Almost all extremal black holes in AdS are singular Gary T. HorowitzaMaciej KolanowskibJorge E. Santosc_2

2025-04-30 0 0 781.66KB 43 页 10玖币

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Almost all extremal black holes in AdS are

singular

Gary T. Horowitz,aMaciej Kolanowski,bJorge E. Santosc

aDepartment of Physics, University of California at Santa Barbara, Santa Barbara, CA

93106, U.S.A.

bInstitute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5,

02-093 Warsaw, Poland

cDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge,

Wilberforce Road, Cambridge, CB3 0WA, UK

E-mail: horowitz@ucsb.edu,maciej.kolanowski@fuw.edu.pl,

jss55@cam.ac.uk

Abstract: We investigate the geometry near the horizon of a generic, four-dimensional

extremal black hole. When the cosmological constant is negative, we show that (in

almost all cases ) tidal forces diverge as one crosses the horizon, and this singularity

is stronger for larger black holes. In particular, this applies to generic nonspherical

black holes, such as those satisfying inhomogeneous boundary conditions. Nevertheless,

all scalar curvature invariants remain ﬁnite. Moreover, we show that nonextremal

black holes have tidal forces that diverge in the extremal limit. Holographically, this

singularity is reﬂected in anomalous scaling of the speciﬁc heat with temperature.

Similar (albeit weaker) eﬀects are present when the cosmological constant is positive,

but not when it vanishes.

arXiv:2210.02473v3 [hep-th] 28 Dec 2022

Contents

1 Introduction 1

2 Simple example 4

3 Einstein–Maxwell: linear theory 6

3.1 General equations 6

3.2 Reissner–Nordstr¨om–AdS 8

3.2.1 Spherical black holes 10

3.2.2 Toroidal black holes 12

3.2.3 Hyperbolic black holes 13

3.3 Kerr–AdS 15

4 Einstein-Maxwell: nonlinear results 17

4.1 Scaling argument 18

4.2 Numerical scheme 20

4.3 Results 23

5 Nonlinear scalar ﬁeld model 25

5.1 Zero temperature results 28

5.2 Non-zero temperature results 29

6 Discussion 33

A Anomalous scaling of the speciﬁc heat 35

1 Introduction

In four-dimensional general relativity, asymptotically ﬂat, stationary black holes have

extremal limits with smooth horizons. This follows from the black hole uniqueness

theorems and the known properties of the Reissner-Nordstr¨om and Kerr solutions.

Over the years, various examples have been found showing that this is not always the

case. A mild lack of smoothness (where the metric is C2but not C3) was ﬁrst noticed

in the static multi-black hole solutions to D“5 Einstein-Maxwell theory [1]. This

became more serious with the discovery that in Dą5, static multi-black hole solutions

– 1 –

have curvature singularities on the horizon [2]. These were null singularities in which

tidal forces on infalling observers diverge, but all curvature scalars remain ﬁnite.

Similar null singularities were also seen in the extremal limit of some black holes

in anti-de Sitter (AdS) space. This includes solutions with less symmetry [3–5], non-

supersymmetric attractor ﬂows [6] and even in some supersymmetric black holes [7].

A natural question to ask is how common are these singular extremal solutions? Are

they exceptional special cases, or indicating a more general phenomenon?

We will show that in AdS they are very common. In fact, almost all extremal black

holes are singular. This is true even in four dimensions (and becomes worse in higher

dimensions). We will focus on four dimensional solutions of Einstein-Maxwell theory

with Λ ă0. The higher dimensional case will be discussed elsewhere [8]. There are

many more stationary black holes in AdS than in asymptotically ﬂat spacetime since

one has the freedom to choose boundary conditions for the metric and vector potential

at inﬁnity. In particular, static nonspherical charged black holes exist, but we will

show they are generically singular. Our results apply whenever rotational symmetry is

broken, so for example, if one puts a cage around a static AdS black hole it becomes

singular.

If the horizon was smooth, it is known that in the extremal limit, the only possible

static near horizon geometry is AdS2ˆS2[9] so the horizon itself remains spherical.

We will see that in four dimensions, even when the horizon becomes singular, a well

deﬁned near horizon geometry exists and remains AdS2ˆS2. Intuitively, this is because

the radial distance (along a static hypersurface) from the horizon to any point outside

is inﬁnite and thus any nonspherical perturbation1should decay before reaching the

horizon. But the key point is how quickly do they decay. The symmetry of AdS2ensures

that all perturbations should have power law behavior near an extremal horizon. If the

exponent is not an integer, the solution is not C8, and if the exponent is too small,

the curvature will diverge. We will show that for AdS black holes with topology S2,

an `“2 perturbation always falls oﬀ slowly enough to produce a singularity on the

horizon. So generic extremal black holes with S2topology are singular. This singularity

is null, and all curvature scalars remain ﬁnite. However, the tidal forces on infalling

particles diverge.

As one increases the charge, this singularity becomes stronger and higher `modes

1One should note that in this work a “perturbation” does not mean any dynamical change, but rather

a change in boundary conditions for the elliptic problem of ﬁnding static black holes. In particular,

our work is diﬀerent from the Aretakis instability of extremal black holes [10] which results from time

dependent perturbations (although both eﬀects originate from the symmetries of the near horizon

AdS2factor).

