Lectures on Quantum Gravity and Black Holes

1 Lectures on Quantum Gravity and Black HolesThomas HartmanCornell UniversityPlease email corrections and suggestions to: are the lecture notes for a one-semester graduate course on blackholes 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 correlationfunctions, Black hole thermodynamics, and entanglement entropy. The emphasis is onsemiclassical Gravity , so topics like string theory, D-branes, and super-Yang Mills arediscussed only very Physics 7661, Spring 2015 PrereqsThis course is aimed at graduate students who have taken 1-2 semesters ofgeneral relativity (including: classical Black Holes , Penrose diagrams, and the Einsteinaction) and 1-2 semesters of Quantum field theory (including: Feynman diagrams, pathintegrals, and gauge symmetry.)

2 No previous knowledge of Quantum Gravity or stringtheory is The problem of Quantum Gravity as an effective field theory .. Quantum Gravity in the Ultraviolet .. Homework ..182 The Laws of Black Hole Quick review of the ordinary laws of thermodynamics .. The Reissner-Nordstrom Black Hole .. The 1st law .. The 2nd law .. Higher curvature corrections .. A look ahead ..283 Rindler Space and Hawking Rindler space .. Near the Black hole horizon .. Periodicity trick for Hawking Temperature .. Unruh radiation .. Hawking radiation ..384 Path integrals, states, and operators in Transition amplitudes .. Wavefunctions .. Cutting the path integral .. Euclidean vs. Lorentzian .. The ground state .. Vacuum correlation functions .. Density matrices .. Thermal partition function .. Thermal correlators.

3 515 Path integral approach to Hawking Rindler Space and Reduced Density Matrices .. Example: Free fields .. Importance of entanglement .. Hartle-Hawking state..636 The Gravitational Path Interpretation of the classical action .. Evaluating the Euclidean action ..697 Thermodynamics of de Sitter Vacuum correlators .. The Static Patch .. Action ..818 Symmetries and the Parameterized Systems .. The ADM Hamiltonian .. Energy .. Other conserved charges .. Asymptotic Symmetry Group .. Example: conserved charges of a rotating body ..939 Symmetries of Exercise: Metric of AdS3.. Exercise: Isometries .. Exercise: Conserved charges .. 10010 Interlude: Preview of the AdS/CFT AdS geometry .. Conformal field theory .. Statement of the AdS/CFT correspondence .. 10411 AdSfrom Near Horizon Near horizon limit of Reissner-Nordstrom.

4 6d Black string .. 11012 Absorption Cross Sections of the Gravity calculation .. 11413 Absorption cross section from the dual Brief Introduction to 2d CFT .. 2d CFT at finite temperature .. Derivation of the absorption cross section .. Decoupling .. 13014 The Statement of The Dictionary .. Example: IIB Strings andN= 4 Super-Yang-Mills .. General requirements .. The Holographic Principle .. 13615 Correlation Functions in Vacuum correlation functions in CFT .. CFT Correlators from AdS Field Theory .. Quantum corrections .. 14116 Black hole thermodynamics Gravitational Free Energy .. Schwarzschild-AdS .. Thermal AdS .. Hawking-Page phase transition .. Large volume limit .. Confinement in CFT .. Free energy at weak and strong coupling .. 15317 Eternal Black Holes and Thermofield double formalism.

5 Holographic dual of the eternal Black hole .. ER=EPR .. Comments in information loss in AdS/CFT .. Maldacena s information paradox .. Entropy in the thermofield double .. 16518 Introduction to Entanglement Definition and Basics .. Geometric entanglement entropy .. Entropy Inequalities .. 17119 Entanglement Entropy in Quantum Field Structure of the Entanglement Entropy .. Lorentz invariance .. 17820 Entanglement Entropy and the Renormalization The space of QFTs .. How to measure degrees of freedom .. Entanglement proof of thec-theorem .. Entanglement proof of theFtheorem .. 18521 Holographic Entanglement The formula .. Example: Vacuum state in 1+1d CFT .. Holographic proof of strong subadditivity .. Some comments about HEE .. 19222 Holographic entanglement at finite Planar limit .. 19723 The Stress Tensor in 2d Infinitessimal coordinate changes.

