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Professor Richard Needs Cavendish Laboratory JJ Thomson Avenue Cambridge CB3 0HE UK [email protected] Eric´ Gourgoulhon

Special Relativity in General Frames

From Particles to Astrophysics

123 Eric´ Gourgoulhon Laboratoire Univers et Theories´ Observatoire de Paris, CNRS, UniversiteParisDiderot´ Meudon, France

Translation from the French language edition of: Relativite´ restreinte: Des particules a` l’astrophysique, c 2010 EDP Sciences, CNRS Edition, France.

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

Library of Congress Control Number: 2013942463

c Springer-Verlag Berlin Heidelberg 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’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. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.

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Foreword

The theory of special relativity holds a distinctive place within physics. Rather than being a specific physical theory, it is (similar to thermodynamics or analytical mechanics) a general theoretical framework within which various dynamical theo- ries can be formulated. In this respect, a modern presentation of special relativity must put forward its essential structures before illustrating them by concrete applications to specific dynamical problems. Such is the challenge (so successfully met!) of the beautiful book by Eric´ Gourgoulhon. Contrary to most textbooks on special relativity, which mix the presentation of this theory with that of its historical development and which sometimes write the specific form of “Lorentz transformations” before indicating that they leave a certain quadratic form invariant, the book by Eric´ Gourgoulhon is centred, from the very beginning, on the essential structure of the theory, i.e. the chrono-geometric structure of the four-dimensional Poincare–Minkowski´ spacetime. The aim is to train the reader to formulate any relativity question in terms of four-dimensional geometry. The word geometry has here the meaning of “synthetic geometry” (a` la Euclid) in contrast with “analytic geometry” (a` la Descartes). Under the expert guidance of Eric´ Gourgoulhon, the reader will learn to set, and to solve, any problem of relativity by drawing spacetime diagrams, made of curves, straight lines, planes, hyperplanes, cones and vectors. He will get accustomed to visualizing the motion of a particle as a line in spacetime, to think about the twin paradox as an application of the “spacetime triangle inequality”, to express the local frame of an observer as a four-dimensional generalization of the Serret–Frenet triad, to compute a spatial distance as a geometric mean of time intervals (via the hyperbolic generalization of the power of a point with respect to a circle) or to understand the Sagnac effect by considering 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. First of all, it is fully up to date and very complete in its coverage of the notions and results where special relativity plays an important role: from Thomas precession to the foundations of general relativity, including tensor calculus, exterior differential calculus, classical electrodynamics, the general notion of energy–momentum tensor

vii viii Foreword and a noteworthy chapter on relativistic hydrodynamics. In addition, this book is sprinkled with enlightening historical notes, in which the author summarizes in a condensed, albeit very informative way the (sometimes very recent) results by historians of science. Finally, the book is richly laden with many examples of applications of special relativity to concrete physical problems. The reader will learn 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-technology experiments: from laser gyrometers to the LHC, including modern replications of the Michelson–Morley experiment, matter wave interferometers, synchrotrons and their radiation, and the comparison of atomic clocks embarked on planes, satellites or the International Space Station. I am sure that the remarkably rich book by Eric´ Gourgoulhon will attract the keen interest of many readers and will enable them to understand and master one of the fundamental 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 very beginning. The mathematical framework is indeed, from the first chapter, that of Minkowski spacetime, and the basic objects are the vectors in this space (often called 4-vectors). Physical laws are translated in terms of geometrical operations (scalar product, orthogonal projection, etc.) on objects of Minkowski spacetime (4-vectors, worldlines, etc.). Many relativity textbooks start rather by a three-dimensional approach, using space + time decompositions based on inertial observers. Only in the second stage they introduce 4-vectors and Minkowski spacetime. In this respect, they are faithful to the historical development of relativity. A more axiomatic approach is adopted here, setting from the very beginning the full mathematical framework as one of the postulates of the theory. From this point of view, the chosen approach is similar to that adopted in classical mechanics or quantum mechanics, where usually the exposition does not follow the history of the theory. The history of relativity is undoubtedly rich and fascinating, but the objective of this book is the learning of special relativity within a consistent and operational setting, from the bases up to advanced topics. The text is, however, enriched with historical notes, which include references to the original works and to the studies by historians of science. Usually, the geometric approach is reserved for general relativity,i.e.forthe incorporation of the gravitational field in relativity theory.1 We employ it here for special relativity, taking into account a geometric structure much simpler than that of general relativity: while the latter is based on the concept of differentiable manifold, special relativity relies entirely on the concept of affine space, which can be identified with the space R4. Consequently, the mathematical prerequisites are relatively limited; they are mostly linear algebra at the level of the first two years of 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) .

