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Eric Gourgoulhon

Special Relativityin General Frames

From Particles to Astrophysics

123

Eric GourgoulhonLaboratoire Univers et TheoriesObservatoire de Paris, CNRS, Universite Paris DiderotMeudon, France

Translation from the French language edition of: Relativite restreinte: Des particules al’astrophysique, c� 2010 EDP Sciences, CNRS Edition, France.

ISSN 1868-4513 ISSN 1868-4521 (electronic)Graduate Texts in PhysicsISBN 978-3-642-37275-9 ISBN 978-3-642-37276-6 (eBook)DOI 10.1007/978-3-642-37276-6Springer Heidelberg New York Dordrecht London

Library of Congress Control Number: 2013942463

c� Springer-Verlag Berlin Heidelberg 2013This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting, reproduction on microfilms or in any other physical way, and transmission or informationstorage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodologynow known or hereafter developed. Exempted from this legal reservation are brief excerpts in connectionwith reviews or scholarly analysis or material supplied specifically for the purpose of being enteredand executed on a computer system, for exclusive use by the purchaser of the work. Duplication ofthis publication or parts thereof is permitted only under the provisions of the Copyright Law of thePublisher’s location, in its current version, and permission for use must always be obtained from Springer.Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violationsare liable to prosecution under the respective Copyright Law.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.While the advice and information in this book are believed to be true and accurate at the date ofpublication, neither the authors nor the editors nor the publisher can accept any legal responsibility forany errors or omissions that may be made. The publisher makes no warranty, express or implied, withrespect to the material contained herein.

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To Valerie and Maxime

Foreword

The theory of special relativity holds a distinctive place within physics. Ratherthan being a specific physical theory, it is (similar to thermodynamics or analyticalmechanics) a general theoretical framework within which various dynamical theo-ries can be formulated. In this respect, a modern presentation of special relativitymust put forward its essential structures before illustrating them by concreteapplications to specific dynamical problems. Such is the challenge (so successfullymet!) of the beautiful book by Eric Gourgoulhon.

Contrary to most textbooks on special relativity, which mix the presentationof this theory with that of its historical development and which sometimes writethe specific form of “Lorentz transformations” before indicating that they leave acertain quadratic form invariant, the book by Eric Gourgoulhon is centred, from thevery beginning, on the essential structure of the theory, i.e. the chrono-geometricstructure of the four-dimensional Poincare–Minkowski spacetime. The aim is totrain the reader to formulate any relativity question in terms of four-dimensionalgeometry. The word geometry has here the meaning of “synthetic geometry” (ala Euclid) in contrast with “analytic geometry” (a la Descartes). Under the expertguidance of Eric Gourgoulhon, the reader will learn to set, and to solve, any problemof relativity by drawing spacetime diagrams, made of curves, straight lines, planes,hyperplanes, cones and vectors. He will get accustomed to visualizing the motionof a particle as a line in spacetime, to think about the twin paradox as an applicationof the “spacetime triangle inequality”, to express the local frame of an observer asa four-dimensional generalization of the Serret–Frenet triad, to compute a spatialdistance as a geometric mean of time intervals (via the hyperbolic generalization ofthe power of a point with respect to a circle) or to understand the Sagnac effect byconsidering two helices in spacetime wound in opposite directions.

Besides the pedagogical characteristic of being centred on a geometric formu-lation, the book by Eric Gourgoulhon is remarkable in many other ways. Firstof all, it is fully up to date and very complete in its coverage of the notions andresults where special relativity plays an important role: from Thomas precession tothe foundations of general relativity, including tensor calculus, exterior differentialcalculus, classical electrodynamics, the general notion of energy–momentum tensor

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and a noteworthy chapter on relativistic hydrodynamics. In addition, this book issprinkled with enlightening historical notes, in which the author summarizes ina condensed, albeit very informative way the (sometimes very recent) results byhistorians of science. Finally, the book is richly laden with many examples ofapplications of special relativity to concrete physical problems. The reader willlearn the role of special relativity in various domains of modern astrophysics(supernova nebulae, relativistic jets, micro-quasars) in the description of the quark-gluon plasma produced in heavy ion collisions, as well as in many high-technologyexperiments: from laser gyrometers to the LHC, including modern replications ofthe Michelson–Morley experiment, matter wave interferometers, synchrotrons andtheir radiation, and the comparison of atomic clocks embarked on planes, satellitesor the International Space Station.

I am sure that the remarkably rich book by Eric Gourgoulhon will attract the keeninterest of many readers and will enable them to understand and master one of thefundamental pillars (with general relativity and quantum theory) of modern physics.

Bures-sur-Yvette, France Thibault Damour

Preface

This book presents a geometrical introduction to special relativity. By geometrical,it is meant that the adopted point of view is four dimensional from the verybeginning. The mathematical framework is indeed, from the first chapter, that ofMinkowski spacetime, and the basic objects are the vectors in this space (often called4-vectors). Physical laws are translated in terms of geometrical operations (scalarproduct, orthogonal projection, etc.) on objects of Minkowski spacetime (4-vectors,worldlines, etc.).

Many relativity textbooks start rather by a three-dimensional approach, usingspace + time decompositions based on inertial observers. Only in the second stagethey introduce 4-vectors and Minkowski spacetime. In this respect, they are faithfulto the historical development of relativity. A more axiomatic approach is adoptedhere, setting from the very beginning the full mathematical framework as one ofthe postulates of the theory. From this point of view, the chosen approach is similarto that adopted in classical mechanics or quantum mechanics, where usually theexposition does not follow the history of the theory. The history of relativity isundoubtedly rich and fascinating, but the objective of this book is the learning ofspecial relativity within a consistent and operational setting, from the bases up toadvanced topics. The text is, however, enriched with historical notes, which includereferences to the original works and to the studies by historians of science.

