The endpoint of the Gregory-Laﬂamme instability of black strings revisited Pau FiguerasTiago Françayand Chenxia Guz School of Mathematical Sciences Queen Mary University of London

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The endpoint of the Gregory-Laﬂamme instability of black strings revisited

Pau Figueras,∗Tiago França,†and Chenxia Gu‡

School of Mathematical Sciences, Queen Mary University of London

Mile End Road, London, E1 4NS, United Kingdom

Tomas Andrade§

Departament de Física Quàntica i Astrofísica, Institut de Ciències del Cosmos, Universitat de Barcelona,

Martí i Franquès 1, E-08028 Barcelona, Spain

We reproduce and extend the previous studies of Lehner and Pretorius of the endpoint of the

Gregory-Laﬂamme instability of black strings in ﬁve space-time dimensions. We consider unstable

black strings of ﬁxed thickness and diﬀerent lengths, and in all cases we conﬁrm that at the interme-

diate stages of the evolution the horizon can be interpreted as a quasistationary self-similar sequence

of black strings connecting spherical black holes on diﬀerent scales. However, we do not ﬁnd any

evidence for a global timescale relating subsequent generations. The endpoint of the instability is

the pinch oﬀ of the horizon in ﬁnite asymptotic time, thus conﬁrming the violation of the weak

cosmic censorship conjecture around black string spacetimes.

I. INTRODUCTION

General Relativity (GR) is the currently accepted

classical theory of gravity. Quite remarkably, so far it

has successfully passed all experimental tests, explaining

gravitational phenomena on an incredibly wide range

of scales, from Solar system scales to cosmology. Fur-

thermore, the detections of gravitational waves by the

LIGO/Virgo/KAGRA collaboration [1, 2] produced in

mergers of compact objects have allowed for new tests

of Einstein’s theory in the strong ﬁeld regime; such tests

should lead to new insights into fundamental aspects of

the theory.

Black holes play a very important role in our under-

standing of GR, and gravity in general, due to their

simplicity, which makes them tractable, and the fact

that they capture key aspects of the theory. One of

the distinguishing features of black holes is the presence

of singularities in their interior, where the description

provided by GR breaks down. Penrose’s famous sin-

gularity theorem [3] establishes that singularities in GR

can occur more generally, as one should expect in a non-

linear theory, and they need not be hidden inside black

holes. The occurrence of singularities limits the predic-

tivity of GR as a classical theory of gravity; to ensure

that GR retains its predictive power in the presence of

certain singularities, Penrose conjectured that the lat-

ter should generically be cloaked by horizons. This is

the Weak Cosmic Censorship Conjecture (WCCC) [4]

(see [5–7] for a modern and mathematically precise for-

mulation), and there are no known counter-examples

in four-dimensional astrophysical settings.1Proving or

disproving this conjecture remains one of the most im-

∗p.ﬁgueras@qmul.ac.uk

†t.e.franca@qmul.ac.uk

‡chenxia.gu@qmul.ac.uk

§tandrade@icc.ub.edu

1Choptuik’s critical collapse [8] is non-generic. In fact,

Christodoulou proved the WCCC for the Einstein-scalar ﬁeld

model in four dimensional asymptotically ﬂat spacetimes in

spherical symmetry [9].

portant open problems in mathematical relativity (see

[10] for recent progress).

While the Kerr black hole is believed to be stable

[11–15], and hence its relevance in astrophysics, higher

dimensional vacuum black holes exhibit much richer dy-

namics. Almost 30 years ago, Gregory and Laﬂamme

(GL) discovered that black strings (and black p-branes

in general) can be unstable to long wavelength pertur-

bations that break the translational symmetry along the

compact extra dimensions [16]. It is fair to say that the

study of the GL instability and its possible endpoints

together with the discovery of the asymptotically ﬂat

black ring in ﬁve dimensions [17] were largely responsi-

ble for the intrinsic interest in understanding the physics

of higher dimensional black holes regardless of string

theory (see [18] for a review). From the extensive work

done in this area over the years it has become clear that

the GL instability is very general and it basically aﬀects

any higher dimensional black hole which is suﬃciently

far from extremality whenever the horizon geometry is

characterized by widely separated length scales. The

latter happens for instance for rapidly spinning asymp-

totically ﬂat black holes [19–22] and black rings [23],

or in anti-de Sitter black holes [24, 25]. Therefore, the

GL instability can teach us very general aspects of the

physics and the possible phases of black holes and their

dynamics in a wide variety of settings.

