GRAVITATION AND COSMOLOGY: PRINCIPLES AND APPLICATIONS OF THE GENERAL THEORY OF RELATIVITY
1 HOW THE THEORY OF RELATIVITY CAME INTO BEING (A BRIEF HISTORICAL SKETCH) 11 SPECIAL VERSUS GENERAL RELATIVITY 12 SPACE AND INERTIA IN NEWTONIAN PHYSICS 13 NEWTON’S THEORY AND THE ORBITS OF PLANETS 14 THE BASIC ASSUMPTIONS OF GENERAL RELATIVITY PART I ELEMENTS OF DIFFERENTIAL GEOMETRY 2 A SHORT SKETCH OF 2-DIMENSIONAL DIFFERENTIAL GEOMETRY 21 CONSTRUCTING PARALLEL STRAIGHT LINES IN A FLAT SPACE 22 GENERALISATION OF THE NOTION OF PARALLELISM TO CURVED SURFACES 3 TENSORS, TENSOR DENSITIES 31 WHAT ARE TENSORS GOOD FOR? 32 DIFFERENTIABLE MANIFOLDS 33 SCALARS 34 CONTRAVARIANT VECTORS 35 COVARIANT VECTORS 36 TENSORS OF SECOND RANK 37 TENSOR DENSITIES 38 TENSOR DENSITIES OF ARBITRARY RANK 39 ALGEBRAIC PROPERTIES OF TENSOR DENSITIES 310 MAPPINGS BETWEEN MANIFOLDS 311 THE LEVI-CIVITA SYMBOL 312 MULTIDIMENSIONAL KRONECKER DELTAS 313 EXAMPLES OF APPLICATIONS OF THE LEVI-CIVITA SYMBOL AND OF THE MULTIDIMENSIONAL KRONECKER DELTA 314 EXERCISES 4 COVARIANT DERIVATIVES 41 DIFFERENTIATION OF TENSORS 42 AXIOMS OF THE COVARIANT DERIVATIVE 43 A FIELD OF BASES ON A MANIFOLD AND SCALAR COMPONENTS OF TENSORS 44 THE AFFINE CONNECTION 45 THE EXPLICIT FORMULA FOR THE COVARIANT DERIVATIVE OF TENSOR DENSITY FIELDS 46 EXERCISES 5 PARALLEL TRANSPORT AND GEODESIC LINES 51 PARALLEL TRANSPORT 52 GEODESIC LINES 53 EXERCISES 6 THE CURVATURE OF A MANIFOLD; FLAT MANIFOLDS 61 THE COMMUTATOR OF SECOND COVARIANT DERIVATIVES 62 THE COMMUTATOR OF DIRECTIONAL COVARIANT DERIVATIVES 63 THE RELATION BETWEEN CURVATURE AND PARALLEL TRANSPORT 64 COVARIANTLY CONSTANT FIELDS OF VECTOR BASES 65 A TORSION-FREE FLAT MANIFOLD 66 PARALLEL TRANSPORT IN A FLAT MANIFOLD 67 GEODESIC DEVIATION 68 ALGEBRAIC AND DIFFERENTIAL IDENTITIES OBEYED BY THE CURVATURE TENSOR 69 EXERCISES 7 RIEMANNIAN GEOMETRY 71 THE METRIC TENSOR 72 RIEMANN SPACES 73 THE SIGNATURE OF A METRIC, DEGENERATE METRICS 74 CHRISTOFFEL SYMBOLS 75 THE CURVATURE OF A RIEMANN SPACE 76 FLAT RIEMANN SPACES 77 SUBSPACES OF A RIEMANN SPACE 78 FLAT RIEMANN SPACES THAT ARE GLOBALLY NON-EUCLIDEAN 79 THE RIEMANN CURVATURE VERSUS THE NORMAL CURVATURE OF A SURFACE 710 THE GEODESIC LINE AS THE LINE OF EXTREMAL DISTANCE 711 MAPPINGS BETWEEN RIEMANN