if governed by the severest empiricist norms, does not at all indicate that coordinates should possess a metrical signiﬁcance, relating to the indications of rods and clocks. Think, for example, of the way coordinates are used in observational astronomy: the essential thing is that the coordinates are assigned to celestial objects in an objective and reproducible way; how the coordinates relate to distances is a matter to be found out subsequently. Coordinates are even routinely attributed to regions of the universe in which rods and clocks could not possibly exist. This is obviously unobjectionable from an empiricist point of view, as long as the method by which the coordinates are assigned is operationally speciﬁed. So, even within the framework of special relativity general coordinate systems that do not reﬂect the indications of rods and clocks are entirely permissible.
What ﬁnally led Einstein to abandon his special relativistic analysis of the meaning of coordinates, he tells us, was the lack of metrical signiﬁcance of coordinates in accelerating frames of reference; the consideration of coordinates on a rotating disc played an important role in reaching this conclusion [2]. But, as we will see, there is confusion here: the metrical signiﬁcance of coordinates in accelerating frames can be determined completely through application of the principles of special relativity, so there can be no need to revise the meaning of the notion of coordinates, or to invoke a new epistemological analysis.
As it turns out, the difference between inertial and non-inertial frames of reference, and between special and general relativity, is not in the epistemological status of the coordinates. Rather, the difference is that chronogeometric characteristics become globally different. This is a physical rather than a philosophical difference, and has nothing to do with the meaning or permissibility of coordinate systems.
The rotating frame of reference nicely illustrates these points. There is no problem in deﬁning operationally meaningful coordinates in a rotating (and therefore accelerating) frame. Furthermore, relating these coordinates to distances and time intervals, and the behavior of moving objects, can be done by the means provided by special relativity. However, the spatial geometry be-

Space, Time and Coordinates in a Rotating World

31

comes non-Euclidean, and local Einstein synchrony does not lead to a global notion of time. These latter features constitute the essential differences from the situation in an inertial frame.
In the course of our discussion we will have occasion to comment on a number of related issues, such as the status of rods and clocks, the behavior of accelerating measuring devices, the conventionality of simultaneity, and the Ehrenfest paradox.

2. The rotating frame of reference
Let us start from Minkowski space-time, coordinatized by inertial coordinates r, ϕ, z and t: r and ϕ are polar coordinates in a plane, z is a Cartesian coordinate orthogonal to this plane, and t is the standard time coordinate. It so happens that r, z, and t can be thought of as representing the indications of rods and clocks, but that is not important for their role as coordinates, which is just to pinpoint events unequivocally. The choice of coordinates is conventional and pragmatic. In this case we choose polar coordinates because we are going to describe a system that possesses axial symmetry: polar coordinates simplify the description.
Once we have laid down coordinates, the metrical aspects should be introduced via further stipulations. This is ordinarily done through the introduction of the ‘line element’ ds2 = c2dt2 − dr2 − r2dϕ2 − dz2, plus a speciﬁcation of what this mathematical expression represents physically. The traditional approach is to invoke standard rods and clocks: ds/c is the time measured b√y a standard clock whose r, ϕ and z coordinates are constant. Furthermore,
−ds2 is the length of a rod with a stationary position in the coordinates and with constant coordinates and differences dr, dϕ, dz between its endpoints, taken at one instant according to standard simultaneity (dt = 0). However, it would be a mistake to think that rods and clocks are indispensable to relate the coordinates to metrical concepts. In section 4 below we will discuss an approach that does not make use of rods and material clocks.
We now introduce alternative coordinates for the events in this Minkowski world: t = t, r = r, ϕ = ϕ − ωt and z = z, with ω a constant. Since rest in the new coordinates obviously means uniform rotation with respect to the old frame, we call the frame of reference deﬁned by these new coordinates the rotating frame of reference.
It is clear that if operational methods are at hand to ﬁx the old coordinates, the same methods can be used to assign values to the new coordinates (we assume ω to be known). So from an empiricist or operational point of view the new coordinates are impeccable. However, from the special theory of relativity we know that material bodies at rest in the new coordinates may not exist (ωr may be greater than c, the velocity of light). It is true, therefore, that the new

32

D. Dieks

coordinates will not always have a direct interpretation in terms of co-moving bodies–but this is something to be distinguished sharply from the more general question of whether they have adequate empirical signiﬁcance at all.
Substitution of the rotating coordinates into the expression for the line element yields ds2 = (c2 − r 2ω2)dt 2 − dr 2 − r 2dϕ 2 − dz 2 − 2ωr 2dϕ dt . As we already mentioned, it is a basic principle of the special theory of relativity that the line element supplies all information about the physics of the situation, as described in the given coordinates. It was also mentioned above that the traditional link between ds and physical concepts makes use of clocks and measuring rods. However, there is another and more fundamental physical interpretation available that only makes use of the basic laws of motion: as long as no disturbing forces act, point particles follow time-like geodesics and light follows null-geodesics in the metric deﬁned by ds2. The relation between these dynamical aspects (how particles and light move) and the metrical aspects (rods and clocks) will be the subject of comments in section 4.

3. Rods and clocks
Let us for the moment stay with the physical interpretation of ds in terms of measurements performed with rods and clocks. Concerning time, the coordinating principle is that ds/c represents proper time, measured by a clock whose world line connects the events between which ds is calculated. This principle entails that a clock at rest in the rotating frame will indicate the proper time

ds/c = (1 − r 2ω2/c2)dt .

(2.1)

Because t = t and t has the physical meaning of the time indicated by a clock

at rest in the old frame, this implies that clocks at rest in the rotating frame are

slow compared to clocks in the original (“laboratory”) frame.

√

With regard to spatial distances, the interpretative principle is that −ds2

gives the length of an inﬁnitesimal rod whose endpoints are simultaneous ac-

cording to standard simultaneity in the rod’s rest frame ( [3], p.187). (A rod

is a three-dimensional object, so we need a stipulation about the instants at

which its endpoints should be considered in order to get a four-dimensional

interval for which ds can be calculated.) When we apply this rule to rods that

are at rest in the rotating frame of reference, we encounter the complication

that dt = 0 does not automatically correspond to standard simultaneity in the

rotating frame. The deﬁnition of standard synchrony of two (inﬁnitesimally

near) clocks A and B is that a light signal sent from A to B and immediately

reﬂected to A, reaches B when B indicates a time that is halfway between the

instants of emission and reception, respectively, as measured by A. Suppose

that A and B, both at rest in the rotating frame, have positions with coordinate

differences dr, dϕ and dz–from now on we drop the primes of the rotating

Space, Time and Coordinates in a Rotating World

33

coordinates. A light signal between A and B follows a null-geodesic: ds2 = (c2 − r2ω2)dt2 − dr2 − r2dϕ2 − dz2 − 2ωr2dϕdt = 0. (2.2)

This equation gives the following solutions for dt when it is applied to the signals from A to B and back, respectively:

dt1,2 = ±ωr2dϕ +

(c2

− ω2r2)(dz2 c2 − ω2r2

+

dr2)

+

c2r2dϕ2

.

(2.3)

If t0 is the time coordinate of the emission event at A, the event at A with time coordinate t0 + 1/2(dt1 + dt2) is standard-simultaneous with the event at B with time coordinate t0 + dt1. It follows that standard synchrony between inﬁnitesimally close events corresponds to the following difference in t-coordinate:

dt = (t0 + dt1) − (t0 + 1/2dt1 + 1/2dt2) = (ωr2dϕ)/(c2 − ω2r2). (2.4)

As was to be expected, it is only for events that differ in their ϕ-coordinates that dt = 0 is not equivalent to standard simultaneity; indeed, the instantaneous velocity of the rotating frame is tangentially directed, and the relativistic dilation and contraction effects only take place in the direction of the velocity.
The spatial distance between two inﬁnitesimally near points, as measured by a rod resting in the rotating frame, is found by substituting the just-derived value of dt, (2.4), in the expression for ds2. The result is the following expression for the 3-dimensional spatial line element:

dl2

=

dr2

+

1

r2dϕ2 − ω2r2/c2

+

dz2.

(2.5)

We could have found (2.1) and (2.5) in a simpler way by making use of the standard expressions for the time dilation and Lorentz contraction undergone by clocks and rods, respectively, that possess the instantaneous velocity ωr. However, the use of the line element as the central theoretical quantity provides us with a unifying framework that makes it easier to discuss the relation between metrical and dynamical concepts.
4. Space and time without rods and clocks
In his Autobiographical Notes, Einstein already points out that from a fundamental point of view it is unsatisfactory to interpret ds via measuring procedures with complicated macroscopic instruments. Indeed, this could create the false impression that rods and clocks are basic entities without which the theory would have no physical content. However, it is clear that rods and clocks themselves consist of more fundamental entities, like atoms and molecules. In

34

D. Dieks

principle it would therefore be better to base the interpretation of the theory directly on what it says about the fundamental constituents of matter. It is only because no complete theory of matter was available, Einstein explains, that it was expedient to introduce the theory through measurements by rods and clocks. In principle they should be eliminated at a later stage.
This desideratum, to do without rods and clocks, becomes even more urgent when accelerated frames of reference are considered, as in the case of our rotating world. Obviously the motions of clocks and rods that are stationary in the rotating frame are not inertial. Centrifugal and Coriolis forces will therefore arise, which will distort the rotating instruments. It is not a priori clear that such deformed instruments will keep on functioning as indicators of ds. Indeed, one could easily think of rods or clocks that would be torn apart by centrifugal forces and would therefore certainly not indicate any length or time intervals.
Fortunately, it is possible to found the space-time description of our rotating world on a more fundamental level than that of macroscopic measuring devices. In fact, in general space-times one can use the basic principles that time-like geodesics are physically realized by inertially moving point-particles and that null-geodesics represent light rays, to deﬁne space-time distances between neighboring events ([4], section 16.4). In our case, Minkowski spacetime, we can start by constructing a set of elementary ‘light clocks’ by letting light signals bounce back and forth between neighboring parallel particle geodesics. If we conﬁne our attention to the plane z = 0, we can take the geodesics deﬁned in the laboratory frame (the inertial system we started with) by constant r, ϕ and r + dr, ϕ, respectively. The thus constructed clock has a constant period (the dt between two ‘ticks’) of 2dr/c. In other words, we have here an elementary process that provides a physical realization of t; and we have come to this conclusion on the basis of the dynamical postulates alone (the only ingredient is that light follows null-geodesics). Length can be determined in a similar way: let a light signal depart from A, with ﬁxed r and ϕ and go to a neighboring position B with r + dr and ϕ + dϕ from which it returns immediately to A. Let the round trip time measured at A be dt. We can now deﬁne the spatial distance dl between A and B as cdt/2. From the postulate that light follows null-geodesics it follows that dl2 = dr2 + r2dϕ2. In this way the laboratory coordinates obtain metrical signiﬁcance, without reliance on macroscopic clocks and rigid rods. When such (complicated) systems are introduced at a later stage, we can study their workings on the basis of the fundamental laws of physics governing their constituents and see, on that basis, whether they are indeed suitable to measure the just-deﬁned intervals.
We now turn our attention to measurements performed within the rotating system, i.e. with instruments resting in the rotating coordinates. From Eq.

Space, Time and Coordinates in a Rotating World

35

(2.3) we see that the round trip time dt needed by a light signal between two

neighboring points that are stationary in the rotating frame of reference is given

by

dt = dt1 + dt2 = 2

(c2

−

ω2r2)dr2 + c2 − ω2r2

c2r2dϕ2

.

If the laboratory coordinate t is used as the measure of time, and if the deﬁni-

tion dl = cdt/2 is used to ﬁx spatial distances, we arrive at the metric

dl2

=

(1

−

ω2r2/c2)dr2 + r2dϕ2

(1 − ω2r2/c2)2

.

