Volume I

Advanced and Modern

General Relativity

Draft version

16th November 2012

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Contents:

• Chapter 1: Einstein’s hole argument Complete, Partial and DiracObservables

• Chapter 2: Introduction to General Relativity and its physicalObservables

• Chapter 3: Black holes - Event, Isolated and Dynamical Horizons• Chapter 4: Cosmology• Chapter 5: Hawking Penrose Singularity Theorems• Chapter 6: Consistent Discrete Classical GR• Chapter 7: Quantum Field Theory on Curved Spacetimes• Chapter 8: Introduction to Quantum General Relativity

• Glossary

• Many Detailed Appendices with Worked Exercises

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Chapter 1: Classical GR, Einstein’s hole argumentand Physical geometry

• Einstein’s hole argument.

• Measurements of geometry.

• Conceptual issues.

• Relational mechanics.

• Partial, complete and Dirac observables.

Chapter 2: Introduction to General Relativity and itsPhysical Observables

• Special Relativity

• Principles of General Relativity

• Spacetime Measurements

• GPS Observables

• Action Principle

• Perturbations Around Exact SolutionsChapter 3: Black holes - Event, Isolated and Dynam-ical Horizons

• Event Horizons and Penrose Diagrams

• Non-Expanding Horizons

• Weakly Isolated Horizons and Generalisations of the Laws of Black HoleMechanics

• Isolated Horizons and Rotating Isolated Horizons

• Dynamical Horizons

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Chapter 4: Cosmology

• Classical Cosmology

• Homogeneous and Isotropic Cosmology

• Outline of the Singularity Theorems

• Gauge Invariant Perturbations

• .

Chapter 5: Hawking Penrose Singularity Theorems.

• Basic Definitions

• Strong Causality

• The Space of Causal Curves

• Conjugate Points

• Singularity Theorems

• Collapse of a Star

• The Big Bang

• Initial Value Problem

Chapter 6: Consistent Discrete Classical GR.

• Introduction.

• .

•• .

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Chapter 7: Quantum Field Theory on Curved Space-times

• Introduction

• Quantum Field Theory on Flat Spacetime

• Quantum Field Theory on Curved Spacetime

• Quantum Fields in an Expanding spacetime

• Quantum Fields During Inflation

• Hawking Radiation

• Algebraic Approach

• Back Reaction

• The Need for Quantum Gravity

• Stochastic Gravity

Chapter 8: Introduction to Quantum Gravity.

• Introduction to Quantum General Relativity

• Ashtekar-Barbero Variables

• Quantum Constraints - the Equations of Canonical Quantum Gravity

• The Loop Representation

• Geometric Operators

• Spin Networks

• The Hamiltonian Constraint and the Modern forumalism

• Spin Foams

• Semi-Classical Limit

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• The Master Constraint Programme

• Physical Applications: Black Hole Entropy, Loop Quantum Cosmology,Quantum Gravity Phenenomology, and Background-independent Scatter-

ing Amplitudes

• The problem of Time

• Other Approaches

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Appendices

• A Physics Glossary

• B Mathematics Glossary• C Mathematics

• D Constrained Hamiltonian Systems, Dirac Observables and the Con-straint Algebra

• E ADM and First Order Formulation of Einstein’s Equations

• F Self-dual Connection Formulation and Ashtekar’s new Variables

• G The Holst Action and Ashtekar-Barbero Real Variables

• H Basic Functional Analysis

• I Qunatum Field Theory

• J Details of Hawking’s Calculation

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Contents

1 Classical GR, Einstein’s hole argument and Physical Geometry 54

1.1 Einstein’s Hole Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

1.2 Background Independence - A Farewell to Spacetime . . . . . . . . . . . . 69

1.2.1 Comparision of GR with the Rubber Sheet Analogy . . . . . . . . . 69

1.2.2 The View of the World that Emerges . . . . . . . . . . . . . . . . . 71

1.2.3 Common Misunderstandings . . . . . . . . . . . . . . . . . . . . . . 74

1.2.4 The Blessing of background independence - Non-Peturbative Quan-tum Gravity Finite and Requires No Renormalization! . . . . . . . 75

1.3 Physical Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

1.3.1 Physical GPS Coordinates . . . . . . . . . . . . . . . . . . . . . . . 79

1.3.2 Physical Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

1.3.3 Description of a Measurement of Area . . . . . . . . . . . . . . . . 81

1.3.4 Einstein’s Field Equations . . . . . . . . . . . . . . . . . . . . . . . 83

1.3.5 The velocity composition law . . . . . . . . . . . . . . . . . . . . . 84

1.3.6 Dust as Matter Reference System . . . . . . . . . . . . . . . . . . . 84

1.4 Some conceptual issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

1.5 Relational Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

1.5.1 Covariant Hamiltonian Formulation . . . . . . . . . . . . . . . . . . 85

1.5.2 Depamerizable Mechanics: Identification of a “time” variable . . . . 88

1.5.3 Fully Constrained Hamiltonian Systems . . . . . . . . . . . . . . . 89

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1.6 Partial, Complete and Dirac Observables . . . . . . . . . . . . . . . . . . . 90

1.6.1 Infinitely Many Constraints . . . . . . . . . . . . . . . . . . . . . . 93

1.6.2 Observables for Canonical General Relativity . . . . . . . . . . . . . 94

1.6.3 Approximate Observables for Canonical General Relativity . . . . . 95

1.7 The Problem of Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

1.8 The problem of Quantum Cosmology . . . . . . . . . . . . . . . . . . . . . 98

2 Introduction to General Relativity and its Physical Observables 99

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

2.2 Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

2.2.1 Simultaneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

2.2.2 Time Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

2.2.3 Length Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

2.2.4 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . 102

2.2.5 Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

2.2.6 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

2.2.7 The Relativistic Doppler Effect . . . . . . . . . . . . . . . . . . . . 106

2.2.8 Relativistic Momentum . . . . . . . . . . . . . . . . . . . . . . . . . 107

2.2.9 Relativistic Mass and Energy . . . . . . . . . . . . . . . . . . . . . 108

2.2.10 The Twin Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

2.2.11 Lorentz group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

2.3 The Principles of General Relativity . . . . . . . . . . . . . . . . . . . . . . 112

2.3.1 The Principle of Equivalence . . . . . . . . . . . . . . . . . . . . . . 112

2.3.2 The Gravitation Red-shift: Warping Time . . . . . . . . . . . . . . 112

2.3.3 The Curvature of Spacetime . . . . . . . . . . . . . . . . . . . . . . 113

2.3.4 Curvature in a Weak Uniform Gravitation field . . . . . . . . . . . 114

2.3.5 The Principle of General Relativity . . . . . . . . . . . . . . . . . . 116

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2.3.6 Background Independent Theories . . . . . . . . . . . . . . . . . . . 116

2.3.7 Einstein’s Hole Argument . . . . . . . . . . . . . . . . . . . . . . . 116

2.4 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

2.5 Tensor Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

2.5.1 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

2.5.2 Covariant Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 121

2.5.3 The Metric Connection . . . . . . . . . . . . . . . . . . . . . . . . . 123

2.5.4 Curvature Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

2.6 Space-Time Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

2.6.1 Measurements of Time Intervals and Space Distances . . . . . . . . 123

2.6.2 Measurement Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

2.6.3 Geodesis Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

2.6.4 World Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

2.6.5 Measurement of Relative Velocities . . . . . . . . . . . . . . . . . . 131

2.6.6 Derivation of Lorentz Transformation Formula in GR . . . . . . . . 132

2.7 Derivation of Vacuum Field Equations . . . . . . . . . . . . . . . . . . . . 132

2.7.1 The Newtonian Equation of Deviation . . . . . . . . . . . . . . . . 132

2.7.2 Equation of geodesic deviation . . . . . . . . . . . . . . . . . . . . . 133

2.7.3 The Newtonian Correspondence . . . . . . . . . . . . . . . . . . . . 137

2.7.4 The Vacuum Field Equations . . . . . . . . . . . . . . . . . . . . . 140

2.8 GPS Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

2.9 Measurement of an Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

2.10 Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

2.10.1 Dust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

2.10.2 Perfect Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

2.10.3 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 150

2.10.4 Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

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2.10.5 Yang-Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