ix x Preface 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 scalar product of two vectors is not the standard scalar product in Euclidean space but is given by a privileged symmetric bilinear form, the so-called metric tensor.Once this is accepted, physical results are obtained faster than by means of the “classic” three-dimensional formulation, and a more profound understanding of relativity is acquired. Moreover, learning general relativity is made much easier, starting from such an approach. In connection with the four-dimensional approach, another characteristic of this monograph is to lay the discussion of physically measurable effects on the most general type of observer, i.e. allowing for accelerated and rotating frames. On the opposite, most of (all?) special relativity treatises are based on a privileged class of observers: the inertial ones. Although it is true that for these observers the perception of physical phenomena is the simplest one (for instance, for an inertial observer, light in vacuum moves along a straight line and at a constant speed), the real world is made of accelerated and rotating observers. Therefore, it seems conceptually clearer to discuss first the measures performed by a generic observer and to treat afterwards the particular case of inertial observers. Conversely, if one restricts first to inertial observers, it becomes cumbersome to extend the discussion to general observers. As a matter of fact, this is to a great extent the source of the various “paradoxes” that appeared in the course of the development of relativity. As mentioned above, the three-dimensional approach to relativity is based on inertial observers, since one may associate with each observer of this kind a global decomposition of spacetime in a “time” part and a “space” part. One of the consequences of the “general observer” approach adopted here is the least weight attributed to the famous Lorentz transformation between the frames of two inertial observers. This transformation, which is usually introduced in the first chapter of a relativity course, appears here only in Chap. 6. In particular, the physical effects of time dilation or aberration of light are derived (geometrically) in Chaps. 2 and 4, without appealing explicitly to the Lorentz transformation. Similarly, the principle of relativity, on which special relativity has been founded at the beginning of the twentieth century (hence its name!), is mentioned here only in Chap. 9, at the occasion of a historical note. The plan of the book is as follows. The full mathematical framework (Minkowski spacetime) is set in Chap. 1. The concepts of worldline and proper time are then introduced (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 and his (local) rest space. This is done in the most general way, taking into account acceleration as well as rotation. The notion of observer being settled, we are in position to address kinematics. This is performed in two steps: (i) by fixing the observer in Chap. 4 (introduction of the Lorentz factor, as well as relative velocity and relative acceleration) and (ii) by discussing all the effects induced by a change of observer in Chap. 5 (laws of velocity composition and acceleration composition, Doppler effect, aberration, image formation, “superluminal” motions in 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 boosts and Thomas rotation, and its Lie group structure (Chap. 7). Chapter 8 focuses on the privileged class of inertial observers, with the introduction of the Poincare´ group and its Lie algebra. The dynamics starts in Chap. 9, where the notion of 4-momentum is 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 to the concepts of centre of inertia and spin. Relativistic dynamics is subsequently reformulated in Chap. 11 by means of a principle of least action. The conservation laws appear then as consequences of Noether theorem. A Hamiltonian formulation of the dynamics of relativistic particles is also presented in this chapter. Chapter 12 focuses on accelerated observers, discussing kinematical aspects (Rindler horizon, clock synchronization, Thomas precession) as well as dynamical ones (spectral shift, motion of free particles). A second type of non-inertial observers is studied in Chap. 13: the rotating ones. This chapter ends with an extensive discussion of the Sagnac effect and its application to laser gyrometers in inertial guidance systems on board airplanes. The second part of the book opens in Chap. 14, where the physical object under focus is no longer a particle but a field. This part starts by three purely mathematical chapters to introduce the notions of tensor (Chap. 14), tensor field (Chap. 15) and integration over a subdomain of spacetime (Chap. 16). Among other things, these chapters present the p-forms and exterior calculus, which are very useful not only for electromagnetism but also for hydrodynamics. We felt necessary to devote an entire chapter to integration in order to introduce with enough 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. Equipped with these mathematical tools, we proceed to electromagnetism in Chap. 17. Here again, the emphasis is put on the four-dimensional aspect: the electromagnetic#» field#» tensor 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 types of particle accelerators are discussed in this chapter. Chapter 18 presents Maxwell equations, here also in a four-dimensional form, which is intrinsically#» #» simpler than the classical set of three-dimensional equations involving E and B.TheLienard–´ Wiechert potentials are derived in this chapter, leading to the electromagnetic field generated by a charged particle in arbitrary motion. Chapter 19 introduces the concept of energy–momentum tensor, a fundamental tool for the dynamics of continuous media in relativity. The principles of conservation of energy–momentum and angular momentum are notably presented in a “continuous” version, as opposed to the “discrete” version considered in Chaps. 9 and 10. The energy–momentum of the electromagnetic field can then be discussed in depth in Chap. 20. In that chapter, the energy and momentum radiated away by a moving charge are computed. A particular case is constituted by synchrotron radiation, whose applications in astrophysics and in synchrotron facilities are discussed. Chapter 21 introduces relativistic hydrodynamics, first in a standard form and next making use of the exterior calculus presented in Chaps. 14–16. The latter approach facilitates greatly xii Preface the derivation of relativistic generalizations of the classical theorems of fluid mechanics. Two particularly important and contemporary applications are explored in this chapter: relativistic jets in astrophysics and the quark-gluon plasma produced in heavy ion colliders. At last, the book ends by the problem of gravitation (Chap. 22): after some discussion about the unsuccessful attempts to incorporate gravitation in special relativity, the theory of general relativity is briefly introduced. Let us point out that the study of accelerated observers performed in Chap. 12 allows one, via the equivalence principle, to treat easily some relativistic effects of gravitation, such as the gravitational redshift or the bending of light rays. The book contains six purely mathematical chapters (Chaps. 1, 6, 7, 14, 15 and 16). The aim is to introduce in a consistent and gradual way all the tools required for special relativity, up to rather advanced topics. As a monograph devoted to a theory whose foundations are more than a hundred years old, the book does not contain any truly original result. One may, however, note the general expression of the 4-acceleration of a particle in terms of its acceleration and velocity both relative to a generic observer (i.e. accelerated or rotating) [Eq. (4.60)]; the composition law of relative accelerations resulting from a change of observer and providing the relativistic generalization of centripetal and Coriolis accelerations [Eq. (5.56)]; the complete classification of restricted Lorentz transformations from a null eigenvector (Sect. 6.4); the elementary and relatively short derivation of Thomas rotation in the most general case (Sect. 6.7.2); the expressions of energy and momentum relative to 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 space of an observer and his simultaneity hypersurface (Sect. 12.3); the expression of the 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 of the 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 quantum mechanics is treated. In particular, spinors and representations of the Poincare´ group are not discussed (see, e.g., Cartan (1966), Naber (2012), Penrose and Rindler (1984), Na¨ımark (1962)). Although these notions are not quantum by themselves, they are mostly used in relativistic quantum theory, notably to write Dirac equation—which we do not address here.