However, it is more natural to link the measure of time intervals in the rotating system to the indications furnished by light clocks that are co-moving, i.e. stationary in the rotating coordinates instead of stationary in the laboratory frame. So let a light ray bounce back and forth between two points that only differ in their r-coordinate, by the amount dr, in the rotating frame. It follo√ws from the expression (2.2) that the period of the thus deﬁned clock is 2dr/ c2 − ω2r2, whereas the period of the similar and instantaneously coinciding clock in the laboratory frame is 2dr/c. The period of the rotating light clock is therefore longer, by a factor 1/ 1 − ω2r2/c2, than the period of the laboratory clock. When we now deﬁne distances as cdτ /2, with τ measured in the new ‘comoving’ time units, we have to multiply the distances we found a moment ago by 1 − ω2r2/c2. The ﬁnal result is
dl2 = dr2 + r2dϕ2/(1 − ω2r2/c2).

This is the same result as we found in Eq. (2.5).

5. Accelerating measuring devices
The above sketch shows how we can achieve a physical implementation of the two systems of coordinates, and give them metrical meaning, by the sole use of point-particles and light. The thus deﬁned space-time distances can be used to calibrate macroscopic measuring rods and clocks. Indeed, it is clear that in general such instruments will be deformed by the rotational motion, and that this will introduce inaccuracies in their readings.
The general effect of accelerations can be illustrated by the consideration of a light-clock of the kind mentioned above: a light signal bouncing back and forth between two particle world-lines. Light travelling to and fro between two mirrors resting in an inertial system, with mutual distance L, deﬁnes a clock with half period T = L/c. When the two mirrors move uniformly with the same velocity →−v , in a direction parallel to their planes, a simple application of the Pythagorean theorem shows that the half period of the moving clock becomes L/(c 1 − v2/c2) = T / 1 − v2/c2. This demonstrates the presence

36

D. Dieks

of time dilation in the case of a moving light-clock (by means of the relativity
principle this result can be extended to other time-keeping devices). Consider
now what happens if the velocity is not uniform but the system starts accelerating when the light leaves the ﬁrst mirror, with a small acceleration →−a in the direction of →−v . As judged from the inertial frame, the light now needs a time T to reach the second mirror; during this time the accelerating mirror system has covered a distance s ≈ vT + 1/2aT 2. Application of Pythagoras now yields c2T 2 = L2 + s2. It follows that

c2T 2 = L2 + v2T 2 + avT 3 + 1/4a2T 4.

(2.6)

The half period T that follows from this equation obviously depends on a. However, it is also obvious that the extent of the change in the period caused by a depends on the magnitude of T itself. If we make T in Eq. (2.6) very small, by reducing L, we ﬁnd in the limiting situation T = T / 1 − v2/c2, just as in the case of the uniformly moving clock. In other words, the acceleration has an effect, but the magnitude of this effect depends on the peculiarities of the speciﬁc clock we are considering (in this case on L). This acceleration-dependent effect can be made as small as we wish, by using suitably constructed clocks (in the example: by reducing L). What remains in all cases is the universal effect caused by the velocity.
This shows in what sense velocities have a universal effect on length and time determinations, but accelerations not. There is no independent postulate involved here; everything can be derived from the dynamical principles of special relativity theory, by considering the inner workings of the measuring devices. It turns out that acceleration-dependent effects are there, but can be varied, and corrected for, by varying the characteristics of the devices. This is the real content of the textbook statement that acceleration has no metrical effects. It should be stressed again that this does not constitute a new hypothesis that has to be added to the dynamical principles of the theory of relativity. Quite to the contrary, the effects of accelerations on any given clock or measuring rod can be computed from the dynamical principles applied to these devices.
Of course, that the magnitudes of distortions will depend on the speciﬁc constitutions of the rods or clocks in question is only to be expected. Robust rods and clocks will be less affected accelerations than fragile ones. One way of correcting for the deformations is to gauge the accelerating instruments against the light measurements results described in section (4). The expressions (2.1) and (2.5) should be understood as applying to the results of space-time measurements performed with thus corrected measuring devices.

Space, Time and Coordinates in a Rotating World

37

6. Space and time in the rotating frame
The spatial geometry deﬁned by the line element (2.5) is non-Euclidean, with a negative r-dependent curvature (see [5], pp. 330-337). One of the notorious characteristics of this geometry is that the circumference of a circle with radius r (in the plane z = 0) is 2πr/(1 − ω2r2/c2)1/2, which is greater than 2πr. The recognition that the geometry in accelerated frames of reference will in general be non-Euclidean, which through the equivalence principle suggests that the presence of gravitation will also cause deviations from Euclidean geometry, played an important role in Einstein’s route to General Relativity. We will restrict ourselves to the special theory, however.
The properties of time in the rotating frame are perhaps even more interesting than the spatial characteristics. Expression (2.4) demonstrates that standard simultaneity between neighboring events in the rotating frame corresponds to a non-zero difference dt. It follows that if we go along a circle with radius r, in the positive φ-direction, while establishing standard simultaneity along the way, we create a ‘time gap’ t = 2πωr2/(c2 − ω2r2) upon completion of the circle. Doing the same thing in the opposite direction results in a time gap of the same absolute value but with opposite sign. So the total time difference generated by synchronizing over a complete circle in one direction, and comparing the result with doing the same thing in the other direction is
t = 4πωr2/(c2 − ω2r2). Now suppose that two light signals are emitted from a source ﬁxed in the rotating frame and start travelling, in opposite directions, along the same circle of constant r. We follow the two signals while locally using standard synchrony; this has the advantage that locally the standard constant velocity c can be attributed to the signals. We therefore conclude that the two signals use the same amount of time in order to complete their circles and return to their source, as calculated by integrating the elapsed time intervals measured in the successive local comoving inertial frames (the signals cover the same distances, with the same velocity c, as judged from these frames). However, because of the just-mentioned time gaps the two signals do not complete their circles simultaneously, in one event. There is a time difference t = 4πωr2/(c2 − ω2r2) between their arrival times, as measured in the coordinate t. This is the celebrated Sagnac effect (see [6], p. 652 for a related derivation). The Sagnac effect directly reﬂects the space-time geometry of the rotating frame; it does not depend on the speciﬁc nature of the signals that propagate in the two directions. Indeed, as long as the two signals have the same velocities in the locally deﬁned inertial frames with standard synchrony, the difference in arrival times is given by the above time gap. So the same Sagnac time difference is there not only for light, but for any two identical signals running

38

D. Dieks

into two directions. The Sagnac experiment directly probes the space-time
relations in the rotating frame.
Because of the difference in arrival times of the two light signals, the ve-
locity of light obviously cannot be everywhere the same in the rotating coordi-
nates. This is a consequence of the fact that in the rotating frame events with equal time coordinate t are not standard simultaneous. So t may appear as an
unnatural time coordinate for the rotating frame: it would be desirable to have
a time coordinate that would reﬂect standard simultaneity everywhere. The question can therefore be asked whether we could deﬁne a coordinate t˜in such a way that dt˜ = 0 would imply standard synchrony in the local inertial frame. Suppose that t˜ = t˜(t, r, ϕ), then we should have that dt˜ = 0 if Eq. (2.4) holds. This implies that ω2r2/(c2 − ω2r2)∂t˜/∂t + ∂t˜/∂ϕ = 0 and ∂t˜/∂r = 0. In view of the axial symmetry in our frame we may assume that ∂t˜/∂ϕ = 0. The only solution of our partial differential equations is therefore that t˜is independent of r, ϕ and t, which clearly is unacceptable. Therefore, it turns out to
be a basic characteristic of the rotating frame that the locally deﬁned Lorentz
frames do not mesh: they cannot be combined into one frame with a globally
deﬁned standard simultaneity. Evidently it is possible to deﬁne global time coordinates, like t; but the description of physical processes in terms of these
coordinates must necessarily differ from the standard description in inertial
systems. The non-constancy of the velocity of light in the rotating system fur-
nishes an example. It should be noted that this peculiarity of the description of
physical processes in the rotating system is not a consequence of the presence
of centrifugal and Coriolis forces: indeed, in our space-time determinations we
have compensated for the effects of such forces. It is the space-time geometry
itself that is at issue.

7. Simultaneity, slow clock transport and conventionality
As we saw in the previous section, the Sagnac effect is independent of the nature of the signals that propagate into the two directions on the rotating disc. So, if we transport two clocks along a circle with radius r around the center of the disk, one clockwise and one counter-clockwise, while keeping their velocities the same in the locally co-moving inertial frames, there will be a difference
t = 4πωr2/(c2 − ω2r2) between their return times (measured in the laboratory time t). It is well known that the indications of the clocks will conform to standard simultaneity in the limiting situation of vanishing velocities. That is, if the clocks are transported very slowly with respect to the rotating disc, they will remain synchronized according to standard simultaneity in the local inertial frames. It follows that slow clock transport cannot be used to deﬁne an unambiguous global time coordinate on the rotating disc: in the just-mentioned case the result will depend on whether a clockwise or counter-clockwise path

Space, Time and Coordinates in a Rotating World

39

is chosen. In general, the result of synchronization by slow clock transport will be path dependent.
With regard to time in inertial frames, there has been a long-standing and notorious debate about whether standard simultaneity (ε = 1/2 according to Reichenbach’s formulation) is conventional or not. One of the arguments often put forward against the conventionality thesis is that the natural procedure of slow clock transport leads to ε = 1/2, thus showing its privileged status. In the case of the rotating world, this argument can only be applied locally. Neither the Einstein light signal procedure, nor the slow transport of clock can be used to establish a global notion of simultaneity on the rotating disc.
More generally, it cannot be denied that in inertial frames standard simultaneity has a special status: it allows a simple formulation of the laws, conforms to slow clock transport and other physically plausible synchronization procedures, and agrees with Minkowski-orthogonality with respect to world lines representing the state of rest [7]. So time coordinates t that correspond to this notion of simultaneity (in the sense that dt = 0 expresses simultaneity) may be said to be privileged. In non-inertial frames this still is so, though now the argument applies only locally. The rotating system illustrates the situation very well: in each point on the disc standard simultaneity can be deﬁned just as in an inertial system, but this does not result in a global time coordinate. This supports the general conclusion of this paper, namely that the difference between the status of coordinates in inertial and non-inertial frames of reference, or special and general relativity, is not so much a matter of epistemology—or philosophical analysis of the meaning of coordinates—but rather a matter of physical facts. In global inertial systems privileged coordinates can be chosen that have a global metrical interpretation. In reference frames that are not globally inertial such privileged coordinates do not exist in general. This is not a matter of a different philosophical status of coordinates, but rather a reﬂection of different global space-time symmetry properties—a factual physical difference rather than a philosophical distinction.
The purpose of coordinates is to label events unambiguously, which can be done in inﬁnitely many different ways. The choice between these different possibilities is a matter of pragmatics; though there may be very good reasons to prefer one choice over another. Thus, in inertial frames of reference time coordinates that reﬂect standard simultaneity lead for many purposes to an especially simple description. In this case there exists a physically signiﬁcant global temporal relation between events, and coordinates that are adapted to this relation inherit its special status. But in the general case no physically signiﬁcant simultaneity relation exists. Global "simultaneity" can then only refer to some global time coordinate, which is chosen conventionally. This is true in non-inertial frames of reference, like the rotating disc, and in generally relativistic space-times in which there are no global temporal symmetries.

40

D. Dieks

These non-inertial frames of reference, and general relativistic space-times, seem an arena where the thesis that (global) simultaneity is conventional can be defended without controversy.

