Lectures On Quantum Gravity And Black Holes
Lectures on Quantum Gravity and Black HolesThomas HartmanCornell UniversityPlease email corrections and suggestions to: hartman@cornell.eduAbstract These 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 Relativity. Next, we rediscover the AdS/CFT correspondence by scattering fields off nearextremal 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 briefly.Course Cornell Physics 7661, Spring 2015Prereqs This 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.) No previous knowledge of quantum gravity or stringtheory is necessary.1
Contents1 The problem of quantum gravity81.1Gravity as an effective field theory. . . . . . . . . . . . . . . . . . . .81.2Quantum gravity in the Ultraviolet . . . . . . . . . . . . . . . . . . . .151.3Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .182 The Laws of Black Hole Thermodynamics192.1Quick review of the ordinary laws of thermodynamics . . . . . . . . . .192.2The Reissner-Nordstrom Black Hole . . . . . . . . . . . . . . . . . . . .202.3The 1st law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .222.4The 2nd law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .252.5Higher curvature corrections . . . . . . . . . . . . . . . . . . . . . . . .272.6A look ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .283 Rindler Space and Hawking Radiation303.1Rindler space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .303.2Near the black hole horizon . . . . . . . . . . . . . . . . . . . . . . . .313.3Periodicity trick for Hawking Temperature . . . . . . . . . . . . . . . .323.4Unruh radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .353.5Hawking radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .384 Path integrals, states, and operators in QFT414.1Transition amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . .414.2Wavefunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .424.3Cutting the path integral . . . . . . . . . . . . . . . . . . . . . . . . . .424.4Euclidean vs. Lorentzian . . . . . . . . . . . . . . . . . . . . . . . . . .444.5The ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .452
4.6Vacuum correlation functions . . . . . . . . . . . . . . . . . . . . . . .464.7Density matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .504.8Thermal partition function . . . . . . . . . . . . . . . . . . . . . . . . .504.9Thermal correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . . .515 Path integral approach to Hawking radiation545.1Rindler Space and Reduced Density Matrices . . . . . . . . . . . . . . .545.2Example: Free fields . . . . . . . . . . . . . . . . . . . . . . . . . . . .565.3Importance of entanglement . . . . . . . . . . . . . . . . . . . . . . . .595.4Hartle-Hawking state63. . . . . . . . . . . . . . . . . . . . . . . . . . .6 The Gravitational Path Integral686.1Interpretation of the classical action . . . . . . . . . . . . . . . . . . . .686.2Evaluating the Euclidean action . . . . . . . . . . . . . . . . . . . . . .697 Thermodynamics of de Sitter space757.1Vacuum correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . . .777.2The Static Patch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .797.3Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .818 Symmetries and the Hamiltonian838.1Parameterized Systems . . . . . . . . . . . . . . . . . . . . . . . . . . .838.2The ADM Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . .858.3Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .888.4Other conserved charges . . . . . . . . . . . . . . . . . . . . . . . . . .898.5Asymptotic Symmetry Group . . . . . . . . . . . . . . . . . . . . . . .908.6Example: conserved charges of a rotating body . . . . . . . . . . . . . .939 Symmetries of AdS3973
9.1Exercise: Metric of AdS3 . . . . . . . . . . . . . . . . . . . . . . . . . .979.2Exercise: Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . .989.3Exercise: Conserved charges . . . . . . . . . . . . . . . . . . . . . . . . 10010 Interlude: Preview of the AdS/CFT correspondence10210.1 AdS geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10210.2 Conformal field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 10310.3 Statement of the AdS/CFT correspondence . . . . . . . . . . . . . . . 10411 AdS from Near Horizon Limits10711.1 Near horizon limit of Reissner-Nordstrom . . . . . . . . . . . . . . . . . 10711.2 6d black string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11012 Absorption Cross Sections of the D1-D5-P11412.1 Gravity calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11413 Absorption cross section from the dual CFT12113.1 Brief Introduction to 2d CFT . . . . . . . . . . . . . . . . . . . . . . . 12113.2 2d CFT at finite temperature . . . . . . . . . . . . . . . . . . . . . . . 12513.3 Derivation of the absorption cross section . . . . . . . . . . . . . . . . . 12713.4 Decoupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13014 The Statement of AdS/CFT13114.1 The Dictionary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13114.2 Example: IIB Strings and N 4 Super-Yang-Mills . . . . . . . . . . . 13314.3 General requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13514.4 The Holographic Principle . . . . . . . . . . . . . . . . . . . . . . . . . 13615 Correlation Functions in AdS/CFT4138
