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Lectures on Quantum Gravity and Black Holes Lectures on Quantum Gravity and Black Holes Thomas Hartman Cornell University Please email corrections and suggestions to: [email protected] Abstract These are the lecture notes for a one-semester graduate course on black holes and quantum gravity. We start with black hole thermodynamics, Rindler space, Hawking radiation, Euclidean path integrals, and conserved quantities in General Rel- ativity. Next, we rediscover the AdS/CFT correspondence by scattering fields off near- extremal black holes. The final third of the course is on AdS/CFT, including correlation functions, black hole thermodynamics, and entanglement entropy. The emphasis is on semiclassical gravity, so topics like string theory, D-branes, and super-Yang Mills are discussed only very briefly. Course Cornell Physics 7661, Spring 2015 Prereqs This course is aimed at graduate students who have taken 1-2 semesters of general relativity (including: classical black holes, Penrose diagrams, and the Einstein action) and 1-2 semesters of quantum field theory (including: Feynman diagrams, path integrals, and gauge symmetry.) No previous knowledge of quantum gravity or string theory is necessary. 1 Contents 1 The problem of quantum gravity8 1.1 Gravity as an effective field theory . .8 1.2 Quantum gravity in the Ultraviolet . 15 1.3 Homework . 18 2 The Laws of Black Hole Thermodynamics 19 2.1 Quick review of the ordinary laws of thermodynamics . 19 2.2 The Reissner-Nordstrom Black Hole . 20 2.3 The 1st law . 22 2.4 The 2nd law . 25 2.5 Higher curvature corrections . 27 2.6 A look ahead . 28 3 Rindler Space and Hawking Radiation 30 3.1 Rindler space . 30 3.2 Near the black hole horizon . 31 3.3 Periodicity trick for Hawking Temperature . 32 3.4 Unruh radiation . 35 3.5 Hawking radiation . 38 4 Path integrals, states, and operators in QFT 41 4.1 Transition amplitudes . 41 4.2 Wavefunctions . 42 4.3 Cutting the path integral . 42 4.4 Euclidean vs. Lorentzian . 44 4.5 The ground state . 45 2 4.6 Vacuum correlation functions . 46 4.7 Density matrices . 50 4.8 Thermal partition function . 50 4.9 Thermal correlators . 51 5 Path integral approach to Hawking radiation 54 5.1 Rindler Space and Reduced Density Matrices . 54 5.2 Example: Free fields . 56 5.3 Importance of entanglement . 59 5.4 Hartle-Hawking state . 63 6 The Gravitational Path Integral 68 6.1 Interpretation of the classical action . 68 6.2 Evaluating the Euclidean action . 69 7 Thermodynamics of de Sitter space 75 7.1 Vacuum correlators . 77 7.2 The Static Patch . 79 7.3 Action . 81 8 Symmetries and the Hamiltonian 83 8.1 Parameterized Systems . 83 8.2 The ADM Hamiltonian . 85 8.3 Energy . 88 8.4 Other conserved charges . 89 8.5 Asymptotic Symmetry Group . 90 8.6 Example: conserved charges of a rotating body . 93 9 Symmetries of AdS3 97 3 9.1 Exercise: Metric of AdS3 .......................... 97 9.2 Exercise: Isometries . 98 9.3 Exercise: Conserved charges . 100 10 Interlude: Preview of the AdS/CFT correspondence 102 10.1 AdS geometry . 102 10.2 Conformal field theory . 103 10.3 Statement of the AdS/CFT correspondence . 104 11 AdS from Near Horizon Limits 107 11.1 Near horizon limit of Reissner-Nordstrom . 107 11.2 6d black string . 110 12 Absorption Cross Sections of the D1-D5-P 114 12.1 Gravity calculation . 114 13 Absorption cross section from the dual CFT 121 13.1 Brief Introduction to 2d CFT . 121 13.2 2d CFT at finite temperature . 125 13.3 Derivation of the absorption cross section . 127 13.4 Decoupling . 130 14 The Statement of AdS/CFT 131 14.1 The Dictionary . 131 14.2 Example: IIB Strings and N = 4 Super-Yang-Mills . 133 14.3 General requirements . 135 14.4 The Holographic Principle . 136 15 Correlation Functions in AdS/CFT 138 4 15.1 Vacuum correlation functions in CFT . 