Special relativity in general frames: from particles to astrophysics
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| Format: | Buch |
| Sprache: | English French |
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Berlin ; Heidelberg
Springer
2013
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| Schriftenreihe: | Graduate texts in physics
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| Beschreibung: | Literaturverzeichnis Seite 741-759 |
| Beschreibung: | XXX, 784 Seiten Illustrationen, Diagramme 235 mm x 155 mm |
| ISBN: | 3642372759 9783642372759 |
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| 240 | 1 | 0 | |a Relativité restreinte <engl.> |
| 245 | 1 | 0 | |a Special relativity in general frames |b from particles to astrophysics |c Éric Gourgoulhon |
| 264 | 1 | |a Berlin ; Heidelberg |b Springer |c 2013 | |
| 300 | |a XXX, 784 Seiten |b Illustrationen, Diagramme |c 235 mm x 155 mm | ||
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| 490 | 0 | |a Graduate texts in physics | |
| 500 | |a Literaturverzeichnis Seite 741-759 | ||
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Datensatz im Suchindex
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CONTENTS
1 MINKOWSKI SPACETIME 1
1.1 INTRODUCTION 1
1.2 THE FOUR DIMENSIONS 1
1.2.1 SPACETIME AS AN AFFINE SPACE 1
1.2.2 A FEW NOTATIONS 3
1.2.3 AFFINE COORDINATE SYSTEM 4
1.2.4 CONSTANT C 4
1.2.5 NEWTONIAN SPACETIME 5
1.3 METRIC TENSOR 6
1.3.1 SCALAR PRODUCT ON SPACETIME 6
1.3.2 MATRIX OF THE METRIC TENSOR 9
1.3.3 ORTHONORMAL BASES 10
1.3.4 CLASSIFICATION OF VECTORS WITH RESPECT TO G 11
1.3.5 NORM OF A VECTOR 11
1.3.6 SPACETIME DIAGRAMS 12
1.4 NULL CONE AND TIME ARROW 15
1.4.1 DEFINITIONS 15
1.4.2 TWO USEFUL LEMMAS 16
1.4.3 CLASSIFICATION OF UNIT VECTORS 17
1.5 SPACETIME ORIENTATION 20
1.6 VECTOR/LINEAR FORM DUALITY 22
1.6.1 LINEAR FORMS AND DUAL SPACE 22
1.6.2 METRIC DUALITY 23
1.7 MINKOWSKI SPACETIME 25
1.8 BEFORE GOING FURTHER 27
2 WORLDLINES AND PROPER TIME 29
2.1 INTRODUCTION 29
2.2 WORLDLINE OF A PARTICLE 29
XVII
HTTP://D-NB.INFO/1031595481
XV
III CONTENTS
2.3 PROPER TIME 31
2.3.1 DEFINITION 31
2.3.2 IDEAL CLOCK 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 NULL GEODESIES 39
2.5.2 LIGHT CONE 39
2.6 LANGEVIN'S TRAVELLER AND TWIN PARADOX 40
2.6.1 TWINS' WORLDLINES 41
2.6.2 PROPER TIME OF EACH TWIN 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 EXPERIMENTAL VERIFICATIONS , 54
2.7 GEOMETRICAL PROPERTIES OF A WORLDLINE 57
2.7.1 TIMELIKE GEODESIES 57
2.7.2 VECTOR FIELD ALONG A WORLDLINE 59
2.7.3 CURVATURE AND TORSIONS 59
3 OBSERVERS 63
3.1 INTRODUCTION 63
3.2 SIMULTANEITY AND MEASURE OF TIME 63
3.2.1 THE PROBLEM 63
3.2.2 EINSTEIN-POINCARE SIMULTANEITY 64