2.10.6 Fermionic Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

2.10.7 Energy-Momentum Tensor . . . . . . . . . . . . . . . . . . . . . . . 155

2.11 Action Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

2.11.1 GHY boundary term . . . . . . . . . . . . . . . . . . . . . . . . . . 157

2.11.2 Introduction to hypersurfaces . . . . . . . . . . . . . . . . . . . . . 158

2.11.3 Proof of main result . . . . . . . . . . . . . . . . . . . . . . . . . . 162

2.11.4 Variation of the Matter action . . . . . . . . . . . . . . . . . . . . . 166

2.11.5 Invariance of the Einstein-Hilbert Action . . . . . . . . . . . . . . . 167

2.12 Palatini Method in the Metric Formulation . . . . . . . . . . . . . . . . . . 168

2.13 Cosmological Definition of Distance . . . . . . . . . . . . . . . . . . . . . . 172

2.14 Relativistic Material Reference Systems . . . . . . . . . . . . . . . . . . . . 173

2.15 Linearized Equations of General Relativity . . . . . . . . . . . . . . . . . . 174

2.15.1 Gauge Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 175

2.15.2 Linearized Einstein Equations in the Temporal Gauge . . . . . . . . 178

2.15.3 Gravitational Wave Solutions . . . . . . . . . . . . . . . . . . . . . 178

2.15.4 Waves Emitted by Oscillating Masses . . . . . . . . . . . . . . . . . 179

2.16 Classical Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

2.16.1 Fluid Flow Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 180

2.16.2 Newtonian Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . 181

2.16.3 Relativistic Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . 183

2.16.4 Spaces of Constant Curvature . . . . . . . . . . . . . . . . . . . . . 183

2.17 Homogeneous and Isotropic Cosmology . . . . . . . . . . . . . . . . . . . . 183

2.17.1 Friedmann’s Equation - Universe with Dust . . . . . . . . . . . . . 183

2.17.2 The Luminosity Distance . . . . . . . . . . . . . . . . . . . . . . . . 186

2.18 Models with a Cosmological Constant . . . . . . . . . . . . . . . . . . . . . 188

2.18.1 Flat Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

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2.18.2 The de Sitter Model . . . . . . . . . . . . . . . . . . . . . . . . . . 189

2.19 Perturbations of Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . 189

2.19.1 Gauge Dependency in Perturbation Theory . . . . . . . . . . . . . . 190

2.20 Cosmological Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . 198

2.20.1 Scalar-Vector-Tensor Decomposition . . . . . . . . . . . . . . . . . 198

2.20.2 Choice of Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

2.21 Perturbations of Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . 200

2.22 Approximation to Observables of the Full Theory . . . . . . . . . . . . . . 200

2.23 Gravity from Gravitons? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

2.24 Some Things of String Theory . . . . . . . . . . . . . . . . . . . . . . . . . 202

2.25 Biblioliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

2.26 Worked Exercises and Details . . . . . . . . . . . . . . . . . . . . . . . . . 203

2.27 Backreaction Issues in Relativistic Cosmology and the Dark Energy Debate 206

2.27.1 Cosmological Perturbation Theory . . . . . . . . . . . . . . . . . . 206

2.28 Gauge Invariant Perturbations Around Symmetry Reduced Sectors of Gen-eral Relativity: Applications to Cosmology . . . . . . . . . . . . . . . . . . 207

2.28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

2.29 Approximate Complete Observables . . . . . . . . . . . . . . . . . . . . . . 209

2.30 Application to Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

3 Isolated and Dynamical Horizons - Generalizations of Stationary BlackHoles 220

3.1 Review of Stationary Black Holes . . . . . . . . . . . . . . . . . . . . . . . 221

3.1.1 Mass and Angular Momentum of Bodies in Newtonian Gravity andSpecial Realtivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

3.2 The Schwarzschild Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

3.3 Schwarzschild Black Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

3.3.1 Eddington-Finkelstein Coordinates . . . . . . . . . . . . . . . . . . 233

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3.4 Internal Schwarzschild Solution . . . . . . . . . . . . . . . . . . . . . . . . 236

3.5 Penrose-Carter diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

3.5.1 Penrose-Carter Diagram for Minkowski Spacetime. . . . . . . . . . 240

3.5.2 Maximal Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 242

3.5.3 The Kruskal Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 242

3.6 Charged Balck Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

3.6.1 Event Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

3.6.2 Analogue of Eddington-Finkelstein Coordinates . . . . . . . . . . . 255

3.6.3 Penrose Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

3.6.4 Double null coordinates . . . . . . . . . . . . . . . . . . . . . . . . 257

3.6.5 Maximal extension . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

3.7 Rotating Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

3.7.1 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

3.7.2 The Kerr Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

3.7.3 Independence of Metric on Angular Variable ϕ . . . . . . . . . . . . 296

3.7.4 Boyer-Lindquist Coordinates . . . . . . . . . . . . . . . . . . . . . . 299

3.7.5 Interptetation as Rotating Body . . . . . . . . . . . . . . . . . . . . 302

3.7.6 Basic Propertire of the Kerr Solution . . . . . . . . . . . . . . . . . 303

3.7.7 Singularities and Event Horizons . . . . . . . . . . . . . . . . . . . 304

3.8 Raychaudhuri equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

3.8.1 Null geodesic congruences . . . . . . . . . . . . . . . . . . . . . . . 305

3.9 Properties of Null Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

3.9.1 geodesics: Expansion, Rotation, and Shear . . . . . . . . . . . . . . 309

3.9.2 Frobenius’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 310

3.9.3 Null Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

3.9.4 Killing Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

3.10 Laws of (Stationary) Black Hole Mechanics . . . . . . . . . . . . . . . . . . 313

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3.10.1 Zeroth law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

3.10.2 First law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

3.10.3 Second law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

3.10.4 Third law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

3.10.5 Quasi-Local Generalizations . . . . . . . . . . . . . . . . . . . . . . 318

3.10.6 Black Hole Thermodynamics . . . . . . . . . . . . . . . . . . . . . . 318

3.11 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

3.12 Non-Expanding Horizons (NEH) . . . . . . . . . . . . . . . . . . . . . . . . 320

3.13 Weakly Isolated Horizons (WIH) and Generalization of the Laws of BlackHole Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

3.13.1 Zeroth law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

3.13.2 First law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

3.13.3 Second law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

3.14 Isolated Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

3.14.1 Boundary Conditions for Isolated Horizons . . . . . . . . . . . . . . 325

3.15 Rotating Isolated Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

3.15.1 Basic Review of Multipoles . . . . . . . . . . . . . . . . . . . . . . . 327

3.15.2 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . 332

3.15.3 Invariant Coordinates on the Horizon . . . . . . . . . . . . . . . . . 333

3.15.4 Definition of Geometric Multipoles . . . . . . . . . . . . . . . . . . 336

3.16 Dynamical Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

3.16.1 Gravitational Energy Flux . . . . . . . . . . . . . . . . . . . . . . . 339

3.16.2 Rotating Dynamical Horizons . . . . . . . . . . . . . . . . . . . . . 340

3.17 Null Tetrads and Spinor Analysis . . . . . . . . . . . . . . . . . . . . . . . 340

3.17.1 Null Tetrads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

3.17.2 Newman-Penrose Formulism . . . . . . . . . . . . . . . . . . . . . . 345

3.17.3 Spinor Analysis in GR . . . . . . . . . . . . . . . . . . . . . . . . . 346

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3.17.4 Curvature Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

3.17.5 Curvature in spinors . . . . . . . . . . . . . . . . . . . . . . . . . . 373

3.17.6 Spinor Form of the Ricci Identies . . . . . . . . . . . . . . . . . . . 377

3.17.7 Einstein’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

3.17.8 Spinor form of the Bianchi identity . . . . . . . . . . . . . . . . . . 381

3.17.9 Newman-Penrose Formalism in Spinor Form . . . . . . . . . . . . . 382

3.17.10Petrov Classification . . . . . . . . . . . . . . . . . . . . . . . . . . 389

3.17.11Equivalence of Petrov Classification Schemes . . . . . . . . . . . . . 397

3.17.12Petrov classification via Eigenbivectors of the Weyl Tensor . . . . . 403

3.17.13Focussing and Shearing of Null Curves . . . . . . . . . . . . . . . . 409

3.17.14Goldberg Sachs Theorem . . . . . . . . . . . . . . . . . . . . . . . . 411

3.17.15Tetrad Formulism and the Cartan Structure Equations . . . . . . . 421

3.17.16Specialisation to Null Tetrads . . . . . . . . . . . . . . . . . . . . . 426

