groen o., hervik s. einstein''s general theory of relativity

Einstein’s General Theory of Relativity Øyvind Grøn and Sigbjørn Hervik ii Version 9th December 2004. c Grøn & Hervik. Contents Preface xv Notation xvii I INTRODUCTION: N EWTONIAN PHYSICS AND SPECIAL RELATIVITY 1 1 Relativity Principles and Gravitation 3 1.1 Newtonian mechanics . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Galilei–Newton’s principle of Relativity . . . . . . . . . . . . . . 4 1.3 The principle of Relativity . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Newton’s law of Gravitation . . . . . . . . . . . . . . . . . . . . . 6 1.5 Local form of Newton’s Gravitational law . . . . . . . . . . . . . 8 1.6 Tidal forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.7 The principle of equivalence . . . . . . . . . . . . . . . . . . . . . 14 1.8 The covariance principle . . . . . . . . . . . . . . . . . . . . . . . 15 1.9 Mach’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 The Special Theory of Relativity 21 2.1 Coordinate systems and Minkowski-diagrams . . . . . . . . . . 21 2.2 Synchronization of clocks . . . . . . . . . . . . . . . . . . . . . . 23 2.3 The Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4 Relativistic time-dilatation . . . . . . . . . . . . . . . . . . . . . . 25 2.5 The relativity of simultaneity . . . . . . . . . . . . . . . . . . . . 26 2.6 The Lorentz-contraction . . . . . . . . . . . . . . . . . . . . . . . 28 2.7 The Lorentz transformation . . . . . . . . . . . . . . . . . . . . . 30 2.8 Lorentz-invariant interval . . . . . . . . . . . . . . . . . . . . . . 32 2.9 The twin-paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.10 Hyperbolic motion . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.11 Energy and mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.12 Relativistic increase of mass . . . . . . . . . . . . . . . . . . . . . 38 2.13 Tachyons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.14 Magnetism as a relativistic second-order effect . . . . . . . . . . 40 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 II T HE MATHEMATICS OF THE GENERAL THEORY OF RELATIVITY 49 3 Vectors, Tensors, and Forms 51 3.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 iv Contents 3.2 Four-vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3 One-forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.4 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.5 Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4 Basis Vector Fields and the Metric Tensor 63 4.1 Manifolds and their coordinate-systems . . . . . . . . . . . . . . 63 4.2 Tangent vector fields and the coordinate basis vector fields . . . 65 4.3 Structure coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.4 General basis transformations . . . . . . . . . . . . . . . . . . . . 71 4.5 The metric tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.6 Orthonormal basis . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.7 Spatial geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.8 The tetrad field of a comoving coordinate system . . . . . . . . . 80 4.9 The volume form . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.10 Dual forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5 Non-inertial Reference Frames 89 5.1 Spatial geometry in rotating reference frames . . . . . . . . . . . 89 5.2 Ehrenfest’s paradox . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.3 The Sagnac effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.4 Gravitational time dilatation . . . . . . . . . . . . . . . . . . . . . 94 5.5 Uniformly accelerated reference frame . . . . . . . . . . . . . . . 95 5.6 Covariant Lagrangian dynamics . . . . . . . . . . . . . . . . . . 98 5.7 A general equation for the Doppler effect . . . . . . . . . . . . . 103 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6 Differentiation, Connections and Integration 109 6.1 Exterior Differentiation of forms . . . . . . . . . . . . . . . . . . 109 6.2 Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.3 Integration of forms . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.4 Covariant differentiation of vectors . . . . . . . . . . . . . . . . . 120 6.5 Covariant differentiation of forms and tensors . . . . . . . . . . 127 6.6 Exterior differentiation of vectors . . . . . . . . . . . . . . . . . . 129 6.7 Covariant exterior derivative . . . . . . . . . . . . . . . . . . . . 133 6.8 Geodesic normal coordinates . . . . . . . . . . . . . . . . . . . . 136 6.9 One-parameter groups of diffeomorphisms . . . . . . . . . . . . 137 6.10 The Lie derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.11 Killing vectors and Symmetries . . . . . . . . . . . . . . . . . . . 