11institutetext: STAG research centre and Mathematical Sciences, University of
Southampton, University Road Southampton SO17 1BJ, UK.22institutetext:
Department of Physics, Rikkyo University, Toshima, Tokyo 171-8501,
Japan33institutetext: Department of Physics, College of Humanities and
Sciences, Nihon University, Sakurajosui, Tokyo 156-8550, Japan44institutetext:
DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce
Road, Cambridge CB3 0WA, UK55institutetext: Departament de Física Quàntica i
Astrofísica, Institut de Ciències del Cosmos
Universitat de Barcelona, Martí i Franquès, 1, E-08028 Barcelona, Spain

# Gregory-Laflamme encounters Superradiance

Óscar J. C. Dias 2 Takaaki Ishii 3 Keiju Murata 4 Jorge E. Santos 5 Benson Way
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We investigate the effect of superradiant scattering of gravitational
perturbations on the stability of rotating black strings, focusing on the six
dimensional equal-spinning Myers-Perry black string. We find that rapidly
rotating black strings are unstable to gravitational superradiant modes within
a bounded range of string lengths. The instability occurs because momentum
along the string direction creates a potential barrier that allows for the
confinement of superradiant modes. Yet, five dimensional Myers-Perry black
holes do not have stable particle orbits so, unlike other known superradiant
systems, these black strings remain stable to perturbations with sufficiently
high azimuthal mode number — this is a ‘finite-$m$’ superradiant instability.
For some parameters, this instability competes with the Gregory-Laflamme
instability, but otherwise exists independently. The onset of this instability
is degenerate and branches to multiple steady-state solutions. This paper is
the first of a trilogy: in the next two, we construct two distinct families of
rotating strings emerging from the superradiant onset (the ‘black resonator
strings’ and ‘helical black strings’). We argue that similar physics is
present in 5-dimensional Kerr black strings, but not in $D>6$ equal-spinning
Myers-Perry black strings.

††preprint: RUP-22-22

##  1 Introduction

Despite its simplicity, the product spacetime of a Schwarzschild black hole
and a circle, known as the Schwarzschild black string Horowitz:1991cd ,
manages to capture a wealth of physical phenomena affecting black holes in
higher dimensions. Notable are the Gregory-Laflamme instability Gregory:1993vy
; Gregory:1994bj , its associated violation of the weak cosmic censorship
conjecture Horowitz:2001cz ; Lehner:2010pn ; Figueras:2022zkg , and its
neighbouring solutions (non-uniform strings and localized black holes) which
violate black hole uniqueness (see e.g. Gubser:2001ac ; Wiseman:2002zc ;
Sorkin:2004qq ; Kudoh:2004hs ; Sorkin:2006wp ; Headrick:2009pv ;
Figueras:2012xj ). It is now known that the same physics is ubiquitous among
black holes with extended horizons, including those of particular interest to
string theory and holography Hanada:2007wn ; Kawahara:2007fn .

It is therefore natural to consider rotating black strings, as they are more
generic generalizations of static black strings and can express behaviours
that require the presence of angular momentum. Of particular interest to us
are superradiant instabilities. Like the Gregory-Laflamme instability, the
superradiant instability often also comes with violations of uniqueness with
time-periodic black holes Dias:2011at ; Dias:2015rxy ; Ishii:2018oms ;
Ishii:2019wfs ; Ishii:2021xmn , a possible violation of the weak cosmic
censorship conjecture Dias:2011at ; Dias:2015rxy ; Niehoff:2015oga ;
Chesler:2018txn ; Chesler:2021ehz , and has wide applications to astrophysics,
string theory, and holography (see review Brito:2015oca ).

But the Gregory-Laflamme and superradiant instabilities are triggered by
different mechanisms. The Gregory-Laflamme instability is driven by horizon
dynamics and the propensity for horizons to increase in area when there is a
hierarchy of length scales. By contrast, the superradiant instability occurs
when waves are amplified by an ergoregion and then reflected back towards the
horizon, leading to an exponentially growing instability. For black strings,
momentum along the string direction produces an effective mass term, which can
potentially provide enough confinement to trigger a superradiant instability
that would not otherwise exist without this extra dimension Marolf:2004fya ;
Cardoso:2004zz ; Cardoso:2005vk ; Dias:2006zv . Indeed, such an instability
exists for scalar field perturbations of Kerr black strings through this
mechanism Cardoso:2004zz ; Cardoso:2005vk .

Yet, the story changes for higher dimensional rotating (Myers-Perry) black
strings Myers:1986un ; Hawking_1999 . Indeed, unlike Kerr black strings,
singly spinning111Recall that in five or higher dimensions there is more than
one plane of rotation. Myers-Perry black strings are stable to scalar field
perturbations Cardoso:2005vk . Furthermore, there are no stable bounded orbits
around a wide variety of Myers-Perry black holes Tangherlini:1963 ;
Frolov:2003en 222Some higher dimensional black holes admit stable bounded
orbits, e.g. 6-dimensional singly spinning Myers-Perry black holes
Igata:2014xca  or 5-dimensional black rings Igata:2010ye ., preventing
particle-like waves from experiencing repeated amplification. Consequently, no
superradiant instability should exist for such black holes in the eikonal
limit (i.e. point-particle limit), where the azimuthal wave number
$m\to\infty$.

Despite the lack of stable orbits, we demonstrate that rotating black strings
might, in certain cases, still be unstable to superradiance in the most
universal sector: gravitational perturbations. For simplicity, we will focus
only on the six-dimensional equal-spinning Myers-Perry black string (i.e. with
equal angular momenta along two rotation planes) Gibbons:2004js ; Gibbons_2005
, for which the associated gravitational perturbation equations reduce to
ODEs. The five-dimensional equal-spinning Myers-Perry black hole lacks stable
orbits Kagramanova:2012hw , which implies that its associated asymptotically
$\mathcal{M}^{1,4}\times S^{1}$ black string cannot be unstable to
superradiant perturbations in the eikonal limit $m\to\infty$. Nevertheless, we
will find that the instability exists for any finite $m$, with the unstable
region approaching measure zero in parameter space as $m\to\infty$. To our
knowledge, this is the first system where such a ‘finite-$m$’ superradiant
instability is observed. Because of this property, this instability is not
expected to progress indefinitely to smaller and smaller length scales and
lead to a violation of the weak cosmic censorship conjecture, unlike what was
conjectured for ‘infinite-$m$’ rotational superradiance seen in other systems
Dias:2011at ; Dias:2015rxy ; Niehoff:2015oga .

As mentioned earlier, the Gregory-Laflamme and superradiant instabilities are
distinct physical phenomena and exist independently. In this paper, we will
also compare both of these instabilites. Depending on the black hole and wave
parameters, none, only one of them, or both instabilities can be present.
Where both instabilities coexist, we find that the Gregory-Laflamme
instability typically has a much higher growth rate and we discuss possible
scenarios for the time evolution of the system.

We further find linear evidence that this superradiant instability, much like
those affecting rotating black holes, leads to a branch of new steady-state
solutions Basu:2010uz ; Bhattacharyya:2010yg ; Dias:2011tj ; Dias:2011at ;
Dias:2013sdc ; Cardoso:2013pza ; Herdeiro:2014goa ; Dias:2015rxy ;
Ishii:2018oms ; Ishii:2020muv ; Ishii:2021xmn  (these new solutions are often
called ‘hairy’ or ‘resonator’ solutions). Like the case for black holes
Dias:2011tj ; Herdeiro:2014goa ; Dias:2015rxy ; Ishii:2018oms , the onset
frequency (where $\mathrm{Im}(\omega)=0$) for superradiant modes has nonzero
real part ($\mathrm{Re}(\omega)\neq 0$), so the branching solutions are
usually time-periodic, but not time independent nor axisymmetric. The onset of
the instability in black strings can be degenerate, implying that multiple
solutions branch from this onset (i.e. more than a one-parameter family). We
will argue that it is natural to coin these solutions ‘black resonator
strings’ and ‘helical black strings’, and we present perturbative evidence for
their existence. Then, in a sequel of two papers DIMSW2 ; DIMSW3 , we will
construct them nonlinearly (using higher order perturbation theory and fully
nonlinear numerical methods) and study their properties in detail.

The rest of the paper is structured as follows. Section 2 reviews the equal-
spinning Myers-Perry black string. Section 3 studies the onset and growth rate
of the Gregory-Laflamme instability. We then investigate the superradiant
instability in section 4, finding its onsets, associated instability regions
and growth rates. We highlight that the onset of this instability provides
perturbative evidence for the existence of novel (resonator and helical) black
strings. We end with a discussion of our results in Section 5. In the
appendices we argue that the gravitational superradiant instability is not
present for spin 0 and 1 fields and that it also does not extend to $D>6$
equal-spinning black strings.

