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Advanced General Relativity and Quantum Field Theory in Curved Spacetimes

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Advanced General Relativity and Quantum Field Theory in Curved Spacetimes Advanced General Relativity and Quantum Field Theory in Curved Spacetimes Lecture Notes Stefano Liberati SISSA, IFPU, INFN January 2020 Preface The following notes are made by students of the course of \Advanced General Relativity and Quantum Field Theory in Curved Spacetimes", which was held at the International School of Advanced Studies (SISSA) of Trieste (Italy) in the year 2017 by professor Stefano Liberati. Being the course directed to PhD students, this work and the notes therein are aimed to inter- ested readers that already have basic knowledge of Special Relativity, General Relativity, Quantum Mechanics and Quantum Field Theory; however, where possible the authors have included all the definitions and concept necessary to understand most of the topics presented. The course is based on different textbooks and papers; in particular, the first part, about Advanced General Relativity, is based on: • \General Relativity" by R. Wald [1] • \Spacetime and Geometry" by S. Carroll [2] • \A Relativist Toolkit" by E. Poisson [3] • \Gravitation" by T. Padmanabhan [4] while the second part, regarding Quantum Field Theory in Curved Spacetime, is based on • \Quantum Fields in Curved Space" by N. C. Birrell and P. C. W. Davies [5] • \Vacuum Effects in strong fields” by A. A. Grib, S. G. Mamayev and V. M. Mostepanenko [6] • \Introduction to Quantum Effects in Gravity", by V. F Mukhanov and S. Winitzki [7] Where necessary, some other details could have been taken from paper and reviews in the standard literature, that will be listed where needed and in the Bibliography. Every possible mistake present in these notes are due to misunderstandings and imprecisions of which only the authors of the following works must be considered responsible. Credits • Andrea Oddo (Chapters 1, 2, 3, 6, 7, Appendix A; revision) • Giovanni Tambalo (Chapters 8, 9, 10, 11) • Lumen Boco (Chapters 4, 5) Also thanks to Giovanni Tricella and Paolo Campeti for useful scientific discussions. 1 Contents I Advanced General Relativity 5 1 Foundations of Relativity 6 1.1 An operational approach to spacetime . .6 1.2 Axiomatic derivation of Special Relativity . .8 1.3 Equivalence Principles . 12 2 Elements of Differential Geometry 15 2.1 Manifolds, Vectors, Tensors . 15 2.1.1 Vectors . 15 2.1.2 Dual vectors . 16 2.1.3 Tensors . 17 2.2 Tensor Densities . 17 2.3 Differential forms . 18 2.4 Lie Derivative . 19 2.5 The Volume element . 21 2.6 Curvature . 22 2.6.1 Intrinsic vs Extrinsic curvature . 25 2.6.2 Example: Intrinsic-Extrinsic (Gaussian) curvature of two surfaces . 26 2.7 Hypersurfaces . 26 2.7.1 Gauss{Codazzi equations . 27 2.7.2 Extrinsic Curvature . 28 2.8 Killing Vectors . 28 2.9 Conserved quantities . 31 3 Kinematics 33 3.1 Geodesic Deviation Equation . 33 3.2 Raychaudhuri Equations . 35 3.2.1 Time-like geodesics congruence . 36 3.2.2 Null geodesics congruence . 38 3.3 Energy Conditions . 39 4 Variational Principle 41 4.1 Lagrangian Formulation of General Relativity . 41 4.1.1 The Gibbons{Hawking{York counterterm . 44 4.1.2 Schr¨odinger action . 46 2 4.1.3 Palatini variation . 47 5 Alternative Theories of Gravity 49 5.1 Holding SEP: GR in D =4 ................................. 49 6 5.1.1 D < 4......................................... 49 5.1.2 D > 4......................................... 50 5.2 Holding SEP: Lanczos-Lovelock theories . 51 5.2.1 More about the Euler characteristic . 54 5.3 Relaxing SEP: F (R) Theories as a first example of higher curvature gravity . 55 5.3.1 Palatini variation . 55 5.4 Relaxing SEP: Generalized Brans{Dicke theories . 57 5.4.1 F (R) theories as generalized Brans{Dicke theories . 57 5.5 Relaxing SEP: General Scalar-Tensor Theories . 58 5.5.1 Jordan frame . 59 5.5.2 Einstein frame . 59 5.5.3 Horndeski theory . 59 5.5.4 DHOST Theories . 60 5.6 Going further . 62 6 Global Methods 63 6.1 Carter{Penrose Diagrams . 63 6.1.1 Minkowski spacetime CP diagram . 64 6.1.2 Schwarzschild spacetime CP diagram . 66 6.2 Black Holes: Singularities and Event Horizon . 70 6.2.1 Singularities . 70 6.2.2 Horizons . 70 6.3 Killing Horizons and Surface Gravity . 73 6.3.1 Surface gravity: alternative definitions . 74 6.4 Cauchy Horizon . 76 6.5 Cosmological Horizon . 78 6.6 Local black hole horizons . 80 6.6.1 Trapped Surfaces and Trapped Regions . 80 6.6.2 Apparent Horizon . 81 6.6.3 Intrinsic local horizons . 82 6.7 Singularity Theorems . 84 7 From Black Hole Mechanics to Black Hole Thermodynamics 88 7.1 Rotating Black Holes . 89 7.1.1 Kerr Singularity . 90 7.1.2 Horizons . 91 7.1.3 Killing vectors . 92 7.1.4 Surface gravity . 93 7.2 Ergosphere and Penrose Process . 93 7.2.1 The Penrose process . 95 7.2.2 Superradiance . 96 7.3 Black Hole Thermodynamics . 98 7.3.1 More on Null hypersurfaces, Surface integrals and other tools . 98 7.3.2 ADM and Bondi Mass . 101 7.3.3 Komar Mass and the Smarr formula . 102 7.3.4 The four laws of black hole mechanics . 104 7.3.5 On the necessity of Hawking radiation . 107 3 7.3.6 Hawking radiation and black hole entropy . 108 II Quantum Field Theory in Curved Spacetimes 110 8 QFT in curved spacetimes 111 8.1 Second quantization in curved spacetimes and the inequivalence of quantum vacua . 111 8.2 Particle production by a time-varying potential . 114 8.3 Coherent and Squeezed states . 117 9 The Casimir and Unruh effects 120 9.1 The Casimir effect . 121 9.1.1 1+1 dimensional case: . 122 9.1.2 Topological Casimir effect . 124 9.1.3 A few considerations . 125 9.2 The Unruh Effect . 125 9.2.1 The Rindler Wedge . 126 9.2.2 Quantum field theory on the Rindler Wedge . 127 10 Evaporating Black Holes 131 10.1 Moving Mirrors . 131 10.1.1 Field theory with moving boundaries . 131 10.1.2 Quantum field theory with moving boundaries . 132 10.1.3 Special case: the exponentially receding mirror . 133 10.2 Hawking Radiation . 135 10.3 Open issues with Hawking radiation . 140 10.3.1 Trans-planckian problem . 140 10.3.2 The information loss problem . 143 10.4 Black Hole Entropy Interpretations . ..
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