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 8 821 F2008 Lecture 22 Black Hole Mechanics is Thermodymanics Lecturer McGreevy December 10 2008 We will begin to cleave some blazing swath through the set of results of classical gravity We will see that black hole mechanics is thermodynamics Our next goal will be to answer the question Thermodynamics of what First though let s tie up some loose ends from the last lecture 1 Loose Ends Last time we wrote down a con ning solution and argued that it had a mass gap There was the question of why this argument fails in AdS In the con ning solution is normalizable so that 2 2 z 2 1 1 1 O 2 4 z 2 are not only greater than zero but less than D In AdS z 2 J qz which as z goes as D with q z 2 J qz z D 1 2 cos z 2 k2 0 so that dz 2 which is the reason there is no mass gap in AdS 1 3 2 What is a Black Hole A rough de nition of a black hole is that it is a region of space from which there is no escape at least classically We can to zeroth order argue the existence of black holes even without GR If a heavy object creates a gravitational potential well the velocity that a test particle would need to escape at distance r is vesc where 2 GM vesc 2 4 r r Under some reasonable assumptions about physics no particle can go faster than c and so if the test particle is within rs 2GM the Schwarzschild radius for the 2 we need GR of the mass c2 it will not be able to escape Thus a BH forms if the massive object has a radius less than rs The boundary of the region of no escape is called the event horizon If what we re interested in is just some sort of classical evolution outside of the horizon we never have to ask what sits behind it An important question which might annoy us when we say escape escape to where This in general depends on the asymptotics of the space For example one might be tempted to say that a massive particle in AdS can never escape since the geodesics just oscillate around the center of AdS and never reach the boundary However we should probably be more careful in our de nition of a BH so that this case is excluded Most of the theorems we re about to talk about were proven for the asymptotically at case i e not AdS so we may have to think about how to apply them The horizon is a global concept To know where the horizon is you actually need to know the whole future evolution of the spacetime The de nition of the horizon in the asymptotically at case is the boundary of the past of future null in nity It makes sense if you stare at it long enough Really 3 The Schwarzchild BH The metric of the Schwarzchild BH is in these particular coordinates ds2 f dt2 dr 2 r 2 d 22 f 5 the emblackening factor As r f 1 and the space becomes with f 1 2GM r asymptotically at Minkowskian In this spacetime there exists a timelike killing vector t which generates time translations At the horizon r rs however this Killing vector becomes null as g 2 f rs 0 6 Behind the horizon there s a real singularity at r 0 but the classical evolution outside the horizon is una ected Which is nice We can use GR even though we don t know what happens behind the horizon This fact is so convenient that the relativists have elevated it to a grand conjecture 4 The Cosmic Censorship Conjecture due to optimistic relativists everywhere The cosmic censorship conjecture states very roughly that naked singularities don t happen That is to say all singularities are hidden behind a horizon More speci cally it says that they don t happen for some set of nice initial data on some spacelike hypersurface where nice has some technical meaning There exists some numerical and analytical evidence for this We usually just talk about this conjecture and forget about it but we should pause to observe how nice this is It means we don t need to ask hard questions about the singularity quantum gravity etc because we can always apply classical GR outside of a horizon But like all good things this conjecture has a ipside the singularity theorems of Hawking and Penrose These state that if at some point in the evolution a horizon forms or more speci cally a trapped surface see Wald if you are into both beauty and pain the the future evolution will always involve a singularity behind the horizon This rules out the suspicion one might have that the singularity is some artifact of the highly idealized Schwarzchild solution and that some realistic asymmetric gravitational collapse might produce a spacetime with no singularity 5 BH Uniqueness In some sense black holes are actually extremely simple The uniqueness theorems or no hair theorems state that there are only a few numbers which parametrize stationary BH s those with a timelike killing vector it s mass M its angular momentum J and its long range gauge charges i e electric or magnetic charges Qi Black holes are the simplest macroscopic objects there are in this respect they are very much like elementary particles Also excitations above stationary solutions evolve back to the stationary solutions by ringdown via quasinormal modes which have Im 0 The decay is due to energy falling into the horizon In asymptotically at space the black holes can also radiate to in nity via gravitational waves You should think of the black hole as some equilibrium which is parametrized by a small number of state variables If you perturb the equilibrium it returns 3 6 Hawking s Area Theorem Assuming cosmic censorship and the null energy condition which states that Tab ka kb 0 null k Hawking s area theorem states that during the evolution of any black hole the area of the horizon can t decrease The idea of the proof is in two parts If the area ever decreases dA dt 0 then the horizon develops crossing points at some nite later time Consider a bundle of null geodesics which span the horizon that is to say every point on the horizon lies on some null geodesic that stays on the horizon Consider the congruence of such a geodesics These curves have some tangent fuctor ka dx d where is an a ne parameter The fact that this is a geodesic with a ne parameter implies that ka a kb 0 It s useful to introduce a quantity known as the convergence of the congruence which roughly speaking measures how the geodesics are spreading out and hence of how the …