3.2.3 LOCAL REST SPACE 66
3.2.4 NONEXISTENCE OF ABSOLUTE TIME 69
3.2.5 ORTHOGONAL PROJECTOR ONTO THE LOCAL REST SPACE 70
3.2.6 EUCLIDEAN CHARACTER OF THE LOCAL REST SPACE 72
3.3 MEASURING SPATIAL DISTANCES 73
3.3.1 SYNGE FORMULA 73
3.3.2 BORA'S RIGIDITY CRITERION 75
3.4 LOCAL FRAME 76
3.4.1 LOCAL FRAME OF AN OBSERVER 76
3.4.2 COORDINATES WITH RESPECT TO AN OBSERVER 78
3.4.3 REFERENCE SPACE OF AN OBSERVER 79
3.5 FOUR-ROTATION OF A LOCAL FRAME 81
3.5.1 VARIATION OF THE LOCAL FRAME ALONG THE WORLDLINE 81
3.5.2 ORTHOGONAL DECOMPOSITION OF ANTISYMMETRIC
BILINEAR FORMS 83
3.5.3 APPLICATION TO THE VARIATION OF THE LOCAL FRAME 86
3.5.4 INERTIAL OBSERVERS 88
CONTENTS XIX
3.6 DERIVATIVE OF A VECTOR FIELD ALONG A WORLDLINE 89
3.6.1 ABSOLUTE DERIVATIVE 89
3.6.2 DERIVATIVE WITH RESPECT TO AN OBSERVER 90
3.6.3 FERMI-WALKER DERIVATIVE 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 LORENTZ FACTOR 95
4.2.1 DEFINITION 95
4.2.2 EXPRESSION IN TERMS OF THE 4-VELOCITY
AND THE 4-ACCELERATION 98
4.2.3 TIME DILATION .' 100
4.3 VELOCITY RELATIVE TO AN OBSERVER 101
4.3.1 DEFINITION 101
4.3.2 4-VELOCITY AND LORENTZ FACTOR IN TERMS
OF THE VELOCITY 103
4.3.3 MAXIMUM RELATIVE VELOCITY 106
4.3.4 COMPONENT EXPRESSIONS 107
4.4 EXPERIMENTAL VERIFICATIONS OF TIME DILATION 108
4.4.1 ATMOSPHERIC MUONS 108
4.4.2 OTHER TESTS 110
4.5 ACCELERATION RELATIVE TO AN OBSERVER ILL
4.5.1 DEFINITION ILL
4.5.2 RELATION TO THE SECONDW DERIVATIVE
OF THE POSITION VECTOR ILL
4.5.3 EXPRESSION OF THE 4-ACCELERATION 114
4.6 PHOTON MOTION 118
4.6.1 PROPAGATION DIRECTION OF A PHOTON 118
4.6.2 VELOCITY OF LIGHT 120
4.6.3 EXPERIMENTAL TESTS OF THE INVARIANCE
OF THE VELOCITY OF LIGHT 123
5 KINEMATICS 2: CHANGE OF OBSERVER 131
5.1 INTRODUCTION 131
5.2 RELATIONS BETWEEN TWO OBSERVERS 131
5.2.1 RECIPROCITY OF THE RELATIVE VELOCITY 131
5.2.2 LENGTH CONTRACTION 134
5.3 LAW OF VELOCITY COMPOSITION 136
5.3.1 GENERAL FORM 136
5.3.2 DECOMPOSITION IN PARALLEL AND TRANSVERSE PARTS 139
5.3.3 COLLINEAR VELOCITIES 142
5.3.4 ALTERNATIVE FORMULA 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 EXPERIMENTAL VERIFICATIONS 151
5.6 ABERRATION 152
5.6.1 THEORETICAL EXPRESSION 152
5.6.2 DISTORTION OF THE CELESTIAL SPHERE 155
5.6.3 EXPERIMENTAL VERIFICATIONS 157
5.7 IMAGES OF MOVING OBJECTS 158
5.7.1 IMAGE AND INSTANTANEOUS POSITION 158
5.7.2 APPARENT ROTATION 158
5.7.3 IMAGE OF A SPHERE 160
5.7.4 SUPERLUMINAL MOTION'S 163
6 LORENTZ GROUP 167
6.1 INTRODUCTION 167
6.2 LORENTZ TRANSFORMATIONS 167
6.2.1 DEFINITION AND CHARACTERIZATION 167
6.2.2 LORENTZ GROUP 169
6.2.3 PROPERTIES OF LORENTZ TRANSFORMATIONS 170
6.3 SUBGROUPS OF 0(3,1) 172
6.3.1 PROPER LORENTZ GROUP SO(3,L) 172
6.3.2 ORTHOCHRONOUS LORENTZ GROUP 173
6.3.3 RESTRICTED LORENTZ GROUP 174
