![Onset of the Gregory-Laflamme instability ¯ k GL = k GL r 0 as a function of n as predicted by different methods for β = 0: first order blackfold approach (black dashed line), second order blackfold approach (black solid line), fourth order large D approach [38] (blue solid line) and numerical points of [37] (orange squares).](https://www.researchgate.net/profile/Jay-Armas/publication/330700311/figure/fig1/AS:720254674141186@1548733504428/Onset-of-the-Gregory-Laflamme-instability-k-GL-k-GL-r-0-as-a-function-of-n-as_Q320.jpg)
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Instabilities of Thin Black Rings:
Closing the Gap
Jay Armas1,and Enrico Parisini1,2
1Institute for Theoretical Physics, University of Amsterdam,
1090 GL Amsterdam, The Netherlands
Dutch Institute for Emergent Phenomena, The Netherlands
2Dipartimento di Fisica e Astronomia, Università di Bologna,
Via Irnerio 46, 40126 Bologna, Italy
j.armas@uva.nl ,enrico.parisini@studio.unibo.it
Abstract
We initiate the study of dynamical instabilities of higher-dimensional black holes using
the blackfold approach, focusing on asymptotically flat boosted black strings and singly-
spinning black rings in
D≥
5. We derive novel analytic expressions for the growth rate
of the Gregory-Laflamme instability for boosted black strings and its onset for arbitrary
boost parameter. In the case of black rings, we study their stability properties in the region
of parameter space that has so far remained inaccessible to numerical approaches. In
particular, we show that very thin (ultraspinning) black rings exhibit a Gregory-Laflamme
instability, giving strong evidence that black rings are unstable in the entire range of
parameter space. For very thin rings, we show that the growth rate of the instability
increases with increasing non-axisymmetric mode
m
while for thicker rings, there is
competition between the different modes. However, up to second order in the blackfold
approximation, we do not observe an elastic instability, in particular for large modes
m
1,
where this approximation has higher accuracy. This suggests that the Gregory-Laflamme
instability is the dominant instability for very thin black rings. Additionally, we find a
long-lived mode that describes a wiggly time-dependent deformation of a black ring. We
comment on disagreements between our results and corresponding ones obtained from a
large Danalysis of black ring instabilities.
arXiv:1901.09369v3 [hep-th] 11 May 2019
Contents
1 Introduction 1
2 Blackfold equations and linearised perturbations 3
2.1 Blackfoldequations.................................. 4
2.2 Linearisedperturbations............................... 5
3 Instabilities of boosted black strings 7
3.1 Idealordermodes................................... 7
3.2 Firstordermodes................................... 8
3.3 Second order modes and comparison with the large Danalysis . . . . . . . . . . 10
4 Instabilities of black rings 13
4.1 Idealordermodes................................... 13
4.2 First order modes and comparison with large Danalysis ............. 15
4.3 Secondordermodes ................................. 17
5 Discussion 19
A Stress tensor and bending moment of perturbed black branes 21
B Linearised equations at second order 24
C Details on hydrodynamic and elastic modes 25
1 Introduction
Black holes in spacetime dimensions
D≥
5can exhibit different types of instabilities compared
to their four dimensional counterparts. One of these instabilities is the Gregory-Laflamme
instability originally found in the context of perturbations of asymptotically flat black
p
-branes
[
1
]. This type of instability was later found to be present in the context of Myers-Perry black
holes [
2
–
5
] and in the case of five-dimensional black rings [
6
]. In fact, according to the arguments
of [
7
] (see also [
8
] for the case of black rings), any neutral black hole solution that admits a
blackfold limit (i.e. an ultraspinning limit) is expected to suffer from a Gregory-Laflamme
instability.
Non-axisymmetric instabilities, such as bar-mode instabilities [
4
,
9
–
11
], are an additional
feature of higher-dimensional rotating black holes. Recently, it was found that a type of
non-axisymmetric instability - the elastic instability - is also present in five-dimensional black
rings with horizon topology
S1×S2
[
12
]. This instability is related to transverse deformations
of the radius
R
of the
S1
that do not significantly affect the size of the radius
r0
of the
S2
.
Studies of the end point of these instabilities suggest a violation of the weak cosmic censorship
conjecture [
11
–
13
]. It is thus important to study these instabilities in more generality and
in particular by means of analytic methods that can probe regimes of parameter space that
numerical methods cannot reach with acceptable accuracy.
1
Besides having proved to be extremely useful in finding new black hole solutions [
14
–
18
]
in asymptotically flat space, we demonstrate here that the blackfold approach [
7
,
19
] is a
powerful tool for studying hydrodynamic (i.e. Gregory-Laflamme) and elastic instabilities of
higher-dimensional black holes in the ultraspinning regime and away from it.
1
In this context,
one first finds a stationary solution, modelled as a fluid confined to a surface, corresponding to
the black hole solution whose stability one wishes to study. The fundamental fluid variables
and the geometric properties of the surface describing the equilibrium configuration of the fluid
are subsequently perturbed and the stability properties of black holes are found by studying
the propagation of hydrodynamic and elastic modes.
Black rings can be classified as thin 0
≤ν <
1
/
2or as fat 1
/
2
≤ν <
1where for very thin
rings
ν
=
r0/R
is a measure of the ring thickness. Studying Penrose inequalities, the fat
branch of black rings in
D
= 5 has been shown to be unstable [
20
–
22
] while for the thin branch
in
D
= 5, the instability of black rings relies on numerical studies [
6
,
12
]. However, these
numerical studies, due to lack of accuracy, have only established the existence of instabilities
for
ν≥
0
.
144 [
6
] and for
ν≥
0
.
15 [
12
]. The region
ν <
0
.
144 is left unknown, with the only
suggestive arguments of [
7
,
8
] being applicable in the strict case of
ν
= 0, for which there
is barely any distinction between the black ring and the boosted black string. Additionally,
the numerical studies of [
6
,
12
] have not consider non-axisymmetric modes with
m >
2
2
and,
moreover, there is currently no knowledge of these instabilities in
D≥
6for which no exact
black ring analytic solution is known.3
This paper deals with the study of black ring instabilities in the very thin regime for
D≥
5and
arbitrary
m
. Its aim is to provide an analytic understanding of some of these instabilities and
to progress in closing the gap in parameter space by showing that some of these instabilities
are present also for some part of the region
ν <
0
.
144. The blackfold approach has shown to
accurately describe stationary thin black rings. In the left plot of fig. 1, it is shown the phase
diagram of
D
= 5 black rings, where the reduced area
aH
and reduced angular momentum
j
were introduced in [
28
]. The black solid line is the curve obtained from the exact black ring
solution of [
27
] while the dashed red line is the blackfold approximation up to first order in
derivatives [
28
].
4
This approximation works relatively well for
j&
2
.
2which is equivalent to
the region 0
≤ν.
0
.
025. In the plot on the right in fig. 1, it is shown the
D
= 7 black ring
solution numerically obtained in [
29
] (black solid line) and the blackfold approximation up to
second order in derivatives (dashed red line) [
30
]. In this case the blackfold approximation
works well for
j&
1
.
2which corresponds to the region 0
≤ν.
0
.
27. Thus our general analysis
of dynamical instabilities of black rings for arbitrary
m
and
D
is expected to be valid at
least in the region 0
≤ν.
0
.
025 for
D
= 5
,
6for which the blackfold approximation is not
1
We note that we are interpreting the non-axisymmetric instability found for black rings in [
12
] as an elastic
instability from the blackfold point of view. The rationale for this interpretation is that, in the context of
blackfolds, elastic instabilities of black rings are related to deformations of the radial direction of the
S1
which
is the type of deformation encountered in [
12
]. In general, we do no expect all other types of non-axisymmetric
instabilities [
4
,
9
–
11
] to be elastic instabilities from the blackfold point of view. In fact, some of them, if visible
within the blackfold approximation, might be of hydrodynamic nature.