Usually, the geometric approach is reserved for general relativity, i.e. for theincorporation of the gravitational field in relativity theory.1 We employ it herefor special relativity, taking into account a geometric structure much simpler thanthat of general relativity: while the latter is based on the concept of differentiablemanifold, special relativity relies entirely on the concept of affine space, which canbe identified with the space R

4. Consequently, the mathematical prerequisites arerelatively limited; they are mostly linear algebra at the level of the first two yearsof university. The mathematics used here is actually the same as those of a course

1Two notable exceptions are the monographs by Costa de Beauregard (1949) and Synge (1956) .

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of classical mechanics, provided one is ready to take into account two things: (i)vectors do not belong to a linear space of dimension three, but four, and (ii) the scalarproduct of two vectors is not the standard scalar product in Euclidean space but isgiven by a privileged symmetric bilinear form, the so-called metric tensor. Oncethis is accepted, physical results are obtained faster than by means of the “classic”three-dimensional formulation, and a more profound understanding of relativity isacquired. Moreover, learning general relativity is made much easier, starting fromsuch an approach.

In connection with the four-dimensional approach, another characteristic of thismonograph is to lay the discussion of physically measurable effects on the mostgeneral type of observer, i.e. allowing for accelerated and rotating frames. Onthe opposite, most of (all?) special relativity treatises are based on a privilegedclass of observers: the inertial ones. Although it is true that for these observersthe perception of physical phenomena is the simplest one (for instance, for aninertial observer, light in vacuum moves along a straight line and at a constantspeed), the real world is made of accelerated and rotating observers. Therefore, itseems conceptually clearer to discuss first the measures performed by a genericobserver and to treat afterwards the particular case of inertial observers. Conversely,if one restricts first to inertial observers, it becomes cumbersome to extend thediscussion to general observers. As a matter of fact, this is to a great extent thesource of the various “paradoxes” that appeared in the course of the developmentof relativity. As mentioned above, the three-dimensional approach to relativity isbased on inertial observers, since one may associate with each observer of this kinda global decomposition of spacetime in a “time” part and a “space” part.

One of the consequences of the “general observer” approach adopted here is theleast weight attributed to the famous Lorentz transformation between the frames oftwo inertial observers. This transformation, which is usually introduced in the firstchapter of a relativity course, appears here only in Chap. 6. In particular, the physicaleffects of time dilation or aberration of light are derived (geometrically) in Chaps. 2and 4, without appealing explicitly to the Lorentz transformation. Similarly, theprinciple of relativity, on which special relativity has been founded at the beginningof the twentieth century (hence its name!), is mentioned here only in Chap. 9, at theoccasion of a historical note.

The plan of the book is as follows. The full mathematical framework (Minkowskispacetime) is set in Chap. 1. The concepts of worldline and proper time are thenintroduced (Chap. 2) and are illustrated by a detailed exposition of the famous“twin paradox”. Chapter 3 is entirely devoted to the definition of an observer andhis (local) rest space. This is done in the most general way, taking into accountacceleration as well as rotation. The notion of observer being settled, we arein position to address kinematics. This is performed in two steps: (i) by fixingthe observer in Chap. 4 (introduction of the Lorentz factor, as well as relativevelocity and relative acceleration) and (ii) by discussing all the effects inducedby a change of observer in Chap. 5 (laws of velocity composition and accelerationcomposition, Doppler effect, aberration, image formation, “superluminal” motionsin astrophysics). The two chapters that follow are entirely devoted to the Lorentz

Preface xi

group, exploring its algebraic structure (Chap. 6), with the introduction of boostsand Thomas rotation, and its Lie group structure (Chap. 7). Chapter 8 focuses on theprivileged class of inertial observers, with the introduction of the Poincare group andits Lie algebra. The dynamics starts in Chap. 9, where the notion of 4-momentumis presented, as well as the principle of its conservation for any isolated system.On its side, Chap. 10 is devoted to the conservation of angular momentum and tothe concepts of centre of inertia and spin. Relativistic dynamics is subsequentlyreformulated in Chap. 11 by means of a principle of least action. The conservationlaws appear then as consequences of Noether theorem. A Hamiltonian formulationof the dynamics of relativistic particles is also presented in this chapter. Chapter 12focuses on accelerated observers, discussing kinematical aspects (Rindler horizon,clock synchronization, Thomas precession) as well as dynamical ones (spectralshift, motion of free particles). A second type of non-inertial observers is studiedin Chap. 13: the rotating ones. This chapter ends with an extensive discussion of theSagnac effect and its application to laser gyrometers in inertial guidance systems onboard airplanes.