SPACES 712 CONFORMALLY RELATED RIEMANN SPACES 713 CONFORMAL CURVATURE 714 TIMELIKE, NULL AND SPACELIKE INTERVALS IN A 4-DIMENSIONAL SPACETIME 715 EMBEDDINGS OF RIEMANN SPACES IN RIEMANN SPACES OF HIGHER DIMENSION 716 THE PETROV CLASSIFICATION 717 EXERCISES 8 SYMMETRIES OF RIEMANN SPACES, INVARIANCE OF TENSORS 81 SYMMETRY TRANSFORMATIONS 82 THE KILLING EQUATIONS 83 THE CONNECTION BETWEEN GENERATORS AND THE INVARIANCE TRANSFORMATIONS 84 FINDING THE KILLING VECTOR FIELDS 85 A KILLING VECTOR FIELD ALONG A GEODESIC IS A GEODESIC DEVIATION FIELD 86 INVARIANCE OF OTHER TENSOR FIELDS 87 THE LIE DERIVATIVE 88 THE ALGEBRA OF KILLING VECTOR FIELDS 89 SURFACE-FORMING VECTOR FIELDS 810 SPHERICALLY SYMMETRIC 4-DIMENSIONAL RIEMANN SPACES 811 * CONFORMAL KILLING FIELDS AND THEIR FINITE BASIS 812 * THE MAXIMAL DIMENSION OF AN INVARIANCE GROUP 813 EXERCISES 9 METHODS TO CALCULATE THE CURVATURE QUICKLY: DIFFERENTIAL FORMS AND ALGEBRAIC COMPUTER PROGRAMS 91 THE BASIS OF DIFFERENTIAL FORMS 92 THE CONNECTION FORMS 93 THE RIEMANN TENSOR 94 USING COMPUTERS TO CALCULATE THE CURVATURE 95 EXERCISES 10 THE SPATIALLY HOMOGENEOUS BIANCHI-TYPE SPACETIMES 101 THE BIANCHI CLASSIFICATION OF 3-DIMENSIONAL LIE ALGEBRAS 102 THE DIMENSION OF THE GROUP VERSUS THE DIMENSION OF THE ORBIT 103 ACTION OF A GROUP ON A MANIFOLD 104 GROUPS ACTING TRANSITIVELY, HOMOGENEOUS SPACES 105 INVARIANT VECTOR FIELDS 106 THE METRICS OF THE BIANCHI-TYPE SPACETIMES 107 THE ISOTROPIC BIANCHI-TYPE (ROBERTSON–WALKER) SPACETIMES 108 EXERCISES 11 * THE PETROV CLASSIFICATION BY THE SPINOR METHOD 111 WHAT IS A SPINOR? 112 TRANSLATING SPINORS TO TENSORS AND VICE VERSA 113 THE SPINOR IMAGE OF THE WEYL TENSOR 114 THE PETROV CLASSIFICATION IN THE SPINOR REPRESENTATION 115 THE WEYL SPINOR REPRESENTED AS A 3 × 3 COMPLEX MATRIX 116 THE EQUIVALENCE OF THE PENROSE CLASSES TO THE PETROV CLASSES 117 THE PETROV CLASSIFICATION BY THE DEBEVER METHOD 118 EXERCISES PART II THE THEORY OF GRAVITATION 12 THE EINSTEIN EQUATIONS AND THE SOURCES OF A GRAVITATIONAL FIELD 121 WHY RIEMANNIAN GEOMETRY? 