Michael Tsamparlis. Special Relativity. An Introduction with 200 Problems and Solutions

Special Relativity

Michael Tsamparlis Special Relativity An Introduction with 200 Problems and Solutions 123

Dr. Michael Tsamparlis Department of Astrophysics, Astronomy and Mechanics University of Athens Panepistimiopolis GR 157 84 ZOGRAFOS Athens Greece mtsampa@phys.uoa.gr Additional material to this book can be downloaded from http://extra.springer.com. Password: 978-3-642-03836-5 ISBN 978-3-642-03836-5 e-isbn 978-3-642-03837-2 DOI 10.1007/978-3-642-03837-2 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009940408 c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Cover design: estudio Calamar S.L. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Omnia mea mecum fero Whatever I possess I bear with me

Preface Writing a new book on the classic subject of Special Relativity, on which numerous important physicists have contributed and many books have already been written, can be like adding another epicycle to the Ptolemaic cosmology. Furthermore, it is our belief that if a book has no new elements, but simply repeats what is written in the existing literature, perhaps with a different style, then this is not enough to justify its publication. However, after having spent a number of years, both in class and research with relativity, I have come to the conclusion that there exists a place for a new book. Since it appears that somewhere along the way, mathematics may have obscured and prevailed to the degree that we tend to teach relativity (and I believe, theoretical physics) simply using heavier mathematics without the inspiration and the mastery of the classic physicists of the last century. Moreover current trends encourage the application of techniques in producing quick results and not tedious conceptual approaches resulting in long-lasting reasoning. On the other hand, physics cannot be done á la carte stripped from philosophy, or, to put it in a simple but dramatic context A building is not an accumulation of stones! As a result of the above, a major aim in the writing of this book has been the distinction between the mathematics of Minkowski space and the physics of relativity. This is necessary for one to understand the physics of the theory and not stay with the geometry, which by itself is a very elegant and attractive tool. Therefore in the first chapter we develop the mathematics needed for the statement and development of the theory. The approach is limited and concise but sufficient for the purposes it is supposed to serve. Having finished with the mathematical concepts we continue with the foundation of the physical theory. Chapter 2 sets the framework on the scope and the structure of a theory of physics. We introduce the principle of relativity and the covariance principle, both principles being keystones in every theory of physics. Subsequently we apply the scenario first to formulate Newtonian Physics (Chap. 3) and then Special Relativity (Chap. 4). The formulation of Newtonian Physics is done in a relativistic way, in order to prepare the ground for a proper understanding of the parallel formulation of Special Relativity. Having founded the theory we continue with its application. The approach is systematic in the sense that we develop the theory by means of a stepwise introduction vii

viii Preface of new physical quantities. Special Relativity being a kinematic theory forces us to consider as the fundamental quantity the position four-vector. This is done in Chap. 5 where we define the relativistic measurement of the position four-vector by means of the process of chronometry. To relate the theory with Newtonian reality, we introduce rules, which identify Newtonian space and Newtonian time in Special Relativity. In Chaps. 6 and 7 we introduce the remaining elements of kinematics, that is, the four-velocity and the four-acceleration. We discuss the well-known relativistic composition law for the three-velocities and show that it is equivalent to the Einstein relativity principle, that is, the Lorentz transformation. In the chapter of fouracceleration we introduce the concept of synchronization which is a key concept in the relativistic description of motion. Finally, we discuss the phenomenon of acceleration redshift which together with some other applications of four-acceleration shows that here the limits of Special Relativity are reached and one must go over to General Relativity. After the presentation of kinematics, in Chap. 8 we discuss various paradoxes, which play an important role in the physical understanding of the theory. We choose to present paradoxes which are not well known, as for example, it is the twin paradox. In Chap. 9 we introduce the (relativistic) mass and the four-momentum by means of which we distinguish the particles in massive particles and luxons (photons). Chapter 10 is the most useful chapter of this book, because it concerns relativistic reactions, where the use of Special Relativity is indispensible. This chapter contains many examples in order to familiarize the student with a tool, that will be necessary to other major courses such as particle physics and high energy physics. In Chap. 11 we commence the dynamics of Special Relativity by the introduction of the four-force. We discuss many practical problems and use the tetrahedron of Frenet Serret to compute the generic form of the four-force. We show how the wellknown four-forces comply with the generic form. In Chap. 12 we introduce the concept of covariant decomposition of a tensor along a vector and give the basic results concerning the 1 + 3 decomposition in Minkowski space. The mathematics of this chapter is necessary in order to understand properly the relativistic physics. It is used extensively in General Relativity but up to now we have not seen its explicit appearance in Special Relativity, even though it is a powerful and natural tool both for the theory and the applications. Chapter 13 is the next pillar of Special Relativity, that is, electromagnetism. We present in a concise way the standard vector form of electromagnetism and subsequently we are led to the four formalism formulation as a natural consequence. After discussing the standard material on the subject (four-potential, electromagnetic field tensor, etc.) we continue with lesser known material, such as the tensor formulation of Ohm s law and the 1 + 3 decomposition of Maxwell s equations. The reason why we introduce these more advanced topics is that we wish to prepare the student for courses on important subjects such as relativistic magnetohydrodynamics (RMHD). The rest of the book concerns topics which, to our knowledge, cannot be found in the existing books on Special Relativity yet. In Chap. 14 we discuss the concept