8. The rotating Ehrenfest cylinder
Not only in its temporal aspects, but also in its spatial physical properties the rotating frame differs globally from an inertial frame. Until now we spoke about a rotating frame of reference as deﬁned by a set of rotating coordinates, without discussing a possible material realization of this frame. It is clear from the outset that the special theory of relativity sets limits to such a realization: objects at rest in the rotating frame should not move faster than light as judged from the inertial laboratory frame. This implies that ωr < c should hold for such an object. In other words, there is an upper bound to the value of r that can be realized materially.
However, even if this condition is satisﬁed there remain interesting questions, as made clear by Ehrenfest in his famous note on the subject [8]. Suppose that a solid cylinder of radius R is gradually put into rotation about its axis; ﬁnally it reaches a state of uniform rotation with angular velocity ω. It would seem that in the ﬁnal state the cylinder has to satisfy contradictory requirements: on the one hand the Lorentz contraction should make the circumference shorter, on the other hand the radial elements should not contract because their motion is normal to their lengths. From symmetry it is clear that the form of a cross section of the moving cylinder remains a circle, as judged from the laboratory frame; but this would apparently mean that the circumference of the circle has become smaller while the radius has stayed the same. This is inconsistent (remember that Euclidean geometry holds in the laboratory frame).
The solution of this paradox is that the various parts of the cylinder, being fastened to each other, cannot move freely and therefore cannot Lorentz contract as freely moving inﬁnitesimal measuring rods would do. What will happen to the cylinder during its acceleration depends on the elastic properties of the material: tensions will develop because the tangential elements want to shrink, whereas the radial elements do not. A possible scenario is that the tangential elements will be stretched as compared to their natural (i.e. Lorentz contracted) lengths. Another possibility, if the material is sufﬁciently strong, is that the radius will contract, allowing the circumference to contract too. However, if ω becomes big enough one would have to expect that the tensions and strains grow to such an extent that they cause the cylinder to explode. This makes it clear that the Lorentz contraction can be responsible for clearly dynamical effects—the contractions are not just a matter of “perspective” (see [9] and [10]). (Of course, this whole discussion is rather academical because cen-

Space, Time and Coordinates in a Rotating World

41

trifugal forces will tear the cylinder apart before the relativistic effects become noticeable.)
As long as the cylinder survives, and keeps its cylindrical shape (as judged from the laboratory frame), not all its elements will be free from deformations, tensions or strains. However, the length determinations by measuring rods at rest in the rotating frame, as discussed in section 3, were supposed to be carried out with freely movable rods that are not hampered in their Lorentz contractions. So measuring rods laid out along the circumference of the circle will have undergone a Lorentz contraction, whereas rods laid out along a radius will have retained their rest length (as judged from the laboratory system). The measurement would reveal that the circumference is longer than 2π times the radius, in conformity with equation (2.5).
The spatial geometry of the disc is therefore non-Euclidean. That means that distance relations must be represented by a metrical tensor that cannot be put into the Euclidean diagonal form everywhere. It remains possible, of course, to choose coordinates locally in such a way that the Euclidean form results at the point in question. The difference from the inertial system concerns global aspects, not local ones. The impossibility to deﬁne a global coordinate system in which the metrical tensor reduces to its Euclidean standard form implies that there cannot be coordinates whose differences correspond to distances everywhere. The situation is analogous to the one we discussed in the context of time coordinates: nothing changes in the status and meaning of coordinates when we go from inertial to non-inertial systems. The things that do change are the global characteristics of the physical geometry, which are coordinateindependent.

Conclusion
The transition from inertial to non-inertial frames of reference, and the transition from special to general relativity, does not imply a change in the status and meaning of coordinate systems. It is therefore a misunderstanding to think that general relativity allows a wider class of coordinate systems than classical physics or special relativity. In classical physics and in relativity theory, both in inertial systems and non-inertial systems, coordinates just serve to label events. The choice for a particular coordinate system from the inﬁnity of possible ones is dictated by pragmatic considerations.
What does change in the transition from inertial to non-inertial systems, and from special to general relativity, are the global aspects of the physical spatial and temporal relations. Pragmatic arguments for choosing one coordinate system over another may therefore lead to different choices in the different situations: if geometrical relations have become different, coordinate systems with different characteristics, adapted to the new geometry, may lead to a simpler

42

D. Dieks

description. But this does not change the conventional nature of the coordinates.

References

[1] Albert Einstein: Philosopher-Scientist, P.A. Schilp (ed.), Open Court, La Salle, 1949.
[2] J. Stachel, “The Rigidly Rotating Disc as the ‘Missing Link’ in the History of General Relativity”, pp. 48-62 in Einstein and the History of General Relativity, D.Howard and J. Stachel, (eds.), Birkhäuser, Basel, 1989.
[3] H. Reichenbach, The Philosophy of Space and Time, Dover, New York, 1957.
[4] C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation, Freeman, San Francisco, 1973.
[5] M.-A. Tonnelat, The Principles of Electromagnetic Theory and of Relativity, Reidel, Dordrecht, 1966.
[6] D. Dieks and G. Nienhuis, “Relativistic Aspects of Nonrelativistic Quantum Mechanics”, American Journal of Physics 58 (1990) 650-655.
[7] D. Malament, “Causal Theories of Time and the Conventionality of Simultaneity”, Noûs 11 (1977) 293-300.
[8] P. Ehrenfest, “Gleichformige Rotation starrer Körper und Relativitätstheorie”, Physikalische Zeitschrift 10 (1909) 918.
[9] D. Dieks, “Time in Special Relativity and its Philosophical Signiﬁcance”, European Journal of Physics 12 (1991) 253-259.
[10] D. Dieks, “The ‘Reality’ of the Lorentz Contraction”, Zeitschrift für allgemeine Wissenschaftstheorie 15 (1984) 330-342.

Chapter 3
THE HYPOTHESIS OF LOCALITY AND ITS LIMITATIONS

Bahram Mashhoon
Department of Physics and Astronomy University of Missouri-Columbia Columbia, Missouri 65211, USA

Abstract

The hypothesis of locality, its origin and consequences are discussed. This supposition is necessary for establishing the local spacetime frame of accelerated observers; in this connection, the measurement of length in a rotating system is considered in detail. Various limitations of the hypothesis of locality are examined.

1. Introduction
The basic laws of microphysics have been formulated with respect to ideal inertial observers. However, all actual observers are accelerated. To interpret the results of experiments, it is therefore necessary to establish a connection between actual and inertial observers. This is achieved in the standard theory of relativity by means of the hypothesis of locality, namely, the assumption that an accelerated observer at each instant along its worldline is physically equivalent to an otherwise identical momentarily comoving inertial observer. In this way a noninertial observer passes through a continuous inﬁnity of hypothetical momentarily comoving inertial observers [1].
The hypothesis of locality stems from Newtonian mechanics, where the state of a particle is given at each instant of time by its position and velocity. Thus the accelerated observer and the hypothetical inertial observer share the same state and are therefore equivalent. Hence, the treatment of accelerated systems in Newtonian mechanics requires no new assumption. More generally, if all physical phenomena could be reduced to pointlike coincidences of classical point particles and electromagnetic rays, then the hypothesis of locality would be exactly valid. However, an electromagnetic wave has intrinsic scales of

43
G. Rizzi and M.L. Ruggiero (eds.), Relativity in Rotating Frames, 43-55. © 2004 Kluwer Academic Publishers.

44

B. Mashhoon

length and time characterized by its wavelength λ and period λ/c. For instance, the measurement of the frequency of the wave necessitates observation of a few oscillations before a reasonable determination can become possible. If the state of the observer does not change appreciably over this period of time, then the hypothesis of locality would be essentially valid. This criterion may be expressed as λ/L << 1, where L is the relevant acceleration length of the observer. That is, the observer has intrinsic scales of length L and time L/c that characterize the degree of variation of its state. For instance, L = c2/a for an observer with translational acceleration a, while L = c/Ω for an observer rotating with frequency Ω [1, 2].
The consistency of these ideas can be seen in the case of an accelerating charged particle. Imagine a particle of mass m and charge q moving under the inﬂuence of an external force Fext. The particle radiates electromagnetic waves that have a characteristic wavelength λ ∼ L, where L is the acceleration length of the particle. Thus the interaction of the particle with the electromagnetic ﬁeld violates the hypothesis of locality since λ/L ∼ 1. The radiating charged particle is therefore not momentarily equivalent to an otherwise identical comoving inertial particle. This agrees with the fact that in the nonrelativistic approximation the Abraham-Lorentz equation of motion of the particle is

dv 2 q2 d2v

m dt

− 3 c3

dt2

+ ... = Fext

,

(3.1)

so that the state of a radiating particle is not determined by its position and velocity alone.
Imagine an accelerated measuring device in Minkowski spacetime. The internal dynamics of the device is then subject to inertial effects that consist of the inertial forces of classical mechanics together with their generalizations to electromagnetic and quantum domains. If the net inﬂuence of these inertial effects integrates — over the relevant length and time scales of a measurement — to perturbations that do not appreciably disturb the result of the measurement and can therefore be neglected, then the hypothesis of locality is valid and the device can be considered standard (or ideal). Consider, for instance, the measurement of time dilation in terms of muon lifetime by observing the decay of muons in a storage ring. It follows from the hypothesis of locality that τμ = γτμ0, where γ is the Lorentz factor and τμ0 is the lifetime of the muon at rest in the background inertial frame. On the other hand, the lifetime of such a muon has been calculated on the basis of quantum theory by assuming that the muon occupies a high-energy Landau level in a constant magnetic ﬁeld [3]. One can show that the result of [3] can be expressed as [4]

The Hypothesis of Locality and its Limitations

45

τμ

γτμ0

2 1+
3

λ2 L

.

(3.2)

Here λ = /(mc) is the Compton wavelength of the muon, m is the muon mass and L = c2/a, where a = γ2v2/r is the effective centripetal acceleration of the muons in the storage ring. The hypothesis of locality is completely adequate for such experiments since λ/L is extremely small. In fact, the hypothesis of locality is clearly valid in many Earth-bound experimental situations since c2/g⊕ 1 lyr and c/Ω⊕ 28 AU.
The hypothesis of locality plays a crucial role in Einstein’s theory of gravitation: Einstein’s principle of equivalence together with the hypothesis of locality implies that an observer in a gravitational ﬁeld is locally inertial. Indeed, the equivalence between an observer in a gravitational ﬁeld and an accelerated observer in Minkowski spacetime is useless operationally unless one speciﬁes what an accelerated observer measures.
The hypothesis of locality was formally introduced in [1] and its limitations were pointed out. To clarify the origin of this conception, some background information is provided in section 2. The implications of this assumption for length determination in rotating systems are pointed out in section 3. Section 4 contains a discussion.

2. Background
Maxwell’s considerations regarding optical phenomena in moving systems implicitly contained the hypothesis of locality [5]. The fundamental form of Maxwell’s theory of electromagnetism, derived from Maxwell’s original electrodynamics of media, is essentially due to Lorentz’s development of the theory of electrons.
Lorentz conceived of an electron as an extremely small charged particle with a certain smooth volume charge density. A free electron at rest was regarded as a spherical material system with certain internal forces that ensured the constancy of its size and form. An electron in translational motion would then be a ﬂattened ellipsoid according to Lorentz, since it would be deformed from its original spherical shape by the Lorentz-FitzGerald contraction in the direction of its motion. The internal dynamics of electrons therefore became a subject of scientiﬁc inquiry and in 1906 Poincaré postulated the existence of a particular type of internal stress that could balance the electrostatic repulsion even in a moving (and hence ﬂattened) electron. These issues are discussed in detail in the ﬁfth chapter (on optical phenomena in moving bodies) of Lorentz’s book [6] on the theory of electrons.
In extending the Lorentz transformations in a pointwise manner to accelerating electrons, Lorentz encountered a problem regarding the dynamical equi-

46

B. Mashhoon

librium of the internal state of the electron. To avoid this problem, Lorentz introduced a basic assumption that is discussed in section 183 of his book [6]:
“... it has been presupposed that in a curvilinear motion the electron constantly has its short axis along the tangent to the path, and that, while the velocity changes, the ratio between the axes of the ellipsoid is changing at the same time.”
To elucidate this assumption, Lorentz explained its approximate validity as follows (§183 of [6]):
“... If the form and the orientation of the electron are determined by forces, we cannot be certain that there exists at every instant a state of equilibrium. Even while the translation is constant, there may be small oscillations of the corpuscle, both in shape and in orientation, and under variable circumstances, i.e. when the velocity of translation is changing either in direction or in magnitude, the lagging behind of which we have just spoken cannot be entirely avoided. The case is similar to that of a pendulum bob acted on by a variable force, whose changes, as is well known, it does not instantaneously follow. The pendulum may, however, approximately be said to do so when the variations of the force are very slow in comparison with its own free vibrations. Similarly, the electron may be regarded as being, at every instant, in the state of equilibrium corresponding to its velocity, provided that the time in which the velocity changes perceptibly be very much longer than the period of the oscillations that can be performed under the inﬂuence of the regulating forces.”
It is therefore clear that the hypothesis of locality and its limitation were discussed by Lorentz for the case of the motion of electrons.
Einstein, in conformity with his general approach of formulating symmetrylike principles that would be independent of the speciﬁc nature of matter, simply adopted the same general assumption for rods and clocks. In fact, in discussing the rotating disk problem, Einstein stated in a footnote on page 60 of [7] that:
“These considerations assume that the behavior of rods and clocks depends only upon velocities, and not upon accelerations, or, at least, that the inﬂuence of acceleration does not counteract that of velocity.”
The modern experimental foundation of Einstein’s theory of gravitation necessitates that this assumption be extended to all (standard) measuring devices; therefore, the hypothesis of locality supersedes the clock hypothesis, etc.
Though the hypothesis of locality originates from Newtonian mechanics, one should point out that the state of a relativistic point particle differs from that in Newtonian mechanics: the magnitude of velocity is always less than c. Moreover, the hypothesis of locality rests on the possibility of deﬁning instantaneous inertial rest frames along the worldline of an arbitrary point particle. In fact, Minkowski raised this possibility and hence the corresponding hypothesis of locality to the level of a fundamental axiom [8].
Another aspect of Lorentz’s presupposition must be mentioned here that involves the extension of the notion of rigid motion to the relativistic domain: the

