Relativity in Modern Physics

Cover......Page 1
Relativity in Modern Physics......Page 4
Copyright......Page 5
Dedication
......Page 6
Foreword......Page 8
Contents......Page 10
BOOK 1. Space, time, and gravity in Newton’s theory......Page 14
Contents......Page 16
Part I. Kinematics......Page 22
1.1 Absolute space and time......Page 24
1.2 The absolute reference frame......Page 25
1.3 Change of Cartesian coordinates......Page 26
The Euler angles......Page 27
1.4 The group of rigid displacements......Page 28
1.5 Kinematics of a point particle (I)......Page 29
Instantaneous rotation of a frame......Page 30
1.6 Cartesian vectors and vector fields......Page 31
2.1 Tensor spaces......Page 33
2.2 Affine and Euclidean planes......Page 34
2.3 Change of basis and frame......Page 36
Basis vectors and derivative operators......Page 37
2.5 Examples of moving frames......Page 38
The Frenet trihedron......Page 39
2.6 The ABCs of vector calculus......Page 40
The Stokes and Gauss gradient theorems......Page 42
The metric in polar coordinates......Page 43
The polar components of a vector field......Page 44
3.2 The covariant derivative......Page 45
Transformation of the connection coefficients......Page 46
3.3 Parallel transport......Page 47
3.4 The covariant derivative and the metric tensor......Page 48
3.5 Kinematics of a point particle (III)......Page 49
Trajectories in polar coordinates and the auto-parallel......Page 50
3.6 Differential operators and integration......Page 51
The area of a sphere in spheroidal coordinates......Page 53
4.1 Tangent spaces, vectors, and tangents......Page 54
4.3 The metric tensor, triads, and frame fields......Page 56
Moving frames and spherical coordinates......Page 57
4.4 Vector fields, form fields, and tensor fields......Page 58
4.5 The covariant derivative (II)......Page 59
4.6 Vector calculus and differential operators......Page 61
The surface element......Page 62
Part II. Dynamics......Page 64
The law of motion in curvilinear coordinates......Page 66
5.2 Properties of forces......Page 67
5.3 The principle of Galilean relativity......Page 68
5.4 Moving frames and inertial forces......Page 69
The Foucault pendulum......Page 70
The deviation toward the east......Page 71
6.1 The equations of motion of a solid......Page 72
The moments of inertia of a homogeneous ellipsoid......Page 73
Symmetric tops and the polhode......Page 74
6.3 The Euler equations of fluid motion......Page 75
The Eulerian and Lagrangian derivatives......Page 76
Newton’s bucket......Page 77
7.1 Momentum and the center of mass......Page 78
7.2 Angular momentum......Page 79
7.3 Energy......Page 80
Velocity-dependent potential energy......Page 82
7.4 Virial theorems......Page 83
Tensorial virial theorem......Page 84
8.1 The Euler–Lagrange equations......Page 86
8.2 The laws of motion (II)......Page 87
Acceleration and the covariant derivative......Page 89
Constraints and Lagrange multipliers......Page 90
Incompressible fluids......Page 91
8.3 Conservation laws (II)......Page 92
Isometries of the Euclidean metric......Page 94
9.1 Hamilton’s equations......Page 96
Legendre transformations......Page 97
9.2 Canonical transformations......Page 98
Example of a canonical transformation......Page 99
The Hamilton–Jacobi method: an example......Page 100
9.4 Poisson brackets......Page 101
Invariance of the Poisson brackets......Page 103
The Poisson bracket and canonical transformation......Page 104
10.1 Liouville’s theorem and equation......Page 105
10.2 The Boltzmann–Vlasov equation......Page 106
10.3 The Jeans equations......Page 108
10.4 The Maxwell distribution and the thermodynamical limit......Page 110
Part III. Gravitation......Page 112
11.1 Gravitational mass and inertial mass......Page 114
11.2 Equality of gravitational and inertial mass......Page 115
Experiments involving torsion balances......Page 116
11.3 Newton’s gravitational force and field......Page 117
Measurement of mass and of Newton’s constant......Page 118
11.4 The Poisson equation and the gravitational Lagrangian......Page 119
Spherical symmetry: Newton’s theorem......Page 120
Newtonian gravity: unanswered questions......Page 122
12.1 The reduced equations of motion......Page 123
12.2 The ellipses of Kepler......Page 124
The geometry of an ellipse......Page 125
Orbital parameters......Page 126
12.3 The Kepler problem in the Lagrangian formalism......Page 127
12.4 Central forces......Page 129
Central forces in the Hamiltonian formalism......Page 130
