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General Relativity

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General Relativity GENERALRELATIVITY t h i m o p r e i s1 17th April 2020 1 [email protected] CONTENTS 1 differential geomtry 3 1.1 Differentiable manifolds 3 1.2 The tangent space 4 1.2.1 Tangent vectors 6 1.3 Dual vectors and tensor 6 1.3.1 Tensor densities 8 1.4 Metric 11 1.5 Connections and covariant derivatives 13 1.5.1 Note on exponential map/Riemannian normal coordinates - TO DO 18 1.6 Geodesics 20 1.6.1 Equivalent deriavtion of the Geodesic Equation - Weinberg 22 1.6.2 Character of geodesic motion is sustained and proper time is extremal on geodesics 24 1.6.3 Another remark on geodesic equation using the principle of general covariance 26 1.6.4 On the parametrization of the path 27 1.7 An equivalent consideration of parallel transport, geodesics 29 1.7.1 Formal solution to the parallel transport equa- tion 31 1.8 Curvature 33 1.8.1 Torsion and metric connection 34 1.8.2 How to get from the connection coefficients to the connection-the metric connection 34 1.8.3 Conceptional flow of how to add structure on our mathematical constructs 36 1.8.4 The curvature 37 1.8.5 Independent components of the Riemann tensor and intuition for curvature 39 1.8.6 The Ricci tensor 41 1.8.7 The Einstein tensor 43 1.9 The Lie derivative 43 1.9.1 Pull-back and Push-forward 43 1.9.2 Connection between coordinate transformations and diffeomorphism 47 1.9.3 The Lie derivative 48 1.10 Symmetric Spaces 50 1.10.1 Killing vectors 50 1.10.2 Maximally Symmetric Spaces and their Unique- ness 54 iii iv co n t e n t s 1.10.3 Maximally symmetric spaces and their construc- tion 56 1.10.4 Tensors in a Maximally symmetric space 59 1.10.5 Spaces with Maximally Symmetric Subspaces 60 1.10.6 Cosmological Lie Algebra - Equivalent deriva- tion of Robertson-Walker metric via Killing fields 63 1.10.7 Conservation laws from Killing vector fields - More in depth 66 1.10.8 Explicit Killing vectors 67 1.11 Differential forms 68 1.11.1 Detour- de Rham Cohomology – TO EXPAND 70 1.11.2 Hodge star operator 70 1.11.3 Codifferential 72 1.11.4 Conventional differential operators via forms 73 1.12 Integrals 73 1.13 Frame fields in GR and the Tetrad formalism, ”tetrad”=”vierbein”=”frame field” 74 1.13.1 On the tetrad formalism - Weinberg 77 1.14 Cartan’s structure equations 81 1.15 Comparison of gauge theories in particle physics with Riemannian geometry 84 2 general relativity - the theory 87 2.1 Introduction to General Relativity 87 2.1.1 Closer Look at the Equivalence Principle 87 2.1.2 Another look at the equivalence principle by Feyn- man 89 2.1.3 Consequences of principle of equivalence 89 2.1.4 From Newton to Einstein 91 2.2 On our Path towards a curved Spacetime 91 2.3 Newtonian limit: 92 2.4 General Covariance 94 2.4.1 Principle of General Covariance represented by Diffeomorphism Invariance in GR 96 2.4.2 The physical meaning of curvature 99 2.5 Physical laws in external gravitational fields - Thus a metric is given, what are the dynamics 101 2.5.1 Relativistic variational principle 101 2.5.2 Proper time in general coordinates 101 2.5.3 Motion of particles 102 2.5.4 Motion of light 103 2.5.5 Energy-momentum ”conservation” 104 2.5.6 The connection between curvatures and matter 105 2.6 The geodesic deviation equation 106 2.6.1 Equivalent treatment of Geodesic deviation - Car- oll 107 2.7 Einstein’s field equations 109 co n t e n t s v 2.7.1 A not successful attempt via Poisson’s equation 109 2.7.2 Heuristic Derivation 110 2.7.3 On the Meaning of the Field Equations 112 2.8 On the action principle - Weinberg 114 2.8.1 General definition of the Energy-Momentum-tensor by Weinberg 114 2.8.2 General Covariance and Energy-Momentum Con- servation - Weinberg 115 2.9 Derivation of Einstein’s Field Equations from Least Ac- tion Principle 118 2.9.1 The full Einstein field equations 123 2.9.2 Coordinate Conditions - A note on gauge free- dom in Einstein’s field equations -Weinberg 125 2.10 The energy-momentum tensor 128 2.10.1 Energy,Momentum, and Angular Momentum of Gravitation - Weinberg 128 2.10.2 The matter sector 132 2.10.3 The action for classical particles in a gravitational field 133 2.11 Modifications of the field