3.18 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

3.19 Biblioliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

3.20 Worked Exercises and Details . . . . . . . . . . . . . . . . . . . . . . . . . 432

3.20.1 Non-Expanding Horizons . . . . . . . . . . . . . . . . . . . . . . . . 435

3.20.2 Weakley Isolated Horizons . . . . . . . . . . . . . . . . . . . . . . . 437

3.20.3 Isolated Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

3.20.4 Rotating Isolated Horizons . . . . . . . . . . . . . . . . . . . . . . . 443

3.20.5 Dynamical Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . 446

4 Classical Cosmology 477

4.1 Classical Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

4.1.1 Fluid Flow Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 477

4.1.2 Newtonian Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . 478

4.1.3 Relativistic Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . 479

15

4.1.4 Spaces of Constant Curvature . . . . . . . . . . . . . . . . . . . . . 480

4.2 Homogeneous and Isotropic Cosmology . . . . . . . . . . . . . . . . . . . . 480

4.2.1 The Luminosity Distance . . . . . . . . . . . . . . . . . . . . . . . . 480

4.3 The Singularity Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

4.3.1 Application of the Singularity Theorem: Cosmological Singularity . 483

4.3.2 Energy Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

4.3.3 Causality and Chronology . . . . . . . . . . . . . . . . . . . . . . . 484

4.3.4 Existence of maximum geodesic . . . . . . . . . . . . . . . . . . . . 486

4.3.5 The Significance of Conjugate Points: The Singularity Theorems . . 489

4.4 Backreaction Issues in Relativistic Cosmology and the Dark Energy Debate 489

4.4.1 Cosmological Perturbation Theory . . . . . . . . . . . . . . . . . . 489

4.5 Gauge Invariant Perturbations Around Symmetry Reduced Sectors of Gen-eral Relativity: Applications to Cosmology . . . . . . . . . . . . . . . . . . 490

4.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

4.6 Approximate Complete Observables . . . . . . . . . . . . . . . . . . . . . . 492

4.7 Application to Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . 496

5 Proof of the Hawking-Penrose Singularity Theorems 502

5.1 Proof of the Hawking-Penrose Singularity Theorem . . . . . . . . . . . . . 502

5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502

5.1.2 Some Basic Terminology . . . . . . . . . . . . . . . . . . . . . . . . 503

5.1.3 The Singularity Theorem of Hawking and Penrose . . . . . . . . . . 507

5.1.4 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

5.1.5 Achronal Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

5.1.6 Strong Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

5.1.7 The Space of Causal Curves . . . . . . . . . . . . . . . . . . . . . . 526

5.1.8 Conjugate Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536

16

5.1.9 Timelike Congruences . . . . . . . . . . . . . . . . . . . . . . . . . 541

5.1.10 The Singularity Theorems . . . . . . . . . . . . . . . . . . . . . . . 559

5.1.11 Implication of the “Displayed” Singularity Theorem from the Es-tablished Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562

5.2 Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588

5.2.1 Collapse of a Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

5.2.2 The Cauchy Problem - Existence and Uniqueness . . . . . . . . . . 593

5.2.3 Non-Linear Hyperbolic Differential Equations . . . . . . . . . . . . 594

5.2.4 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . 598

5.2.5 Cauchy-Kowalewski Theorem . . . . . . . . . . . . . . . . . . . . . 599

5.2.6 Reduced Einstein Equations . . . . . . . . . . . . . . . . . . . . . . 600

5.2.7 Sobolev Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

5.2.8 Developments for the Empty Space Einstein Equations . . . . . . . 604

5.2.9 Stability of Closed Trapped Surfaces . . . . . . . . . . . . . . . . . 605

5.2.10 Apparent Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

5.3 The Big Bang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

5.4 Biblioliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606

5.5 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606

5.5.1 Review of LP Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 606

5.5.2 Hardy-Littlewood-Sobolev Inequality . . . . . . . . . . . . . . . . . 608

5.5.3 Sobolev Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 611

5.5.4 Classical Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . 617

5.5.5 Fractional Hs - Sobolev Spaces . . . . . . . . . . . . . . . . . . . . 620

5.6 Worked Exercies and Details . . . . . . . . . . . . . . . . . . . . . . . . . . 620

6 Consistent Discrete Classical GR 625

6.0.1 “Dirac’s” canonical approach to general discrete systems . . . . . . 627

17

7 Quantum Field Theory on Curved Spacetimes 629

7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629

7.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629

7.3 Quantum Field Theory in Flat Spacetime . . . . . . . . . . . . . . . . . . . 630

7.3.1 The Simple Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . 630

7.3.2 Quantisation of the Klein-Gordon field in Flat Spacetime . . . . . . 631

7.3.3 Mode Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632

7.3.4 Behaviour of Fock Basis Under Lorentz Transformations . . . . . . 632

7.3.5 Fock Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632

7.4 Quantum Field Theory on Curved Spacetimes . . . . . . . . . . . . . . . . 633

7.4.1 Bogolyubov Transformations . . . . . . . . . . . . . . . . . . . . . . 633

7.4.2 a-Particles in the b-Vacuum . . . . . . . . . . . . . . . . . . . . . . 634

7.5 Quantum Fields in an Expanding Universe . . . . . . . . . . . . . . . . . . 634

7.5.1 Particle Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . 634

7.6 Quantum Fields During Inflation . . . . . . . . . . . . . . . . . . . . . . . 635

7.7 The Unruh Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635

7.8 Hawking Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635

7.8.1 Hand Wavy Calculation . . . . . . . . . . . . . . . . . . . . . . . . 635

7.8.2 Hawking’s Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 636

7.8.3 Rotating Black Holes and Higher Spin Fields . . . . . . . . . . . . . 639

7.8.4 Black Hole Thermodynamics . . . . . . . . . . . . . . . . . . . . . . 640

7.8.5 Information Loss Paradox . . . . . . . . . . . . . . . . . . . . . . . 640

7.9 Backreaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640

7.10 The Algebraic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640

7.10.1 Algebraic Quantum Theory . . . . . . . . . . . . . . . . . . . . . . 640

7.10.2 Wightman Axioms in Minkowski Spacetime . . . . . . . . . . . . . 640

7.10.3 Revision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641

18

7.10.4 Operator Product Expansion . . . . . . . . . . . . . . . . . . . . . 641

7.10.5 Algebraic Quantum Field Theory . . . . . . . . . . . . . . . . . . . 641

7.10.6 Viewpoint on QFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 641

7.10.7 Global and Local Particles . . . . . . . . . . . . . . . . . . . . . . . 641

7.11 The Need For Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . . . 642

7.11.1 Backreaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642

7.11.2 Information Loss Paradox . . . . . . . . . . . . . . . . . . . . . . . 642

7.12 Stochastic Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642

8 Introduction to Quantum General Relativity 643

8.1 The Problem of Quantising General Relativity . . . . . . . . . . . . . . . . 643

8.1.1 The Problem of Time in Canonical Quantum Gravity . . . . . . . . 644

8.2 Introduction to Loop Qauntum Gravity (LQG) . . . . . . . . . . . . . . . 644

8.3 ADM Metric 3+1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 644

8.3.1 Relation of Hamiltonian Formulation to Einstein’s Equations . . . . 646

8.4 The New Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646

8.4.1 Triad and connection formulation of GR . . . . . . . . . . . . . . . 646

8.4.2 Ashtekar’s new variables . . . . . . . . . . . . . . . . . . . . . . . . 649

8.4.3 Derivation of Ashtekar’s Formalism from the Self-dual Action . . . 650

8.5 Ashtekar-Barbero Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 654

8.5.1 The Holst Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654

8.5.2 3+1 Decomposition of the Holst Action . . . . . . . . . . . . . . . . 654

8.5.3 Introduction to Canonical Transformations . . . . . . . . . . . . . . 656

8.5.4 Simple Analogies to Canonical Transformations for the FollowingSection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658

8.5.5 Canonical Transformation on Extended Phase Spase: Obtaining theGauss Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662

8.5.6 Poisson Brackets for New Variables . . . . . . . . . . . . . . . . . . 665

19

8.5.7 The constraints in the New Variables . . . . . . . . . . . . . . . . . 666

8.5.8 The Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666

8.5.9 The Poisson bracket algebra . . . . . . . . . . . . . . . . . . . . . . 668

8.5.10 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669

8.6 Quantisation of the Constraints - the Equations of Quantum General Rel-ativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670

8.6.1 Quantum Constraints as the Equations of Quantum General relativity671

8.7 The Loop Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671

8.7.1 The loop transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 672

8.7.2 Solutions to all the Constraints . . . . . . . . . . . . . . . . . . . . 673

8.8 Geometric operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674

8.8.1 The Area Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 674

8.8.2 The Volume Operator . . . . . . . . . . . . . . . . . . . . . . . . . 675

8.8.3 Physical Meaning of these Results . . . . . . . . . . . . . . . . . . . 675

8.9 Spin Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675

8.10 Hamiltonian Constraint and the Modern Formulism . . . . . . . . . . . . . 676

8.10.1 Deriving Thiemann’s Identity and Other equations . . . . . . . . . 678

8.11 Quantising the Hamiltonian Constraint . . . . . . . . . . . . . . . . . . . . 681

8.12 Inclusion of Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683

8.12.1 Scalar Field - Higgs . . . . . . . . . . . . . . . . . . . . . . . . . . . 683

8.12.2 Yang-Milss Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 683

8.12.3 Dirac Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683

8.13 Spin Foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683

8.13.1 Spin Foam from the Hamiltonian Constraint . . . . . . . . . . . . . 683

8.13.2 Spin Foam from BF Theory . . . . . . . . . . . . . . . . . . . . . . 684

8.14 Semiclassical Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685

20

8.14.1 Why might LQG not have General Relativity as its Semiclassicallimit? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685

