INTRODUCTION TO RELATIVITY
CONTENTS 1 HOW THE THEORY OF RELATIVITY CAME INTO BEING A BRIEF HISTORICAL SKETCH 1 1 1 SPECIAL VERSUS GENERAL RELATIVITY 1 1 2 SPACE AND INERTIA IN NEWTONIAN PHYSICS 1 1 3 NEWTON S THEORY AND THE ORBITS OF PLANETS 2 1 4 THE BASIC ASSUMPTIONS OF GENERAL RELATIVITY 3 PART I ELEMENTS OF DI ERENTIAL GEOMETRY 7 2 A SHORT SKETCH OF 2 DIMENSIONAL DI ERENTIAL GEOMETRY 9 2 1 CONSTRUCTING PARALLEL STRAIGHT LINES IN A AT SPACE 9 2 2 GENERALISATION OF THE NOTION OF PARALLELISM TO CURVED SURFACES 10 3 TENSORS TENSOR DENSITIES 12 3 1 WHAT ARE TENSORS GOOD FOR 12 3 2 DI ERENTIABLE MANIFOLDS 12 3 3 SCALARS 14 3 4 CONTRAVARIANT VECTORS 14 3 5 COVARIANT VECTORS 14 3 6 TENSORS OF SECOND RANK 15 3 7 TENSOR DENSITIES 16 3 8 TENSOR DENSITIES OF ARBITRARY RANK 16 3 9 ALGEBRAIC PROPERTIES OF TENSOR DENSITIES 16 3 10 MAPPINGS BETWEEN MANIFOLDS 18 3 11 THE LEVI CIVITA SYMBOL 20 3 12 MULTIDIMENSIONAL KRONECKER DELTAS 21 3 13 EXAMPLES OF APPLICATIONS OF THE LEVI CIVITA SYMBOL AND OF THE MULTIDIMENSIONAL KRONECKER DELTA 22 3 14 EXERCISES 23 4 COVARIANT DERIVATIVES 24 4 1 DI ERENTIATION OF TENSORS 24 4 2 AXIOMS OF THE COVARIANT DERIVATIVE 26 4 3 A ELD OF BASES ON A MANIFOLD AND SCALAR COMPONENTS OF TENSORS 27 4 4 THE A NE CONNECTION 28 4 5 THE EXPLICIT FORMULA FOR THE COVARIANT DERIVATIVE OF TENSOR DENSITY ELDS 28 4 6 EXERCISES 30 5 PARALLEL TRANSPORT AND GEODESIC LINES 31 5 1 PARALLEL TRANSPORT 31 5 2 GEODESIC LINES 32 5 3 EXERCISES 33 6 THE CURVATURE OF A MANIFOLD AT MANIFOLDS 34 6 1 THE COMMUTATOR OF SECOND COVARIANT DERIVATIVES 34 6 2 THE COMMUTATOR OF DIRECTIONAL COVARIANT DERIVATIVES 35 6 3 THE RELATION BETWEEN CURVATURE AND PARALLEL TRANSPORT 36 6 4 COVARIANTLY CONSTANT ELDS OF VECTOR BASES 41 6 5 A TORSION FREE AT MANIFOLD 41 6 6 PARALLEL TRANSPORT IN A AT MANIFOLD 41 6 7 GEODESIC DEVIATION 42 6 8 ALGEBRAIC AND DI ERENTIAL IDENTITIES OBEYED BY THE CURVATURE TENSOR 43 6 9 EXERCISES 44 7 RIEMANNIAN GEOMETRY 45 7 1 THE METRIC TENSOR 45 7 2 RIEMANN SPACES 45 7 3 THE SIGNATURE OF A METRIC DEGENERATE METRICS 46 7 4 CHRISTO EL SYMBOLS 47 7 5 THE CURVATURE OF A RIEMANN SPACE 48 7 6 FLAT RIEMANN SPACES 49 7 7 SUBSPACES OF A RIEMANN SPACE 49 7 8 FLAT RIEMANN SPACES THAT ARE GLOBALLY