15.1 Vacuum correlation functions in CFT . . . . . . . . . . . . . . . . . . . 13815.2 CFT Correlators from AdS Field Theory . . . . . . . . . . . . . . . . . 14015.3 Quantum corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14116 Black hole thermodynamics in AdS514316.1 Gravitational Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . 14416.1.1 Schwarzschild-AdS . . . . . . . . . . . . . . . . . . . . . . . . . 14416.1.2 Thermal AdS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14816.1.3 Hawking-Page phase transition . . . . . . . . . . . . . . . . . . 14916.1.4 Large volume limit . . . . . . . . . . . . . . . . . . . . . . . . . 15116.2 Confinement in CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15116.3 Free energy at weak and strong coupling . . . . . . . . . . . . . . . . . 15317 Eternal Black Holes and Entanglement15717.1 Thermofield double formalism . . . . . . . . . . . . . . . . . . . . . . . 15717.2 Holographic dual of the eternal black hole . . . . . . . . . . . . . . . . 15917.3 ER EPR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16217.4 Comments in information loss in AdS/CFT. . . . . . . . . . . . . . . 16317.5 Maldacena’s information paradox . . . . . . . . . . . . . . . . . . . . . 16317.6 Entropy in the thermofield double . . . . . . . . . . . . . . . . . . . . . 16518 Introduction to Entanglement Entropy16618.1 Definition and Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16618.2 Geometric entanglement entropy . . . . . . . . . . . . . . . . . . . . . . 16918.3 Entropy Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17119 Entanglement Entropy in Quantum Field Theory17519.1 Structure of the Entanglement Entropy . . . . . . . . . . . . . . . . . . 1765
19.2 Lorentz invariance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17820 Entanglement Entropy and the Renormalization Group18020.1 The space of QFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18020.2 How to measure degrees of freedom . . . . . . . . . . . . . . . . . . . . 18120.3 Entanglement proof of the c-theorem . . . . . . . . . . . . . . . . . . . 18320.4 Entanglement proof of the F theorem . . . . . . . . . . . . . . . . . . . 18521 Holographic Entanglement Entropy18721.1 The formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18721.2 Example: Vacuum state in 1 1d CFT . . . . . . . . . . . . . . . . . . 18821.3 Holographic proof of strong subadditivity . . . . . . . . . . . . . . . . . 19121.4 Some comments about HEE . . . . . . . . . . . . . . . . . . . . . . . . 19222 Holographic entanglement at finite temperature19422.1 Planar limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19723 The Stress Tensor in 2d CFT19823.1 Infinitessimal coordinate changes . . . . . . . . . . . . . . . . . . . . . 19823.2 The Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19923.3 Ward identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20223.4 Operator product expansion . . . . . . . . . . . . . . . . . . . . . . . . 20423.5 The Central Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20523.6 Casimir Energy on the Circle . . . . . . . . . . . . . . . . . . . . . . . 20824 The stress tensor in 3d gravity21024.1 Brown-York tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21024.2 Conformal transformations and the Brown-Henneaux central charge . . 21124.3 Casimir energy on the circle . . . . . . . . . . . . . . . . . . . . . . . . 2126
25 Thermodynamics of 2d CFT21425.1 A first look at the S transformation . . . . . . . . . . . . . . . . . . . . 21425.2 SL(2, Z) transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 21725.3 Thermodynamics at high temperature26 Black hole microstate counting. . . . . . . . . . . . . . . . . . 21922126.1 From the Cardy formula . . . . . . . . . . . . . . . . . . . . . . . . . . 22126.2 Strominger-Vafa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2227
1The 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 is1S 16πGNZ g (R 2Λ Lmatter ) .(1.1)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 haveto be careful about when it is reliable and when it isn’t. 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 completion.1.1Gravity 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 kept.This 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. Donaghue gr-qc/9512024.8