143 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7 Curvature 149 7.1 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.2 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.3 The Riemann Curvature Tensor . . . . . . . . . . . . . . . . . . . 153 7.4 Extrinsic and Intrinsic Curvature . . . . . . . . . . . . . . . . . . 159 7.5 The equation of geodesic deviation . . . . . . . . . . . . . . . . . 162 7.6 Spaces of constant curvature . . . . . . . . . . . . . . . . . . . . . 163 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Contents v III EINSTEIN’S FIELD EQUATIONS 175 8 Einstein’s Field Equations 177 8.1 Deduction of Einstein’s vacuum field equations from Hilbert’s variational principle . . . . . . . . . . . . . . . . . . . . . . . . . 177 8.2 The field equations in the presence of matter and energy . . . . 180 8.3 Energy-momentum conservation . . . . . . . . . . . . . . . . . . 181 8.4 Energy-momentum tensors . . . . . . . . . . . . . . . . . . . . . 182 8.5 Some particular fluids . . . . . . . . . . . . . . . . . . . . . . . . 184 8.6 The paths of free point particles . . . . . . . . . . . . . . . . . . . 188 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 9 The Linear Field Approximation 191 9.1 The linearised field equations . . . . . . . . . . . . . . . . . . . . 191 9.2 The Newtonian limit of general relativity . . . . . . . . . . . . . 194 9.3 Solutions to the linearised field equations . . . . . . . . . . . . . 195 9.4 Gravitoelectromagnetism . . . . . . . . . . . . . . . . . . . . . . 197 9.5 Gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . 199 9.6 Gravitational radiation from sources . . . . . . . . . . . . . . . . 202 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 10 The Schwarzschild Solution and Black Holes 211 10.1 The Schwarzschild solution for empty space . . . . . . . . . . . 211 10.2 Radial free fall in Schwarzschild spacetime . . . . . . . . . . . . 216 10.3 The light-cone in a Schwarzschild spacetime . . . . . . . . . . . 217 10.4 Particle trajectories in Schwarzschild spacetime . . . . . . . . . . 221 10.5 Analytical extension of the Schwarzschild spacetime . . . . . . . 226 10.6 Charged and rotating black holes . . . . . . . . . . . . . . . . . . 229 10.7 Black Hole thermodynamics . . . . . . . . . . . . . . . . . . . . . 241 10.8 The Tolman-Oppenheimer-Volkoff equation . . . . . . . . . . . . 248 10.9 The interior Schwarzschild solution . . . . . . . . . . . . . . . . 249 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 IV C OSMOLOGY 259 11 Homogeneous and Isotropic Universe Models 261 11.1 The cosmological principles . . . . . . . . . . . . . . . . . . . . . 261 11.2 Friedmann-Robertson-Walker models . . . . . . . . . . . . . . . 262 11.3 Dynamics of Homogeneous and Isotropic cosmologies . . . . . 265 11.4 Cosmological redshift and the Hubble law . . . . . . . . . . . . 267 11.5 Radiation dominated universe models . . . . . . . . . . . . . . . 272 11.6 Matter dominated universe models . . . . . . . . . . . . . . . . . 275 11.7 The gravitational lens effect . . . . . . . . . . . . . . . . . . . . . 277 11.8 Redshift-luminosity relation . . . . . . . . . . . . . . . . . . . . . 283 11.9 Cosmological horizons . . . . . . . . . . . . . . . . . . . . . . . . 287 11.10Big Bang in an infinite Universe . . . . . . . . . . . . . . . . . . . 288 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 vi Contents 12 Universe Models with Vacuum Energy 297 12.1 Einstein’s static universe . . . . . . . . . . . . . . . . . . . . . . . 297 12.2 de Sitter’s solution . . . . . . . . . . . . . . . . . . . . . . . . . . 298 12.3 The de Sitter hyperboloid . . . . . . . . . . . . . . . . . . . . . . 301 12.4 The horizon problem and the flatness problem . . . . . . . . . . 302 12.5 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 12.6 The Friedmann-Lemaître model . . . . . . . . . . . . . . . . . . . 311 12.7 Universe models with quintessence energy . . . . . . . . . . . . 317 12.8 Dark energy and the statefinder diagnostic . . . . . . . . . . . . 320 12.9 Cosmic density perturbations . . . . . . . . . . . . . . . . . . . . 327 12.10Temperature fluctuations in the CMB . . . . . . . . . . . . . . . . 331 12.11Mach’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 12.12The History of our Universe . . . . . . . . . . . . . . . . . . . . . 341 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 13 An Anisotropic Universe 359 13.1 The Bianchi type I universe model . . . . . . . . . . . . . . . . . 359 13.2 The Kasner solutions . . . . . . . . . . . . . . . . . . . . . . . . . 