Relativity, gravitation, and cosmology : a basic introduction

Einstein's general theory of relativity is introduced in this advanced undergraduate and beginning graduate level textbook. Topics include special relativity in the formalism of Minkowski's four-dimensional space-time, the principle of equivalence, Riemannian geometry and tensor analysis, Einstein's field equation and cosmology.The author presents the subject from the very beginning with an emphasis on physical examples and simple applications without the full tensor apparatus. One first learns how to describe curved spacetime. At this mathematically more accessible level, the reader can already study the many interesting phenomena such as gravitational lensing, precession of Mercury's perihelion, black holes, as well as cosmology. The full tensor formulation is presented later, when the Einstein equation is solved for a few symmetric cases. Many modern topics in cosmology are discussed in this book: from inflation and cosmic microwave anisotropy to the "dark energy" that propels an accelerating universe.Mathematical accessibility, together with the various pedagogical devices (e.g., worked-out solutions of chapter-end problems), make it practical for interested readers to use the book to study general relativity, gravitation and cosmology on their own.

Author Notes

Ta-Pei Cheng is Professor of Physics at the University of Missouri - St. Louis, USA.

Reviews 1

Choice Review

Cheng's book is intended as an introductory work for advanced undergraduate students; it begins with a physics-based development of the key concepts in special and then general relativity. Cheng (Univ. of Missouri) introduces the equivalence principle and shows how it leads to the concepts of time dilation, length contraction, and curved space-time. This introduction is very accessible to students who have had freshman- and sophomore-level courses in Newtonian physics and electrodynamics. Cheng develops basic concepts in differential geometry and finally the full field equations for general relativity. With this groundwork laid, the remainder of the book discusses modern cosmology and relativistic astrophysics, including black-hole physics and inflationary cosmology. The book is well written and illustrated and has a satisfactory bibliography. It also includes answers to selected homework problems. Instructors will find it useful and libraries will want to have it on their shelves as a reference for students interested in learning these subjects on their own. ^BSumming Up: Recommended. Upper-division undergraduates; graduate students. A. Spero formerly, University of California, Lawrence Livermore National Laboratory