2
Here
m
is a discrete number characterising the mode of non-axisymmetric perturbations of the form
ei(−ωτ +mφ/R)where ωis the frequency, τthe time direction and φthe angular direction along the S1.
3
Albeit the work of refs. [
23
,
24
] which use large
D
techniques that we will comment upon and ref. [
25
], which
has considered the evolution of the Gregory-Laflamme instability for black rings at large
D
for
m∼ O
(
√D
)
and found evidence that the end point of the instability is a non-uniform, non-stationary, black ring. The
behaviour of the end point is expected to depend on the dimension Das in the case of black strings [26].
4
We note that the dashed red line in the left plot of fig. 1is the curve obtained at ideal order in the
approximation, since the first order approximation does not produce corrections to stationary black rings [
28
].
2
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
j0.0
0.2
0.4
0.6
0.8
1.0
aH
0.8 1.0 1.2 1.4 1.6 1.8 2.0
j0.80
0.85
0.90
0.95
1.00
1.05
1.10
aH
Figure 1:
On the left we show the reduced area
aH
as a function of the reduced angular momentum
j
for
D
= 5 where the black line is the exact curve of the black ring solution [
27
] and the dashed red
curve is the blackfold approximation up to first order in derivatives [
28
]. On the right we show the
behaviour of the same quantities for black rings in
D
= 7 where the black line is the numerical solution
of [29] and the dashed red line the blackfold approximation up to second order in derivatives [30].
under control beyond first order, and in the region 0
≤ν.
0
.
27 for
D≥
7for which the
approximation is under control up to second order in r0/R.
In order to proceed with this analysis, we first introduce the blackfold effective theory in
sec. 2and derive novel variational formulae required to study perturbations around equilibrium
configurations. In sec. 3we study instabilities of boosted black strings as a way of calibrating
our method, since in this case our results, besides providing a check of the
R→ ∞
limit
of black ring instabilities, can be compared against existent numerical and analytic results.
In this context, we provide novel expressions for the growth rates of the Gregory-Laflamme
instability and its onset for arbitrary boost parameter. In sec. 4we study the instabilities of
black rings and identify the Gregory-Laflamme instability in
D≥
5, providing analytic results
for the growth rates of the instability and its onset. We do not find an elastic instability at this
order in the blackfold approximation and hence our results for the black ring contradict the
corresponding large
D
analysis [
24
], which is shown to be incorrect. In sec. 5we summarise
our main results and comment on open research directions. In app. Awe provide the corrected
stress tensor and bending of perturbed black branes in asymptotically flat space, which contains
the identification of new transport coefficients. In app. Bwe provide details on the perturbed
equations at second order, while in app. C, supplemented by the ancillary Mathematica file,
we give further details on hydrodynamic and elastic modes.
2 Blackfold equations and linearised perturbations
In this section we briefly review the essential aspects of the blackfold approach required
for the purposes of this work. We discuss the blackfold equations up to second order in a
long-wavelength expansion which determine the equilibrium configurations that we wish to
perturb. Subsequently, we derive new general formulae for linearised perturbations of the
equilibrium blackfold equations, ultimately focusing on the case of 2-dimensional worldvolumes
which describe black strings and black rings. These results will then be used in the remaining
sections in order to study the hydrodynamic and elastic stability of these later two cases.
3
2.1 Blackfold equations
The blackfold approach consists of wrapping black branes on weakly curved (
p
+ 1)-dimensional
submanifolds
Wp+1
embedded in a
D
=
n
+
p
+ 3-dimensional spacetime endowed with metric
gµν
(
x
)and coordinates
xµ
[
7
,
19
]. The location of the submanifold in the ambient spacetime is
determined by the embedding map
Xµ
(
σ
), where
σa
are coordinates on
Wp+1
. The submanifold
inherits the induced metric
γab
=
eµaeνbgµν
where
eµa
=
∂aXµ
are a set of (
p
+ 1) tangent
vectors and
a, b, c, ...
are surface indices. The set of (
n
+ 2) normal vectors
nµi
are defined
implicitly by the relations
nµieµa
= 0 and
nµinµj
=
δij
where
i, j, k, ...
are normal indices. The
extrinsic curvature of the submanifold is defined as
Kabρ
=
∇aeρb
where
∇a
is the covariant
derivative compatible with both
gµν
and
γab
. It is also useful to define its projection along
the normal vector, i.e.
Kabi
=
nρi∇aeρb
and mean extrinsic curvature
Ki
=
γabKab i
(or
equivalently Kρ≡γabKab ρ).
In vacuum, this approach is generically applicable if the horizon size of the black brane
r0
satisfies the hierarchy of scales
r0R
where
R
is the smallest intrinsic or extrinsic scale
associated with the submanifold or to variations of the fluid degrees of freedom that live on it.
In the case of singly-spinning black holes, this implies that the black hole must be ultraspinning,
i.e. in appropriate units the black hole angular momentum is much larger than its mass. In
the case of black rings with horizon topology
S1×S2
, the ultra-spinning limit is commonly
referred to as the thin limit, since in this case the radius
R
of the
S1
must be much larger than
the horizon radius r0of the S2.
In the context of vacuum General Relativity, the starting point is the boosted Schwarzschild
black brane. The process of "wrapping" the black brane on a weakly curved submanifold
translates into a small, long-wavelength, perturbation of the black brane geometry that must
satisfy Einstein equations order-by-order in a derivative expansion. Typically, the expansion
parameter
ε
1is defined as
ε
=
r0/R
or
ε
=
kr0
where
k
is the wavenumber of the
perturbation being performed. A subset of the Einstein equations (constraint equations) up to
order O(ε)has been identified to be [28,31,32]
∇aTab = 0 , T abKabi= 0 ,(2.1)
where we have ignored the existence of edges on the submanifold. Here the stress tensor
Tab
up to first order in derivatives is given in terms of a viscous fluid [31]
Tab =Tab
(0) +Tab
(1) , T ab
(0) =uaub+P P ab , T ab
(1) =−2ησab −ζθP ab ,(2.2)
where
ua
is the normalised fluid velocity
uaua
=
−
1and
Pab
=
γab
+
uaub
is a perpendicular
projector to
ua
. The thermodynamic quantities
and
P
denote the energy density and pressure
respectively while
η
and
ζ
denote shear and bulk viscosity. All these quantities are a function
of the local temperature
T
. Their specific dependence and form in terms of the black brane
radius
r0
is given in app. A, together with the definition of the shear tensor
σab
and the fluid
expansion θ.
At one higher order, the stress tensor
Tab
receives additional corrections that depend on
derivatives of the intrinsic and extrinsic geometry as well as on second derivatives of the
fundamental fluid variables. If
n≥
3, these corrections are dominant compared to backreaction
corrections and in this case, the equations of motion
(2.1)
are modified at order
O
(
ε2
)to
[33]
∇aTab =eµb∇a∇bDabµ , T abKabi=niµ∇a∇bDabµ ,(2.3)
4
where we have assumed that the background metric is flat (i.e. the associated Riemann tensor
vanishes) and that the brane is not spinning in transverse directions to
Wp+1
. In eq.
(2.3)
,
Tab
receives an additional correction
Tab
(2)
and
Dabµ
is the brane bending moment that encodes
the response of the black brane due to extrinsic deformations. The bending moment can be
written as
Dabµ
=
YabcdKcdµ
where
Yabcd
is the Young modulus [
34
]. The explicit form of
these structures is detailed in app. A.