362 13.3 The energy-momentum conservation law in an anisotropic uni- verse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 13.4 Models with a perfect fluid . . . . . . . . . . . . . . . . . . . . . 365 13.5 Inflation through bulk viscosity . . . . . . . . . . . . . . . . . . . 368 13.6 A universe with a dissipative fluid . . . . . . . . . . . . . . . . . 369 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 V A DVANCED TOPICS 375 14 Covariant decomposition, Singularities, and Canonical Cosmology 377 14.1 Covariant decomposition . . . . . . . . . . . . . . . . . . . . . . 377 14.2 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . 380 14.3 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 14.4 Lagrangian formulation of General Relativity . . . . . . . . . . . 387 14.5 Hamiltonian formulation . . . . . . . . . . . . . . . . . . . . . . . 390 14.6 Canonical formulation with matter and energy . . . . . . . . . . 392 14.7 The space of three-metrics: Superspace . . . . . . . . . . . . . . . 394 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 15 Homogeneous Spaces 401 15.1 Lie groups and Lie algebras . . . . . . . . . . . . . . . . . . . . . 401 15.2 Homogeneous spaces . . . . . . . . . . . . . . . . . . . . . . . . . 404 15.3 The Bianchi models . . . . . . . . . . . . . . . . . . . . . . . . . . 407 15.4 The orthonormal frame approach to the Bianchi models . . . . . 411 15.5 The 8 model geometries . . . . . . . . . . . . . . . . . . . . . . . 416 15.6 Constructing compact quotients . . . . . . . . . . . . . . . . . . . 418 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 16 Israel’s Formalism: The metric junction method 427 16.1 The relativistic theory of surface layers . . . . . . . . . . . . . . . 427 16.2 Einstein’s field equations . . . . . . . . . . . . . . . . . . . . . . . 429 16.3 Surface layers and boundary surfaces . . . . . . . . . . . . . . . 431 16.4 Spherical shell of dust in vacuum . . . . . . . . . . . . . . . . . . 433 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 Contents vii 17 Brane-worlds 441 17.1 Field equations on the brane . . . . . . . . . . . . . . . . . . . . . 441 17.2 Five-dimensional brane cosmology . . . . . . . . . . . . . . . . . 444 17.3 Problem with perfect fluid brane world in an empty bulk . . . . 447 17.4 Solutions in the bulk . . . . . . . . . . . . . . . . . . . . . . . . . 447 17.5 Towards a realistic brane cosmology . . . . . . . . . . . . . . . . 449 17.6 Inflation in the brane . . . . . . . . . . . . . . . . . . . . . . . . . 452 17.7 Dynamics of two branes . . . . . . . . . . . . . . . . . . . . . . . 455 17.8 The hierarchy problem and the weakness of gravity . . . . . . . 457 17.9 The Randall-Sundrum models . . . . . . . . . . . . . . . . . . . . 459 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 18 Kaluza-Klein Theory 465 18.1 A fifth extra dimension . . . . . . . . . . . . . . . . . . . . . . . . 465 18.2 The Kaluza-Klein action . . . . . . . . . . . . . . . . . . . . . . . 467 18.3 Implications of a fifth extra dimension . . . . . . . . . . . . . . . 471 18.4 Conformal transformations . . . . . . . . . . . . . . . . . . . . . 474 18.5 Conformal transformation of the Kaluza-Klein action . . . . . . 478 18.6 Kaluza-Klein cosmology . . . . . . . . . . . . . . . . . . . . . . . 480 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 VI A PPENDICES 487 A Constants of Nature 489 B Penrose diagrams 491 B.1 Conformal transformations and causal structure . . . . . . . . . 491 B.2 Schwarzschild spacetime . . . . . . . . . . . . . . . . . . . . . . . 493 B.3 de Sitter spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . 493 C Anti-de Sitter spacetime 497 C.1 The anti-de Sitter hyperboloid . . . . . . . . . . . . . . . . . . . . 497 C.2 Foliations of AdS n . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 C.3 Geodesics in AdS n . . . . . . . . . . . . . . . . . . . . . . . . . . 499 C.4 The BTZ black hole . . . . . . . . . . . . . . . . . . . . . . . . . . 500 C.5 AdS 3 as the group SL(2, R) . . . . . . . . . . . . . . . . . . . . . 501 D Suggested further reading 503 Bibliography 507 Index 515 List of Problems Chapter 1 17 1.1 The strength of gravity compared to the Coulomb force . . . . 17 1.2 Falling objects in the gravitational field of the Earth . . . . . . . 17 1.3 Newtonian potentials for spherically symmetric bodies . . . . 17 1.4 The Earth-Moon system . . . . . . . . . . . . . . . . . . . . . . . 18 1.5 The Roche-limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.6 A Newtonian Black Hole . . . . . . . . . . . . . . . . . . . . . . 18 1.7 Non-relativistic Kepler orbits . . . . . . . . . . . . . . . . . . . . 19 Chapter 2 42 2.1 Two successive boosts in different directions . . . . . . . . . . . 