One Step Forward and One Step Back (In Understanding Quantum Black Holes) Gary Horowitz UC Santa Barbara

Transcription

1 One Step Forward and One Step Back (In Understanding Quantum Black Holes) Gary Horowitz UC Santa Barbara

2 Commun. Math. Phys. 88, (1983) Communications in Mathematical Physics Springer-Verlag 1983 Positive Mass Theorems for Black Holes G. W. Gibbons 1, S. W. Hawking 1, Gary T. Horowitz 2 '*, and Malcolm J. Perry 3 '** 1 D.A.M.TP, University of Cambridge, Silver Street, Cambridge CB39EW, England 2 Institute for Advanced Study, Princeton, NJ 08540, USA 3 Dept. of Physics, Princeton University, NJ 08544, USA Abstract. We extend Witten's proof of the positive mass theorem at spacelike infinity to show that the mass is positive for initial data on an asymptotically flat spatial hypersurface which is regular outside an apparent horizon H. In addition, we prove that if a black hole has electromagnetic charge, then the mass is greater than the modulus of the charge. These results are also valid for the Bondi mass at null infinity. Finally, in the case of the Einstein equation with a negative cosmological constant, we show that a suitably defined mass is positive for data on an asymptotically anti-de Sitter surface which is regular outside an apparent horizon.

3 First the good news: One step forward (based on work with N. Engelhardt, and )

4 Holography Early moqvaqon came from the Bekenstein- Hawking entropy of black holes: S BH = A / 4 Holography is not just a property of black holes, but should be a general property of quantum gravity ( t HooZ and Susskind): Everything that happens in a region of space can be described by degrees of freedom living on the boundary.

5 Gauge/gravity duality (Maldacena; Gubser, Klebanov, Polyakov; Wi\en) With anq- de Si\er boundary condiqons, string theory (which includes gravity) is completely equivalent to a (nongravitaqonal) gauge theory living on the boundary at infinity. When string theory is weakly coupled, gauge theory is strongly coupled, and vice versa.

6 A powerful feature of gauge/gravity duality is that statements that are easy to establish on one side ozen imply highly nontrivial results about the dual theory. For example, the fact that black hole evaporaqon must be unitary follows immediately from unitary evoluqon of the dual gauge theory.

7 Cosmic censorship Classical GR conjecture: Generic, asymptoqcally flat, iniqal data has a maximal evoluqon that contains a complete null infinity. Part of I + i.e. this can t happen. IniQal data If cosmic censorship fails, it was hoped that quantum gravity would resolve the singularity so evoluqon conqnues.

8 In holography we know that this is true! Regardless of what happens in a localized region in the interior, evoluqon in the QFT on the boundary conqnues.? QG QFT

9 When can two QFTs communicate? Usually, two QFTs on separate spaceqmes cannot send signals to one another QFT 1 QFT 2

10 Two copies of a CFT on Minkowski space can be mapped either to one staqc cylinder or two.

11 We will consider CFTs on S n x R where this problem doesn t arise: Two CFT s on S n x R cannot be conformally mapped into a single larger spaceqme, since S n x R is conformally maximally extended. No Transmission Principle (NTP): If two CFTs on S n x R have gravity duals, then no signals can be transmi\ed between their bulk duals.

12 No evoluqon through black holes CFT2 Could quantum gravity resolve the singularity and allow signals to emerge in another asymptoqcally AdS spaceqme? CFT1

13 No evoluqon through black holes CFT2 CFT1 Could quantum gravity resolve the singularity and allow signals to emerge in another asymptoqcally AdS spaceqme? No. This would violate the NTP.

14 CFT2 A charged (or rotaqng) AdS black hole seems to violate the NTP even classically. inner horizon outer horizon inner horizon outer horizon CFT1 But the inner horizon is known to be unstable. Signals cannot get through classically. NTP implies that signals cannot get through even in full quantum gravity.

15 ApplicaQon to singular CFTs Some CFTs cannot be evolved past a certain Qme. If evoluqon on the boundary stops, then evoluqon in the bulk must stop as well. There must be a cosmological singularity classically, which quantum gravity cannot resolve into a bounce.