The second part of the book opens in Chap. 14, where the physical objectunder focus is no longer a particle but a field. This part starts by three purelymathematical chapters to introduce the notions of tensor (Chap. 14), tensor field(Chap. 15) and integration over a subdomain of spacetime (Chap. 16). Amongother things, these chapters present the p-forms and exterior calculus, which arevery useful not only for electromagnetism but also for hydrodynamics. We feltnecessary to devote an entire chapter to integration in order to introduce withenough details and examples the notions of submanifold of Minkowski spacetime,area and volume element; integral of a scalar or vector field; and flux integral.The chapter ends by the famous Stokes’ theorem and its applications. Equippedwith these mathematical tools, we proceed to electromagnetism in Chap. 17. Hereagain, the emphasis is put on the four-dimensional aspect: the electromagnetic fieldtensor F is introduced first, and the electric and magnetic field vectors

#»

E and#»

B

appear in a second stage. The motion of charged particles and the various typesof particle accelerators are discussed in this chapter. Chapter 18 presents Maxwellequations, here also in a four-dimensional form, which is intrinsically simpler thanthe classical set of three-dimensional equations involving

#»

E and#»

B. The Lienard–Wiechert potentials are derived in this chapter, leading to the electromagneticfield generated by a charged particle in arbitrary motion. Chapter 19 introducesthe concept of energy–momentum tensor, a fundamental tool for the dynamics ofcontinuous media in relativity. The principles of conservation of energy–momentumand angular momentum are notably presented in a “continuous” version, as opposedto the “discrete” version considered in Chaps. 9 and 10. The energy–momentumof the electromagnetic field can then be discussed in depth in Chap. 20. In thatchapter, the energy and momentum radiated away by a moving charge are computed.A particular case is constituted by synchrotron radiation, whose applications inastrophysics and in synchrotron facilities are discussed. Chapter 21 introducesrelativistic hydrodynamics, first in a standard form and next making use of theexterior calculus presented in Chaps. 14–16. The latter approach facilitates greatly

xii Preface

the derivation of relativistic generalizations of the classical theorems of fluidmechanics. Two particularly important and contemporary applications are exploredin this chapter: relativistic jets in astrophysics and the quark-gluon plasma producedin heavy ion colliders. At last, the book ends by the problem of gravitation(Chap. 22): after some discussion about the unsuccessful attempts to incorporategravitation in special relativity, the theory of general relativity is briefly introduced.Let us point out that the study of accelerated observers performed in Chap. 12allows one, via the equivalence principle, to treat easily some relativistic effectsof gravitation, such as the gravitational redshift or the bending of light rays.

The book contains six purely mathematical chapters (Chaps. 1, 6, 7, 14, 15and 16). The aim is to introduce in a consistent and gradual way all the tools requiredfor special relativity, up to rather advanced topics. As a monograph devoted to atheory whose foundations are more than a hundred years old, the book does notcontain any truly original result. One may, however, note the general expression ofthe 4-acceleration of a particle in terms of its acceleration and velocity both relativeto a generic observer (i.e. accelerated or rotating) [Eq. (4.60)]; the compositionlaw of relative accelerations resulting from a change of observer and providing therelativistic generalization of centripetal and Coriolis accelerations [Eq. (5.56)]; thecomplete classification of restricted Lorentz transformations from a null eigenvector(Sect. 6.4); the elementary and relatively short derivation of Thomas rotation in themost general case (Sect. 6.7.2); the expressions of energy and momentum relativeto an observer, taking into account the acceleration and rotation of that observer[Eqs. (9.12) and (9.13)]; the computation of the discrepancy between the rest spaceof an observer and his simultaneity hypersurface (Sect. 12.3); the expression ofthe 4-acceleration of an observer in terms of physically measurable quantities[Eq. (12.73)]; the equation of motion of a free particle in Rindler coordinates[Eqs. (12.75) and (12.82)]; and the demonstration that the nonrelativistic limit ofthe canonical equation of fluid dynamics is the Crocco equation (Sect. 21.5.4).

One of the book’s limitations is the classical domain: no topic related to quantummechanics is treated. In particular, spinors and representations of the Poincaregroup are not discussed (see, e.g., Cartan (1966), Naber (2012), Penrose andRindler (1984), Naımark (1962)). Although these notions are not quantum bythemselves, they are mostly used in relativistic quantum theory, notably to writeDirac equation—which we do not address here.

Notes

Notations: In order to facilitate the reading, mathematical notations and symbolsintroduced in the course of the text are collected in the notation index (p. 761).Throughout the text, the abbreviation iff stands for if, and only if.

Web page: The page http://relativite.obspm.fr/sperel is devoted to the book. Itcontains the errata, the clickable list of bibliographic references, all the links

Preface xiii

listed in Appendix B, as well as various complements. The reader is invited touse this page to report any error that he/she may find in the text.

This book has been first published in French language by EDP Sciences & CNRSEditions in 2010 (Gourgoulhon 2010). The differences with respect to that versionare rather minor: they regard some improvements in the presentation and in thefigures, as well as some updates in the bibliography.

Meudon, France Eric Gourgoulhon

Acknowledgements

I have benefited enormously from exchanges with many colleagues during theredaction of this book, among them Miguel Angel Aloy, Silvano Bonazzola,Christian Bracco, Brandon Carter, Piotr Chrusciel, Bartolome Coll, Jean-LouisCornou, Thibault Damour, Olivier Darrigol, Nathalie Deruelle, Philippe Droz-Vincent, Guillaume Faye, Thierry Grandou, Jean Eisenstaedt, Gilles Esposito-Farese, Jose Marıa Ibanez, Marianne Imperor-Clerc, Jose Luis Jaramillo, ArnaudLandragin, Jean-Philippe Lenain, Gregory Malykin, Fabrice Mottez, Petar Mimica,Jean-Philippe Nicolas, Jerome Novak, Micaela Oertel, Jean-Pierre Provost, AlainRiazuelo, Matteo Luca Ruggiero, Christophe Sauty, Helene Sol, Pierre Teyssandier,Nicolas Vasset, Christiane Vilain, Loıc Villain, Frederic Vincent, Scott Walter andAndreas Zech.