Central forces and Newton’s theorem......Page 131
13.1 The Laplace effect......Page 132
13.2 The restricted three-body problem......Page 134
The Lagrange points......Page 135
13.3 Gauss equations of perturbations......Page 136
The example of a radial perturbation......Page 138
Advance of the perihelion of Mercury......Page 139
14.1 Quadrupole expansion of the potential......Page 141
The precession of the equinoxes......Page 142
14.2 Causes of non-sphericity of bodies......Page 144
Tides in the fluid model of the Earth......Page 145
The Roche limit......Page 147
14.3 The figure of the Earth......Page 148
The Earth’s rotation and the recession of the Moon......Page 150
Synchronous rotation of the Moon......Page 151
15.2 Static models with spherical symmetry......Page 152
A ‘star’ of constant density......Page 153
15.3 Polytropes and the Lane–Emden equation......Page 154
15.4 The isothermal sphere......Page 156
Kinetic theory and the isothermal sphere......Page 157
15.5 Maclaurin spheroids......Page 158
Jacobi ellipsoids......Page 160
16.1 The model of an expanding sphere......Page 161
16.2 The pitfalls of the infinite Newtonian universe......Page 162
16.3 The ‘Friedmann’ equation......Page 163
16.4 The evolution of perturbations......Page 164
16.5 Olbers’s paradox......Page 166
Resolution of Olbers’s paradox......Page 167
R¨omer and the speed of light......Page 168
17.2 Stellar aberration......Page 169
17.3 Wave propagation......Page 170
The Doppler effect......Page 172
Optical interference......Page 173
17.5 The Michelson–Morley experiment......Page 175
List of books and articles cited in the text......Page 176
BOOK 2. Special relativity and Maxwell’s theory......Page 180
Contents......Page 182
A note on the units......Page 188
Part I: Kinematics......Page 190
The two founding principles......Page 192
Review of vector geometry......Page 193
“c=1”......Page 195
1.3 Lorentz transformations......Page 196
The muon lifetime (I)......Page 199
1.5 Thomas rotation......Page 200
2.1 World lines......Page 203
Uniform circular motion......Page 204
Extremization of the proper time and the geodesic......Page 205
The muon lifetime (II)......Page 206
2.3 The Langevin twins......Page 207
The Langevin twins in a closed space......Page 208
2.4 Transformation of velocities and accelerations......Page 209
The Fresnel formula and the Fizeau experiment......Page 211
The 3-velocity and 3-acceleration of a light line......Page 213
3.2 The Sagnac effect......Page 214
The Michelson–Gale–Pearson experiment......Page 215
3.3 Aberration formulas......Page 216
‘Superluminal’ jets......Page 217
Reflection on a moving mirror......Page 219
4.1 The wave vector and spectral shifts......Page 220
4.2 Light signals and spectral shifts......Page 221
4.3 An example of a particle horizon......Page 222
Light signals and wave vectors......Page 224
Review of Newtonian spacetime......Page 226
5.2 The example of Rindler coordinates......Page 228
Review of curvilinear coordinates......Page 229
The Rindler frame and spectral shifts......Page 231
5.3 Rotating reference frames and the geometrization of inertia......Page 232
A uniformly rotating disk......Page 233
Review of covariant differentiation......Page 234
The ‘rigidity’ of the Rindler frame......Page 236
Part II. Dynamics......Page 238
6.1 Free particles......Page 240
6.2 Interactions......Page 241
6.3 The momentum conservation law......Page 243
6.4 Collisions......Page 245
6.5 Compton scattering......Page 247
The Klein–Nishina and Thomson formulas......Page 249
7.1 Angular momentum and center of mass......Page 251
World lines of centers of mass......Page 252
7.2 Intrinsic angular momentum......Page 253
The Levi-Civita symbol......Page 254
Relativistic ‘tops’......Page 255
7.4 Thomas precession......Page 256
Thomas precession and a rotating frame......Page 257
Thomas precession and the Gravity Probe B experiment......Page 258
8.1 Equations of motion of a free field......Page 259
8.2 The energy–momentum tensor of a free field......Page 260
Transformation of the 4-momentum......Page 263
8.3 Fluid equations of motion......Page 264
The equation of state of a relativistic fluid......Page 265
8.4 The particle energy–momentum tensor......Page 266
8.5 Conservation laws in an accelerated frame......Page 267
Isometries and Killing vectors of M_4......Page 268
9.1 The Klein–Gordon equation......Page 270
The variational derivative and Hamilton’s equations......Page 271
The Fourier transform......Page 272
The Dirac distribution......Page 273
