Special Relativity

Michael Tsamparlis

Special Relativity

An Introduction with 200 Problemsand Solutions

123

Dr. Michael TsamparlisDepartment of Astrophysics, Astronomy and MechanicsUniversity of AthensPanepistimiopolisGR 157 84 ZOGRAFOSAthensGreecemtsampa@phys.uoa.gr

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Preface

Writing a new book on the classic subject of Special Relativity, on which numerousimportant physicists have contributed and many books have already been written,can be like adding another epicycle to the Ptolemaic cosmology. Furthermore, it isour belief that if a book has no new elements, but simply repeats what is writtenin the existing literature, perhaps with a different style, then this is not enoughto justify its publication. However, after having spent a number of years, both inclass and research with relativity, I have come to the conclusion that there exists aplace for a new book. Since it appears that somewhere along the way, mathemat-ics may have obscured and prevailed to the degree that we tend to teach relativity(and I believe, theoretical physics) simply using “heavier” mathematics without theinspiration and the mastery of the classic physicists of the last century. Moreovercurrent trends encourage the application of techniques in producing quick resultsand not tedious conceptual approaches resulting in long-lasting reasoning. On theother hand, physics cannot be done a la carte stripped from philosophy, or, to put itin 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 thedistinction between the mathematics of Minkowski space and the physics of rel-ativity. This is necessary for one to understand the physics of the theory and notstay with the geometry, which by itself is a very elegant and attractive tool. There-fore in the first chapter we develop the mathematics needed for the statement anddevelopment of the theory. The approach is limited and concise but sufficient for thepurposes it is supposed to serve. Having finished with the mathematical concepts wecontinue with the foundation of the physical theory. Chapter 2 sets the frameworkon the scope and the structure of a theory of physics. We introduce the principleof relativity and the covariance principle, both principles being keystones in everytheory of physics. Subsequently we apply the scenario first to formulate NewtonianPhysics (Chap. 3) and then Special Relativity (Chap. 4). The formulation of Newto-nian Physics is done in a relativistic way, in order to prepare the ground for a properunderstanding of the parallel formulation of Special Relativity.

Having founded the theory we continue with its application. The approach is sys-tematic in the sense that we develop the theory by means of a stepwise introduction

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viii Preface

of new physical quantities. Special Relativity being a kinematic theory forces usto consider as the fundamental quantity the position four-vector. This is done inChap. 5 where we define the relativistic measurement of the position four-vector bymeans of the process of chronometry. To relate the theory with Newtonian reality,we introduce rules, which identify Newtonian space and Newtonian time in SpecialRelativity.

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 relativisticcomposition law for the three-velocities and show that it is equivalent to the Ein-stein relativity principle, that is, the Lorentz transformation. In the chapter of four-acceleration we introduce the concept of synchronization which is a key concept inthe relativistic description of motion. Finally, we discuss the phenomenon of accel-eration redshift which together with some other applications of four-accelerationshows that here the limits of Special Relativity are reached and one must go over toGeneral 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 chooseto present paradoxes which are not well known, as for example, it is the twin para-dox.

In Chap. 9 we introduce the (relativistic) mass and the four-momentum by meansof which we distinguish the particles in massive particles and luxons (photons).

Chapter 10 is the most useful chapter of this book, because it concerns relativisticreactions, where the use of Special Relativity is indispensible. This chapter containsmany examples in order to familiarize the student with a tool, that will be necessaryto other major courses such as particle physics and high energy physics.

In Chap. 11 we commence the dynamics of Special Relativity by the introductionof the four-force. We discuss many practical problems and use the tetrahedron ofFrenet–Serret to compute the generic form of the four-force. We show how the well-known four-forces comply with the generic form.