NON EUCLIDEAN 50 7 9 THE RIEMANN CURVATURE VERSUS THE NORMAL CURVATURE OF A SURFACE 51 7 10 THE GEODESIC LINE AS THE LINE OF EXTREMAL DISTANCE 51 7 11 MAPPINGS BETWEEN RIEMANN SPACES 52 7 12 CONFORMALLY RELATED RIEMANN SPACES 53 7 13 CONFORMAL CURVATURE 54 7 14 TIMELIKE NULL AND SPACELIKE INTERVALS IN A 4 DIMENSIONAL SPACETIME 56 7 15 EMBEDDINGS OF RIEMANN SPACES IN RIEMANN SPACES OF HIGHER DIMENSION 58 7 16 THE PETROV CLASSI CATION 65 7 17 EXERCISES 67 8 SYMMETRIES OF RIEMANN SPACES INVARIANCE OF TENSORS 69 8 1 SYMMETRY TRANSFORMATIONS 69 8 2 THE KILLING EQUATIONS 69 8 3 THE CONNECTION BETWEEN GENERATORS AND THE INVARIANCE TRANSFORMATIONS 72 8 4 FINDING THE KILLING VECTOR ELDS 72 8 5 A KILLING VECTOR ELD ALONG A GEODESIC IS A GEODESIC DEVIATION ELD 74 8 6 INVARIANCE OF OTHER TENSOR ELDS 74 8 7 THE LIE DERIVATIVE 75 8 8 THE ALGEBRA OF KILLING VECTOR ELDS 76 8 9 SURFACE FORMING VECTOR ELDS 76 8 10 SPHERICALLY SYMMETRIC 4 DIMENSIONAL RIEMANN SPACES 77 8 11 CONFORMAL KILLING ELDS AND THEIR NITE BASIS 81 8 12 THE MAXIMAL DIMENSION OF AN INVARIANCE GROUP 83 8 13 EXERCISES 85 9 METHODS TO CALCULATE THE CURVATURE QUICKLY DI ERENTIAL FORMS AND ALGEBRAIC COMPUTER PROGRAMS 89 9 1 THE BASIS OF DI ERENTIAL FORMS 89 9 2 THE CONNECTION FORMS 90 9 3 THE RIEMANN TENSOR 91 9 4 USING COMPUTERS TO CALCULATE THE CURVATURE 92 9 5 EXERCISES 93 10 THE SPATIALLY HOMOGENEOUS BIANCHI TYPE SPACETIMES 94 10 1 THE BIANCHI CLASSI CATION OF 3 DIMENSIONAL LIE ALGEBRAS 94 10 2 THE DIMENSION OF THE GROUP VERSUS THE DIMENSION OF THE ORBIT 99 10 3 ACTION OF A GROUP ON A MANIFOLD 99 10 4 GROUPS ACTING TRANSITIVELY HOMOGENEOUS SPACES 100 10 5 INVARIANT VECTOR ELDS 100 10 6 THE METRICS OF THE BIANCHI TYPE SPACETIMES 102 10 7 THE ISOTROPIC BIANCHI TYPE ROBERTSON WALKER SPACETIMES 103 10 8 EXERCISES 106 11 THE PETROV CLASSI CATION BY THE SPINOR METHOD 108 11 1 WHAT IS A SPINOR 108 11 2 TRANSLATING SPINORS TO TENSORS AND VICE VERSA 109 11 3 THE SPINOR IMAGE OF THE WEYL TENSOR 111 11 4 THE PETROV CLASSI CATION IN THE SPINOR REPRESENTATION 111 11 5 THE WEYL SPINOR REPRESENTED AS A 3 3 COMPLEX MATRIX 112 11 6 THE EQUIVALENCE OF THE PENROSE CLASSES TO THE PETROV CLASSES 114 11 7 THE PETROV CLASSI CATION BY THE DEBEVER METHOD 