The Hypothesis of Locality and its Limitations

47

electron moves rigidly as it is always undeformed in its momentary rest frame. The notion of rigid motion in the special and general theories of relativity has been discussed by a number of authors [9, 10, 11, 12, 13]. It is important to note that the concept of an inﬁnitesimal rigid rod is indispensable in the theory of relativity (cf. section 3).
In some expositions of relativity theory, such as [10] and [14], the hypothesis of locality is completely implicit. For instance, in Robertson’s paper on “Postulate versus Observation in the Special Theory of Relativity” [14], attention is simply conﬁned to “the kinematics im kleinen of physical spacetime” [14]. However, when interpreting the observational foundations of special relativity, one must recognize that actual observers are all accelerated and that the difference between accelerated and inertial observers must be investigated; in fact, this problem is ignored in [14] by simply asserting that physics is essentially local.

3. Length measurement
To illustrate the nature of the hypothesis of locality, it is interesting to consider spatial measurements of rotating observers. Imagine observers A and B moving on a circle of radius r about the origin in the (x, y)-plane of a background global inertial frame with coordinates (t, x, y, z). Expressed in terms of the azimuthal angle ϕ, the location of A and B at t = 0 can be chosen such that ϕA = 0 and ϕB = Δ with no loss in generality. The motion of A and B is then assumed to be such that for t > 0 they rotate in exactly the same way along the circle with angular frequency Ωˆ 0(t) > 0. Thus for t > 0 observers A and B can be characterized by the azimuthal angles

t

t

ϕA(t) = Ωˆ 0(t )dt , ϕB(t) = Δ + Ωˆ 0(t )dt .

0

0

(3.3)

According to the static inertial observers in the background global frame, the angular separation of A and B is constant at any time t > 0 and is given by ϕB(t) − ϕA(t) = Δ; moreover, the spatial separation of the two observers along the circular arc at time t > 0 is (t) = rΔ.
Consider now a class of observers O populating the whole arc from A to B and moving exactly the same way as A and B. At any time t > 0, it
appears to inertial observers at rest in the background frame that these rotating observers are all at rest in the (x , y , z ) system that is obtained from (x, y, z) by a simple rotation about the z-axis with frequency Ωˆ 0(t). What is the length of the arc according to these rotating observers? It follows from an application of the hypothesis of locality that for t > 0 the spatial separation between A and B as measured by the rotating observers is = γˆ (t), where γˆ is the

48

B. Mashhoon

Lorentz factor corresponding to vˆ = rΩˆ 0(t). Units are chosen here such that c = 1 throughout this section. Indeed at any time t > 0 in the inertial frame, each observer O is momentarily equivalent to a comoving inertial observer and the corresponding inﬁnitesimal element of the arc δ has a rest length δ in the momentarily comoving inertial frame such that from the Lorentz transformation between this local inertial frame and the global background inertial frame one obtains

1 − vˆ2 δ = δ

(3.4)

in accordance with the Lorentz-FitzGerald contraction. Deﬁning

=Σδ ,

(3.5)

where each δ is the inﬁnitesimal length at rest in a different local inertial frame, one arrives at = γˆ , since vˆ(t) is the same for the class of observers O at time t. The same result is obtained if length is measured using light travel time over inﬁnitesimal distances between observers O, since in each local inertial frame the two methods give the same answer. As is well known, the light signals could also be used for the synchronization of standard clocks carried by observers O.
It is important to remark here that equation (3.5) is far from a proper geometric deﬁnition of length and one must question whether it is even physically reasonable, since each δ in equation (3.5) refers to a different local Lorentz frame. In any case, in this approach the length of the arc as measured by the accelerated observers is

= γˆ(t)rΔ .

(3.6)

The sum in equation (3.5) involves inﬁnitesimal rest segments each from a separate local inertial frame. Perhaps the situation could be improved by combining these inﬁnite disjoint local inertial rest frames into one continuous accelerated frame of reference. The most natural way to accomplish this would involve choosing one of the noninertial observers on the arc and establishing a geodesic coordinate system along its worldline. In such a system, the measure of separation along the worldline (proper time) and away from it (proper length) would also be determined by the hypothesis of locality. That is, at any instant of proper time the rules of Euclidean geometry are applicable as the accelerated observer is instantaneously inertial. It turns out that the length of the arc determined in this way would in general be different from and would depend on which reference observer O : A → B is chosen for this purpose [15]. To illustrate this state of affairs and for the sake of concreteness, in the rest of

The Hypothesis of Locality and its Limitations

49

this section the length of the arc will be determined in a geodesic coordinate system along the worldline of observer A and the result will be compared with equation (3.6).
In the background inertial frame, the coordinates of observer A are

xμA = (t, r cos ϕA, r sin ϕA, 0) ,

(3.7)

and the proper time along the worldline of A is given by

t

τ=

1 − vˆ2(t ) dt ,

0

(3.8)

where τ = 0 at t = 0 by assumption. It is further assumed that τ = τ (t) has
an inverse and the inverse function is denoted by t = F (τ ). Thus dt/dτ = dF/dτ = γ(τ ) = (1 − v2)−1/2 is the Lorentz factor along the worldline of A, so that v(τ ) := vˆ(t) and γ(τ ) := γˆ(t). Moreover, it is useful to deﬁne φ(τ ) := ϕA(t) and dφ/dτ = γΩ0(τ ), where Ω0(τ ) := Ωˆ 0(t). With these deﬁnitions, the natural orthonormal tetrad frame along the worldline of A for
τ > 0 is given by

λμ(0) = γ(1, −v sin φ, v cos φ, 0) , λμ(1) = (0, cos φ, sin φ, 0) , λμ(2) = γ(v, − sin φ, cos φ, 0) , λμ(3) = (0 , 0 , 0 , 1) ,

(3.9) (3.10) (3.11) (3.12)

where λμ(0) = dxμA/dτ is the temporal axis and the spatial triad corresponds to the natural spatial frame of the rotating observer. To obtain this tetrad in a simple fashion, ﬁrst note that by setting r = 0 and hence v = 0 and γ = 1 in equations (3.9) - (3.12) one has the natural tetrad of the ﬁxed noninertial observer at the spatial origin — as well as the class of noninertial observers at rest in the background inertial frame — that refers its observations to the axes of the (x , y , z ) coordinate system alluded to before; then, boosting this tetrad with speed v along the second spatial axis tangent to the circle of radius r results in equations (3.9) - (3.12).
It follows from the orthonormality of the tetrad system (3.9) - (3.12) that the acceleration tensor Aαβ deﬁned by

dλμ(α) dτ

= Aαβ λμ(β)

(3.13)

is antisymmetric. The translational acceleration of observer A, which is the “electric” part of the acceleration tensor (ai = A0i), is given by

50

B. Mashhoon

a = (−γ2vΩ0

,

γ2 dv dτ

,

0)

(3.14)

with respect to the tetrad frame and similarly the rotational frequency of A,

which is

the “magnetic” part of the acceleration tensor (Ωi

=

1 2

ij k Aj k ),

is

given by

Ω = (0, 0, γ2Ω0) .

(3.15)

Moreover, in close analogy with electrodynamics, one can deﬁne the invariants of the acceleration tensor as

I = −a2 + Ω2 = γ2Ω20 − γ4

dv dτ

2

(3.16)

and I∗ = −a · Ω = 0. The analogue of a null electromagnetic ﬁeld is in this

case a null acceleration tensor; that is, an acceleration tensor is null if both I and I∗ vanish. A rotating observer with a null acceleration tensor is discussed

in the appendix.

The translational acceleration a consists of the well-known centripetal acceleration γ2v2/r and the tangential acceleration γ2dv/dτ . The latter formula

is consistent with the corresponding result in the case of linear acceleration

along a ﬁxed direction. To interpret equation (3.15) as the frequency of rotation

of the spatial frame with respect to a local nonrotating frame, it is necessary

to construct a nonrotating, i.e. Fermi-Walker transported, orthonormal tetrad frame λ˜μ(α) along the worldline of observer A. Let λ˜μ(0) = λμ(0) , λ˜μ(3) = λμ(3) and

λ˜μ(1) = cos Φ λμ(1) − sin Φ λμ(2) , λ˜μ(2) = sin Φ λμ(1) + cos Φ λμ(2) ,
where the angle Φ is deﬁned by

(3.17) (3.18)

τ

Φ = Ω(τ )dτ ,

(3.19)

0

so that dΦ/dτ = γ2Ω0. It remains to show that λ˜μ(i) , i = 1, 2, 3, correspond to local ideal gyroscope directions. This can be demonstrated explicitly using

equations (3.17) - (3.19) and one ﬁnds that

dλ˜μ(i) dτ

= a˜i λ˜μ(0)

,

(3.20)

where ˜a is the translational acceleration with respect to the nonrotating frame,

as expected. It is straightforward to study the average motion of the spatial

The Hypothesis of Locality and its Limitations

51

frame λ˜μ(i) with respect to the background inertial axes and illustrate Thomas precession with frequency (1 − γˆ)Ωˆ 0 per unit time t. That is, the frame of the accelerated observer rotates with frequency Ωˆ 0(t) about the background
inertial axes, while the Fermi-Walker transported frame rotates with frequency −γˆΩˆ 0 per unit time t with respect to the frame of the accelerated observer
according to equations (3.17) - (3.19); therefore, the unit gyroscope directions precess with respect to the background inertial frame with frequency (1−γˆ)Ωˆ 0
as measured by the static background inertial observers.
Along the worldline of observer A, the geodesic coordinates can be intro-
duced as follows: At a proper time τ , consider the straight spacelike geodesics that span the hyperplane orthogonal to the worldline. An event xμ = (t, x, y, z) on this hyperplane is assigned geodesic coordinates X μ = (T, X) such that

xμ = xμA(τ ) + Xiλμ(i)(τ ) , τ = T .

(3.21)

Let X = (X, Y, Z) and recall that along the worldline of A, t = F (τ ) and ϕA(t) = φ(τ ); then, the transformation to the new coordinates is given by

t = F (T ) + γ(T )v(T )Y , x = (X + r) cos φ(T ) − γ(T )Y sin φ(T ) , y = (X + r) sin φ(T ) + γ(T )Y cos φ(T ) , z = Z.

(3.22) (3.23) (3.24) (3.25)

For r = 0, the geodesic coordinate system reduces to (t , x , y , z ), where t = t; that is, the standard rotating coordinate system is simply the geodesic coordinate system constructed along the worldline of the noninertial observer at rest at the origin of spatial coordinates.
The form of the metric tensor in the geodesic coordinate system has been discussed in [1, 15, 16]. It turns out that in the case under consideration here the geodesic coordinates are admissible within a cylindrical region [16]. The boundary of this region is a real elliptic cylinder for I > 0, a parabolic cylinder for I = 0 or a hyperbolic cylinder for I < 0, where the acceleration invariant I is given by equation (3.16).
The class of observers O : A → B lies on an arc of the circle x2+y2 = r2 in the background coordinate system; therefore, it follows from equations (3.23) and (3.24) that in the geodesic coordinate system the corresponding ﬁgure is an ellipse

(X + r)2

Y2

r2 + (rγ−1)2 = 1

(3.26)

52

B. Mashhoon

√ with semimajor axis r, semiminor axis r 1 − v2 and eccentricity v. The latter quantities are in general dependent upon time T , hence at a given time t each
observer lies on a different ellipse. It is natural to think of the ellipse (3.26) as a circle of radius r that has suffered Lorentz-FitzGerald contraction along the
direction of motion [1, 15]. The measurement of the length from A to B in the new system involves the
integration of dL, dL2 = dX2 + dY 2, along the curve from A : (TA, 0, 0, 0) to B : (TB, XB, YB, 0) corresponding to A : (t, r cos ϕA, r sin ϕA, 0) and B : (t, r cos ϕB, r sin ϕB, 0) in the background inertial frame. To clarify the situation, it is useful to introduce — in analogy with the elliptic motion in the Kepler problem — the eccentric anomaly θ by

X + r = r cos θ , Y = r 1 − v2 sin θ .