In Chap. 12 we introduce the concept of covariant decomposition of a tensoralong a vector and give the basic results concerning the 1 + 3 decomposition inMinkowski space. The mathematics of this chapter is necessary in order to under-stand properly the relativistic physics. It is used extensively in General Relativitybut up to now we have not seen its explicit appearance in Special Relativity, eventhough 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. Wepresent in a concise way the standard vector form of electromagnetism and subse-quently we are led to the four formalism formulation as a natural consequence. Afterdiscussing the standard material on the subject (four-potential, electromagnetic fieldtensor, etc.) we continue with lesser known material, such as the tensor formulationof Ohm’s law and the 1+3 decomposition of Maxwell’s equations. The reason whywe introduce these more advanced topics is that we wish to prepare the student forcourses on important subjects such as relativistic magnetohydrodynamics (RMHD).

The rest of the book concerns topics which, to our knowledge, cannot be foundin the existing books on Special Relativity yet. In Chap. 14 we discuss the concept

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of spin as a natural result of the generalization of the angular momentum tensorin Special Relativity. We follow a formal mathematical procedure, which revealswhat “the spin is” without the use of the quantum field theory. As an application, wediscuss the motion of a charged particle with spin in a homogeneous electromagneticfield and recover the well-known results in the literature.

Chapter 15 deals with the covariant Lorentz transformation, a form which is notwidely known. All four types of Lorentz transformations are produced in covariantform and the results are applied to applications involving the geometry of three-velocity space, the composition of Lorentz transformations, etc.

Finally, in Chap. 16 we study the reaction A+ B −→ C + D in a fully covariantform. The results are generic and can be used to develop software which will solvesuch reactions directly, provided one introduces the right data.

The book includes numerous exercises and solved problems, plenty of whichsupplement the theory and can be useful to the reader on many occasions. In addi-tion, a large number of problems, carefully classified in all topics accompany thebook.

The above does not cover all topics we would like to consider. One such topicis relativistic waves, which leads to the introduction of De Broglie waves and sub-sequently to the foundation of quantum mechanics. A second topic is relativistichydrodynamics and its extension to RMHD. However, one has to draw a line some-where 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 forthe preparation of this book, I cannot help feeling that, perhaps, I should not haveundertaken the project. However, I feel that it would be unfair to all the students andcolleagues, who for more that 30 years have helped me to understand and developthe relativistic ideas, to find and solve problems, and in general to keep my interestalive. Therefore the present book is a collective work and my role has been simplyto compile these experiences. I do not mention specific names – the list would betoo long, and I will certainly forget quite a few – but they know and I know, and thatis 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 themfor their continuous support and understanding.

Athens, Greece Michael TsamparlisOctober 2009

Contents

1 Mathematical Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Elements From the Theory of Linear Spaces . . . . . . . . . . . . . . . . . . 2

1.2.1 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . 21.3 Inner Product – Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4.1 Operations of Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.5 The Case of Euclidean Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.6 The Lorentz Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.6.1 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 181.7 Algebraic Determination of the General Vector Lorentz

Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.8 The Kinematic Interpretation of the General Lorentz

Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.8.1 Relativistic Parallelism of Space Axes . . . . . . . . . . . . . . . 401.8.2 The Kinematic Interpretation of Lorentz Transformation 42

1.9 The Geometry of the Boost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431.10 Characteristic Frames of Four-Vectors . . . . . . . . . . . . . . . . . . . . . . . 48

1.10.1 Proper Frame of a Timelike Four-Vector . . . . . . . . . . . . . 481.10.2 Characteristic Frame of a Spacelike Four-Vector . . . . . . 49

1.11 Particle Four-Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501.12 The Center System (CS) of a System of Particle Four-Vectors . . . . 52

2 The Structure of the Theories of Physics . . . . . . . . . . . . . . . . . . . . . . . . . . 552.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.2 The Role of Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.3 The Structure of a Theory of Physics . . . . . . . . . . . . . . . . . . . . . . . . 592.4 Physical Quantities and Reality of a Theory of Physics . . . . . . . . . 602.5 Inertial Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.6 Geometrization of the Principle of Relativity . . . . . . . . . . . . . . . . . . 63

