Introduction to General Relativity

A student-friendly style, over 100 illustrations, and numerous exercises are brought together in this textbook for advanced undergraduate and beginning graduate students in physics and mathematics. Lewis Ryder develops the theory of general relativity in detail. Covering the core topics of black holes, gravitational radiation, and cosmology, he provides an overview of general relativity and its modern ramifications. The book contains chapters on gravitational radiation, cosmology, and connections between general relativity and the fundamental physics of the microworld. It explains the geometry of curved spaces and contains key solutions of Einstein's equations - the Schwarzschild and Kerr solutions. Mathematical calculations are worked out in detail, so students can develop an intuitive understanding of the subject, as well as learn how to perform calculations. The book also includes topics concerned with the relation between general relativity and other areas of fundamental physics. Password protected solutions for instructors are available at www.cambridge.org/9780521845632. Cover 1 Half-title 3 Title 5 Copyright 6 Dedication 7 Contents 11 Preface 15 Notation, important formulae and physical constants 16 1 Introduction 19 1.1 The need for a theory of gravity 19 1.2 Gravitation and inertia: the Equivalence Principle in mechanics 21 1.1.1 A remark on inertial mass 25 1.1.2 Tidal forces 26 1.3 The Equivalence Principle and optics 27 1.4 Curved surfaces 32 Further reading 34 Problems 34 2 Special Relativity, non-inertial effects and electromagnetism 36 2.1 Special Relativity: Einstein’s train 36 2.1.1 Minkowski space-time 39 2.1.2 Lorentz transformations 40 2.2 Twin paradox: accelerations 44 2.3 Rotating frames: the Sagnac effect 47 2.3.1 Clock synchronisation 50 2.4 Inertia: Newton versus Mach 52 2.5 Thomas precession 54 2.6 Electromagnetism 58 2.6.1 Maxwell’s equations 61 2.7 Principle of General Covariance 61 Further reading 63 Problems 64 3 Differential geometry I: vectors, differential forms and absolute differentiation 65 3.1 Space-time as a differentiable manifold 65 3.2 Vectors and vector fields 67 3.2.1 Holonomic and anholonomic bases 72 3.3 One-forms 73 3.3.1 Transformation rules 78 3.3.2 A note on orthogonal coordinate systems 78 3.4 Tensors 79 3.4.1 Contraction 81 3.4.2 Symmetry and antisymmetry 81 3.4.3 Quotient theorem 82 3.5 Differential forms: Hodge duality 83 3.5.1 Remarks on the algebra of p-forms 88 3.5.2 A note on orientation 89 3.6 Exterior derivative operator: generalised Stokes’ theorem 90 3.6.1 Generalised Stokes’ theorem 93 3.6.2 Closed and exact forms 94 3.7 Maxwell’s equations and differential forms 95 3.8 Metric tensor 97 3.8.1 Holonomic and anholonomic (coordinate and non-coordinate) bases 99 3.8.2 Tensor densities: volume elements 101 3.9 Absolute differentiation: connection forms 104 3.9.1 Tensors 109 3.10 Parallel transport 111 3.11 Some relations involving connection coefficients 115 3.11.1 Derivatives of scalar and tensor densities 117 3.11.2 Note on torsion and curvature 119 3.12 Examples 120 3.12.1 Plane E: coordinate basis 121 3.12.2 Sphere S 123 3.13 General formula for connection coefficients 125 Further reading 128 Problems 128 4 Differential geometry II: geodesics and curvature 130 4.1 Autoparallel curves and geodesics 130 4.1.1 Autoparallel curves 132 4.1.2 Geodesics 132 4.1.3 Examples 135 4.2 Geodesic coordinates 137 4.3 Curvature 139 4.3.1 Round trips by parallel transport 141 4.4 Symmetries of the Riemann tensor 143 4.5 Ricci tensor and curvature scalar 144 4.5.1 Plane and sphere: holonomic basis 144 4.5.2 Plane and sphere: anholonomic basis 146 4.6 Curvature 2-form 147 4.7 Geodesic deviation 150 4.8 Bianchi identities 152 Further reading 153 Problems 153 5 Einstein field equations, the Schwarzschild solution and experimental tests of General Relativity 155 5.1 Newtonian limit 155 5.2 Einstein field equations 157 5.2.1 Vacuum field equations 157 5.2.2 Energy-momentum tensor 158 5.2.3 Matter field equations 161 5.3 Schwarzschild solution 164 5.3.1 Apparent ‘singularity’ at r = 2m 167 5.3.2 Isotropic form of the Schwarzschild solution 169 5.4 Time dependence and spherical symmetry: Birkhoff’s theorem 169 5.5 Gravitational red-shift 172 5.6 Geodesics in Schwarzschild space-time 176 5.7 