(3.27)

Then, for a typical rotating observer O : (t, r cos ϕ, r sin ϕ, 0) on the arc from A → B with

t
ϕ = δ + Ωˆ 0(t )dt
0

(3.28)

one has in geodesic coordinates O : (T, X, Y, 0), where X and Y are given by

equations (3.27), and equations (3.22) - (3.24) imply that

t = F (T ) + rv(T ) sin θ ,

(3.29)

ϕ = θ + φ(T ) .

(3.30)

As O ranges from A to B, δ : 0 → Δ in equation (3.28) and hence θ : 0 → Θ. For a ﬁxed t, t = F (TA), equation (3.29) can be solved to give T as a function of θ; then, a detailed calculation involving equations (3.27) - (3.30) shows that

Θ
L=r
0
Here W is deﬁned by

1 − v2W cos2 θ dθ .

(3.31)

W

=

γ2

1 − r2v˙2 sin2 θ (γ + rv˙ sin θ)2

,

(3.32)

v˙ = dv/dT and Θ can be found in terms of Δ by solving equations (3.29) and (3.30) at B:

t = F (TB) + rv(TB) sin Θ ,

(3.33)

The Hypothesis of Locality and its Limitations

53

t
Δ + Ωˆ 0(t )dt = Θ + φ(TB) .
0

(3.34)

In practice, the explicit calculation of L can be rather complicated; therefore, for the sake of simplicity only the case of constant v (i.e. uniform rotation) will be considered further here [1, 15]. Then, W = 1 and equation (3.31) simply refers to the arc of a constant ellipse for which a Kepler-like equation

θ − v2 sin θ = δ

(3.35)

follows from equations (3.29) and (3.30). Furthermore, the proper acceleration length of the uniformly rotating observer A is given by L = I −1/2 = (γΩ0)−1. The case of uniform rotation, where L and are independent of time and L = in general, has been treated in detail in [1, 15] and it is clear that irrespective of the magnitude of Δ, L/ → 1 as r/L = vγ → 0; on the other hand for Δ → 0, L/ → 1 irrespective of v < 1. That is, consistency can be achieved only if the length under consideration is negligibly small compared
to the acceleration length of the observer.

4. Discussion
It is important to recognize that the hypothesis of locality is an essential element of the theories of special and general relativity. In particular, it is indispensable for the measurement of spatial and temporal intervals by accelerated observers. Therefore, relativistic measurement theory must take this basic assumption and its limitations into account. This has been done for the measurement of time in [17]. In connection with the measurement of distance, it has been shown that there is a lack of uniqueness; however, this problem can be resolved if the distance under consideration is much smaller than the relevant acceleration length of the observer [1, 15]. This means that from a basic standpoint the signiﬁcance of noninertial reference frames is rather limited [16]. In practice, however, the difference between L and (discussed in section 3) is usually rather small; for instance, in the case of the equatorial circumference of the Earth this difference amounts to about 10−2 cm [15].
The application of these concepts to standard accelerated measuring devices that are by deﬁnition consistent with the hypothesis of locality results in a certain maximal acceleration [18, 19] that is imposed by the quantum theory. For a classical device of mass M , the dimensions of the device must be much larger than /(M c) according to the quantum theory of measurement [20, 21]. On the other hand, the dimensions of the device must be much smaller than its acceleration length L . It follows that L >> /(M c) for any standard classical measuring device [2, 4]. Thus for L = c2/a, the translational acceleration a must be much smaller than M c3/ , while for L = c/Ω, the rotational fre-

54

B. Mashhoon

quency Ω must be much smaller than M c2/ . Further discussion of the notion of maximal acceleration is contained in [22].
The hypothesis of locality is compatible with wave phenomena only when the latter are considered in the ray limit (λ/L → 0). To go beyond the basic limitations inherent in the hypothesis of locality regarding the treatment of wave phenomena, a nonlocal theory of accelerated observers has been developed [23, 24, 25]. In this theory, the amplitude of a radiation ﬁeld as measured by an accelerated observer depends on its history, namely, its past worldline in Minkowski spacetime. This acceleration-induced nonlocality constitutes the ﬁrst step in the program of developing a nonlocal theory of gravitation.

Appendix: Null acceleration
The relativistic theory of an observer in arbitrary circular motion is treated in section 3. In this case, the proper acceleration length of the observer is deﬁned to be |I|−1/2, where I is given by equation (3.16). It is interesting to study the circular motion of an observer with a constant prescribed magnitude of I. In fact, equation (3.16) can be written as

dvˆ dt

2

=

1 r2

vˆ2(1 − vˆ2)2

− I(1 − vˆ2)3

,

which for constant I can be simply integrated. For the null acceleration case I = 0, the solution is

vˆ−2

=

1

+

η

e∓2

t r

for η > 0. The upper sign refers to motion that asymptotically (t → ∞) approaches the speed of light, while the lower sign corresponds to an asymptotic state of rest.

References
[1] B. Mashhoon, Phys. Lett. A 145 (1990) 147. [2] B. Mashhoon, Phys. Lett. A 143 (1990) 176. [3] A. M. Eisele, Helv. Phys. Acta 60 (1987) 1024. [4] B. Mashhoon, in: Black Holes: Theory and Observation, Lecture Notes
in Physics 514, edited by F. W. Hehl, C. Kiefer and R. J. K. Metzler (Springer, Heidelberg, 1998) pp. 269-284. [5] J. C. Maxwell, Nature 21 (1880) 314. [6] H. A. Lorentz, The Theory of Electrons (Dover, New York, 1952). [7] A. Einstein, The Meaning of Relativity (Princeton University Press, Princeton, 1950). [8] H. Minkowski, The Principle of Relativity, by H. A. Lorentz, A. Einstein, H. Minkowski and H. Weyl (Dover, New York, 1952) p. 80. [9] M. Born, Ann. Phys. (Leipzig) 30 (1909) 1.

REFERENCES

55

[10] W. Pauli, Theory of Relativity (Dover, New York, 1981). [11] J. L. Synge, Relativity: The Special Theory, 2nd edition (North-Holland,
Amsterdam, 1965). [12] W. Rindler, Introduction to Special Relativity, 2nd edition (Clarendon
Press, Oxford, 1991). [13] G. Salzman and A. H. Taub, Phys. Rev. 95 (1954) 1659. [14] H. P. Robertson, Rev. Mod. Phys. 21 (1949) 378. [15] B. Mashhoon and U. Muench, Ann. Phys. (Leipzig) 11 (2002) 532. [16] B. Mashhoon, in: Advances in General Relativity and Cosmology, edited
by G. Ferrarese (Pitagora, Bologna, 2003). [17] S. R. Mainwaring and G. E. Stedman, Phys. Rev. A 47 (1993) 3611. [18] E. R. Caianiello, Lett. Nuovo Cimento 41 (1984) 370. [19] E. R. Caianiello, Riv. Nuovo Cimento 15 (1992) no. 4. [20] E. Schrödinger, Preuss. Akad. Wiss. Berlin Ber. 12 (1931) 238. [21] H. Salecker and E. P. Wigner, Phys. Rev. 109 (1958) 571. [22] G. Papini, Phys. Lett. A 305 (2002) 359. [23] B. Mashhoon, Phys. Rev. A 47 (1993) 4498. [24] C. Chicone and B. Mashhoon, Ann. Phys. (Leipzig) 11 (2002) 309. [25] C. Chicone and B. Mashhoon, Phys. Lett. A 298 (2002) 229.

Chapter 4
SAGNAC EFFECT: END OF THE MYSTERY

Franco Selleri
Università di Bari - Dipartimento di Fisica INFN - Sezione di Bari I 70126 Bari, Italy selleri@ba.infn.it

Abstract

Transformations of space and time depending on a synchronization parameter , e1, indicate the existence of a privileged inertial system S0. The Lorentz transformations are obtained for a particular e1 = 0. No classical experiment on inertial frames depends on the choice of e1, but if accelerations are considered only e1 = 0 remains possible. The choice e1 = 0 provides a rational resolution of the long standing mystery connected with the relativistic interpretation of the
Sagnac effect.

1. History: 1913 - 2003
In the Sagnac 1913 experiment a circular platform was made to rotate uniformly around a vertical axis at a rate of 1-2 full rotations per second. In an interferometer mounted on the platform, two interfering light beams, reﬂected by four mirrors, propagated in opposite directions along a closed horizontal circuit deﬁning a certain area A. The rotating system included also the luminous source and a detector (a photographic plate recording the interference fringes). On the pictures obtained during a clockwise and a counterclockwise rotation with the same frequency, the interference fringes were observed to be in different positions. Sagnac measured the relative displacement Δz by overlapping the two ﬁgures.
This displacement Δz is strictly tied to the time delay with which a light beam reaches the detector with respect to the other one and turned out to depend on the disk angular velocity. Sagnac observed a shift of the interference fringes every time the rotation was modiﬁed. Considering his experiment

57
G. Rizzi and M.L. Ruggiero (eds.), Relativity in Rotating Frames, 57-77. © 2004 Kluwer Academic Publishers.

58

F. Selleri

Figure 4.1. Simpliﬁed conﬁguration of the Sagnac apparatus. Light from a source S is divided in two parts by the semitransparent mirror A. The ﬁrst part follows the path ABCDAO concordant with the platform rotation, the second part follows ADCBAO discordant from rotation. The interference fringes are observed in O.
conceptually similar to the Michelson-Morley one, he informed the scientiﬁc community with two papers (in French) bearing the titles "The existence of the luminiferous ether demonstrated by means of the effect of a relative ether wind in an uniformly rotating interferometer"[1] and "On the proof of reality of the luminiferous ether with the experiment of the rotating interferometer"[2].
The experiment was repeated many times in different ways, with the full conﬁrmation of the results obtained by Sagnac. Famous is the 1925 repetition by Michelson and Gale[3] for the very large dimensions of the optical interference system (a rectangle about 650m x 360m); in this case the disk was the Earth itself at the latitude concerned. The light propagation times were not the same, as evidenced by the resulting fringe shift. Full consistency was found with the Sagnac formula [Eq. (4.3) below] if the angular velocity of the Earth rotation was used.
An important question is the following: can light propagate with the usual velocity c relatively to the rotating platform? The question was directly faced in the 1942 experiment by Dufour e Prunier[4], in which the mirrors deﬁning the paths of the interfering light beams were partly ﬁxed in the laboratory (directly above the disk) and partly in the spinning disk. The fringe shifts were the same as in a repetition of the test with all mirrors ﬁxed on the disk, conﬁrming that the light does not adapt to the movement of the disk, and that it is physically connected with some other reference system, in all probability inertial.
Surprisingly theoreticians were little interested in the Sagnac effect, as if it did not pose a conceptual challenge. As far as I know Einstein’s publications never mentioned it, for example. A ﬁrst discussion by Langevin came only 7-