2.6.1 Principle of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.6.2 The Covariance Principle . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.7 Relativity and the Predictions of a Theory . . . . . . . . . . . . . . . . . . . . 66

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3 Newtonian Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.2 Newtonian Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2.1 Mass Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.2.2 Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.2.3 Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.3 Newtonian Inertial Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.3.1 Determination of Newtonian Inertial Observers . . . . . . . 753.3.2 Measurement of the Position Vector . . . . . . . . . . . . . . . . 77

3.4 Galileo Principle of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.5 Galileo Transformations for Space and Time – Newtonian

Physical Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.5.1 Galileo Covariant Principle: Part I . . . . . . . . . . . . . . . . . . 793.5.2 Galileo Principle of Communication . . . . . . . . . . . . . . . . 80

3.6 Newtonian Physical Quantities. The Covariance Principle . . . . . . . 813.6.1 Galileo Covariance Principle: Part II . . . . . . . . . . . . . . . . 81

3.7 Newtonian Composition Law of Vectors . . . . . . . . . . . . . . . . . . . . . 823.8 Newtonian Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.8.1 Law of Conservation of Linear Momentum . . . . . . . . . . 84

4 The Foundation of Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.2 Light and the Galileo Principle of Relativity . . . . . . . . . . . . . . . . . . 88

4.2.1 The Existence of Non-Newtonian Physical Quantities . . 884.2.2 The Limit of Special Relativity to Newtonian Physics . . 89

4.3 The Physical Role of the Speed of Light . . . . . . . . . . . . . . . . . . . . . . 924.4 The Physical Definition of Spacetime . . . . . . . . . . . . . . . . . . . . . . . . 93

4.4.1 The Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.4.2 The Geometry of Spacetime . . . . . . . . . . . . . . . . . . . . . . . 94

4.5 Structures in Minkowski Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.5.1 The Light Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.5.2 World Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.5.3 Curves in Minkowski Space . . . . . . . . . . . . . . . . . . . . . . . 984.5.4 Geometric Definition of Relativistic Inertial

Observers (RIO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.5.5 Proper Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.5.6 The Proper Frame of a RIO . . . . . . . . . . . . . . . . . . . . . . . . 1004.5.7 Proper or Rest Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.6 Spacetime Description of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.6.1 The Physical Definition of a RIO . . . . . . . . . . . . . . . . . . . 1034.6.2 Relativistic Measurement of the Position Vector . . . . . . 1044.6.3 The Physical Definition of an LRIO. . . . . . . . . . . . . . . . . 105

4.7 The Einstein Principle of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.7.1 The Equation of Lorentz Isometry . . . . . . . . . . . . . . . . . . 106

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4.8 The Lorentz Covariance Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.8.1 Rules for Constructing Lorentz Tensors . . . . . . . . . . . . . 1094.8.2 Potential Relativistic Physical Quantities . . . . . . . . . . . . 110

4.9 Universal Speeds and the Lorentz Transformation . . . . . . . . . . . . . 110

5 The Physics of the Position Four-Vector . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.2 The Concepts of Space and Time in Special Relativity . . . . . . . . . . 1175.3 Measurement of Spatial and Temporal Distance

in Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.4 Relativistic Definition of Spatial and Temporal Distances . . . . . . . 1205.5 Timelike Position Four-Vector – Measurement

of Temporal Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.6 Spacelike Position Four-Vector – Measurement

of Spatial Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.7 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.8 The Reality of Length Contraction and Time Dilation . . . . . . . . . . 1305.9 The Rigid Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1325.10 Optical Images in Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . 1345.11 How to Solve Problems Involving Spatial and Temporal Distance 141