Counting 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 metric gµν has†10 components 4 diffeos 4 non-dynamical 2 dof .(1.2)In D dimensions, we count dof of a massless particle by looking at how the particlestates transform under SO(D 2), the group of rotations that preserve a null ray.‡ Aspin-2 particle transforms in the symmetric traceless tensor rep of SO(D 2), whichhas dimension 21 (D 2)(D 1) 1 12 D(D 3). Similarly, assuming diffeomorphisminvariance, the metric has1D(D2 1) D D 21 D(D 3)(1.3)degrees of freedom.Note that in D 3, the metric has no (local) dof. It turns out that it does have somenonlocal dof; this will be useful later in the course.Back 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 tensor Rµνρσ and covariant derivatives µ . Each Riemanncontains g, so the derivative expansion is an expansion in the number of R’s and ’s. Up to 4th order in derivatives,1S 16πGNZ g 2Λ R c1 R2 c2 Rµν Rµν c3 Rµνρσ Rµνρσ · · · (1.4)So the general theory is the Einstein-Hilbert term plus higher curvature corrections.We have ignored the matter terms Lmatter and matter-curvature couplings, like φR. See Weinberg QFT V1, section 5.9, 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 40functions worth of diffeomorphisms, xµ xµ (xµ ). And g̈0µ 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).‡See Weinberg QFT V1, section 2.5.9
Step 3: Coefficients scale of new physicsCoefficients should be fixed by dimensional analysis, up to O(1) factors. This doesn’twork for the cosmological constant: experiment (ie the fact the universe is not Plancksized) indicates that Λ is unnaturally small. This is the cosmological constant problem.We will just sweep this under the rug, take this fine-tuning as an experimental fact,and proceed to higher order.For these purposes let’s take the coordinates to have dimensions of length, so the metricis dimensionless, and R has 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,1 (MP )D 2 .GN(1.5)In D 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 for E & MP . The rulesof effective field theory tell us that we must include the R2 terms, with coefficientsc1,2,3 1/MP2 . Higher curvature terms should also be included, suppressed by morepowers of MP . More generally, the rule is that these coefficients should be suppressedby the scale of new physics, which we will call Ms . New physics must appear at orbelow the Planck to save unitarity, so Ms MP , but it’s possible that Ms MP . Soto allow for this possibility, we setc1,2,3 1.Ms2(1.6)In string theory Ms would be the mass of excited string states:Ms 1, sc1,2,3 α0(1.7)where s is the string length and α0 2s is the string tension. In this context the R2and higher curvature terms in the action are called ‘stringy corrections.’Steps 4 and 5: Do quantum field theory, but be careful what’s reliableIn this section to be concrete we will work in D 4. To do perturbation theory (about10
flat space) with the action (1.4), we setgµν ηµν 1hµνMP(1.8)and expand in h, or equivalently in 1/MP . The factor of MP is inserted here so thatthe quadratic action is canonically normalized; schematically, the perturbative actionlooks likeZS 11 h h h h h · · · 2MPMs 1 h h h 2 h 2 h · · ·MP22 (1.9)where the first terms come from expanding the Einstein action, and the other termscome from the higher curvature corrections. In curved space, the higher curvatureterms would also contribute to the terms like h h h since R const 2 h · · · .Scattering and the strong coupling scaleAs expected in a non-renormalizable theory, the perturbative expansion breaks downat high energies. First consider the case where we set Ms MP (or, we keep only theEinstein term in the Lagrangian) and calculate the amplitude for graviton scatteringin perturbation theory: GN GN crosses (1.10) ··· This is an expansion in the coupling constantGNGN 1/MP ; but this is dimensionful, soit must really be an expansion in E/MP . That is, each diagram contributes something The term written here comes from a term in the action g 1 δg, R 2 δg, then rescale δg 1MPh.11 MsMP 2R2 , where we pick off the terms