Part I Relativity Metric Description of Spacetime
1 Introduction and overview	p. 3
1.1 Relativity as a coordinate symmetry	p. 5
1.1.1 From Newtonian relativity to aether	p. 5
1.1.2 Einsteinian relativity	p. 6
1.1.3 Coordinate symmetry transformations	p. 7
1.1.4 New kinematics and dynamics	p. 7
1.2 GR as a gravitational field theory	p. 8
1.2.1 Einstein's motivations for the general theory	p. 8
1.2.2 Geometry as gravity	p. 10
1.2.3 Mathematical language of relativity	p. 11
1.2.4 GR is the framework for cosmology	p. 12
Review questions	p. 12
2 Special relativity and the flat spacetime	p. 14
2.1 Coordinate symmetries	p. 14
2.1.1 Rotational symmetry	p. 14
2.1.2 Newtonian physics and Galilean symmetry	p. 16
2.1.3 Electrodynamics and Lorentz symmetry	p. 17
2.1.4 Velocity addition rule amended	p. 18
2.2 The new kinematics of space and time	p. 19
2.2.1 Relativity of spatial equilocality	p. 20
2.2.2 Relativity of simultaneity-the new kinematics	p. 20
2.2.3 The invariant space-time interval	p. 22
2.3 Geometric formulation of SR	p. 24
2.3.1 General coordinates and the metric tensor	p. 24
2.3.2 Derivation of Lorentz transformation	p. 28
2.3.3 The spacetime diagram	p. 30
2.3.4 Time-dilation and length contraction	p. 32
Review questions	p. 35
Problems	p. 35
3 The principle of equivalence	p. 38
3.1 Newtonian gravitation potential-a review	p. 38
3.2 EP introduced	p. 39
3.2.1 Inertial mass vs. gravitational mass	p. 40
3.2.2 EP and its significance	p. 41
3.3 Implications of the strong EP	p. 43
3.3.1 Gravitational redshift and time dilation	p. 43
3.3.2 Light ray deflection calculated	p. 48
3.3.3 Energy considerations of a gravitating light pulse	p. 51
3.3.4 Einstein's inference of a curved spacetime	p. 52
Review questions	p. 53
Problems	p. 53
4 Metric description of a curved space	p. 55
4.1 Gaussian coordinates	p. 56
4.2 Metric tensor	p. 57
4.2.1 Geodesic as the shortest curve	p. 59
4.2.2 Local Euclidean coordinates	p. 61
4.3 Curvature	p. 63
4.3.1 Gaussian curvature	p. 63
4.3.2 Spaces with constant curvature	p. 64
4.3.3 Curvature measures deviation from Euclidean relations	p. 66
Review questions	p. 68
Problems	p. 69
5 GR as a geometric theory of gravity - I	p. 71
5.1 Geometry as gravity	p. 71
5.1.1 EP physics and a warped spacetime	p. 73
5.1.2 Curved spacetime as gravitational field	p. 74
5.2 Geodesic equation as GR equation of motion	p. 75
5.2.1 The Newtonian limit	p. 76
5.2.2 Gravitational redshift revisited	p. 78
5.3 The curvature of spacetime	p. 79
5.3.1 Tidal force as the curvature of spacetime	p. 80
5.3.2 The GR field equation described	p. 83
Review questions	p. 85
Problems	p. 85
6 Spacetime outside a spherical star	p. 87
6.1 Description of Schwarzschild spacetime	p. 87
6.1.1 Spherically symmetric metric tensor	p. 88
6.1.2 Schwarzschild geometry	p. 90
6.2 Gravitational lensing	p. 92
6.2.1 Light ray deflection revisited	p. 93
6.2.2 The lens equation	p. 93
6.3 Precession of Mercury's perihelion	p. 97
6.4 Black holes	p. 102
6.4.1 Singularities of the Schwarzschild metric	p. 102
6.4.2 Time measurements in the Schwarzschild spacetime	p. 102
6.4.3 Lightcones of the Schwarzschild black hole	p. 105
6.4.4 Orbit of an object around a black hole	p. 108
6.4.5 Physical reality of black holes	p. 108
Review questions	p. 111
Problems	p. 112
Part II Cosmology
7 The homogeneous and isotropic universe	p. 115
7.1 The cosmos observed	p. 116
7.1.1 Matter distribution on the cosmic distance scale	p. 116
7.1.2 Cosmological redshift: Hubble's law	p. 116
7.1.3 Age of the universe	p. 120
7.1.4 Dark matter and mass density of the universe	p. 121
7.2 The cosmological principle	p. 125
7.3 The Robertson-Walker metric	p. 127
7.3.1 Proper distance in the RW geometry	p. 129
7.3.2 Redshift and luminosity distance	p. 130
Review questions	p. 133
Problems	p. 134
8 The expanding universe and thermal relics	p. 136
8.1 Friedmann equations	p. 137
8.1.1 The quasi-Newtonian interpretation	p. 139
8.2 Time evolution of model universes	p. 142