The endpoint of the GL instability of black strings

was ﬁnally spelled out in a famous paper by Lehner and

Pretorius [26] (see [27] for a reivew), who used numeri-

cal relativity techniques to solve the Einstein equations

numerically in the highly dynamical and fully non-linear

regime. They found that the horizon evolves into a

sequence of spherical black holes joined by string seg-

ments; these string segments are themselves GL unsta-

ble, triggering a cascade of instabilities that give rise

to new generations of black holes and black strings on

ever smaller scales, eventually leading to the pinch oﬀ

of horizon somewhere along the string. In particular,

[26] argue that the dynamics of the GL instability is

arXiv:2210.13501v2 [hep-th] 14 Feb 2023

(globally) self-similar.2Using this observation they es-

timate that the pinch oﬀ time (as measured by asymp-

totic observers) is given by a geometric series and hence

ﬁnite. Since horizons cannot bifurcate in a smooth man-

ner (see e.g., Proposition 9.2.5 in [28]), [26] conclude

that a naked singularity must form at the pinch oﬀ. In-

deed, the simulations of [26] show that the spacetime

curvature invariants at the horizon of the black string

blow up as the system approaches the pinch oﬀ. Fur-

thermore, since no ﬁne tuning is required, one concludes

that the endpoint of the GL instability constitutes a vi-

olation of the WCCC in higher dimensional asymptoti-

cally Kaluza-Klein (KK) spaces. Following the original

work of [26], further studies of certain higher dimen-

sional asymptotically ﬂat rotating black holes and black

rings that also suﬀer from the GL instability have pro-

vided further support for this picture in ﬁnite number

of spacetime dimensions [29–32] and in the large Dlimit

of GR [33–36].

The seminal paper of [26] is more than ten years old

and, as important as it is for the current understand-

ing of the WCCC and its potential violations, it has not

been independently reproduced in the literature. There-

fore, the ﬁrst goal of the present article is to reproduce

the main results of [26], using completely independent

methods and a diﬀerent code: While [26] solve the Ein-

stein equations using harmonic coordinates and excision

with the PAMR/AMRD libraries,3here we solve the Ein-

stein equations using the CCZ4 formulation [37, 38] in

singularity avoiding coordinates and the GRChombo code

[39, 40]. The second goal of this paper is to extend the

results of [26] by evolving the system closer to the singu-

larity than ever before and by considering black strings

of diﬀerent lengths to obtain a more general picture of

the evolution and the endpoint of the GL instability of

black strings. While we reproduce the main result of

[26], namely that the GL unstable black string evolves

into a sequence of ever thinner strings connecting black

holes leading to a pinch oﬀ in ﬁnite asymptotic time,

we do not ﬁnd any evidence of a global timescale relat-

ing subsequent generations. This implies that the pinch

oﬀ time is not given by a geometric series; instead, the

local dynamics on the string segments plays an impor-

tant role beyond the third generation, leading to a faster

approach to the singularity. The distinct role that the

third generation plays here is due to our choice of initial

data (as well as in [26]). Furthermore, our simulations

indicate that the dynamics near the singularity seems to

be independent of macroscopic details of the string, sug-

gesting the existence of a universal local solution that

controls the pinch oﬀ, as in certain ﬂuids [41, 42].

The rest of this article is organised as follows: In

Section II we describe in detail the numerical methods

that we have used and our choice of initial conditions.

Section III contains the main results of the article. In

2This should not be confused with local self-similarity near the

singularity, which would manifest itself as a scaling solution

describing the approach to the singularity.

3http://laplace.physics.ubc.ca/Group/Software.html.