I am infinitely grateful to Luc Blanchet, Thibault Damour, Olivier Darrigol,Thierry Grandou, Valerie Le Boulch, Micaela Oertel, Alain Riazuelo, PierreTeyssandier, Loıc Villain, Frederic Vincent and Scott Walter for the detailed lectureof a preliminary draft of the French version of the book. Their numerous correctionsand suggestions have been of a great value! I would also like to thank AndreValentin, who gave a precious help in tracking errors and typos in the French edition.Parts of the English version have been read by Michał Bejger, Isabel Cordero-Carrion, Fabian Laudenbach, Luciano Rezzolla, Pierre Spagnou, Francisco Uibleinand Jean-Bernard Zuber, who provided valuable remarks and corrections. I alsothank Piotr Chrusciel for his help and Ute Kraus and Daniel Weiskopf for their kindpermission to reproduce Figs. 5.12 and 5.15. I warmly thank Thibault Damour whomade me the honour of writing the foreword.

My gratitude goes also to the staff of the library of Observatoire de Paris(Meudon campus) for their kindness and efficiency. Furthermore, I have the luckto work in a laboratory with an administrative and technical staff who are both niceand competent. Thanks then to Jean-Yves Giot, Virginie Hababou, David Lepine,Stephane Mene, Nathalie Ollivier and Stephane Thomas. This book partly arisesfrom the general relativity lectures that I am giving at the Master of Astronomy andAstrophysics at Observatoire de Paris and Universities Paris 6, 7 and 11. I would liketo express here my gratitude to the students for the exchanges during the courses,

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which are an unequalled source of stimulation. I have also a thought for the authorsof the free softwares LATEX, LibreOffice Draw, Inkscape, Gnuplot, Xmgrace andSubversion. Without these extraordinary tools, the writing of the book would havebeen certainly more difficult or even impossible within a reasonable delay.

Finally, I thank most sincerely Michele Leduc and Michel Le Bellac for theirconfidence, their advices and their encouragements all along the redaction of theFrench version. I also warmly thank Ramon Khanna at Springer for having madethe English version possible.

Contents

1 Minkowski Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Four Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2.1 Spacetime as an Affine Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 A Few Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.3 Affine Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.4 Constant c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.5 Newtonian Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.1 Scalar Product on Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.2 Matrix of the Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.3 Orthonormal Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.4 Classification of Vectors with Respect to g . . . . . . . . . . . . . . . 111.3.5 Norm of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3.6 Spacetime Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Null Cone and Time Arrow .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4.2 Two Useful Lemmas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4.3 Classification of Unit Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.5 Spacetime Orientation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.6 Vector/Linear Form Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.6.1 Linear Forms and Dual Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.6.2 Metric Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.7 Minkowski Spacetime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.8 Before Going Further. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2 Worldlines and Proper Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 Worldline of a Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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2.3 Proper Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.2 Ideal Clock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4 Four-Velocity and Four-Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.4.1 Four-Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.4.2 Four-Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.5 Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.5.1 Null Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.5.2 Light Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.6 Langevin’s Traveller and Twin Paradox .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.6.1 Twins’ Worldlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.6.2 Proper Time of Each Twin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.6.3 The “Paradox” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.6.4 4-Velocity and 4-Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.6.5 A Round Trip to the Galactic Centre . . . . . . . . . . . . . . . . . . . . . . 512.6.6 Experimental Verifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.7 Geometrical Properties of a Worldline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.7.1 Timelike Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.7.2 Vector Field Along a Worldline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.7.3 Curvature and Torsions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3 Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.2 Simultaneity and Measure of Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.2.2 Einstein–Poincare Simultaneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.2.3 Local Rest Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.2.4 Nonexistence of Absolute Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.2.5 Orthogonal Projector Onto the Local Rest Space . . . . . . . . . 703.2.6 Euclidean Character of the Local Rest Space . . . . . . . . . . . . . 72

3.3 Measuring Spatial Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.3.1 Synge Formula .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.3.2 Born’s Rigidity Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.4 Local Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.4.1 Local Frame of an Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.4.2 Coordinates with Respect to an Observer . . . . . . . . . . . . . . . . . 783.4.3 Reference Space of an Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.5 Four-Rotation of a Local Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.5.1 Variation of the Local Frame Along the Worldline . . . . . . . 813.5.2 Orthogonal Decomposition of Antisymmetric

Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.5.3 Application to the Variation of the Local Frame . . . . . . . . . . 863.5.4 Inertial Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

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3.6 Derivative of a Vector Field Along a Worldline . . . . . . . . . . . . . . . . . . . . 893.6.1 Absolute Derivative.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.6.2 Derivative with Respect to an Observer . . . . . . . . . . . . . . . . . . . 903.6.3 Fermi–Walker Derivative .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.7 Locality of an Observer’s Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4 Kinematics 1: Motion with Respect to an Observer . . . . . . . . . . . . . . . . . . . . 954.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.2 Lorentz Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.2.2 Expression in Terms of the 4-Velocity

and the 4-Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.2.3 Time Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.3 Velocity Relative to an Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.3.2 4-Velocity and Lorentz Factor in Terms

of the Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.3.3 Maximum Relative Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.3.4 Component Expressions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.4 Experimental Verifications of Time Dilation. . . . . . . . . . . . . . . . . . . . . . . . 1084.4.1 Atmospheric Muons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.4.2 Other Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.5 Acceleration Relative to an Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.5.2 Relation to the Secondw Derivative