Precession of planetary orbits 178 5.8 Deflection of light 180 5.9 Note on PPN formalism 182 5.10 Gravitational lenses 183 5.11 Radar echoes from planets 187 5.12 Radial motion in a Schwarzschild field: black holes – frozen stars 191 5.13 A gravitational clock effect 194 Further reading 196 Problems 196 6 Gravitomagnetic effects: gyroscopes and clocks 198 6.1 Linear approximation 198 6.1.1 Static case: mass 203 6.1.2 Rotating body: angular momentum 205 6.2 Precession of gyroscopes: the Lense–Thirring effect 209 6.2.1 Gravity Probe B 216 6.2.2 ‘Inertial drag’ 217 6.2.3 Lense–Thirring effect and Mach’s Principle 217 6.3 Gravitomagnetism 218 6.4 Gravitomagnetic clock effect 222 6.5 Fermi–Walker transport: tetrad formalism 225 6.6 Lie derivatives, Killing vectors and groups of motion 229 6.6.1 Groups of motion 234 6.7 Static and stationary space-times 237 6.8 Killing vectors and conservation laws 241 Further reading 243 Problems 244 7 Gravitational collapse and black holes 245 7.1 The interior Schwarzschild solution and the Tolman–Oppenheimer–Volkoff equation 246 7.2 Energy density and binding energy 255 7.3 Degenerate stars: white dwarfs and neutron stars 261 7.4 Schwarzschild orbits: Eddington–Finkelstein coordinates 269 7.5 Kruskal–Szekeres coordinates 273 7.6 Einstein–Rosen bridge and wormholes 277 7.7 Conformal treatment of infinity: Penrose diagrams 279 7.8 Rotating black holes: Kerr solution 283 7.9 The ergosphere and energy extraction from a black hole 289 7.10 Surface gravity 298 7.11 Thermodynamics of black holes and further observations 305 7.12 Global matters: singularities, trapped surfaces and Cosmic Censorship 309 Further reading 311 Problems 312 8 Action principle, conservation laws and the Cauchy problem 313 8.1 Gravitational action and field equations 313 8.2 Energy-momentum pseudotensor 318 8.3 Cauchy problem 322 Further reading 327 Problems 327 9 Gravitational radiation 328 9.1 Weak field approximation 328 9.1.1 Spin 2 graviton 331 9.1.2 The effect of gravitational waves 333 9.2 Radiation from a rotating binary source 335 9.2.1 Flux 338 9.2.2 Radiated energy 340 9.2.3 Spin-up and the binary pulsar PSR 1913+16 343 9.2.4 Search for gravitational waves 345 9.3 Parallels between electrodynamics and General Relativity: Petrov classification 346 9.3.1 A geometric approach to electrodynamics 346 9.3.2 Petrov classification 351 Further reading 358 Problems 358 10 Cosmology 359 10.1 Brief description of the Universe 359 10.2 Robertson–Walker metric 362 10.3 Hubble’s law and the cosmological red-shift 373 10.4 Horizons 375 10.5 Luminosity–red-shift relation 378 10.6 Dynamical equations of cosmology 381 10.6.1 Newtonian interpretation 385 10.6.2 Critical density 386 10.7 Friedmann models and the cosmological constant 389 10.8 Cosmic background radiation 393 10.9 Brief sketch of the early Universe 395 10.10 The inflationary universe and the Higgs mechanism 401 Further reading 409 Problems 409 11 Gravitation and field theory 410 11.1 Electrodynamics as an abelian gauge theory 412 11.2 Non-abelian gauge theories 418 11.3 Gauging Lorentz symmetry: torsion 427 11.4 Dirac equation in Schwarzschild space-time 434 11.5 Five dimensions: gravity plus electromagnetism 436 Further reading 441 Problems 442 References 443 Index 457 A student-friendly style, over 100 illustrations, and numerous exercises are brought together in this textbook for advanced undergraduate and beginning graduate students in physics and mathematics. Lewis Ryder develops the theory of general relativity in detail. Covering the core topics of black holes, gravitational radiation, and cosmology, he provides an overview of general relativity and its modern ramifications. The book contains chapters on gravitational radiation, cosmology, and connections between general relativity and the fundamental physics of the microworld. It explains the geometry of curved spaces and contains key solutions of Einstein's equations - the Schwarzschild and Kerr solutions. Mathematical calculations are worked out in detail, so students can develop an intuitive understanding of the subject, as well as learn how to perform calculations. The book also includes topics concerned with the relation between general relativity and other areas of fundamental physics. Selected solutions for instructors are available under Resources.