Preface ix of spin as a natural result of the generalization of the angular momentum tensor in Special Relativity. We follow a formal mathematical procedure, which reveals what the spin is without the use of the quantum field theory. As an application, we discuss the motion of a charged particle with spin in a homogeneous electromagnetic field and recover the well-known results in the literature. Chapter 15 deals with the covariant Lorentz transformation, a form which is not widely known. All four types of Lorentz transformations are produced in covariant form and the results are applied to applications involving the geometry of threevelocity space, the composition of Lorentz transformations, etc. Finally, in Chap. 16 we study the reaction A + B C + D in a fully covariant form. The results are generic and can be used to develop software which will solve such reactions directly, provided one introduces the right data. The book includes numerous exercises and solved problems, plenty of which supplement the theory and can be useful to the reader on many occasions. In addition, a large number of problems, carefully classified in all topics accompany the book. The above does not cover all topics we would like to consider. One such topic is relativistic waves, which leads to the introduction of De Broglie waves and subsequently to the foundation of quantum mechanics. A second topic is relativistic hydrodynamics and its extension to RMHD. However, one has to draw a line somewhere and leave the future to take care of things to be done. Looking back at the long hours over the many years which were necessary for the preparation of this book, I cannot help feeling that, perhaps, I should not have undertaken the project. However, I feel that it would be unfair to all the students and colleagues, who for more that 30 years have helped me to understand and develop the relativistic ideas, to find and solve problems, and in general to keep my interest alive. Therefore the present book is a collective work and my role has been simply to compile these experiences. I do not mention specific names the list would be too long, and I will certainly forget quite a few but they know and I know, and that is enough. I close this preface, with an apology to my family for the long working hours; that I was kept away from them for writing this book and I would like to thank them for their continuous support and understanding. Athens, Greece October 2009 Michael Tsamparlis

Contents 1 Mathematical Part... 1 1.1 Introduction...... 1 1.2 Elements From the Theory of Linear Spaces........ 2 1.2.1 CoordinateTransformations... 2 1.3 Inner Product Metric.... 6 1.4 Tensors... 10 1.4.1 OperationsofTensors... 13 1.5 The Case of Euclidean Geometry...... 14 1.6 TheLorentzGeometry... 17 1.6.1 LorentzTransformations... 18 1.7 Algebraic Determination of the General Vector Lorentz Transformation... 26 1.8 The Kinematic Interpretation of the General Lorentz Transformation... 40 1.8.1 Relativistic Parallelism of Space Axes..... 40 1.8.2 The Kinematic Interpretation of Lorentz Transformation 42 1.9 TheGeometryoftheBoost... 43 1.10 CharacteristicFramesofFour-Vectors... 48 1.10.1 Proper Frame of a Timelike Four-Vector.......... 48 1.10.2 Characteristic Frame of a Spacelike Four-Vector... 49 1.11 ParticleFour-Vectors... 50 1.12 The Center System (CS) of a System of Particle Four-Vectors.... 52 2 The Structure of the Theories of Physics... 55 2.1 Introduction...... 55 2.2 TheRoleofPhysics... 56 2.3 The Structure of a Theory of Physics... 59 2.4 Physical Quantities and Reality of a Theory of Physics...... 60 2.5 InertialObservers... 62 2.6 GeometrizationofthePrincipleofRelativity... 63 2.6.1 PrincipleofInertia... 63 2.6.2 The Covariance Principle.... 64 2.7 Relativity and the Predictions of a Theory... 66 xi