122 LOCAL INERTIAL FRAMES 123 TRAJECTORIES OF FREE MOTION IN EINSTEIN’S THEORY 124 SPECIAL RELATIVITY VERSUS GRAVITATION THEORY 125 THE NEWTONIAN LIMIT OF GENERAL RELATIVITY 126 SOURCES OF THE GRAVITATIONAL FIELD 127 THE EINSTEIN EQUATIONS 128 HILBERT’S DERIVATION OF THE EINSTEIN EQUATIONS 129 THE PALATINI VARIATIONAL PRINCIPLE 1210 THE ASYMPTOTICALLY CARTESIAN COORDINATES AND THE ASYMPTOTICALLY FLAT SPACETIME 1211 THE NEWTONIAN LIMIT OF EINSTEIN’S EQUATIONS 1212 EXAMPLES OF SOURCES IN THE EINSTEIN EQUATIONS: PERFECT FLUID AND DUST 1213 EQUATIONS OF MOTION OF A PERFECT FLUID 1214 THE COSMOLOGICAL CONSTANT 1215 AN EXAMPLE OF AN EXACT SOLUTION OF EINSTEIN’S EQUATIONS: A BIANCHI TYPE I SPACETIME WITH DUST SOURCE 1216 * OTHER GRAVITATION THEORIES 12161 THE BRANS–DICKE THEORY 12162 THE BERGMANN–WAGONER THEORY 12163 THE EINSTEIN–CARTAN THEORY 12164 THE BI-METRIC ROSEN THEORY 1217 MATCHING SOLUTIONS OF EINSTEIN’S EQUATIONS 1218 THE WEAK-FIELD APPROXIMATION TO GENERAL RELATIVITY 1219 EXERCISES 13 THE MAXWELL AND EINSTEIN–MAXWELL EQUATIONS AND THE KALUZA–KLEIN THEORY 131 THE LORENTZ-COVARIANT DESCRIPTION OF ELECTROMAGNETIC FIELD 132 THE COVARIANT FORM OF THE MAXWELL EQUATIONS 133 THE ENERGY-MOMENTUM TENSOR OF ELECTROMAGNETIC FIELD 134 THE EINSTEIN–MAXWELL EQUATIONS 135 * THE VARIATIONAL PRINCIPLE FOR THE MAXWELL AND EINSTEIN–MAXWELL EQUATIONS 136 * THE KALUZA–KLEIN THEORY 137 EXERCISES 14 SPHERICALLY SYMMETRIC GRAVITATIONAL FIELDS OF ISOLATED OBJECTS 141 THE CURVATURE COORDINATES 142 SYMMETRY INHERITANCE 143 SPHERICALLY SYMMETRIC ELECTROMAGNETIC FIELD IN VACUUM 144 THE SCHWARZSCHILD AND REISSNER–NORDSTRO?M SOLUTIONS 145 ORBITS OF PLANETS IN THE GRAVITATIONAL FIELD OF THE SUN 146 DEFLECTION OF LIGHT RAYS IN THE SCHWARZSCHILD FIELD 147 MEASURING THE DEFLECTION OF LIGHT RAYS 148 GRAVITATIONAL LENSES 149 THE SPURIOUS SINGULARITY OF THE SCHWARZSCHILD SOLUTION AT R = 2M 1410 * EMBEDDING THE SCHWARZSCHILD SPACETIME IN A FLAT RIEMANNIAN SPACE 1411 INTERPRETATION OF THE SPURIOUS SINGULARITY AT R = 2M; BLACK HOLES 1412 THE SCHWARZSCHILD METRIC IN OTHER COORDINATE SYSTEMS 1413 THE EQUATION OF HYDROSTATIC EQUILIBRIUM 1414 THE ‘INTERIOR SCHWARZSCHILD SOLUTION’ 1415 * THE MAXIMAL ANALYTIC EXTENSION OF THE REISSNER–NORDSTRO?M METRIC 1416 MOTION OF PARTICLES IN THE REISSNER–NORDSTRO?M SPACETIME WITH E2 < M2 1417 EXERCISES 15 RELATIVISTIC HYDRODYNAMICS AND THERMODYNAMICS 151 MOTION OF A CONTINUOUS MEDIUM IN NEWTONIAN HYDRODYNAMICS 152 MOTION OF A CONTINUOUS MEDIUM IN RELATIVISTIC