5.11.1 A Brief Summary of the Lorentz Transformation . . . . . . 1415.11.2 Parallel and Normal Decomposition of Lorentz

Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1425.11.3 Methodologies of Solving Problems Involving Boosts . 1435.11.4 The Algebraic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1465.11.5 The Geometric Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6 Relativistic Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1556.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1556.2 Relativistic Mass Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1556.3 Relativistic Composition of Three-Vectors . . . . . . . . . . . . . . . . . . . . 1596.4 Relative Four-Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1666.5 The three-Velocity Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1746.6 Thomas Precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

7 Four-Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1857.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1857.2 The Four-Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1867.3 Calculating Accelerated Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1937.4 Hyperbolic Motion of a Relativistic Mass Particle . . . . . . . . . . . . . 197

7.4.1 Geometric Representation of Hyperbolic Motion . . . . . . 2007.5 Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

7.5.1 Einstein Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . 2047.6 Rigid Motion of Many Relativistic Mass Points . . . . . . . . . . . . . . . 205

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7.7 Rigid Motion and Hyperbolic Motion . . . . . . . . . . . . . . . . . . . . . . . . 2067.7.1 The Synchronization of LRIO . . . . . . . . . . . . . . . . . . . . . 2087.7.2 Synchronization of Chronometry . . . . . . . . . . . . . . . . . . . 2097.7.3 The Kinematics in the LCF Σ . . . . . . . . . . . . . . . . . . . . . . 2117.7.4 The Case of the Gravitational Field . . . . . . . . . . . . . . . . . 214

7.8 General One-Dimensional Rigid Motion . . . . . . . . . . . . . . . . . . . . . 2167.8.1 The Case of Hyperbolic Motion . . . . . . . . . . . . . . . . . . . . 217

7.9 Rotational Rigid Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2197.9.1 The Transitive Property of the Rigid Rotational Motion 222

7.10 The Rotating Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2247.10.1 The Kinematics of Relativistic Observers . . . . . . . . . . . . 2247.10.2 Chronometry and the Spatial Line Element . . . . . . . . . . . 2257.10.3 The Rotating Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2287.10.4 Definition of the Rotating Disk for a RIO . . . . . . . . . . . . 2297.10.5 The Locally Relativistic Inertial Observer (LRIO) . . . . . 2307.10.6 The Accelerated Observer . . . . . . . . . . . . . . . . . . . . . . . . . 235

7.11 The Generalization of Lorentz Transformationand the Accelerated Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2397.11.1 The Generalized Lorentz Transformation . . . . . . . . . . . . 2407.11.2 The Special Case u0(l ′, x ′) = u1(l ′, x ′) = u(x ′) . . . . . . . 2427.11.3 Equation of Motion in a Gravitational Field . . . . . . . . . . 247

7.12 The Limits of Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2487.12.1 Experiment 1: The Gravitational Redshift . . . . . . . . . . . . 2497.12.2 Experiment 2: The Gravitational Time Dilation . . . . . . . 2517.12.3 Experiment 3: The Curvature of Spacetime . . . . . . . . . . 252

8 Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2538.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2538.2 Various Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

9 Mass – Four-Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2659.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2659.2 The (Relativistic) Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2669.3 The Four-Momentum of a ReMaP . . . . . . . . . . . . . . . . . . . . . . . . . . . 2679.4 The Four-Momentum of Photons (Luxons) . . . . . . . . . . . . . . . . . . . 2759.5 The Four-Momentum of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . 2789.6 The System of Natural Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

10 Relativistic Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28310.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28310.2 Representation of Particle Reactions . . . . . . . . . . . . . . . . . . . . . . . . . 28410.3 Relativistic Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

10.3.1 The Sum of Particle Four-Vectors . . . . . . . . . . . . . . . . . . 28610.3.2 The Relativistic Triangle Inequality . . . . . . . . . . . . . . . . . 288

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10.4 Working with Four-Momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28910.5 Special Coordinate Frames in the Study of Relativistic Collisions 29110.6 The Generic Reaction A + B → C . . . . . . . . . . . . . . . . . . . . . . . . . . 292