of the Position Vector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.5.3 Expression of the 4-Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.6 Photon Motion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.6.1 Propagation Direction of a Photon . . . . . . . . . . . . . . . . . . . . . . . . . 1184.6.2 Velocity of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.6.3 Experimental Tests of the Invariance

of the Velocity of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5 Kinematics 2: Change of Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.2 Relations Between Two Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.2.1 Reciprocity of the Relative Velocity . . . . . . . . . . . . . . . . . . . . . . . 1315.2.2 Length Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.3 Law of Velocity Composition.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.3.1 General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.3.2 Decomposition in Parallel and Transverse Parts . . . . . . . . . . 1395.3.3 Collinear Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1425.3.4 Alternative Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1435.3.5 Experimental Verification: Fizeau Experiment .. . . . . . . . . . . 144

5.4 Law of Acceleration Composition .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

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5.5 Doppler Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485.5.1 Derivation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485.5.2 Experimental Verifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.6 Aberration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1525.6.1 Theoretical Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1525.6.2 Distortion of the Celestial Sphere. . . . . . . . . . . . . . . . . . . . . . . . . . 1555.6.3 Experimental Verifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

5.7 Images of Moving Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1585.7.1 Image and Instantaneous Position . . . . . . . . . . . . . . . . . . . . . . . . . 1585.7.2 Apparent Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1585.7.3 Image of a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1605.7.4 Superluminal Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6 Lorentz Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1676.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1676.2 Lorentz Transformations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.2.1 Definition and Characterization .. . . . . . . . . . . . . . . . . . . . . . . . . . . 1676.2.2 Lorentz Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1696.2.3 Properties of Lorentz Transformations . . . . . . . . . . . . . . . . . . . . 170

6.3 Subgroups of O(3,1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1726.3.1 Proper Lorentz Group SO(3,1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1726.3.2 Orthochronous Lorentz Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1736.3.3 Restricted Lorentz Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1746.3.4 Reduction of the Lorentz Group to SOo.3; 1/ . . . . . . . . . . . . . 174

6.4 Classification of Restricted Lorentz Transformations . . . . . . . . . . . . . . 1766.4.1 Invariant Null Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1766.4.2 Decomposition with Respect to an Invariant

Null Direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1786.4.3 Spatial Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1816.4.4 Lorentz Boosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1836.4.5 Null Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1856.4.6 Four-Screws .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1886.4.7 Eigenvectors of a Restricted Lorentz Transformation . . . . 1896.4.8 Summary .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

6.5 Polar Decomposition .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1916.5.1 Statement and Demonstration .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1916.5.2 Explicit Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

6.6 Properties of Lorentz Boosts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1956.6.1 Kinematical Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1956.6.2 Expression in a General Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1986.6.3 Rapidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1996.6.4 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

6.7 Composition of Boosts and Thomas Rotation . . . . . . . . . . . . . . . . . . . . . . 2026.7.1 Coplanar Boosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2046.7.2 Thomas Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

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6.7.3 Thomas Rotation Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2126.7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

7 Lorentz Group as a Lie Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2177.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2177.2 Lie Group Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

7.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2177.2.2 Dimension of the Lorentz group . . . . . . . . . . . . . . . . . . . . . . . . . . . 2197.2.3 Topology of the Lorentz Group .. . . . . . . . . . . . . . . . . . . . . . . . . . . 220

7.3 Generators and Lie Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2217.3.1 Infinitesimal Lorentz Transformations . . . . . . . . . . . . . . . . . . . . 2217.3.2 Structure of Lie Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2227.3.3 Generators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2247.3.4 Link with the Variation of a Local Frame . . . . . . . . . . . . . . . . . 227

7.4 Reduction of O(3,1) to Its Lie Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2287.4.1 Exponential Map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2287.4.2 Generation of Lorentz Boosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2317.4.3 Generation of Spatial Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2337.4.4 Structure Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

7.5 Relations Between the Lorentz Group and SL(2,C) . . . . . . . . . . . . . . . . 2377.5.1 Spinor Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2377.5.2 The Spinor Map from SU(2) to SO(3) . . . . . . . . . . . . . . . . . . . . . 2437.5.3 The Spinor Map and Lorentz Boosts . . . . . . . . . . . . . . . . . . . . . . 2477.5.4 Covering of the Restricted Lorentz Group by SL(2,C) . . . 2487.5.5 Existence of Null Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2497.5.6 Lie Algebra of SL(2,C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2507.5.7 Exponential Map on sl(2,C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

8 Inertial Observers and Poincare Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2578.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2578.2 Characterization of Inertial Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

8.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2578.2.2 Worldline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2588.2.3 Globality of the Local Rest Space . . . . . . . . . . . . . . . . . . . . . . . . . 2598.2.4 Rigid Array of Inertial Observers . . . . . . . . . . . . . . . . . . . . . . . . . . 260

8.3 Poincare Group .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2618.3.1 Change of Inertial Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2618.3.2 Active Poincare Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 2638.3.3 Group Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2648.3.4 The Poincare Group as a Lie Group . . . . . . . . . . . . . . . . . . . . . . . 266

9 Energy and Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2719.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2719.2 Four-Momentum, Mass and Energy .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

9.2.1 Four-Momentum and Mass of a Particle . . . . . . . . . . . . . . . . . . 2719.2.2 Energy and Momentum Relative to an Observer . . . . . . . . . . 273