Notes

Notations: In order to facilitate the reading, mathematical notations and symbols introduced 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. It contains 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 to use 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 & CNRS Editions in 2010 (Gourgoulhon 2010). The differences with respect to that version are rather minor: they regard some improvements in the presentation and in the figures, as well as some updates in the bibliography.

Meudon, France Eric´ Gourgoulhon

Acknowledgements

I have benefited enormously from exchanges with many colleagues during the redaction of this book, among them Miguel Angel Aloy, Silvano Bonazzola, Christian Bracco, Brandon Carter, Piotr Chrusciel, Bartolome´ Coll, Jean-Louis Cornou, Thibault Damour, Olivier Darrigol, Nathalie Deruelle, Philippe Droz- Vincent, Guillaume Faye, Thierry Grandou, Jean Eisenstaedt, Gilles Esposito- Farese,` JoseMar´ ´ıa Ibanez,˜ Marianne Imperor-Clerc,´ Jose´ Luis Jaramillo, Arnaud Landragin, Jean-Philippe Lenain, Gregory Malykin, Fabrice Mottez, Petar Mimica, Jean-Philippe Nicolas, Jer´ omeˆ Novak, Micaela Oertel, Jean-Pierre Provost, Alain Riazuelo, Matteo Luca Ruggiero, Christophe Sauty, Hel´ ene` Sol, Pierre Teyssandier, Nicolas Vasset, Christiane Vilain, Lo¨ıc Villain, Fred´ eric´ Vincent, Scott Walter and Andreas Zech. I am infinitely grateful to Luc Blanchet, Thibault Damour, Olivier Darrigol, Thierry Grandou, Valerie´ Le Boulch, Micaela Oertel, Alain Riazuelo, Pierre Teyssandier, Lo¨ıc Villain, Fred´ eric´ Vincent and Scott Walter for the detailed lecture of a preliminary draft of the French version of the book. Their numerous corrections and suggestions have been of a great value! I would also like to thank Andre´ Valentin, 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 Uiblein and Jean-Bernard Zuber, who provided valuable remarks and corrections. I also thank Piotr Chrusciel for his help and Ute Kraus and Daniel Weiskopf for their kind permission to reproduce Figs. 5.12 and 5.15. I warmly thank Thibault Damour who made 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 luck to work in a laboratory with an administrative and technical staff who are both nice and competent. Thanks then to Jean-Yves Giot, Virginie Hababou, David Lepine,´ Stephane´ Men´ e,´ Nathalie Ollivier and Stephane´ Thomas. This book partly arises from the general relativity lectures that I am giving at the Master of Astronomy and Astrophysics at Observatoire de Paris and Universities Paris 6, 7 and 11. I would like to express here my gratitude to the students for the exchanges during the courses,

xv xvi Acknowledgements which are an unequalled source of stimulation. I have also a thought for the authors of the free softwares LATEX, LibreOffice Draw, Inkscape, Gnuplot, Xmgrace and Subversion. Without these extraordinary tools, the writing of the book would have been certainly more difficult or even impossible within a reasonable delay. Finally, I thank most sincerely Michele` Leduc and Michel Le Bellac for their confidence, their advices and their encouragements all along the redaction of the French version. I also warmly thank Ramon Khanna at Springer for having made the English version possible. Contents

1 Minkowski Spacetime ...... 1 1.1 Introduction...... 1 1.2 TheFourDimensions ...... 1 1.2.1 SpacetimeasanAffineSpace...... 1 1.2.2 A FewNotations...... 3 1.2.3 AffineCoordinateSystem...... 4 1.2.4 Constant c ...... 4 1.2.5 NewtonianSpacetime ...... 5 1.3 MetricTensor...... 6 1.3.1 Scalar Product on Spacetime ...... 6 1.3.2 MatrixoftheMetricTensor...... 9 1.3.3 Orthonormal Bases ...... 10 1.3.4 Classification of Vectors with Respect to g ...... 11 1.3.5 Normofa Vector ...... 11 1.3.6 SpacetimeDiagrams...... 12 1.4 NullConeandTimeArrow...... 15 1.4.1 Definitions ...... 15 1.4.2 TwoUsefulLemmas...... 16 1.4.3 ClassificationofUnitVectors...... 17 1.5 SpacetimeOrientation...... 20 1.6 Vector/Linear Form Duality ...... 22 1.6.1 LinearFormsandDualSpace...... 22 1.6.2 Metric Duality ...... 23 1.7 MinkowskiSpacetime...... 25 1.8 BeforeGoingFurther...... 27 2 Worldlines and Proper Time ...... 29 2.1 Introduction...... 29 2.2 Worldlineofa Particle ...... 29