9.3 Complex fields, charge, and symmetry breaking......Page 275
9.4 The BEH mechanism......Page 277
On the ‘renormalizability’ of the theory......Page 278
10.1 The coupling of a scalar field to a particle......Page 279
Short-range interactions......Page 281
10.2 The field–matter system......Page 282
Light deflection......Page 284
Advance of the perihelion......Page 285
10.4 Inertial and gravitational masses of a two-body system......Page 286
Inertia and gravitation......Page 289
Part III. Electromagnetism......Page 292
11.1 The electromagnetic potential and field......Page 294
Review of three-dimensional vector calculus......Page 295
11.2 Transformation of the field under a Lorentz rotation......Page 296
11.3 The equation of motion of a charge......Page 297
‘Fictitious time’ vs. ‘pure and simple’ time......Page 298
11.4 Charge in a uniform and constant field......Page 299
Charge in a critical electric field......Page 300
Synchrotron motion......Page 301
12.2 Current and charge conservation......Page 302
12.3 The second group of Maxwell equations......Page 304
Lorentz invariance......Page 305
Duality and magnetic charges......Page 306
12.4 The field energy–momentum tensor......Page 307
Transformation of the energy–momentum tensor......Page 310
13.1 Coulomb’s law......Page 311
The ‘classical’ electron radius......Page 312
13.3 A charge in a Coulomb field......Page 313
Bound motion in a Coulomb field......Page 314
The Rutherford effective cross section......Page 315
Sommerfeld quantization......Page 317
13.4 Spin in a Coulomb field......Page 318
13.5 The Biot–Savart law......Page 319
The magnetic moment......Page 321
14.1 Two degrees of freedom......Page 322
On the Cauchy problem......Page 323
14.2 The gauge-invariant action......Page 324
14.3 Hamiltonian formalisms......Page 325
The non-gauge-invariant kinetic term......Page 328
The Proca Hamiltonian......Page 329
15.1 Monochromatic plane waves: propagation......Page 330
Fourier decomposition of the Hamiltonian......Page 331
15.2 Monochromatic plane waves: polarization......Page 333
Spherical waves......Page 335
15.4 Motion of a charge in a plane wave......Page 336
A charge in a linearly polarized wave......Page 337
A charge in a circularly polarized wave......Page 338
A Gaussian wave packet......Page 339
16.1 The Maxwell equations in a medium......Page 341
Waves in a homogeneous medium......Page 342
16.2 Matching conditions and the Snell–Descartes laws......Page 343
The Fresnel laws......Page 345
Reflection and transmission coefficients......Page 346
16.3 The geometrical optics approximation......Page 347
16.4 Fermat’s principle......Page 348
The Snell–Descartes laws revisited......Page 349
16.5 The various descriptions of light......Page 350
Part IV. Electrodynamics......Page 352
17.1 The field of a charge in uniform motion......Page 354
Derivatives of retarded quantities......Page 355
17.2 The field of an accelerated charge......Page 357
The retarded propagator and a continuous distribution......Page 358
The field of a uniformly accelerated charge......Page 359
The field of a charge in circular motion......Page 360
17.3 The radiation field......Page 361
The radiation field of a uniformly accelerated charge......Page 362
The radiation field of a charge in circular motion......Page 363
The radiation field of a charge accelerated by a wave......Page 364
18.1 The Larmor formulas......Page 365
Calculation of spatial averages......Page 366
18.2 Radiation by a linearly accelerated charge......Page 368
Radiation losses in a linear accelerator......Page 369
18.3 Radiation of a charge in circular motion......Page 370
Radiation of the hydrogen atom......Page 371
18.4 Thomson scattering......Page 372
Thomson scattering and the Compton effect......Page 373
19.1 The Abraham–Lorentz–Dirac force......Page 374
Laurent expansion of retarded quantities......Page 375
19.2 The reaction force and the Larmor formulas......Page 377
19.3 Caveats......Page 378
20.1 The dipole field and radiation......Page 380
20.2 The quadrupole field and radiation......Page 381
Asymptotic expansion of retarded quantities......Page 383
20.3 The charge equations of motion......Page 384
Higher orders and Lagrange series......Page 386
21.1 The Darwin Lagrangian and conserved quantities......Page 388
The center-of-mass system......Page 390
21.2 Radiation reaction......Page 391
The problem of the stability of the hydrogen atom......Page 392
21.3 The exact equations of motion......Page 393