10.6.1 The Physics of the Generic Reaction . . . . . . . . . . . . . . . . 29310.6.2 Threshold of a Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

10.7 Transformation of Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30410.7.1 Radiative Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30810.7.2 Reactions With Two-Photon Final State . . . . . . . . . . . . . 31210.7.3 Elastic Collisions – Scattering . . . . . . . . . . . . . . . . . . . . . 317

11 Four-Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32511.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32511.2 The Four-Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32511.3 Inertial Four-Force and Four-Potential . . . . . . . . . . . . . . . . . . . . . . . 340

11.3.1 The Vector Four-Potential . . . . . . . . . . . . . . . . . . . . . . . . . 34211.4 The Lagrangian Formalism for Inertial Four-Forces . . . . . . . . . . . . 34311.5 Motion in a Central Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35011.6 Motion of a Rocket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35511.7 The Frenet–Serret Frame in Minkowski Space . . . . . . . . . . . . . . . . 363

11.7.1 The Physical Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36811.7.2 The Generic Inertial Four-Force . . . . . . . . . . . . . . . . . . . . 372

12 Irreducible Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37712.1 Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

12.1.1 Writing a Tensor of Valence (0,2) as a Matrix . . . . . . . . 37812.2 The Irreducible Decomposition wrt a Non-null Vector . . . . . . . . . . 379

12.2.1 Decomposition in a Euclidean Space En . . . . . . . . . . . . . 37912.2.2 1+ 3 Decomposition in Minkowski Space . . . . . . . . . . . 383

12.3 1+1+2 Decomposition wrt a Pair of Timelike Vectors . . . . . . . . . . 389

13 The Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39513.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39513.2 Maxwell Equations in Newtonian Physics . . . . . . . . . . . . . . . . . . . . 39613.3 The Electromagnetic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39913.4 The Equation of Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40513.5 The Electromagnetic Four-Potential . . . . . . . . . . . . . . . . . . . . . . . . . 41213.6 The Electromagnetic Field Tensor Fi j . . . . . . . . . . . . . . . . . . . . . . . . 415

13.6.1 The Transformation of the Fields . . . . . . . . . . . . . . . . . . . 41513.6.2 Maxwell Equations in Terms of Fi j . . . . . . . . . . . . . . . . . 41713.6.3 The Invariants of the Electromagnetic Field . . . . . . . . . . 418

13.7 The Physical Significance of the Electromagnetic Invariants . . . . . 42113.7.1 The Case Y = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42213.7.2 The Case Y �= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

xvi Contents

13.8 Motion of a Charge in an Electromagnetic Field – The LorentzForce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

13.9 Motion of a Charge in a Homogeneous Electromagnetic Field . . . 42913.9.1 The Case of a Homogeneous Electric Field . . . . . . . . . . 43013.9.2 The Case of a Homogeneous Magnetic Field . . . . . . . . . 43413.9.3 The Case of Two Homogeneous Fields of Equal

Strength and Perpendicular Directions . . . . . . . . . . . . . . 43613.9.4 The Case of Homogeneous and Parallel Fields E ‖ B . . 438

13.10 The Relativistic Electric and Magnetic Fields . . . . . . . . . . . . . . . . . 44013.10.1 The Levi-Civita Tensor Density . . . . . . . . . . . . . . . . . . . . 44013.10.2 The Case of Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44213.10.3 The Electromagnetic Theory for a General Medium . . . 44513.10.4 The Electric and Magnetic Moments . . . . . . . . . . . . . . . . 44813.10.5 Maxwell Equations for a General Medium . . . . . . . . . . . 44813.10.6 The 1+ 3 Decomposition of Maxwell Equations . . . . . . 449

13.11 The Four-Current of Conductivity and Ohm’s Law . . . . . . . . . . . . . 45413.11.1 The Continuity Equation J a

;a = 0for an Isotropic Material . . . . . . . . . . . . . . . . . . . . . . . . . . 458