42 2.2 Length-contraction and time-dilatation . . . . . . . . . . . . . . 43 2.3 Faster than the speed of light? . . . . . . . . . . . . . . . . . . . 44 2.4 Reflection angles off moving mirrors . . . . . . . . . . . . . . . 44 2.5 Minkowski-diagram . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.6 Robb’s Lorentz invariant spacetime interval formula . . . . . . 45 2.7 The Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.8 Abberation and Doppler effect . . . . . . . . . . . . . . . . . . . 45 2.9 A traffic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.10 The twin-paradox . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.11 Work and rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.12 Muon experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.13 Cerenkov radiation . . . . . . . . . . . . . . . . . . . . . . . . . 47 Chapter 3 60 3.1 The tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.2 Contractions of tensors . . . . . . . . . . . . . . . . . . . . . . . 60 3.3 Four-vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.4 The Lorentz-Abraham-Dirac equation . . . . . . . . . . . . . . . 62 Chapter 4 85 4.1 Coordinate-transformations in a two-dimensional Euclidean plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.2 Covariant and contravariant components . . . . . . . . . . . . . 86 4.3 The Levi-Civitá symbol . . . . . . . . . . . . . . . . . . . . . . . 86 4.4 Dual forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Chapter 5 107 5.1 Geodetic curves in space . . . . . . . . . . . . . . . . . . . . . . 107 5.2 Free particle in a hyperbolic reference frame . . . . . . . . . . . 107 5.3 Spatial geodesics in a rotating RF . . . . . . . . . . . . . . . . . 108 x List of Problems Chapter 6 147 6.1 Loop integral of a closed form . . . . . . . . . . . . . . . . . . . 147 6.2 The covariant derivative . . . . . . . . . . . . . . . . . . . . . . 147 6.3 The Poincaré half-plane . . . . . . . . . . . . . . . . . . . . . . . 148 6.4 The Christoffel symbols in a rotating reference frame with plane polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Chapter 7 170 7.1 Rotation matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 7.2 Inverse metric on S n . . . . . . . . . . . . . . . . . . . . . . . . . 170 7.3 The curvature of a curve . . . . . . . . . . . . . . . . . . . . . . 170 7.4 The Gauss-Codazzi equations . . . . . . . . . . . . . . . . . . . 171 7.5 The Poincaré half-space . . . . . . . . . . . . . . . . . . . . . . . 171 7.6 The pseudo-sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 172 7.7 A non-Cartesian coordinate system in two dimensions . . . . . 172 7.8 The curvature tensor of a sphere . . . . . . . . . . . . . . . . . . 172 7.9 The curvature scalar of a surface of simultaneity . . . . . . . . . 172 7.10 The tidal force pendulum and the curvature of space . . . . . . 172 7.11 The Weyl tensor vanishes for spaces of constant curvature . . . 173 7.12 Frobenius’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 173 Chapter 8 189 8.1 Lorentz transformation of a perfect fluid . . . . . . . . . . . . . 189 8.2 Geodesic equation and constants of motion . . . . . . . . . . . 189 Chapter 9 206 9.1 The Linearised Einstein Field Equations . . . . . . . . . . . . . 206 9.2 Gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . 208 9.3 The spacetime inside and outside a rotating spherical shell . . 209 Chapter 10 251 10.1 The Schwarzschild metric in Isotropic coordinates . . . . . . . 251 10.2 Embedding of the interior Schwarzschild metric . . . . . . . . . 252 10.3 The Schwarzschild-de Sitter metric . . . . . . . . . . . . . . . . 252 10.4 The life time of a black hole . . . . . . . . . . . . . . . . . . . . . 252 10.5 A spaceship falling into a black hole . . . . . . . . . . . . . . . . 252 10.6 The GPS Navigation System . . . . . . . . . . . . . . . . . . . . 253 10.7 Physical interpretation of the Kerr metric . . . . . . . . . . . . . 253 10.8 A gravitomagnetic clock effect . . . . . . . . . . . . . . . . . . . 253 10.9 The photon sphere radius of a Reissner-Nordström black hole . 254 10.10 Curvature of 3-space and 2-surfaces of the internal and the external Schwarzschild spacetimes . . . . . . . . . . . . . . . . . 254 10.11 Proper radial distance in the external Schwarzschild space . . . 255 10.12 Gravitational redshift in the Schwarzschild spacetime . . . . . 255 10.13 The Reissner-Nordström repulsion . . . . . . . . . . . . . . . . 256 10.14 Light-like geodesics in the Reissner-Nordström spacetime . . . 256 10.15 Birkhoff’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 256 10.16 Gravitational mass . . . . . . . . . . . . . . . . . . . . . . . . . . 257 [...]