Spacetime, Geometry and Gravitation

by Sharan, Pankaj

• ISBN13: 9783764399702
• ISBN10: 3764399708
• Format: Hardcover
• Copyright: 2009-09-01
• Publisher: Birkhauser

Summary

This introductory textbook on the general theory of relativity presents a solid foundation for those who want to learn about relativity. The subject is presented in a physically intuitive, but mathematically rigorous style. The topic of relativity is covered in a broad and deep manner. Besides, the aim is that after reading the book a student should not feel discouraged when she opens advanced texts on general relativity for further reading. The book consists of three parts: (1) An introduction to the general theory of relativity, (2) geometrical mathematical background material, and (3) topics that include the action principle, weak gravitational fields and gravitational waves, Schwarzschild and Kerr solution, and the Friedman equation in cosmology.The book is suitable for advanced graduates and graduates, but also for established researchers wishing to be educated about the field.

Table of Contents

Preface	p. xi
Notation and Conventions	p. xiii
Spacetime	p. 1
Introduction	p. 3
Inertial and Non-Inertial Frames	p. 4
Space and Time	p. 11
Linearly Accelerated Frame	p. 16
Need for the Riemannian Geometry	p. 19
General Theory of Relativity	p. 20
Tutorial	p. 23
Literature	p. 33
What is Curvature?	p. 39
Concept of Curvature	p. 39
"Theorema Egregium" of Gauss	p. 42
The Gauss Equation	p. 44
The Geodesic Equation	p. 46
Historical Note on Riemann	p. 48
Tutorial on Surfaces	p. 49
General Relativity Basics	p. 55
Riemannian Space	p. 55
General Relativity	p. 57
Solving the Einstein Equation	p. 58
Particle Trajectories	p. 59
Path of Light Rays	p. 59
Weak Field and Newtonian Limit	p. 59
Tutorial on Indexed Quantities	p. 61
Spherically Symmetric Gravitational Field	p. 65
The Schwarzschild Solution	p. 65
Conserved Quantities	p. 67
Planetary Motion	p. 68
Deflection of Light in a Gravitational Field	p. 73
Gravitational Lensing	p. 76
Tutorial	p. 77
Geometry	p. 81
Vectors and Tensors	p. 83
Vector Spaces	p. 83
Tensor Product	p. 86
Wedge or Exterior Product	p. 91
Tutorial	p. 95
Inner Product	p. 97
Definition	p. 97
Orthonormal Bases	p. 98
Correspondence Between V and V*	p. 100
Inner Product in V*	p. 102
Orientation and Cartan Tensor	p. 104
Hodge *-Operator	p. 105
Minkowski Space	p. 109
Tutorial	p. 111
Elementary Differential Geometry	p. 115
Coordinates and Functions	p. 115
Curves and Tangent Vectors	p. 117
Tangent Space	p. 119
Vector Fields on a Manifold	p. 120
Local Basis Fields	p. 122
Lie Bracket	p. 123
Cotangent Space	p. 123
Tensor Fields	p. 125
Defining Tensors Fields	p. 125
Differential Forms and Exterior Derivative	p. 126
Closed and Exact Differential Forms	p. 128
Tutorial	p. 129
Connection and Curvature	p. 133
Directional Derivative	p. 133
Transformation Formula for ¿ijk	p. 136
Geodesics	p. 138
Covariant Derivative	p. 139
Abstract Definition	p. 140
Torsion Tensor	p. 141
Cartan Equations	p. 142
Curvature 2-Form	p. 145
Riemann-Christoffel Curvature Tensor	p. 146
Components of the Curvature Tensor	p. 148
Covariant Derivative of Tensor Fields	p. 148
Transport Round a Closed Curve	p. 152
Tutorial	p. 152
Riemannian Geometry	p. 155
Riemannian Space	p. 155
Levi-Civita Connection	p. 158
Bianchi Identity in Components	p. 161
Symmetry Properties of the Curvature Tensor	p. 163
Ricci, Einstein and Weyl Tensors	p. 165
Geodesies	p. 167
Calculating Connection Matrix	p. 169
Covariant Riemann Tensor R(W,Z;X,Y)p171
Isometries and Killing Vector Fields	p. 172
Tutorial	p. 175
Additional Topics in Geometry	p. 181
Mappings Between Manifolds	p. 181
Integral Curves of a Vector Field	p. 185
Lie Derivative	p. 186
Submanifolds	p. 190
Frobenius Theorem	p. 191
Induced Metric	p. 192
Hypersurface	p. 192
Homogeneous and Isotropic Spaces	p. 193
Maximally Symmetric Spaces	p. 195
Integration	p. 200
Integration on a Riemannian Manifold	p. 203
Tutorial	p. 206
Gravitation	p. 209
The Einstein Equation	p. 211
Stress-Energy-Momentum Tensor	p. 212
Relativists Perfect Fluid	p. 216
Interpretation of T¿¿;¿=0	p. 219
Electromagnetic Fields	p. 219
Action Principle	p. 221
Diffeomorphic Invariance	p. 228
Tutorial	p. 230
General Features of Spacetime	p. 237
Signature and Time Orientability	p. 237
Local Flatness	p. 238
Static and Stationary Spacetimes	p. 240
Fermi Transport	p. 241
Fermi-Walker Transport	p. 244
Penrose Diagrams	p. 246
Solutions of Einstein Equations	p. 249
Tutorial	p. 250
Weak Gravitational Fields	p. 257
Einstein Tensor for Weak Fields	p. 257
'Fixing a Gauge'	p. 259
The Solution	p. 260
Static Mass Distribution	p. 260
Slowly Rotating Mass Distribution	p. 264
Gravi-Magnetic Effects	p. 266
Energy and Momentum	p. 269
Energy Psuedo-Tensor	p. 270
Energy-Momentum for an Isolated System	p. 272
¿¿ up to Second-Order	p. 273
Gravitational Waves	p. 274
Detection of Gravitational Waves	p. 280
Tutorial	p. 282
Schwarzschild and Kerr Solutions	p. 287
The Schwarzschild Solution	p. 288
Kruskal-Szekeres Coordinates	p. 290
Extension of Schwarzschild Spacetime	p. 298
Spherical Mass Distribution: Interior Solution	p. 299
The Kerr Solution	p. 303
Tutorial	p. 312
Cosmology	p. 317
The Universe	p. 317
Friedman Equations	p. 320
Cosmological Constant	p. 322
Models of the Universe	p. 323
History of the Universe	p. 325
Tutorial on the FRW Metric	p. 325
Special Topics	p. 329
The Gauss Equation	p. 329
The Gauss and Codacci Equations	p. 333
Bases on M and S	p. 335
The Raychaudhuri Equation	p. 341
Index	p. 351
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