The ‘predictivization’ of hereditary equations......Page 394
22.1 p-forms and the exterior product......Page 396
22.2 The dual of a p-form......Page 397
22.3 The exterior derivative......Page 398
22.4 A rewriting of the Maxwell equations......Page 399
22.5 Differential operators and the exterior derivative......Page 400
22.6 Integration and the Stokes theorem......Page 401
List of books and articles cited in the text......Page 403
BOOK 3. General relativity and gravitation......Page 408
Contents......Page 410
Part I. Curved spacetime and gravitation......Page 416
1.1 A ‘general’ relativity......Page 418
1.3 A mosaic of pieces of M_4......Page 420
1.4 The reference ‘mollusk’......Page 421
1.5 A curved spacetime......Page 422
1.6 Gravitational redshift......Page 423
The Hafele–Keating and Alley et al. experiments......Page 425
2.1 The connection, parallel transport, and curvature......Page 426
2.2 Commutation of derivatives, torsion, and curvature......Page 428
2.3 ‘Geodesic’ deviation and curvature......Page 429
2.4 The metric tensor and the Levi-Civita connection......Page 430
2.5 Locally inertial frames......Page 432
2.6 Properties of the Riemann tensor......Page 433
The Euler–Lagrange equations and the geodesic......Page 435
3.2 Equations of motion of fluids......Page 436
The particle energy–momentum tensor......Page 437
3.3 The coupling of a field to gravity......Page 438
The divergence theorem......Page 439
The Klein–Gordon and Maxwell equations......Page 440
The case of scalar and electromagnetic fields......Page 441
Examples of Noether tensors......Page 443
Ten equations for ten unknowns......Page 444
The ‘cosmological constant’......Page 445
4.2 The 1+3 decomposition......Page 446
Propagation of the constraints......Page 447
4.3 Matching conditions......Page 449
4.4 The Hilbert action and the Einstein equations......Page 450
The Gibbons–Hawking–York boundary term......Page 451
The Palatini variation......Page 452
A ‘special relativistic’ interpretation of gravitation......Page 453
4.5 The gravitational Hamiltonian......Page 454
5.1 Isometries and Killing vectors......Page 456
5.2 First integrals of the geodesic equation......Page 457
5.3 Isometries and energy–momentum......Page 458
5.4 Noether charges......Page 459
The Katz superpotential......Page 461
The Freud superpotential and Einstein pseudo-tensor......Page 462
The Komar integrals......Page 463
The strong equivalence principle......Page 464
Part II. The Schwarzschild solution and black holes......Page 466
Static and spherically symmetric spaces......Page 468
6.2 Static, spherically symmetric stars......Page 470
The proper mass of a star......Page 472
The inertial mass of a star......Page 473
The concept of critical mass......Page 474
6.4 Gravitational collapse and black holes......Page 475
6.5 The Lemaıtre–Tolman–Bondi solution......Page 477
6.6 The interior Friedmann solution......Page 478
7.1 The Lemaıtre coordinates......Page 480
7.2 The Kruskal–Szekeres extension......Page 481
From Rindler spacetime to M4......Page 482
Regularization of a metric......Page 484
7.3 The Penrose–Carter diagram......Page 486
The various notions of horizon......Page 489
7.4 The Reissner–Nordstr¨om black hole......Page 490
The case q2 = m2......Page 492
8.1 Kerr–Schild metrics......Page 494
Spheroidal coordinates......Page 495
8.3 The geodesic equation......Page 498
Schwarzschild geodesics......Page 500
8.4 The Kerr black hole......Page 501
The ratio a/m......Page 502
Mass and angular momentum......Page 504
9.1 The Penrose process and irreducible mass......Page 505
The surface gravity......Page 507
9.2 Superradiance......Page 508
On the stability of black holes......Page 511
The Schwinger effect......Page 512
The Unruh effect......Page 514
10.2 On the thermodynamics of black holes......Page 518
10.3 The Israel uniqueness theorem......Page 519
The uniqueness of the Kerr black hole......Page 522
Part III. General relativity and experiment......Page 524
Harmonic coordinates......Page 526
11.2 The geodesic equation......Page 527
11.3 The bending of light......Page 528
11.4 The Shapiro effect......Page 530
Measurements of the Shapiro effect......Page 531
11.5 Advance of the perihelion......Page 532
11.6 Post-Keplerian geodesics......Page 534
11.7 Spin in a gravitational field......Page 536
The gravitomagnetic field......Page 538
12.1 The metric at post-Newtonian order......Page 540
The metric at Newtonian order......Page 541
The asymptotic metric of a stationary system......Page 542