8.14.2 Difficulties Checking the Semiclassical Limit of LQG . . . . . . . . 686

8.14.3 Progress in demonstrating LQG has the Correct Semiclassical Limit 686

8.15 Master Constrain Programme . . . . . . . . . . . . . . . . . . . . . . . . . 687

8.15.1 The Master Constraint . . . . . . . . . . . . . . . . . . . . . . . . . 687

8.15.2 Quantising the Master Constraint . . . . . . . . . . . . . . . . . . . 688

8.15.3 Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689

8.15.4 Solving the Master Constraint . . . . . . . . . . . . . . . . . . . . . 689

8.15.5 Testing the Master Constraint . . . . . . . . . . . . . . . . . . . . . 689

8.15.6 Applications of the Master Constraint . . . . . . . . . . . . . . . . 690

8.16 Black Hole Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690

8.16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690

8.16.2 The LQG Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 690

8.17 Loop Quantum Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . 692

8.17.1 Traditional Wheeler-De Witt Quantization . . . . . . . . . . . . . . 692

8.17.2 Introduction to Loop Quantum cosmology . . . . . . . . . . . . . . 692

8.18 Loop Quantum Cosmology Phenomenology . . . . . . . . . . . . . . . . . . 692

8.19 Background Independent Scattering Amplitudes . . . . . . . . . . . . . . . 692

8.20 The Problem of Time in Quantum Gravity . . . . . . . . . . . . . . . . . . 693

8.21 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693

8.22 Other Approaches to Quantum Gravity . . . . . . . . . . . . . . . . . . . . 693

8.22.1 String Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693

8.22.2 Causal Dynamical Triangulations . . . . . . . . . . . . . . . . . . . 694

8.22.3 Consistent Discrete Quantum Gravity . . . . . . . . . . . . . . . . . 694

8.22.4 Noncommutative Geometry . . . . . . . . . . . . . . . . . . . . . . 694

8.22.5 Twistor Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694

21

A Physics Glossary 695

B Mathematics Glossary 763

C Mathematics 824

C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824

C.2 Action Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824

C.2.1 Variational Derivative and the Euler-Lagrange Equation . . . . . . 824

C.2.2 Action Principle for a Particle . . . . . . . . . . . . . . . . . . . . . 825

C.2.3 Action Principle for one Independent Variable . . . . . . . . . . . . 826

C.2.4 Action Principle for Several Dependent variables . . . . . . . . . . . 828

C.2.5 Action Principle for Several Independent variables . . . . . . . . . . 829

C.2.6 Action Principle for Several Dependent variables and Several Inde-pendent variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829

C.2.7 Complex dependent variables . . . . . . . . . . . . . . . . . . . . . 830

C.2.8 Fucntional Derivatives and the Fundamental Lemma of Calculus ofVariations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834

C.2.9 Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 840

C.2.10 Action Principle with Variantion of Boundary . . . . . . . . . . . . 842

C.2.11 Lagrange multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . 843

C.2.12 Action principle subject to constraints . . . . . . . . . . . . . . . . 846

C.3 Some Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847

C.4 Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849

C.5 Totally Antisymmetric symbol and Determinates . . . . . . . . . . . . . . 850

C.5.1 First Definition of Determinants . . . . . . . . . . . . . . . . . . . . 850

C.5.2 The Levi-Civita Tensor . . . . . . . . . . . . . . . . . . . . . . . . . 851

C.5.3 Inverse Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855

C.5.4 Cross Product, Area and Volume Vector Equations . . . . . . . . . 857

22

C.5.5 Determinant as Volume Factor . . . . . . . . . . . . . . . . . . . . . 859

C.5.6 Generalisation to n−dimensions . . . . . . . . . . . . . . . . . . . . 861

C.5.7 Recursive Definition of Volume . . . . . . . . . . . . . . . . . . . . 863

C.5.8 Linear Mappings and Volume Change . . . . . . . . . . . . . . . . . 866

C.5.9 Jacobian of Coordinate transformations . . . . . . . . . . . . . . . . 866

C.5.10 Summary of Transformations . . . . . . . . . . . . . . . . . . . . . 868

C.5.11 Variation of a Determinate . . . . . . . . . . . . . . . . . . . . . . . 871

C.5.12 Tensor Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872

C.5.13 Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 873

C.5.14 Summary of Tensor Calculus . . . . . . . . . . . . . . . . . . . . . . 874

C.5.15 Linear operators and Matrices . . . . . . . . . . . . . . . . . . . . . 874

C.6 Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875

C.6.1 Examples of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 876

C.6.2 Unitary Representations of Groups . . . . . . . . . . . . . . . . . . 881

C.6.3 Schur’s First Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 882

C.6.4 Schur’s Second Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 885

C.6.5 Orthogonality relations . . . . . . . . . . . . . . . . . . . . . . . . . 887

C.6.6 The Characters of a Representation . . . . . . . . . . . . . . . . . . 890

C.6.7 Direct Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 891

C.7 Continuous Groups, Lie Groups and Lie algebras . . . . . . . . . . . . . . 893

C.7.1 Infinitesmal Generating Technique . . . . . . . . . . . . . . . . . . . 893

C.7.2 General Structure of Lie Groups . . . . . . . . . . . . . . . . . . . . 897

C.7.3 Rotations SO(3) and SU(2) . . . . . . . . . . . . . . . . . . . . . . 898

C.7.4 Spin Direct Products . . . . . . . . . . . . . . . . . . . . . . . . . . 900

C.7.5 Direct Products and Clebsch-Gordan Coefficients . . . . . . . . . . 907

C.7.6 Recoupling Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 910

C.7.7 SO(3,1) and SL(2,C) . . . . . . . . . . . . . . . . . . . . . . . . . . 910

23

C.7.8 SO(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912

C.7.9 Conformal Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913

C.7.10 Group Integration: The Haar Measure . . . . . . . . . . . . . . . . 914

C.7.11 Peter-Weyl theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 919

C.7.12 Analogies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 921

C.7.13 Clebsch-Gordan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923

C.7.14 Semi-direct Products . . . . . . . . . . . . . . . . . . . . . . . . . . 923

C.8 Infinite-Dimensional Group Representations . . . . . . . . . . . . . . . . . 927

C.8.1 Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927

C.8.2 Countable and Locally Compact Topological Groups . . . . . . . . 927

C.8.3 Haar Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 928

C.8.4 Summary of Group theory . . . . . . . . . . . . . . . . . . . . . . . 928

C.9 Manifolds and Elementary Topology . . . . . . . . . . . . . . . . . . . . . 929

C.9.1 Sets and Mappings Between Sets . . . . . . . . . . . . . . . . . . . 929

C.9.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 930

C.10 Elementary Tensor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 932

C.10.1 Affine Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934

C.10.2 Affine Geodesic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936

C.10.3 The Metric Connection . . . . . . . . . . . . . . . . . . . . . . . . . 937

C.10.4 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 939

C.10.5 Gaussian Normal Coordinates . . . . . . . . . . . . . . . . . . . . . 941

C.10.6 Bianchi Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942

C.10.7 Conformal Tensor, Ricci tensor and Ricci Scalar . . . . . . . . . . . 943

C.10.8 The Weyl Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943

C.10.9 Index Free Formulism . . . . . . . . . . . . . . . . . . . . . . . . . 943

C.11 Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946

C.11.1 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946

24

C.11.2 Covectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 948

C.11.3 Induced Metric and Other Objects on Sub-manifolds . . . . . . . . 949

C.12 Active Diffeomorphisms and the Lie Derivative . . . . . . . . . . . . . . . . 951

C.12.1 Mapping a Manifold to Itself Along Integral Curves . . . . . . . . . 953

C.12.2 The Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 954

C.12.3 Pull-back and Lie Derivative of a co-vector . . . . . . . . . . . . . . 961

C.12.4 More on Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . 962

C.12.5 Isometries and Killing Vector Fields . . . . . . . . . . . . . . . . . . 963

C.12.6 Conserved Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . 965

C.12.7 Adapted Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 966

C.12.8 Properties of Killing Fields . . . . . . . . . . . . . . . . . . . . . . . 967

C.12.9 Diffeomorphism Gauge Group - Symmetry of GR Under Active Dif-feomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967