Of particular importance is the class of solutions that describes the equilibrium sector of
(2.1)
and
(2.3)
. In this case, the fluid velocity must be aligned with a worldvolume Killing vector
field kasuch that
ua=ka
k,T=T
k,k=| − γabkakb|1/2,(2.4)
where
T
is the constant global temperature of the fluid and
k
is the modulus of the timelike
Killing vector field. The worldvolume Killing vector field is subjected to the constraint that
its pushforward onto the ambient spacetime coincides with a background Killing vector field
kµ
, i.e.
kµ
=
eµaka
. For this particular class of solutions, the first equation in
(2.1)
and
(2.3)
,
which is the set of hydrodynamic equations for the stress tensor
Tab
, is automatically satisfied,
regardless of the choice of embedding map Xµ. The second equation in (2.1) and (2.3) is the
remaining non-trivial elastic equation that determines conditions on the embedding map given
a particular choice of
ka
. Solutions that satisfy
(2.4)
have
σab
=
θ
= 0 by virtue of the Killing
equation and hence
Tab
(1)
= 0. It is this equilibrium class of solutions that we wish to perturb
around in order to study the stability properties of given configurations.
2.2 Linearised perturbations
The purpose of this section is to derive variational formulae that describe linear perturbations
of equilibrium configurations that solve
(2.1)
and
(2.3)
following the machinery developed in
[
35
].
5
This will form the basis for studying the stability of propagating sound and elastic modes
in the next sections.
The fluid degrees of freedom consist of a scalar degree of freedom, which we choose to be the
energy density
, and
p
independent components of the fluid velocity
ua
supplemented by
n
+ 2 transverse components of the embedding map which we denote by
Xµ
⊥
(
σ
) =
⊥µνXν
(
σ
)
where
⊥µν
=
nµiniν
.
6
Our intent is to perform a slight perturbation of these variables around
equilibrium solutions such that
→+δ , ua→ua+δua, Xµ
⊥(σ)→Xµ
⊥(σ) + δXµ
⊥(σ).(2.5)
Under these perturbations all geometric quantities transform, for instance
δγab
= 2
KabρδXρ
⊥
[
35
]. The normalisation condition
uaua
=
−
1, implies the constraint on the variations of the
fluid velocity
uaδua=uaubKabρδXρ
⊥,(2.6)
which is the statement that only
p
components of the fluid velocity are independent. These
variations are sufficient to characterise the deformations of the ideal order stress tensor, which
5
When describing perturbations of equilibrium configurations, the scale of the problem is set by 1
/T
. Thus,
by means of eq. (2.4) and app. A, when writing ε=r0k, it is really meant ε=k/T .
6
The remaining
p
+ 1 components of the embedding map
Xµ
can always be chosen as the coordinates
on
Wp+1
since
σa
=
eaµXµ
. Hence, when
Wp+1
has no edges, variations of these components coincide with
worldvolume reparametrisations and can be ignored.
5
take the form
δT ab
(0) =
n+ 1 nuaub−γab δ
+ 2nu(aδub)−δγab,(2.7)
where we have used the specific equation of state
=
−
(
n
+ 1)
P
provided in app. A. In order
to determine the variation of the equations of motion
(2.1)
up to first order, one also requires
the variation of the first order stress tensor
δT ab
(1) =−2ηδσab −ζδ θP ab , δθ =∇aδua−ua∇a(KρδXρ),(2.8)
where we have used that in equilibrium
θ
=
σab
= 0. As we are interested in the
p
= 1
case, we have not written explicitly the variation
δσab
. This is sufficient for obtaining linear
perturbations of (2.1). Defining δT ab =δT ab
(0) +δT ab
(1), these take the general form
∇aδT ab −Tcb∇cKρδXρ
⊥−2Tac∇ahKbcρδXρ
⊥i+Tac∇bKacρ⊥ρλ δXλ
⊥= 0 ,
δT abKabi+Tabniµ∇a∇bδXµ
⊥= 0 ,
(2.9)
where we have used that for equilibrium solutions
TabKabi
= 0 up to first order and focused on
backgrounds with vanishing Riemann tensor. At second order and for
n≥
3, eqs.
(2.9)
receive
modifications due to the right hand side of
(2.3)
. These modifications are cumbersome and are
detailed in app. B. Eqs.
(2.9)
and
(B.3)
are the equations that we wish to solve for different
initial configurations in terms of the perturbed fields
(2.5)
, in particular we wish to analyse
the vanishing of the determinant of eqs.
(2.9)
and
(B.3)
which provide sufficient conditions for
the existence of solutions.
2.2.1 Two-dimensional worldvolumes
In the next sections we focus on two-dimensional worldvolumes (
p
= 1) which can describe
boosted black strings and black rings in
D≥
5. In this case, the analysis simplifies considerably
since, for instance, δσab = 0 and hence
δT ab
(1) =−ζ δθP ab .(2.10)
In turn, the effect of the first order corrections to the stress tensor in
(2.9)
only depends on
the bulk viscosity in the form
∇aδT ab
(1) =−ζ∇aPabδθ,
δT ab
(1)Kabi=−ζδθ Ki+uaubKabi=−ζ δθ n+ 1
nKi+Oε3,
(2.11)
where in the last equality we have used
(2.1)
and neglected
Oε3
terms which we do not
consider in this paper. If in addition we focus on the case of boosted black strings for which
Kabi= 0, eqs. (2.9) further simplify to
∇aδT ab = 0 , T abniµ∇a∇bδXµ
⊥= 0 ,(2.12)
up to first order in derivatives. This shows that in this situation, the extrinsic perturbations
δXµ
⊥
decouple from the intrinsic perturbations
δ
and
δua
. At second order, these equations
receive non-trivial corrections, as explained in app. Band for some configurations, such as
black rings, the perturbations begin to couple. In the next section, we use the variational
formulae derived here to study the stability of boosted black strings.
6
3 Instabilities of boosted black strings
Gregory-Laflamme instabilities of static black strings using the blackfold approach were
considered in [
7
,
31
,
36
,
37
]. Here we consider both fluid and elastic perturbations of boosted
black strings up to second order in the derivative expansion. We compare our results with
the static and boosted cases with the large
D
analysis performed in [
23
,
24
,
38
], showing the
relevance of the Young modulus of black strings (defined in app. A) in the dispersion relation of
elastic modes. Elastic modes are shown to always be stable. We also derive novel expressions
for
kGL
, which describes the onset of the Gregory-Laflamme instability for arbitrary boost
parameter and compare with the numerical analysis of [
39
], finding good agreement when
D≥7. The results of this exercise are extremely useful to study perturbations of black rings
in sec. 4as they must be recovered at large ring radius.
3.1 Ideal order modes
We consider the equilibrium solution of
(2.1)
that represents a boosted black string. To begin
with, we introduce the background Minkowksi metric in the form
ds2≡gµν dxµdxν=−dt2+
D−1
X
i=1 dxi2,(3.1)
and we embed the string via the map
Xt
=
τ
,
X1
=
z
and
Xi
= 0
, i
= 2
, ..., D −
1, leading
to the induced string metric
ds2≡γabdσadσb=−dτ2+dz2.(3.2)
In addition, a boosted string is characterised by the Killing vector field
ka∂a
=
∂τ
+
β∂z
with
modulus
k
=
p1−β2
, where 0
≤β <
1is the boost parameter. The case
β
= 0 describes the
static black string. This embbeding is a minimal surface (a two-dimensional plane in (
t, x1
))
and hence has vanishing extrinsic curvature, i.e. Kabi= 0.
In order to study potential instabilities of these configurations we consider linearised perturba-
tions around these equilibrium configurations. Due to the variational constraint
(2.6)
, these
can be parametrised by small perturbations of the energy density
δ
(
σ
), the
z
-component of
the fluid velocity
δuz
(
σ
)and of the (
n
+ 2) transverse components of embedding map
δXµ
(
σ
)
⊥
.
