Einstein's General Theory of Relativity: With Modern Applications in Cosmology (Google eBook)

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Springer, Aug 24, 2007 - Science - 560 pages
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The book introduces the general theory of relativity and includes applications to cosmology. The book contains a thorough introduction to tensor calculus and curved manifolds. After the necessary mathematical tools are introduced, we give a thorough presentation of the theory of relativity. Also, some advanced topics not previously covered by textbooks; e.g. Kaluza-Klein theory, Israel's formalism and branes. Anisotropic cosmological models are also included. The book contains a large number of new exercises and examples, each with separate headings. The reader will get an updated introduction to general relativity including the most recent developments in cosmology.
  

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Contents

108 The TolmanOppenheimerVolkoff equation
252
109 The interior Schwarzschild solution
253
1010 Relativistic gravitation versus Newtonian gravitation
256
Problems
257
COSMOLOGY
264
Homogeneous and Isotropic Universe Models
267
112 FriedmannRobertsonWalker models
268
113 Dynamics of Homogeneous and Isotropic cosmologies
271

19 Machs principle
16
Problems
17
The Special Theory of Relativity
21
22 Synchronization of clocks
23
24 Relativistic timedilatation
25
25 The relativity of simultaneity
26
26 The Lorentzcontraction
28
27 The Lorentz transformation
30
28 Lorentzinvariant interval
32
29 The twinparadox
34
210 Hyperbolic motion
35
211 Energy and mass
37
212 Relativistic increase of mass
38
213 Tachyons
39
214 Magnetism as a relativistic secondorder effect
40
Problems
42
THE MATHEMATICS OF THE GENERAL THEORY OF RELATIVITY
48
Vectors Tensors and Forms
51
32 Fourvectors
52
33 Oneforms
54
34 Tensors
55
35 Forms
57
Problems
60
Basis Vector Fields and the Metric Tensor
63
42 Tangent vector fields and the coordinate basis vector fields
65
43 Structure coefficients
71
44 General basis transformations
72
45 The metric tensor
74
46 Orthonormal basis
76
47 Spatial geometry
78
48 The tetrad field of a comoving coordinate system
80
49 The volume form
81
410 Dual forms
82
Noninertial Reference Frames
89
52 Ehrenfests paradox
90
53 The Sagnac effect
93
54 Gravitational time dilatation
94
55 Uniformly accelerated reference frame
95
56 Covariant Lagrangian dynamics
98
57 A general equation for the Doppler effect
103
Problems
107
and Integration 6
109
62 Electromagnetism
113
63 Integration of forms
115
64 Covariant differentiation of vectors
120
65 Covariant differentiation of forms and tensors
127
66 Exterior differentiation of vectors
129
67 Covariant exterior derivative
133
68 Geodesic normal coordinates
136
69 Oneparameter groups of diffeomorphisms
137
610 The Lie derivative
140
611 Killing vectors and Symmetries
144
Problems
147
Curvature
150
72 Surfaces
153
73 The Riemann Curvature Tensor
155
74 Extrinsic and Intrinsic Curvature
161
75 The equation of geodesic deviation
164
76 Spaces of constant curvature
166
EINSTEINS FIELD EQUATIONS
177
Einsteins Field Equations
179
82 Deduction of Einsteins vacuum field equations from Hilberts variational principle
180
83 The field equations in the presence of matter and energy
183
84 Energymomentum conservation
185
86 Some particular fluids
187
87 The paths of free point particles
191
The Linear Field Approximation
195
92 The Newtonian limit of general relativity
198
93 Solutions of the linearised field equations
199
94 Gravitoelectromagnetism
201
95 Gravitational waves
203
96 Gravitational radiation from sources
206
The Schwarzschild Solution and Black Holes
215
102 Radial free fall in Schwarzschild spacetime
220
103 The lightcone in a Schwarzschild spacetime
221
104 Particle trajectoriesin Schwarzschild spacetime
225
105 Analytical extension of the Schwarzschild spacetime
230
106 Charged and rotating black holes
233
107 Black Hole thermodynamics
245
114 Cosmological redshift and the Hubble law
273
115 Radiation dominated universe models
278
116 Matter dominated universe models
281
117 The gravitational lens effect
283
118 Redshiftluminosity relation
289
119 Cosmological horizons
293
1110 Big Bang in an infinite Universe
294
Problems
296
Universe Models with Vacuum Energy
304
122 de Sitters solution
306
The many guises of the de Sitter spacetime
309
124 The horizon problem and the flatness problem
310
125 Inflation
312
126 The FriedmannLemaître model
319
127 Universe models with quintessence energy
325
128 Dark energy explored by means of supernova observations and the statefinder diagnostic
328
129 Cosmic density perturbations
334
1210 Temperaturefluctuations in the cosmic microwave background CMB
338
1211 Machs principle
345
1212 The History of our Universe
348
Anisotropic and Inhomogeneous Universe Models
367
132 The Kasner solutions
370
133 The energymomentum conservation law in an anisotropic universe
371
134 Models with a perfect fluid
373
135 Inflation through bulk viscosity
376
136 A universe with a dissipative fluid
377
137 The LemaîtreTolmanBondi universe models
379
Problems
383
ADVANCED TOPICS
386
Covariant Decomposition Singularities and Canonical Cosmology
389
142 Equations of motion
392
143 Singularities
394
144 Lagrangian formulation of General Relativity
399
145 Hamiltonian formulation
402
146 Canonical formulation with matter and energy
404
Superspace
406
Problems
410
Spatially Homogeneous Universe Models
413
152 Homogeneous spaces
416
153 The Bianchi models
420
154 The orthonormal frame approach to the nontilted Bianchi models
423
155 The 8 model geometries
428
156 Constructing compact quotients
430
Problems
433
Israels Formalism The Metric Junction Method
438
162 Einsteins field equations
441
163 Surface layers and boundary surfaces
443
164 Spherical shell of dust in vacuum
445
Problems
450
Braneworlds
453
172 Fivedimensional brane cosmology
456
173 Solutions in the bulk
459
175 Inflation in the brane
464
176 Dynamics of two branes
467
177 of gravity The hierarchy problem and the weakness
469
178 The RandallSundrum models
471
Problems
474
KaluzaKlein Theory
478
182 The KaluzaKlein action
481
183 Implications of a fifth extra dimension
485
184 Conformal transformations
488
185 Conformal transformation of the KaluzaKlein action
491
186 KaluzaKlein cosmology
494
Problems
497
APPENDICES
500
Constants of Nature
501
B
505
B2 Schwarzschild spacetime
507
C
511
C2 Foliations of AdSn
512
C3 Geodesies in AdS
513
C4 The BTZ black hole
514
C5 AdS3 as the group SL2R
515
D
517
E
519
Bibliography
523
Index
533
Copyright

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