C.13 Frame Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 968

C.14 The Spin Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969

C.14.1 Curvature Associated with the Spin Connection. . . . . . . . . . . . 971

C.15 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973

C.15.1 Exterior Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976

C.15.2 Exterior Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 976

C.15.3 Maxwell’s equations in differential forms . . . . . . . . . . . . . . . 982

C.15.4 Integration on a Manifold . . . . . . . . . . . . . . . . . . . . . . . 988

C.15.5 Cartan Structure Equations . . . . . . . . . . . . . . . . . . . . . . 991

C.15.6 A Differential Geometry Translator . . . . . . . . . . . . . . . . . . 993

C.16 More on Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994

C.16.1 Discrete Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996

C.16.2 Universal Covering Group . . . . . . . . . . . . . . . . . . . . . . . 996

C.17 Group Actions on Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997

25

C.17.1 Transitive Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . 999

C.17.2 Faithful Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999

C.17.3 Free Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1000

C.17.4 Introduction to Gauge Invariance of the Yang-Mills Equations . . . 1000

C.18 Principle Bundles and Connections . . . . . . . . . . . . . . . . . . . . . . 1003

C.19 Fibre Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004

C.19.1 The Structure Group of a Bundle . . . . . . . . . . . . . . . . . . . 1010

C.19.2 Frame Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1010

C.19.3 The Idea of a Principal Bundle . . . . . . . . . . . . . . . . . . . . 1011

C.19.4 Principal Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012

C.19.5 Action of the structure Group on a Principal Bundle . . . . . . . . 1013

C.19.6 Connections on Principal Bundles . . . . . . . . . . . . . . . . . . . 1013

C.19.7 Gauge Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017

C.19.8 Parallel Transport in a Principal Bundle . . . . . . . . . . . . . . . 1023

C.19.9 Curvature on a Principal Bundle . . . . . . . . . . . . . . . . . . . 1023

C.19.10Extension and Reduction of Principal Bundles . . . . . . . . . . . . 1024

C.19.11The Complex Line Bundle . . . . . . . . . . . . . . . . . . . . . . . 1026

C.20 Summary of Differential Geometry . . . . . . . . . . . . . . . . . . . . . . 1026

C.21 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026

C.22 Biblioliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027

C.23 Worked Exercises and Details . . . . . . . . . . . . . . . . . . . . . . . . . 1027

C.23.1 Dynamical and Non-Dynamical Symmetries . . . . . . . . . . . . . 1027

D Constrained Hamiltonian Systems, Dirac observables and the ConstraintAlgebra 1031

D.0.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031

D.1 Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032

26

D.1.1 Poisson Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034

D.1.2 Sympletic Geometry and Phase Space . . . . . . . . . . . . . . . . . 1036

D.1.3 Canonical Transformations . . . . . . . . . . . . . . . . . . . . . . . 1039

D.1.4 Infinitesimal Contact Transformations . . . . . . . . . . . . . . . . 1043

D.1.5 Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045

D.2 Geometry of Configuration Space and Phase Space . . . . . . . . . . . . . 1046

D.2.1 Vector Fields on Configuration Space and Phase Space . . . . . . . 1046

D.2.2 The Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046

D.2.3 Definition of Hamiltonian System . . . . . . . . . . . . . . . . . . . 1047

D.2.4 Sympletic Geometry of Phase Space . . . . . . . . . . . . . . . . . . 1048

D.2.5 Canonical Transformations . . . . . . . . . . . . . . . . . . . . . . . 1050

D.2.6 The Hamiltoian Framework: Resumé . . . . . . . . . . . . . . . . . 1051

D.2.7 Connection to quantum mechanics . . . . . . . . . . . . . . . . . . 1051

D.3 Covarient Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052

D.3.1 Space of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052

D.3.2 Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1053

D.3.3 Hamiltonian-Jacobi Theory . . . . . . . . . . . . . . . . . . . . . . 1054

D.3.4 Hamilton principal function . . . . . . . . . . . . . . . . . . . . . . 1057

D.4 Solving for the Dynamics using the HJ Equation . . . . . . . . . . . . . . . 1059

D.4.1 1. Free particle (one-dimension) . . . . . . . . . . . . . . . . . . . . 1059

D.4.2 2. The Harmonic oscillator (one-dimension) . . . . . . . . . . . . . 1063

D.4.3 Hamiltonian Charateristic Function . . . . . . . . . . . . . . . . . . 1065

D.4.4 ‘Derivation’ of Schrodinger’s Equation . . . . . . . . . . . . . . . . 1067

D.5 Constrained Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . 1069

D.5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073

D.5.2 Dirac’s Procedure for Constrained Hamiltonian Systems . . . . . . 1075

D.5.3 First Class Constraints and Gauge Symmetries . . . . . . . . . . . . 1079

27

D.5.4 Dirac Method and Electrodynamics . . . . . . . . . . . . . . . . . . 1080

D.5.5 Quantization of Constrained Hamiltonian Systems . . . . . . . . . . 1081

D.5.6 Dirac Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081

D.5.7 Darboux’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081

D.5.8 Symplectic Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1083

D.5.9 Poisson Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099

D.5.10 Symplectic Group Actions . . . . . . . . . . . . . . . . . . . . . . . 1099

D.6 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1100

D.7 Open Constraint Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102

D.7.1 The Master Constraint . . . . . . . . . . . . . . . . . . . . . . . . . 1102

D.8 Partial, Complete and Dirac Observables for Hamiltonian Constrained Sys-tems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102

D.8.1 Weak Dirac Observables . . . . . . . . . . . . . . . . . . . . . . . . 1104

D.8.2 Backreaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106

D.8.3 Different Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106

D.8.4 Systems with Several Constraints . . . . . . . . . . . . . . . . . . . 1106

D.8.5 Partial Differential Equations for Complete Observables . . . . . . . 1107

D.8.6 Partially Invariant Partial Observables . . . . . . . . . . . . . . . . 1111

D.9 A Perturbative Approach to Dirac Observables and Their Spacetime Algebra1113

D.9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113

D.9.2 The Approximate Dirac Observable . . . . . . . . . . . . . . . . . . 1113

D.9.3 Abelianization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115

D.9.4 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116

D.9.5 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1120

D.9.6 The second Order Approximation . . . . . . . . . . . . . . . . . . . 1121

D.9.7 Application to General Relativity . . . . . . . . . . . . . . . . . . . 1122

D.9.8 ADM Clock Variables . . . . . . . . . . . . . . . . . . . . . . . . . 1122

28

D.9.9 Gravity Coupled to a Scalar Field . . . . . . . . . . . . . . . . . . . 1123

D.9.10 Control of Gauge Dependence . . . . . . . . . . . . . . . . . . . . . 1124

D.9.11 Outlook and Summary . . . . . . . . . . . . . . . . . . . . . . . . . 1124

D.10 Reduced Phase Space Quantization of Constrained Theories . . . . . . . . 1125

D.10.1 Reduced Phase Space Quantization with Dirac Observables . . . . . 1125

D.11 Biblioliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126

D.12 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127

D.12.1 Poisson Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127

D.12.2 Worked Examples: Dittrich . . . . . . . . . . . . . . . . . . . . . . 1131

D.12.3 Worked Examples: Brute force Thiemann . . . . . . . . . . . . . . 1144

E ADM and First order Formalism of Einstein’s Theory 1160

E.1 Intrinsic and Extrinsic Curvature . . . . . . . . . . . . . . . . . . . . . . . 1160

E.2 ADM Metric Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1161

E.2.1 The Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164

E.3 The Hamiltonian Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 1164

E.4 Stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166

E.5 The Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166

E.6 Gravitational Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 1167

E.6.1 Boundary Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1169

E.6.2 Constraint Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 1169

E.7 First Order Formulation of Einstein Equations . . . . . . . . . . . . . . . . 1173

E.8 Palatini Method in the Connection Formulation . . . . . . . . . . . . . . . 1173

E.8.1 Method I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1173

E.8.2 Method II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175

E.9 Inclusion of Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1180

E.9.1 Yang-Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1180

29

E.9.2 Klein-Gordan - Scalar Matter Field . . . . . . . . . . . . . . . . . . 1180

E.9.3 Fermionic Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1181

E.9.4 In the Language of Differential Geometry . . . . . . . . . . . . . . . 1181

E.10 Worked Exercises and Details . . . . . . . . . . . . . . . . . . . . . . . . . 1182

F Self-dual Connection Formulation and Ashtekar’s new Variables 1186

F.1 Self-dual Connection Formulation . . . . . . . . . . . . . . . . . . . . . . . 1186

F.1.1 Self-dual Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186

F.1.2 Self-dual Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193

F.2 Ashtekar’s Canonical Formalism . . . . . . . . . . . . . . . . . . . . . . . . 1195

F.3 Generators of Symmetry Transformations . . . . . . . . . . . . . . . . . . . 1195

