Relativity SPECIAL, GENERAL, AND COSMOLOGICAL SECOND EDITION. Wolfgang Rindler. Professor of Physics The University of Texas at Dallas

Transcription

1 Relativity SPECIAL, GENERAL, AND COSMOLOGICAL SECOND EDITION Wolfgang Rindler Professor of Physics The University of Texas at Dallas OXPORD UNIVERSITY PRESS

2 Contents Introduction l 1 From absolute space and time to influenceable spacetime: an overview Definition of relativity Newton's laws and inertial frames The Galilean transformation Newtonian relativity Objections to absolute space; Mach's principle The ether Michelson and Morley's search for the ether Lorentz's ether theory Origins of special relativity Further arguments for Einstein's two postulates Cosmology and first doubts about inertial frames Inertial and gravitational mass Einstein's equivalence principle Preview of general relativity Caveats on the equivalence principle Gravitational frequency shift and light bending 24 Exercises Special Relativity 31 2 Foundations of special relativity; The Lorentz transformation On the nature of physical theories Basic features of special relativity Relativistic problem solving Relativity of simultaneity, time dilation and length contraction: a preview The relativity principle and the homogeneity and isotropy of inertial frames The coordinate lattice; Definitions of simultaneity Derivation of the Lorentz transformation 43

3 xii Contents 2.8 Properties of the Lorentz transformation 2.9 Graphical representation of the Lorentz transformation 2.10 The relativistic speed limit 2.11 Which transformations are allowed by the relativity principle? Exercises 2 3 Relativistic kinematics Introduction World-picture and world-map Length contraction Length contraction paradox Time dilation; The twin paradox Velocity transformation; Relative and mutual velocity Acceleration transformation; Hyperbolic motion Rigid motion and the uniformly accelerated rod 71 Exercises Relativistic optics Introduction The drag effect The Doppler effect Aberration The visual appearance of moving objects 82 Exercises Spacetime and four-vectors The discovery of Minkowski space Three-dimensional Minkowski diagrams Light cones and intervals Three-vectors Four-vectors The geometry of four-vectors Plane waves 103 Exercises Relativistic particle mechanics Domain of sufficient validity of Newtonian mechanics The axioms of the new mechanics The equivalence of mass and energy Four-momentum identities Relativistic billiards The zero-momentum frame ' Threshold energies Light quanta and de Broglie waves 119

4 Contents xiii 6.9 The Compton effect Four-force and three-force 123 Exercises Four-tensors; Electromagnetism in vacuum Tensors: Preliminary ideas and notations Tensors: Definition and properties Maxwell's equations in tensor form The four-potential Transformation of e and b; The dual field The field of a uniformly moving point charge The field of an infinite straight current The energy tensor of the electromagnetic field From the mechanics of the field to the mechanics of material continua 154 Exercises II General Relativity Curved spaces and the basic ideas of general relativity Curved surfaces Curved spaces of higher dimensions Riemannian spaces A plan for general relativity 177 Exercises Static and stationary spacetimes The coordinate lattice Synchronization of clocks First standard form of the metric Newtonian support for the geodesic law of motion Symmetries and the geometric characterization of static and stationary spacetimes Canonical metric and relativistic potentials The uniformly rotating lattice in Minkowski space 198 Exercises Geodesies, curvature tensor and vacuum field equations Tensors for general relativity Geodesies Geodesic coordinates Covariant and absolute differentiation The Riemann curvature tensor Einstein's vacuum field equations 221 Exercises

5 xiv Contents 11 The Schwarzschild metric Derivation of the metric Properties of the metric The geometry of the Schwarzschild lattice Contributions of the spatial curvature to post-newtonian effects Coordinates and measurements The gravitational frequency shift Isotropic metric and Shapiro time delay Particle orbits in Schwarzschild space The precession of Mercury's orbit Photon orbits Deflection of light by a spherical mass Gravitational lenses de Sitter precession via rotating coordinates 252 Exercises Black holes and Kruskal space Schwarzschild black holes Potential energy; A general-relativistic 'proof of E = me The extendibility of Schwarzschild spacetime The uniformly accelerated lattice Kruskal space Black-hole thermodynamics and related topics 279 Exercises An exact plane gravitational wave Introduction The plane-wave metric When wave meets dust Inertial coordinates behind the wave When wave meets light The Penrose topology Solving the field equation 293 Exercises The full field equations; de Sitter space The laws of physics in curved spacetime At last, the full field equations The cosmological constant Modified Schwarzschild space de Sitter space Anti-de Sitter space 312 Exercises

6 Contents xv 15 Linearized general relativity 15.1 The basic equations 15.2 Gravitational waves; The TT gauge 15.3 Some physics of plane waves 15.4 Generation and detection of gravitational waves 15.5 The electromagnetic analogy in linearized GR Exercises 15 m Cosmology 16 Cosmological spacetimes 16.1 The basic facts 16.2 Beginning to construct the model 16.3 Milne's model 16.4 The Friedman-Robertson-Walker metric 16.5 Robertson and Walker's theorem Exercises Light propagation in FRW universes Representation of FRW universes by subuniverses The cosmological frequency shift Cosmological horizons The apparent horizon Observables Exercises Dynamics of FRW universes 18.1 Applying the field equations 18.2 What the field equations tell us 18.3 The Friedman models 18.4 Once again, comparison with observation Inflation 18.6 The anthropic principle Exercises Appendix: Curvature tensor components for the diagonal metric 419 Index 423