Special Theory of Relativity 0828516650, 9780828516655

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SPECIAL THEORY OF RELATIVITY VA. UGAROV

CnEUV!AJlbHA51 TEOPVI51 OTHOCVITEJlbHOCTY1 B. A. YfAPOB

H3,nATEJibCTBO «HAYKA:. MOCKBA

SPECIAL THEORY OF RELATIVITY V.A.UGAROV TRA:-.ISLATED

FROM THE IWSSIA"l'

BY YURI ATANOV

MIR PUBLISHERS MOSCOW

Firs! published 1979 Revised from the 1977 Russian edition

Ha QN2AUIICKOM RSI>!Ke

@

f.IIIBH&Il peAIHUH11 $H3HKO•M&TeMITH'IeCKOII: JIBTepaT)'pW H3AITellbCTBa cHayKu, 1977

© English 1ranslal ion, Mir Publishers, 1979

PREFACE It gives me pleasure to thank B. M. Bolotovsky and S. N. Sto-

lyarov who have writtC'n §§ 6.14, 6.15 of this book. I wish to express my special gratitude to V. L. Ginzburg. This book quotes many things which I learned at the seminar led by him. A few

questions were discussed with him directly; in particular. the problem of an energy-momentum-tension tensor should be mentioned. Finally, V. L. Ginzburg has written the article "Who Developed the Special Theory or Relativity, and How?" to he published in this book (Supplement 1). In my opinion, this article gh·es very precise answers to questions which would be met by

anyone interested in the history of the STR evolution. I feel myselr honoured to have this article included in the book. The author

CONTENTS !Preface

Chapter I. CLASSICAL MECHANICS AND THE PRINCIPLE OF RELATIVITY • • • II § § § § § § § § §

1.1. 1.2. I 3. 1.4. I 5. 1.6. 1.7. 1.8. 1.9

A coordinate system and a reference frame in classical mechanics The choice nf a reference lr3me. Tile Galilean transformation . The Galilean principlt:> of relativity. Newlon's second law. Newton's laws and inertial frames of reference. Absolute lime and absolule space How physics was approaching the theory of relativity • • The generalization of lhe Galilean principle of relativity • The velocity of light in vacuo . .

II 14 15 19 24 29 30 33 36

Chapter 2 THE EINSTEIN POSTULATES THE INTERVAL BETWEEN EVENTS. THE LORENTZ TRANSFORMATION .

38

§ 2.1. Einstein's postulates 38 § 2 2. The relalivislic frame of reference 41 § 2.3. The direct consequences of Einstein's postulates (a few imaginary 45 experiments) § 2 4 The relalivily of synchronization of clocks belonging to two inertial fran1es of reference. The direct derivation of the Lorentz 52 transfornlation § 2.5. The Lorentz transformation as a consequence of Einstein's pos~~

M

§ 2.6. The propagation of the light wave profile. An interval between -§ 2.7. The-Lorentz transformalion as a consequence of the invariance M • . • , . • • of the interval between events . • 63 -§ 2.8. Complex values in the STR. Symmetric designations . . • . . 65 '§ 2.9. A geometric illustration of the Lorentz transformation • • , , 69

Contents

Chapter 3. CONSEQUENCES OF THE LORENTZ TRANSFORMATION. THE CLASSIFICATION OF INTERVALS AND THE PRINCI· PLE OF CAUSALITY. THE K CALCULUS . 71

§ 3.1. On the measurement of lengths and time inlen·als. The relativity 71 of simultaneity § 32. Relativity of length of moving rulers (scales). A vis1ble shape of obj&ts moving at relativistic velocihes . 74 § 3.3. Relativity of lime inte1 vals between events . 83 § 3.4. The classification of intervals and the principle of causahty 90§ 3 5. The transformation of velocity components of a particle on Iran· sition from one inertial frame of reference to another . . . . . 94 § 3.6. The transformation of an absolute value and the direction of the velocity of a particle. , !Cl § 3.7. The K calculus (the radar method) . • 105Chapter 4. THE FOUR-DIMENSIONAL SPACE-TIME

• 117

§ 4.1. Three-dimensional and four-dimensional Euclidean spaces. , § 4..2. The 4-space-lime, or the four-dimensional pseudo-Euclidean space § 4.3. 4-vectors and 4-tensors • § 4.4. A pseudo-Euclidean plane . . . • . . . • Chapter 5. RELATIVISTIC MECHANICS OF A PARTICLE

, 133

§ 5.1. A 4-velocity and 4-acceleration. . . § 52. A 4-force and a four-dimensional equahon of motion . . § 53. A three-dimensional relativistic equation of motion of a particle . (the second law of Newton in a relativistic form) . § 5.4. The relativistic express1on for a particle's energy . . § 55. A 4-vector of energy-momentum . • § 5.6. The rest mass of a system. The binding energy . • § 5.7. Some problems of relativistic DJechanics of a particle . § 5.8. The conservation laws of relativistic mechanics. ,

Chapter 6. THE MAXWELL THEORY IN A RELATIVISTIC FORM.