12.2 The post-Newtonian field of compact bodies......Page 543
12.3 The EIH equations of motion......Page 544
Lorentz invariance of the Lagrangian......Page 547
12.5 Post-Keplerian trajectories......Page 548
12.6 The timing formula of a binary system......Page 550
13.1 Linearization of the Einstein equations......Page 553
13.2 Gravitational waves......Page 554
Fourier decomposition of SVT perturbations......Page 555
The Hamiltonian of gravitational waves......Page 557
13.3 The motion of a particle in a wave......Page 558
Detection of a gravitational wave......Page 559
13.4 The first quadrupole formula......Page 560
The harmonic condition and conservation law......Page 561
Orders of magnitude......Page 563
14.1 The Einstein equations at post-linear order......Page 564
14.2 The second quadrupole formula......Page 565
The pseudo-tensor and energy of the system......Page 566
14.3 Radiation of a self-gravitating system......Page 567
Retarded quantities and their derivatives......Page 568
Expansion of retarded quantities......Page 569
Back reaction of the radiation on the motion......Page 574
15.1 The gravitational field in the post-Minkowski approximation......Page 575
Regularization of the field equations......Page 576
The relativistic Fock function......Page 578
The need for a third iteration......Page 580
15.3 Conservation laws and energy balance......Page 583
The binary pulsar PSR 1913+16......Page 585
16.1 The 2PN Hamiltonian......Page 587
The Fokker Lagrangian......Page 588
16.2 The equations of motion of a test particle in a SSS metric......Page 591
16.3 The EOB mapping......Page 592
16.4 The EOB Hamiltonian and the resummed dynamics......Page 596
16.5 EOB dynamics including the radiation reaction force......Page 597
16.6 EOB waveform of two coalescing black holes......Page 599
Part IV. Friedmann–Lemaıtre solutions and cosmology......Page 602
The Copernican and cosmological principles......Page 604
17.2 Spacetimes with homogeneous and isotropic sections......Page 606
17.3 Milne spacetime......Page 607
17.4 de Sitter spacetime......Page 608
Foliations of de Sitter spacetime......Page 609
The Weyl postulate......Page 611
Scale factors and Hubble’s law: examples......Page 613
18.3 The Friedmann–Lemaıtre equations......Page 614
18.4 The first models of the universe (1917–1960)......Page 615
19.1 The matter content of the universe and its evolution......Page 618
19.2 Evolution of the scale factor......Page 619
Einstein–de Sitter models......Page 620
19.3 Parameter values......Page 621
20.1 The hot Big Bang model: unanswered questions......Page 624
The horizon vs. the Hubble radius......Page 626
20.2 Inflation......Page 627
20.3 ‘Chaotic’ inflation......Page 630
Other models......Page 631
Exiting inflation......Page 632
21.1 Perturbations of the geometry......Page 633
Linearization of the Einstein tensor......Page 634
21.2 Matter perturbations......Page 635
The Stewart–Walker lemma......Page 636
Bessel functions......Page 638
21.4 Evolution of scalar perturbations......Page 639
Matching of the perturbations in different eras......Page 641
21.5 The sub-Hubble limit......Page 642
22.1 Perturbations during inflation......Page 645
22.2 The action of the perturbations......Page 647
The action and evolution equations of the scalar modes......Page 648
22.3 Determination of the initial conditions......Page 649
Perturbation spectrum in the adiabatic approximation......Page 651
Perturbations after inflation......Page 654
22.5 Predictions of slow-roll inflation......Page 655
The standard model of cosmology: current status......Page 657
Part V. Elements of Riemannian geometry......Page 658
23.1 Tangent spaces of a non-connected manifold......Page 660
23.2 The exterior derivative......Page 662
The Lie bracket and closed paths......Page 663
‘Passive’ definition of the Lie derivative......Page 665
23.4 The covariant derivative and connected manifold......Page 666
23.6 Curvature of a covariant derivative......Page 667
24.1 The metric manifold and Levi-Civita connection......Page 669
The geodesic and geodesic deviation equations......Page 670
24.2 Properties of the curvature tensor......Page 671
24.3 Variation of the Hilbert action......Page 672
25.2 The connection and torsion forms......Page 675
25.4 The Levi-Civita connection......Page 676
25.5 Components of the Riemann tensor......Page 677
25.6 The Riemann tensor of the Schwarzschild metric......Page 678
List of books and articles cited in the text......Page 681
Index......Page 692