xvii xviii Contents

2.3 ProperTime...... 31 2.3.1 Definition ...... 31 2.3.2 IdealClock...... 33 2.4 Four-Velocity and Four-Acceleration ...... 35 2.4.1 Four-Velocity ...... 35 2.4.2 Four-Acceleration ...... 37 2.5 Photons ...... 39 2.5.1 NullGeodesics...... 39 2.5.2 LightCone...... 39 2.6 Langevin’sTravellerandTwinParadox...... 40 2.6.1 Twins’Worldlines ...... 41 2.6.2 ProperTimeofEachTwin ...... 43 2.6.3 The “Paradox” ...... 44 2.6.4 4-Velocity and 4-Acceleration ...... 47 2.6.5 A Round Trip to the Galactic Centre ...... 51 2.6.6 ExperimentalVerifications...... 54 2.7 GeometricalPropertiesofa Worldline...... 57 2.7.1 TimelikeGeodesics...... 57 2.7.2 VectorFieldAlonga Worldline...... 59 2.7.3 CurvatureandTorsions...... 59 3 Observers ...... 63 3.1 Introduction...... 63 3.2 SimultaneityandMeasureofTime...... 63 3.2.1 TheProblem ...... 63 3.2.2 Einstein–PoincareSimultaneity...... ´ 64 3.2.3 LocalRestSpace ...... 66 3.2.4 NonexistenceofAbsoluteTime ...... 69 3.2.5 Orthogonal Projector Onto the Local Rest Space ...... 70 3.2.6 EuclideanCharacteroftheLocalRestSpace...... 72 3.3 MeasuringSpatialDistances...... 73 3.3.1 Synge Formula...... 73 3.3.2 Born’sRigidityCriterion...... 75 3.4 LocalFrame ...... 76 3.4.1 LocalFrameofanObserver ...... 76 3.4.2 CoordinateswithRespecttoanObserver ...... 78 3.4.3 ReferenceSpaceofanObserver...... 79 3.5 Four-Rotationofa LocalFrame...... 81 3.5.1 VariationoftheLocalFrameAlongtheWorldline ...... 81 3.5.2 Orthogonal Decomposition of Antisymmetric Bilinear Forms ...... 83 3.5.3 ApplicationtotheVariationoftheLocalFrame...... 86 3.5.4 InertialObservers...... 88 Contents xix

3.6 Derivativeofa VectorFieldAlonga Worldline ...... 89 3.6.1 AbsoluteDerivative...... 89 3.6.2 DerivativewithRespecttoanObserver...... 90 3.6.3 Fermi–WalkerDerivative...... 91 3.7 Locality of an Observer’s Frame ...... 92 4 Kinematics 1: Motion with Respect to an Observer ...... 95 4.1 Introduction...... 95 4.2 LorentzFactor...... 95 4.2.1 Definition ...... 95 4.2.2 Expression in Terms of the 4-Velocity and the 4-Acceleration ...... 98 4.2.3 TimeDilation ...... 100 4.3 VelocityRelativetoanObserver ...... 101 4.3.1 Definition ...... 101 4.3.2 4-Velocity and Lorentz Factor in Terms oftheVelocity ...... 103 4.3.3 MaximumRelativeVelocity ...... 106 4.3.4 Component Expressions...... 107 4.4 ExperimentalVerificationsofTimeDilation...... 108 4.4.1 Atmospheric Muons ...... 108 4.4.2 OtherTests...... 110 4.5 Acceleration Relative to an Observer ...... 111 4.5.1 Definition ...... 111 4.5.2 Relation to the Secondw Derivative of the Position Vector...... 111 4.5.3 Expression of the 4-Acceleration ...... 114 4.6 PhotonMotion...... 118 4.6.1 PropagationDirectionofa Photon...... 118 4.6.2 VelocityofLight...... 120 4.6.3 Experimental Tests of the Invariance oftheVelocityofLight...... 123 5 Kinematics 2: Change of Observer ...... 131 5.1 Introduction...... 131 5.2 RelationsBetweenTwoObservers...... 131 5.2.1 ReciprocityoftheRelativeVelocity...... 131 5.2.2 LengthContraction...... 134 5.3 Law of Velocity Composition...... 136 5.3.1 GeneralForm ...... 136 5.3.2 Decomposition in Parallel and Transverse Parts ...... 139 5.3.3 Collinear Velocities ...... 142 5.3.4 AlternativeFormula ...... 143 5.3.5 Experimental Verification: Fizeau Experiment...... 144 5.4 Law of Acceleration Composition...... 146 xx Contents