6.3.4 REDUCTION OF THE LORENTZ GROUP TO S0
0
(3, 1) 174
6.4 CLASSIFICATION OF RESTRICTED LORENTZ TRANSFORMATIONS 176
6.4.1 INVARIANT NULL DIRECTION 176
6.4.2 DECOMPOSITION WITH RESPECT TO AN INVARIANT
NULL DIRECTION 178
6.4.3 SPATIAL ROTATIONS 181
6.4.4 LORENTZ BOOSTS 183
6.4.5 NULL ROTATIONS 185
6.4.6 FOUR-SCREWS 188
6.4.7 EIGENVECTORS OF A RESTRICTED LORENTZ TRANSFORMATION 189
6.4.8 SUMMARY 190
6.5 POLAR DECOMPOSITION 191
6.5.1 STATEMENT AND DEMONSTRATION 191
6.5.2 EXPLICIT FORMS 194
6.6 PROPERTIES OF LORENTZ BOOSTS 195
6.6.1 KINEMATICAL INTERPRETATION 195
6.6.2 EXPRESSION IN A GENERAL BASIS 198
6.6.3 RAPIDITY 199
6.6.4 EIGENVALUES 202
6.7 COMPOSITION OF BOOSTS AND THOMAS ROTATION 202
6.7.1 COPLANAR BOOSTS 204
6.7.2 THOMAS ROTATION 206
CONTENTS XXI
6.7.3 THOMAS ROTATION ANGLE 212
6.7.4 CONCLUSION 216
7 LORENTZ GROUP AS A LIE GROUP 217
7.1 INTRODUCTION 217
7.2 LIE GROUP STRUCTURE 217
7.2.1 DEFINITIONS 217
7.2.2 DIMENSION OF THE LORENTZ GROUP 219
7.2.3 TOPOLOGY OF THE LORENTZ GROUP 220
7.3 GENERATORS AND LIE ALGEBRA 221
7.3.1 INFINITESIMAL LORENTZ TRANSFORMATIONS 221
7.3.2 STRUCTURE OF LIE ALGEBRA 222
7.3.3 GENERATORS 224
7.3.4 LINK WITH THE VARIATION OF A LOCAL FRAME 227
7.4 REDUCTION OF 0(3,1) TO ITS LIE ALGEBRA 228
7.4.1 EXPONENTIAL MAP 228
7.4.2 GENERATION OF LORENTZ BOOSTS 231
7.4.3 GENERATION OF SPATIAL ROTATIONS 233
7.4.4 STRUCTURE CONSTANTS 234
7.5 RELATIONS BETWEEN THE LORENTZ GROUP AND SL(2,C) 237
7.5.1 SPINORMAP 237
7.5.2 THE SPINOR MAP FROM SU(2) TO SO(3) 243
7.5.3 THE SPINOR MAP AND LORENTZ BOOSTS 247
7.5.4 COVERING OF THE RESTRICTED LORENTZ GROUP BY SL(2,C) . 248
7.5.5 EXISTENCE OF NULL EIGENVECTORS 249
7.5.6 LIE ALGEBRA OF SL(2,C) 250
7.5.7 EXPONENTIAL MAP ON SL(2,C) 254
8 INERTIAL OBSERVERS AND POINCARE GROUP 257
8.1 INTRODUCTION 257
8.2 CHARACTERIZATION OF INERTIAL OBSERVERS 257
8.2.1 DEFINITION 257
8.2.2 WORLDLINE 258
8.2.3 GLOBALITY OF THE LOCAL REST SPACE 259
8.2.4 RIGID ARRAY OF INERTIAL OBSERVERS 260
8.3 POINCARE GROUP 261
8.3.1 CHANGE OF INERTIAL COORDINATES 261
8.3.2 ACTIVE POINCARE TRANSFORMATIONS 263
8.3.3 GROUP STRUCTURE 264
8.3.4 THE POINCARE GROUP AS A LIE GROUP 266
9 ENERGY AND MOMENTUM 271
9.1 INTRODUCTION 271
9.2 FOUR-MOMENTUM, MASS AND ENERGY 271
9.2.1 FOUR-MOMENTUM AND MASS OF A PARTICLE 271
9.2.2 ENERGY AND MOMENTUM RELATIVE TO AN OBSERVER 273
XXII CONTENTS
9.2.3 CASE OF A MASSIVE PARTICLE 276
9.2.4 ENERGY AND MOMENTUM OF A PHOTON 280
9.2.5 RELATION BETWEEN P, E AND THE RELATIVE VELOCITY 281
9.2.6 COMPONENTS OF THE 4-MOMENTUM 281
9.3 CONSERVATION OF 4-MOMENTUM 282
9.3.1 4-MOMENTUM OF A PARTICLE SYSTEM 282
9.3.2 ISOLATED SYSTEM AND PARTICLE COLLISIONS 284
9.3.3 PRINCIPLE OF 4-MOMENTUM CONSERVATION 285