HYDRODYNAMICS 153 THE EQUATIONS OF EVOLUTION OF ?, ??? , ??? AND U?? ; THE RAYCHAUDHURI EQUATION 154 SINGULARITIES AND SINGULARITY THEOREMS 155 RELATIVISTIC THERMODYNAMICS 156 EXERCISES 16 RELATIVISTIC COSMOLOGY I: GENERAL GEOMETRY 161 A CONTINUOUS MEDIUM AS A MODEL OF THE UNIVERSE 162 THE GEOMETRIC OPTICS APPROXIMATION 163 THE REDSHIFT 164 THE OPTICAL TENSORS 165 THE APPARENT HORIZON 166 * THE DOUBLE-NULL TETRAD 167 * THE EQUATIONS OF PROPAGATION OF THE OPTICAL SCALARS 168 * THE GOLDBERG–SACHS THEOREM 169 * THE AREA DISTANCE 1610 * THE RECIPROCITY THEOREM 1611 OTHER OBSERVABLE QUANTITIES 1612 THE FERMI–WALKER TRANSPORT 1613 POSITION DRIFT OF LIGHT SOURCES 1614 EXERCISES 17 RELATIVISTIC COSMOLOGY II: THE ROBERTSON–WALKER GEOMETRY 171 THE ROBERTSON–WALKER METRICS AS MODELS OF THE UNIVERSE 172 OPTICAL OBSERVATIONS IN AN R–W UNIVERSE 1721 THE REDSHIFT 1722 THE REDSHIFT–DISTANCE RELATION 1723 NUMBER COUNTS 173 THE FRIEDMANN EQUATION 174 THE FRIEDMANN SOLUTIONS WITH ? = 0 175 THE REDSHIFT–DISTANCE RELATION IN THE ? = 0 FRIEDMANN MODELS 176 THE NEWTONIAN COSMOLOGY 177 THE FRIEDMANN SOLUTIONS WITH THE COSMOLOGICAL CONSTANT 178 THE ?CDM MODEL 179 THE REDSHIFT–DISTANCE RELATION IN THE ? 6= 0 FRIEDMANN MODELS 1710 THE REDSHIFT DRIFT: A TEST FOR ACCELERATING EXPANSION 1711 HORIZONS IN THE ROBERTSON–WALKER MODELS 1712 THE INFLATIONARY MODELS AND THE ‘PROBLEMS’ THEY SOLVED 1713 THE VALUE OF THE COSMOLOGICAL CONSTANT 1714 THE ‘HISTORY OF THE UNIVERSE’ 1715 INVARIANT DEFINITIONS OF THE ROBERTSON–WALKER MODELS 1716 DIFFERENT REPRESENTATIONS OF THE R–W METRICS 1717 EXERCISES 18 RELATIVISTIC COSMOLOGY III: THE LEMAI?TRE–TOLMAN GEOMETRY 181 THE COMOVING-SYNCHRONOUS COORDINATES 182 THE SPHERICALLY SYMMETRIC INHOMOGENEOUS MODELS 183 THE LEMAI?TRE–TOLMAN MODEL 184 CONDITIONS OF REGULARITY AT THE CENTRE 185 FORMATION OF VOIDS IN THE UNIVERSE 186 FORMATION OF OTHER STRUCTURES IN THE UNIVERSE 1861 DENSITY TO DENSITY EVOLUTION 1862 VELOCITY TO DENSITY EVOLUTION 1863 VELOCITY TO VELOCITY EVOLUTION 187 THE INFLUENCE OF COSMIC EXPANSION ON PLANETARY ORBITS 188 * THE APPARENT HORIZONS FOR A CENTRAL OBSERVER IN L–T MODELS 189 * BLACK HOLES IN THE EVOLVING UNIVERSE 1810 * SHELL CROSSINGS AND NECKS/WORMHOLES 18101 E < 0 18102 E = 0 18103 E > 0 18104 FINAL COMMENT 