##  2 Equal-spinning Myers-Perry black strings

The Myers-Perry black hole Myers:1986un ; Hawking_1999  is an exact solution
describing an asymptotically flat, rotating black hole in any dimension. In
odd dimensions and with all angular momenta set equal, the Myers-Perry black
hole has an enhanced symmetry group permitting the solution to be written in a
cohomogeneity-1 way (i.e. depending nontrivially only on a single coordinate)
Gibbons:2004js ; Gibbons_2005 . We will be working with the six-dimensional
($D=6$) black string given by the product of an equal-spinning five-
dimensional Myers-Perry black hole and a circle. Hereafter, unless otherwise
specified, the Myers-Perry (MP) black string refers to this specific solution.
Its metric can be written333The radial coordinate used here can be converted
to the standard Boyer-Lindquist radial coordinate in Myers:1986un  through
$r^{2}\to r^{2}+a^{2}$.

$\displaystyle{\mathrm{d}}s^{2}_{\rm
MP\,string}=-\frac{F}{H}\,{\mathrm{d}}t^{2}+\frac{{\mathrm{d}}r^{2}}{F}+r^{2}\left[H\left(\frac{\sigma_{3}}{2}-\frac{\Omega}{H}\,{\mathrm{d}}t\right)^{2}+\frac{1}{4}\left(\sigma_{1}^{2}+\sigma_{2}^{2}\right)\right]+\mathrm{d}z^{2}\,,$
(1)

where

$F(r)=1-\frac{r_{0}^{2}}{r^{2}}+\frac{a^{2}r_{0}^{2}}{r^{4}}\,,\qquad
H(r)=1+\frac{a^{2}r_{0}^{2}}{r^{4}}\,,\qquad\Omega(r)=\frac{a\,r_{0}^{2}}{r^{4}}\,,$
(2)

and

$\begin{split}\sigma_{1}&=-\sin(2\psi)\,\mathrm{d}\theta+\cos(2\psi)\sin\theta\,\mathrm{d}\phi\
,\\\
\sigma_{2}&=\cos(2\psi)\,\mathrm{d}\theta+\sin(2\psi)\sin\theta\,\mathrm{d}\phi\
,\\\ \sigma_{3}&=2\,\mathrm{d}\psi+\cos\theta\,\mathrm{d}\phi\end{split}$ (3)

are left-invariant one-forms that satisfy the Maurer-Cartan equation
$\mathrm{d}\sigma_{i}=\frac{1}{2}\epsilon_{ijk}\sigma_{j}\wedge\sigma_{k}$.
The Euler angles $(\theta,\phi,\psi)$ are coordinates on a squashed $S^{3}$
with ranges $0\leq\theta<\pi$, $0\leq\phi<2\pi$, and $0\leq\psi<2\pi$. We take
the coordinate $z\in(0,L)$ to be periodic with circle length $L$. Killing
vectors of this spacetime are given by

$\begin{cases}\partial_{t},\>\partial_{z},\>\partial_{\psi}\,,&\\\
\xi_{1}=\cos\phi\,\partial_{\theta}+\frac{1}{2}\frac{\sin\phi}{\sin\theta}\,\partial_{\psi}-\cot\theta\sin\phi\,\partial_{\phi}\,,&\\\
\xi_{2}=-\sin\phi\,\partial_{\theta}+\frac{1}{2}\frac{\cos\phi}{\sin\theta}\,\partial_{\psi}-\cot\theta\cos\phi\,\partial_{\phi}\,,&\\\
\xi_{3}=\partial_{\phi}\,,&\end{cases}\\\ $ (4)

and thus the isometry group of this spacetime is $\mathbb{R}_{t}\times
U(1)_{\psi}\times SU(2)\times U(1)_{z}$, with $\xi_{i}$ ($i=1,2,3$) being the
generators of the $SU(2)$-symmetry group,
$[\xi_{i},\xi_{j}]=\epsilon_{ijk}\xi_{k}$. The one-forms $\sigma_{i}$ are
invariant under $\xi_{j}$, _i.e._ $\pounds_{\xi_{j}}\,\sigma_{i}=0$ (where
$\pounds$ stands for Lie derivative).

The solution has 3 dimensionful parameters: the mass radius $r_{0}$, the
rotation parameter $a$, and the circle length $L$. The horizon radius $r_{+}$
is defined as the largest real root of $F$ which can be used to express the
mass radius as $r_{0}=r_{+}/\sqrt{1-(a/r_{+})^{2}}$. The energy, angular
momentum, tension along $z$, temperature, angular velocity and entropy of the
Myers-Perry string are, respectively:444 Note that we have defined $J$ and
$\Omega_{H}$ with respect to a canonically normalised periodic variable
$\psi\sim\psi+2\pi$, in accordance with the work of Myers and Perry
Myers:1986un  (another option that is often taken would be
$\psi\sim\psi+4\pi$); in these conventions, the superradiant factor (also
known as resonant or syncronization factor) is given by $\omega-2m\Omega_{H}$.
The non-standard factor of 2 reflects the fact that we have equal rotation
along the two planes and we are using the period of $2\pi$ for $\psi$.

$\begin{split}E&=\frac{3\pi r_{0}^{2}L}{8G_{6}}\,,\qquad J=\frac{\pi
r_{0}^{2}aL}{4G_{6}}\,,\qquad T_{z}=\frac{\pi r_{0}^{2}L}{8G_{6}}\,,\\\
T_{H}&=\frac{1}{2\pi
r_{+}}\frac{1-2(a/r_{+})^{2}}{\sqrt{1-(a/r_{+})^{2}}}\,,\qquad
S=\frac{L}{2G_{6}}\frac{\pi^{2}r_{+}^{3}}{\sqrt{1-(a/r_{+})^{2}}}\,,\qquad\Omega_{H}=\frac{a}{r_{+}^{2}}\,,\\\
\end{split}$ (5)

where $G_{6}$ is the six-dimensional Newton’s constant. In horizon radius
units (equivalent to set $r_{+}\equiv 1$), Myers-Perry black strings have two
dimensionless parameters, which we will take for convenience to be
$\Omega_{H}r_{+}$ and $kr_{+}$, where the wavenumber $k=2\pi/L$. Note that the
temperature vanishes at $a/r_{+}=1/\sqrt{2}$, where the Myers-Perry black
string is extremal with angular frequency
$\Omega^{\mathrm{ext}}_{H}r_{+}=1/\sqrt{2}$.

We will show that the Myers-Perry black string is unstable to at least two
sectors of gravitational perturbations. One is the familiar Gregory-Laflamme
instability, first studied in the context of Schwarzschild black
strings/branes Gregory:1993vy ; Gregory:1994bj  and then extended to rotating
black strings in Kleihaus:2007dg ; Dias:2009iu ; Dias:2010eu ; Dias:2010maa ;
Dias:2011jg . The other is the superradiant instability whereby the Kaluza-
Klein momentum along the string direction provides an effective mass term that
confines superradiant bound states. This confinement mechanism for the
instability was first proposed in Marolf:2004fya  and the perturbative
analyses of the associated timescale, for Kerr strings, were done in
Cardoso:2004zz ; Dias:2006zv . Numerical superradiant analysis of Kerr and
single-spinning Myers-Perry black strings for scalar fields where also done in
Cardoso:2005vk .

Our linear mode stability analysis requires studying linearised gravitational
perturbations. Let $h_{AB}$ be a linear metric perturbation about the Myers-
Perry black string. We will choose to work in the traceless-transverse gauge
defined by555Greek indices $\mu,\nu,\ldots$ run only over the 5-dimensional
coordinates $(t,r,\theta,\phi,\psi)$, capital Latin indices $A,B,\ldots$ run
over these plus the extended direction $z$, and the lower case indices
$i,j,k,\ldots$ run over the Euler angles $\\{\theta,\phi,\psi\\}$.

$h^{A}_{\phantom{A}A}=0\;,\qquad\nabla^{A}h_{AB}=0\,.$ (6)

In this gauge, the linearised Einstein equation reduces to

$(\Delta_{L}h)_{AB}\equiv-\nabla^{C}\nabla_{C}h_{AB}-2R_{ACBD}h^{CD}=0\,,$ (7)

where $\Delta_{L}$ is the Lichnerowicz operator, and all curvatures and
covariant derivatives are computed with respect to the Myers-Perry black
string background.

##  3 The Gregory-Laflamme instability on Myers-Perry black strings

We now review the Gregory-Laflamme (GL) instability and study its properties
in the (equal-spinning) Myers-Perry black string. Generally, the Gregory-
Laflamme instability is present when there are one or more directions where
the horizon is elongated. Intuitively, the Gregory-Laflamme instability is
driven by the horizon’s tendency to maximise its area (entropy) when there is
a hierarchy of scales, while keeping the mass and angular momentum of the
black hole fixed. Often, the entropically preferred configuration with the
same mass and angular momenta (and fixed Kaluza-Klein circle length) is a
‘localized’ black hole with spherical horizon topology (or an array of such
spherical black holes). Moreover, the dynamical transition from the black
string into the localized black hole configuration necessarily comes with a
change in horizon topology, which violates the weak cosmic censorship
conjecture. These are all fairly general qualitative features of gravitational
systems with a hierarchy of horizon length scales, though specific details can
differ.