9.3.4 APPLICATION TO AN ISOLATED PARTICLE: LAW OF INERTIA 286
9.3.5 4-MOMENTUM OF AN ISOLATED SYSTEM 288
9.3.6 ENERGY AND LINEAR MOMENTUM OF A SYSTEM 291
9.3.7 APPLICATION: DOPPLER EFFECT 293
9.4 PARTICLE COLLISIONS 294
9.4.1 LOCALIZED INTERACTIONS 294
9.4.2 COLLISION BETWEEN TWO PARTICLES 294
9.4.3 ELASTIC COLLISION 295
9.4.4 COMPTON EFFECT 301
9.4.5 INVERSE COMPTON SCATTERING 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 FORCE MEASURED BY AN OBSERVER 314
9.5.4 RELATIVISTIC VERSION OF NEWTON'S SECOND LAW 316
9.5.5 EVOLUTION OF ENERGY 317
9.5.6 EXPRESSION OF THE 4-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 CHANGE OF ORIGIN 324
10.3.3 ANGULAR MOMENTUM VECTOR AND MASS-ENERGY
DIPOLE MOMENT 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
VECTOR RELATIVE TO AN INERTIAL OBSERVER 328
CONTENTS XXIII
10.5 CENTRE OF INERTIA AND SPIN 329
10.5.1 CENTROID OF A SYSTEM 329
10.5.2 CENTRE OF INERTIA OF AN ISOLATED SYSTEM 330
10.5.3 SPIN OF AN ISOLATED SYSTEM 333
10.5.4 KONIG THEOREM 334
10.5.5 MINIMAL SIZE OF A SYSTEM WITH SPIN 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 PARTICLE WITH SPIN 342
10.7.1 DEFINITION 342
10.7.2 SPIN EVOLUTION 345
10.7.3 FREE GYROSCOPE 346
10.7.4 BMT EQUATION 347
11 PRINCIPLE OF LEAST ACTION 349
11.1 INTRODUCTION 349
11.2 PRINCIPLE OF LEAST ACTION FOR A PARTICLE 349
11.2.1 REMINDER OF NONRELATIVISTIC LAGRANGIAN MECHANICS 349
11.2.2 RELATIVISTIC GENERALIZATION 350
11.2.3 LAGRANGIAN AND ACTION FOR A PARTICLE 351
11.2.4 PRINCIPLE OF LEAST ACTION 352
11.2.5 ACTION OF A FREE PARTICLE 354
11.2.6 PARTICLE IN A VECTOR FIELD 357
11.2.7 OTHER EXAMPLES OF LAGRANGIANS 358
11.3 NOETHER THEOREM 360
11.3.1 NOETHER THEOREM FOR A PARTICLE 360
11.3.2 APPLICATION TO A FREE PARTICLE 362
11.4 HAMILTONIAN FORMULATION 365
11.4.1 REMINDER OF NONRELATIVISTIC HAMILTONIAN MECHANICS . 365
11.4.2 GENERALIZED FOUR-MOMENTUM OF A RELATIVISTIC
PARTICLE 369
11.4.3 HAMILTONIAN OF A RELATIVISTIC PARTICLE 371
11.5 SYSTEMS OF PARTICLES 374
11.5.1 PRINCIPLE OF LEAST ACTION 375
11.5.2 HAMILTONIAN FORMULATION 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 CHANGE OF THE REFERENCE INERTIAL OBSERVER 386
12.2.4 MOTION PERCEIVED BY THE INERTIAL OBSERVER 388
12.2.5 LOCAL REST SPACES 389
XXIV
CONTENTS
12.2.6 RINDLER HORIZON 391
12.2.7 LOCAL FRAME OF THE UNIFORMLY ACCELERATED OBSERVER . 393
12.3 DIFFERENCE BETWEEN THE LOCAL REST SPACE
AND THE SIMULTANEITY HYPERSURFACE 397
12.3.1 CASE OF A GENERIC OBSERVER 397
12.3.2 CASE OF A UNIFORMLY ACCELERATED OBSERVER 400
12.4 PHYSICS IN AN ACCELERATED FRAME 400
12.4.1 CLOCK SYNCHRONIZATION 400
12.4.2 4-ACCELERATION OF COMOVING OBSERVERS 404
12.4.3 RIGID RULER IN ACCELERATED MOTION 405
12.4.4 PHOTON TRAJECTORIES 408
12.4.5 SPECTRAL SHIFT 409