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9.2.3 Case of a Massive Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2769.2.4 Energy and Momentum of a Photon . . . . . . . . . . . . . . . . . . . . . . . 2809.2.5 Relation Between P , E and the Relative Velocity . . . . . . . . 2819.2.6 Components of the 4-Momentum .. . . . . . . . . . . . . . . . . . . . . . . . . 281

9.3 Conservation of 4-Momentum .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2829.3.1 4-Momentum of a Particle System. . . . . . . . . . . . . . . . . . . . . . . . . 2829.3.2 Isolated System and Particle Collisions . . . . . . . . . . . . . . . . . . . 2849.3.3 Principle of 4-Momentum Conservation . . . . . . . . . . . . . . . . . . 2859.3.4 Application to an Isolated Particle: Law of Inertia . . . . . . . . 2869.3.5 4-Momentum of an Isolated System . . . . . . . . . . . . . . . . . . . . . . . 2889.3.6 Energy and Linear Momentum of a System .. . . . . . . . . . . . . . 2919.3.7 Application: Doppler Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

9.4 Particle Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2949.4.1 Localized Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2949.4.2 Collision Between Two Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 2949.4.3 Elastic Collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2959.4.4 Compton Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3019.4.5 Inverse Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3049.4.6 Inelastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

9.5 Four-Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3129.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3129.5.2 Orthogonal Decomposition of the 4-Force . . . . . . . . . . . . . . . . 3139.5.3 Force Measured by an Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . 3149.5.4 Relativistic Version of Newton’s Second Law . . . . . . . . . . . . 3169.5.5 Evolution of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3179.5.6 Expression of the 4-Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

10 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31910.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31910.2 Angular Momentum of a Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

10.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31910.2.2 Angular Momentum Vector Relative to an Observer . . . . . 32010.2.3 Components of the Angular Momentum . . . . . . . . . . . . . . . . . . 322

10.3 Angular Momentum of a System .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32310.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32310.3.2 Change of Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32410.3.3 Angular Momentum Vector and Mass-Energy

Dipole Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32410.4 Conservation of Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

10.4.1 Principle of Angular Momentum Conservation .. . . . . . . . . . 32610.4.2 Angular Momentum of an Isolated System . . . . . . . . . . . . . . . 32710.4.3 Conservation of the Angular Momentum

Vector Relative to an Inertial Observer . . . . . . . . . . . . . . . . . . . . 328

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10.5 Centre of Inertia and Spin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32910.5.1 Centroid of a System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32910.5.2 Centre of Inertia of an Isolated System .. . . . . . . . . . . . . . . . . . . 33010.5.3 Spin of an Isolated System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33310.5.4 Konig Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33410.5.5 Minimal Size of a System with Spin . . . . . . . . . . . . . . . . . . . . . . 336

10.6 Angular Momentum Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33910.6.1 Four-Torque .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33910.6.2 Evolution of the Angular Momentum Vector . . . . . . . . . . . . . 340

10.7 Particle with Spin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34210.7.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34210.7.2 Spin Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34510.7.3 Free Gyroscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34610.7.4 BMT Equation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

11 Principle of Least Action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34911.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34911.2 Principle of Least Action for a Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

11.2.1 Reminder of Nonrelativistic Lagrangian Mechanics . . . . . . 34911.2.2 Relativistic Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35011.2.3 Lagrangian and Action for a Particle . . . . . . . . . . . . . . . . . . . . . . 35111.2.4 Principle of Least Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35211.2.5 Action of a Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35411.2.6 Particle in a Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35711.2.7 Other Examples of Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

11.3 Noether Theorem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36011.3.1 Noether Theorem for a Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36011.3.2 Application to a Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

11.4 Hamiltonian Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36511.4.1 Reminder of Nonrelativistic Hamiltonian Mechanics . . . . 36511.4.2 Generalized Four-Momentum of a Relativistic

Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36911.4.3 Hamiltonian of a Relativistic Particle. . . . . . . . . . . . . . . . . . . . . . 371

11.5 Systems of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37411.5.1 Principle of Least Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37511.5.2 Hamiltonian Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

12 Accelerated Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38112.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38112.2 Uniformly Accelerated Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

12.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38112.2.2 Worldline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38212.2.3 Change of the Reference Inertial Observer . . . . . . . . . . . . . . . . 38612.2.4 Motion Perceived by the Inertial Observer . . . . . . . . . . . . . . . . 38812.2.5 Local Rest Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

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12.2.6 Rindler Horizon.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39112.2.7 Local Frame of the Uniformly Accelerated Observer . . . . 393

12.3 Difference Between the Local Rest Spaceand the Simultaneity Hypersurface .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39712.3.1 Case of a Generic Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39712.3.2 Case of a Uniformly Accelerated Observer . . . . . . . . . . . . . . . 400

12.4 Physics in an Accelerated Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40012.4.1 Clock Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40012.4.2 4-Acceleration of Comoving Observers . . . . . . . . . . . . . . . . . . . 40412.4.3 Rigid Ruler in Accelerated Motion . . . . . . . . . . . . . . . . . . . . . . . . 40512.4.4 Photon Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40812.4.5 Spectral Shift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40912.4.6 Motion of Free Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

12.5 Thomas Precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41512.5.1 Derivation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41512.5.2 Application to a Gyroscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42112.5.3 Gyroscope in Circular Orbit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42212.5.4 Thomas Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

13 Rotating Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42713.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42713.2 Rotation Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

13.2.1 Physical Realization of a Nonrotating Observer . . . . . . . . . . 42713.2.2 Measurement of the Rotation Velocity . . . . . . . . . . . . . . . . . . . . 428