– 2 –

also become singular. Similar results hold for static black holes of diﬀerent topolo-

gies (with the exception of small toroidal ones) and for Kerr-AdS. In fact, for large

hyperbolic black holes, the singularity is so strong that some perturbations diverge at

the horizon. Thus, we see that almost all extremal black holes in AdS are singular.

Smoothness of the known exact solutions is an artifact of the symmetry rather than a

basic physical feature. Solutions with extremal AdS black holes in nonspherical back-

grounds have been constructed before [11]. Although it was not noticed at the time,

the current analysis shows that these “hovering” black holes also have diverging tidal

forces on their horizon.

A natural question is whether our assumption that the cosmological constant is

negative is needed at all. Even when Λ ě0, if the horizon was smooth, the only static

near horizon extremal black hole geometry would be AdS2ˆS2, so all perturbations

must fall oﬀ like a power law. However when Λ “0, one ﬁnds that all exponents are

positive integers and so the metric is indeed smooth. In this case, since there is no

other scale and the exponents are dimensionless, they cannot depend on the charge.

They turn out to be integers in four dimensions, but not in higher dimensions. When

Λ is positive, the exponents are no longer integers and small black holes are singular.

The program of systematic investigations of the spacetime near extremal horizons

was ﬁrst proposed in [12] and was continued in [13–16]. Unfortunately, the starting

point of that analysis was the Taylor expansion in the distance from the horizon. This

clearly assumes smoothness and is generically not allowed. Thus, one should rather see

these results (at least with Λ ‰0) as a search for very special, ﬁne-tuned solutions. This

clariﬁes the conclusion of [16] where it was shown that the transversal deformation of

the extremal Reissner–Nordstr¨om-(A)dS horizon are spherically symmetric unless the

charge takes a special value (depending on the cosmological constant).

If one considers a nonextremal black hole with temperature T, these singularities

are always removed. Thus, it is tempting to simply ignore them as an artifact of TÑ0

limit. However, as we will show, even in this case tidal forces at the horizon grow as an

inverse power of the temperature and diverge in the limit. Thus even a tiny, symmetry

breaking perturbation at inﬁnity becomes arbitrarily large near the horizon as we lower

T. This large curvature may lead to quantum corrections near the horizon, but we do

not currently understand the form of these corrections. In a holographic theory, we

will show that there is a clear signal of the singularity for large black holes: the speciﬁc

heat (and other quantities) has anomalous scaling with Tnear T“0.2

The reason these singularities exist and some of their properties can already be

seen by looking at a massless scalar ﬁeld in an extremal black hole background. So we

2We thank Sean Hartnoll for suggesting this might occur.

– 3 –

start by discussing this simple example in the next section. In Sec. 3, we begin our

main analysis of Einstein-Maxwell solutions, by studying linearized gravitational and

electromagnetic perturbations of the near horizon geometry of extremal black holes.

Sec. 4 contains a discussion of the full nonlinear story, and shows that the singularities

indicated by the linearized analysis indeed arise in the full solutions as TÑ0. To

see the anomalous scaling of the speciﬁc heat, one needs to go to very low T, which is

diﬃcult to reach in the Einstein-Maxwell theory. So in Sec. 5 we introduce a simpler

theory in which this eﬀect can be clearly demonstrated. We conclude in Sec. 6 with a

brief discussion.

2 Simple example

Before we get into the technical details, let us consider a very simple toy model which

will illustrate the main ideas. We will consider a massless scalar ﬁeld on an extremal

Reissner–Nordstr¨om-AdS (RN ASdS) black hole. Recall that the RN AdS metric is

ds2“ ´fprqdt2`dr2

fprq`r2dΩ2(2.1)

where dΩ2is the line element on a unit radius round two-sphere,

fprq “ r2

L2`1´2M

r`Q2

r2(2.2)

and Lis the AdS radius. In the extremal limit, the horizon is at

r`“d2Q2

1`a1`12Q2{L2(2.3)

and

f2pr`q “ 6

L2`2Q2

(2.4)

We now perturb this spacetime by adding a static, massless scalar ﬁeld φ. Since the

background is spherically symmetric, we may expand φinto the spherical harmonics:

φ“ÿ

`,m

φ`mY`m.(2.5)

Then, the Klein–Gordon equation reads

pfφ1

`mq1`2fφ1

r´`p``1q

r2φ`m “0 (2.6)

– 4 –

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摘要：

AlmostallextremalblackholesinAdSaresingularGaryT.Horowitz,aMaciejKolanowski,bJorgeE.SantoscaDepartmentofPhysics,UniversityofCaliforniaatSantaBarbara,SantaBarbara,CA93106,U.S.A.bInstituteofTheoreticalPhysics,FacultyofPhysics,UniversityofWarsaw,Pasteura5,02-093Warsaw,PolandcDepartmentofAppliedMathemat...

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Almost all extremal black holes in AdS are singular Gary T. HorowitzaMaciej KolanowskibJorge E. Santosc_2

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