A working knowledge of Einstein's theory of general relativity is an essential tool for every physicist today. This self-contained book is an introductory text on the subject aimed at first-year graduate students, or advanced undergraduates, in physics that assumes only a basic understanding of classical Lagrangian mechanics. The mechanics problem of a point mass constrained to move without friction on a two-dimensional surface of arbitrary shape serves as a paradigm for the development of the mathematics and physics of general relativity. After reviewing special relativity, the basic principles of general relativity are presented, and the most important applications are discussed. The final special topics section guides the reader through a few important areas of current research.This book will allow the reader to approach the more advanced texts and monographs, as well as the continual influx of fascinating new experimental results, with a deeper understanding and sense of appreciation.
ISBN-13: 9789812705853
Media Type: Paperback
Publisher: World Scientific Publishing Company - Incorporated
Publication Date: 04-20-2007
Pages: 356
Product Dimensions: 5.90(w) x 8.90(h) x 0.70(d)
Table of Contents
Preface vii
Introduction 1
Particle on a Two-Dimensional Surface 9
Infinitesimal Displacements 10
Lagrange's Equations 12
Reciprocal Basis 14
Geodesic Motion 19
Role of Coordinates 24
Curvilinear Coordinate Systems 27
Line Element 27
Reciprocal Basis 29
Metric 29
Vectors 30
Tensors 31
Change of Basis 33
Transformation Law 36
Affine Connection 39
Example: Polar Coordinates 41
Particle on a Two-Dimensional Surface-Revisited 43
Motion in Three-Dimensional Euclidian Space 43
Parallel Displacement 46
Some Tensor Analysis 51
Covariant Derivative 51
The Riemann Curvature Tensor 54
Second Covariant Derivative 60
Covariant Differentiation 64
Symmetry Properties of the Riemann Tensor 67
Bianchi Identities 68
The Einstein Tensor 69
Example-Surface of a Sphere 72
Volume element 76
Special Relativity 87
Basic Principles 87
Four-Vectors 88
Relativistic Particle Motion 91
Lorentz Transformations 98
General Tensor Transformation Law 109
Relativistic Hydrodynamics 111
Transition to General Relativity 120
General Relativity 123
Einstein's Theory 123
Newtonian Limit 126
The Equivalence Principle 131
Local Freely Falling Frame (LF[superscript 3]) 133
Spherically Symmetric Solution to Field Equations 135
Solution in Vacuum 143
Interpretation of Schwarzschild Metric 148
A Few Applications 152
Precession of Perihelion 155
Lagrangian 156
Equations of Motion 159
Equilibrium Circular Orbits 163
Small Oscillations About Circular Orbits 164
Some Numbers and Comparison with Experiment 169
Deflection of Light 171
Gravitational Redshift 173
Basic Observation 174
Frequency Shift 174
Propagation of Light 176
A Cyclic Process 178
Neutron Stars 181
Hydrodynamics in General Relativity 181
Tolman-Oppenheimer-Volkoff (TOV) Equations 190
Relativistic Mean Field Theory of Nuclear Matter (RMFT) 201
Neutron Stars and Black Holes 210
Cosmology 217
Uniform Mass Background 218
Robertson-Walker Metric with k = 0 220
Solution to Einstein's Equations 225
Interpretation 229
Cosmological Redshift 233
Horizon 237
Some Comments 239
Gravitational Radiation 241
Linearized Theory 241
Auxiliary (Lorentz) Condition 245
Plane-Wave Solution to the Einstein Field Equations 250
Interpretation 253
Detection 257
Special Topics 263
Einstein-Hilbert Lagrangian 264
Cosmological Constant 267
Additional Scalar Field 268
Robertson-Walker Metric with k [not equal] 0 270
Inflation 275
Problems 281
Reduction of g[superscript mu nu delta]R[subscript mu nu] to covariant divergences 321
Robertson-Walker Metric with k [not equal] 325
Bibliography 327
Index 331
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