5.5 Doppler Effect ...... 148 5.5.1 Derivation...... 148 5.5.2 ExperimentalVerifications...... 151 5.6 Aberration ...... 152 5.6.1 TheoreticalExpression...... 152 5.6.2 DistortionoftheCelestialSphere...... 155 5.6.3 ExperimentalVerifications...... 157 5.7 ImagesofMovingObjects...... 158 5.7.1 Image and Instantaneous Position ...... 158 5.7.2 ApparentRotation ...... 158 5.7.3 Imageofa Sphere ...... 160 5.7.4 SuperluminalMotions...... 163 6 Lorentz Group ...... 167 6.1 Introduction...... 167 6.2 LorentzTransformations...... 167 6.2.1 Definition and Characterization ...... 167 6.2.2 LorentzGroup ...... 169 6.2.3 PropertiesofLorentzTransformations...... 170 6.3 Subgroups of O(3,1)...... 172 6.3.1 ProperLorentzGroupSO(3,1)...... 172 6.3.2 Orthochronous Lorentz Group ...... 173 6.3.3 RestrictedLorentzGroup ...... 174 6.3.4 Reduction of the Lorentz Group to SOo.3; 1/ ...... 174 6.4 ClassificationofRestrictedLorentzTransformations...... 176 6.4.1 InvariantNullDirection...... 176 6.4.2 Decomposition with Respect to an Invariant NullDirection...... 178 6.4.3 SpatialRotations...... 181 6.4.4 LorentzBoosts...... 183 6.4.5 NullRotations ...... 185 6.4.6 Four-Screws...... 188 6.4.7 Eigenvectorsofa RestrictedLorentzTransformation .... 189 6.4.8 Summary...... 190 6.5 Polar Decomposition ...... 191 6.5.1 StatementandDemonstration...... 191 6.5.2 ExplicitForms ...... 194 6.6 PropertiesofLorentzBoosts...... 195 6.6.1 KinematicalInterpretation ...... 195 6.6.2 Expressionina GeneralBasis ...... 198 6.6.3 Rapidity...... 199 6.6.4 Eigenvalues ...... 202 6.7 Composition of Boosts and Thomas Rotation ...... 202 6.7.1 CoplanarBoosts ...... 204 6.7.2 ThomasRotation ...... 206 Contents xxi

6.7.3 ThomasRotationAngle...... 212 6.7.4 Conclusion...... 216 7 Lorentz Group as a Lie Group ...... 217 7.1 Introduction...... 217 7.2 LieGroupStructure...... 217 7.2.1 Definitions ...... 217 7.2.2 DimensionoftheLorentzgroup...... 219 7.2.3 Topology of the Lorentz Group ...... 220 7.3 GeneratorsandLieAlgebra...... 221 7.3.1 InfinitesimalLorentzTransformations ...... 221 7.3.2 StructureofLieAlgebra...... 222 7.3.3 Generators...... 224 7.3.4 LinkwiththeVariationofa LocalFrame ...... 227 7.4 ReductionofO(3,1)toItsLieAlgebra...... 228 7.4.1 Exponential Map...... 228 7.4.2 GenerationofLorentzBoosts...... 231 7.4.3 GenerationofSpatialRotations...... 233 7.4.4 StructureConstants...... 234 7.5 RelationsBetweentheLorentzGroupandSL(2,C)...... 237 7.5.1 SpinorMap ...... 237 7.5.2 TheSpinorMapfromSU(2)toSO(3)...... 243 7.5.3 TheSpinorMapandLorentzBoosts ...... 247 7.5.4 Covering of the Restricted Lorentz Group by SL(2,C) ... 248 7.5.5 ExistenceofNullEigenvectors ...... 249 7.5.6 LieAlgebraofSL(2,C)...... 250 7.5.7 Exponential Map on sl(2,C) ...... 254 8 Inertial Observers and PoincareGroup´ ...... 257 8.1 Introduction...... 257 8.2 CharacterizationofInertialObservers ...... 257 8.2.1 Definition ...... 257 8.2.2 Worldline...... 258 8.2.3 Globality of the Local Rest Space ...... 259 8.2.4 RigidArrayofInertialObservers...... 260 8.3 PoincareGroup...... ´ 261 8.3.1 ChangeofInertialCoordinates ...... 261 8.3.2 Active PoincareTransformations...... ´ 263 8.3.3 GroupStructure...... 264 8.3.4 The PoincareGroupasa´ LieGroup...... 266 9 Energy and Momentum...... 271 9.1 Introduction...... 271 9.2 Four-Momentum,MassandEnergy...... 271 9.2.1 Four-MomentumandMassofa Particle ...... 271 9.2.2 EnergyandMomentumRelativetoanObserver...... 273 xxii Contents

9.2.3 Caseofa MassiveParticle ...... 276 9.2.4 EnergyandMomentumofa Photon...... 280 9.2.5 Relation Between P , E andtheRelativeVelocity...... 281 9.2.6 Components of the 4-Momentum...... 281 9.3 Conservationof4-Momentum...... 282 9.3.1 4-Momentumofa ParticleSystem...... 282 9.3.2 Isolated System and Particle Collisions ...... 284 9.3.3 Principleof4-MomentumConservation ...... 285 9.3.4 ApplicationtoanIsolatedParticle:LawofInertia...... 286 9.3.5 4-MomentumofanIsolatedSystem...... 288 9.3.6 EnergyandLinearMomentumofa System...... 291 9.3.7 Application: Doppler Effect...... 293 9.4 Particle Collisions ...... 294 9.4.1 LocalizedInteractions...... 294 9.4.2 Collision Between Two Particles ...... 294 9.4.3 Elastic Collision ...... 295 9.4.4 ComptonEffect...... 301 9.4.5 InverseComptonScattering...... 304 9.4.6 Inelastic Collisions ...... 307 9.5 Four-Force...... 312 9.5.1 Definition ...... 312 9.5.2 Orthogonal Decomposition of the 4-Force ...... 313 9.5.3 ForceMeasuredbyanObserver ...... 314 9.5.4 RelativisticVersionofNewton’sSecondLaw ...... 316 9.5.5 EvolutionofEnergy...... 317 9.5.6 Expressionofthe4-Force...... 318 10 Angular Momentum...... 319 10.1 Introduction...... 319 10.2 Angular Momentum of a Particle...... 319 10.2.1 Definition ...... 319 10.2.2 Angular Momentum Vector Relative to an Observer ..... 320 10.2.3 Components of the Angular Momentum ...... 322 10.3 Angular Momentum of a System ...... 323 10.3.1 Definition ...... 323 10.3.2 ChangeofOrigin ...... 324 10.3.3 Angular Momentum Vector and Mass-Energy DipoleMoment...... 324 10.4 Conservation of Angular Momentum ...... 326 10.4.1 Principle of Angular Momentum Conservation ...... 326 10.4.2 Angular Momentum of an Isolated System ...... 327 10.4.3 Conservation of the Angular Momentum VectorRelativetoanInertialObserver...... 328 Contents xxiii