12.4.6 MOTION OF FREE PARTICLES 412
12.5 THOMAS PRECESSION 415
12.5.1 DERIVATION 415
12.5.2 APPLICATION TO A GYROSCOPE 421
12.5.3 GYROSCOPE IN CIRCULAR ORBIT 422
12.5.4 THOMAS EQUATION 423
13 ROTATING OBSERVERS 427
13.1 INTRODUCTION 427
13.2 ROTATION VELOCITY 427
13.2.1 PHYSICAL REALIZATION OF A NONROTATING OBSERVER 427
13.2.2 MEASUREMENT OF THE ROTATION VELOCITY 428
13.3 ROTATING DISK 429
13.3.1 UNIFORMLY ROTATING OBSERVER 429
13.3.2 COROTATING OBSERVERS 431
13.3.3 4-ACCELERATION AND 4-ROTATION
OF THE COROTATING OBSERVER 433
13.3.4 SIMULTANEITY FOR A COROTATING OBSERVER 436
13.4 CLOCK DESYNCHRONIZATION 439
13.4.1 INTRODUCTION 439
13.4.2 LOCAL SYNCHRONIZATION 440
13.4.3 IMPOSSIBILITY OF A GLOBAL SYNCHRONIZATION 442
13.4.4 CLOCK TRANSPORT ON THE ROTATING DISK 446
13.4.5 EXPERIMENTAL MEASURES OF THE DESYNCHRONIZATION 450
13.5 EHRENFEST PARADOX 453
13.5.1 CIRCUMFERENCE OF THE ROTATING DISK 453
13.5.2 DISK RADIUS 453
13.5.3 THE "PARADOX" 454
13.5.4 SETTING THE DISK INTO ROTATION 455
13.6 SAGNAC EFFECT 458
13.6.1 SAGNAC DELAY 459
13.6.2 ALTERNATIVE DERIVATION 461
13.6.3 PROPER TRAVELLING TIME FOR EACH SIGNAL 463
CONTENTS XXV
13.6.4 OPTICAL SAGNAC INTERFEROMETER 464
13.6.5 MATTER-WAVE SAGNAC INTERFEROMETER 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 TENSORS ALREADY MET 474
14.3 OPERATIONS ON TENSORS 475
14.3.1 TENSOR PRODUCT 475
14.3.2 COMPONENTS IN A VECTOR BASIS 476
14.3.3 CHANGE OF BASIS 477
14.3.4 COMPONENTS AND METRIC DUALITY 479
14.3.5 CONTRACTION 480
14.4 ALTERNATE FORMS 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 TENSORS ASSOCIATED WITH THE LEVI-CIVITA TENSOR 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 ARBITRARY COORDINATES ON SPACETIME 495
15.2.1 COORDINATE SYSTEM 495
15.2.2 COORDINATE BASIS 496
15.2.3 COMPONENTS OF THE METRIC TENSOR 498
15.3 TENSOR FIELDS 502
15.3.1 DEFINITIONS 502
15.3.2 SCALAR FIELD AND GRADIENT 503
15.3.3 GRADIENTS OF COORDINATES 504
15.4 COVARIANT DERIVATIVE 505
15.4.1 COVARIANT DERIVATIVE OF A VECTOR 505
15.4.2 GENERALIZATION TO ALL TENSORS 506
15.4.3 CONNECTION COEFFICIENTS 508
15.4.4 CHRISTOFFEL SYMBOLS 510
15.4.5 DIVERGENCE OF A VECTOR FIELD 512
15.4.6 DIVERGENCE OF A TENSOR FIELD 513
15.5 DIFFERENTIAL FORMS 513
15.5.1 DEFINITION ' 513
15.5.2 EXTERIOR DERIVATIVE 514
XXVI CONTENTS
15.5.3 PROPERTIES OF THE EXTERIOR DERIVATIVE 517
15.5.4 EXPANSION WITH RESPECT TO A COORDINATE SYSTEM 518
15.5.5 EXTERIOR DERIVATIVE OF A 3-FORM
AND DIVERGENCE OF A VECTOR FIELD 519
16 INTEGRATION IN SPACETIME 521
16.1 INTRODUCTION 521
16.2 INTEGRATION OVER A FOUR-DIMENSIONAL VOLUME 521
16.2.1 VOLUME ELEMENT 521
16.2.2 FOUR-VOLUME OF A PART OF SPACETIME 522
16.2.3 INTEGRAL OF A DIFFERENTIAL 4-FORM 523
16.3 SUBMANIFOLDS OF § 524