of order E2MP2 1 number of loops.(1.11)So the strong coupling scale, where loop diagrams are the same size as tree diagrams,isEstrong MP .(1.12)Below the strong coupling scale, this is a perfectly good quantum theory. We can useit to make reliable predictions about graviton-graviton scattering, including calculableloop corrections.If there are two different scales Ms and MP with Ms MP the situation is slightlymore complicated. The Einstein term is strongly coupled at the Planck scale, butlooking at (1.9), the higher curvature terms become strongly coupled at a lower scalesomewhere between Ms and MP ,Estrong MPx Ms1 x(1.13)for some x (0, 1). (This is a known number that you can find by examining all thediagrams.) It is important, however, that interactions in the higher curvature termsstill come with powers of 1/MP , so even in this case Estrong contains some factor of MP(ie, x 0): the theory is still weakly coupled at the scale of new physics, E Ms .Classical corrections to the Newtonian potential and ghostsReturning to the classical theory, consider for example the theory with just the firstterm in (1.4), so c1 (Ms ) 2 and c2 c3 0. The equations of motion derived fromthis action are schematically h 1Ms 2 h 8πGN T .(1.14)Going to momentum space E 2 , so clearly the higher curvature term is negligibleat low energies E Ms . The propagator looks likeq2111 2 2 24 Ms qqq Ms212(1.15)
The 2nd term looks like a massive field with mass Ms , but the wrong sign. It is a new,non-unitary degree of freedom, or ‘ghost’, in the classical theory. Although the onlyfield is still the metric, it makes sense that we’ve added a degree of freedom becausewe need more than 2 functions worth of initial data to solve the 4th order equation ofmotion (1.14).The ‘ghost’ should not bother us, because it appears at the scale Ms . This is the scaleof new physics where we should not trust our effective field theory anyway. And atenergies E Ms , the ghost has no effect on classical gravity. To see this, let’s computethe classical potential between two massive objects. The first term in (1.15) gives theNewtonian 1/r potential. The second term looks like a massive Yukawa force, so theclassical potential is V (r) GN m1 m21 e rMs rr .(1.16)This tiny for distances r 1/Ms .Loop corrections to the Newtonian potentialThe 2pt function has calculable, reliable quantum corrections: ··· 111 a 4q2 1 qlog ···q 2 Ms2 q 2 MP2Λ2 q 2(1.17)The first two terms are the classical part,q2111 2 2 ··· , 24 Ms qqMs(1.18)and the log term is the loop diagram (including external legs!). a is an order 1 numberthat can been calculated from this diagram, and we’ve dropped some terms to simplifythe discussion (see Donaghue for details). To calculate the attractive potential between13
two stationary masses, we set the frequency to zero q 0 0 and go to position space, Z 111d q 2 2 log q 2 · · ·2 qMsMP3 ei q· x 111 2 δ(r) 2 3r MsMP r(1.19)
Lectures on Quantum Gravity and Black Holes Thomas Hartman Cornell University Please email corrections and suggestions to: hartman@cornell.edu Abstract These are the lecture notes for a one-semester graduate course on black holes and quantum gravity. We File Size: 1MB
An indirect way of observing quantum gravity e ects is via the gauge / gravity corre-spondence, which relates quantum eld theories and quantum gravity. 1.3 Approaches to quantum gravity List of the largest existing research programmes. Semiclassical gravity Energy-momentum-t
spondence, which relates quantum eld theories and quantum gravity. 1.3 Approaches to quantum gravity List of the largest existing research programmes. Semiclassical gravity Energy-momentum-tensor is expectation value. Need self-consistent solution First step towards quantum gravity, ma
According to the quantum model, an electron can be given a name with the use of quantum numbers. Four types of quantum numbers are used in this; Principle quantum number, n Angular momentum quantum number, I Magnetic quantum number, m l Spin quantum number, m s The principle quantum
1. Quantum bits In quantum computing, a qubit or quantum bit is the basic unit of quantum information—the quantum version of the classical binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics.
Lectures on loop quantum gravity . It is still not clear what could be the role of quantum gravity in the solution of this problem . The Einstein-Hilbert action: The action that leads to the Einstein equations is, and variati
by quantum eld theories. 4. Quantum gravity: if you don’t care about boiling water, perhaps you like quantum gravity. The so-called AdS/CFT correspondence tells us that quantum gravity in Ddimensions is precisely equivalent to a quantum eld theory in D 1 dimensions (in its most we
For example, quantum cryptography is a direct application of quantum uncertainty and both quantum teleportation and quantum computation are direct applications of quantum entanglement, the con-cept underlying quantum nonlocality (Schro dinger, 1935). I will discuss a number of fundamental concepts in quantum physics with direct reference to .
Cracknell, P Carlisle : Historic Building Survey and Archaeological Illustration (HBSAI), 2005, 21pp, colour pls, fi gs, refs Work undertaken by: Historic Building Survey and Archaeological Illustration (HBSAI) SMR primary record number: 1593 Archaeological periods represented: PM. Archaeological Investigations Project 2005 Building Survey North West (G.16.2118) {EC17F9C4-61F0-4672-B70D .