117 I 18 120 123-

134 140 143 149 153 157 161 175-

, 180

6.1. The three-dimensional system of Maxwell's equations. A 4-potential and 4-current . . 6 2. The transformation of a 4-potential and 4-current . • • • • , 6 3. An electromagnetic !ield tensor • 6 4. The transformation of electric and magnetic !ield components , 6 5. The electromagnetic 6eld invariants . . , 66. The Lorentz force, , •• , , 6.7. Covariance of the S}'stem of the Maxwell equations • , • 6.8 The Minkowski equations for moving media (the transformation . oJ material equations) , • , . , , • • • , • • • , • •

181 184 188 192 193 199 205 208-

Contents

§ 6.9 The transformation of electric and magnetic moments . . § 6 10 Some problems involving the transformation of an electromag. . . . . . • netic field § 611. An en.ergy-momentum-tension tensor ol an electromagnetic field in vacuo . . . . . . . . . § 6 12. An energy-momentum-tension tensor of an electromagnetiC field in a medium. The Minkowski tensor and Abraham tensor . . § 6 t3. An energy-momentum-tension tensor of a spherically symmetric

-~

214 216 222 233

.m medium . . . 210

§ 6.14. The field potentials in a moving non-conducting § 6.t5. The field potentials in a moving conducting medium

. . . . 246

Chapter 7. OPTICAL PHENOMENA AND THE SPECIAL THEORY OF . :153 R,ELATIVITY

§ 7.1. Properties ol plane light waves . § 7.2. A 4-wave vector. The Doppler effect. Aberration of § 7.3. A plane wave limited in space. The transformation . wave energy and amplitude . . . • . § 7.4. The pressure exerted by an electromagnetic wave • • . . . • • . surface § 7.5. The light frequency variation on renection from a

-~~

• • . light . of the plane . (light) on a . moving sur-

258 2fi1 2 5 210

.m

§ 7.6. Light quanta (photons) as relativistic particles . 276 § 7.7. Light quanta in a medium. The Vavilov-Cherenkov effect. The anomalous Doppler effect • . . • • , • • . • • • • • . 280

Chapler 8. ON CERTAIN PARADOXES OF THE SPECIAL THEORY OF . 286 RELATIVITY

§ 8 I. § 82 § 8 3. § 8-1. § 8.5.

Faster-than-light veloC'ities • . The thread-and-lever paradox • • • • Tho! tachyons • . The clock paradox • . . . . The "equivalence" of mass and energy. The zero rest mass . .

SUPPLE.'o1ENT

287 292 297 303 3t0

. 3t7

I. Who

developed the special theory of relativity, and how, L. GmzbJr·)) . The unsuccessful senrch for a medium for the propagation of light Was Michelson's experiment "decisive" for the creation of the special theory of relativity? . . . . . . . . . Why shouldn't the mass-velocity dependence, or the relativtslic mass, be introduced? . . , . Non-inertial fram~s of reference. The :;pecial theory ot relativity and the advance to gravitational theury (the general theory of (~

II. III. IV. V.

--

3t7 328 345

350

to

Contents

MAIN EVENTS RELATED TO THE HISTORY OF THE STR .

• 361

Appendix 1.

• 362

• • • •

§ 1. The symmetric notation. The summation rules • § 2. The transformation of coordinates in the case of a rotation of the . Cartesian system of coordinates • • § 3. The tensors • § 4. The invariance of a 4-divergence and d'Alembert's operator. . § 5. The convolution ("rejuvenation") of tensor indices . . § 6. Some data on determinants. The dual tensors . . § 7. The stress tensor . • • • • • • . • § 8. The rectilinear oblique-angted systems of coordinates . • § 9. The definition of the hyperbolic functions and some relationships between them • • , , • Bibliography to Appendix I • • •

362 364368. 373375377 38338&

392"

. • • . • • 393-

Appendix II. The basic formulae of electrodynamics in the Gaussian system 394 Bibliography Index