Sagnac effect: end of the mystery

59

8 years later[5] and was as much formally self-assured as substantially weak. One of the opening statements is this: "I will show how the theory of general relativity explains the results of Sagnac’s experiment in a quantitative way". Langevin argues that Sagnac’s is a ﬁrst order experiment, on which all theories (relativistic or prerelativistic) must agree qualitatively and quantitatively, given that the experimental precision does not allow one to detect second order effects: therefore it cannot produce evidence for or against any theory. Then he goes on to show that an application of Galilean kinematics explains the empirical observations! In fact his approach is only slightly veiled in relativistic form by some words and symbols, but is really 100% Galilean.
The impression that Langevin, beyond words, could not be satisﬁed with his explanation is reinforced by his second article of 1937[6] in which two(!) relativistic treatments are presented. The ﬁrst one is still that of 1921, this time deduced from the strange idea that the time to be adopted everywhere on the platform is that of the rotational centre (which is motionless in the laboratory). The second one is to deﬁne "time" in such a way as to enforce a velocity of light constant and equal to c by starting from a non total differential, falling so ﬂatly in the problem of the discontinuity for a tour around the disk that we will discuss later.
In 1963 was published the very inﬂuential review paper by Post[7], who seems to agree with the idea that two relativistic proofs of the Sagnac effect are better than one. The ﬁrst proof (in the main text) uses arbitrarily the laboratory to platform transformation of time t = t R where R is the usual square root factor of relativity, here written with the rotational velocity. The second proof (in an appendix) starts from the Lorentz transformation t = t + v · r/c2 /R, but it hastens to make the second term disappear with the (arbitrary) choice of r perpendicular to v.
The tendency to cancel the spatial variables in the transformation of time is thus common to Langevin and Post and shows once more the great difﬁculty in explaining the physics of the rotating platform with the TSR. The ﬁnal result can only be a great confusion, to the point that Hasselbach and Nicklaus, describing their own experiment[8], list about twenty different explanations of the Sagnac effect and comment: "This great variety (if not disparity) in the derivation of the Sagnac phase shift constitutes one of the several controversies ... that have been surrounding the Sagnac effect since the earliest days of studying interferences in rotating frames of reference".
In the present paper we will show that the problems concerning the Sagnac effect are overcome by adopting on the rotating platform the one way velocity of light given by

c

c1(θ) = 1 + β cos θ

(4.1)

60

F. Selleri

with β = ω r/c, where r is the distance from the platform rotation centre, ω the angular velocity and θ the angle between the light propagation direction and the rotational velocity in the point where light is moving. In general c1(θ) varies from point to point of the light path, but it equals constantly either c1(0) or c1(π) if light is moving on a circle centered in the platform rotation centre.
Equation (4.1) is valid in inertial systems, where the "inertial transformations" give the best description of the empirical evidence[9]: see Appendix 4.B for a short review. It applies also to the rotating platform, a small segment of which for a short time can be considered as practically belonging to a comoving inertial system.

2. The Sagnac Correction on the Earth Surface
As recounted by Kelly[10], in 1980 the CCDS (Comitè Consultatif pour la Dèﬁnition de la Seconde) and in 1990 the CCIR (International Radio Consultative Committee) suggested rules - later universally adopted - for synchronizing clocks in different points of the globe. Two are the methods used to accomplish this task. The ﬁrst one is to transport a clock from one site to another and to regulate clocks at rest in the second site with the time reading of the transported clock. The second method is to send an electromagnetic signal informing the second site of the time reading in the ﬁrst site. The rules of the committee establish that three corrections should be applied before comparing clock readings:

(a) the ﬁrst correction keeps into account the velocity effect of the theory of special relativity (TSR). It is proportional to v2/2c2, where v is the velocity of
the airplane, and corresponds to a slower timing of the transported clock;
(b) the second correction keeps into account the gravitational effect of the theory of general relativity (TGR). It is proportional to g(φ)h/c2 where g is
the total acceleration (gravitational and centrifugal) at sea level at the latitude φ and h is the height over sea level. It corresponds to a faster timing of the
transported clock; (c) the "Sagnac correction" is assumed proportional to 2AEω/c2, where
AE is the equatorial projection of the area enclosed by the path of travel of the clock (or of the electromagnetic signal) and the lines connecting the two clock sites to the centre of the Earth, and ω is the angular velocity of the Earth.

There are no doubts about nature and need of the ﬁrst two corrections, but the justiﬁcation of the third one is unconvincing. I agree completely with Kelly [11] when he says that the only possible reason to include (c) is that the eastward velocity of light relative to the Earth is different from the westward.

Sagnac effect: end of the mystery

61

Figure 4.2. An electromagnetic signal travels between two points W and T on the Earth via a geostationary satellite S (seen from the North pole).
In fact we will next deduce, for a real experiment, the "Sagnac correction" from Eq. (4.1) applied to a geostationary satellite, for which the satellite itself and the Earth surface can be thought to be at rest on the same rotating platform.
Saburi et al. carried out their experiment in 1976, before the CCDS and CCIR deliberations, and made clear that "corrections" were indeed necessary already in the title of their paper[12] ("High-Precision Time Comparison via Satellite and Observed Discrepancy of Synchronization"). They had two atomic clocks, not quite synchronous, one in a ﬁrst station W (near Washington, USA) the other one in a second station T (near Tokyo, Japan) practically on the same parallel of the two cities. The time difference between the two clocks at 02h 34m UTC on August 27, 1976 was measured with two different methods:
(i) by sending an airplane carrying a third clock (initially synchronous with the one in W ) from W to T , via Hawaii (westward);
(ii) by sending an electromagnetic signal, via a geostationary satellite, from W to T , again westward.
The uncorrected airplane clock found the T clock 9.42 μs fast with respect to the W clock. The velocity correction and the gravitational correction together were estimated to be about 0.080 μs (to be subtracted to the time shown

62

F. Selleri

by the transported clock). By applying such a correction the T − W time difference increased to 9.50 μs.
The electromagnetic signal carried with itself the time shown by the clock of the transmitting station. Assuming that the signal velocity was c, it was found that the T clock was 9.11 μs fast with respect to the W clock.
Thus, the discrepancy between the two measurements was about 0.39 μs.

Let LW S and LST the Washington-satellite and Tokyo-satellite distances, respectively (see ﬁgure 4.2). As most physicists in similar experiments, Saburi and collaborators synchronized clocks by imposing that the velocity of light is c, that is in such a way that

tT

− tW

=

LW S

+ LST c

(4.2)

tW and tT being the times of signal departure from W and arrival in T as marked by the respective clocks. In order to ensure that Eq. (4.2) applied to their clocks they had to apply the so called "Sagnac correction" to the clock of the receiving station. Such a correction is given by

ΔtT

=

2 ω AE c2

(4.3)

where AE is the area of the quadrangle OW ST O of ﬁgure 4.2. By adopting (4.2) Saburi and collaborators made an error because, as we
now know (see Appendix 4.B), the correct velocity of light is that given by the inertial transformations, which in the appropriate directions is

c

c

cW S = 1 + β cos αW S ; cST = 1 + β cos αST

(4.4)

where β = ω r/c (r is the radius of the W − T parallel and ω is the Earth angular velocity), αW S is the angle between the line W S and the local velocity (normal to the radius OW ), αST is the angle between the line ST and the normal to the radius OT in ﬁgure 4.2. Therefore

π

π

αW S = θW − 2 ; αST = θT − 2

(4.5)

where θW and θT are the angles OWˆ S and OTˆS of ﬁgure 4.2, respectively. But (4.4) is not the velocity adopted in this experiment. Having imposed the
impossible condition (4.2) the quoted authors had now to apply the mysterious "Sagnac correction" ΔtT on the time of arrival in T . Such a correction, from our point of view, can be calculated by replacing c with cW S and cST as follows

ΔtT

=

LW S cW S

−

LW S c

+

LST cST

−

LST c

(4.6)

Sagnac effect: end of the mystery

63

which is positive because c > cW S, cST . One can also write

ΔtT

=

β (LW S cos αW S + LST cos αST ) c

whence, using (4.5)

(4.7)

ΔtT

=

ωr

(LW S sinθW + c2

LST sinθT )

(4.8)

But rLW S sin θW +rLST sin θT = 2AE, where AE is the area of the quadrangle OW ST O of ﬁgure 4.2. We have thus provided a full physical justiﬁcation of (4.3). We see that the mystery of the "Sagnac correction" of Earth physics is fully eliminated by adopting the inertial transformations. The procedure which we can suggest to experimentalists is to avoid using a wrong velocity of light and correcting the result with an ad hoc term, but rather to use from the beginning the velocity of light (4.1) of the inertial transformations.
Our present results conﬁrm the following qualitative observation of Hayden [13]: electromagnetic signals need more time for a full tour around our planet toward east than toward west and this can only mean that relatively to the Earth the velocity of light in the two senses is not the same.

3. Rotating Platforms
In this section we review earlier results and show that the comparison between the relativistic descriptions of rotating platforms and inertial reference systems points out to the existence of a fundamental difﬁculty. Furthermore in the next section we show that this difﬁculty can be overcome only by substituting the Lorentz transformations between inertial systems with the "inertial" ones[14]. The problem is tightly bound to the Sagnac effect.
It is well known that no perfectly inertial frame exists in practice because of Earth rotation, of orbital motion around the Sun, of Galactic rotation. All knowledge about inertial systems has therefore been obtained in frames having small but non zero acceleration a. For this reason the mathematical limit a → 0 taken in the theoretical schemes should be smooth and no discontinuities should arise between systems with small acceleration and inertial systems. From such a point of view the existing relativistic theory will be shown to be inconsistent.
Consider an inertial reference system S0 and assume that it is isotropic so that the one-way velocity of light relative to S0 has the usual value c in all directions. In relativity the latter assumption is true in all inertial frames, while in other theories only one such frame exists.
In this system there is a circular platform having radius r and centre constantly at rest in S0 which rotates around its axis with constant angular velocity ω and peripheral velocity v = ω r. On its rim, consider a single clock CΣ

64

F. Selleri

(marking the time t) and assume it to be set as follows: When a clock of the

laboratory momentarily very near CΣ shows time t0 = 0 then also CΣ is set at time t = 0. When the platform is not rotating, CΣ constantly shows the same time as the laboratory clocks. When it rotates, however, motion modiﬁes the

pace of CΣ and the relationship between the times t and t0 is taken to have the general form

t0 = t F (v, ...)

(4.9)

where F is a function of velocity v and eventually acceleration and higher derivatives of position (not shown). Eq. (4.9) is a consequence of the isotropy of S0. Its validity can be shown in three steps:

1. In the inertial system S0 all directions are physically equivalent. If a clock is moving on a straight line with a certain speed v relative to S0, the modiﬁcation of the rate of advancement of its hands cannot depend on the orientation of .
2. A similar case is the clock CΣ at rest on the rim of a platform, whose centre is at rest in S0, rotating with constant angular velocity. If space is isotropical the speed of its hands cannot depend on the angle between the clock instantaneous velocity vector and any given direction in S0 but only on speed v and eventually acceleration.
3. This conclusion, clearly correct by symmetry reasons, was conﬁrmed experimentally by the 1977 CERN measurements of the anomalous magnetic moment of the muon[15]. The decay of muons was followed very closely in different parts of the storage ring and the results showed a decay rate constant in the different points of the trajectory.

Thus we have every reason to believe (4.9) to be correct. We are of course far from ignorant about the function F . There are strong experimental indications [15] that the dependence on the acceleration is totally absent and that:

1 F (v, ...) =
1 − v2/c2

(4.10)

Important as it is, Eq. (4.10) is however irrelevant for our present needs, because the results obtained below hold for all possible factors F .
On the rim of the platform besides clock CΣ there is a light source Σ placed in a ﬁxed position very near CΣ. Two light ﬂashes leave Σ at the time t1 of CΣ and are forced to move on a circumference, by "sliding" on the internal surface of a cylindrical mirror placed at rest on the platform, all around it and very near its border. Mirror apart, the light ﬂashes propagate in the vacuum. The motion of the mirror cannot modify the velocity of light, because the mirror behaves like a source ("virtual") and a source motion never changes the veloc-

Sagnac effect: end of the mystery

65

ity of the emitted light signals. Thus, relative to the laboratory, the light ﬂashes

propagate with the usual velocity c.