10.5 CentreofInertiaandSpin...... 329 10.5.1 Centroidofa System ...... 329 10.5.2 CentreofInertiaofanIsolatedSystem...... 330 10.5.3 SpinofanIsolatedSystem...... 333 10.5.4 KonigTheorem...... ¨ 334 10.5.5 MinimalSizeofa SystemwithSpin ...... 336 10.6 Angular Momentum Evolution ...... 339 10.6.1 Four-Torque...... 339 10.6.2 Evolution of the Angular Momentum Vector ...... 340 10.7 ParticlewithSpin...... 342 10.7.1 Definition ...... 342 10.7.2 SpinEvolution...... 345 10.7.3 FreeGyroscope...... 346 10.7.4 BMTEquation...... 347 11 Principle of Least Action...... 349 11.1 Introduction...... 349 11.2 PrincipleofLeastActionfora Particle ...... 349 11.2.1 ReminderofNonrelativisticLagrangianMechanics...... 349 11.2.2 RelativisticGeneralization...... 350 11.2.3 LagrangianandActionfora Particle ...... 351 11.2.4 PrincipleofLeastAction...... 352 11.2.5 Actionofa FreeParticle ...... 354 11.2.6 Particleina VectorField ...... 357 11.2.7 OtherExamplesofLagrangians ...... 358 11.3 NoetherTheorem...... 360 11.3.1 NoetherTheoremfora Particle...... 360 11.3.2 Applicationtoa FreeParticle...... 362 11.4 HamiltonianFormulation...... 365 11.4.1 ReminderofNonrelativisticHamiltonianMechanics .... 365 11.4.2 Generalized Four-Momentum of a Relativistic Particle ...... 369 11.4.3 Hamiltonianofa RelativisticParticle...... 371 11.5 SystemsofParticles...... 374 11.5.1 PrincipleofLeastAction...... 375 11.5.2 HamiltonianFormulation ...... 378 12 Accelerated Observers ...... 381 12.1 Introduction...... 381 12.2 Uniformly Accelerated Observer ...... 381 12.2.1 Definition ...... 381 12.2.2 Worldline...... 382 12.2.3 ChangeoftheReferenceInertialObserver...... 386 12.2.4 MotionPerceivedbytheInertialObserver...... 388 12.2.5 Local Rest Spaces ...... 389 xxiv Contents

12.2.6 RindlerHorizon...... 391 12.2.7 Local Frame of the Uniformly Accelerated Observer .... 393 12.3 Difference Between the Local Rest Space andtheSimultaneityHypersurface...... 397 12.3.1 Caseofa GenericObserver...... 397 12.3.2 Case of a Uniformly Accelerated Observer ...... 400 12.4 Physics in an Accelerated Frame ...... 400 12.4.1 ClockSynchronization...... 400 12.4.2 4-Acceleration of Comoving Observers ...... 404 12.4.3 Rigid Ruler in Accelerated Motion ...... 405 12.4.4 PhotonTrajectories...... 408 12.4.5 SpectralShift...... 409 12.4.6 MotionofFreeParticles...... 412 12.5 ThomasPrecession...... 415 12.5.1 Derivation...... 415 12.5.2 Applicationtoa Gyroscope...... 421 12.5.3 GyroscopeinCircularOrbit...... 422 12.5.4 ThomasEquation...... 423 13 Rotating Observers ...... 427 13.1 Introduction...... 427 13.2 RotationVelocity...... 427 13.2.1 PhysicalRealizationofa NonrotatingObserver...... 427 13.2.2 MeasurementoftheRotationVelocity ...... 428 13.3 RotatingDisk...... 429 13.3.1 UniformlyRotatingObserver...... 429 13.3.2 CorotatingObservers...... 431 13.3.3 4-Acceleration and 4-Rotation oftheCorotatingObserver...... 433 13.3.4 Simultaneityfora CorotatingObserver ...... 436 13.4 ClockDesynchronization...... 439 13.4.1 Introduction ...... 439 13.4.2 LocalSynchronization...... 440 13.4.3 Impossibility of a Global Synchronization ...... 442 13.4.4 ClockTransportontheRotatingDisk...... 446 13.4.5 ExperimentalMeasuresoftheDesynchronization...... 450 13.5 EhrenfestParadox...... 453 13.5.1 CircumferenceoftheRotatingDisk...... 453 13.5.2 DiskRadius...... 453 13.5.3 The “Paradox” ...... 454 13.5.4 Setting the Disk into Rotation...... 455 13.6 SagnacEffect...... 458 13.6.1 SagnacDelay ...... 459 13.6.2 AlternativeDerivation...... 461 13.6.3 Proper Travelling Time for Each Signal ...... 463 Contents xxv

