Relativity Demystified - A Self-Teaching Guide

370 Pages·2005·1.646 MB·English

by  David McMahon, Paul M. Alsing

RelativityDemystified
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RelativityDemystified
DAVID McMAHON
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Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Manufactured in the United States of America. Except as
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DOI: 10.1036/0071455450
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CONTENTS
Preface xi
CHAPTER 1 A Quick Review of Special Relativity 1
Frame of Reference 5
Clock Synchronization 5
Inertial Frames 6
Galilean Transformations 7
Events 7
The Interval 8
Postulates of Special Relativity 9
Three Basic Physical Implications 13
Light Cones and Spacetime Diagrams 17
Four Vectors 19
Relativistic Mass and Energy 20
Quiz 21
CHAPTER 2 Vectors, One Forms, and the Metric 23
Vectors 23
New Notation 25
Four Vectors 27
The Einstein Summation Convention 28
Tangent Vectors, One Forms, and the
Coordinate Basis 29
Coordinate Transformations 31
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CONTENTS
vi
The Metric 32
The Signature of a Metric 36
The Flat Space Metric 37
The Metric as a Tensor 37
Index Raising and Lowering 38
Index Gymnastics 41
The Dot Product 42
Passing Arguments to the Metric 43
Null Vectors 45
The Metric Determinant 45
Quiz 45
CHAPTER 3 More on Tensors 47
Manifolds 47
Parameterized Curves 49
Tangent Vectors and One Forms, Again 50
Tensors as Functions 53
Tensor Operations 54
The Levi-Cevita Tensor 59
Quiz 59
CHAPTER 4 Tensor Calculus 60
Testing Tensor Character 60
The Importance of Tensor Equations 61
The Covariant Derivative 62
The Torsion Tensor 72
The Metric and Christoffel Symbols 72
The Exterior Derivative 79
The Lie Derivative 81
The Absolute Derivative and Geodesics 82
The Riemann Tensor 85
The Ricci Tensor and Ricci Scalar 88
The Weyl Tensor and Conformal Metrics 90
Quiz 91
CONTENTS
vii
CHAPTER 5 Cartan’s Structure Equations 93
Introduction 93
Holonomic (Coordinate) Bases 94
Nonholonomic Bases 95
Commutation Coefficients 96
Commutation Coefficients and Basis
One Forms 98
Transforming between Bases 100
A Note on Notation 103
Cartan’s First Structure Equation and the
Ricci Rotation Coefficients 104
Computing Curvature 112
Quiz 120
CHAPTER 6 The Einstein Field Equations 122
Equivalence of Mass in Newtonian Theory 123
Test Particles 126
The Einstein Lift Experiments 126
The Weak Equivalence Principle 130
The Strong Equivalence Principle 130
The Principle of General Covariance 131
Geodesic Deviation 131
The Einstein Equations 136
The Einstein Equations with Cosmological
Constant 138
An Example Solving Einstein’s Equations
in 2+1 Dimensions 139
Energy Conditions 152
Quiz 152
CHAPTER 7 The Energy-Momentum Tensor 155
Energy Density 156
Momentum Density and Energy Flux 156
Stress 156
Conservation Equations 157
CONTENTS
viii
Dust 158
Perfect Fluids 160
Relativistic Effects on Number Density 163
More Complicated Fluids 164
Quiz 165
CHAPTER 8 Killing Vectors 167
Introduction 167
Derivatives of Killing Vectors 177
Constructing a Conserved Current
with Killing Vectors 178
Quiz 178
CHAPTER 9 Null Tetrads and the Petrov
Classification 180
Null Vectors 182
A Null Tetrad 184
Extending the Formalism 190
Physical Interpretation and the Petrov
Classification 193
Quiz 201
CHAPTER 10 The Schwarzschild Solution 203
The Vacuum Equations 204
A Static, Spherically Symmetric Spacetime 204
The Curvature One Forms 206
Solving for the Curvature Tensor 209
The Vacuum Equations 211
The Meaning of the Integration Constant 214
The Schwarzschild Metric 215
The Time Coordinate 215
The Schwarzschild Radius 215
Geodesics in the Schwarzschild Spacetime 216
Particle Orbits in the Schwarzschild
Spacetime 218
CONTENTS
ix
The Deflection of Light Rays 224
Time Delay 229
Quiz 230
CHAPTER 11 Black Holes 233
Redshift in a Gravitational Field 234
Coordinate Singularities 235
Eddington-Finkelstein Coordinates 236
The Path of a Radially Infalling Particle 238
Eddington-Finkelstein Coordinates 239
Kruskal Coordinates 242
The Kerr Black Hole 244
Frame Dragging 249
The Singularity 252
A Summary of the Orbital Equations
for the Kerr Metric 252
Further Reading 253
Quiz 254
CHAPTER 12 Cosmology 256
The Cosmological Principle 257
A Metric Incorporating Spatial
Homogeneity and Isotropy 257
Spaces of Positive, Negative, and
Zero Curvature 262
Useful Definitions 264
The Robertson-Walker Metric and the
Friedmann Equations 267
Different Models of the Universe 271
Quiz 276
CHAPTER 13 Gravitational Waves 279
The Linearized Metric 280
Traveling Wave Solutions 284
The Canonical Form and Plane Waves 287