# Basic Relativity

Springer Science & Business Media, Nov 1, 2001 - Science - 452 pages
This is a comprehensive textbook for advanced undergraduates and beginning graduate students in physics or astrophysics, developing both the formalism and the physical ideas of special and general relativity in a logical and coherent way. The book is in two parts. Part one focuses on the special theory and begins with the study of relativistic kinematics from three points of view: the physical (the classic gedanken experiments), the algebraic (the Lorentz transformations), and the graphic (the Minkowski diagrams). Part one concludes with chapters on relativistic dynamics and electrodynamics. Part two begins with a chapter introducing differential geometry to set the mathematical background for general relativity. The physical basis for the theory is begun in the chapter on uniform accelerations. Subsequent chapters cover rotation, the electromagnetic field, and material media. A second chapter on differential geometry provides the background for Einstein's gravitational-field equation and Schwarzschild's solution. The physical significance of this solution is examined together with the challenges to the theory that have been successfully met inside the solar system. Other applications follow in the final chapters on astronomy and cosmology: These include black holes, quasars, and gravity waves as well as the relativistic features of an expanding universe ż including a section on the inflationary model.

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### Contents

 Principle of Relativity 3 12 A Century of Electricity and Magnetism 5 13 Maxwells Equations 7 14 Stellar Aberration 8 16 The TroutonNoble Experiment 13 The Physical Arguments 18 22 Some Applications 27 23 Velocity Addition 35
 Summary of Metric Relationships 234 86 Kinematic Characteristics of the System 235 87 Falling Bodies 241 88 Geodesic Paths 244 89 Falling Clocks 249 810 A Supported Object 252 811 Local Coordinates 253 Summary of Kinematic Relationships 256

 24 The Twin Paradox 37 25 The Pole in the Barn Paradox 40 26 Coordinate Frames of Reference 42 Problems 44 The Algebraic and Graphic Arguments 48 32 Other Applications 51 33 Velocity Addition 55 34 The Invariant Interval 57 35 The Minkowski Diagram 61 36 Use of the Minkowski Diagram 66 37 FourVectors 71 38 Velocity and Acceleration FourVectors 72 39 The Propagation FourVector 75 310 Doppler Effect 78 311 Experimental EvidenceKinematics 81 Problems 84 Mathematical Tools 91 42 The Lorentz Transformation 98 43 Vector Operators 100 44 Tensors 103 45 The Metric Inequality 107 Summary 109 Problems 110 Dynamics 113 51 The Physical Assumptions 114 52 The EulerLagrange Formalism 121 53 The Momentum FourVector 125 54 The FourForce 127 55 Torque 134 56 Collisions 136 57 Experimental EvidenceDynamics 142 Problems 144 Electromagnetic Theory 148 62 Lorentz Force 153 63 Moving Magnet Problem 157 64 TroutonNoble Experiment 161 65 Maxwells Equations 163 66 Electromagnetic Potentials 166 67 EnergyMomentum Tensor 168 Problems 171 Part II 175 Differential Geometry I 177 72 The Metric Tensor 178 73 Vectors 181 74 The Rectilinear Case 183 75 The Polar Case 185 76 Contravariant Metric Tensor 190 77 Tensors 191 Summary of Tensor Algebra 195 78 Parallel Displacement 196 79 The Geodesic Path 204 710 Parallel Displacement of Covariant Vectors 208 711 Covariant Derivatives 209 712 SpaceTime Differential Geometry 213 Summary of Four Vectors 217 Problems 218 Uniform Acceleration 221 82 Accelerating a Point Mass 224 83 A Uniformly Accelerated Frame 228 84 Uniformly Accelerated Coordinates 231 85 The Matter of Metric 232
 812 Dynamics 257 813 Gravitational Force and Constants of Motion 261 Problems 265 Rotation and the Electromagnetic Field 269 92 Physical Interpretation 271 93 The Geodesic Equation 273 94 Dynamics 274 95 General Electromagnetic Fields 278 96 NonGeodesic Paths 281 97 Generally Covariant Field Equations 283 Problems 284 The Material Medium 287 102 Dust Particles 288 103 Ideal Gas 290 105 The Total Tensor 295 Problems 296 Differential Geometry II Curved Surfaces 298 112 A Curvature Criterion 304 113 Curvature Tensor on a Sphere 306 114 Ricci Tensor and the Scalar Curvature 307 Problems 309 General Relativity 312 121 The Principle of Equivalence 313 122 Einsteins Field Equation 315 123 Evaluation of the Constant 318 124 The Schwarzschild Solution 321 125 Kinematic Characteristics of the Field 324 126 Falling Bodies 328 127 FourVelocity 330 129 Theory as Construct 338 1210 Three Tests of General Relativity 339 1211 New Tests and Challenges 344 Problems 346 Astrophysics 349 132 Black Holes 351 133 Rotating Black Holes 358 134 Evidence for Compact Objects 368 135 Gravity Waves 377 Problems 388 Cosmology 392 142 The Cosmological Constant 393 143 ThreeDimensional Hypersurface 394 144 General Solution of the Field Equation 398 145 Einstein and de Sitter Solutions 400 146 The MatterDominated Universe 402 147 Critical Mass 405 148 Measuring a Flat MatterDominated Einsteinde Sitter Universe 407 149 The Inflationary Universe 416 Problems 423 Appendixes 425 B Calculus of Variations 427 C The Geodesic Equation 430 D The Geodesic Equation in Coordinate Form 431 E Uniformly Accelerated Transformation Equations 432 F The RiemannChristoffel Curvature Tensor 434 G Transformation to the Tangent Plane 436 H General Lorentz Transformation and the Stress Tensor 438 Answers to Selected Problems 439 References 443 Index 445 Copyright