The description of light propagation given by the laboratory observers is

the following: two light ﬂashes leave Σ at time t01. The ﬁrst one propagates on a circumference, in the sense discordant from the platform rotation, and

comes back to S at time t02 after a full circle around the platform. The second ﬂash propagates on the same circumference, in the sense concordant with the

platform rotation, and comes back to Σ at time t03 after a full circle around the platform. These laboratory times, all relative to events taking place in a ﬁxed

point of the platform very near CΣ, are related to the corresponding platform times via (4.9):

t0i = ti F (v, ...) (i = 1,2,3)

(4.11)

The circumference length is assumed to be L0 and L, measured in the laboratory S0 and on the platform, respectively. Light propagating in the direction opposite to the disk rotation, must cover a distance smaller than L0 by x = ω r (t02 − t01) , the shift of Σ during the time t02 − t01 taken by light to reach Σ. Therefore

L0 − x = c ( t02 − t01) ; x = ω r ( t02 − t01)

(4.12)

From these equations it follows:

t02

−

t01

=

L0 c (1 +

β)

(4.13)

with β = ω r /c. Light propagating in the rotational direction of the disk, must instead cover a distance larger than the disk circumference length L0 by a quantity y = ω r (t03−t01) equalling the shift of Σ during the time t03−t01 taken by light to reach Σ. Therefore

L0 + y = c (t03 − t01) ; y = ω r ( t03 − t01)

(4.14)

One now gets

t03

−

t01

=

L0 c (1 −

β)

(4.15)

The difference between (4.15) and (4.13) gives the delay between the arrivals of the two light ﬂashes back in Σ which is

t03 −

t02

=

2 L0 β c (1 − β2)

(4.16)

This is the well known delay time for the Sagnac effect calculated in the laboratory. We show next that these relations ﬁx to some extent the velocity of light relative to the disk. In fact Eq. (4.11) applied to (4.13) and (4.15) gives

(t2

− t1) F

=

L0 c (1 +

β)

;

(t3

− t1) F

=

L0 c (1 − β)

(4.17)

66

F. Selleri

Figure 4.3. By symmetry reasons, the velocity of light relative to the rotating disk between two nearby points A and B does not depend on the angle φ ﬁxing the position of the segment AB on the rim of the disk.

If c˜(0) and c˜(π) are the light velocities, relative to the disk, for the ﬂash propagating in the direction of the disk rotation and in the opposite direction, respectively, we have from the very deﬁnition of velocity

1 = t2 − t1 =

L0/L

1 ;

= t3 − t1 =

L0/L

c˜ (π)

L

F c (1 + β) c˜(0)

L

F c (1 − β)

(4.18)

From (4.18) it follows :

c˜(π) = 1 + β c˜(0) 1 − β

(4.19)

Notice that the function F has disappeared in the ratio (4.19).

Next comes an important remark. Clearly, Eq. (4.19) gives us not only the

ratio of the two global light velocities for full trips around the platform, but the

ratio of the instantaneous velocities as well. In fact the isotropy of the inertial system S0 ensures, by symmetry, that the instantaneous velocities of light are the same in all points of the rim of the rotating circular disk whose centre is at rest in S0. There is no reason why the light instantaneous velocities relative to the disk in the different points of the rim should not be equal to one another.

With reference to ﬁgure 4.3 we can therefore write the equations

c˜φ1 (0) = c˜φ2 (0) ; c˜φ1 (π) = c˜φ2 (π)
where φ1 and φ2 are arbitrary values of the angle φ. Therefore the light instantaneous velocities relative to the disk will also co-
incide with the average velocities c˜(0) and c˜(π), and Eq. (4.19) will apply also to the ratio of the instantaneous velocities [thus we do not need a different symbol for the instantaneous velocities].

Sagnac effect: end of the mystery

67

The consequences of (4.19) applied to instantaneous velocities will be discussed in the next section.

4. Absolute simultaneity in inertial systems

The result (4.19) holds with the same numerical value for platforms having different radius, but the same peripheral velocity v. Let a set of circular platforms be given with centres at rest in S0. Let their radii be r1, r2, ... ri, ..., with r1 < r2 < ... < ri < ..., and suppose they are made to spin with angular velocities ω1, ω2, ... ωi, ... such that

ω1r1 = ω2r2 = ... = ωiri = ... = v

(4.20)

where v is constant. Obviously, then, (4.19) applies to all such platforms with the same β (β = v/c). The centripetal accelerations decrease regularly with increasing ri. Therefore, a small part AB of the rim of a platform, having peripheral velocity v and large radius, for a short time is completely equivalent to a small part of a "comoving" inertial reference frame (endowed with the same velocity). For all practical purposes the segment AB will belong to that inertial reference frame. But the velocities of light in the two directions AB and BA has to satisfy (4.19). It follows that the one way velocity of light relative to the comoving inertial frame cannot be c and must instead satisfy

c1(π) = 1 + β c1(0) 1 − β

(4.21)

As shown in Appendix 4.A the equivalent transformations (of which the Lorentz transformations are a particular case) predict the inverse one way velocity of light relative to the comoving system S:

1

1β

=+ c1(θ) c

c + e1 R cos θ

(4.22)

where θ is the angle between the light propagation direction and the absolute velocity v of S. Eq. (4.22) applied to the cases θ = 0 and θ = π gives

1

1

β

1

1β

=+ c1(0) c

c + e1R

;

=−

c1(π) c

c + e1R

whence

c1(π) = 1 + β + c e1R c1(0) 1 − β − c e1R

Clearly Eq. (4.24) is compatible with (4.21) only if

(4.23) (4.24)

e1 = 0

(4.25)

68

F. Selleri

Figure 4.4. The ratio ρ = c˜(π)/c˜(0) plotted as a function of acceleration for rotating platforms of constant peripheral velocity and decreasing radius (increasing acceleration). The prediction of the TSR is 1 (black dot on the ρ axis) and is not continuous with the ρ value of the rotating platforms.

We thus see that our fundamental result (4.21) is consistent with the physics of the inertial systems only if absolute simultaneity is adopted.

For a better understanding of the reasons why the TSR does not work consider again the ratio

c˜(π) ρ ≡ c˜(0)

(4.26)

which, owing to (4.19), is larger than unity. Therefore the light velocities parallel and antiparallel to the disk peripheral velocity are different. For the TSR this conclusion is unacceptable, because a set of platforms, all endowed with the same peripheral velocity locally approximates an inertial system better and better with increasing radius. The logical situation is shown in ﬁgure 4.4.
Thus the TSR predicts for ρ a discontinuity at zero acceleration. While all the experiments are performed in the real physical world [where, of course, a = 0, ρ = (1 + β)/(1 − β)], the theory has gone out of the world (a = 0, ρ = 1)! This discontinuity is the origin of the problems met with clock synchronization on or near the Earth surface we discussed in section 2. This is not a surprise: after all also the Earth is some kind of rotating platform! Notice that the velocity of light given by Eq. (4.22) with e1 = 0 is required for all inertial systems but one, the isotropic system S0. In fact, for every small region AB of every such system it is possible to imagine a large rotating platform with center at rest in S0 and rim locally comoving with AB and the result

Sagnac effect: end of the mystery

69

(4.25) can be applied. Therefore the velocity of light depends on direction in all inertial systems with the sole exception of the privileged one S0.

5. The impossible defense of orthodoxy

It was pointed out by T. Van Flandern[16] that an "orthodox" way of dealing with the rotating platform problem should assume a position dependent desynchronization, with respect to the laboratory clocks, as an objective phenomenon, concretely applicable to the clocks placed in different ﬁxed points of the rim of the platform. The Lorentz transformation of time

t R − t0 = − x0 v/c2

(4.27)

can be read as follows: the difference between the time t of the "moving" frame

S (corrected by a R factor in order to cancel the time dilation effect) and the

time t0 of the "stationary" frame S0 has a linear dependence on the coordinate

x0 of S0 as given by (4.27). This difference is called "desynchronization".

Applied to the rim of the circular platform of radius r Eq. (4.27) would

become

t R − t0 = − r θ0 v/c2

(4.28)

where θ0 is the angle between the radius on which the given clock is placed and any ﬁxed direction.
Why the orthodox idea cannot work? Well, for at least three reasons. First of all because the whole argument leading to (4.19) and then to (4.21) was based on a single clock in a ﬁxed position, for which it does not make sense to assume a position dependent desynchronization. Secondly because experimental evidence shows that many small clocks (muons) injected in different points of the CERN muon storage ring behave in the same way, independently of their position: in such a case it is certainly not possible to conceive a human intervention desynchronizing the muons! Thirdly because, anyway, such an approach would end in a mess.
Let us see how. Assume that the disk rotation is counterclockwise in ﬁgure 4.5. Two ﬂashes of light are emitted at laboratory time t0 = 0 by the source Σ in opposite directions along the platform border. Assuming Eq. (4.28) to be applicable, when the right-moving (left-moving) ﬂash reaches point A (point B) at lab. time t0A (t0B) after covering a distance x0A (x0B) (measured in the lab. along the platform border), it will ﬁnd a local clock desynchronized by ΔtA > 0 (ΔtB < 0) with respect to the laboratory clocks given by

ΔtA = x0Av/c2 + α ; ΔtB = − x0Bv/c2 + α

(4.29)

where α represents whatever desynchronization the clock placed in Σ might have had at time t0 = 0 with respect to the laboratory clocks. We will now show that the desynchronization (4.29), far from eliminating the discontinuity

70

F. Selleri

Figure 4.5. Two light ﬂashes emitted by Σ in opposite directions along the platform border at the laboratory time t0 = 0 meet in the point P in which the local platform clock should mark two different times.

of ﬁgure 4.4, gives rise to a further discontinuity in the time shown by clocks placed on the platform rim.
From the point of view of the laboratory observers the space between Σ and the right-moving (left-moving) ﬂash opens at a rate c − v (c + v). Therefore:

(c − v) t0A = x0A ; (c + v) t0B = x0B
From Eq. (4.29) we then get ΔtA = (c − v) t0A v/c2 + α ; ΔtB = − (c + v) t0B v/c2 + α (4.30)
There will be a time t0P in which the two ﬂashes meet in point P after describing different paths. When this happens one should have

t0A = t0B = t0P

(4.31)

The problem is that Eqs. (4.30) should both apply to the same clock at the common time t0P , but they are instead incompatible if (4.31) is satisﬁed, as ΔtA = ΔtB would then give c = −c.

We can add that Eq. (4.29) is in sharp contradiction with the rotational invariance assumed above to prove the existence of the discontinuity of ﬁgure 4.4: if the inertial system S0 (in which the centre of the circular platform is at rest) is isotropic and if the platform is set in rotation in a regular way, no difference between clocks on its border can ever arise. It is impossible to understand why the clock in point A should be desynchronized differently from the clock in B, unless this is achieved artiﬁcially by some observer.

Sagnac effect: end of the mystery

71

In conclusion this "orthodox" way of dealing with the rotating platform problem is only a useless complication. The only rational possibility remains the adoption of our inertial transformations.