xii Contents 3 Newtonian Physics... 67 3.1 Introduction...... 67 3.2 NewtonianKinematics... 68 3.2.1 MassPoint... 68 3.2.2 Space... 69 3.2.3 Time... 71 3.3 NewtonianInertialObservers... 74 3.3.1 DeterminationofNewtonianInertialObservers... 75 3.3.2 Measurement of the Position Vector...... 77 3.4 Galileo Principle of Relativity......... 78 3.5 Galileo Transformations for Space and Time Newtonian Physical Quantities......... 79 3.5.1 Galileo Covariant Principle: Part I........ 79 3.5.2 Galileo Principle of Communication...... 80 3.6 Newtonian Physical Quantities. The Covariance Principle... 81 3.6.1 Galileo Covariance Principle: Part II...... 81 3.7 Newtonian Composition Law of Vectors.... 82 3.8 Newtonian Dynamics..... 83 3.8.1 Law of Conservation of Linear Momentum....... 84 4 The Foundation of Special Relativity... 87 4.1 Introduction...... 87 4.2 Light and the Galileo Principle of Relativity........ 88 4.2.1 The Existence of Non-Newtonian Physical Quantities.. 88 4.2.2 The Limit of Special Relativity to Newtonian Physics.. 89 4.3 The Physical Role of the Speed of Light..... 92 4.4 The Physical Definition of Spacetime....... 93 4.4.1 TheEvents... 94 4.4.2 The Geometry of Spacetime...... 94 4.5 Structures in Minkowski Space........ 96 4.5.1 TheLightCone... 96 4.5.2 WorldLines... 97 4.5.3 Curves in Minkowski Space...... 98 4.5.4 Geometric Definition of Relativistic Inertial Observers(RIO)... 99 4.5.5 Proper Time.... 99 4.5.6 The Proper Frame of a RIO.......100 4.5.7 Proper or Rest Space........101 4.6 Spacetime Description of Motion...... 102 4.6.1 The Physical Definition of a RIO......... 103 4.6.2 Relativistic Measurement of the Position Vector...104 4.6.3 The Physical Definition of an LRIO...105 4.7 TheEinsteinPrincipleofRelativity...105 4.7.1 TheEquationofLorentzIsometry...106

Contents xiii 4.8 The Lorentz Covariance Principle...... 108 4.8.1 RulesforConstructingLorentzTensors...109 4.8.2 Potential Relativistic Physical Quantities......... 110 4.9 Universal Speeds and the Lorentz Transformation...110 5 The Physics of the Position Four-Vector...117 5.1 Introduction......117 5.2 The Concepts of Space and Time in Special Relativity.......117 5.3 Measurement of Spatial and Temporal Distance in Special Relativity........118 5.4 Relativistic Definition of Spatial and Temporal Distances....120 5.5 Timelike Position Four-Vector Measurement of Temporal Distance.......121 5.6 Spacelike Position Four-Vector Measurement of Spatial Distance.........126 5.7 The General Case........129 5.8 The Reality of Length Contraction and Time Dilation.......130 5.9 TheRigidRod...132 5.10 Optical Images in Special Relativity....134 5.11 How to Solve Problems Involving Spatial and Temporal Distance 141 5.11.1 A Brief Summary of the Lorentz Transformation...... 141 5.11.2 Parallel and Normal Decomposition of Lorentz Transformation... 142 5.11.3 Methodologies of Solving Problems Involving Boosts. 143 5.11.4 The Algebraic Method....... 146 5.11.5 TheGeometricMethod...150 6 Relativistic Kinematics...155 6.1 Introduction......155 6.2 RelativisticMassPoint...155 6.3 Relativistic Composition of Three-Vectors.......... 159 6.4 RelativeFour-Vectors...166 6.5 The three-velocity Space...... 174 6.6 ThomasPrecession...177 7 Four-Acceleration...185 7.1 Introduction......185 7.2 The Four-Acceleration....186 7.3 Calculating Accelerated Motions...193 7.4 Hyperbolic Motion of a Relativistic Mass Particle...197 7.4.1 Geometric Representation of Hyperbolic Motion...200 7.5 Synchronization..........204 7.5.1 Einstein Synchronization..... 204 7.6 RigidMotionofManyRelativisticMassPoints...205