In the case of the (equal-spinning) Myers-Perry black string, the two
lengthscales are $r_{+}$ and $L$ and we expect the instability to occur at
least for $L\gg r_{+}$. The resulting Gregory-Laflamme instability was already
briefly studied in Kleihaus:2007dg ; Dias:2010eu , although in Dias:2010eu
more emphasis was given to the so-called ultraspinning instability of the
equal angular momenta Myers-Perry black hole.666The ultraspinning instability
affects rapidly rotating black holes in $D\geq 6$ and their associated black
strings in $D\geq 7$, and do not affect the Myers-Perry black string studied
here. We study the Gregory-Laflamme mode that is unstable for all values of
the rotation, including in the Schwarszchild string limit. In the rotating
Myers-Perry strings, there are axisymmetric Gregory-Laflamme instabilities
that are only present for rotations above a critical value Dias:2010eu  that
we do not discuss here.  We also note that the Gregory-Laflamme instabilities
of single-spinning and arbitrarily spinning Myers-Perry black strings were
also discussed in Dias:2009iu ; Dias:2010maa  and Dias:2011jg , respectively.

###  3.1 Linear perturbations

As stated above, we will work in the transverse-traceless gauge (6) which, in
the present case, amounts to working in the gauge $h_{Az}=0$. Then for
convenience, we will only consider the components $h_{\mu\nu}$ with
$(\mu,\nu)$ running only over the 5-dimensional coordinates
$(t,r,\theta,\phi,\psi)$777To make contact with Dias:2010eu , note that the
transverse-traceless gauge conditions (6) reduce to
$h^{\mu}{}_{\mu}=0\;,\nabla^{\mu}h_{\mu\nu}=0$ and the linearized equation
reads
$(\Delta_{L}h)_{\mu\nu}\equiv-\nabla^{\rho}\nabla_{\rho}h_{\mu\nu}-2R_{\mu\rho\nu\sigma}h^{\rho\sigma}=-k^{2}h_{\mu\nu}$.
.

The Gregory-Laflamme perturbations of the Myers-Perry string (1) that we
investigate have the form $\mathrm{d}s^{2}=\mathrm{d}s^{2}_{\mathrm{\rm
MP\,string}}+\delta\mathrm{d}s^{2}_{\mathrm{\hbox{\tiny GL}}}$ with

$\displaystyle\delta\mathrm{d}s^{2}_{\mathrm{\hbox{\tiny GL}}}$
$\displaystyle=h_{\mu\nu}\mathrm{d}X^{\mu}\mathrm{d}X^{\nu}$ (8)
$\displaystyle=e^{{\rm i}kz}e^{-{\rm
i}\,\omega\,t}\Bigg{(}-\frac{F}{H}\,\Big{(}Q_{1}{\mathrm{d}}t+2Q_{6}{\mathrm{d}}r\Big{)}{\mathrm{d}}t+\frac{{\mathrm{d}}r}{F}\Big{(}Q_{2}{\mathrm{d}}r+Q_{7}\sigma_{3}\Big{)}$
$\displaystyle\qquad+r^{2}\left[H\left(\frac{\sigma_{3}}{2}-\frac{\Omega}{H}{\mathrm{d}}t\right)\bigg{(}Q_{3}\left(\frac{\sigma_{3}}{2}-\frac{\Omega}{H}{\mathrm{d}}t\right)-2\frac{\Omega}{H}Q_{5}{\mathrm{d}}t\bigg{)}+\frac{Q_{4}}{4}\left(\sigma_{1}^{2}+\sigma_{2}^{2}\right)\right]\Bigg{)}\,,$

where $Q_{i}$ ($i=1,\ldots,7$) are unknown functions of $r$ and we have
introduced the mode frequency $\omega$ and the Kaluza-Klein momentum or
wavenumber $k=2\pi/L$.

Note that this ansatz does not encompass the most general set of gravitational
perturbations that trigger a Gregory-Laflamme instability. We have set the
azimuthal wave number $m$ associated to the Killing vector $\partial_{\psi}$
of the background to zero, as the Gregory-Laflamme instability is strongest in
the axisymmetric sector of perturbations. We have also not included any
$\theta$ and $\phi$ dependence.888These can be introduced by decomposing the
perturbations into charged harmonics on $\mathbb{C}\mathrm{P}^{1}\simeq S^{2}$
Hoxha:2000jf ; Kunduri:2006qa ; Martin:2008pf ; Dias:2010eu . There are
scalar, vector and tensor harmonics on $\mathbb{C}\mathrm{P}^{1}$ but only the
scalar describes Gregory-Laflamme instabilities Hoxha:2000jf ; Kunduri:2006qa
; Martin:2008pf ; Dias:2010eu ; Ishii:2018oms .  Gregory-Laflamme
instabilities with such angular dependence (i.e. with $m\neq 0$) Dias:2010eu
only exist above a critical rotation and typically have a smaller growth rate.
The modes that we study exist even without rotation.

The linearized Einstein equation (7) and the gauge conditions (6) can be
reduced to a system of three second order equations for
$\\{Q_{2},Q_{3},Q_{5}\\}$. This can be accomplished as follows. Using the
gauge conditions (6), $Q_{1}$ and $Q_{4}$ can be expressed algebraically in
terms of the other functions and their derivatives. The linearised Einstein
equation (7) then reduces to a second order equation for $Q_{2}$ and four
first order equations for $Q_{3,5,6,7}$. We can further use the first order
equations for $Q_{3}$ and $Q_{5}$ to get two algebraic equations for $Q_{6}$
and $Q_{7}$, which finally converts the equations to a system of three second
order equations for $\\{Q_{2},Q_{3},Q_{5}\\}$ (when $\omega=0$, we get a
similar system of 3 coupled ODEs but with $Q_{6}=Q_{7}=0$).

We now discuss boundary conditions. The asymptotic behaviour of the functions
is $Q_{i}|_{r\to\infty}\sim e^{\pm\sqrt{k^{2}-\omega^{2}}\,r}$. We keep only
the exponentially decaying solution to preserve Kaluza-Klein asymptotics
($\mathcal{M}^{1,4}\times S^{1}$). In more detail, the Gregory-Laflamme modes
decay as $Q_{2,3,5}|_{r\to\infty}\sim
e^{-\sqrt{k^{2}-\omega^{2}}\,r}r^{\,\alpha_{2,3,5}}$ where $\alpha_{i}$ are
constants.999For time dependent perturbations ($\omega\neq 0$) one has
$\alpha_{2}=\alpha_{3}=-3/2$ and $\alpha_{5}=3/2$ while for the zero mode
problem ($\omega=0$) one has $\alpha_{2}=-5/2$, $\alpha_{3}=-3/2$ and
$\alpha_{5}=3/2$. At the horizon, we have $Q_{i}|_{\cal
H}\sim(r-r_{+})^{\pm{\rm i}\,\frac{\omega}{4\pi T_{H}}}$. We keep the solution
with the minus sign as it is the one that is regular in ingoing Eddington-
Finkelstein coordinates. In more detail, the Gregory-Laflamme modes behave as
$Q_{2,3,5}|_{r\to r_{+}}\sim(r-r_{+})^{\beta_{2,3,5}-{\rm
i}\,\frac{\omega}{4\pi T_{H}}}$ with $\beta_{2}=-1$ and
$\beta_{3}=\beta_{5}=0$.101010The Myers-Perry string background is smooth
across the event horizon if we use ingoing Eddington-Finkelstein coordinates
$(v,r,\theta,\phi,\Psi)$ given by: $v=t+\int
H^{1/2}F^{-1}\,\mathrm{d}r\,,\Psi=\psi+2\int\Omega\,H^{-1/2}F^{-1}\,\mathrm{d}r\,$.
For time-dependent perturbations ($\omega\neq 0$) one has $\beta_{2}=-1$ and
$\beta_{3}=\beta_{5}=0$, while for the zero mode problem ($\omega=0$) one has
$\beta_{2}=\beta_{3}=0$ and $\beta_{5}=1$.

For numerics, it is convenient to introduce a radial coordinate

$y=\sqrt{1-\frac{r_{+}}{r}},$ (9)

which ranges $y\in(0,1)$ with the horizon $r=r_{+}$ being at $y=0$ and the
asymptotic region $r\to\infty$ at $y=1$.

We will work in units of the horizon radius (equivalent to $r_{+}\equiv 1$),
where the dimensionless quantities are given by

$\widetilde{a}=a/r_{+}\,,\quad\widetilde{L}=L/r_{+}\quad\widetilde{\Omega}_{H}=\Omega_{H}r_{+}\,,\quad\widetilde{T}_{H}=T_{H}r_{+}\,,\quad\widetilde{k}=kr_{+}\,,\quad\widetilde{\omega}=\omega
r_{+}\,.$ (10)

In these units, the Myers-Perry black string can be parametrised by
$\widetilde{\Omega}_{H}$ and $\widetilde{k}$.

Our numerical study of the Gregory-Laflamme instability proceeds in two parts,
each of which require separate numerical computations. First, while scanning
through $\widetilde{\Omega}_{H}$, we obtain a critical wavenumber
$\widetilde{k}=\widetilde{k}_{(0)}|_{\hbox{\tiny GL}}$ that yields the onset
zero mode value $\widetilde{\omega}=0$. Second, we solve the general problem
for $\widetilde{\omega}\neq 0$ while scanning through parameter space
$(\widetilde{\Omega}_{H},\widetilde{k}$).