Section III A we describe the evolution of the apparent

horizon area; Section III B studies the dynamics of the

apparent horizon and Section III C contains the details

of the approach to the singularity. In Section IV we

summarize our main results and outline directions for

future research. Convergence tests are presented in Ap-

pendix A.

In this article we use the following conventions: G=

c= 1. Greek letters µ, ν, . . . denote spacetime indices

while Latin letters i, j, . . . denote indices on the spatial

hypersurfaces.

II. NUMERICAL METHODS

A. Evolution

We solve the Einstein vacuum equations in 4+1 di-

mensions in the CCZ4 formulation [37, 38] using the

GRChombo code [39, 40]. We use Cartesian coordinates

and impose SO(3) symmetry along the Minkowski di-

rections at the level of the equations of motion using

the modiﬁed cartoon method [43–45], thus reducing the

eﬀective dimensionality of the problem to 2+1. The di-

mensionally reduced equations of motion in the BSSN

formulation can be found in [45]; the generalization to

CCZ4 is straightforward.

To stably simulate black string spacetimes, we rede-

ﬁne the constraint damping parameter κ1→κ1/α as

in [38], where αis the lapse function. In the results re-

ported in Section III, we used κ1= 0.37 and κ2=−0.8.

We use 6th order ﬁnite diﬀerences to discretise the spa-

tial derivatives and a standard RK4 time integrator to

step forward in time. Since the overall convergence or-

der cannot be higher than four, we use 6th order Kreiss-

Oliger dissipation. In Appendix A we show that the

order of convergence that we achieve is roughly three,

as expected in a typical AMR code as GRChombo. As

in similar settings [29–31], to control the gradients near

the coordinate singularity present in the computational

domain inside black holes, we add diﬀusion terms well

inside the apparent horizon (AH) to the right hand side

of the equations of motion for those variables that ap-

pear with second order spatial derivatives. We place

the outer boundary along the Minkowski directions at

Louter = 256r0, where r0is the mass parameter of the

black string, see equation (3) in Section II B. At the

outer boundary x=Louter we impose either Sommer-

feld or periodic boundary conditions, while the direction

along the string, z, is periodic with period L.4The grid

spacing in the coarsest level typically is dx = 0.25 r0

and we add another 12 levels of reﬁnement (so 13 levels

in total) with a reﬁnement ratio of 2:1.

We evolve the lapse αwith the standard 1+log slicing

4Since Louter is not in causal contact with the black string for

the entire duration of our simulations (see Section III), the par-

ticular choice of boundary conditions at x=Louter makes no

diﬀerence in practice.

condition,

(∂t−βi∂i)α=cαα(K−2 Θ) ,(1)

with cα= 1.3. Here Kis the trace of the extrinsic

curvature of the spatial slices and Θis another of the

CCZ4 evolution variables. We evolve the shift vector βi

with the integrated Gamma-driver,

(∂t−βj∂j)βi=cβˆ

Γi−η βi,(2)

where ˆ

Γiis the usual CCZ4 evolution variable and

cβ= 0.6; these choices of gauge parameters have proven

to work well in numerical simulations of higher dimen-

sional black hole spacetimes [29–31, 44, 46].5Notice

that unlike [29–31], we have not included an extra term

in (2) corresponding to the contracted Christoﬀel sym-

bols of the (conformally rescaled) initial spatial metric.

The reason is that such term vanishes for our choice of

initial conditions, see Section II B.

B. Initial data

We start with an unperturbed 5Dblack string written

in Gullstrand-Painlevé coordinates [47, 48],

ds2=−1−r0

rdt2+2rr0

rdt dr+dr2+r2dΩ2

(2) +dz2,

(3)

where r0is the usual Schwarzschild mass parameter,

z∼z+Lis the KK compact direction and dΩ2

(2) is

the standard metric on the unit round two-sphere. The

ADM mass of (3) is

M=1

2L r0.(4)

