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MIT 8 821 - Thermodymanics

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8.821 F2008 Lecture 22: Black Hole Mechanics is ThermodymanicsLecturer: McGreevy Scribe: David GuarreraDecember 10, 2008We will begin to “cleave some blazing swath” through the set of results of classical gravity. We willsee 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 EndsLast time we wrote down a confining solu tion and argu ed that it had a mass gap . There was thequestion of why this argument fails in AdS.In the confining solution Φωis normalizable so thatω2Zφ2=Z(∂zφ)2(1)are not only greater than zero, but less than ∞.In AdS, Φω= zD2Jν(qz), which, as z −→ ∞, goes aszD2Jν(qz) ∼ zD−12cos(z +πν2−π4)(1 + O(1z)) (2)with q =√ω2− k2> 0 so thatZdz φ2= ∞ (3)which is the reason there is no mass gap in AdS.12 What is a Black Hole?A rough definition of a black hole is that it is a “region of sp ace from which there is no escape (atleast classically).” We can, to zeroth order, argue the existence of black holes even without GR. Ifa heavy object creates a gravitational potential well, the velocity that a test particle would needto escape at distance r is vesc, wherev2escr∼GMr2. (4)Under some reasonable assumptions about physics, no particle can go faster than c, and so if thetest particle is within rs=2GMc2(the Schwarzschild radius (for the ’2’ we need GR)) of the mass,it will not be able to escape.Thu s , a BH forms if the massive object h as a radius less than rs. The boundary of th e region of noescape is called the event horizon. If wh at we’re interested in is j ust some sort of classical evolutionoutside of the horizon, we never have to ask what sits behind it.An important question which might annoy us: when we say “escap e”, escape to where? T his, ingeneral, depends on the asymptotics of the space. For example, one might be tempted to say thata massive particle in AdS can never escape, since the geodesics just oscillate around the center ofAdS and never reach the boundary. However, we should probably be more careful in our definitionof a BH so that this case is excluded. Most of the theorems we’re about to talk about were provenfor the asymptotically flat 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 thewhole fu tu re evolution of the spacetime. The definition of the horizon in the asymptotically flatcase is “the boundary of the past of future null infinity”. It makes sense if you stare at it longenough. Really.3 The Schwarzchild BHThe metric of the Schwarzchild BH is (in these particular coordinates),ds2= −f dt2+dr2f+ r2dΩ22(5)with f ≡ 1 −2GMr(the “emblackening factor”). As r −→ ∞, f −→ 1 and the space becomesasymptotically flat (Minkowskian). In this spacetime, there exists a timelike killing vector ξ =∂∂twhich generates time translations. At the horizon r = rs, however, this Killing vector becomesnull as|ξ| =pξµξνgµν=pf(rs) = 0 (6)2Behind the horizon, there’s a real singularity at r = 0, but the classical evolution outside the horizonis unaffected. Which is nice. We can use GR even though we don’t know what happens behind thehorizon. This fact is so convenient that the relativists have elevated it to a grand conjecture.....4 The Cosmic Censorship Conjecture (due to optimistic relativistseverywhere)The cosmic censorship conjecture states, very roughly, that naked singularities don’t happen. Thatis to say, all singularities are hidden behind a horizon. More specifically, it says that th ey don’thappen for some set of “nice” initial d ata on some spacelike hypersurface, where “nice” h as sometechnical meaning. There exists some numerical and analytical evidence for this. We us ually justtalk about this conjecture and forget about it, but we should pause to observe how nice this is. Itmeans we don’t n eed to ask h ard questions about the singularity, quantum gravity, etc., becausewe can always apply classical GR outside of a horizon.But, like all good things, this conjecture h as a flipside–the singularity theorems of Hawking andPenrose. These state that if at some point in the evolution a horizon forms (or more specificallya “trapped surface”–see Wald if you are into both beauty and pain), the the future evolution willalways involve a singularity behind the horizon. Th is rules out the suspicion one might have thatthe singularity is some artifact of the highly idealized Schwarzchild solution, and that s ome realistic,asymmetric gravitational collapse might produce a spacetime with no singularity.5 BH UniquenessIn some sense, black holes are actually extremely simple. The uniqueness theorems (or “no hair”theorems) state th at th ere are only a few numbers which parametrize stationary BH’s (those witha timelike killing vector): it’s mass M , its angular momentu m J, and its long range gauge ch arges(i.e. electric or magnetic charges), Qi. Black holes are the simplest macroscopic objects thereare–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 (wh ich have Im ω > 0). The d ecay is due to energy falling into the h orizon.In asymptotically flat space, the black holes can also radiate to infinity via gravitational waves.You should think of the black hole as some equilibrium which is parametrized by a small numberof state variab les. If you perturb the equilibrium, it returns.36 Hawking’s Area TheoremAssuming cosmic censorship, and the null energy condition (which s tates that Tabkakb≥ 0 ∀ null k),Hawking’s area theorem states that during the evolution of any black hole, th e area of the horizoncan’t decrease. The idea of the proof is in two parts.If the area ever decreases (dAdt< 0), then th e horizon d evelops “crossing points” at some finite latertime. Consider a bundle of null geodesics which span the horizon–that is to say, every point onthe horizon lies on some null geodesic that stays on th e horizon. Consider the congruence of suchgeodesics. These curves have some tangent fuctor ka=dxadλwhere λ is an affine parameter. Thefact that this is a geodesic with affine parameter implies that ka∇akb= 0. It’s useful to introducea quantity


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