MIT OpenCourseWarehttp://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 22: Black Hole Mechanics is Thermodymanics Lecturer: McGreevy December 10, 2008 We will begin to “cleave s ome 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. Loose Ends Last time we w rote down a confining solution and argued that it had a mass gap. There was the question of why this argument fails in AdS. In the confining s olution Φω is normalizable so that ω2 φ2 = (∂zφ)2 (1) are not only greater than zero, but less than ∞. D In AdS, Φω = z 2 Jν(qz), which, as z −→ ∞, goes as z D 2 Jν(qz) ∼ z D−1 2 cos(z + πν 2 − π 4)(1 + O( 1 z )) with q = √ω2 − k2 > 0 so that (2) � dz φ2 = ∞ (3) which is the reason there is no mass gap in AdS. 1� � 2 What is a Black Hole? A rough definition 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 obj ect creates a gravitational potential well, the velocity that a test particle would need to escape at distance r is vesc, where v2 GM esc r ∼ r. (4) 2 Under some reasonable assumptions about physics, no particle can go faster than c, and so if the 2GM test particle is within rs = c2 (the Schwarzschild radius (for the ’2’ we need GR)) of the mass, it will not be able to escape. Thus, a BH forms if the massive object has a radius less th an rs. The boundary of the region of no escape is called the event horizon. If what we’r e 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 ann oy 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 definition of a BH so that th is case is excluded. Most of th e theorems we’re about to talk about were proven for the asymptotically flat case (i.e. not Ad S), 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 futu re evolution of the spacetime. The definition of the horizon in the asymptotically flat case is “the boundary of the past of future null infinity”. 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 coord inates), ds2 = −fdt2 + drf 2 + r 2dΩ22 (5) with f ≡2GM (the “emblackening factor”). As r −→ ∞, f −→ 1 and the space becomes 1 −r∂asymptotically flat (Minkowskian). In this spacetime, there exists a timelike killing vector ξ = ∂t which generates time translations. At the horizon r = rs , however, this Killing vector becomes null as |ξ| = ξµξνgµν = f(rs) = 0 (6) 2Behind the horizon, there’s a real singularity at r = 0, but the classical evolution outside the horizon is unaffected. 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 s tates, very roughly, that naked sin gularities don’t happen . That is to say, all singularities are hidden behind a horizon. More specifically, 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, bu t 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 flipside–the singularity theorems of Hawking and Penrose. These state that if at some point in the evolution a horizon forms (or more specifically a “trapped surface”–see Wald if you are into both beauty and pain), the the futur e 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 id ealized 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 w ith 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 obj ects 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 ω > 0). The decay is due to energy falling into the h orizon. In asymptotically flat space, the black holes can also radiate to infin ity via gravitational waves. You should think of the