1811 THE REDSHIFT ALONG RADIAL RAYS 1812 THE BLUESHIFT 1813 * APPARENT HORIZONS FOR NONCENTRAL OBSERVERS 1814 THE INFLUENCE OF INHOMOGENEITIES IN MATTER DISTRIBUTION ON THE COSMIC MICROWAVE BACKGROUND RADIATION 1815 MATCHING THE L–T MODELS TO THE SCHWARZSCHILD AND FRIEDMANN SOLUTIONS 1816 * THE SHELL FOCUSSING SINGULARITY 1817 * EXTENDING AN L–T SPACETIME THROUGH A SHELL CROSSING SINGULARITY 1818 * SINGULARITIES AND COSMIC CENSORSHIP 1819 SOLVING THE ‘HORIZON PROBLEM’ WITHOUT INFLATION 1820 * THE EVOLUTION OF R(T, M ) VERSUS THE EVOLUTION OF ?(T, M ) 1821 * INCREASING AND DECREASING DENSITY PERTURBATIONS 1822 MIMICKING ACCELERATING EXPANSION OF THE UNIVERSE BY INHOMOGENEITIES IN MATTER DISTRIBUTION 1823 DRIFT OF LIGHT RAYS 1824 * L&T CURIO SHOP 18241 LAGGING CORES OF THE BIG BANG 18242 STRANGE OR NONINTUITIVE PROPERTIES OF THE L–T MODEL 18243 CHANCES TO FIT AN L–T MODEL TO OBSERVATIONS 18244 AN ‘IN ONE EAR AND OUT THE OTHER’ UNIVERSE 18245 A ‘STRING OF BEADS’ UNIVERSE 18246 UNCERTAINTIES IN INFERRING THE SPATIAL DISTRIBUTION OF MATTER 18247 IS THE DISTRIBUTION OF MATTER IN OUR UNIVERSE FRACTAL? 18248 GENERAL RESULTS RELATED TO THE L–T MODELS 1825 EXERCISES 19 RELATIVISTIC COSMOLOGY IV: SIMPLE GENERALISATIONS OF L–T AND RELATED GEOMETRIES 191 THE PLANE- AND HYPERBOLICALLY SYMMETRIC SPACETIMES 192 G3 /S2 -SYMMETRIC DUST SOLUTIONS WITH R,R 6= 0 193 PLANE SYMMETRIC DUST SOLUTIONS WITH R,R 6= 0 194 G3 /S2 -SYMMETRIC DUST IN ELECTROMAGNETIC FIELD, THE CASE R,R 6= 0 1941 INTEGRALS OF THE FIELD EQUATIONS 1942 MATCHING THE CHARGED DUST METRIC TO THE REISSNER–NORDSTRO?M METRIC 1943 PREVENTION OF THE BIG CRUNCH SINGULARITY BY ELECTRIC CHARGE 1944 * CHARGED DUST IN CURVATURE AND MASS-CURVATURE COORDINATES 1945 REGULARITY CONDITIONS AT THE CENTRE 1946 * SHELL CROSSINGS IN CHARGED DUST 195 THE DATT–RUBAN SOLUTION 196 EXERCISES 20 RELATIVISTIC COSMOLOGY V: THE SZEKERES GEOMETRIES 201 THE SZEKERES–SZAFRON FAMILY OF METRICS 2011 THE ?,Z = 0 SUBFAMILY 2012 THE ?,Z 6= 0 SUBFAMILY 2013 INTERPRETATION OF THE SZEKERES–SZAFRON COORDINATES 2014 COMMON PROPERTIES OF THE TWO SUBFAMILIES 2015 * THE INVARIANT DEFINITIONS OF THE SZEKERES–SZAFRON METRICS 202 THE SZEKERES SOLUTIONS AND THEIR PROPERTIES 2021 THE ?,Z = 0 SUBFAMILY 2022 THE ?,Z 6= 0 SUBFAMILY 2023 * THE ?,Z = 0 FAMILY AS A LIMIT OF THE ?,Z 6= 0 FAMILY 203 PROPERTIES OF THE QUASI-SPHERICAL SZEKERES SOLUTIONS WITH ?,Z 6= 0 = ? 