One Step Forward and One Step Back (In Understanding Quantum Black Holes) Gary Horowitz UC Santa Barbara Communications in Commun. Math. Phys. 88, 295-308 (1983) Mathematical Physics © Springer-Verlag 1983

Positive Mass Theorems for Black Holes G. W. Gibbons1, S. W. Hawking1, Gary T. Horowitz2'*, 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.

1. Introduction The gravitational potential energy of any system is always negative because gravity is an attractive force. In Newtonian theory one can shrink any system to an arbitrarily small size and make the total energy indefinitely negative. However it appears that one cannot do this according to the general theory of relativity. As one considers smaller and smaller configurations for the system, the potential energy becomes more negative but the total energy, i.e. the rest mass plus potential energy plus kinetic energy seems to remain positive. At a certain critical size an outer future trapped surface appears [1, 2]. This is a closed spacelike 2-surface which is in such a strong gravitational field that the outgoing future directed light rays or null geodesies orthogonal to it are converging, i.e. they are being dragged back by the gravitational field. The outer boundary of the region on a spacelike hypersurface which contains outer future trapped surfaces is called the future apparent horizon [2, 3]. By the singularity theorems (see [2]) the system must collapse to produce a spacetime singularity provided that certain physically reasonable conditions hold. According to the unproved but very plausible cosmic

* Albert Einstein Fellow ** Supported by National Science Foundation Grant PHY 80-19754 First the good news:

One step forward

(based on work with N. Engelhardt, 1509.07509 and 1605.04335)

Holography

Early movaon came from the Bekenstein- Hawking entropy of black holes:

SBH = A / 4

Holography is not just a property of black holes, but should be a general property of quantum gravity (‘t Hoo and Susskind):

Everything that happens in a region of space can be described by degrees of freedom living on the boundary. Gauge/gravity duality (Maldacena; Gubser, Klebanov, Polyakov; Wien)

With an-de Sier boundary condions, string theory (which includes gravity) is completely equivalent to a (nongravitaonal) gauge theory living on the boundary at infinity. When string theory is weakly coupled, gauge theory is strongly coupled, and vice versa. A powerful feature of gauge/gravity duality is that statements that are easy to establish on one side oen imply highly nontrivial results about the dual theory.

For example, the fact that black hole evaporaon must be unitary follows immediately from unitary evoluon of the dual gauge theory. Cosmic censorship

Classical GR conjecture: Generic, asymptocally flat, inial data has a maximal evoluon that contains a complete null infinity.

i.e. this can’t happen. If cosmic censorship fails, it was hoped that Part of I+ quantum gravity would resolve the singularity Inial data so evoluon connues. In holography we know that this is true! Regardless of what happens in a localized region in the interior, evoluon in the QFT on the boundary connues.

? QG QFT When can two QFTs communicate?

Usually, two QFTs on separate spacemes cannot send signals to one another

QFT1 QFT2 Two copies of a CFT on Minkowski space can be mapped either to one stac cylinder or two. We will consider CFTs on Sn x R where this problem doesn’t arise:

Two CFT’s on Sn x R cannot be conformally mapped into a single larger spaceme, since Sn x R is conformally maximally extended.

No Transmission Principle (NTP): If two CFTs on Sn x R have gravity duals, then no signals can be transmied between their bulk duals. No evoluon through black holes

Could quantum gravity resolve the singularity and allow signals CFT2 to emerge in another asymptocally AdS spaceme?

CFT1 No evoluon through black holes

Could quantum gravity resolve the singularity and allow signals CFT2 to emerge in another asymptocally AdS spaceme?

No. This would violate the NTP. CFT1 A charged (or rotang) AdS black hole seems to violate CFT2 the NTP even classically.

But the inner horizon is known to be unstable. Signals in n n zo er ri h o o cannot get through classically. h ri r zo ne n in o u n te zo r ri h o NTP implies that signals o h r r iz e o ut n o CFT1 cannot get through even in full quantum gravity. Applicaon to singular CFTs

Some CFTs cannot be evolved past a certain me.

If evoluon on the boundary stops, then evoluon in the bulk must stop as well.

There must be a cosmological singularity classically, which quantum gravity cannot resolve into a bounce. Comments

1) One cannot avoid our conclusions by adding couplings between the CFTs associated with different asymptoc regions, since that would violate causality.

2) One could make up a rule to idenfy a state in CFT1 with one in CFT2, 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 idenfy the states. Now the bad news:

One step back

(based on work with Kunduri and Lucie, 1704.04071) Jan. 1996: Strominger and Vafa reproduce the Bekenstein-Hawking entropy of a stac, extremal 5D black hole by counng microstates of string theory.

Feb. 1996: This is generalized to a rotang 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. Recently, Kunduri and Lucie found a new family of extreme rotang black holes in 5D supergravity.

Like BMPV, it is asymptocally flat and can have the same asymptoc charges.

Unlike BMPV, there is nontrivial topology outside the horizon.

These new black holes can have greater entropy than BMPV.

Puzzle

The 1996 microstate counng was thought to include all bound states of strings and branes with the given total charges.

Why did the original counng of microstates agree with the BMPV black hole entropy?

There are other examples of soluons with greater entropy than BMPV, but they either involve more than one horizon or are not asymptocally flat.

These are the first examples of asymptocally flat, single horizon black holes with the same charges but greater entropy than BMPV. The new black hole soluons

5D minimal supergravity acon:

5 mn 2 mnpqr S = d x p g(R FmnF ) ✏ AmFnpFqr 3p3 Z  There is a large class of staonary, nonsingular soluons called “bubbling geometries”. They

1) Have nontrivial topology 2) Are supersymmetric 3) Determined by harmonic funcons

(Likely to be nonlinearly unstable: Eperon, Reall, Santos) One can add a spherical extremal black hole to these geometries keeping nontrivial topology outside.

Consider the simplest case with one nontrivial S2.

2 Parameters: Q, J1, J2, q (flux through S ) M is determined by Q.

BMPV BH has J1 = J2 so we impose this also. To remove scaling symmetry, we work with

η = |J |/Q3/2, ν = |q|/Q1/2, a = A/Q3/2 1 H BMPV extremal limit is η = 1. The allowed range of these parameters is inside the triangular region: p 1.05

aH = 0 η 1.00

0.95 Regularity at 0.90 aH = 0 “centers”

0.85

0.26 0.28 0.30 0.32 0.34 ν

The point p corresponds to a smooth bubbling geometry. The allowed range of these parameters is inside the triangular region: p 1.05

aH = 0 η 1.00

0.95 Regularity at 0.90 aH = 0 “centers”

0.85

0.26 0.28 0.30 0.32 0.34 ν

Note that this includes angular momentum larger than the extremal limit for BMPV. 2 1/2 For BMPV: aH = (1-η )

The horizon area for BMPV (blue) and new black holes (yellow): η

aH

ν There is a simple gravitaonal explanaon for both exceeding the BMPV extremal limit and having greater entropy:

SBMPV = 0 in extremal limit. The new soluons have structure outside the horizon which can carry angular momentum. So when J approaches Jmax 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. Where are the microstates of these black holes?

Most of the counng of microstates in the 1990’s was based on a certain configuraon of branes and strings in flat space.

There must be more complicated bound states of branes and strings that have greater entropy. 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 counng microstates in string theory. 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 applicaons will be considered this fall, for a posion to start in Fall 2018. Preference will be given for applicants working in quantum gravity.

See UCSB Physics Department website (or contact me) for details in September.