115 11 8 EXERCISES 116 PART II THE THEORY OF GRAVITATION 121 12 THE EINSTEIN EQUATIONS AND THE SOURCES OF A GRAVITATIONAL ELD 123 12 1 WHY RIEMANNIAN GEOMETRY 123 12 2 LOCAL INERTIAL FRAMES 123 12 3 TRAJECTORIES OF FREE MOTION IN EINSTEIN S THEORY 124 12 4 SPECIAL RELATIVITY VERSUS GRAVITATION THEORY 127 12 5 THE NEWTONIAN LIMIT OF GENERAL RELATIVITY 127 12 6 SOURCES OF THE GRAVITATIONAL ELD 128 12 7 THE EINSTEIN EQUATIONS 128 12 8 HILBERT S DERIVATION OF THE EINSTEIN EQUATIONS 129 12 9 THE PALATINI VARIATIONAL PRINCIPLE 133 12 10 THE ASYMPTOTICALLY CARTESIAN COORDINATES AND THE ASYMPTOTICALLY AT SPACETIME 133 12 11 THE NEWTONIAN LIMIT OF EINSTEIN S EQUATIONS 133 12 12 EXAMPLES OF SOURCES IN THE EINSTEIN EQUATIONS PERFECT UID AND DUST 137 12 13 EQUATIONS OF MOTION OF A PERFECT UID 139 12 14 THE COSMOLOGICAL CONSTANT 140 12 15 AN EXAMPLE OF AN EXACT SOLUTION OF EINSTEIN S EQUATIONS A BIANCHI TYPE I SPACETIME WITH DUST SOURCE 141 12 16 OTHER GRAVITATION THEORIES 144 12 16 1 THE BRANS DICKE THEORY 144 12 16 2 THE BERGMANN WAGONER THEORY 145 12 16 3 THE EINSTEIN CARTAN THEORY 145 12 16 4 THE BI METRIC ROSEN THEORY 146 12 17 MATCHING SOLUTIONS OF EINSTEIN S EQUATIONS 146 12 18 THE WEAK ELD APPROXIMATION TO GENERAL RELATIVITY 148 12 19 EXERCISES 154 13 THE MAXWELL AND EINSTEIN MAXWELL EQUATIONS AND THE KALUZA KLEIN THEORY 156 13 1 THE LORENTZ COVARIANT DESCRIPTION OF ELECTROMAGNETIC ELD 156 13 2 THE COVARIANT FORM OF THE MAXWELL EQUATIONS 156 13 3 THE ENERGY MOMENTUM TENSOR OF ELECTROMAGNETIC ELD 157 13 4 THE EINSTEIN MAXWELL EQUATIONS 158 13 5 THE VARIATIONAL PRINCIPLE FOR THE MAXWELL AND EINSTEIN MAXWELL EQUATIONS 158 13 6 THE KALUZA KLEIN THEORY 159 13 7 EXERCISES 161 14 SPHERICALLY SYMMETRIC GRAVITATIONAL ELDS OF ISOLATED OBJECTS 162 14 1 THE CURVATURE COORDINATES 162 14 2 SYMMETRY INHERITANCE 165 14 3 SPHERICALLY SYMMETRIC ELECTROMAGNETIC ELD IN VACUUM 166 14 4 THE SCHWARZSCHILD AND REISSNER NORDSTR OM SOLUTIONS 166 14 5 ORBITS OF PLANETS IN THE GRAVITATIONAL ELD OF THE SUN 169 14 6 DE ECTION OF LIGHT RAYS IN THE SCHWARZSCHILD ELD 175 14 7 MEASURING THE DE ECTION OF LIGHT RAYS 178 14 8 GRAVITATIONAL LENSES 180 14 9 THE SPURIOUS SINGULARITY OF THE SCHWARZSCHILD SOLUTION AT R 2M 