Special Theory of Relativity

410 Pages·1979·18.212 MB·English

by  Vladimir A. Ugarov

SPECIAL
THEORY
OF
RELATIVITY
VA. UGAROV

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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
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© 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 ex
press 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 men
tioned. 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 RELA-
TIVITY • • • II
§ 1.1. A coordinate system and a reference frame in classical mechanics II
§ 1.2. The choice nf a reference lr3me. 14
§ I 3. Tile Galilean transformation . 15
§ 1.4. The Galilean principlt:> of relativity. Newlon's second law. 19
§ I 5. Newton's laws and inertial frames of reference. 24
§ 1.6. Absolute lime and absolule space 29
§ 1.7. How physics was approaching the theory of relativity • • 30
§ 1.8. The generalization of lhe Galilean principle of relativity • 33
§ 1.9 The velocity of light in vacuo . . 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
experiments) 45
§ 2 4 The relalivily of synchronization of clocks belonging to two iner-
tial fran1es of reference. The direct derivation of the Lorentz
transfornlation 52
§ 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
of simultaneity 71
§ 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) . • 105-
Chapter 4. THE FOUR-DIMENSIONAL SPACE-TIME • 117
§ 4.1. Three-dimensional and four-dimensional Euclidean spaces. , 117
§ 4..2. The 4-space-lime, or the four-dimensional pseudo-Euclidean space I 18
§ 4.3. 4-vectors and 4-tensors • 120
§ 4.4. A pseudo-Euclidean plane . . . • . . . • 123-
Chapter 5. RELATIVISTIC MECHANICS OF A PARTICLE , 133
§ 5.1. A 4-velocity and 4-acceleration. . . 134
§ 52. A 4-force and a four-dimensional equahon of motion . . 140
§ 53. A three-dimensional relativistic equation of motion of a particle
(the second law of Newton in a relativistic form) . . 143
§ 5.4. The relativistic express1on for a particle's energy . . 149
§ 55. A 4-vector of energy-momentum . • 153
§ 5.6. The rest mass of a system. The binding energy . • 157
§ 5.7. Some problems of relativistic DJechanics of a particle . 161
§ 5.8. The conservation laws of relativistic mechanics. , 175-
Chapter 6. THE MAXWELL THEORY IN A RELATIVISTIC FORM. , 180
6.1. The three-dimensional system of Maxwell's equations. A 4-poten-
tial and 4-current . . 181
6 2. The transformation of a 4-potential and 4-current . • • • • , 184
6 3. An electromagnetic !ield tensor • 188
6 4. The transformation of electric and magnetic !ield components , 192
6 5. The electromagnetic 6eld invariants . . , 193
66. The Lorentz force, , •• , , 199
6.7. Covariance of the S}'stem of the Maxwell equations • , • 205
6.8 The Minkowski equations for moving media (the transformation
. oJ material equations) , • , . , , • • • , • • • , • • 208-
Contents
§ 6.9 The transformation of electric and magnetic moments . . 214
§ 6 10 Some problems involving the transformation of an electromag-
netic field . . . . . . • 216
§ 611. An en.ergy-momentum-tension tensor ol an electromagnetic field
in vacuo . . . . . . . . . 222
§ 6 12. An energy-momentum-tension tensor of an electromagnetiC field
in a medium. The Minkowski tensor and Abraham tensor . . 233
§ 6 t3. An energy-momentum-tension tensor of a spherically symmetric
-~ .m
§ 6.14. The field potentials in a moving non-conducting medium ... 210
§ 6.t5. The field potentials in a moving conducting medium .... 246
Chapter 7. OPTICAL PHENOMENA AND THE SPECIAL THEORY OF
R,ELATIVITY . :153
§ 7.1. Properties ol plane light waves . • • . 258
§ 7.2. A 4-wave vector. The Doppler effect. Aberration of light . 2fi1
§ 7.3. A plane wave limited in space. The transformation of the plane
wave energy and amplitude . . . • . . . 2 5
§ 7.4. The pressure exerted by an electromagnetic wave (light) on a
surface • • . . . • • . . 210
§ 7.5. The light frequency variation on renection from a moving sur-
-~~ .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
RELATIVITY . 286
§ 8 I. Faster-than-light veloC'ities • . 287
§ 8 2 The thread-and-lever paradox • • • • 292
§ 8 3. Tho! tachyons • . 297
§ 8-1. The clock paradox • . . . . 303
§ 8.5. The "equivalence" of mass and energy. The zero rest mass . . 3t0
SUPPLE.'o1ENT . 3t7
I. Who developed the special theory of relativity, and how,
(~ L. GmzbJr·)) . 3t7
II. The unsuccessful senrch for a medium for the propagation of light 328
III. Was Michelson's experiment "decisive" for the creation of the spe-
-cial -theory of relativity? . . . . . . . . . 345
IV. Why shouldn't the mass-velocity dependence, or the relativtslic -
mass, be introduced? . . , . 350
V. Non-inertial fram~s of reference. The :;pecial theory ot relativity
and the advance to gravitational theury (the general theory of