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The Einstein Equations RelativityDemystified i Demystified Series Advanced Statistics Demystified Math Proofs Demystified Algebra Demystified Math Word Problems Demystified Anatomy Demystified Medical Terminology Demystified asp.net Demystified Meteorology Demystified Astronomy Demystified Microbiology Demystified Biology Demystified OOP Demystified Business Calculus Demystified Options Demystified Business Statistics Demystified Organic Chemistry Demystified C++ Demystified Personal Computing Demystified Calculus Demystified Pharmacology Demystified Chemistry Demystified Physics Demystified College Algebra Demystified Physiology Demystified Databases Demystified Pre-Algebra Demystified Data Structures Demystified Precalculus Demystified Differential Equations Demystified Probability Demystified Digital Electronics Demystified Project Management Demystified Earth Science Demystified Quality Management Demystified Electricity Demystified Quantum Mechanics Demystified Electronics Demystified Relativity Demystified Environmental Science Demystified Robotics Demystified Everyday Math Demystified Six Sigma Demystified Genetics Demystified sql Demystified Geometry Demystified Statistics Demystified Home Networking Demystified Trigonometry Demystified Investing Demystified uml Demystified Java Demystified Visual Basic 2005 Demystified JavaScript Demystified Visual C # 2005 Demystified Linear Algebra Demystified xml Demystified Macroeconomics Demystified ii RelativityDemystified DAVID McMAHON McGRAW-HILL New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto iii Copyright © 2006 by The McGraw-Hill Companies, Inc. 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DOI: 10.1036/0071455450 For more information about this title, click here 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 v vi CONTENTS 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 viii CONTENTS 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 x CONTENTS The Behavior of Particles as a Gravitational Wave Passes 291 The Weyl Scalars 294 Review: Petrov Types and the Optical Scalars 295 pp Gravity Waves 297 Plane Waves 301 The Aichelburg-Sexl Solution 303 Colliding Gravity Waves 304 The Effects of Collision 311 More General Collisions 312 Nonzero Cosmological Constant 318 Further Reading 321 Quiz 322 Final Exam 323 Quiz and Exam Solutions 329 References and Bibliography 333 Index 337 PREFACE The theory of relativity stands out as one of the greatest achievements in science. The “special theory”, which did not include gravity, was put forward by Einstein in 1905 to explain many troubling facts that had arisen
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