182 14 10 EMBEDDING THE SCHWARZSCHILD SPACETIME IN A AT RIEMANNIAN SPACE 187 14 11 INTERPRETATION OF THE SPURIOUS SINGULARITY AT R 2M BLACK HOLES 191 14 12 THE SCHWARZSCHILD METRIC IN OTHER COORDINATE SYSTEMS 193 14 13 THE EQUATION OF HYDROSTATIC EQUILIBRIUM 194 14 14 THE INTERIOR SCHWARZSCHILD SOLUTION 196 14 15 THE MAXIMAL ANALYTIC EXTENSION OF THE REISSNER NORDSTR OM METRIC 197 14 16 MOTION OF PARTICLES IN THE REISSNER NORDSTR OM SPACETIME WITH E2 M2 205 14 17 EXERCISES 209 15 RELATIVISTIC HYDRODYNAMICS AND THERMODYNAMICS 211 15 1 MOTION OF A CONTINUOUS MEDIUM IN NEWTONIAN HYDRODYNAMICS 211 15 2 MOTION OF A CONTINUOUS MEDIUM IN RELATIVISTIC HYDRODYNAMICS 213 15 3 THE EQUATIONS OF EVOLUTION OF AND U THE RAYCHAUDHURI EQUATION 216 15 4 SINGULARITIES AND SINGULARITY THEOREMS 218 15 5 RELATIVISTIC THERMODYNAMICS 219 15 6 EXERCISES 222 16 RELATIVISTIC COSMOLOGY I GENERAL GEOMETRY 223 16 1 A CONTINUOUS MEDIUM AS A MODEL OF THE UNIVERSE 223 16 2 THE GEOMETRIC OPTICS APPROXIMATION 224 16 3 THE REDSHIFT 226 16 4 THE OPTICAL TENSORS 228 16 5 THE APPARENT HORIZON 230 16 6 THE DOUBLE NULL TETRAD 231 16 7 THE EQUATIONS OF PROPAGATION OF THE OPTICAL SCALARS 233 16 8 THE GOLDBERG SACHS THEOREM 235 16 9 THE AREA DISTANCE 240 16 10 THE RECIPROCITY THEOREM 243 16 11 OTHER OBSERVABLE QUANTITIES 246 16 12 THE FERMI WALKER TRANSPORT 246 16 13 POSITION DRIFT OF LIGHT SOURCES 248 16 14 EXERCISES 253 17 RELATIVISTIC COSMOLOGY II THE ROBERTSON WALKER GEOMETRY 255 17 1 THE ROBERTSON WALKER METRICS AS MODELS OF THE UNIVERSE 255 17 2 OPTICAL OBSERVATIONS IN AN R W UNIVERSE 257 17 2 1 THE REDSHIFT 257 17 2 2 THE REDSHIFT DISTANCE RELATION 258 17 2 3 NUMBER COUNTS 259 17 3 THE FRIEDMANN EQUATION 260 17 4 THE FRIEDMANN SOLUTIONS WITH 0 262 17 5 THE REDSHIFT DISTANCE RELATION IN THE 0 FRIEDMANN MODELS 263 17 6 THE NEWTONIAN COSMOLOGY 264 17 7 THE FRIEDMANN SOLUTIONS WITH THE COSMOLOGICAL CONSTANT 266 17 8 THE CDM MODEL 269 17 9 THE REDSHIFT DISTANCE RELATION IN THE 0 FRIEDMANN MODELS 271 17 10 THE REDSHIFT DRIFT A TEST FOR ACCELERATING EXPANSION 272 17 11 HORIZONS IN THE ROBERTSON WALKER MODELS 273 17 12 THE IN ATIONARY MODELS AND THE PROBLEMS THEY SOLVED 277 17 13 THE VALUE OF THE COSMOLOGICAL CONSTANT 282