16.3.1 DEFINITION OF A SUBMANIFOLD 524
16.3.2 SUBMANIFOLD WITH BOUNDARY 526
16.3.3 ORIENTATION OF A SUBMANIFOLD 527
16.4 INTEGRATION ON A SUBMANIFOLD OF § 527
16.4.1 INTEGRAL OF ANY DIFFERENTIAL FORM 527
16.4.2 VOLUME ELEMENT OF A HYPERSURFACE 530
16.4.3 AREA ELEMENT OF A SURFACE 532
16.4.4 LENGTH-ELEMENT OF A CURVE 534
16.4.5 INTEGRAL OF A SCALAR FIELD ON A SUBMANIFOLD 535
16.4.6 INTEGRAL OF A TENSOR FIELD 536
16.4.7 FLUX INTEGRALS 536
16.5 STOKES' THEOREM 538
16.5.1 STATEMENT AND EXAMPLES 538
16.5.2 APPLICATIONS 540
17 ELECTROMAGNETIC FIELD 545
17.1 INTRODUCTION 545
17.2 ELECTROMAGNETIC FIELD TENSOR 545
17.2.1 ELECTROMAGNETIC FIELD AND LORENTZ 4-FORCE 545
17.2.2 THE ELECTROMAGNETIC FIELD AS A 2-FORM 547
17.2.3 ELECTRIC AND MAGNETIC FIELDS 547
17.2.4 LORENTZ FORCE RELATIVE TO AN OBSERVER 549
17.2.5 METRIC DUAL AND HODGE DUAL 550
17.3 CHANGE OF OBSERVER 552
17.3.1 TRANSFORMATION LAW OF THE ELECTRIC
AND MAGNETIC FIELDS 552
17.3.2 ELECTROMAGNETIC FIELD INVARIANTS 555
17.3.3 REDUCTION TO PARALLEL ELECTRIC AND MAGNETIC FIELDS 557
17.3.4 FIELD CREATED BY A CHARGE IN TRANSLATION 559
17.4 PARTICLE IN AN ELECTROMAGNETIC FIELD 562
17.4.1 UNIFORM ELECTROMAGNETIC FIELD: NON-NULL CASE 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 STORAGE RINGS 583
18 MAXWELL EQUATIONS 585
18.1 INTRODUCTION 585
18.2 ELECTRIC FOUR-CURRENT 586
18.2.1 ELECTRIC FOUR-CURRENT VECTOR 586
18.2.2 ELECTRIC INTENSITY 588
18.2.3 CHARGE DENSITY AND CURRENT DENSITY 591
18.2.4 FOUR-CURRENT OF A CONTINUOUS MEDIA 592
18.3 MAXWELL EQUATIONS 592
18.3.1 STATEMENT 592
18.3.2 ALTERNATIVE FORMS 593
18.3.3 EXPRESSION IN TERMS OF ELECTRIC AND MAGNETIC FIELDS* 595
18.4 ELECTRIC CHARGE CONSERVATION 598
18.4.1 DERIVATION FROM MAXWELL EQUATIONS 598
18.4.2 EXPRESSION IN TERMS OF CHARGE AND CURRENT DENSITIES . 601
18.4.3 GAUSS THEOREM 601
18.5 SOLVING MAXWELL EQUATIONS 603
18.5.1 FOUR-POTENTIAL 603
18.5.2 ELECTRIC AND MAGNETIC POTENTIALS 604
18.5.3 GAUGE CHOICE 606
18.5.4 ELECTROMAGNETIC WAVES 607
18.5.5 SOLUTION FOR THE 4-POTENTIAL IN LORENZ GAUGE 608
18.6 FIELD CREATED BY A MOVING CHARGE 611
18.6.1 LIENARD-WIECHERT 4-POTENTIAL 611
18.6.2 ELECTROMAGNETIC FIELD 615
18.6.3 ELECTRIC AND MAGNETIC FIELDS 617
18.6.4 CHARGE IN INERTIAL MOTION 618
18.6.5 RADIATIVE PART 620
18.7 MAXWELL EQUATIONS FROM A PRINCIPLE OF LEAST ACTION 622
18.7.1 PRINCIPLE OF LEAST ACTION IN A CLASSICAL FIELD THEORY * 622
18.7.2 CASE OF THE ELECTROMAGNETIC FIELD 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 LOCAL VERSION 637
19.3.3 FOUR-FORCE DENSITY 638
19.3.4 CONSERVATION OF ENERGY AND MOMENTUM
WITH RESPECT TO AN OBSERVER 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 RADIATED ENERGY 650
20.3.3 RADIATED 4-MOMENTUM 652