• 399t

• . • . • . • • . . . • • . • . • . • . • • . • • • 40&

CHAPTER

r

CLASSICAL MECHANICS AND THE PRINCIPLE OF RELATIVITY

§ t.l. A coordinate system and a reference frame in classical mechanics. All natural phenomena happen in space and in the .course of time, and an element of any phenomenon is something Qccurring at a given moment of time and at a given point in space. In the special theory of relativity* it is customary to refer to that "something" taking place at a given point and at a given moment .of time (in fact, something concentrated in a sufficiently small volume of space and limited by a small time interval) as an event. This definition shows that concrete features of an event may be very different. That is why it is usual to indicate that "the event .consists in ... ". The examples of events can be the emission of a light signal from a certain point in space at a certain moment of time, or the presence of a moving particle (a material point) at .a given point in space and at a given moment of time. When an event is realized, one says that it "happened" (or is happening, or will happen). Any physical phenomenon represents a sequence of events. A description of a separate event serves as a basis for the description of any phenomenon and therefore we begin with the description of a separate event. To characterize a point in space where an event occurred, every point in space has to be labelled before specific physical phenomena arc analysed. But space is uniform and isotropic and this implies that all points in space and all directions in it are equal. It should be pointed out at once that we deal here with the free space, or vacuum. The investigation of physical phenomena in uacuo is of prime importance for the special theory of relativit}'. Ewn though ,·acuum is a complex physical system, it is sufficient for our purpose to assume that in the space domain which we take for vacuum, no substance possessing a finite rest mass is practically present and gravitational and electric fields are not too strong. But even when all points in space are equal, one can still smgle out a certain point by placing a material object, i.e. an

~fter

the complete term uspedal theory of relativity" will be some· LLmes abbreviated as STR.

12

Spedal Theory of Relativity

object having a finite rest mass, in it. Points in space are usually labelled by means of a coordinate system. With the help of the material object we distinguish a point which is the origin of co· ordinates The simplest coordinate system is the Cartesian syo;. tern. Its construction begins with the tracing of three mutu.t!ly perpendicular straight lines, i.e. the coordmnte X, Y, Z axes. !11 terms of physics, however, these are not just abstract

strarg~rt

lines. Theoretically, the coordinate axes are rigid non-deformable • solids. By the way, instruments, standards and other objects of a given reference frame \\"ill be always fixed to them and therefore it should be borne in mind that a physical coordinate system is always a material object. In the Cartesian coordinate system points are quite easy to label. From any point M in space one can construct the perpen· diculars to the X, Y, Z axes or, in other words, project this point on the coordinate axes. Having measured the distances of the point projections from the origin along the X, Y, Z axes by means of the chosen scale, 've obtain the numbers x, y, z, which are called the Cartesian coordinates of the point. The distances can be mea':>· ured via the step-by-step transposition of a unit scale along the axis from the origin to the point projection on the axis. In fad, such a procedure used for length measurements in everyday life can also be used for determining the length of a stretch or an object if it is at rest in a given coordinate system. As we shall see later, the special theory of relativity furnishes a very convenient method of measuring distances without recourse to rigid scales and their step-by-step transposition (see Chapter 2). Both methods are equivalent, of course. Thus through the introduction of the Cartesian coordinate S\ stem every point in space acquires three numbers, that is the thi-ee Cartesian coordinates x, y, z. The principal objective of physics, however, is to study motion. Although mechanical motion is th•.! simplest type of motion, its description requires time measurements and therefore the coordinate system has to be of necesstly supplemented by a clock. This clock is needed to register tile occurrence of events at various points in space. How many clocl-s are needed? In classical mechanics they do not usually hesitate over the answer to this question and l· sociated with an interaction between bodies. Let us examine one useful, even though very plain example. Let a body be at rest in an inertial frame of reference K. Th~:n according to the second law of Newton no forces act on this body. Without touching it let us consider it from the viewpoint of an observer moving relative to the frame K with an acceleration a. This observer will note that the body in question moves relative to him with an acceleration -a. If the second law of Newton were valid in his frame, he could say that the body experiences th~ force -ma. But we know from the observer in an inertial frame of reference that there is no force acting on the body. Therefore, the second law of Newton is merely not valid in the reference frame moving relative to the inertial frame with an acceleration. Many readers have already realized, of course, that passing into the reference frame moving at an accelerating velocity, we detect "a force of inertia" which is not adually a force in Newtonian me· chanics (see Supplement V). Since the Jaws of Newton are not valid in all reference frames, Newton had to point out that a cer· lain reference frame was available in which all these laws were

Special Theory of Relativity

26

valid. And the first law of Newton is, in fact, equivalent to thi