In the zero mode case with $\omega=0$, the boundary conditions can be enforced
by the field redefinitions

$\displaystyle
Q_{2}(y)=e^{-\frac{\widetilde{k}}{1-y^{2}}}\left(1-y^{2}\right)^{5/2}q^{(0)}_{1}(y)\,,$
$\displaystyle
Q_{3}(y)=e^{-\frac{\widetilde{k}}{1-y^{2}}}\left(1-y^{2}\right)^{3/2}q^{(0)}_{2}(y)\,,$
(11) $\displaystyle
Q_{5}(y)=e^{-\frac{\widetilde{k}}{1-y^{2}}}\left(1-y^{2}\right)^{-3/2}y^{2}q^{(0)}_{3}(y)\,,$

and requiring that the $q^{(0)}_{i}$’s are smooth functions. From the
equations of motion, it follows that the $q^{(0)}_{i}$’s now obey Neumann
conditions at the horizon and Robin conditions at infinity.

With $\widetilde{\omega}=0$, the critical wavenumber
$\widetilde{k}=\widetilde{k}_{(0)}|_{\hbox{\tiny GL}}$ appears as a quadratic
eigenvalue problem. For any given rotation below extremality
$0\leq\widetilde{\Omega}_{H}\leq 1/\sqrt{2}$, there is a unique solution with
real, positive $\widetilde{k}=\widetilde{k}_{(0)}|_{\hbox{\tiny GL}}$, which
we identify as the critical wavenumber for the onset of the Gregory-Laflamme
instability. This gives us the onset curve $\widetilde{k}_{(0)}|_{\hbox{\tiny
GL}}(\widetilde{\Omega}_{H})$. Note there is also a trivial $\widetilde{k}=0$
solution.

As a result of this analysis, we will find that unstable Myers-Perry black
strings lie in the rectangular region of parameter space given by
$0\leq\widetilde{\Omega}_{H}\leq 1/\sqrt{2}$ and
$0<\widetilde{k}<\widetilde{k}_{(0)}|_{\hbox{\tiny
GL}}(\widetilde{\Omega}_{H})$.

With the zero mode solution in hand, we can then proceed to the second part of
our numerical study, this time with $\widetilde{\omega}\neq 0$. For this case,
boundary conditions can be imposed by the field redefinitions

$\displaystyle
Q_{2}(y)=e^{-\frac{\sqrt{\widetilde{k}^{2}-\widetilde{\omega}^{2}}}{1-y^{2}}}\left(1-y^{2}\right)^{3/2}y^{-2-\frac{{\rm
i}\,\widetilde{\omega}}{2\pi\widetilde{T}_{H}}}q_{1}(y)\,,$ $\displaystyle
Q_{3}(y)=e^{-\frac{\sqrt{\widetilde{k}^{2}-\widetilde{\omega}^{2}}}{1-y^{2}}}\left(1-y^{2}\right)^{3/2}y^{-\frac{{\rm
i}\,\widetilde{\omega}}{2\pi\widetilde{T}_{H}}}q_{2}(y)\,,$ (12)
$\displaystyle
Q_{5}(y)=e^{-\frac{\sqrt{\widetilde{k}^{2}-\widetilde{\omega}^{2}}}{1-y^{2}}}\left(1-y^{2}\right)^{-3/2}y^{-\frac{{\rm
i}\,\widetilde{\omega}}{2\pi\widetilde{T}_{H}}}q_{3}(y)\,,$

again with the requirement that the $q_{i}$’s are smooth functions.

Now, the aim is to compute the frequency $\widetilde{\omega}$ for any given
$\widetilde{k}$ and $\widetilde{\Omega}_{H}$ in the unstable region. This is a
non-polynomial eigenvalue problem in $\widetilde{\omega}$, which can be
approached by using a Newton-Raphson algorithm with the zero mode solution as
an initial seed. To fix the normalization of the eigenvalue problem, we set
$Q_{5}|_{y=0}=1$. (See e.g. Cardoso:2013pza ; Dias:2015nua  for details on
using the Newton-Raphson method for non-polynomial eigenvalue problems).

Since Gregory-Laflamme modes have purely imaginary frequencies, ${\rm
Re}\,\widetilde{\omega}=0$, it is convenient to redefine the frequency as
$\widetilde{\Gamma}={\rm i}\,\widetilde{\omega}$, which reduces the domain of
the eigenvalue problem from the complex plane to the real line. Further note
that one then has
$e^{-\frac{\sqrt{\widetilde{k}^{2}-\widetilde{\omega}^{2}}}{1-y^{2}}}=e^{-\frac{\sqrt{\widetilde{k}^{2}+\widetilde{\Gamma}^{2}}}{1-y^{2}}}$
and asymptotic decaying modes should exist for any $\widetilde{k}$, although
only long wavelengths with $\widetilde{k}\leq\widetilde{k}_{(0)}|_{\hbox{\tiny
GL}}$, should be unstable.

Numerically, we discretise the differential equations using pseudo-spectral
methods on a Chebyshev grid. As expected for pseudospectral method, our
results converge exponentially with the number of grid points. All the
frequencies that will be presented in our plots are accurate up to at least
the eighth decimal digit. Moreover, our results match those of Dias:2010eu
111111The numerical codes used were developed independently..

###  3.2 Gregory-Laflamme unstable region and growth rates

Figure 1: Onset mode wavenumber (left) and growth rate (right) of the
axisymmetric ($m=0$) Gregory-Laflamme instability for Myers-Perry black
strings. In the left panel, unstable modes have
$\widetilde{k}<\widetilde{k}_{(0)}$ (i.e.
$\widetilde{L}>2\pi/\widetilde{k}_{(0)}$ are unstable). The vertical dashed
line marks extremality with $\widetilde{\Omega}_{H}=1/\sqrt{2}\simeq
0.707107$. Critical wavenumbers are $\widetilde{k}_{(0)}\simeq 1.268916$ at
$\widetilde{\Omega}_{H}=0$ Gregory:1993vy  and $\widetilde{k}_{(0)}\simeq
3.767342$ at $\Omega_{H}\simeq 0.999996\,\Omega^{\mathrm{ext}}_{H}\simeq
0.707104$ (extrapolates to $\widetilde{k}_{(0)}\simeq 3.783201$ at
$\Omega_{H}=\Omega^{\mathrm{ext}}_{H}$). In the right panel, we only show
solutions up to $\widetilde{\Omega}_{H}\sim 0.6<1/\sqrt{2}$. In this range,
the growth rate increases with increasing $\widetilde{\Omega}_{H}$ and fixed
$\widetilde{k}$. For higher $\widetilde{\Omega}_{H}$, the growth rate falls:
see discussion of right panel of Fig. 2 for further details). All modes here
have purely imaginary frequency, ${\rm Re}\,\omega=0$.

We now present our results for the Gregory-Laflamme instability. The zero mode
wavenumber $\widetilde{k}_{(0)}\equiv\widetilde{k}_{(0)}|_{\hbox{\tiny GL}}$
and growth rates $\mathrm{Im}\,\widetilde{\omega}$ are shown in the left and
right panels of Fig. 1, respectively.

From the left panel, one sees that the critical wavenumber
$\widetilde{k}_{(0)}$ exists for all values of the angular velocity up to
extremality, $0\leq\widetilde{\Omega}_{H}\leq 1/\sqrt{2}$. For zero rotation,
the unstable region is $0<\widetilde{k}\lesssim 1.268916$, which agrees with
the instability of the Schwarzschild-Tangherlini black string Gregory:1993vy .
As rotation is increased towards extremality, the unstable region grows to
$0<\widetilde{k}\lesssim 3.783$. In this sense, we can state that rotating
black strings are more prone to the Gregory-Laflamme instability.

The right panel of Fig. 1 displays the Gregory-Laflamme growth rate
$\widetilde{\Gamma}={\rm Im}(\omega r_{+})$ across the dimensionless
parameters $\widetilde{\Omega}_{H}$ and $\widetilde{k}$. The onset zero mode
curve with $\widetilde{k}=\widetilde{k}_{(0)}|_{\hbox{\tiny
GL}}(\widetilde{\Omega}_{H})$ and ${\rm Im}\,\widetilde{\omega}=0$, is also
displayed in the right panel of Fig. 1 as the orange curve. The fact that the
growth rate computation returns ${\rm Im}\,\widetilde{\omega}=0$ along the
same line $\widetilde{k}=\widetilde{k}_{(0)}|_{\hbox{\tiny
GL}}(\widetilde{\Omega}_{H})$ as the independent zero mode numerical code is a
non-trivial check of our computations.