6. New proofs of absolute simultaneity
We will now present a more general proof of the absolute simultaneity condition e1 = 0, by deducing it in the broader context of the "general transformations" [Eqs. (4.32) below]. In this way absolute simultaneity will be shown to be necessary in all theories avoiding the discontinuity between inertial and accelerated systems.
Given the inertial frames S0 and S one can set up Cartesian coordinates and make the following usual assumptions:

(i) Space is homogeneous and isotropic and time homogeneous, at least if
judged by observers at rest in S0, so that relatively to S0 the velocity of light is the same (”c”) in all directions, clocks can be synchronized in S0 with Einstein’s method, and the one way velocities can be measured in S0 ;
(ii) The origin of S, observed from S0, is seen to move with velocity v < c parallel to the +x0 axis, that is according to the equation x0 = v t0;
(iii) The axes of S and S0 coincide at t = t0 = 0;

The

general

transformations ⎧

from

S0

to

S

are

then

necessarily

⎨

x = f1(x0 − vt0)

⎩

y

= g2 y0 ; z = g2 z0 t = e1 x0 + e4t0

(4.32)

where f1, g2, e4 and e1 are v dependent parameters. The transformations in-

verse of (4.32) can easily be shown to be

⎧

⎪⎨ ⎪⎩

x0 =

y0

=

1 g2

(e4/f1)x + v t

e4+e1 v
y ; z0 =

1 g2

z

t0

=

t − (e1/f1) x e4+e1 v

(4.33)

The one way velocity of light relative to the moving system S, c1(θ), can be found by applying (4.33) to the equation

x20 + y02 + z02 = c2t20

(4.34)

describing a spherical wave front born at time t0 = 0 in the origin of the axes of the isotropic system S0. A calculation lengthy but devoid of conceptual difﬁculties leads to the following result:

1 c1(θ)

=

1 f1 (1 − β2)

e1

+

e4 c

β

cosθ +

e4 c

+ e1 β

cos2θ + γ2sin2θ 1/2
(4.35)

72

F. Selleri

where β = v/c, θ is the angle between the light propagation direction and the absolute velocity v of S, and

γ2 = f12

1 − β2 g22

(4.36)

From Eq. (4.35) one obviously gets

1 c1(0)

=

1
f1 (1−β2)

e4 c

+ e1

(1 + β)

1 c1(π)

=

1
f1 (1−β2)

e4 c

− e1

(1 − β)

(4.37)

whence

c1(π) = c1(0)

e4 c

+ e1

(1 + β)

e4 c

− e1

(1 − β)

(4.38)

As we know, the assumed continuity between rotating platforms and inertial

systems leads to Eq. (4.21) which we repeat here:

c1(π) = 1 + β c1(0) 1 − β By comparing (4.38) and (4.39) it follows

(4.39)

e1 = 0

(4.40)

Therefore the most general transformations of space and time between inertial

systems allowed by continuity are

⎧

⎨

x = f1(x0 − vt0)

⎩

y

=

g2 y0 ; z = t = e4t0

g2 z0

(4.41)

and imply the necessary existence of absolute simultaneity. In fact, two pointlike events with coordinates x10 and x20 (x01 = x02) taking place at the same time t0, according to the fourth of (4.41) are judged simultaneous also in S. Once more the absolute simultaneity is seen to be unavoidable. With e1 = 0 the velocity of light becomes

1 c1(θ)

=

c f1

e4 (1 − β2)

cos2θ + γ2sin2θ 1/2 + β cosθ

(4.42)

showing that the β cosθ term in the one way velocity of light is a ﬁxed ingredient in all theories of inertial systems satisfying the continuity condition with the accelerated ones.
The Galilean transformations are of the type (4.32) with

f1G = g2G = eG4 = 1 ; eG1 = 0

(4.43)

Sagnac effect: end of the mystery

73

Using (4.43) the one way velocity of light of the Galilean theory can be obtained as a particular case from (4.42) and turns out to be given by

1

1

cG1 (θ) = c (1 − β2)

1 − β2sin2θ 1/2 + β cosθ

(4.44)

Naturally Eq. (4.44) contains the characteristic term β cosθ of all theories treating inertial systems in a way continuous with the accelerated ones. The absence of this term in the TSR, in which c1 = c is isotropic, gives rise to the discontinuity of ﬁgure 4.4 which, as we saw, can be eliminated only by adopting e1 = 0.
We can conclude that the famous synchronization problem is solved by nature itself: it is not true that the synchronization procedure can be chosen freely as the usually adopted convention leads to an unacceptable discontinuity in the physical theory.

Appendix: A - The Equivalent Transformations
According to Poincarè[17], Reichenbach[18], Jammer[19] and Mansouri and Sexl[20] the clock synchronization in inertial systems is conventional, but the choice based on the invariance of the one way velocity of light made in the TSR was legitimate on reasons of simplicity. In [21] I showed that a suitable parameter e1 can be introduced to allow for different synchronizations in the transformations of the space and time variables. The TSR is obtained for a particular nonzero value of e1.
These developments are brieﬂy reviewed in the present section. I also found, however, that the choice e1 = 0 is the only one allowing for a treatment of accelerations rationally connected with the physics of inertial systems. This result is deduced once more in the main text as far as centripetal accelerations are concerned. Another proof of e1 = 0 is reviewed in Appendix 4.B.
Given the inertial frames S0 and S one can set up Cartesian coordinates and make the following assumptions:
(i) Space is homogeneous and isotropic and time homogeneous, at least if judged by observers at rest in S0;
(ii) In the isotropic system S0 the velocity of light is ”c” in all directions, so that clocks can be synchronized in S0 and one way velocities relative to S0 can be measured;
(iii) The origin of S, observed from S0, is seento move with velocity v < c parallel to the +x0 axis, that is according to the equation x0 = v t0;
(iv) The axes of S and S0 coincide for t = t0 = 0;
The system S0 turns out to have a privileged status in all theories satisfying the assumptions (i) and (ii), with only one exception, the TSR. Two further assumptions based on direct experimental evidence can be added:
(v) The two way velocity of light is the same in all directions and in all inertial systems[22];

74

F. Selleri

(vi) Clock retardation takes place with the usual velocity dependent factor when clocks move with respect to the isotropic reference frame S0 ([23]–[26]).

These conditions were shown[21] to imply for the transformations of the space and time

variables from S0 to S;

⎧ ⎪⎪⎨

x

=

x0 − v t0 R

y = y0 ; z = z0

⎪⎪⎩ t = R t0 + e1 (x0 − v t0)

(4.A.1)

where

R = 1 − v2/c2

(4.A.2)

Eqs. (4.A.1) and the assumption (ii) imply that relative to the moving system S the one way velocity of light propagating at an angle θ from the velocity v of S relative to S0 ("absolute velocity") is[21]:

c c1(θ) = 1 + Γ cos θ

(4.A.3)

with

v

Γ = c2 + e1 R

while, of course, the two way velocity of light is c in all directions.

The inverse transformations of (4.A.1) are

⎧ ⎨

x0

=

(R − e1v) x

+

vt R

⎩

y0 = y ; z0 = z

t0

=

t − R e1 x R

(4.A.4) (4.A.5)

All theories with different values of e1 imply the existence of a privileged inertial system, S0, in which the velocity of light is isotropic, as it is clear also from (4.A.3)-(4.A.4) since c1(θ) → c if v → 0, given that in this limit the transformations (4.A.1) must become identities and therefore e1 → 0 and Γ → 0. This is very important. If our theory describes correctly the physical reality a particular inertial system has to exist in which simultaneity and time are not conventional but truly physical. This should be the system in which the Lorentz ether is at rest, of course.
The TSR is a particular case of the previous theory, obtained for

e1

=

−

v c2R

(4.A.6)

giving Γ = 0 and c1(θ) = c and reducing (4.A.1) to the Lorentz form.

Appendix: B - The Inertial Transformations

The inertial transformations are obtained from the equivalent transformations, by setting

e1 = 0:

⎧ ⎪⎪⎨

x

=

x0 − v t0 R

y = y0 ; z = z0

⎪⎪⎩

t = R t0

(4.B.1)

Sagnac effect: end of the mystery

75

where R is the usual square root given by (4.A.2). Eqs. (4.B.1) imply that relative to the moving system S the one way velocity of light propagating at an angle θ from the velocity v of S relative to S0 ("absolute velocity") is[21]:

c1(θ)

=

1

c + β cos θ

(4.B.2)

with

β = v/c

(4.B.3)

while, of course, the two way velocity of light is c in all directions. The inverse transformations of (4.B.1) are

⎧

⎨ x0 = R

x

+

vt R2

⎩

y0 = y ; z0 = z

t0

=

t R

(4.B.4)

The theory of the inertial transformations implies the existence of a privileged inertial system, S0, in which the velocity of light is isotropic, as it is clear from (4.B.2) if β = 0.
The usually assumed indifference of the physical reality about clock synchronization exists only insofar as one neglects accelerations. When these come into play every inertial system exists, so to say, only for a vanishingly small time interval and it is physically impossible in the accelerated frame to adopt any time-consuming procedure for the synchronization of distant clocks (such as Einstein’s procedure based on light signals). Yet physical events take place and synchronization must somehow be ﬁxed by nature itself. In the text we saw how this happens for rotating platforms. Here we will review the argument for linear accelerations.

The accelerating spaceships In the isotropic system S0 clocks have been synchronized with the Einstein method, by using light signals. Two identical spaceships A and B initially at rest on the x0 axis of S0 have internal clocks synchronized with those of S0. At time t0 = 0 the spaceships start accelerating in the direction +x0, and they do so in exactly the same way, so that they have the same velocity v(t0) at every time t0 of S0. At time t¯0 they reach a preestablished velocity v = v(t¯0) and their acceleration ends. For t0 ≥ t¯0 the spaceships are at rest in a different inertial system S (which they concretely determine) in motion with velocity v with respect to S0. The relationship between the coordinates of S0 and S is given by the transformations (1) with e1 = 0 (not by the Lorentz transformations), because the delay between the times marked by clocks on board of A and B and those in S0 does not depend on position: since A and B had at every time exactly the same velocity, their clocks accumulated exactly the same delay with respect to S0. Therefore two events simultaneous in S0, taking place in points of space near which A and B are passing, must be simultaneous also for the travellers in A and B, and thus also in the rest system of the spaceships, S. This is clearly a
situation of absolute simultaneity which cannot be accounted for if the Lorentz transformations
are applied, but is obtained from (4.B.1) and (4.B.4). Not only the absolute simultaneity arises spontaneously in S, but it provides the only rea-
sonable description of the physical reality. To see this, suppose that in A and B there are two
passengers who are homozygous twins. Naturally nothing can stop them from re-synchronizing
their clocks, once the acceleration has ceased. However, if they do so, they discover to have

76

F. Selleri

different biological ages at the same time of S, as they cannot re-synchronize their bodies! Everything is regular, instead, if they do not modify the times of their clocks.

References
[1] M.G. Sagnac, Compt. Rend. 157, 708, 1410 (1913). [2] M.G. Sagnac, J. de Phys. 4, 177 (1914). [3] A.A. Michelson and H.G. Gale, Astroph. J. 61, 137 (1925). [4] A. Dufour and F. Prunier, Compt. Rend. 204, 1925 (1937); A. Dufour and
F. Prunier, J. de Phys. 3, 153 (1942) [5] P. Langevin, Compt. Rend. 173, 831 (1921); [6] P. Langevin, Compt. Rend. 205, 304 (1937). [7] E.J. Post, Rev. Mod. Phys. 39, 475 (1967). [8] F. Hasselbach and M. Nicklaus, Phys. Rev. A 48, 143 (1993). [9] F. Selleri, Found. Phys. 26, 641 (1996); F. Selleri, Chin. J. Syst. Eng.
Electr. 6, 25 (1995). [10] A.G. Kelly in: Open Questions in Relativistic Physics, F. Selleri, ed.
(Apeiron Press, Montreal, 1998), p. 25. [11] A.G. Kelly, Electronics World, September 2000, p. 722. [12] Y. Saburi et al., IEEE Trans. IM25, 473 (1976). [13] H.C. Hayden, Physics Essays, 8, 366 (1995). [14] F. Selleri, in New Developments on Fundamental Problems in Quantum
Physics, M. Ferrero et al., eds. (Kluwer, Dordrecht, 1997), p. 381; F. Goy and F. Selleri, Found. Phys. Lett. 10, 17 (1997); F. Selleri, Found. Phys. Lett. 10, 73 (1997). [15] J. Bailey et al., Nature 268, 301 (1977) . [16] Tom Van Flandern, private communication. [17] H. Poincarè, Rev. Metaphys. Morale 6, 1 (1898). [18] H. Reichenbach, The Philosophy of Space and Time, (Dover, New York, 1958). [19] M. Jammer, Some fundamental problems in the special theory of relativity, in: G. Toraldo di Francia, ed., Problems in The Foundations of Physics, pp. 202-236 (Società Italiana di Fisica, Bologna and North Holland, Amsterdam, 1979). [20] R. Mansouri and R. Sexl, General Relat. Gravit. 8, 497, 515, 809 (1977). [21] F. Selleri, Found. Phys. 26, 641 (1996); Found. Phys. Lett. 9, 43 (1996). [22] Presently the two way velocity of light in vacuum is known with an error of 20 cm/s. See: P.T. Woods, K.C. Sholton, W.R.C. Rowley, Appl. Opt. 17, 1048 (1978); D.A. Jennings, R.E. Drullinger, K.M. Evenson, C.R. Pollock, J.S Wells, J. Res. Natl. Bur. Stand. 92, 11 (1987). [23] B. Rossi and D.B. Hall, Phys. Rev. 59, 223 (1941).