6 The Stress Tensor .. Ward identities .. Operator product expansion .. The Central Charge .. Casimir Energy on the Circle .. 20824 The stress tensor in 3d Brown-York tensor .. Conformal transformations and the Brown-Henneaux central charge .. Casimir energy on the circle .. 212625 Thermodynamics of 2d A first look at theStransformation .. (2,Z) transformations .. Thermodynamics at high temperature .. 21926 Black hole microstate From the Cardy formula .. Strominger-Vafa .. 22271 The problem of Quantum gravityThese are Lectures on Quantum Gravity . To start, we better understand clearly whatproblem we are trying to solve when we say Quantum Gravity . At low energies, theclassical action isS=116 GN g(R 2 +Lmatter).( )Why not just quantize this action? The answer of course is that it is not does not mean it is useless to understand Quantum Gravity , it just means we haveto be careful about when it is reliable and when it isn t.

7 In this first lecture we willconsider Gravity as a low-energy effective field theory, see when it breaks down, andmake some general observations about what we should expect or not expect from theUV Gravity as an effective field theoryThe 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 aremuch smaller than the terms you works for renormalizable or non-renormalizable theories. Let s follow the stepsfor Gravity . Our starting assumption is that nature has a graviton a massless spin-2field. This theory can be consistent only if it is diffeomorphism invariant.

8 Donaghue gr- metric degrees of freedomThis can be argued various ways ; we ll just count degrees of freedom. In 4D, a masslessparticle has two degrees of freedom (2 helicities). Similarly, the metricg has 10 components 4 diffeos 4 non-dynamical = 2 dof.( )InDdimensions, we count dof of a massless particle by looking at how the particlestates transform underSO(D 2), the group of rotations that preserve a null ray. Aspin-2 particle transforms in the symmetric traceless tensor rep ofSO(D 2), whichhas dimension12(D 2)(D 1) 1 =12D(D 3). Similarly, assuming diffeomorphisminvariance, the metric has12D(D+ 1) D D=12D(D 3)( )degrees of that inD= 3, the metric has no (local) dof. It turns out that it does have somenonlocal dof; this will be useful later in the to effective field theory: Steps 1 and 2, the derivative expansionThe only things that can appear in a diff-invariant Lagrangian for the metric are objectsbuilt out of the Riemann tensorR and covariant derivatives.

9 Each Riemanncontains g, so the derivative expansion is an expansion in the number ofR s and s. Up to 4th order in derivatives,S=116 GN g( 2 +R+c1R2+c2R R +c3R R + )( )So the general theory is the Einstein-Hilbert term plus higher curvature have ignored the matter termsLmatterand matter-curvature couplings, like R. See Weinberg QFT V1, section , and the discussion of the Weinberg-Witten theorem below,and Weinberg Phys. Rev. 135, B1049 (1964). In more detail: a 4x4 symmetric matrix has 10 independent components. In 4D we have 4functions worth of diffeomorphisms,x x (x ). And g0 cannot appear in a 2nd order diff-invariantequation of motion, so these components are non-dynamical. For more details, see the discussion ofgravitational waves in any introductory GR textbook, which should show that in transverse-tracelessgauge the linearized Einstein equation have two independent solutions (the + and polarizations).

10 See Weinberg QFT V1, section 3: Coefficients = scale of new physicsCoefficients should be fixed by dimensional analysis, up toO(1) factors. This doesn twork for the cosmological constant: experiment (ie the fact the universe is not Planck-sized) indicates that is unnaturally small. This is the cosmological constant will just sweep this under the rug, take this fine-tuning as an experimental fact,and proceed to higher these purposes let s take the coordinates to have dimensions of length, so the metricis dimensionless, andRhas mass dimension 2. The action should be dimensionless(since~= 1). Looking at the Einstein-Hilbert term, that means [GN] = 2 D, so interms of the Planck scale,1GN (MP)D 2.( )InD >2, this term is not renormalizable. This means that the theory is stronglycoupled at the Planck scale. If we try to compute scattering amplitudes using Feynmandiagrams, we would find non-sensical, non-unitary answers forE&MP.