8.3 Big bang cosmology	p. 145
8.3.1 Scale-dependence of radiation temperature	p. 145
8.3.2 Different thermal equilibrium stages	p. 147
8.4 Primordial nucleosynthesis	p. 149
8.5 Photon decoupling and the CMB	p. 152
8.5.1 Universe became transparent to photons	p. 153
8.5.2 The discovery of CMB radiation	p. 154
8.5.3 Photons, neutrinos, and the radiation-matter equality time	p. 155
8.5.4 CMB temperature fluctuation	p. 159
Review questions	p. 162
Problems	p. 163
9 Inflation and the accelerating universe	p. 165
9.1 The cosmological constant	p. 166
9.1.1 Vacuum-energy as source of gravitational repulsion	p. 167
9.1.2 The static universe	p. 168
9.2 The inflationary epoch	p. 170
9.2.1 Initial conditions for the standard big bang model	p. 171
9.2.2 The inflation scenario	p. 173
9.2.3 Inflation and the conditions it left behind	p. 175
9.3 CMB anisotropy and evidence for k = 0	p. 178
9.3.1 Three regions of the angular power spectrum	p. 179
9.3.2 The primary peak and spatial geometry of the universe	p. 181
9.4 The accelerating universe in the present epoch	p. 183
9.4.1 Distant supernovae and the 1998 discovery	p. 184
9.4.2 Transition from deceleration to acceleration	p. 187
9.5 The concordant picture	p. 189
Review questions	p. 193
Problems	p. 193
Part III Relativity Full Tensor Formulation
10 Tensors in special relativity	p. 197
10.1 General coordinate systems	p. 197
10.2 Four-vectors in Minkowski spacetime	p. 200
10.3 Manifestly covariant formalism for E&M	p. 205
10.3.1 The electromagnetic field tensor	p. 205
10.3.2 Electric charge conservation	p. 208
10.4 Energy-momentum tensors	p. 208
Review questions	p. 213
Problems	p. 213
11 Tensors in general relativity	p. 215
11.1 Derivatives in a curved space	p. 215
11.1.1 General coordinate transformations	p. 216
11.1.2 Covariant differentiation	p. 218
11.1.3 Christoffel symbols and metric tensor	p. 220
11.2 Parallel transport	p. 222
11.2.1 Component changes under parallel transport	p. 222
11.2.2 The geodesic as the straightest possible curve	p. 224
11.3 Riemannian curvature tensor	p. 225
11.3.1 The curvature tensor in an n-dimensional space	p. 226
11.3.2 Symmetries and contractions of the curvature tensor	p. 228
Review questions	p. 230
Problems	p. 231
12 GR as a geometric theory of gravity - II	p. 233
12.1 The principle of general covariance	p. 233
12.1.1 Geodesic equation from SR equation of motion	p. 235
12.2 Einstein field equation	p. 236
12.2.1 Finding the relativistic gravitational field equation	p. 236
12.2.2 Newtonian limit of the Einstein equation	p. 237
12.3 The Schwarzschild exterior solution	p. 239
12.4 The Einstein equation for cosmology	p. 244
12.4.1 Solution for a homogeneous and isotropic 3D space	p. 244
12.4.2 Friedmann equations	p. 246
12.4.3 Einstein equation with a cosmological constant term	p. 247
Review questions	p. 248
Problems	p. 248
13 Linearized theory and gravitational waves	p. 250
13.1 The linearized Einstein theory	p. 251
13.1.1 The coordinate change called gauge transformation	p. 252
13.1.2 The wave equation in the Lorentz gauge	p. 253
13.2 Plane waves and the polarization tensor	p. 254
13.3 Gravitational wave detection	p. 255
13.3.1 Effect of gravitational waves on test particles	p. 255
13.3.2 Gravitational wave interferometers	p. 257
13.4 Evidence for gravitational wave	p. 259
13.4.1 Energy flux in linearized gravitational waves	p. 260
13.4.2 Emission of gravitational radiation	p. 262
13.4.3 Binary pulsar PSR 1913+16	p. 264
Review questions	p. 268
Problems	p. 269
A Supplementary notes	p. 271
A.1 The twin paradox (Section 2.3.4)	p. 271
A.2 A glimpse of advanced topics in black hole physics (Section 6.4)	p. 275
A.3 False vacuum and hidden symmetry (Section 9.2.2)	p. 279
A.4 The problem of quantum vacuum energy as A (Section 9.4)	p. 280
B Answer keys to review questions	p. 283
C Solutions of selected problems	p. 293
References	p. 330
Bibliography	p. 333
Index	p. 335

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