8 821 F2008 Lecture 23 Black Hole Thermodynamics Lecturer McGreevy Scribe Tongyan Lin 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 1 Laws of Thermodynamics Recall from last time that for a black hole Area Entropy T 1 where was the surface gravity The near horizon metric is ds2 2 2 dt2 d 2 2 2 d 2 d 2 2 when we go to Euclidean time it If has periodicity 2 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 equivalent to periodic euclidean time with period 1 T so we identify with temperature T The laws of stationary black hole thermodynamics analogous to the usual laws of thermodynamics are 0th thermal equilibrium is constant over the event horizon This means temperature is constant in space and time Thus stationary black holes are in thermal equilibrium with constant temperature John thinks the proof of the 0th law doesn t depend on the shape of the black hole as long as its a stationary solution 1 1st conservation of energy dE dM dJ dQ dA P dV 8 G 4 dJ is the change in rotational energy dQ is the electrical energy and 8 G dA T dS is heat exchange This law relates 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 A Proof of the exact relation between S and A in a later lecture A 0 since S 4 G 3rd absolute zero entropy or rather T cannot taken to zero in a finite number of steps This doesn t mean that S T 0 0 but it does probably mean at T 0 there is a minimum in entropy These laws follow 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 the 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 eQ where Q is the charge and is some power So S T 0 ln degeneracies Q The Kerr Newman black hole is another counterexample Here are some facts about the KN black hole that you can easily derive or look up A 4 2M M Q2 p M 2 Q2 J 2 M 2 4 A 5 The black hole is extremal when 0 This is also the BPS bound when the black hole is supersymmetric If 0 then there is a naked singularity Note that that when 0 T 0 but S A 6 0 As for the 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 non extremal RN black hole so 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 hole becomes extremal namely M m Q q 6 How the mass m is attracted to the beautiful black hole by a force F M m r 2 but the charge q is repulsed by a force F Qq r 2 Thus for the matter to fall in freely M m 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 from cooling the black hole Or you have 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 tre H T e F T 7 1 where the S 1 has radius 1 T 1 T with free energy F on a space with geometry Sth 3 and 3 is some 3 manifold We can give 3 finite volume as an IR regulator This is a deformation of the IR physics modes with T EKK don t notice For large V3 V ol 3 then F cV3 T 4 which is clear from extensivity of F and dimensional analysis 3 AdS black holes This object goes by many names such as planar black hole Poincare black hole black brane This is a black hole in AdSD 1 but probably many of the equations below mean D 4 The metric is dz 2 L2 2 2 2 f dt d x ds z2 f 4 z f 1 4 8 zm We again put the x coordinates on a finite volume space for example in box of volume V3 x 1 3 x V3 periodic BCs Notice that if f 1 we get the Poincare AdS metric and in fact f only deviates from 1 at larger z representing the fact that this is an IR deformation Whence 3 and asymptotes It solves Einstein s equations with a cosmological constant D 1 D 2 2L2 to Poincare 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 solution from the near horizon limit of D3 branes it s the near horizon limit of black 3 branes in R9 1 in particular the near extremal RR soliton with geometry 2 f dt2 d x2 p dr 2 2 2 p H r r d 5 ds f r H r L 4 r4 r4 f r 1 H4 r H r 1 9 Again f 1 gives the usual RR soliton Note that there also exists a black hole which asymptotes to GLOBAL AdS with boundary S 1 S 3 which is known as AdS Schwarzchild which describes a CFT on S 3 at finite temperature T Let s check out the horizon properties so we can find the usual thermodynamic quantities we re interested in The near horizon metric is ds2 2 2 d 2 d 2 L2 2 d x 2 zm 10 where 2 zm and the temperature is T 2 1 zm Meanwhile the area of horizon is A Z 3 gd x z zm fixedt L zm 3 …