16 Comments 1) One cannot avoid our conclusions by adding couplings between the CFTs associated with different asymptoqc regions, since that would violate causality. 2) One could make up a rule to idenqfy a state in CFT 1 with one in CFT 2, but it would be extra input not contained in the original CFTs bulk and boundary theories would not be equivalent. 3) There is no natural way to idenqfy the states.

17 Now the bad news: One step back (based on work with Kunduri and Lucien, )

18 Jan. 1996: Strominger and Vafa reproduce the Bekenstein- Hawking entropy of a staqc, extremal 5D black hole by counqng microstates of string theory. Feb. 1996: This is generalized to a rotaqng 5D extremal black hole by Brekenridge, Myers, Peet, and Vafa (BMPV). Within months, this is generalized to near extremal black holes in both 4 and 5 dimensions.

19 Recently, Kunduri and Lucien found a new family of extreme rotaqng black holes in 5D supergravity. Like BMPV, it is asymptoqcally flat and can have the same asymptoqc charges. Unlike BMPV, there is nontrivial topology outside the horizon. These new black holes can have greater entropy than BMPV.

20 Puzzle The 1996 microstate counqng was thought to include all bound states of strings and branes with the given total charges. Why did the original counqng of microstates agree with the BMPV black hole entropy?

21 There are other examples of soluqons with greater entropy than BMPV, but they either involve more than one horizon or are not asymptoqcally flat. These are the first examples of asymptoqcally flat, single horizon black holes with the same charges but greater entropy than BMPV.

22 The new black hole soluqons 5D minimal supergravity acqon: Z apple p S = d 5 x g(r Fmn F mn ) 2 3 p 3 mnpqr A m F np F qr There is a large class of staqonary, nonsingular soluqons called bubbling geometries. They 1) Have nontrivial topology 2) Are supersymmetric 3) Determined by harmonic funcqons (Likely to be nonlinearly unstable: Eperon, Reall, Santos)

23 One can add a spherical extremal black hole to these geometries keeping nontrivial topology outside. Consider the simplest case with one nontrivial S 2. Parameters: Q, J 1, J 2, q (flux through S 2 ) M is determined by Q. BMPV BH has J 1 = J 2 so we impose this also. To remove scaling symmetry, we work with η = J 1 /Q 3/2, ν = q /Q 1/2, a H = A/Q 3/2 BMPV extremal limit is η = 1.

24 The allowed range of these parameters is inside the triangular region: η p a H = a H = 0 Regularity at centers ν The point p corresponds to a smooth bubbling geometry.

25 The allowed range of these parameters is inside the triangular region: η p a H = a H = 0 Regularity at centers ν Note that this includes angular momentum larger than the extremal limit for BMPV.

26 For BMPV: a H = (1- η 2 ) 1/2 The horizon area for BMPV (blue) and new black holes (yellow): η a H ν

27 There is a simple gravitaqonal explanaqon for both exceeding the BMPV extremal limit and having greater entropy: S BMPV = 0 in extremal limit. The new soluqons have structure outside the horizon which can carry angular momentum. So when J approaches J max for BMPV, the angular momentum carried by the black hole is less than this. So the entropy remains nonzero and larger angular momentum is possible.

28 Where are the microstates of these black holes? Most of the counqng of microstates in the 1990 s was based on a certain configuraqon of branes and strings in flat space. There must be more complicated bound states of branes and strings that have greater entropy.

29 Quantum BH Summary One step forward: Gauge/gravity duality implies that quantum gravity cannot resolve the singularity inside black holes. One step back: There are gaps in our understanding of the entropy of certain extremal black holes in terms of counqng microstates in string theory.

30 Due to the generosity of an anonymous donor, we have a new postdoctoral fellowship series at University of California, Santa Barbara called Fundamental Physics Fellows First applicaqons will be considered this fall, for a posiqon to start in Fall Preference will be given for applicants working in quantum gravity. See UCSB Physics Department website (or contact me) for details in September.

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