In particular, we consider plane wave solutions to the perturbation equations
(2.9)
that we
parametrise as
δ(σ) = δei(−ωτ+kz), δuz(σ) = δuzei(−ωτ+kz), δXµ
⊥(σ) = δXµ
⊥ei(−ωτ +kz),(3.3)
where
ω
is the oscillation frequency,
k
the wavenumber and
δ, δuz
and
δXµ
⊥
are small pertur-
bation amplitudes. Introducing these variations into the perturbed equations at ideal order
(2.9)
and demanding the determinant of the system of 3 equations to vanish leads to two elastic
modes (due to perturbations of second equation in
(2.1)
) and two hydrodynamic modes (due
to perturbations of the first equation in
(2.1)
), which are solved perturbatively such that
ω=ω(0) +ω(1)kr0+ω(2)(kr0)2+... , (3.4)
where it is assumed that
ε
=
kr0
1. Under this approximation, the ideal order frequencies
read
ω(0)
1,2(k) = nβ ±√n+ 1 1−β2
n+ 1 −β2k , ω(0)
3,4(k) = β(n+ 2) ±i√n+ 1 1−β2
n+1+β2k . (3.5)
7
These two sets of frequencies are obtained independently from each of the equations in
(2.9)
since
intrinsic and extrinsic perturbations decouple for the black string and are valid for any
n≥
1.
In particular, the frequencies
ω1,2
are the elastic frequencies associated with perturbations
of the second equation in
(2.1)
and
ω3,4
the hydrodynamic frequencies, associated with the
first equation in
(2.1)
. The frequency
ω3
will be interpreted as being associated with the
Gregory-Laflamme instability. In the static case
β
= 0,
(3.5)
had been obtained in [
7
]. Contrary
to the case
β
= 0, the hydrodynamic frequencies with
β >
0are not purely imaginary. A case
of particular interest is when
β
= 1
/√n+ 1
, which corresponds to the value of the boost that
locally characterises the black ring of sec. 4. In this case the modes (3.5) become
ω(0)
1(k) = 0 , ω(0)
2(k) = 2√n+ 1
n+ 2 k , ω(0)
3,4(k)=(n+ 2 ±in)√n+ 1
n2+ 2n+ 2k , (3.6)
and hence have a zero-frequency mode. It is worth noticing that the elastic modes
ω1,2
in
(3.5)
are always real and positive for all values of 0
≤β <
1and thus the perturbations
(3.3)
describe
oscillating but stable solutions. The hydrodynamic mode
ω4
has a negative imaginary part and
hence the perturbations
(3.3)
describe stable solutions which are damped in time. On the other
hand, the hydrodynamic mode ω3has a positive imaginary part and hence the perturbations
will grow exponentially in time, signalling the existence of the well-known Gregory-Laflamme
instability of black strings [
1
], as spelled out in [
7
] for the case
β
= 0. This instability grows
faster for lower values of
n
and smaller values of
β
, and becomes attenuated as one approaches
n→ ∞ or β→1.
Given that
ω4
has a negative imaginary part that does not vanish for any value of
β
, any
higher order correction to
ω4
cannot make it unstable for small values of
kr0
. However, the
elastic modes in
(3.5)
have real frequencies and thus it is plausible that higher order corrections
could introduce a small but imaginary part. As we will show below, this is however not the
case.
3.2 First order modes
In finding high-order derivative corrections to
(3.5)
, our aim is to understand whether other
types of instabilities appear (such as elastic instabilities) as the black string effectively becomes
less thin. We also wish to understand the onset of the instability described by
kGL
, i.e. the
maximum value of
k
for which the instability is present. For
p
= 1, as mentioned in sec. 2.2,
first order corrections are controlled only by the bulk viscosity
ζ
. Evaluating the perturbative
equations accounting for the first order corrections in the stress tensor and using
(2.9)
leads to
the following correction to the hydrodynamic frequencies
ω(1)
3,4(k) = ∓(n+ 2)k3β±i√n+ 1n+ 1 ∓i√n+ 1β2
n√n+ 1 (n+1+β2)3k , (3.7)
where we have ignored corrections of the order of
O(kr0)2
. The elastic frequencies remain
the same as in eq.
(3.5)
and thus they remain stable under linear perturbations. The correction
to
ω(1)
3
has been obtained in [
31
] when
β
= 0 but not explicitly for
ω(1)
4
. We observe here that
when
β
= 0 the corrections become equal and purely imaginary. Hence the hydrodynamic
modes up to first order become
ω3,4(k) = i
√n+ 1 ±k−(n+ 2)
np(n+ 1)r0k2!+O(kr0)2,(3.8)
8
0.2 0.4 0.6 0.8 1.0
k
0.05
0.10
0.15
0.20
0.25
0.30
Im(ω)
0.02 0.04 0.06 0.08 0.10
k
0.02
0.04
0.06
0.08
Im(ω)
Figure 2:
On the left we show the dimensionless imaginary part of
ω3
, defined as
¯ω
=
Imω3r0
, as a
function of
¯
k
=
kr0/√n
for
D
= 5 (
n
= 1). The black solid line represents
β
= 0, the grey solid line
β
= 1
/√2
and the red solid line
β
= 9
/
10 while the grey dashed line is the imaginary part of the ideal
order result
(3.5)
. We have shown these curves up to
¯
k
= 1, but we only expect them to be strictly
valid for small
¯
k
. On the right plot we show the behaviour of the imaginary part of
ω3
for
β
= 1
/
10
and n= 1 (black), n= 2 (red), n= 3 (blue) and n= 4 (purple).
where the "+" sign corresponds to the solution of [
31
]. Overall, the mode
ω3
is always unstable
while
ω4
is always stable. In fig. 2on the left, we show the behaviour of the growth rate of the
Gregory-Laflamme instability (i.e. the imaginary part of
ω3
) for
D
= 5 and different values
of
β
. As
β
increases, the behaviour of the dimensionless frequency
¯ω
becomes increasingly
linear as a function of
¯
k
. The dashed and solid grey lines exhibit the improvement of first
order corrections to the hydrodynamic modes and deviations from the linear ideal order result
(3.6)
. On the right plot of fig. 2, we exhibit the behaviour of the growth rate of the instability
for
β
= 1
/
10 and for different values of
n
starting with
n
= 1 (black line) and ending in the
n
= 4 (purple line). The plot shows that the growth rate increases with increasing dimension
for small ¯
k.
The case
β
= 0 was explicitly compared against numerical simulations in [
31
] and agreement
was found in the entire range of
¯
k
at large
n
while for small
n
it is only a good approximation
for smaller values of
¯
k
. In the case of
β6
= 0, a numerical study was performed in [
8
] and in
particular it was found a finite value of
kGLr0<
2for all values of
β
for
n
= 1. We observe
that
ω3
in
(3.7)
is characterised by a value of
kGLr0
that increases significantly as
β→
1for
n= 1. In particular, using (3.7) we can find the analytic value of kGLr0to be
kGLr0=n+1+β22qn+1
1−β2
(n2+ 3n+ 2) (n+ 1 −3β2),(3.9)
and it leads to no solution for
β≥p(n+ 1)/3
. The result
(3.9)
is in fact a new analytic result,
not present in the literature but not an extremely useful one for small
n
or
β≥p(n+ 1)/3
.
However, at large
n
and
β
= 0 this result is approximately valid as already noted in [
31
].