2031 BASIC PHYSICAL RESTRICTIONS 2032 THE SIGNIFICANCE OF E 2033 CONDITIONS OF REGULARITY AT THE ORIGIN 2034 SHELL CROSSINGS 2035 REGULAR MAXIMA AND MINIMA 2036 THE MASS DIPOLE 2037 * THE ABSOLUTE APPARENT HORIZON 2038 * THE APPARENT HORIZON AND ITS RELATION TO THE AAH 2039 * WHICH IS THE TRUE HORIZON – THE AH OR THE AAH? 204 * THE GOODE–WAINWRIGHT REPRESENTATION OF THE SZEKERES SOLUTIONS 205 SELECTED INTERESTING SUBCASES OF THE SZEKERES–SZAFRON FAMILY 2051 THE SZAFRON–WAINWRIGHT MODEL 2052 THE TOROIDAL UNIVERSE OF SENIN 206 SELECTED FURTHER READING ON THE SZEKERES MODELS 207 EXERCISES 21 THE KERR METRIC 211 THE KERR–SCHILD METRICS 212 THE DERIVATION OF THE KERR METRIC BY THE ORIGINAL METHOD 213 BASIC PROPERTIES 214 * DERIVATION OF THE KERR METRIC BY CARTER’S METHOD – FROM THE SEPARABILITY OF THE KLEIN–GORDON EQUATION 215 THE EVENT HORIZONS AND THE STATIONARY LIMIT HYPERSURFACES 216 THE HAMILTONIAN AND THE POISSON BRACKET 217 GENERAL GEODESICS 218 GEODESICS IN THE EQUATORIAL PLANE 219 *THE MAXIMAL ANALYTIC EXTENSION OF THE KERR METRIC 2110 * THE PENROSE PROCESS 2111 STATIONARY–AXISYMMETRIC SPACETIMES AND LOCALLY NONROTATING OBSERVERS 2112 * ELLIPSOIDAL SPACETIMES 2113 A NEWTONIAN ANALOGUE OF THE KERR SOLUTION 2114 A SOURCE OF THE KERR FIELD? 2115 EXERCISES 22 RELATIVITY ENTERS TECHNOLOGY: THE GLOBAL POSITIONING SYSTEM 221 PURPOSE AND SETUP 222 THE PRINCIPLE OF POSITION DETERMINATION 223 THE REFERENCE FRAMES AND THE SAGNAC EFFECT 224 EARTH’S GRAVITATION AND THE SI TIME UNITS 225 SELECTED CORRECTIONS OF THE ORBITS OF THE GPS SATELLITES 2251 CORRECTIONS FOR GRAVITY AND VELOCITY 2252 THE ECCENTRICITY CORRECTION 226 THE 9 LARGEST RELATIVISTIC EFFECTS IN THE GPS 227 EXERCISES 23 SUBJECTS OMITTED FROM THIS BOOK 24 COMMENTS TO SELECTED EXERCISES AND CALCULATIONS 241 EXERCISE 1 TO CHAPTER 14 242 EXERCISE 14 TO CHAPTER 14 243 VERIFYING EQS (1935) WITH (1931) AND (1932) WITH (1928) 244 VERIFYING THE EINSTEIN EQUATIONS (202), (209) AND (2011) 245 EQUATION (20179) DEFINES ? AT THE AAH UNIQUELY 246 THE FOUR CURVES IN FIG 204 MEET AT ONE POINT 247 THE DISCARDED CASE IN EQS (202)–(2011) 248 HINTS FOR VERIFYING EQ (2128)