From Fig. 1, we see that Myers-Perry black strings are Gregory-Laflamme
unstable for the whole rotation range $0\leq\widetilde{\Omega}_{H}\leq
1/\sqrt{2}$ whenever the wavenumber is below critical
$0<\widetilde{k}<\widetilde{k}_{(0)}|_{\hbox{\tiny
GL}}(\widetilde{\Omega}_{H})$, that is, whenever the length is higher than the
zero mode critical length, $\widetilde{L}\geq\widetilde{L}_{(0)}|_{\hbox{\tiny
GL}}$. For most range of parameters (aside from those close to extremality
which we will discuss later in Fig. 2), the growth rate of the Gregory-
Laflamme instability increases as the rotation increases at fixed
$\widetilde{L}$. Moreover, for a given $\widetilde{\Omega}_{H}$ the maximum
growth rate is higher than the one for the Schwarzschild-Tangherlini black
string. Thus, in this sense, we see that the Gregory-Laflamme instability can
get stronger with rotation.

Figure 2: Growth rate for the Gregory-Laflamme instability of the Myers-Perry
black string with fixed $\widetilde{\Omega}_{H}=0.66$ (left), and with
$\widetilde{k}=\frac{1}{2}\,\widetilde{k}^{\mathrm{GL}}_{(0)}$ (right). Note
that in the right plot, $\widetilde{k}^{\mathrm{GL}}_{(0)}$ changes with
$\widetilde{\Omega}_{H}$ according to Fig. 1, so $\widetilde{k}$ varies as
well. The yellow diamond in the right panel at
$(\widetilde{\Omega}_{H},\widetilde{\omega})\simeq(1/\sqrt{2},0.101756)$ is
the extrapolated value of the growth rate at extremality (vertical dashed
line).

For completeness, in the left panel of Fig. 2 we focus our attention on Myers-
Perry black strings with $\widetilde{\Omega}_{H}=0.66$ and describe how the
Gregory-Laflamme growth rate changes as
$0\leq\widetilde{k}\leq\widetilde{k}_{(0)}|_{\hbox{\tiny GL}}\simeq 2.533565$.
This is a curve that is qualitatively similar to that of the Schwarzschild
string. On the right panel, we select Myers-Perry strings that have a length
that is twice as large as the critical onset length,
$\widetilde{L}=2\widetilde{L}_{(0)}|_{\hbox{\tiny GL}}$, for each rotation
(note $\widetilde{L}$ is not constant as $\widetilde{\Omega}_{H}$ changes). We
see that starting at zero rotation, the growth rate first increases, attains a
maximum, and then decreases substantially in a short window of angular
velocity as extremality is approached. Note that in the right panel of Fig. 1
this sharp decrease is not visible because we do not show the solutions too
close to extremality where the fall occurs, i.e. we just display solutions up
to $\widetilde{\Omega}_{H}\sim 0.6<1/\sqrt{2}$.

Summarizing our results, as long as we are not too close to extremality,
adding rotation tends to increase the parameter range of unstable black
strings, as well as increase the unstable growth rate. In both these senses,
the Gregory-Laflamme instability tends to get stronger with added rotation.
Our results are qualitatively consistent with previous results found in
Dias:2010eu .

##  4 The superradiant instability of Myers-Perry black strings

In this section, we will show that the Myers-Perry black string is unstable to
gravitational superradiance and compare the associated unstable region and
growth rate to those of the Gregory-Laflamme instability.

Superradiant instabilities require two primary ingredients. The first is the
existence of superradiant scattering, i.e. an amplification mechanism of
incident waves that enter an ergoregion. This is present in the Myers-Perry
string. The second is a confining mechanism whereby the amplified waves are
reflected back to the ergoregion and can continue to extract energy and
angular momenta until they back-react on the spacetime geometry. At the
outset, it is unclear whether the Myers-Perry black string satisfies this
second criterion.

Heuristically, non-trivial Kaluza-Klein modes (i.e. modes with frequency
$\omega$ and momentum $k$ along the string direction) decay asymptotically as
$e^{-\sqrt{k^{2}-\omega^{2}}\,r}$, with the momentum $k$ providing an
effective mass proportional to $\sqrt{k^{2}-\omega^{2}}$. This effective mass
creates a potential barrier along the radial direction that might provide
enough of a confining mechanism, as first proposed in Marolf:2004fya  (see
also Cardoso:2004zz ; Cardoso:2005vk ; Dias:2006zv ). Though this argument
applies for generic (integer spin) perturbations, the fact is that no such
instability has been found for scalar and Maxwell fields on the (equal-
spinning) Myers-Perry black string. Indeed, we have explicitly checked that
there is no superradiant instability for these perturbations and, actually, we
can provide an analytical argument for the absence of the instability as
discussed in Appendix A. For the scalar field on the single spinning Myers-
Perry black string, Cardoso:2005vk  also observed the lack of a superradiant
instability. Furthermore, the lack of stable orbits around 5-dimensional
equal-spinning Myers-Perry black holes also strongly suggests that there
should not be a superradiant instability in the associated six-dimensional
Myers-Perry black string in the eikonal limit (where the azimuthal mode number
$m\to\infty$).

Yet, despite the above negative results, we will nevertheless find a
superradiant instability for gravitational perturbations in six-dimensional
equal-spinning Myers-Perry black strings, for any finite azimuthal number $m$.
Perhaps surprisingly, this result appears to be valid only for $D=6$ equal-
spinning Myers-Perry black strings. Indeed, in Appendix B, we provide an
argument suggesting the absence of the instability in $D=2N+4$ dimensional
equal-spinning Myers-Perry black strings for $N\geq 2$ (i.e. for $D\geq 8$).
So $D=6$ is very special.

###  4.1 Wigner D-matrices

Unlike the Gregory-Laflamme instability, superradiant instabilities that are
sourced by rotation require non-axisymmetric ($m\neq 0$) modes. It is known on
fairly general grounds that superradiant modes satisfy
$\mathrm{Re}(\omega)<2m\Omega_{H}$, where $m$ is the azimuthal mode number, in
this case for the coordinate $\psi$ (recall footnote 4). We seek special
sectors of perturbations that are decoupled and induce the superradiant
instability in the Myers-Perry black string.

To describe decoupled perturbations in these conditions, we use the Wigner
D-matrices $D^{j}_{\ell^{\prime}\ell}(\theta,\phi,\psi)$. We ask the reader to
see Ishii:2021xmn  for a detailed review of the Wigner D-matrices in the
context of black hole perturbations and here we just highlight key aspects
that are fundamental for our study.

Let us define the angular momentum operators $L_{i}\equiv\mathrm{i}\xi_{i}$
and $R_{3}\equiv\mathrm{i}\frac{1}{2}\partial_{\psi}$. They satisfy the
commutation relations, $[L_{i},L_{j}]=\mathrm{i}\epsilon_{ijk}L_{k}$ and
$[L_{i},R_{3}]=0$. We can take the set of mutually commutative operators
$(L^{2},L_{3},R_{3})$ where $L^{2}\equiv L_{1}^{2}+L_{2}^{2}+L_{3}^{2}$.
Wigner D-matrices are defined as eigenfunctions of these operators:

$L^{2}D^{j}_{\ell^{\prime}\ell}=j(j+1)D^{j}_{\ell^{\prime}\ell}\ ,\quad
L_{3}D^{j}_{\ell^{\prime}\ell}=\ell^{\prime}D^{j}_{\ell^{\prime}\ell}\ ,\quad
R_{3}D^{j}_{\ell^{\prime}\ell}=\ell D^{j}_{\ell^{\prime}\ell}\ .$ (13)

The quantum numbers can take the values

$j=0,\frac{1}{2},1,\frac{3}{2},\dots\
,\qquad\ell^{\prime},\ell=-j,-j+1,\dots,j\ .$ (14)

There are convenient formulae for the derivatives of Wigner D-matrices:

$\begin{split}\partial_{\theta}D_{\ell}&=-\frac{\mathrm{i}}{2}(\epsilon_{-}e^{-2\mathrm{i}\psi}D_{\ell-1}+\epsilon_{+}e^{2\mathrm{i}\psi}D_{\ell+1})\,,\\\
\partial_{\phi}D_{\ell}&=-\mathrm{i}\ell\cos\theta
D_{\ell}+\frac{1}{2}\epsilon_{-}\sin\theta\,e^{-2\mathrm{i}\psi}D_{\ell-1}-\frac{1}{2}\epsilon_{+}\sin\theta\,e^{2\mathrm{i}\psi}D_{\ell+1}\,,\\\
\partial_{\psi}D_{\ell}&=-2\,\mathrm{i}\ell D_{\ell}\,,\end{split}$ (15)

where $\epsilon_{\pm}\equiv\sqrt{(j\mp\ell)(j\pm\ell+1)}$ and the common
indices $(j,\ell^{\prime})$ were suppressed to simplify the notation. We ask
the reader to see Hu:1974hh ; Ishii:2020muv  for a derivation of these
results.

For the following perturbation analysis, it is convenient to introduce 1-forms
$\sigma_{\pm}$ defined by

$\sigma_{\pm}=\frac{1}{2}(\sigma_{1}\mp\mathrm{i}\,\sigma_{2})=\frac{1}{2}e^{\mp
2\,\mathrm{i}\,\psi}(\mp\mathrm{i}\,\mathrm{d}\theta+\sin\theta\mathrm{d}\phi)\,,$
(16)

where $\sigma_{1,2}$ were defined in (3). The one-forms $\sigma_{\pm}$ are
“eigen 1-forms” of $R_{3}$, i.e.
$\pounds_{R_{3}}\sigma_{\pm}=\pm\sigma_{\pm}$.