13.6.4 OpticalSagnacInterferometer...... 464 13.6.5 Matter-WaveSagnacInterferometer...... 468 13.6.6 Application:Gyrometers...... 469 14 Tensors and Alternate Forms ...... 473 14.1 Introduction...... 473 14.2 Tensors: Definition and Examples ...... 473 14.2.1 Definition ...... 473 14.2.2 TensorsAlreadyMet ...... 474 14.3 OperationsonTensors ...... 475 14.3.1 Tensor Product ...... 475 14.3.2 Components in a Vector Basis ...... 476 14.3.3 ChangeofBasis...... 477 14.3.4 Components and Metric Duality ...... 479 14.3.5 Contraction...... 480 14.4 AlternateForms ...... 481 14.4.1 Definition and Examples ...... 481 14.4.2 Exterior Product ...... 483 14.4.3 Basis of the Space of p-Forms...... 484 14.4.4 Components of the Levi–Civita Tensor...... 485 14.5 Hodge Duality ...... 487 14.5.1 TensorsAssociatedwiththeLevi–CivitaTensor...... 487 14.5.2 Hodge Star ...... 490 14.5.3 Hodge Star and Exterior Product ...... 492 14.5.4 Orthogonal Decomposition of 2-Forms ...... 493 15 Fields on Spacetime ...... 495 15.1 Introduction...... 495 15.2 ArbitraryCoordinatesonSpacetime ...... 495 15.2.1 CoordinateSystem ...... 495 15.2.2 CoordinateBasis...... 496 15.2.3 Components of the Metric Tensor ...... 498 15.3 TensorFields ...... 502 15.3.1 Definitions ...... 502 15.3.2 ScalarFieldandGradient ...... 503 15.3.3 GradientsofCoordinates...... 504 15.4 CovariantDerivative ...... 505 15.4.1 CovariantDerivativeofa Vector...... 505 15.4.2 GeneralizationtoAllTensors...... 506 15.4.3 ConnectionCoefficients...... 508 15.4.4 ChristoffelSymbols ...... 510 15.4.5 Divergenceofa VectorField...... 512 15.4.6 Divergenceofa TensorField...... 513 15.5 DifferentialForms...... 513 15.5.1 Definition ...... 513 15.5.2 ExteriorDerivative ...... 514 xxvi Contents

15.5.3 PropertiesoftheExteriorDerivative ...... 517 15.5.4 ExpansionwithRespecttoa CoordinateSystem ...... 518 15.5.5 Exterior Derivative of a 3-Form andDivergenceofa VectorField ...... 519 16 Integration in Spacetime ...... 521 16.1 Introduction...... 521 16.2 IntegrationOvera Four-DimensionalVolume...... 521 16.2.1 VolumeElement ...... 521 16.2.2 Four-Volumeofa PartofSpacetime...... 522 16.2.3 Integralofa Differential4-Form...... 523 16.3 Submanifolds of E ...... 524 16.3.1 Definition of a Submanifold ...... 524 16.3.2 Submanifold with Boundary ...... 526 16.3.3 Orientationofa Submanifold...... 527 16.4 Integration on a Submanifold of E ...... 527 16.4.1 IntegralofAnyDifferentialForm ...... 527 16.4.2 VolumeElementofa Hypersurface...... 530 16.4.3 AreaElementofa Surface ...... 532 16.4.4 Length-Elementofa Curve ...... 534 16.4.5 Integralofa ScalarFieldona Submanifold...... 535 16.4.6 Integralofa TensorField...... 536 16.4.7 FluxIntegrals ...... 536 16.5 Stokes’Theorem ...... 538 16.5.1 StatementandExamples ...... 538 16.5.2 Applications...... 540 17 Electromagnetic Field ...... 545 17.1 Introduction...... 545 17.2 ElectromagneticFieldTensor...... 545 17.2.1 ElectromagneticFieldandLorentz4-Force...... 545 17.2.2 TheElectromagneticFieldasa 2-Form ...... 547 17.2.3 ElectricandMagneticFields...... 547 17.2.4 LorentzForceRelativetoanObserver ...... 549 17.2.5 Metric Dual and Hodge Dual ...... 550 17.3 ChangeofObserver...... 552 17.3.1 Transformation Law of the Electric andMagneticFields...... 552 17.3.2 ElectromagneticFieldInvariants...... 555 17.3.3 ReductiontoParallelElectricandMagneticFields...... 557 17.3.4 FieldCreatedbya ChargeinTranslation...... 559 17.4 ParticleinanElectromagneticField...... 562 17.4.1 UniformElectromagneticField:Non-NullCase...... 563 17.4.2 Orthogonal Electric and Magnetic Fields ...... 568 Contents xxvii