xiv Contents 7.7 Rigid Motion and Hyperbolic Motion....... 206 7.7.1 The Synchronization of LRIO....208 7.7.2 Synchronization of Chronometry......... 209 7.7.3 The Kinematics in the LCF Σ...211 7.7.4 The Case of the Gravitational Field.......214 7.8 General One-Dimensional Rigid Motion....216 7.8.1 The Case of Hyperbolic Motion...217 7.9 Rotational Rigid Motion...219 7.9.1 The Transitive Property of the Rigid Rotational Motion 222 7.10 TheRotatingDisk...224 7.10.1 TheKinematicsofRelativisticObservers...224 7.10.2 Chronometry and the Spatial Line Element........ 225 7.10.3 TheRotatingDisk...228 7.10.4 Definition of the Rotating Disk for a RIO.........229 7.10.5 The Locally Relativistic Inertial Observer (LRIO)..... 230 7.10.6 The Accelerated Observer....235 7.11 The Generalization of Lorentz Transformation and the Accelerated Observers...239 7.11.1 The Generalized Lorentz Transformation......... 240 7.11.2 The Special Case u 0 (l, x ) = u 1 (l, x ) = u(x )...242 7.11.3 Equation of Motion in a Gravitational Field.......247 7.12 The Limits of Special Relativity.......248 7.12.1 Experiment 1: The Gravitational Redshift......... 249 7.12.2 Experiment 2: The Gravitational Time Dilation....251 7.12.3 Experiment 3: The Curvature of Spacetime.......252 8 Paradoxes...253 8.1 Introduction......253 8.2 Various Paradoxes........254 9 Mass Four-Momentum...265 9.1 Introduction......265 9.2 The(Relativistic)Mass...266 9.3 TheFour-MomentumofaReMaP...267 9.4 The Four-Momentum of Photons (Luxons)......... 275 9.5 TheFour-MomentumofParticles...278 9.6 TheSystemofNaturalUnits...278 10 Relativistic Reactions...283 10.1 Introduction......283 10.2 RepresentationofParticleReactions...284 10.3 RelativisticReactions...285 10.3.1 TheSumofParticleFour-Vectors...286 10.3.2 The Relativistic Triangle Inequality....... 288

Contents xv 10.4 WorkingwithFour-Momenta...289 10.5 Special Coordinate Frames in the Study of Relativistic Collisions 291 10.6 The Generic Reaction A + B C...292 10.6.1 The Physics of the Generic Reaction......293 10.6.2 ThresholdofaReaction...297 10.7 TransformationofAngles...304 10.7.1 RadiativeTransitions...308 10.7.2 Reactions With Two-Photon Final State...312 10.7.3 Elastic Collisions Scattering....317 11 Four-Force...325 11.1 Introduction......325 11.2 TheFour-Force...325 11.3 InertialFour-ForceandFour-Potential...340 11.3.1 TheVectorFour-Potential...342 11.4 The Lagrangian Formalism for Inertial Four-Forces......... 343 11.5 MotioninaCentralPotential...350 11.6 MotionofaRocket...355 11.7 The Frenet Serret Frame in Minkowski Space......363 11.7.1 ThePhysicalBasis...368 11.7.2 The Generic Inertial Four-Force.......... 372 12 Irreducible Decompositions...377 12.1 Decompositions..........377 12.1.1 Writing a Tensor of Valence (0,2) as a Matrix.....378 12.2 The Irreducible Decomposition wrt a Non-null Vector.......379 12.2.1 Decomposition in a Euclidean Space E n...379 12.2.2 1 + 3 Decomposition in Minkowski Space........ 383 12.3 1+1+2 Decomposition wrt a Pair of Timelike Vectors...389 13 The Electromagnetic Field...395 13.1 Introduction......395 13.2 MaxwellEquationsinNewtonianPhysics...396 13.3 The Electromagnetic Potential......... 399 13.4 TheEquationofContinuity...405 13.5 The Electromagnetic Four-Potential....412 13.6 The Electromagnetic Field Tensor F ij...415 13.6.1 TheTransformationoftheFields...415 13.6.2 Maxwell Equations in Terms of F ij...417 13.6.3 The Invariants of the Electromagnetic Field.......418 13.7 The Physical Significance of the Electromagnetic Invariants..... 421 13.7.1 The Case Y = 0...422 13.7.2 The Case Y 0...423