F.3.1 The Gauss-law Constraint Generates Gauge Transformations . . . . 1196

F.3.2 Incorperating Matter in the Quantum Theory . . . . . . . . . . . . 1197

F.4 Toy Model: Free Particle described using Half-Complex Coordinates. . . . 1197

F.4.1 Complex Variables and Reality Conditions . . . . . . . . . . . . . . 1197

F.4.2 Quantization in Compex Coordinates. . . . . . . . . . . . . . . . . 1198

G The Holst Action and Ashtekar-Barbero Real Variables 1199

G.1 The Holst Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1199

G.1.1 3+1 Decomposition of the Holst Action . . . . . . . . . . . . . . . . 1199

G.1.2 The Diffeomorphism Constraint . . . . . . . . . . . . . . . . . . . . 1200

G.1.3 The Hamiltonian Constraint . . . . . . . . . . . . . . . . . . . . . . 1201

G.1.4 Addition Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 1202

G.1.5 Final Total Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 1203

G.2 Inclusion of Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204

G.2.1 Yang-Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204

G.2.2 Klein-Gordan - Scalar Matter . . . . . . . . . . . . . . . . . . . . . 1204

G.2.3 Fermionic Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204

30

G.3 Biblioliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204

G.4 Worked Exercises and Details . . . . . . . . . . . . . . . . . . . . . . . . . 1204

H Basic Functional Analysis 1208

H.1 Finite Hilbert Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1208

H.1.1 The Hamilton-Cayley Theorem. . . . . . . . . . . . . . . . . . . . . 1208

H.1.2 Projection Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 1209

H.1.3 Spectral Theorem for Finite Spaces . . . . . . . . . . . . . . . . . . 1212

I Quantum Field Theory 1216

I.1 Elementary Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 1216

I.1.1 Path Integrals and Functional Integrals . . . . . . . . . . . . . . . . 1216

I.1.2 Semi-Classical Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 1217

I.1.3 Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . 1218

I.1.4 N Real variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1218

I.1.5 Complex variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 1220

I.2 Grassmann Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1221

I.3 Quantization on the Space of Classical Solutions . . . . . . . . . . . . . . . 1229

I.3.1 Harmonic Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . 1231

I.3.2 ‘Fock Space’ Quantization . . . . . . . . . . . . . . . . . . . . . . . 1233

I.3.3 The Fock Representation of Field Theory . . . . . . . . . . . . . . . 1241

I.3.4 The Fock Representation of a Free Scalar Field . . . . . . . . . . . 1242

I.3.5 The Fock Representation of the Maxwell Field . . . . . . . . . . . . 1243

I.3.6 Quantum Field Theory on Curved Spacetime - The Basics . . . . . 1243

J Details of Hawking’s Calculation 1244

J.1 Decomposition into Complete Basis . . . . . . . . . . . . . . . . . . . . . . 1244

J.2 Solution of Klein-Gordon Equation in Schwarzschild Spacetime . . . . . . . 1244

31

J.3 Bogoliubov Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1247

32

List of Figures

1.1 Rubbersheet simulation of geodesic moton in special relativity. . . . . . . 55

1.2 rubbersheet. It doesnot matter that the coordinates are time-dependent -it still serves as a physical refference system. . . . . . . . . . . . . . . . . 55

1.3 E(x) = Q/x2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

1.4 E(y) = Q/y2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

1.5 E(y) = Q/y2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

1.6 Passive spatial difeomorphism f : M → M refers to invariance underchange of coordinates. The same object in a different coordinate system.Any theory of nature is invariant under passive difeomorphisms. . . . . . . 63

1.7 An active difeomorphism f : M → M draggs fields on the manifold whileremaining in the same coordinate system. f is viewed as a map that asso-ciates one point in the manifold to another one. . . . . . . . . . . . . . . . 64

1.8 The value of φ̃(P ) at P is equated to the value of φ(P0) at P0, i.e. φ̃(P ) =φ(P0). Under this transformation f we indentify one point of the manifoldP0 to another point P f : Po → P . . . . . . . . . . . . . . . . . . . . . . . 64

1.9 The value of the metric function g̃ab at P is defined by the value of themetric function gab at P0, i.e. g̃ab(P ) = gab(P0). We go to a new coordinatesystem which assigns P the same coordinate values that P0 has in the x-coordinates, so that g̃ab(y1 = u1, y1 = u2) = gab(x1 = u1, x1 = u2), compareto (1.31). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

1.10 (a) An active diffeomorphism in which we indentify one point of the man-ifold to another point. (b) We then go to a coordinate system that asignsthe newly identified points the origonal coordinate values. . . . . . . . . . . 66

33

1.11 (a) An active diffeomorphism in which we actively drag the tensor functionover the, in doing so indentify one point of the manifold to another. (b)We then go to a coordinate system which asigns the newly identified pointstheir origonal coordinate values. That is to say - we carry the tensorfunction over the manifold, keeping the coordinate lines ‘attached’. . . . . . 67

1.12 Einstein’s hole argument. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

1.13 Resolution of Einstein’s hole argument. . . . . . . . . . . . . . . . . . . . . 68

1.14 Illustration of smearing. operator valued distributions. . . . . . . . . . . . 77

1.15 Regime where gravity is very strong so that the non-perturbative and back-ground independence of GR must be taken into account. That spacetimepoints have no independent physical reality casts doubt on the hand-waveyargument I gave above. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

1.16 Labortary walls exemplify Newton’s absolute space and the clock absolutetime. We can define positions relative to the wall. . . . . . . . . . . . . . . 78

1.17 GPS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

1.18 GPS3D A spacetime point in Minkoski spacetime can be expressed as arelation amogst 4? measurable variables. This definition of spacetimelocation retains meaning in the jump to GR. . . . . . . . . . . . . . . . . . 81

1.19 clock time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

1.20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

1.21 measLocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

1.22 measVelocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

1.23 partialobs. τ is an unphysical parammeter labelling different possible cor-relations between the time reading t of the clock and the elongation x ofthe pendulm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

1.24 partComptDitt1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

1.25 partComptDitt3. (a) t = t1 when the clock function T (αtC(x)) assumes

the value τ . (b) The function F[f,T ](τ, x) gives the value that the function

f(αtC(x)) assumes if the function T (αtC(x)) assumes the value τ . F[f,T ](τ, x)

is a complete observable generated from the partial observables T (x) andf(x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

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2.1 timedilF. (a) In system S ′ a light pulse is emmitted from a source at O′

and is reflected back along the same line, and takes a time ∆t0 to performa round trip. (b) Path of the same light pulse, as observed in the systemS. The speed of the light pulse is the same as in system S ′, but the pathis longer, and hence the moving clock takes a longer amount of time, ∆t,to perform one tick. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

2.2 Say we have a clock at rest with respect to the system S located at theorigin of S. We also have a clock at rest with respect to the “moving”system S ′ located at the origin of S ′. We assume that the origins coincideat an initial time t = t′ = 0. Recall this assumption was made whenderiving the Lorentz transformation formula. . . . . . . . . . . . . . . . . . 104

2.3 rocketEarth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

2.4 rocketEarth2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

2.5 rocket. The clock at the top seems to run faster than the one on the bottom.113

2.6 rocketaccel. The clock at the top seems to run faster than the one on thebottom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

2.7 lightdeflec. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

2.8 WeakGrav. geodesic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

2.9 Hole. Einstein’s hole argument. . . . . . . . . . . . . . . . . . . . . . . . . 117

2.10 Hole3. Einstein’s hole argument.Φ : M → M′ → M . . . . . . . . . . . . . 118

2.11 Hole4. Einstein’s hole argument. A gauge transformation which does notchange the coordinate label system but moves the points on the manifold,and then evaluate the coordinates of the new point . . . . . . . . . . . . . 118

2.12 We display the geometric interpretation of the curvature tensor. Carry athird vector Z, by parallel transport from p to s via q, comparing this withtransporting this from p to s′ via r we find a discrepancy between the twovectors given in terms of the curvature tensor components R dabc by theformula ǫ2XaY bZcR dabc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

2.13 geodesic deviation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

2.14 clock time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

2.15 measLocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

2.16 tidalforceF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

2.17 worldfunc1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

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2.18 measVelocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

2.19 η is the orthogonal connecting vector. . . . . . . . . . . . . . . . . . . . . . 135

2.20 We find the spatial frame components ηα of the orthogonal connectingvector by projecting onto a spatial frame field. This is the precise analogueof the Newtonian connecting vector. . . . . . . . . . . . . . . . . . . . . . . 138

2.21 Different world line passing through P corresponds to different observerwith different va. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