The eigenvalue of $-R_{3}$ is the azimuthal mode number $m$. For example, we
say that $D^{j}_{\ell^{\prime}\ell}$ has azimuthal mode number $m=-\ell$. We
can also define the azimuthal mode number for tensors. For example, tensors
$D^{j}_{\ell^{\prime}\ell}\sigma_{-}^{2}$,
$D^{j}_{\ell^{\prime}\ell}\sigma_{+}\sigma_{-}$ and
$D^{j}_{\ell^{\prime}\ell}\sigma_{+}^{2}$ have $m=-\ell+2,-\ell,-\ell-2$,
respectively.

###  4.2 Linear perturbations

This time we focus our attention on a traceless-transverse perturbation of (1)
that has the form $\mathrm{d}s^{2}=\mathrm{d}s^{2}_{\mathrm{\rm
MP\,string}}+\delta\mathrm{d}s^{2}_{\hbox{\tiny SR}}$ where

$\displaystyle\delta\mathrm{d}s^{2}_{\hbox{\tiny SR}}$
$\displaystyle=h_{MN}\mathrm{d}x^{M}\mathrm{d}x^{N}=r^{2}e^{{\rm
i}\,k\,z}e^{-{\rm
i}\,\omega\,t}D^{j}_{\ell^{\prime}\ell}\,Q(r)\sigma_{-}^{2}\quad\text{with}\quad\ell^{\prime}=\ell=-j\,,$
$\displaystyle=\frac{1}{4}e^{{\rm i}\,k\,z}e^{-{\rm i}\,\omega\,t}e^{{\rm
i}\,(m-2)\,(\phi+2\psi)}\cos^{2(m-2)}\left(\frac{\theta}{2}\right)\,Q(r)\left(\sigma_{1}^{2}-\sigma_{2}^{2}+2\,{\rm
i}\,\sigma_{1}\,\sigma_{2}\right)$ (17)

which describes a superradiant perturbation with a charged scalar
$\mathbb{C}\mathrm{P}^{1}$ harmonic dependence in $\psi$ corresponding to
azimuthal quantum number $m=j+2$, and thus $m\in\\{2,5/2,3,7/2,\ldots\\}$; see
(14). Since this is the maximum azimuthal wavenumber for a given $j$, this
perturbation decouples from other perturbations and it is the simplest
disturbance that permits superradiance in the Myers-Perry black string.

The perturbation (4.2) is trivially in the traceless-transverse gauge (6) and
reduces the Lichnerowicz equation (7) to a second order ODE for $Q(r)$:

$\left(r^{3}FQ^{\prime}\right)^{\prime}+r\left[\frac{Hr^{2}}{F}\left(\omega-2m\frac{\Omega}{H}\right)^{2}-\frac{4m^{2}}{H}+4m-k^{2}r^{2}\right]Q=0\,.$
(18)

Using the tortoise coordinate, $dr_{\ast}=\sqrt{H}/Fdr$, and redefining the
perturbation variable as $\Phi=(rH)^{1/4}Q$, (18) can be rewritten in the
Schrödinger form

$\frac{d^{2}\Phi}{dr_{\ast}^{2}}-V\Phi=0\ ,$ (19)

where its potential reads:

$V\equiv\frac{F}{H}k^{2}-\left(\omega-2m\frac{\Omega}{H}\right)^{2}+\frac{4F}{r^{2}H}\left[(m-1)m-m^{2}\left(1-\frac{1}{H}\right)\right]+\frac{1}{r^{3/2}H^{1/4}}\frac{\mathrm{d}^{2}}{\mathrm{d}r_{\ast}^{2}}\left(r^{3/2}H^{1/4}\right)\
.$ (20)

Near infinity, the Schrödinger potential $V$ behaves as

$V|_{r\to\infty}\simeq
k^{2}-\omega^{2}+\left[4m(m-1)+\frac{3}{4}-\frac{k^{2}r_{+}^{2}}{1-\Omega_{H}^{2}r_{+}^{2}}\right]\frac{1}{r^{2}}+\mathcal{O}\left(r^{-4}\right).$
(21)

Note that the coefficient of $1/r^{2}$ becomes negative for a sufficiently
large $k$. This suggests that the gravitational perturbation might get
confined near the horizon (inside the ergoregion) and thus there might be room
for a superradiant instability Cardoso:2005vk  (note that in the scalar field
perturbation case, there is no Schrödinger potential barrier that confines the
superradiant modes which is consistent with the fact that there is no
superradiant instability; see appendix A). Moreover, for $m\to\infty$, the
$1/r^{2}$ term in (21) becomes positive. This is related to the fact that
there are no stable particle orbits in the five-dimensional Myers-Perry black
hole Tangherlini:1963 ; Frolov:2003en . So if the superradiant instability
happens to be present for the Myers-Perry black string at finite $m$, it
should definitely shut down in the eikonal ($m\to\infty$) limit.

The analysis above suggests the possible existence of a superradiant
instability for finite $m$, which we now attempt to find. To search for
possible unstable modes of (19), we need to impose the relevant physical
boundary conditions. At infinity, an asymptotic analysis of the ODE (18)
indicates the existence of two independent solutions: $Q\sim
r^{-3/2}e^{\pm\sqrt{k^{2}-\omega^{2}}\,r}$. We discard the exponentially
growing mode to preserve the Kaluza-Klein $\mathcal{M}^{1,4}\times S_{z}^{1}$
asymptotics. Near the horizon, the ODE has again two independent solutions:
$Q\sim(r-r_{+})^{\pm{\rm i}\,\frac{\omega-2m\Omega_{H}}{4\pi T_{H}}}$. We
impose boundary conditions that retain only the solution with the minus sign
to maintain regularity in ingoing Eddington-Finkelstein coordinates (see
footnote 10). Introducing, as we have for the Gregory-Laflamme instability,
the compact radial coordinate $y=\sqrt{1-\frac{r_{+}}{r}}$ and working in
units of $r_{+}$ as in (10), these boundary conditions can be imposed by
setting

$Q(y)=e^{-\frac{\sqrt{\widetilde{k}^{2}-\widetilde{\omega}^{2}}}{1-y^{2}}}\left(1-y^{2}\right)^{3/2}\Big{[}\big{(}2-y^{2}\big{)}y^{2}\Big{]}^{-{\rm
i}\,\frac{\widetilde{\omega}-2m\widetilde{\Omega}_{H}}{4\pi\widetilde{T}_{H}}}q(y)$
(22)

and requiring that $q$ is a smooth function. Smoothness of $q$ and the
equations of motion now yield a Neumann condition at the horizon and a Robin
condition at infinity for $q(y)$. Given parameters $\widetilde{\Omega}_{H}$
and $\widetilde{k}$, our ODE and its boundary conditions describe a quadratic
eigenvalue problem for $\widetilde{\omega}$.

To find the onset of the instability we will set
$\widetilde{\omega}=2m\widetilde{\Omega}_{H}$ and
$\widetilde{k}=\widetilde{k}_{(0)}$ (recall footnote 4). The above discussion
on boundary conditions still holds in this particular case, but the problem
becomes a quadratic eigenvalue problem for
$\widetilde{\kappa}\equiv\sqrt{\widetilde{k}_{(0)}^{2}-4m^{2}\widetilde{\Omega}_{H}^{2}}$.

Our numerical computation now proceeds as follows. As before, we find the
onset curve first, and then find the growth rates. We first fix
$m\in\\{2,5/2,3,7/2,\ldots\\}$ (we will start with $m=2$ and later consider
higher $m$). Then, scanning over $\widetilde{\Omega}_{H}$, we attempt to find
the onset modes by setting $\widetilde{\omega}=2m\widetilde{\Omega}_{H}$ (the
condition for the onset of superradiance), relabelling
$\widetilde{k}=\widetilde{k}_{(0)}$, then solving the the corresponding
quadratic eigenvalue problem for $\widetilde{\kappa}$ on a Myers-Perry black
string family, which also ultimately gives us a value for
$\widetilde{k}_{(0)}$. We expect real, positive solutions for
$\widetilde{\kappa}$ to exist for a range of dimensionless rotations
$\widetilde{\Omega}_{H}|_{c}<\widetilde{\Omega}_{H}<\widetilde{\Omega}^{\mathrm{ext}}_{H}=1/\sqrt{2}$,
which gives us an onset curve $\widetilde{k}_{(0)}|_{\hbox{\tiny
SR}}(\widetilde{\Omega}_{H})$ for this range of $\widetilde{\Omega}_{H}$.