20.3.4 ANGULAR DISTRIBUTION OF RADIATION 655
20.4 SYNCHROTRON RADIATION 659
20.4.1 INTRODUCTION 659
20.4.2 SPECTRUM OF SYNCHROTRON RADIATION 661
20.4.3 APPLICATIONS 663
21 RELATIVISTIC HYDRODYNAMICS 667
21.1 INTRODUCTION 667
21.2 THE PERFECT FLUID MODEL 668
21.2.1 ENERGY-MOMENTUM TENSOR 668
21.2.2 QUANTITIES RELATIVE TO AN ARBITRARY OBSERVER 670
21.2.3 PRESSURELESS FLUID (DUST) 671
21.2.4 EQUATION OF STATE AND THERMODYNAMIC RELATIONS 672
21.2.5 SIMPLE FLUIDS 674
21.3 BARYON NUMBER CONSERVATION 676
21.3.1 BARYON FOUR-CURRENT 676
21.3.2 PRINCIPLE OF BARYON NUMBER CONSERVATION 677
21.3.3 EXPRESSION WITH RESPECT TO AN INERTIAL OBSERVER 679
21.4 ENERGY-MOMENTUM CONSERVATION 680
21.4.1 INTRODUCTION 680
21.4.2 PROJECTION ONTO THE FLUID 4-VELOCITY 681
21.4.3 PART ORTHOGONAL TO THE FLUID 4-VELOCITY 682
21.4.4 EVOLUTION OF THE FLUID ENERGY RELATIVE
TO SOME OBSERVER 683
CONTENTS XXIX
21.4.5 RELATIVISTIC EULER EQUATION 684
21.4.6 SPEED OF SOUND 685
21.4.7 RELATIVISTIC HYDRODYNAMICS AS A SYSTEM
OF CONSERVATION LAWS 686
21.5 FORMULATION BASED ON EXTERIOR CALCULUS 687
21.5.1 EQUATION OF MOTION 688
21.5.2 VORTICITY OF A SIMPLE FLUID 689
21.5.3 CANONICAL FORM OF THE EQUATION OF MOTION 690
21.5.4 NONRELATIVISTIC LIMIT: CROCCO EQUATION 692
21.6 CONSERVATION LAWS ; 694
21.6.1 BERNOULLI'S THEOREM 694
21.6.2 IRROTATIONAL FLOW 696
21.6.3 KELVIN'S CIRCULATION THEOREM 698
21.7 APPLICATIONS 701
21.7.1 ASTROPHYSICS: JETS AND GAMMA-RAY BURSTS 701
21.7.2 QUARK-GLUON PLASMA AT RHIC AND AT LHC 703
21.8 TO GO FURTHER 709
22 WHAT ABOUT RELATIVISTIC GRAVITATION? 711
22.1 INTRODUCTION 711
22.2 GRAVITATION IN MINKOWSKI SPACETIME 711
22.2.1 NORDSTROM'S SCALAR THEORY 712
22.2.2 INCOMPATIBILITY WITH OBSERVATIONS 719
22.2.3 VECTOR THEORY 720
22.2.4 TENSOR THEORY 722
22.3 EQUIVALENCE PRINCIPLE 723
22.3.1 THE PRINCIPLE 723
22.3.2 GRAVITATIONAL REDSHIFT AND INCOMPATIBILITY
WITH THE MINKOWSKI METRIC 724
22.3.3 EXPERIMENTAL VERIFICATIONS
OF THE GRAVITATIONAL REDSHIFT 726
22.3.4 LIGHT DEFLECTION 729
22.4 GENERAL RELATIVITY 729
A BASIC ALGEBRA 733
A.L BASIC STRUCTURES 733
A. 1.1 GROUP 733
A. 1.2 FIELDS 734
A.2 LINEAR ALGEBRA 735
A.2.1 VECTOR SPACE 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 |
| any_adam_object | 1 |
| author | Gourgoulhon, Éric |
| author_facet | Gourgoulhon, Éric |
| author_role | aut |
| author_sort | Gourgoulhon, Éric |
| author_variant | é g ég |
| building | Verbundindex |
| bvnumber | BV041284664 |
| classification_rvk | UH 8200 |
| classification_tum | PHY 041f |
| ctrlnum | (OCoLC)861772309 (DE-599)DNB1031595481 |
| dewey-full | 530.11 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.11 |
| dewey-search | 530.11 |
| dewey-sort | 3530.11 |
| dewey-tens | 530 - Physics |
| discipline | Physik |