17.5 Application: Particle Accelerators...... 576 17.5.1 Acceleration by an Electric Field ...... 576 17.5.2 Linear Accelerators ...... 577 17.5.3 Cyclotrons ...... 578 17.5.4 Synchrotrons...... 580 17.5.5 StorageRings...... 583 18 Maxwell Equations ...... 585 18.1 Introduction...... 585 18.2 ElectricFour-Current...... 586 18.2.1 ElectricFour-CurrentVector...... 586 18.2.2 ElectricIntensity...... 588 18.2.3 ChargeDensityandCurrentDensity ...... 591 18.2.4 Four-Current of a Continuous Media ...... 592 18.3 MaxwellEquations...... 592 18.3.1 Statement...... 592 18.3.2 AlternativeForms...... 593 18.3.3 ExpressioninTermsofElectricandMagneticFields .... 595 18.4 ElectricChargeConservation...... 598 18.4.1 DerivationfromMaxwellequations...... 598 18.4.2 Expression in Terms of Charge and Current Densities ... 601 18.4.3 GaussTheorem ...... 601 18.5 SolvingMaxwellEquations ...... 603 18.5.1 Four-Potential...... 603 18.5.2 ElectricandMagneticPotentials...... 604 18.5.3 GaugeChoice...... 606 18.5.4 ElectromagneticWaves...... 607 18.5.5 Solutionforthe4-PotentialinLorenzGauge...... 608 18.6 FieldCreatedbya MovingCharge...... 611 18.6.1 Lienard–Wiechert4-Potential...... ´ 611 18.6.2 ElectromagneticField...... 615 18.6.3 ElectricandMagneticFields...... 617 18.6.4 ChargeinInertialMotion ...... 618 18.6.5 RadiativePart...... 620 18.7 MaxwellEquationsfroma PrincipleofLeastAction...... 622 18.7.1 PrincipleofLeastActionina ClassicalFieldTheory .... 622 18.7.2 CaseoftheElectromagneticField...... 626 19 Energy–Momentum Tensor ...... 629 19.1 Introduction...... 629 19.2 Energy–Momentum Tensor ...... 629 19.2.1 Definition ...... 629 19.2.2 Interpretation...... 632 19.2.3 Symmetry of the Energy–Momentum Tensor ...... 635 xxviii Contents

19.3 Energy–Momentum Conservation ...... 636 19.3.1 Statement...... 637 19.3.2 LocalVersion ...... 637 19.3.3 Four-ForceDensity...... 638 19.3.4 Conservation of Energy and Momentum withRespecttoanObserver ...... 640 19.4 Angular Momentum...... 641 19.4.1 Definition ...... 641 19.4.2 Angular Momentum Conservation...... 642 20 Energy–Momentum of the Electromagnetic Field ...... 645 20.1 Introduction...... 645 20.2 Energy–Momentum Tensor of the Electromagnetic Field...... 645 20.2.1 Introduction ...... 645 20.2.2 Quantities Relative to an Observer...... 648 20.3 Radiation by an Accelerated Charge ...... 649 20.3.1 Electromagnetic Energy–Momentum Tensor ...... 649 20.3.2 RadiatedEnergy...... 650 20.3.3 Radiated4-Momentum...... 652 20.3.4 Angular Distribution of Radiation ...... 655 20.4 SynchrotronRadiation ...... 659 20.4.1 Introduction ...... 659 20.4.2 SpectrumofSynchrotronRadiation ...... 661 20.4.3 Applications...... 663 21 Relativistic Hydrodynamics ...... 667 21.1 Introduction...... 667 21.2 ThePerfectFluidModel...... 668 21.2.1 Energy–Momentum Tensor ...... 668 21.2.2 Quantities Relative to an Arbitrary Observer...... 670 21.2.3 PressurelessFluid(Dust)...... 671 21.2.4 Equation of State and Thermodynamic Relations...... 672 21.2.5 SimpleFluids ...... 674 21.3 BaryonNumberConservation...... 676 21.3.1 BaryonFour-Current ...... 676 21.3.2 PrincipleofBaryonNumberConservation...... 677 21.3.3 ExpressionwithRespecttoanInertialObserver...... 679 21.4 Energy–Momentum Conservation ...... 680 21.4.1 Introduction ...... 680 21.4.2 ProjectionontotheFluid4-Velocity...... 681 21.4.3 Part Orthogonal to the Fluid 4-Velocity ...... 682 21.4.4 Evolution of the Fluid Energy Relative toSomeObserver...... 683 Contents xxix

21.4.5 RelativisticEulerEquation...... 684 21.4.6 Speed of Sound ...... 685 21.4.7 Relativistic Hydrodynamics as a System ofConservationLaws ...... 686 21.5 FormulationBasedonExteriorCalculus...... 687 21.5.1 EquationofMotion...... 688 21.5.2 Vorticityofa SimpleFluid...... 689 21.5.3 Canonical Form of the Equation of Motion ...... 690 21.5.4 NonrelativisticLimit:CroccoEquation ...... 692 21.6 ConservationLaws...... 694 21.6.1 Bernoulli’s Theorem...... 694 21.6.2 Irrotational Flow ...... 696 21.6.3 Kelvin’sCirculationTheorem ...... 698 21.7 Applications ...... 701 21.7.1 Astrophysics: Jets and Gamma-Ray Bursts ...... 701 21.7.2 Quark-GluonPlasmaatRHICandatLHC...... 703 21.8 ToGoFurther...... 709 22 What About Relativistic Gravitation?...... 711 22.1 Introduction...... 711 22.2 GravitationinMinkowskiSpacetime ...... 711 22.2.1 Nordstrom’sScalarTheory...... ¨ 712 22.2.2 Incompatibility with Observations ...... 719 22.2.3 VectorTheory...... 720 22.2.4 TensorTheory ...... 722 22.3 EquivalencePrinciple...... 723 22.3.1 ThePrinciple...... 723 22.3.2 Gravitational Redshift and Incompatibility withtheMinkowskiMetric ...... 724 22.3.3 Experimental Verifications oftheGravitationalRedshift...... 726 22.3.4 LightDeflection ...... 729 22.4 GeneralRelativity...... 729

A Basic Algebra ...... 733 A.1 BasicStructures ...... 733 A.1.1 Group...... 733 A.1.2 Fields...... 734 A.2 LinearAlgebra...... 735 A.2.1 VectorSpace...... 735 A.2.2 Algebra...... 736 xxx Contents

B Web Pages ...... 737

C Special Relativity Books ...... 739

References...... 741

List of Symbols ...... 761

Index ...... 765