MIT OpenCourseWare http://ocw.mit.edu 8.821 String Theory Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.1 8.821 F2008 Lecture 23: Black Hole Thermodynamics Lecturer: McGreevy December 2, 2008 In today’s lecture we’ll discuss the laws of black hole thermodynamics and how AdS black holes are related to finite temperature CFTs, and Koushik will give a related presentation. Laws of Thermodynamics Recall from last time that for a black hole Area ∼ Entropy κ ∼ T (1) where κ was the surface gravity. The near-h orizon metric is ds2 ∼ −κ2ρ2dt2 + dρ2 + ... = κ2ρ2dτ2 + dρ2 + ... (2) when we go to Euclidean time τ ≡ it. If τ has periodicity τ ∼ τ +2π /κ then the euclidean geometry is regular. Recall the canonical ensemble thermal partition function is Zth = tre −H/T (3) where e−H/T propagates the system with imaginary time t = 1/iT . Thermal equilibrium is equiv-alent to periodic euclidean time with period 1/T, so we identify κ with temperature T . The laws of (stationary) black hole thermodynamics, analogous to the usual laws of thermodynam-ics, are: • 0th (thermal equilibrium): κ is constant over the event h orizon. This means temperature is constant in space and time. Thu s stationary black holes are in thermal equilibrium with constant temperature. John thinks the proof of the 0th law doesn’t depend on the s hape of the black hole, as long as its a stationary solution. 1� • 1st (conservation of energy): κ dE = dM = ΩdJ + ΦdQ + dA (+P dV ) (4) 8πG κΩdJ is the change in rotational energy, ΦdQ is the electrical energy, and 8πGdA = T dS is heat exchange. This law r elates the change in the energy (or equivalently mass) to changes in various properties of the black hole. The last term describing mechanical work P dV isn’t present for black holes but IS for black branes... 2nd (entropy increases): This is the area theorem for a black hole we proved last lecture, • ˙ AA ≥ 0, since S = (Proof of the exact relation between S and A in a later lecture.) 4�G. • 3rd (absolute zero entropy): κ (or rather T ) cannot taken to zero in a finite number of steps. This doesn’t mean that S(T = 0) = 0, but it do es probably mean at T = 0 there is a minimum in entr opy. These laws f ollow from Einstein’s Equation, the energy condition we discussed last class, and assuming we have stationary black holes. 1.1 3rd law Since we discussed the 2nd law last time, and th e 0th and 1st laws are pretty convincing, we now provide some evidence for the validity of the 3rd law. First, why isn’t it true that S(T = 0) = 0? Counterexamples are everywhere if you just open your eyes to them: It is well known to some people that there exist supersymmetric theories with LARGE ground • state degeneracies, ∝ eQα where Q is the charge and α is some power. So, S(T = 0) = ln(degeneracies) ∼ Qα . • The Kerr-Newman black hole is another counterexample. Here are some f acts about the KN black h ole that you can easily derive or look up: A = 4π(2M(M + µ ) −Q2) µ = M2 − Q2 − J2/M2 κ = 4πµ/A (5) The black hole is extremal when µ = 0. (This is also the BPS bound when the black hole is supersymmetric.) If µ < 0 then there is a naked singularity. Note that that when µ = 0, κ ∝ T = 0 but S ∝ A =� 0. As for th e claims of the 3rd law, we have some anecdotal evidence. Let’s consider a non-extremal KN black hole with J = 0, in other words a n on-extremal RN black hole, s o Q < M. (Everywhere Q is really Q ). | |2� � What can we do to try to make this black hole extremal? We need to throw on some charge q and mass m , such that the black h ole becomes extremal, namely M + m = Q + q (6) How, the mass m is attracted to the beautiful black hole by a force F ∼ Mm/r2 but the charge q is repulsed by a force F ∼ Qq/r2 . Thus for the matter to fall in freely, Mm > Qq. According to some mysterious algebra, this relation along with Q < M actually implies Q + q < M + m. Therefore you have to force the matter onto the black hole, which somehow adds heat and prevents you fr om cooling the black hole. Or you h ave to throw in infinitesimal little bits which takes FOREVER. 2 CFT at finite temperature We’re going to use the power of AdSCFT to describe CFTs at finite temperature with black holes. In particular we mean a 3 + 1 dimensional relativistic CFT. The partition function is Z(τ) = tre −H/T = e −F/T (7) with free energy F , on a space with geometry S1 th × Σ3 where the S1 has radius 1/T, τ ∼ τ + 1/T and Σ3 is some 3 manifold. We can give Σ3 finite volume as an IR regulator. This is a d eformation of the IR physics (modes with ω ≫ T = EKK don’t notice). For large V3 = V ol(Σ3), then F = cV3T4 which is clear from extensivity of F and dimensional analysis. 3 AdS black holes This object goes by many names, su ch as planar black hole, Poincare black hole, black brane... This is a b lack hole in AdSD+1, but p robably m any of the equations below mean D = 4. The metric is ds2 = L2 −fdt2 + d�x 2 + dz2 z2 f 4zf = 1 −z4 (8) m We again put the �x coordinates on a finite volume space, for example in box of volum e V3, x ∼ x + V31/3, periodic BCs. Notice that if f = 1 we get th e Poincar e AdS metric, and in fact f only deviates from 1 at larger z representing the fact th at this is an IR deformation. Whence: 3� � � It solves Einstein’s equations with a cosmological constant Λ = (D+1)(D+2) and asymptotes • 2L2 to Poincar e AdS, differing only in the IR region with a horizon at z = zm, fixed t. • It’s the double Wick rotation of the confining solution with t = iythere, y = −itthere. • Analogous to how we got the AdS