The advantage of using these coordinates is that they

are horizon-penetrating while the metric on the spatial

sections is ﬂat. The latter makes it particularly conve-

nient for constructing perturbed initial data that mini-

mizes the amount of initial constraint violations, as we

explain below. We regularize the physical singularity

present in the initial data slice using the “turduckening"

approach [49, 50] and cutting oﬀ by hand the range of

the radial coordinate. In terms of the radial cartoon

coordinate x, if x<ε, then we evaluate the initial data

quantities derived from (3) at x=ε; for the results

presented below, we typically use ε= 0.1r0.6

From the initial data (3), we read oﬀ the 4+1quan-

tities, noting that in Cartesian coordinates det γ= 1

and hence the unperturbed conformal factor satisﬁes

5Note that the values of the gauge parameters cαand cβthat

we use in (1) and (2) diﬀer from the typical values used in

black holes binary mergers in astrophysical scenarios, which are

cα= 2 and cβ= 0.75.

6We should emphasize that we only employ this regularization

procedure at the level of the initial data; at the later stages in

the evolution, the xcoordinate takes values in 0<x<xmax.

χ≡(det γ)−1

4= 1. Furthermore, the Christoﬀel sym-

bols associated to the spatial conformal metric ˜γij triv-

ially vanish. We introduce a constraint violating pertur-

bation on the conformal factor χthat triggers the GL

instability:

χ= 1 + sin 2πnz

Le−x

r0−r0

x2

,(5)

where is the amplitude of the perturbation, n∈Nse-

lects the GL harmonic to be excited and the exponential

factor in (5) ensures that the perturbation is localized

near the horizon. Therefore, our perturbation (5) does

not change the ADM mass (4) of the spacetime. For the

results reported in Section III, we chose r0= 1,= 0.01

and n= 1. We keep track of the constraint violations

introduced by our perturbation and we verify that for

.0.01 they are exponentially suppressed by the damp-

ing terms in the CCZ4 equations on a timescale that is

much faster than any other timescale in the problem.

The remaining 4 + 1 quantities are left unperturbed; for

the initial lapse αand shift vector βi, we set α= 1 and

βi= 0.7

C. Grid hierarchy and AMR

The location along the string where the ﬁrst genera-

tion forms is sensitive to the initial conditions, see Sec-

tion III. Beyond this point and for initial data with zero

total momentum, as in our case and in [26], the evo-

lution should respect the Z2reﬂection symmetry about

the centre of the ﬁrst generation string segment. The

reason is that the n= 1 GL harmonic, which is the one

that governs the subsequent universal evolution of the

strings with the thicknesses that we have considered,

has this symmetry. However, truncation errors due to

the asymmetry of the various levels of reﬁnement that

are automatically added as the evolution of the insta-

bility unfolds can break this physical symmetry of the

problem. This eﬀect is visible from the third generation

onwards in the animation of the evolution of the insta-

bility produced by [26] and in Fig. 2 of this reference.

In this subsection we provide details of the grid hierar-

chy that we used in our simulations that ensures that

our numerical method respects this Z2symmetry; we

emphasise that we did not explicitly enforce this sym-

metry in our simulations, so the fact that it is respected

is a sign that the truncation errors do not interfere with

the physics. The reader interested in the results of our

simulations can safely skip the rest of this subsection.

In similar problems that some of us addressed in the

past using GRChombo, e.g., [29–32], a reﬁnement crite-

rion based on the gradients of the conformal factor χ

7This choice of lapse and shift implies that initially there is a

period of strong gauge dynamics superposed with the physical

evolution of the GL instability. This period of gauge adjustment

typically lasts t/r0∼10, whilst with our perturbation the onset

of the GL instability occurs at t/r0∼70 −100 or even later,

see Fig. 2.

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TheendpointoftheGregory-LaammeinstabilityofblackstringsrevisitedPauFigueras,TiagoFrança,yandChenxiaGuzSchoolofMathematicalSciences,QueenMaryUniversityofLondonMileEndRoad,London,E14NS,UnitedKingdomTomasAndradexDepartamentdeFísicaQuànticaiAstrofísica,InstitutdeCiènciesdelCosmos,UniversitatdeBarcelon...

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The endpoint of the Gregory-Laﬂamme instability of black strings revisited Pau FiguerasTiago Françayand Chenxia Guz School of Mathematical Sciences Queen Mary University of London

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