Introduction to General Relativity

by Walecka, John Dirk

• ISBN13: 9789812705853
• ISBN10: 9812705856
• eBook ISBN(s): 9789813101616
• Format: Paperback
• Copyright: 2007-07-30
• Publisher: World Scientific Pub Co Inc

Summary

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.

Table of Contents

Preface	p. vii
Introduction	p. 1
Particle on a Two-Dimensional Surface	p. 9
Infinitesimal Displacements	p. 10
Lagrange's Equations	p. 12
Reciprocal Basis	p. 14
Geodesic Motion	p. 19
Role of Coordinates	p. 24
Curvilinear Coordinate Systems	p. 27
Line Element	p. 27
Reciprocal Basis	p. 29
Metric	p. 29
Vectors	p. 30
Tensors	p. 31
Change of Basis	p. 33
Transformation Law	p. 36
Affine Connection	p. 39
Example: Polar Coordinates	p. 41
Particle on a Two-Dimensional Surface-Revisited	p. 43
Motion in Three-Dimensional Euclidian Space	p. 43
Parallel Displacement	p. 46
Some Tensor Analysis	p. 51
Covariant Derivative	p. 51
The Riemann Curvature Tensor	p. 54
Second Covariant Derivative	p. 60
Covariant Differentiation	p. 64
Symmetry Properties of the Riemann Tensor	p. 67
Bianchi Identities	p. 68
The Einstein Tensor	p. 69
Example-Surface of a Sphere	p. 72
Volume element	p. 76
Special Relativity	p. 87
Basic Principles	p. 87
Four-Vectors	p. 88
Relativistic Particle Motion	p. 91
Lorentz Transformations	p. 98
General Tensor Transformation Law	p. 109
Relativistic Hydrodynamics	p. 111
Transition to General Relativity	p. 120
General Relativity	p. 123
Einstein's Theory	p. 123
Newtonian Limit	p. 126
The Equivalence Principle	p. 131
Local Freely Falling Frame (LF[superscript 3])	p. 133
Spherically Symmetric Solution to Field Equations	p. 135
Solution in Vacuum	p. 143
Interpretation of Schwarzschild Metric	p. 148
A Few Applications	p. 152
Precession of Perihelion	p. 155
Lagrangian	p. 156
Equations of Motion	p. 159
Equilibrium Circular Orbits	p. 163
Small Oscillations About Circular Orbits	p. 164
Some Numbers and Comparison with Experiment	p. 169
Deflection of Light	p. 171
Gravitational Redshift	p. 173
Basic Observation	p. 174
Frequency Shift	p. 174
Propagation of Light	p. 176
A Cyclic Process	p. 178
Neutron Stars	p. 181
Hydrodynamics in General Relativity	p. 181
Tolman-Oppenheimer-Volkoff (TOV) Equations	p. 190
Relativistic Mean Field Theory of Nuclear Matter (RMFT)	p. 201
Neutron Stars and Black Holes	p. 210
Cosmology	p. 217
Uniform Mass Background	p. 218
Robertson-Walker Metric with k = 0	p. 220
Solution to Einstein's Equations	p. 225
Interpretation	p. 229
Cosmological Redshift	p. 233
Horizon	p. 237
Some Comments	p. 239
Gravitational Radiation	p. 241
Linearized Theory	p. 241
Auxiliary (Lorentz) Condition	p. 245
Plane-Wave Solution to the Einstein Field Equations	p. 250
Interpretation	p. 253
Detection	p. 257
Special Topics	p. 263
Einstein-Hilbert Lagrangian	p. 264
Cosmological Constant	p. 267
Additional Scalar Field	p. 268
Robertson-Walker Metric with k [not equal] 0	p. 270
Inflation	p. 275
Problems	p. 281
Reduction of g[superscript mu nu delta] R[superscript mu nu] to covariant divergences	p. 321
Robertson-Walker Metric with k [not equal] 0	p. 335
Bibliography	p. 327
Index	p. 331
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