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.
Product Details
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
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