Having found the critical $\widetilde{k}_{(0)}(\widetilde{\Omega}_{H})$ for a
given $m$, we then expect that for values of $\widetilde{k}$ just critical
value $\widetilde{k}\lesssim\widetilde{k}_{{0}}$, the mode will be unstable
with frequency ${\rm Im}\,\widetilde{\omega}>0$. We can find this frequency by
solving the aforementioned quadratic eigenvalue problem for
$\widetilde{\omega}$. We can then continue to vary $\widetilde{\Omega}_{H}$
and $\widetilde{k}$ to explore the regions where ${\rm
Im}\,\widetilde{\omega}>0$. Numerically, we again rely on pseudospectral
methods with a Chebyshev grid and use a Newton-Raphson method to track modes
as we vary parameters. We follow the strategy described in Cardoso:2013pza ;
Dias:2015nua  to solve eigenvalue problems using Newton-Raphson, fixing the
normalization by setting $q|_{y=0}=1$.

###  4.3 Superradiant unstable regions and growth rates

Figure 3: The blue disk curves with endpoints $A$ and $c$ corresponds to the
$m=2$ onset mode (with $\widetilde{\omega}=2m\widetilde{\Omega}_{H}$) of the
superradiant instability of the Myers-Perry black string. This curve exists in
the window
$\widetilde{\Omega}_{H}|_{c}\leq\widetilde{\Omega}_{H}\leq\widetilde{\Omega}_{H}|_{A}$
where $\widetilde{\Omega}_{H}|_{c}=3/5$ and
$\widetilde{\Omega}_{H}|_{A}\equiv\widetilde{\Omega}^{\mathrm{ext}}_{H}=1/\sqrt{2}$
is the angular velocity at extremality (the vertical dashed line identifies
this extremality). Left: Myers-Perry black strings are superradiant unstable
in the triangular region $ABc$ bounded by the shown curves. Besides the onset
curve ($Ac$), we also show the green curve
$\widetilde{k}_{\star}(\widetilde{\Omega}_{H})$, which describes modes with
${\rm Re\,\widetilde{\omega}}=\widetilde{k}$ (and ${\rm
Im}\,\widetilde{\omega}=0$). This green curve
$\widetilde{k}_{\star}(\widetilde{\Omega}_{H})$ begins at point $c$ where it
intersects the magenta dashed line $\widetilde{k}=2m\widetilde{\Omega}_{H}$
(with $m=2$), and ends at point $B$ where
$\widetilde{\Omega}_{H}=\widetilde{\Omega}^{\mathrm{ext}}_{H}=1/\sqrt{2}$. For
reference, point $A$ has
$(\widetilde{\Omega}_{H},\widetilde{k}_{(0)},\widetilde{L}_{(0)})_{A}\simeq(1/\sqrt{2},3.805857,1.650925)$,
point $B$ has
$(\widetilde{\Omega}_{H},\widetilde{k}_{(0)},\widetilde{L}_{(0)})_{B}=(1/\sqrt{2},3/\sqrt{2},2\sqrt{2}\pi/3)$,
and point $c$ has
$(\widetilde{\Omega}_{H},\widetilde{k}_{(0)},\widetilde{L}_{(0)})_{c}=(3/5,12/5,5\pi/6)$,
where $\widetilde{L}_{(0)}=2\pi/\widetilde{k}_{(0)}$. Right: This plot
demonstrates that the onset curve $Ac$ satisfies
$\widetilde{k}_{(0)}^{2}-4m^{2}\widetilde{\Omega}_{H}^{2}\geq 0$, which
implies that these onset modes decay exponentially in the radial direction.

Let us discuss the results in detail for $m=2$ before proceeding with higher
values of $m$. We begin with the critical superradiant onset curve
$\widetilde{k}_{(0)}(\widetilde{\Omega}_{H})$, where
$\widetilde{\omega}=2m\widetilde{\Omega}_{H}$ (and is critical because ${\rm
Im}\,\widetilde{\omega}=0$). This curve is the blue $cA$ curve in Fig. 3, and
exists for a range of rotation
$\widetilde{\Omega}_{H}|_{c}<\widetilde{\Omega}_{H}<\widetilde{\Omega}^{\mathrm{ext}}_{H}=1/\sqrt{2}$.
Within this range, wavenumbers just below critical
$\widetilde{k}\lesssim\widetilde{k}_{{0}}$ are unstable.

For reasons that will later become clear, we wish to demonstrate that the
onset modes at $\widetilde{k}_{(0)}(\widetilde{\Omega}_{H})$ decay
exponentially in the radial direction (as argued previously when discussing
the asymptotic boundary condition of the problem). Since, by the known
asymptotic behaviour and our boundary conditions, the solutions have a falloff
given by $e^{-\sqrt{k^{2}-\omega^{2}}\,r}$ and this onset curve $cA$ satisfies
$\widetilde{\omega}=2m\widetilde{\Omega}_{H}$, we simply need to show that
$\widetilde{k}^{2}-4m^{2}\widetilde{\Omega}_{H}^{2}>0$. We can easily see from
the right panel of Fig. 3 that this condition is satisfied within the given
range
$\widetilde{\Omega}_{H}|_{c}<\widetilde{\Omega}_{H}\leq\widetilde{\Omega}^{\mathrm{ext}}_{H}=1/\sqrt{2}$,
and appears to be marginal, $\widetilde{k}\to 2m\widetilde{\Omega}_{H}$, at
$\widetilde{\Omega}_{H}=\widetilde{\Omega}_{H}|_{c}$.

After finding this onset curve $\widetilde{k}_{(0)}(\widetilde{\Omega}_{H})$,
we can then explore the range of parameters where an instability exists (i.e.
where ${\rm Im}\,\widetilde{\omega}>0$). The real and imaginary parts of
$\widetilde{\omega}$ for parameters with such an instability are shown in Fig.
4. We see that the unstable region in $(\widetilde{\Omega}_{H},\widetilde{k})$
parameter space is triangular, and lies between the blue onset curve
$\widetilde{k}_{(0)}(\widetilde{\Omega}_{H})$, extremality
$\widetilde{\Omega}_{H}=\widetilde{\Omega}^{\mathrm{ext}}_{H}=1/\sqrt{2}$, and
a new green curve which we will call
$\widetilde{k}_{\star}(\widetilde{\Omega}_{H})$. This unstable triangular
region is the same as the triangular region $ABc$ shown in the left panel of
Fig. 3. The blue and green curves (both with ${\rm Im}\,\widetilde{\omega}=0$)
of Fig. 4 correspond, respectively, to the blue onset curve $cA$ and green
$cB$ curve of Fig. 3. On other hand the extremal $AB$ curve in Fig. 3
corresponds to a curve with ${\rm Im}\,\widetilde{\omega}>0$ at
$\widetilde{\Omega}_{H}=\widetilde{\Omega}_{H}^{\rm ext}$.

Figure 4: Frequency for $m=2$ unstable superradiant modes, as a function of
parameters of Myers-Perry black strings. The instability is present (i.e.
${\rm Im}\,\widetilde{\omega}>0$) for
$\widetilde{\Omega}_{H}|_{c}\leq\widetilde{\Omega}_{H}\leq 1/\sqrt{2}$, with
$\widetilde{\Omega}_{H}|_{c}=3/5$, and
$\widetilde{k}_{\star}(\widetilde{\Omega}_{H})\leq\widetilde{k}\leq\widetilde{k}_{(0)}(\widetilde{\Omega}_{H})$.
The critical curve $\widetilde{k}_{(0)}(\widetilde{\Omega}_{H})$ (and ${\rm
Im}\,\widetilde{\omega}=0$) was also computed independently in Fig. 3 and we
show it as a continuous blue line in the ${\rm Im}\,\widetilde{\omega}=0$
plane. On the other hand, the green curve represents the line
$\widetilde{k}_{\star}(\widetilde{\Omega}_{H})$ as defined parametrically in
(24). It meets the merger blue curve at $\widetilde{\Omega}_{H}|_{c}=3/5$. The
${\rm Im}\,\widetilde{\omega}=0$ plane matches the left panel of Fig. 3.

Let us now explain the physical origin of this new green curve
$\widetilde{k}_{\star}(\widetilde{\Omega}_{H})$. As we have mentioned earlier,
superradiant instabilities require two ingredients: amplification from an
ergoregion, and a confining mechanism. The blue onset curve
$\widetilde{k}_{(0)}(\widetilde{\Omega}_{H})$ we have found satisfies the
superradiant condition $\widetilde{\omega}=2m\widetilde{\Omega}_{H}$, which
corresponds to the place where the amplification mechanism shuts off. It is
natural, then, to suggest that the green curve
$\widetilde{k}_{\star}(\widetilde{\Omega}_{H})$ corresponds to where the
confining mechanism shuts off. This occurs when the exponential fall-off of
$e^{-\sqrt{k^{2}-\omega^{2}}\,r}$ becomes marginal with
$\widetilde{k}=\widetilde{\omega}$ (recall that the curve
$\widetilde{k}_{\star}(\widetilde{\Omega}_{H})$ already has
$\mathrm{Im}\,\widetilde{\omega}=0$).

We have already seen evidence for this idea from the right panel of Fig. 3 and
the discussion a few paragraphs ago, where we show that the modes on the onset
$Ac$ curve $\widetilde{k}_{(0)}(\widetilde{\Omega}_{H})$ have this exponential
fall-off and that this fall-off becomes marginal at
$\widetilde{\Omega}_{H}=\widetilde{\Omega}_{H}|_{c}$, which is precisely the
location where the onset curve $\widetilde{k}_{(0)}(\widetilde{\Omega}_{H})$
intersects this new green $cB$ curve
$\widetilde{k}_{\star}(\widetilde{\Omega}_{H})$.

