Contents

RELATIVITY,GRAVITATION ANDCOSMOLOGY

Introduction9 Chapter 1Special relativity and spacetime 11 Introduction11 1.1 Basicconcepts of special relativity 12 1.1.1Events, framesofreference andobservers 12 1.1.2 Thepostulates of special relativity 14 1.2 Coordinate transformations16 1.2.1The Galilean transformations16 1.2.2The Lorentztransformations18 1.2.3Aderivationofthe Lorentztransformations21 1.2.4Intervals andtheir transformation rules 23 1.3 Consequences of theLorentztransformations24 1.3.1Timedilation24 1.3.2Lengthcontraction26 1.3.3The relativity of simultaneity 27 1.3.4The Doppler effect 28 1.3.5The velocity transformation 29 1.4 Minkowskispacetime31 1.4.1Spacetimediagrams, lightconesand causality 31 1.4.2Spacetimeseparation andthe Minkowskimetric 35 1.4.3The twin effect 38 Chapter 2Special relativity and physical laws 45 Introduction45 2.1 Invariants andphysical laws 46 2.1.1The invariance of physical quantities 46 2.1.2The invariance of physical laws 47 2.2 Thelawsofmechanics 49 2.2.1Relativistic momentum 49 2.2.2Relativistic kinetic energy 52 2.2.3Total relativistic energy andmassenergy54 2.2.4Four-momentum 56 2.2.5The energy–momentum relation58 2.2.6The conservationofenergyand momentum 60 2.2.7Four-force 61 2.2.8Four-vectors62 2.3 Thelawsofelectromagnetism67 5 Contents

2.3.1 Theconservationofcharge67 2.3.2The Lorentzforce law68 2.3.3The transformation of electric andmagnetic fields73 2.3.4The Maxwell equations 74 2.3.5Four-tensors 75 Chapter 3Geometryand curvedspacetime 80 Introduction80 3.1 Line elements anddifferential geometry82 3.1.1Lineelements in aplane82 3.1.2Curved surfaces 85 3.2 Metricsand connections90 3.2.1Metrics andRiemannian geometry90 3.2.2Connectionsand parallel transport 92 3.3 Geodesics 97 3.3.1Mostdirect route between twopoints97 3.3.2Shortest distance between twopoints98 3.4 Curvature100 3.4.1Curvatureofacurveinaplane101 3.4.2Gaussian curvatureofatwo-dimensionalsurface 102 3.4.3Curvatureinspaces of higherdimensions 104 3.4.4Curvatureofspacetime106 Chapter 4General relativity and gravitation110 Introduction110 4.1 Thefounding principles of general relativity 111 4.1.1The principle of equivalence 112 4.1.2The principle of general covariance 116 4.1.3The principle of consistency 124 4.2 Thebasic ingredients of general relativity 126 4.2.1The energy–momentum tensor 126 4.2.2The Einstein tensor132 4.3 Einstein’s field equations andgeodesicmotion133 4.3.1The Einstein field equations 134 4.3.2Geodesicmotion136 4.3.3The Newtonian limit of Einstein’s field equations 138 4.3.4The cosmological constant139 Chapter 5Schwarzschild spacetime 144 Introduction144 5.1 Themetric of Schwarzschild spacetime145 6 Contents

5.1.1 TheSchwarzschild metric 145 5.1.2Derivation of theSchwarzschild metric 146 5.2 Properties of Schwarzschild spacetime151 5.2.1Spherical symmetry151 5.2.2Asymptotic flatness152 5.2.3Time-independence152 5.2.4Singularity 153 5.2.5Generality 154 5.3 Coordinates andmeasurements in Schwarzschild spacetime154 5.3.1Framesand observers 155 5.3.2 Proper timeand gravitationaltimedilation156 5.3.3Proper distance 159 5.4 GeodesicmotioninSchwarzschild spacetime160 5.4.1The geodesicequations 161 5.4.2Constants of themotioninSchwarzschild spacetime162 5.4.3Orbital motioninSchwarzschild spacetime166 Chapter 6Black holes171 Introduction171 6.1 Introducingblack holes 171 6.1.1Ablack holeand its eventhorizon 171 6.1.2Abriefhistory of black holes 172 6.1.3The classification of black holes 175 6.2 Non-rotating black holes 176 6.2.1Falling into anon-rotating black hole177 6.2.2Observing afall from faraway179 6.2.3Tidal effects near anon-rotating black hole183 6.2.4The deflection of light near anon-rotating black hole186 6.2.5The eventhorizon andbeyond 187 6.3 Rotating black holes 192 6.3.1The Kerr solutionand rotatingblack holes 192 6.3.2Motionnear arotating black hole194 6.4 Quantum physics andblack holes 198 6.4.1Hawking radiation 198 6.4.2Singularities andquantum physics 200 Chapter 7Testinggeneral relativity 204 Introduction204 7.1 Theclassictests of general relativity 204 7.1.1Precession of theperihelionofMercury 204 7 Contents

7.1.2 Deflection of light by theSun 205 7.1.3Gravitationalredshiftand gravitationaltimedilation206 7.1.4Timedelay of signals passing theSun 211 7.2 Satellite-basedtests 213 7.2.1Geodesicgyroscope precession 213 7.2.2Frame dragging 214 7.2.3The LAGEOS satellites 215 7.2.4Gravity Probe B216 7.3 Astronomical observations 217 7.3.1Black holes 217 7.3.2Gravitationallensing 223 7.4 Gravitationalwaves 226 7.4.1Gravitationalwaves andthe Einstein field equations 226 7.4.2Methods of detectinggravitationalwaves 229 7.4.3Likely sourcesofgravitationalwaves 231 Chapter 8Relativistic cosmology234 Introduction234 8.1 Basicprinciples andsupporting observations 235 8.1.1The applicability of general relativity 235 8.1.2The cosmological principle 236 8.1.3Weyl’spostulate 240 8.2 Robertson–Walker spacetime242 8.2.1The Robertson–Walker metric 243 8.2.2Proper distances andvelocities in cosmicspacetime245 8.2.3The cosmicgeometryofspace andspacetime247 8.3 TheFriedmann equations andcosmicevolution251 8.3.1The energy–momentum tensor of thecosmos251 8.3.2The Friedmann equations 254 8.3.3Three cosmological models with k =0 256 8.3.4Friedmann–Robertson–Walker models in general 259 8.4 Friedmann–Robertson–Walker models andobservations 263 8.4.1Cosmological redshift andcosmicexpansion263 8.4.2Density parametersand theage of theUniverse269 8.4.3Horizons andlimits 270 Appendix277 Solutions279 Acknowledgements308 Index309 8