13.12 The Electromagnetic Field in a Homogeneousand Isotropic Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

13.13 Electric Conductivity and the Propagation Equation for Ea . . . . . . 46313.14 The Generalized Ohm’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46513.15 The Energy Momentum Tensor of the Electromagnetic Field . . . . 46713.16 The Electromagnetic Field of a Moving Charge . . . . . . . . . . . . . . . 475

13.16.1 The Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47713.16.2 The Fields Ei , Bi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47813.16.3 The Lienard–Wiechert Potentials and the Fields E, B . . 478

13.17 Special Relativity and Practical Applications . . . . . . . . . . . . . . . . . . 48913.18 The Systems of Units SI and Gauss in Electromagnetism . . . . . . . 492

14 Relativistic Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49514.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49514.2 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495

14.2.1 1+ 3 Decomposition of a Bivector Xab . . . . . . . . . . . . . 49514.3 The Derivative of Xab Along the Vector pa . . . . . . . . . . . . . . . . . . . 49814.4 The Angular Momentum in Special Relativity . . . . . . . . . . . . . . . . . 500

14.4.1 The Angular Momentum in Newtonian Theory . . . . . . . 50014.4.2 The Angular Momentum of a Particle in Special

Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50214.5 The Intrinsic Angular Momentum – The Spin Vector . . . . . . . . . . . 506

14.5.1 The Magnetic Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50614.5.2 The Relativistic Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51014.5.3 Motion of a Particle with Spin in a Homogeneous

Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51514.5.4 Transformation of Motion in Σ . . . . . . . . . . . . . . . . . . . . 517

Contents xvii

15 The Covariant Lorentz Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 52115.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52115.2 The Covariant Lorentz Transformation . . . . . . . . . . . . . . . . . . . . . . . 523

15.2.1 Definition of the Lorentz Transformation . . . . . . . . . . . . 52315.2.2 Computation of the Covariant Lorentz Transformation . 52415.2.3 The Action of the Covariant Lorentz Transformation

on the Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52915.2.4 The Invariant Length of a Four-Vector . . . . . . . . . . . . . . . 534

15.3 The Four Types of the Lorentz Transformation Viewedas Spacetime Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534

15.4 Relativistic Composition Rule of Four-Vectors . . . . . . . . . . . . . . . . 53715.4.1 Computation of the Composite Four-Vector . . . . . . . . . . 54015.4.2 The Relativistic Composition Rule for Three-Velocities 54215.4.3 Riemannian Geometry and Special Relativity . . . . . . . . 54415.4.4 The Relativistic Rule for the Composition

of Three-Accelerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 54815.5 The Composition of Lorentz Transformations . . . . . . . . . . . . . . . . . 550

16 Geometric Description of Relativistic Interactions . . . . . . . . . . . . . . . . . 55516.1 Collisions and Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55516.2 Geometric Description of Collisions in Newtonian Physics . . . . . . 55616.3 Geometric Description of Relativistic Reactions . . . . . . . . . . . . . . . 55816.4 The General Geometric Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559

16.4.1 The 1+3 Decomposition of a Particle Four-Vectorwrt a Timelike Four-Vector . . . . . . . . . . . . . . . . . . . . . . . 561

16.5 The System of Two to One Particle Four-Vectors . . . . . . . . . . . . . . 56316.5.1 The Triangle Function of a System of Two Particle

Four-Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56516.5.2 Extreme Values of the Four-Vectors (A ± B)2 . . . . . . . . 56716.5.3 The System Aa, Ba, (A + B)a of Particle

Four-Vectors in CS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56816.5.4 The System Aa, Ba, (A + B)a in the Lab . . . . . . . . . . . . 570

16.6 The Relativistic System Aa + Ba → Ca + Da . . . . . . . . . . . . . . . . 57416.6.1 The Reaction B −→ C + D . . . . . . . . . . . . . . . . . . . . . . . 587

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591