xvi Contents 13.8 Motion of a Charge in an Electromagnetic Field The Lorentz Force...426 13.9 Motion of a Charge in a Homogeneous Electromagnetic Field... 429 13.9.1 The Case of a Homogeneous Electric Field.......430 13.9.2 The Case of a Homogeneous Magnetic Field......434 13.9.3 The Case of Two Homogeneous Fields of Equal Strength and Perpendicular Directions... 436 13.9.4 The Case of Homogeneous and Parallel Fields E B.. 438 13.10 The Relativistic Electric and Magnetic Fields.......440 13.10.1 TheLevi-CivitaTensorDensity...440 13.10.2 The Case of Vacuum........442 13.10.3 The Electromagnetic Theory for a General Medium... 445 13.10.4 The Electric and Magnetic Moments......448 13.10.5 Maxwell Equations for a General Medium........ 448 13.10.6 The 1 + 3 Decomposition of Maxwell Equations...... 449 13.11 The Four-Current of Conductivity and Ohm s Law..........454 13.11.1 The Continuity Equation J a ;a = 0 foranisotropicmaterial... 458 13.12 The Electromagnetic Field in a Homogeneous andisotropicmedium...459 13.13 Electric Conductivity and the Propagation Equation for E a...463 13.14 The Generalized Ohm s Law...465 13.15 The Energy Momentum Tensor of the Electromagnetic Field.... 467 13.16 The Electromagnetic Field of a Moving Charge.....475 13.16.1 TheInvariants...477 13.16.2 The Fields E i, B i...478 13.16.3 The Liénard Wiechert Potentials and the Fields E, B.. 478 13.17 Special Relativity and Practical Applications........ 489 13.18 The Systems of Units SI and Gauss in Electromagnetism....492 14 Relativistic Angular Momentum...495 14.1 Introduction......495 14.2 MathematicalPreliminaries...495 14.2.1 1 + 3 Decomposition of a Bivector X ab...495 14.3 The Derivative of X ab Along the Vector p a...498 14.4 The Angular Momentum in Special Relativity...500 14.4.1 The Angular Momentum in Newtonian Theory....500 14.4.2 The Angular Momentum of a Particle in Special Relativity... 502 14.5 The Intrinsic Angular Momentum The Spin Vector........506 14.5.1 The Magnetic Dipole........ 506 14.5.2 TheRelativisticSpin...510 14.5.3 Motion of a Particle with Spin in a Homogeneous Electromagnetic Field...... 515 14.5.4 Transformation of Motion in Σ...517

Contents xvii 15 The Covariant Lorentz Transformation...521 15.1 Introduction......521 15.2 TheCovariantLorentzTransformation...523 15.2.1 Definition of the Lorentz Transformation......... 523 15.2.2 Computation of the Covariant Lorentz Transformation. 524 15.2.3 The Action of the Covariant Lorentz Transformation onthecoordinates... 529 15.2.4 The Invariant Length of a Four-Vector.....534 15.3 The Four Types of the Lorentz Transformation Viewed as Spacetime Reflections....534 15.4 Relativistic Composition Rule of Four-Vectors......537 15.4.1 ComputationoftheCompositeFour-Vector...540 15.4.2 The Relativistic Composition Rule for Three-Velocities 542 15.4.3 Riemannian Geometry and Special Relativity.....544 15.4.4 The Relativistic Rule for the Composition of Three-Accelerations..... 548 15.5 The Composition of Lorentz Transformations....... 550 16 Geometric Description of Relativistic Interactions...555 16.1 Collisions and Geometry...... 555 16.2 Geometric Description of Collisions in Newtonian Physics...556 16.3 GeometricDescriptionofRelativisticReactions...558 16.4 The General Geometric Results........ 559 16.4.1 The 1+3 Decomposition of a Particle Four-Vector wrtatimelikefour-vector... 561 16.5 TheSystemofTwotoOneParticleFour-Vectors...563 16.5.1 The Triangle Function of a System of Two Particle Four-Vectors... 565 16.5.2 Extreme Values of the Four-Vectors (A ± B) 2...567 16.5.3 The System A a, B a, (A + B) a of Particle Four-VectorsinCS... 568 16.5.4 The System A a, B a, (A + B) a inthelab...570 16.6 The Relativistic System A a + B a C a + D a...574 16.6.1 The Reaction B C + D...587 Bibliography...589 Index...591