equations - TO EXPAND 135 2.12 Initial Value Problem in GR 136 2.13 Existence of Solutions to the field equations 138 2.13.1 Hawking-Penrose Singularity Theorems: TO DO 140 3 applications of general relativity 143 3.1 Weak Gravitational Fields 143 3.2 Gauge transformations in general relativity 143 3.2.1 Firstly, remark on the difference to diffeomorph- ism invariance, aka general covariance 143 3.2.2 On General Covariance and gauge symmetry by Weinberg 144 3.2.3 Finding the gauge class 146 3.2.4 Linearised theory of gravity 149 3.2.5 Nearly Newtonian gravity 151 3.3 Gravitational Waves - The propagation of the metric per- turbations 156 3.4 The Schwarzschild solution 158 3.4.1 Stationary and static space-times 158 3.4.2 The Schwarzschild solution 158 3.4.3 Solution of the field equations 160 3.4.4 Birkhoff’s theorem 162 3.5 Physics in the Schwarzschild Space-Time 165 3.5.1 A note on the Killing vectors of a Schwarzschild Space-Time 165 3.5.2 Orbits in the Schwarzschild Space-Time 166 3.5.3 Perihelion shift 169 3.5.4 Light deflection 170 vi co n t e n t s 3.5.5 Redshift in Schwarzschild Spacetime 172 3.6 Black Holes 173 3.6.1 Schwarzschild Black Holes 173 3.6.2 Charged, Rotating Black Holes 184 3.6.3 Entropy and Temperature of a Black Hole 197 3.7 Cosmology 199 4 co s m o lo gy 201 4.1 Homogeneous and Isotropic Cosmology - GR perspect- ive 201 4.1.1 Spherically-symmetric space-times 201 4.1.2 Homogeneous and Isotropic Space-times 203 4.1.3 Friedmann’s equations 208 4.1.4 On Cosmology by Weinberg 212 4.1.5 Notes on Cosmology from GR lecture, to include into Cosmology write-up 213 4.1.6 Comoving Coordinates Weinberg 215 4.1.7 Cosmological redshift- TO DO 215 4.2 Friedmann models - A physical approach – streamline with the above 216 4.2.1 Symmetry assumptions and consequences 216 4.2.2 Friedmann equations 217 4.2.3 Parameters 219 4.2.4 Redshift (cosmological) 222 4.2.5 Age and Expansion of the Universe 223 4.2.6 Distance measures 225 4.2.7 Horizons 229 4.3 Age of the Universe and the cosmic expansion 231 4.3.1 Nuclear cosmo-chronology 231 4.3.2 Stellar ages 231 4.3.3 Cooling of white dwarfs 231 4.3.4 The Hubble constant 231 4.3.5 Supernovae of type Ia 231 4.4 Thermal evolution of the universe 231 4.4.1 Assumptions and quantum statistics 231 4.4.2 Recombination and Nucleosynthesis 237 4.5 Cosmological Inflation 246 4.5.1 The horizon and flatness problem 246 4.5.2 The idea of inflation 248 4.5.3 Dark Energy 255 4.6 Evolution of cosmic structures 256 4.6.1 Linear perturbation theory 256 4.6.2 Linear growth factor, perturbations within and outside the horizon 258 4.6.3 Velocity perturbations and Zel’dovich approxim- ation 258 4.6.4 Nonlinear evolution and power spectra 258 co n t e n t s vii 4.6.5 The Cosmic Microwave Background 258 4.7 Gravitational Lensing 260 4.7.1 Gravitational Lensing Equation from GR 260 4.7.2 Thin gravitational lenses 260 4.7.3 Correlation functions (no thin lens approxima- tion) 260 4.8 Theoretical Astrophysics 260 4.9 Two examples of Relativistic Astrophysics 261 4.9.1 Gravitational Lensing - TO COMPLETE FROM SCRIPT 261 4.10 Tolman-Oppenheimer-Volkoff solution 265 5 generalrelativityasaquantumfieldtheory 271 5.1 Conceptional differences- TO DO 271 5.2 On physical theories, remarks by Feynman 271 5.3 How Feynman derived GR from a field theoretical ap- proach 272 5.3.1 The action for matter fields in a gravitational field 276 5.3.2 Final formulation of Gravity on path to quantum gravity 277 6 possible exam problems 279 7 important equations of gr - to do 281 7.1 Differential Geomtetry 281 7.2 Statements from problem sheets 281 7.3 Statements to incorporate 282 LISTOFFIGURES Figure 1.1 28 Figure 1.2 30 Figure 1.3 32 Figure 1.4 37 Figure 1.5 I.e. cylinder is not curved since it can be glued to- gether by flat spaces. 41 Figure 1.6 E.g. integral curves on the two-sphere, defining the rotations by a given angle. The tangent vectors infin- itesimmaly generate the rotation. Compare treatment in 1.2.1. 44 Figure 1.7 64 Figure 1.8 65 Figure 2.1 113 Figure 2.2 113 Figure 2.3 139 Figure 2.4 140 Figure 3.1 147 Figure 3.2 147 Figure 3.3 Foliation of the static spacetime along the fiducial time direction given by global Killing vector field.
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