13.3 Rotating Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42913.3.1 Uniformly Rotating Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42913.3.2 Corotating Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43113.3.3 4-Acceleration and 4-Rotation

of the Corotating Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43313.3.4 Simultaneity for a Corotating Observer . . . . . . . . . . . . . . . . . . . 436

13.4 Clock Desynchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43913.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43913.4.2 Local Synchronization.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44013.4.3 Impossibility of a Global Synchronization . . . . . . . . . . . . . . . . 44213.4.4 Clock Transport on the Rotating Disk . . . . . . . . . . . . . . . . . . . . . 44613.4.5 Experimental Measures of the Desynchronization .. . . . . . . 450

13.5 Ehrenfest Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45313.5.1 Circumference of the Rotating Disk . . . . . . . . . . . . . . . . . . . . . . . 45313.5.2 Disk Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45313.5.3 The “Paradox” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45413.5.4 Setting the Disk into Rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

13.6 Sagnac Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45813.6.1 Sagnac Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45913.6.2 Alternative Derivation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46113.6.3 Proper Travelling Time for Each Signal . . . . . . . . . . . . . . . . . . . 463

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13.6.4 Optical Sagnac Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46413.6.5 Matter-Wave Sagnac Interferometer .. . . . . . . . . . . . . . . . . . . . . . 46813.6.6 Application: Gyrometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469

14 Tensors and Alternate Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47314.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47314.2 Tensors: Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

14.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47314.2.2 Tensors Already Met . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

14.3 Operations on Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47514.3.1 Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47514.3.2 Components in a Vector Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47614.3.3 Change of Basis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47714.3.4 Components and Metric Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 47914.3.5 Contraction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

14.4 Alternate Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48114.4.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48114.4.2 Exterior Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48314.4.3 Basis of the Space of p-Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48414.4.4 Components of the Levi–Civita Tensor .. . . . . . . . . . . . . . . . . . . 485

14.5 Hodge Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48714.5.1 Tensors Associated with the Levi–Civita Tensor. . . . . . . . . . 48714.5.2 Hodge Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49014.5.3 Hodge Star and Exterior Product . . . . . . . . . . . . . . . . . . . . . . . . . . 49214.5.4 Orthogonal Decomposition of 2-Forms . . . . . . . . . . . . . . . . . . . 493

15 Fields on Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49515.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49515.2 Arbitrary Coordinates on Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495

15.2.1 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49515.2.2 Coordinate Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49615.2.3 Components of the Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . 498

15.3 Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50215.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50215.3.2 Scalar Field and Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50315.3.3 Gradients of Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504

15.4 Covariant Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50515.4.1 Covariant Derivative of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 50515.4.2 Generalization to All Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50615.4.3 Connection Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50815.4.4 Christoffel Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51015.4.5 Divergence of a Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51215.4.6 Divergence of a Tensor Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

15.5 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51315.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51315.5.2 Exterior Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514

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15.5.3 Properties of the Exterior Derivative . . . . . . . . . . . . . . . . . . . . . . 51715.5.4 Expansion with Respect to a Coordinate System . . . . . . . . . 51815.5.5 Exterior Derivative of a 3-Form

and Divergence of a Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . 519

16 Integration in Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52116.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52116.2 Integration Over a Four-Dimensional Volume . . . . . . . . . . . . . . . . . . . . . . 521

16.2.1 Volume Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52116.2.2 Four-Volume of a Part of Spacetime . . . . . . . . . . . . . . . . . . . . . . . 52216.2.3 Integral of a Differential 4-Form .. . . . . . . . . . . . . . . . . . . . . . . . . . 523

16.3 Submanifolds of E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52416.3.1 Definition of a Submanifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52416.3.2 Submanifold with Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52616.3.3 Orientation of a Submanifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

16.4 Integration on a Submanifold of E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52716.4.1 Integral of Any Differential Form . . . . . . . . . . . . . . . . . . . . . . . . . 52716.4.2 Volume Element of a Hypersurface .. . . . . . . . . . . . . . . . . . . . . . . 53016.4.3 Area Element of a Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53216.4.4 Length-Element of a Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53416.4.5 Integral of a Scalar Field on a Submanifold .. . . . . . . . . . . . . . 53516.4.6 Integral of a Tensor Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53616.4.7 Flux Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536

16.5 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53816.5.1 Statement and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53816.5.2 Applications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540

17 Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54517.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54517.2 Electromagnetic Field Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545

17.2.1 Electromagnetic Field and Lorentz 4-Force .. . . . . . . . . . . . . . 54517.2.2 The Electromagnetic Field as a 2-Form . . . . . . . . . . . . . . . . . . . 54717.2.3 Electric and Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54717.2.4 Lorentz Force Relative to an Observer . . . . . . . . . . . . . . . . . . . . 54917.2.5 Metric Dual and Hodge Dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550

17.3 Change of Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55217.3.1 Transformation Law of the Electric

and Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55217.3.2 Electromagnetic Field Invariants. . . . . . . . . . . . . . . . . . . . . . . . . . . 55517.3.3 Reduction to Parallel Electric and Magnetic Fields . . . . . . . 55717.3.4 Field Created by a Charge in Translation.. . . . . . . . . . . . . . . . . 559

17.4 Particle in an Electromagnetic Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56217.4.1 Uniform Electromagnetic Field: Non-Null Case . . . . . . . . . . 56317.4.2 Orthogonal Electric and Magnetic Fields . . . . . . . . . . . . . . . . . 568

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17.5 Application: Particle Accelerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57617.5.1 Acceleration by an Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . 57617.5.2 Linear Accelerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57717.5.3 Cyclotrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57817.5.4 Synchrotrons .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58017.5.5 Storage Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583