| format | Book |
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| id | DE-604.BV041284664 |
| illustrated | Illustrated |
| indexdate | 2024-08-03T00:57:05Z |
| institution | BVB |
| isbn | 3642372759 9783642372759 |
| language | English French |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-026733777 |
| oclc_num | 861772309 |
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| owner_facet | DE-19 DE-BY-UBM DE-11 DE-20 DE-703 DE-83 DE-91G DE-BY-TUM DE-706 |
| physical | XXX, 784 Seiten Illustrationen, Diagramme 235 mm x 155 mm |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | Springer |
| record_format | marc |
| series2 | Graduate texts in physics |
| spelling | Gourgoulhon, Éric Verfasser aut Relativité restreinte <engl.> Special relativity in general frames from particles to astrophysics Éric Gourgoulhon Berlin ; Heidelberg Springer 2013 XXX, 784 Seiten Illustrationen, Diagramme 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Graduate texts in physics Literaturverzeichnis Seite 741-759 Spezielle Relativitätstheorie (DE-588)4182215-8 gnd rswk-swf Spezielle Relativitätstheorie (DE-588)4182215-8 s DE-604 Erscheint auch als Online-Ausgabe 978-3-642-37276-6 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4258631&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026733777&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gourgoulhon, Éric Special relativity in general frames from particles to astrophysics Spezielle Relativitätstheorie (DE-588)4182215-8 gnd |
| subject_GND | (DE-588)4182215-8 |
| title | Special relativity in general frames from particles to astrophysics |
| title_alt | Relativité restreinte <engl.> |
| title_auth | Special relativity in general frames from particles to astrophysics |
| title_exact_search | Special relativity in general frames from particles to astrophysics |
| title_full | Special relativity in general frames from particles to astrophysics Éric Gourgoulhon |
| title_fullStr | Special relativity in general frames from particles to astrophysics Éric Gourgoulhon |
| title_full_unstemmed | Special relativity in general frames from particles to astrophysics Éric Gourgoulhon |
| title_short | Special relativity in general frames |
| title_sort | special relativity in general frames from particles to astrophysics |
| title_sub | from particles to astrophysics |
| topic | Spezielle Relativitätstheorie (DE-588)4182215-8 gnd |
| topic_facet | Spezielle Relativitätstheorie |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=4258631&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026733777&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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