138 15.2 CFT Correlators from AdS Field Theory . 140 15.3 Quantum corrections . 141 16 Black hole thermodynamics in AdS5 143 16.1 Gravitational Free Energy . 144 16.1.1 Schwarzschild-AdS . 144 16.1.2 Thermal AdS . 148 16.1.3 Hawking-Page phase transition . 149 16.1.4 Large volume limit . 151 16.2 Confinement in CFT . 151 16.3 Free energy at weak and strong coupling . 153 17 Eternal Black Holes and Entanglement 157 17.1 Thermofield double formalism . 157 17.2 Holographic dual of the eternal black hole . 159 17.3 ER=EPR . 162 17.4 Comments in information loss in AdS/CFT . 163 17.5 Maldacena's information paradox . 163 17.6 Entropy in the thermofield double . 165 18 Introduction to Entanglement Entropy 166 18.1 Definition and Basics . 166 18.2 Geometric entanglement entropy . 169 18.3 Entropy Inequalities . 171 19 Entanglement Entropy in Quantum Field Theory 175 19.1 Structure of the Entanglement Entropy . 176 5 19.2 Lorentz invariance . 178 20 Entanglement Entropy and the Renormalization Group 180 20.1 The space of QFTs . 180 20.2 How to measure degrees of freedom . 181 20.3 Entanglement proof of the c-theorem . 183 20.4 Entanglement proof of the F theorem . 185 21 Holographic Entanglement Entropy 187 21.1 The formula . 187 21.2 Example: Vacuum state in 1+1d CFT . 188 21.3 Holographic proof of strong subadditivity . 191 21.4 Some comments about HEE . 192 22 Holographic entanglement at finite temperature 194 22.1 Planar limit . 197 23 The Stress Tensor in 2d CFT 198 23.1 Infinitessimal coordinate changes . 198 23.2 The Stress Tensor . 199 23.3 Ward identities . 202 23.4 Operator product expansion . 204 23.5 The Central Charge . 205 23.6 Casimir Energy on the Circle . 208 24 The stress tensor in 3d gravity 210 24.1 Brown-York tensor . 210 24.2 Conformal transformations and the Brown-Henneaux central charge . 211 24.3 Casimir energy on the circle . 212 6 25 Thermodynamics of 2d CFT 214 25.1 A first look at the S transformation . 214 25.2 SL(2;Z) transformations . 217 25.3 Thermodynamics at high temperature . 219 26 Black hole microstate counting 221 26.1 From the Cardy formula . 221 26.2 Strominger-Vafa . 222 7 1 The problem of quantum gravity These are lectures on quantum gravity. To start, we better understand clearly what problem we are trying to solve when we say `quantum gravity.' At low energies, the classical action is 1 Z p S = −g (R − 2Λ + Lmatter) : (1.1) 16πGN Why not just quantize this action? The answer of course is that it is not renormalizable. This does not mean it is useless to understand quantum gravity, it just means we have to be careful about when it is reliable and when it isn't. In this first lecture we will consider gravity as a low-energy effective field theory,∗ see when it breaks down, and make some general observations about what we should expect or not expect from the UV completion. 1.1 Gravity as an effective field theory The rules of effective field theory are: 1. Write down the most general possible action consistent with the symmetries; 2. Keep all terms up to some fixed order in derivatives; 3. Coefficients are fixed by dimensional analysis, up to unknown order 1 factors (unless you have a good reason to think otherwise); 4. Do quantum field theory using this action, including loops; 5. Trust your answer only if the neglected terms in the derivative expansion are much smaller than the terms you kept. This works for renormalizable or non-renormalizable theories. Let's follow the steps for gravity. Our starting assumption is that nature has a graviton | a massless spin-2 field. This theory can be consistent only if it is diffeomorphism invariant. ∗Donaghue gr-qc/9512024. 8 Counting metric degrees of freedom This can be argued various ways∗; we'll just count degrees of freedom..
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