Additionally, the lack of predictability as
β→
1is expected since for fixed
r0∝k/T ∼
1, as
β→
1we need
T→
0and hence for the approximation to be valid we require
kT
, thus
k∼
0. This means that as
β→
1we expect our approximation to be appropriate only near
k∼
0. These considerations and the result
(3.9)
largely improve once we consider second order
corrections as will be shown in the next section.
Finally, in the special case
β
= 1
/√n+ 1
which describes the critical boost of black rings at
9
large radius, the hydrodynamic frequencies become
ω3,4(k)=(n+ 2 ±in)√n+ 1
n2+ 2n+ 2k−i√n(n+ 1)5
2
2 ((1 ±i) + n)3
ζ
k2+O(r0k)2,(3.10)
where
ζ
was defined in app. A. To summarise, up to first order in derivatives we find the existence
of a Gregory-Laflamme instability for arbitrary boost βand no elastic instability.
3.3 Second order modes and comparison with the large Danalysis
At second order in derivatives, the derivation of the modes is more intricate as explained in
sec. 2.2 due to the non-trivial modification to the equations of motion and the appearance
of the Young modulus as a response to bending. Additionally, at second order in derivatives,
hydrodynamic and elastic corrections are dominant compared to backreaction corrections only
if n≥3[34]. This means that the analysis here is only useful for D≥7.
Solving for the second order correction in
(3.4)
using the modified linearised equations
(B.3)
,
one obtains corrections to the elastic modes
ω(2)
1,2(k) = ∓λ1√n+ 1n2k4n2+ 2n+ 1 ±4√n+ 1βn+1+β2+β2(6(n+ 1) + β2)k
P r2
0n+k24,
(3.11)
which are purely real and where
λ1
was introduced in app. A. Thus black strings are elastically
stable within the blackfold approximation up to second order in derivatives. In addition, the
hydrodynamic modes also receive corrections and are given in eq.
(C.1)
. In the case
β
= 0, the
correction
ω(2)
3
agrees with that obtained in [
36
,
37
]. In the case of the critical boost for black
rings, these corrections read
ω(2)
1(k) = −λ1√n+ 1
P r2
0
k , ω(2)
2(k) = λ1n4√n+ 1
P r2
0(n+ 2)4k ,
ω(2)
3,4(k) = ±√n+ 1(n+ 2)(n(2i(n+ (1 ±i))(τω/r0)−in + (i±5)) + (2i±6))
2(n+ (1 ±i))5k ,
(3.12)
where
τω
was introduced in app. A. Using the full expressions for arbitrary
β
given in app. C
we exhibit in the left plot of fig. 3the behaviour of the imaginary part of
ω3
for different values
of
β
for
D
= 7. It is observed a strong modification of the behaviour of
ω3
at the critical
boost when comparing first order (grey dashed line) and second order (grey solid line) in the
figure on the left. Thus, for instance, the first order result
(3.7)
for
β
= 1
/
2only accurately
describes the behaviour of the instability up to values of
¯
k∼
0
.
15. Hence, the behaviour of the
instability for boosted black strings becomes qualitatively similar to the static case (black solid
line) and to the behaviour for
n
= 1 [
8
] as one includes higher-order corrections. Therefore, we
expect these results to be approximately valid for ¯
k∼1and β∼1.
The study of hydrodynamic and elastic instabilities of static and boosted black strings has been
performed in [
23
,
24
,
38
] using large
D
methods. In order to compare our results with those
for the boosted black string at large
D
in [
24
], we redefine the boost parameter
β
such that
α
=
√nβ
and we perform the expansion of our hydrodynamic modes at large
n
(equivalent
to large
D
given that
p
= 1 is fixed). This leads to the following expansions for the elastic
10
0.5 1.0 1.5 2.0
k
0.05
0.10
0.15
Im(ω)
0.2 0.4 0.6 0.8 1.0
k
0.05
0.10
0.15
0.20
0.25
0.30
Im(ω)
Figure 3:
The figure on the left exhibits the imaginary part of
ω3
at second order using
(C.1)
as a
function of
¯
k
for
D
= 7 for different values of the boost parameter:
β
= 0 (black solid line),
β
= 1
/
2
(grey solid line) and
β
= 9
/
10 (red solid line). The grey dashed line represents the first order result for
β
= 1
/
2obtained in
(3.7)
. The figure on the right provides a comparison between blackfold and large
D
results for
D
= 5 and
D
= 7 for static strings and at the critical boost. The blue solid line is the
blackfold result at second order for
β
= 0 and
D
= 7 while the dashed line is the corresponding large
D
result at fourth order derived in [
40
]. The black solid line represents the imaginary part of
ω3
at
first order as in
(3.7)
while the black dashed line is the corresponding result at large
D
[
24
]. The grey
solid line is the imaginary part of
ω3
as in
(C.1)
while the grey dashed line is the corresponding large
Dresult (C.2).
modes
ω1,2(k) =(α∓1) k
√n±1
21∓2α+ 2α2+ 3k2r2
0k
√n3
∓1
83∓8α+ 12α2∓8α3+ 2k213 ∓24α+ 12α2r2
0k
√n5+O1
√n7,
(3.13)
while the hydrodynamic modes at large Dexhibit the following behaviour
ω3,4(k) =(α±i)k
√n−ik2r0
n∓i
21±2iα + 2α2k
√n3
+ik2−2±6iα + 3α2r0
2n2+1
8±3i−8α∓4iα2−8α3+ 8k2(±i+ 2α)r2
0k
√n5
+k28i∓12α+ 60iα2±36α3−3iα4r0
8n3+O1
√n7.
(3.14)
Comparing
(3.13)
and
(3.14)
with the corresponding results for
`
= 0 and
`
= 1 modes in
eqs. (B.12)-(B.13) of [
24
], we find complete agreement provided we ignore the terms of order
r3
0k3
in the elastic modes of [
24
], which are of higher order in the brane thickness.
7
In the case
of the hydrodynamic modes, the two results agree exactly, without any approximation.
7
In order to compare with the results of [
24
] we have set
r0
= 1 and used the large
D
behaviour
τω
=
r0
(1
/
2
−π2/
(3
n2
)
−
4
ζ
(3)
/n3−
4
π4/
(45
n4
) +
O
(
n−5
). Also note that eq. (B.13) of [
24
] contains several typos.
We have used the ancillary file provided with [
24
] to recover eqs. (B.12)-(B.13) in order to identify them. The
correct solution, up to
r2
0k2
terms, is
(3.13)
-
(3.14)
, while the full solution is provided in eqs.
(C.2)
-
(C.3)
for
completeness.
11
At finite values of
D
we provide a comparison between blackfold and large
D
results in the
right plot of fig. 3. The black solid line in the figure on the right corresponds to the first order
result
(3.7)
while the black dashed line is the large
D(C.2)
result at
D
= 5. The grey lines
provide the same comparison with the second order result
(C.1)
for
D
= 7 at the critical boost.
The blue solid line is the blackfold result for
β
= 0 and the blue dashed line is the large
D
fourth order result obtained in [
40
] for
D
= 7. The two approaches give similar results for
small (though not too small) values of
¯
k
when
D
= 7 but differ at very small
¯
k
where the
blackfold approach is more accurate. We also see that in
D
= 7 for the static case, where the
large
D
approach has been pushed to fourth order, and for values of
¯
k&
0
.
3the large
D
result
is very inaccurate with the growth rate of the instability increasing without bound for higher
values of
¯
k
, whereas the blackfold approach has shown to be approximately accurate in the
entire range of
¯
k
[
36
,
37
]. Additionally, for
D
= 5 the large
D
result is highly inaccurate for all
¯
k, only becoming better for larger values of D.