18 Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58518.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58518.2 Electric Four-Current.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

18.2.1 Electric Four-Current Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58618.2.2 Electric Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58818.2.3 Charge Density and Current Density . . . . . . . . . . . . . . . . . . . . . . 59118.2.4 Four-Current of a Continuous Media . . . . . . . . . . . . . . . . . . . . . . 592

18.3 Maxwell Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59218.3.1 Statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59218.3.2 Alternative Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59318.3.3 Expression in Terms of Electric and Magnetic Fields . . . . 595

18.4 Electric Charge Conservation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59818.4.1 Derivation from Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . 59818.4.2 Expression in Terms of Charge and Current Densities . . . 60118.4.3 Gauss Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601

18.5 Solving Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60318.5.1 Four-Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60318.5.2 Electric and Magnetic Potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . 60418.5.3 Gauge Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60618.5.4 Electromagnetic Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60718.5.5 Solution for the 4-Potential in Lorenz Gauge . . . . . . . . . . . . . 608

18.6 Field Created by a Moving Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61118.6.1 Lienard–Wiechert 4-Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61118.6.2 Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61518.6.3 Electric and Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61718.6.4 Charge in Inertial Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61818.6.5 Radiative Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620

18.7 Maxwell Equations from a Principle of Least Action . . . . . . . . . . . . . . 62218.7.1 Principle of Least Action in a Classical Field Theory . . . . 62218.7.2 Case of the Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . 626

19 Energy–Momentum Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62919.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62919.2 Energy–Momentum Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629

19.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62919.2.2 Interpretation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63219.2.3 Symmetry of the Energy–Momentum Tensor . . . . . . . . . . . . . 635

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19.3 Energy–Momentum Conservation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63619.3.1 Statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63719.3.2 Local Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63719.3.3 Four-Force Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63819.3.4 Conservation of Energy and Momentum

with Respect to an Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64019.4 Angular Momentum .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641

19.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64119.4.2 Angular Momentum Conservation.. . . . . . . . . . . . . . . . . . . . . . . . 642

20 Energy–Momentum of the Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . 64520.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64520.2 Energy–Momentum Tensor of the Electromagnetic Field . . . . . . . . . . 645

20.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64520.2.2 Quantities Relative to an Observer. . . . . . . . . . . . . . . . . . . . . . . . . 648

20.3 Radiation by an Accelerated Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64920.3.1 Electromagnetic Energy–Momentum Tensor . . . . . . . . . . . . . 64920.3.2 Radiated Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65020.3.3 Radiated 4-Momentum .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65220.3.4 Angular Distribution of Radiation . . . . . . . . . . . . . . . . . . . . . . . . . 655

20.4 Synchrotron Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65920.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65920.4.2 Spectrum of Synchrotron Radiation . . . . . . . . . . . . . . . . . . . . . . . 66120.4.3 Applications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663

21 Relativistic Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66721.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66721.2 The Perfect Fluid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668

21.2.1 Energy–Momentum Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66821.2.2 Quantities Relative to an Arbitrary Observer.. . . . . . . . . . . . . 67021.2.3 Pressureless Fluid (Dust) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67121.2.4 Equation of State and Thermodynamic Relations. . . . . . . . . 67221.2.5 Simple Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674

21.3 Baryon Number Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67621.3.1 Baryon Four-Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67621.3.2 Principle of Baryon Number Conservation .. . . . . . . . . . . . . . . 67721.3.3 Expression with Respect to an Inertial Observer . . . . . . . . . . 679

21.4 Energy–Momentum Conservation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68021.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68021.4.2 Projection onto the Fluid 4-Velocity . . . . . . . . . . . . . . . . . . . . . . . 68121.4.3 Part Orthogonal to the Fluid 4-Velocity . . . . . . . . . . . . . . . . . . . 68221.4.4 Evolution of the Fluid Energy Relative

to Some Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683

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21.4.5 Relativistic Euler Equation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68421.4.6 Speed of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68521.4.7 Relativistic Hydrodynamics as a System

of Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68621.5 Formulation Based on Exterior Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687

21.5.1 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68821.5.2 Vorticity of a Simple Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68921.5.3 Canonical Form of the Equation of Motion . . . . . . . . . . . . . . . 69021.5.4 Nonrelativistic Limit: Crocco Equation . . . . . . . . . . . . . . . . . . . 692

21.6 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69421.6.1 Bernoulli’s Theorem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69421.6.2 Irrotational Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69621.6.3 Kelvin’s Circulation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698

21.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70121.7.1 Astrophysics: Jets and Gamma-Ray Bursts . . . . . . . . . . . . . . . 70121.7.2 Quark-Gluon Plasma at RHIC and at LHC. . . . . . . . . . . . . . . . 703

21.8 To Go Further. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709

22 What About Relativistic Gravitation? .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71122.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71122.2 Gravitation in Minkowski Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711

22.2.1 Nordstrom’s Scalar Theory.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71222.2.2 Incompatibility with Observations .. . . . . . . . . . . . . . . . . . . . . . . . 71922.2.3 Vector Theory .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72022.2.4 Tensor Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722

22.3 Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72322.3.1 The Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72322.3.2 Gravitational Redshift and Incompatibility

with the Minkowski Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72422.3.3 Experimental Verifications

of the Gravitational Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72622.3.4 Light Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729

22.4 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729

A Basic Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733A.1 Basic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733

A.1.1 Group.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733A.1.2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734

A.2 Linear Algebra.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735A.2.1 Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735A.2.2 Algebra .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736

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B Web Pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737

C Special Relativity Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765