2.22 GPScoord. s1 and s2 are the GPS coordinates of the point p. Σ is a Cauchysurface with p in its future domain of dependence. . . . . . . . . . . . . . . 143

2.23 crossParea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

2.24 continuityEM Y and Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

2.25 Lcfluid. The Lorentz contraction of a fluid element. . . . . . . . . . . . . . 147

2.26 LumDist. Luminosity distance . . . . . . . . . . . . . . . . . . . . . . . . 187

2.27 pertCosGauge. A diffeomorphism on the perturbed manifold M inducesa change in coordinates of the background manifold M0. The issue ofperturbative gauge invariance is closely related, though not equivalent to,the coordinate independence of General Relativity. . . . . . . . . . . . . . . 191

2.28 pertManifolds. 5-dimensional manifold N containing a 1-parameter familyof smooth non-intersecting 4-manifolds Mǫ. N = M× R . . . . . . . . . . 192

2.29 pertManFlow. The diffeomorphism ϕ. . . . . . . . . . . . . . . . . . . . . . 195

2.30 pertManPush. The push-forward ϕλ∗|p is the natural linear map betweenthe tangent spaces TpM0 and Tϕ(p)Mλ induced by the diffeomorphism ϕ.The push-forward ϕ∗λ|p is the linear map between the co-tangent spacesT ∗pM0 and T ∗ϕ(p)Mλ. Push-forwrads and pull-backs are related by . . . . . 196

3.1 eventhorizon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

3.2 Penrose diagram of Minkowski spacetime . . . . . . . . . . . . . . . . . . . 242

3.3 Penrose diagram of the Kruskal solution . . . . . . . . . . . . . . . . . . . 243

3.4 Penrose diagram of a black hole. . . . . . . . . . . . . . . . . . . . . . . . . 243

3.5 Penrose diagram of a black hole. . . . . . . . . . . . . . . . . . . . . . . . . 260

3.6 Rotating blackhole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

3.7 Normal and tangent vector to a tangent . . . . . . . . . . . . . . . . . . . 305

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3.8 (a) Normal to a timelike surface, (b) Normal to a spacelike surface, (c)Normal to a null surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

3.9 Coordinatizing a null surface in Minkowsian spacetime - λ, θA. . . . . . . . 306

3.10 Each two sphere. Coordinates λ, θ, φ. . . . . . . . . . . . . . . . . . . . . . 306

3.11 A spacial two-sphere S embedded in the spacial slice Σ (which in turn isembedded in spacetime M), with two sets of orthogonal null vector fields.The vector field na is the unit timelike normal to Σ, Ra is the unit spacialnormal to S, and na and ℓa are, respevtively, the outgoing and ingoing nullvectors orthogonal to S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

3.12 A spacial two-sphere S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

3.13 In Schwarzschild black hole the horizon is generated by the radial lightrays, which meet at the center. . . . . . . . . . . . . . . . . . . . . . . . . 314

3.14 (a) An open interval of the real line is the set of points between a and bexcluding a and b. (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

3.15 q lies in the chronological future of z. . . . . . . . . . . . . . . . . . . . . . 316

3.16 The chronological future I+(p) of p is an open set; given any point q ∈I+(p), there exists a sufficiently small neighbourhood V (q) contained inI+(p). Similar statements hold for the p in the chronological past I−(q) of q.317

3.17 If two points on the event horizon are timelike separated, we can produce atimelike curve starting inside the black hole which joins to a point outsidethe event horizon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

3.18 When neighbouring null geodesics have conjugate points there exists a tim-like curve joining the two conjugate points. The dashed line represents atimelike curve joining to the null geodesic. The points q and q′ are timelikeseparated - rounding off the corner. We make this argument more rigouressin the appendix M. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

3.19 In flat spacetime, when a null geodesic curve joins onto a timlike curve,there exists a timelike curve between p and q. . . . . . . . . . . . . . . . . 319

3.20 There exists a timlike curve joining the two conjugate points. A timelikecurve joining to the null geodesic. (b) Contiuing in this way, we “peel”away a timelike curve that joins r and p. . . . . . . . . . . . . . . . . . . . 320

3.21 There exists a timlike curve joining the two is way, we a timelike curve thatjoins r and p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

3.22 There exists a timlike curve joining the two is way, we a timelike curve thatjoins r and p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

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3.23 Classical bourndary conditions for weakly isolated horizons. . . . . . . . . 322

3.24 Conformal spacetime diagram of a WHI. . . . . . . . . . . . . . . . . . . . 322

3.25 Construction of a null tetrad for a NEH. Any spatial two surface S deter-mines uniquely, up to rescaling, two null vectors orthogonal to S. . . . . . 323

3.26 multipole is the position of the mass density source and ~r is the requestedposition for the potential Φ(~r). . . . . . . . . . . . . . . . . . . . . . . . . . 327

3.27 dipolemass (a) monpole. (b) dipole (c) quadrapole . . . . . . . . . . . . . . 328

3.28 magmulty is the position of the current density and ~r is the requestedposition for the magnetic potential Φm(~r). . . . . . . . . . . . . . . . . . . 330

3.29 polarcoo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

3.30 axicoords. In addaptive coordinates . . . . . . . . . . . . . . . . . . . . . . 334

3.31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

3.32 visualflownull. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

3.33 Horizons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

4.1 LumDist. Luminosity distance . . . . . . . . . . . . . . . . . . . . . . . . 481

4.2 nonmingeodesic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482

4.3 trapped.two-surface such that the areas of pulses of light emmited fromeach little element of surface decrease in both directions. . . . . . . . . . . 483

4.4 illustrating a trapped surface. . . . . . . . . . . . . . . . . . . . . . . . . . 483

4.5 illustrating a past-directed trapped surface, correspondimg to a cosmolog-ical singularity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

4.6 I+(p) is open. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

4.7 trivialsing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

4.8 The points yn converge to the point q in the boundary of L+(S). From each

yn there is a past directed timelike curve λn to S. These curves convergeto the past directed null geodesic segment γ through q. . . . . . . . . . . . 486

4.9 K = J+(S) ∩ J−(p) K is compact. . . . . . . . . . . . . . . . . . . . . . . 486

4.10 trivialsing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

4.11 A timelike curve can be approximated by null geodesic segments. Thisapproximating null curve fails to have a well defined tangent vector anywhere.488

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4.12 The global hyperbolicity of M is closely related to the future or past de-velopment of initial data from a given spacelike hypersurface. . . . . . . . . 489

5.1 Sir Roger Penrose and Stephen Hawking. Initiated by Penrose, Penrose andHawking, together with Robert Geroch, contributed much of the work onthe existence of spacetime singularities with the use of point-set topologicalmethods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503

5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503

5.3 An example of a non-orientable space. . . . . . . . . . . . . . . . . . . . . 504

5.4 (a) A closed achronal set with edge. (b) A closed achronal set without edge.504

5.5 The h−null cone contains more timelike vectors than the g−null cone sothere is more likelyhood to find closed timelike curves in (M, h) than in(M, g). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

5.6 Domains of dependence. (a) The future domains of dependence of theachronal set S. (b) The past domain of dependence of the set S. (c) Thetotal domain of dependence of S. . . . . . . . . . . . . . . . . . . . . . . . 506

5.7 closedtrapArea. A closed trapped surface is when the outgoing light conealso converges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

5.8 Focal points. (b) Focal point to a surface. . . . . . . . . . . . . . . . . . . 507

5.9 A path is a connected set in R to the space-time manifold M. . . . . . . . 509

5.10 Examples in Minkowski spacetime. (a) The curve α is taken to containits own past end point a and future end point b. (b) The curve α′ is nowfuture-inextendable . (c) The curve α′′ is now past-inextendable (d) Thecurve γ is not future-inextendable because it cannot be prolonged any further.510

5.11 (a) A curve in Minkowski spacetime with point removed. (b) A curve is zigzag that it fails to have a well defined tangent vector at a point. (c) Herewe have a time like curve which is null at its end point. . . . . . . . . . . . 510

5.12 In flat spacetime, when a null geodesic curve joins onto a timlike curve,there exists a timelike curve between p and q. . . . . . . . . . . . . . . . . 511

5.13 Say there are two points p and q connected by a null curve and a point rwhich is connected to q by a timelike curve. A timelike curve joining tothe null geodesic. (b) Contiuing in this way, (c), we “peel” away a timelikecurve that joins r and p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

5.14 Artificial example of how the causal future of a point p will not neccessarilycoincide with the closure of the chronological future of p. . . . . . . . . . . 512

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5.15 The point q is conjugate to p along null geodesics, so a null geodesic γ thatjoins p to q will leave the bounadry of the uture of p at q. . . . . . . . . . . 512