Onset of the instability
It is possible to obtain a refined expression for the onset of the instability
(3.9)
using the
results of app. C. In particular we find that
kGL
can be expressed in closed form as in
(C.4)
and provides a finite
kGLr0
for the entire range 0
≤β <
1. As clear from the comparison
between first and second order results (grey solid and dashed lines) in the left plot in fig. 3,
kGLr0
is highly sensitive to higher-order corrections. In fig. 4, we exhibit the behaviour of the
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
2 4 6 8 10 12 14
n
1
2
3
4
kGL
Figure 4:
Onset of the Gregory-Laflamme instability
¯
kGL
=
kGLr0
as a function of
n
as predicted by
different methods for
β
= 0: first order blackfold approach (black dashed line), second order blackfold
approach (black solid line), fourth order large
D
approach [
40
] (blue solid line) and numerical points of
[39] (orange squares).
onset of the instability as a function of
n
for
β
= 0 as predicted by the second order blackfold
approach (black solid line) compared to the first order result
(3.9)
(black dashed line) and the
large
D
result (blue solid line) obtained in [
40
], together with the numerically obtained points
(orange squares) of [
39
]. The first striking thing to note is that the large
D
approach is highly
accurate for
n≥
2even though, as we have noted above and explicitly shown in fig. 3, the
growth rate of the instability according to the same method increases without bound for
n≤
5,
i.e. for instance, the large
D
result does not predict the existence of a finite
kGLr0
when
n <
6.
Nevertheless, when extrapolating the large
D
result valid for
n≥
6to lower values of
n
the
12
agreement with the numerical values is excellent for
n >
1, visible from the solid blue curve in
fig. 4.
The other interesting feature of fig. 4is that the first order blackfold result more accurately
predicts
kGLr0
for
n≤
4than the second order blackfold result (though when
n
= 1 it is off
by a factor of 5 compared to the numerical result). When
n≥
5, the second order blackfold
result becomes more accurate than the first order result and approaches the large
D
result as
n
increases. This is not surprising since, as stated earlier, the frequency
ω3
reproduces exactly
the corresponding large
D
result. In general, we do not have the right to expect that the
blackfold approach accurately describes the growth rate of the instability for values of
¯
k&
1
and it is already remarkable that in many cases it approximately does so.
A comment on ω4
We note that
ω4
at second order acquires an imaginary part for larger values of
¯
k
. For instance
for
n
= 3, it does so for
¯
k&
5. This threshold value for
ω4
is always more than twice that of
¯
kGL
of
ω3
.
8
This feature is also visible in the large
D
results of [
40
]. We do not expect this to
be a smoking gun for another hydrodynamic instability of black strings since these high values
of ¯
kare a priori outside the regime of validity of both methods.
4 Instabilities of black rings
In this section we focus on the instabilities of asymptotically flat singly-spinning black rings
in
D≥
5by following the same approach as in the previous section. We show that at ideal
order (i.e. ultraspinning) black rings are Gregory-Laflamme unstable under small linearised
perturbations but elastically stable. Including higher derivative corrections yields a similar
behaviour for the dispersion relations of the unstable perturbation as that found numerically
in [
6
] for the non-axisymmetric quantised mode
m
= 2 and
D
= 5. The analysis here has
higher accuracy for large modes
m
1and by including corrections up to second order in the
thickness of the ring, we show that no elastic instability appears, thus contradicting large
D
results [
24
]. We obtain analytic expressions for the onset of the Gregory-Laflamme instability
for black rings and study its behaviour as a function of
m
. We also find a long-lived mode
describing a slowly oscillating wiggly black ring.
4.1 Ideal order modes
The black ring solution, up to first order in derivatives, is an equilibrium solution of
(2.1)
where
the spatial worldvolume geometry is closed and the fluid elements living on it are rotating. It
is useful to write the flat Minkowski background in the form
ds2=−dt2+dr2+r2dϕ2+
D−3
X
i=1 dxi2,(4.1)
8
In fact we also observe that the imaginary part of
ω3
at second order becomes positive again at a higher
value of ¯
k. We also consider this feature to be outside the regime of validity of the method employed here.
13
where we have isolated a two-dimensional spatial plane written in polar coordinates. The ring is
embedded in this background by choosing
Xt
=
τ, X r
=
R, Xϕ
=
φ
and
Xi
= 0
, i
= 1
, ..., D−
3
such that the induced metric and rotating Killing vector field are
ds2=−dτ2+R2dφ2,ka∂a=∂τ+ Ω∂φ,k2= 1 −Ω2R2,(4.2)
where 0
≤φ≤
2
π
and Ωis a constant angular velocity which admits the following expan-
sion
Ω=Ω(0) + Ω(1)ε+ Ω(2) ε2+... , ε =r0
R.(4.3)
At ideal order, eq.
(2.1)
fixes Ω
(0)
= 1
/
(
R√n+ 1
). The difference between
(4.2)
and the
geometry of the black string of the previous section is the closed spatial topology. The geometry
and Killing vector field of the boosted black string in sec. 3are recovered at large radius
R→ ∞
by defining the coordinate
z
=
φR
and the boost velocity
β
= Ω
R
= 1
/√n+ 1
.
As in the case of boosted black strings, we perform small perturbations of the energy density,
fluid velocity and embedding scalars, in particular along the ring radial direction
δ(σ) = δei(−ωτ+kRφ), δ¯uφ(σ) = δ¯uφei(−ωτ +kRφ), δR(σ) = δRei(−ωτ +kRφ),(4.4)
where we have defined
¯
δuφ
=
Rδuφ
which remains finite as
R→ ∞
. In this case, eqs.
(2.9)
couple to each other and hence hydrodynamic and elastic perturbations cannot be studied
individually. This means that
δR
perturbations are necessarily accompanied by
δ
and
δ¯ua
perturbations and vice-versa.
9
Since the spatial topology is closed,
k
is quantised such that
m
=
kR
for discrete
m
. In this context, the vanishing of the determinant of eqs.
(2.9)
leads
to two elastic modes which remain the same as for the boosted black string
(3.5)
with boost
β= 1/√n+ 1 and hence stable, while the hydrodynamic modes read
ω(0)
3,4=√n+ 1
(n2+ 2n+ 2)R(n+ 2)m±p2(n2+ 2n+ 2) −n2m2.(4.5)
At large radius
R→ ∞
(i.e. at large
m→ ∞
), these frequencies reduce to those of the boosted
black string with
β
= 1
/√n+ 1
given in
(3.6)
, as expected. It can be observed that the
frequencies ω(0)
3,4have an imaginary part, with ω(0)
3being unstable only if
m>mmin =√2
npn2+ 2n+ 2 ,(4.6)
while
ω(0)
4
is always stable. In particular
mmin
=
√10
for
n
= 1 and, for
m
= 1, the frequencies
ω(0)
3,4are always real for any nwhile for m≥2complex frequencies are attained only if n≥3.
In any case, for each
n
there is always a sufficiently large enough
m
that makes
ω3
unstable.
This implies that, besides also being unstable in the fat branch 1
/
2
≤r0/R <
1[
20
–
22
], black
rings are also Gregory-Laflamme unstable in the thin regime
r0/R
1in particular in the
regime 0≤r0/R .0.025 as mentioned in sec. 1.
It is worth noting that, for instance, for
m
= 1
,
2the frequency
ω3
is real for
n
= 1. This is
not in contradiction with [
6
,
12
] since the numerical analysis for
m
= 2 has not been carried
9
It is also possible to consider perturbations along the remaining
n
+ 1 components of the embedding map,
which decouple from
δR
perturbations for the black ring even at second order in derivatives. At ideal and first
order, the modes coincide with those of the boosted black string with
β
= 1
/√n+ 1
while at second order the
elastic modes receive 1
/R
corrections which we provide in
(C.6)
. These perturbations do not lead to an elastic
instability.
