8 821 F2008 Lecture 24 Blackhole Thermodynamics Con d Lecturer McGreevy Scribe Wing Ho Ko December 2 2008 1 Classical Blackhole Thermodynamics Con d Previously we have stated the third law for blackhole thermodynamics To rephrase the result 3rd Law of blackhole thermodynamics Extremal blackholes are unnatural In spite of this extremal blackholes are beloved by string theorists because the computations are easier In particular Sextr BH T 0 ln of ground states and SUSY implies that this is independent of couplings and hence can be computed in the weak coupling limit Previously we have also stated the second law but in which we only considered blackholes alone We now consider a more general version the second law in which both blackholes and normal stuff are included Generalized 2rd Law GSL of blackhole thermodynamics Bekenstein PRD 9 3292 1974 A GN 0 for physical processes Stotal Snormal stuff Stotal where is a number yet to be determined The idea behind the GSL is that it will be possible to build perpetual machines if it is false However there is a problem with the lowering the box energy extraction scheme described in a previous lecture To be more specific suppose we lower an infinitesimal box of entropy from to the event horizon and extract energy along the way as described in the previous lecture When the box is at the event horizon we know that Ebox r rs 0 and if we let go at that point A 0 for the blackhole according to the first law Hence the GSL is violated The other laws of classical blackhole thermodynamics also contain flaws 1 Flaw 1 Classically a blackhole cannot radiate so TBH 0 Flaw 3 Classically the area of each individual blackhole must be non decreasing But thermodynamics should require only that Stotal 0 Thus the blackhole thermodynamics and ordinary thermodynamics are not in good parallel 1 The apparent contradictions in the blackhole thermodynamic laws can be resolved by realizing that the blackhole entropy SSek A GN contains a factor of hence SBek in the classical limit 0 when A and GN are fixed In particular for the box lowering scenario even though the change in area A is infinitesimal it can still lead to a finite change S in entropy Hence there is no contradiction with the GSL Similarly from the 1st Law dM TBek dSBek As 0 TBek 8 0 Hence flaw 1 does not lead to a contradiction in the classical limit Also as 0 any finite A for any blackhole will lead to S so flaw 3 does not lead to a contradiction in the classical limit 2 Quantum Blackhole Thermodynamics Our major result is the Hawking radiation law Hawking Radiation A blackhole radiates like a blackbody with T TBH 2 Let s first consider some consequences before considering the explanation 1 This fixes the numerical constant to be 1 4 i e SBH A 4 GN 2 The area of individual blackhole can decrease thus fixes flaw 3 of GSL 3 The resulting dA satisfies GSL for reasonable equation of state E O S For example consider radiation in 4d We have E 1 4 aT 4 and S 1 3 aT 3 Treating blackhole as a while energy conservation implies that dM dE blackbody in flat space dSrad 34 TdE BH dE and hence dSBH TBH Summing up2 dStotal d Srad SBH 13 TdE 0 BH 4 Let s return to the box lowering puzzle a Bekenstein bound Conjecture Entropy in a box of linear size R must satisfy Sbox 2 ER b Buoyancy force Consider a box near event horizon so that it s top side is at r2 and the bottom side is at r1 see Fig 1 The local temperature T1 T2 on the top bottom 1 2 Perhaps this is not a big deal since individual blackholes can merge Consequently this says that the speed of sound is c2sound 1 3 2 p p side satisfies T2 T1 gtt r1 gtt r2 Hence T1 T2 Consequently there is more radiation from the bottom than from the top3 which produces a buoyancy force displaced radiation As a result there is a critical rcrit rs for which E displaced radiation Ebox rcrit By dropping the box at rcrit SBH E TBH but Sbox Sdisplaced radiation E TBH Now let s return to the question of why TBH takes this particular form For a generic possibly non extremal blackhole the event horizon is located at a zero of the emblackening factor f r T2 T1 For example for Schwarzchild blackhole ds2Sch eve dr 2 f dt2 r 2 d 2 f 2 R2 dt2 dR2 r R 2 d 2 r2 r1 nt ho riz on Figure 1 Buoyancy force on a box R2 d 2 dR2 dT 2 dZ 2 dX 2 dY 2 where in the above we have used 4 f 1 rH r rH 2GM R r p r r 2GM 1 4GM t T R sinh and Z R cosh Notice that after the transformation we obtain the Minkowski space R3 1 In other words the Rindler space is a Minkowski space in the coordinate frame of a uniformly accelerating observer Graphically what we get is Fig 2 Notice that the two lines defined by r 2GM and r 2GM divide the space into four regions I II III IV The key observation is that region I is self contained i e region II and III can t communicate with it while information from region IV passes through the line r 2GM and hence corresponds to initial data Therefore at T 0 the Z 0 marked by L on the figure degree of freedom d o f are totally irrelevant and useless for region I Thus we should trace over them when computing stuff in region I Now Claim R trL g s ihg s where HR is the riddler Hamiltonian 5 HR Z 1 2 HR e Z Tab const T surface and Z trR e 2 HR 3 analogous to more water at the bottom of sea is called the Rindler time and plays the role of rapidity 5 Note that HR 1 since the riddler time 1 4 3 a nb 2G M II to singularity T r r 2GM I L to asymptotically flat space R III Z r fixed R M 2G fixed IV Figure 2 The geometry of the Rindler space Notice that the density matrix we obtained is not a pure state but is entangled with the L d o f hence S tr R ln R 6 0 This is a thermal density matrix with temperature Triddler 1 2 Proof of Claim Unruh 1976 Consider a scalar field in Rindler space A complete set of commuting d o f s at T 0 is R x y z for Z 0 x y z L x y z for Z 0 R and L commute because they are at spacelike separated points Any wavefunctional of the field can then be written as L R h L R i The ground state is just the Minkowski space vacuum The Feynman Kac formula …