Richard A. Mould. Basic Relativity. With 144 Figures. Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest
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1 Richard A. Mould Basic Relativity With 144 Figures Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest
2 Contents Preface vii PARTI 1. Principles of Relativity Galileo's Principle A Century of Electricity and Magnetism Maxwell's Equations Stellar Aberration The Michelson-Morley Experiment The Trouton-Noble Experiment 13 Problems The Physical Arguments Physical Ideas Some Applications Velocity Addition The Twin Paradox The Pole in the Barn Paradox Coordinate Frames of Reference 42 Problems The Algebraic and Graphic Arguments The Lorentz Transformation Other Applications Velocity Addition The Invariant Interval The Minkowski Diagram Use of the Minkowski Diagram Four-Vectors Velocity and Acceleration Four-Vectors The Propagation Four-Vector Doppler Effect Experimental Evidence Kinematics 81 Problems Mathematical Tools Matrices The Lorentz Transformation Vector Operators Tensors 103 XI
3 xii Contents 4.5 The Metric Inequality 107 Summary 109 Problems Dynamics The Physical Assumptions The Euler-Lagrange Formalism The Momentum Four-Vector The Four-Force Torque Collisions Experimental Evidence Dynamics 142 Problems Electromagnetic Theory ^^^_ Electric and Magnetic Fields Lorentz Force Moving Magnet Problem Trouton-Noble Experiment Maxwell's Equations Electromagnetic Potentials Energy-Momentum Tensor 168 Problems 170 PART II 7. Differential Geometry I The Scalar Invariant The Metric Tensor Vectors The Rectilinear Case The Polar Case Contravariant Metric Tensor Tensors 191 Summary of Tensor Algebra Parallel Displacement The Geodesic Path Parallel Displacement of Covariant Vectors Covariant Derivatives Space-Time Differential Geometry 213 Summary of Four-Vectors 217 Problems Uniform Acceleration Nonrigid Bodies Accelerating a Point Mass 224
4 Contents xiii 8.3 A Uniformly Accelerated Frame Uniformly Accelerated Coordinates The Matter of Metric 232 Summary of Metric Relationships Kinematic Characteristics of the System Falling Bodies Geodesic Paths Falling Clocks A Supported Object Local Coordinates 253 Summary of Kinematic Relationships Dynamics Gravitational Force and Constants of Motion 261 Problems Rotation and the Electromagnetic Field The Rotation Transformation Physical Interpretation The Geodesic Equation Dynamics General Electromagnetic Fields Nongeodesic Paths Generally Covariant Field Equations 283 Problems The Material Medium The Energy-Momentum Tensor Dust Particles Ideal Gas Internal Forces The Total Tensor 295 Problems Differential Geometry II: Curved Surfaces A Spherical Surface A Curvature Criterion Curvature Tensor on a Sphere Ricci Tensor and the Scalar Curvature 307 Problems General Relativity The Principle of Equivalence Einstein's Field Equation Evaluation of the Constant The Schwarzschild Solution Kinematic Characteristics of the Field 324
5 xiv Contents 12.6 Falling Bodies Four-Velocity Dynamics Theory as Construct Three Tests of General Relativity New Tests and Challenges 344 Problems Astrophysics Compact Objects Black Holes Rotating Black Holes Evidence for Compact Objects Gravity Waves 377 Problems Cosmology The Cosmological Principle The Cosmological Constant Three-Dimensional Hypersurface General Solution of the Field Equation Einstein and de Sitter Solutions The Matter-Dominated Universe Critical Mass Measuring a Flat, Matter-Dominated Universe The Inflationary Universe 416 Problems 423 Appendixes A. The Lorentz Transformation 425 B. Calculus of Variations 427 С The Geodesic Equations 430 D. The Geodesic Equation in Coordinate Form 431 E. Uniformly Accelerated Transformation Equations 432 F. The Riemann-Christoffel Curvature Tensor 434 G. Transformation to the Tangent Plane 436 H. General Lorentz Transformation and the Stress Tensor Answers to Selected Problems 439 References 443 Index 445
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