14
out in the region
ν <
0
.
144 and it is possible to speculate whether the Gregory-Laflamme
instability is present for
ν∼
0or whether it ceases to exist at some small value of
ν
. It is
unclear at the present moment if a real
ω3
is a prediction in the infinitely thin limit or whether
the blackfold approach is not valid for
m
= 1
,
2. In fact, as we shall see when studying first
order corrections, these frequencies acquire an imaginary part but do not have the expected
qualitative behaviour, while the elastic frequencies can develop poles at such low values of
m
. However, it is clear that the method employed here is more accurate when
m
1. The
perturbation wavelength λ∼k−1∼R/m must satisfy
λr0⇒r0
R1
m.(4.7)
At fixed global temperature
T
(i.e. fixed
r0
) the boosted black string limit is attained when
R→ ∞
and hence
m→ ∞
such that
m/R
is finite. Since the method employed here describes
the dynamics of very thin (ultraspinning) rings, according to
(4.7)
the larger
m
is, the smaller
r0/R
must be. In particular, in the regime
m
1the dynamics of black rings is described by
a mild deformation of the dynamics of boosted black strings. Specifically, we may expand the
unstable frequency in (4.5) in powers of 1/m, giving
ω(0)
3=(1 + i)m√n+ 1
(n+1+i)R−i√n+ 1
mnR −i√n+ 1(n(n+ 2) + 2)
2m3n3R+O1
m5,(4.8)
which makes the connection with
(3.6)
explicit in the limit
R→ ∞
and where the second and
third terms represent deviations in 1
/R
away from the boosted black string. It is expected
that these results will provide a good approximate description for higher values of
m
but
only a comparison with a numerical analysis, which is not currently available, will settle this
issue.
The elastic modes for black rings are the same as for boosted black strings with critical boost
(3.6)
and hence purely real. It is plausible that these modes could acquire a positive imaginary
part as we move away from the thin limit. This turns out not to be the case as we will show
below. However, corrections to the dispersion relations are still useful as not only they represent
long/short lived time-dependent black hole solutions but also allows to understand better the
behaviour of dominant instabilities.
4.2 First order modes and comparison with large Danalysis
At first order in derivatives the equation of motion
(2.1)
set Ω
(1)
= 0 for black rings. The
stress tensor receives viscous corrections which for
p
= 1 only depend on the bulk viscosity.
The vanishing of the determinant of the system
(2.9)
now requires that the frequencies of the
elastic modes take the form
ω1= 0 + Oε2, ω2=2m√n+ 1
(n+ 2)R1 + i2m√n(n+ 2)
4(n+ 1) −(m2−1) n2ε+Oε2,(4.9)
where
ε
=
r0/R
makes clear that the ring is not necessarily infinitely thin. We note that
ω2
receives an imaginary contribution that is positive for
m
= 1
∀n
and for
n
= 1 with
m
= 2, otherwise it is a negative contribution. However, the correction to
ω2
has a pole when
˜m
= (
n
+ 2)
/n
, which is maximal when
n
= 1 for which
˜m
= 3 while it is
˜m
= 2 for
n
= 2 and
only has another integer value at
˜m
= 1 when
n→ ∞
. We interpret this divergence as a signal
15
that we should not trust
(4.9)
(and the method in general) for
m≤˜m
. In particular, when
m= ˜mthe expansion manifestly breaks down.
In the more accurate regime m1the non-trivial elastic frequency (4.9) becomes
ω2=2m√n+ 1
(n+ 2)R−4i√n+ 1
n3/2Rε+O1
m2+Oε2.(4.10)
Thus, at first order in derivatives, the blackfold approach is not able to identify an elastic
instability for
m
1. On the other hand, using the large
D
approach ref. [
24
] has claimed the
existence of an elastic instability. In order to provide a comparison
10
, we expand
(4.9)
at large
nand find
ω1= 0 + Oε2, ω2=2m
R
1
√n−4im2
(m2−1) nRε+O1
√n7+Oε2,(4.11)
while the same frequencies in [24] expanded in the thin radius regime ε=r0/R read
ω(D)
1=im2
4 (m2−1) nRε+... , ω(D)
2=2m
R
1
√n−19im2
4 (m2−1) nRε+... . (4.12)
We see that
ω1
and
ω(D)
1
disagree and that
ω2
and
ω(D)
2
only agree at ideal order. In particular,
ω(D)
1
is the frequency responsible for the elastic instability in [
24
], due to its positive imaginary
part for any
m >
1. This disagreement indicates that
ω(D)
1
and
ω(D)
2
are not correct and hence
the results of [24] have not identified an elastic instability.11
0.2 0.4 0.6 0.8 1.0
ν=r0
R
-1.5
-1.0
-0.5
0.5
Im(ω)
0.02 0.04 0.06 0.08 0.10
ν=r0
R
0.05
0.10
0.15
0.20
Im(ω)
Figure 5:
On the left, we show the imaginary part of frequencies
ω2
(red solid line),
ω3
(black solid
line) and
ω4
(grey solid line) for black rings up to first order in derivatives as a function of the ring
thickness
ν
=
r0/R
for
D
= 5 and
m
= 10 using the full expressions provided in the ancillary file. The
dashed lines are the corresponding frequencies for boosted black strings in
(3.7)
with critical boost. On
the right plot we show the behaviour of the imaginary part of
ω3
as a function of
m
for
m
= 6 (red),
m= 8 (blue), m= 10 (black) and m= 12 (purple) for D= 5.
10
We remark that we are not imposing a priori constraints on the form of the mode number
m
as a function
of the parameters of the theory, besides requiring that
m˜m
. Consequently, we expect the comparison with
the large
D
approach to be reasonable under the more general assumption of
mΦ
=
O
(1) as considered in [
24
].
As in [
24
], we are also perturbing the stationary black ring configuration along the physical angular coordinate
Φof the large Dapproach.
11
The author of [
24
] does not think that his results are correct and has not been able to reproduce them at
a later stage. This is why the author has never sent the paper for publication (e-mail correspondence).
16
The remaining sound modes receive the following corrections at large n
ω(1)
3=mm2−3√m2−2−im2−2m2+ 1
m2+i√m2−2m−2nR
+O1
√n3,
ω(1)
4=i(1 −m4+ 2m2)+2√m2−2m
(m2−1) nR +O1
√n3,
(4.13)
which also disagree with [
24
] at first order in the thickness and reduce to
(3.10)
at large
n
. The
results for arbitrary
n
and
m
are provided in the ancillary Mathematica file. We note that the
mode
ω3
is now also unstable for
m
= 2, as previously advertised. However, comparison with
the numerical results of [
6
] for
m
= 2 hints towards the fact that
m
= 2 is outside the regime
of validity of the method employed here. Similarly, comparison of the mode
ω2
, which also
developed an imaginary part at first order for
n
= 1, with the results for the elastic instability
found in [
12
] for
m
= 2 seems to reiterate this point.
12
Additionally, it is clear from
(4.13)
that the expansion also breaks down for
m
= 1 for any
n
as
ω3,4
develop a pole. This gives
additional evidence that the expansion should not be trusted for m≤˜m.
In the left plot of fig. 5we show the imaginary part of
ω2,3,4
for
D
= 5 and
m
= 10. The plot
shows that only the frequency
ω3
(black solid line) has a positive imaginary part and hence
signals a hydrodynamical instability. The dashed lines are the corresponding boosted black
string results of sec. 3at critical boost. As the thickness
ν
increases, the behaviour of the black
ring frequencies increasingly differs from the boosted black string frequencies. It is expected
that the results presented here will be valid for small
ν.
0
.
025. In the left plot of fig. 5we
have clearly extrapolated the curves beyond the regime of validity.