... offers a rigorous introduction to Einstein’s general theory of relativity We start out from the first principles of relativity and present to Einstein’s theory in a self-contained way After introducing Einstein’s field equations, we go onto the most important chapter in this book which contains the three classical tests of the theory and introduces the notion of black holes Recently, cosmology has also... range of validity of the equivalence of all Galilean frames While Galilei and Newton had demanded that the laws of mechanics are the same in all Galilean frames, Einstein postulated that all the physical laws governing the behavior of the material world can be formulated in the same way in all Galilean frames This is Einstein’s special principle of relativity (Note that in the special theory of relativity. .. accordance with the negative result of the Michelson–Morley experiment [MM87] In this experiment one did not succeed in measuring the velocity of the Earth through the so-called ‘ether’, which was thought of as a ‘materialization’ of Newton’s absolute space 6 Relativity Principles and Gravitation However, Einstein retained, in his special theory of relativity, the Newtonian idea of the privileged observers... small 1.7 The principle of equivalence Galilei experimentally investigated the motion of freely falling bodies He found that they moved in the same way, regardless of mass and of composition In Newton’s theory of gravitation, mass appears in two different ways: 1 in the law of gravitation as gravitational mass, mg ; 2 in Newton’s 2nd law as inertial mass, mi The equation of motion of a freely falling... would be no stretching of the string Another example makes use of a turnabout If we stay on this while it rotates, we feel that the centrifugal force leads us outwards At the same time we observe that the heavenly bodies rotate Einstein was impressed by Mach’s arguments, which likely influenced Einstein’s construction of the general theory of relativity Yet it is clear that general relativity does not... strength of gravity compared to the Coulomb force (a) Determine the difference in strength between the Newtonian gravitational attraction and the Coulomb force of the interaction of the proton and the electron in a hydrogen atom (b) What is the gravitational force of attraction of two objects of 1 kg at a separation of 1 m Compare with the corresponding electrostatic force of two charges of 1 C at... the center of the Earth A small solid ball is then dropped into the tube from the surface of the Earth Find the position of the ball as a function of time What is the period of the oscillations of the ball? (d) We now assume that the tube is not passing through the centre of the Earth, but at a closest distance s from the centre Find how the period of the oscillations vary as a function of s Assume... the special principle of relativity should be valid also for Maxwell’s electromagnetic theory This was obtained by replacing the Galilean kinematics by that of the special theory of relativity (see Ch 2), since Maxwell’s equations and Lorentz’s force law is invariant under the Lorentz transformations In particular this implies that the velocity of electromagnetic waves, i.e of light, is the same in... frame, moving with respect to the first This is due to the relativity of simultaneity (see Ch 2) Instantaneous action at a distance can only exist in a theory with absolute simultaneity As a first step towards a relativistically valid theory of gravitation, we shall give a local form of Newton’s law of gravitation We shall now show how Newton’s law of gravitation leads to a field equation for gravity Consider... thermodynamics Black hole thermodynamics is a quantum feature of black holes, but we chose to include it because the study of black holes would have been incomplete without it There are several people who we wish to thank First of all, we would xvi List of Examples like to thank Finn Ravndal who gave a thorough introduction to the theory of relativity in a series of lectures during the late seventies This laid the . these concepts will be presented in this book. The book offers a rigorous introduction to Einstein s general theory of rel- ativity. We start out from the first principles of relativity and present. Unless one just accepts the cosmological princi- ples as a fact, one is unavoidably led to the study of such anisotropic universe models. As an introductory course in general relativity, it is suitable. GRØN Oslo, Norway SIGBJØRN HERVIK Cambridge, United Kingdom Notation We have tried to be as homogeneous as possible when it comes to notation in this book. There are some exceptions, but as a general