5.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

5.17 The future set cannot be bounded by timelike curves. . . . . . . . . . . . . 514

5.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514

5.19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

5.20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

5.21 The lines ℓ1 and ℓ2 have been removed. This is a space-time which is causalbut fails to be strongly causal. . . . . . . . . . . . . . . . . . . . . . . . . . 517

5.22 (a) . (b) Suitable causal basis. . . . . . . . . . . . . . . . . . . . . . . . . . 517

5.23 Minkowski space-time: b is joined to a by a null geodesic we considerx ∈ I+(a) and y ∈ I−(b). In the first in case (a) y ≪ x. In case (b) y < xbut y 6≪ x. In case (c) x and y are not causally related, in particular y 6≪ x.518

5.24 b is joined to a by a null geodesic we consider x ∈ I+(a) and y ∈ I−(b). Inthe first case (a) y ≪ x there are no closed timelike curves. In case (c) Forsome x and y. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518

5.25 converse. (b) That y ≫ x to be true, there must be a timelike curve thatrenters U(a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

5.26 proof of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

5.27 For i ≥ i0, Ui ∈ I+(x), from which we can conclude x ≪ ai. For i ≥ i′0,ci ∈ I−(y). If we choose i > i0, i′0 then we have x ≪ ai and ci ≪ y . . . . . 520

5.28 b a x ∈ I+(a) and y ∈ I−(b). In the first case (a) y ≪ x timelike curves.(b) Points in the open set of b are also in I+(x) ∩ I−(y) In case (c) . . . . . 521

5.29 a ∈ I−(y)∩V (y) and b ∈ I+(y)∩V (y). In the first case (a) y ≪ x timelikecurves. (b) Points in the open set of b are also in I+(x) ∩ I−(y). . . . . . . 521

5.30 a ∈ I−(y)∩V (y) and b ∈ I+(y)∩V (y). In the first case (a) y ≪ x timelikecurves. (b) Points in the open set of b are also in I+(x) ∩ I−(y). . . . . . . 522

5.31 (a) There is an endless null geodesic along which strong causality is violated.(b) Strong causality is violated everywhere in R. . . . . . . . . . . . . . . . 523

5.32 F := I+(Q) and P := I−(Q). Q = F ∩ P , ∂Q = F∂ + ∂P . . . . . . . . . . 523

5.33 The lines ℓ1, ℓ2 and ℓ3 have been removed. . . . . . . . . . . . . . . . . . . 524

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5.34 The future domain of dependence, D+(Σ), of Σ. p is in D+(Σ), q isn’t be-cause ther are past-inextendable causal curve through q that don’t intersectΣ, e.g. the curve γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

5.35 The future domain of dependence, D+(Σ), of a clsoed Σ in Minkowskispacetime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

5.36 Domains of dependence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

5.37 The future domain of dependence. . . . . . . . . . . . . . . . . . . . . . . . 526

5.38 There exists a subsequence γm of past inextendble causal curves that donot meet S that converges to a past inextendble C0 null geodesic γ startingat p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526

5.39 The edge of a closed achronal surface Σ. . . . . . . . . . . . . . . . . . . . 527

5.40 A simple example of a closed achronal surface without edge can be givenby considering the spacetime R × S with light cones locally at 45 degrees.For the open neighbourhood there is no r ∈ I−(p) and q ∈ I+(p) with atimelike curve between them that doesn’t intersect S. The closed achronalset Σ has no edge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

5.41 We can only get convergence to a cluster point in K if the space wasn’tlocally finite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528

5.42 K = J+(S) ∩ J−(p) K is compact. . . . . . . . . . . . . . . . . . . . . . . 528

5.43 K = J+(S) ∩ J−(p) K is compact. . . . . . . . . . . . . . . . . . . . . . . 529

5.44 G is an open set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

5.45 approximate a causal curve by a causal trip. . . . . . . . . . . . . . . . . . 530

5.46 Imposition of the condition I+[Aj ]∩Aj+2 6= ∅ avoids cases such as the above.533

5.47 The geodesic λ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534

5.48 The sequence is a Cauchy sequence in C(a, b), but it is not convergent, sincethere is a point missing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

5.49 Minkowski space time with a point removed is not globally hyperbolic. Thepoint q is not in D+(S) as there are non-spacelike curves like λ which donot meet S in the past. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536

5.50 (a) An upper semi-continuous function. (b) A lower semi-continuous function.537

5.51 Two points joined by a timelike curve can be connected by a broken nullgeodesic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541

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5.52 A sequence of null curves may converge to a timelike curve. . . . . . . . . . 541

5.53 If A and B are parallel in Minkowskia spacetime then γ1 and γ2 are max-imal. (b) Here there is only one element in CK(A,B) which is necessarilymaximal. It is not a geoesic. (c) Here the maximal eleement is a trip. . . . 542

5.54 Displacement vectors for a hypersurface. . . . . . . . . . . . . . . . . . . . 555

5.55 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559

5.56 Future horozmos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559

5.57 “doubling” future horozmos. E+[S] is compact . . . . . . . . . . . . . . . . 560

5.58 From the fact that H is a Cauchy horizon it follows that through everypoint of H there passes a maximally extened past-directed null geodesicthat remains in H . Since H is compact, such a curve would have to comeback arbitrarily close to itself - reentering some Alexandoff neighbourhoodand so violating strong causality. . . . . . . . . . . . . . . . . . . . . . . . 561

5.59 E+[S] is compact by defintion, however its Cauchy horizon is non-compact.As the two subsets can not homeomorphic there must be at least one tra-jectory γ which remains in intD+(E+[S]). . . . . . . . . . . . . . . . . . . 561

5.60 γ is a future endless causal geodesic in intD+(E+(S)). H = H+(E+[S]).As the intersection of the closed set J̇−(γ) with a compact set generatedby null geodesic segments from T of some bounded affine length, E−[T ] iscompact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562

5.61 The geodesic λ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

5.62 The limit geodesic γ contains conjugate points. . . . . . . . . . . . . . . . 564

5.63 Since bn → b, tr(bn) < tr(b)/2 for all n > N . . . . . . . . . . . . . . . . . . 582

5.64 η(S) is contained in the bounded segment from γ(s1) to γ(s5). . . . . . . . 584

5.65 The Jacobi fields which are zero at r must have expansion θ which is positiveat p otherwise r would lie in the bounded interval from γ(s1) to γ(s5). . . . 586

5.66 With RabcdVbV c 6= 0. Non-positive expansion at p (θ > 0) implies there is

a point q conjugate to r, in the past of p. This is just the time reversedversion of the focussing theorem. . . . . . . . . . . . . . . . . . . . . . . . 586

5.67 The null expansion scalar θ̂. . . . . . . . . . . . . . . . . . . . . . . . . . . 587

5.68 Singularity theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

5.69 Diagram of collapse of a star. . . . . . . . . . . . . . . . . . . . . . . . . . 590

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5.70 Penrose diagram of collapse of a star. . . . . . . . . . . . . . . . . . . . . . 591

5.71 starCollaps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592

5.72 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593

5.73 characteristic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595

5.74 EnergyIneq1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602

5.75 EnergyCond. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

5.76 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604

7.1 Reflects forward in time by the strong gravitaional field outside the eventhorizon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636

7.2 timereflection2. The antiparticle mode falling into the black hole can beinterpreted as a particle travelling backwards in time, form the singularitydown to the horizon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637

7.3 Penrose diagram of a star that collapses to form a black hole. . . . . . . . 638

8.1 Graphical representation of the Mandelstam identity (8.148) relating dif-ferent Wilson loops. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676

8.2 The action of the Hamiltonian constraint transalted to the ‘path-integral’or spin foam description. Where N(xn) is the value of N at the vertex and

Hnop are the matrix elements of the operator Ĥ . . . . . . . . . . . . . . . . 684

8.3 a) A spherical star of mass M undergoes collapse. b) Later, a sphericalshell of mass δM falls into the resulting black hole. With ∆1 and ∆2 areboth isolated horizons, only ∆2 is part of the event horizon. . . . . . . . . 691

8.4 Quantum Horizon. Polymer exicitations in the bulk puncture the horizon,endowing it with quantized area. Intrinsically, the horizon is flat except atpunctures where it aquires a quantized deficit angle. These angles add upto 4π. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691

A.1 Abhay Ashtekar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696

A.2 Julian Barbour. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698

A.3 LebRien. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699

A.4 Bondi coordinates at future null infinity. . . . . . . . . . . . . . . . . . . . 701

A.5 Cauchy Horizon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702

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A.6 We display the geometric interpretation of the curvature tensor. Carry athird vector Z, by parallel transport from p to s via q, and compare thiswith transporting this from p to s′ via r. We find that th