In the right plot of fig. 5we exhibit the growth rates of the instability
ω3
for different values of
m
starting with
m
= 6 (red line) and ending with
m
= 12 (purple line) for
D
= 5. The curves
show that the instability grows faster for increasing
m
. Thus, the large
m
modes dominate the
dynamics of very thin black rings.
4.3 Second order modes
At second order in derivatives for
D≥
7, the stability analysis of black rings becomes more
involved due to the additional non-trivial contributions to the equations of motion
(2.3)
. At
this order equilibrium (2.3) requires that13
Ω(2) = Ω(0)
n2+ 3n+ 4
2n2(n+ 2) ξ(n).(4.14)
Thus the ideal order stress tensor will contribute with extra terms due to the second order
correction to Ω. As explained in sec. 1we expect this analysis to be valid for small values of
the thickness ν, in particular for ν.0.27 for D= 7.
Given
(4.14)
, requiring the determinant of
(B.3)
to vanish leads to the purely real second order
12
The comparison of our results with those of [
6
,
12
] is not exact since the latter results are valid for
ν≥
0
.
144
while the former are expected to be valid for
ν.
0
.
025. However, the qualitative behaviour of our results is far
from what is expected, as it does not approximate the results of [
6
,
12
] when extrapolated to larger values of
ν
.
13This result is related to the one obtained in [30] via the field redefinition R→R−Rξ(n)ε2/n.
17
correction to the first elastic mode14
ω1= 0 + mm2−2m2+ 1√n+ 1(3n+ 4)
2 (m2−1) n2(n+ 2)Rξ(n)ε2+Oε3,(4.15)
which is purely real and reduces to
(3.12)
. Thus,
ω1
acquires time-dependent behaviour as
expected, since the fluid velocity is not aligned with a Killing vector field, but no unstable
behaviour. Interestingly, up to this order this mode does not attenuate and thus represents a
long lived time-dependent modulation of a black ring. Given that
ω1
is real we conclude that
the blackfold approach is not able to detect an elastic instability in asymptotically flat black
rings at this given order in the expansion for D≥7and for any value of m≥2.
0.1 0.2 0.3 0.4 0.5
ν=r0
R
-4
-2
2
4
Im(ω)
0.05 0.10 0.15 0.20 0.25
ν=r0
R
0.05
0.10
0.15
Im(ω)
Figure 6:
The left plot exhibits the behaviour of the imaginary part of the frequencies
ω2
(red solid
line),
ω3
(black solid line) and
ω4
(grey solid line) for black rings up to second order in derivatives as a
function of the ring thickness
ν
=
r0/R
for
D
= 7 and
m
= 10. The dashed lines are the corresponding
frequencies for boosted black strings in
(3.12)
with critical boost. The right plot exhibits the growth
rate of the instability associated to
ω3
as a function of
ν
for
m
= 6 (red line),
m
= 8 (blue line),
m
= 10
(black line) and m= 12 (purple line).
The remaining modes acquire non-trivial corrections at second order, whose explicit expression
we have provided in the ancillary Mathematica file. In the left plot of fig. 6we exhibit the
behaviour of the imaginary parts of the frequencies
ω2,3,4
in
D
= 7 for
m
= 10 as a function
of
ν
. As it can be seen from fig. 6, the frequency
ω3
(black solid line) acquires a positive
imaginary part in the region
ν.
0
.
27 and we thus expect to accurately describe the onset of
the Gregory-Laflamme instability. We note that the frequency
ω2
never acquires a positive
imaginary part but that
ω4
does. For
m
= 10 and
D
= 7, as seen from fig. 6, the imaginary
part of
ω4
becomes positive for
ν >
0
.
3. The origin of this positive imaginary part is rooted in
the comment we made at the end of sec. 3.3 about the same behaviour of
ω4
for the boosted
black string. As explained there, the imaginary part of
ω4
lies outside the regime of validity of
the method. If
m
increases, both the imaginary part of
ω3
and
ω4
are pushed to lower values
of
ν
but
ω4
remains outside the regime of validity due to
(4.7)
. Thus, this does not signal a
new instability.
In the right plot of fig. 6we exhibit the growth rates of the instability associated with
ω3
as a function of
m
in
D
= 7 starting with
m
= 6 (red line) and ending in
m
= 12 (purple
line). For small values of
ν
the growth rate increases with increasing
m
as already noted in
fig. 5. As
ν
increases further, the growth rate eventually decreases to zero, analogous to the
14It is worth mentioning that, like ω3,4in (4.13), ω1develops a pole at m= 1 for any n.
18
0 10 20 30 40 50
0.0
0.2
0.4
0.6
0.8
1.0
n
νGL
Figure 7:
Onset of the Gregory-Laflamme instability
νGL
for black rings as a function of
n
for
m
= 8
(black lines),
m
= 12 (blue lines),
m
= 20 (purple lines) and
m
= 50 (red lines) using first order
blackfold approach (dashed lines) and second order blackfold approach (solid lines).
behaviour of boosted black strings and to the numerical results of [
6
] for
m
= 2. It is possible
to determine the onset of the instability analytically. At large
m
and
n
, the onset can be
written in a compact form
νGL =n(4n(2n−3) −53) −60
8mn3/2+n(n(131 −12n(2n+ 1)) + 51) −769
4m3n3/2+O1
m5,1
n3.
(4.16)
The full expression for the onset is provided in the ancillary Mathematica file. In fig. 7we
exhibit the onset of the instability
νGL
as a function of
n
for different values of
m
, in particular
m
= 8 (black line) up to
m
= 50 (red line) as predicted by the first order approximation
(dashed lines) and second order approximation (solid lines). It is clear from fig. 7that the
behaviour of the onset is qualitatively similar to that of the boosted black string of fig. 4. One
observes that as
m
increases, the onset ends at thiner and thiner rings, in agreement with fig. 6.
These analytic results consist of the first analytic determination of νGL.
5 Discussion
In this paper we initiated a systematic study of the dynamical stability of black holes in
D≥
5
in the blackfold limit (ultraspinning limit) and applied it to asymptotically flat boosted black
strings and black rings. In the context of boosted black strings, though studied numerically in
[
1
,
8
], we have provided new analytic results such as the growth rate of the Gregory-Laflamme
instability for arbitrary boost
β
and analytic expressions for the onset of the instability for
arbitrary boost and spacetime dimension. In the context of black rings, we have provided
the first correct analytic expressions for the growth rate of the Gregory-Laflamme instability
as a function of the axisymmetric mode
m
and for the onset of the instability. In
D
= 5,
our analysis is valid for at least
ν.
0
.
025 and in
D
= 7 for
ν.
0
.
27. This thus progresses
in closing the gap in parameter space where black rings were found to be unstable (i.e. for
ν≥
0
.
144 in [
6
] and
ν≥
0
.
15 in [
12
] in
D
= 5) by showing explicitly the instability for very
thin rings, and for large non-axisymmetric modes, where numerical methods are not precise
enough.
19
Despite our analysis including second order corrections to the blackfold approximation, we
have not been able to identify an elastic instability of black rings, as that found numerically
in [
12
] for
m
= 2 and
D
= 5.
15
We have identified divergences in the dispersion relations for
hydrodynamic and elastic modes that manifestly break the expansion when
m
= 3 for
D
= 5,
m
= 2 for
D
= 6 and
m
= 1 for all
D≥
5. We have interpreted these divergences as signalling
that the blackfold approximation breaks down for
m≤˜m
= (
n
+ 2)
/n
. In fact, a qualitative
comparison of the growth rates of potential Gregory-Laflamme and elastic instabilities for
D
= 5 and
m
= 2 found here with those numerically obtained in [
6
,
12