Figure 5: Frequency for the unstable superradiant mode of Myers-Perry black
strings with fixed $\widetilde{\Omega}_{H}=0.66$, as a function of wavenumber
$kr_{+}$. Left: The imaginary part of the frequency ${\rm Im}(\omega r_{+})$
is positive for $kr_{+}\in(\widetilde{k}_{\star},\widetilde{k}_{(0)})$ where
the system is unstable to superradiance. The blue diamond on the right with
$kr_{+}=\widetilde{k}_{(0)}\simeq 2.673761$ corresponds to the superradiance
onset found in Fig. 3. One has ${\rm Im}(\omega r_{+})\to 0$ as
$kr_{+}\to\widetilde{k}_{\star}=\frac{3\sqrt{1411}}{50}$ (green diamond on the
left side of the plot). Right: The magenta dashed line describes the curve
Re$(\omega r_{+})=kr_{+}$. We see that ${\rm Re}(\omega r_{+})<kr_{+}$, and
that ${\rm Re}(\omega r_{+})\to kr_{+}$ as $kr_{+}\to\widetilde{k}_{\star}$
from above (this is best seen in the inset plot which displays
$\widetilde{k}^{2}-(\mathrm{Re}\,\widetilde{\omega})^{2}$ as a function of
$\widetilde{k}$). This demonstrates that when the instability shuts down at
$kr_{+}=\widetilde{k}_{\star}$, the fall-off $e^{-\sqrt{k^{2}-\omega^{2}}\,r}$
ceases to be exponential.

To gather more evidence, we focus our attention on Myers-Perry black strings
with a fixed $\widetilde{\Omega}_{H}$ and compute the frequency
$\widetilde{\omega}$ while scanning over $\widetilde{k}$. The results for
$\widetilde{\Omega}_{H}=0.66$ are shown in Fig. 5. The left panel demonstrates
that the instability is shutting off as $\widetilde{k}$ approaches
$\widetilde{k}_{\star}(0.66)=\frac{3\sqrt{1411}}{50}\simeq 2.253797$ from
above (which is the point in the curve $cB$ of Fig. 3 with
$\widetilde{\Omega}_{H}=0.66$). The right panel demonstrates that as this
occurs, the exponential falloff $e^{-\sqrt{k^{2}-\omega^{2}}\,r}$ is becoming
marginal.

With this evidence at hand, we can now attempt to find the green curve
(hereafter referred to as the confining cutoff curve)
$\widetilde{k}_{\star}(\widetilde{\Omega}_{H})$ directly by enforcing
$\widetilde{\omega}=\widetilde{k}$ in our ODE. This requires that we revisit
the asymptotic Frobenius analysis of (18). Indeed, from (22), we know that
generic superradiant modes decay as
$e^{-\frac{\sqrt{\widetilde{k}^{2}-\widetilde{\omega}^{2}}}{1-y^{2}}}$.
However, when ${\rm Re}\,\widetilde{\omega}=\widetilde{k}$ and ${\rm
Im}\,\widetilde{\omega}=0$, the argument of this exponential vanishes. For
generic $m$, a Frobenius analysis now gives the asymptotic decays
$Q\sim(1-y)^{1\pm\gamma}$ with

$\gamma=\sqrt{(2m-1)^{2}-\frac{\widetilde{k}^{2}}{1-\widetilde{\Omega}_{H}^{2}}}\
.$ (23)

It is not clear which combination of these modes we should take as a boundary
condition. However, note that there is a further special case: if $\gamma=0$,
then the two decays above degenerate to $Q\sim(1-y)$ and the second
independent solution now decays logarithmically. For reasons that are not
completely understood, the curve
$\widetilde{k}^{(m)}_{\star}(\widetilde{\Omega}_{H})$ satisfies this
degenerate case.That is to say, apparently, $\gamma=0$ is the borderline
behaviour for whether the asymptotic behaviour is oscillatory or not and turns
out to be the condition for
$\widetilde{k}^{(m)}_{\star}(\widetilde{\Omega}_{H})$. We have verified this
empirically by directly solving for such modes, and by verifying that such
modes are approached as $\widetilde{k}\to\widetilde{k}^{(m)}_{\star}$ (see
again Fig. 5). From this evidence, and solving for $\widetilde{k}$ from the
condition $\gamma=0$, we conclude that

$\widetilde{k}^{(m)}_{\star}=(2m-1)\sqrt{1-\widetilde{\Omega}_{H}^{2}}\;,$
(24)

which is an explicit analytic expression for
$\widetilde{k}^{(m)}_{\star}(\widetilde{\Omega}_{H})$ that is fully consistent
with our numerical results. For $m=2$ this gives the cutoff curve $cB$ of Fig.
3 (and for $m=2,3,4$ this gives the curves $cB$, $c^{\prime}B^{\prime}$ and
$c^{\prime\prime}B^{\prime\prime}$, respectively, of later Fig. 6). Note,
though that the analytic expression (24) is based on an assumption on the
asymptotic behaviour of the linear solution for which we can only justify
empirically. Furthermore, obtaining this result does not actually require us
to solve a full boundary value problem, for which the boundary conditions
remain unclear. It remains a possibility that the actual linear solution in
the limit of the cutoff curve might not be regular. Indeed, to obtain (24) we
did not have to require, as a boundary condition, that the logarithmic
solution must vanish.

Now we can also determine the frequency $\widetilde{\Omega}^{(m)}_{H}$ where
$\widetilde{k}^{(m)}_{(0)}$ and $\widetilde{k}^{(m)}_{\star}$ intersect; for
$m=2$ this is point $c$ in Fig. 3 where the zero mode $cA$ and cutoff $cB$
curves meet. Recall that $\widetilde{k}^{(m)}_{(0)}$ satisfies
$\widetilde{\omega}=2m\widetilde{\Omega}_{H}$ and
$\widetilde{k}^{(m)}_{\star}$ satisfies $\widetilde{\omega}=\widetilde{k}$.
Combining these with (24), we find that these curves meet at

$\widetilde{\Omega}_{H}^{(m)}=\frac{(2m-1)}{\sqrt{8m^{2}-4m+1}}\quad\Leftrightarrow\quad\frac{{\Omega}_{H}^{(m)}}{\Omega^{\mathrm{ext}}_{H}}=\frac{\sqrt{2}(2m-1)}{\sqrt{8m^{2}-4m+1}}\,.$
(25)

The superradiant onset in the extremal Myers-Perry limit can be also obtained
analytically (for $m=2$, this is point $A$ in Fig. 3). In the extremal Myers-
Perry limit
$\widetilde{\Omega}_{H}=\widetilde{\Omega}^{\mathrm{ext}}_{H}=1/\sqrt{2}$ and
at the onset of the superradiant instability
$\widetilde{\omega}=2m\widetilde{\Omega}_{H}$, the perturbation equation (18)
becomes

$Q^{\prime\prime}+\frac{3r^{2}+r_{+}^{2}}{r(r^{2}-r_{+}^{2})}Q^{\prime}-\frac{r^{2}\\{(k^{2}r_{+}^{2}-2m^{2})r^{2}-4mr_{+}^{2}\\}}{r_{+}^{2}(r^{2}-r_{+}^{2})^{2}}Q=0\
.$ (26)

At the horizon, $Q$ behaves as $Q=(r-r_{+})^{\eta}$ with
$\eta=\left(-1\pm\sqrt{\widetilde{k}^{2}-(2m^{2}+4m-1)}\,\right)/2$. For
reasons that are again not completely understood, the onset of instability at
extremality seems emperically to be located at the boundary of the oscillating
and non-oscillating behaviors, when we have the degenerate case $\eta=-1$We
therefore have

$\widetilde{k}_{(0)}^{(m)}|_{\hbox{\tiny ext}}=\sqrt{2m^{2}+4m-1}\,,$ (27)

which gives an analytic expression for the onset mode wavenumber at
extremality. For $m=2$, we have $\widetilde{k}=\sqrt{15}\simeq 3.873$ which is
point $A$ in Fig. 3 where we also see that our numerical results for the onset
curve $\widetilde{k}^{(m)}_{(0)}(\widetilde{\Omega}_{H})$ approach $A$ as
given by (27) in the extremal limit. Moreover, for $m=3$ and $m=4$ equation
(27) also gives the points $A^{\prime}$ and $A^{\prime\prime}$ in later Fig.
6.

Figure 6:  Left panel: Regions $ABc$, $A^{\prime}B^{\prime}c^{\prime}$, and
$A^{\prime\prime}B^{\prime\prime}c^{\prime\prime}$ where Myers-Perry black
strings are unstable to $m=2,3,4$ superradiant modes, respectively. The orange
squares correspond to the onset of the Gregory-Laflamme instability already
shown in Figs. 1 and 3 (the region below this curve is unstable). Right panel:
Zoom-in of region unstable to both Gregory-Laflamme and superradiance.
$\alpha$ and $\beta$ mark where the Gregory-Laflamme onset curve intersects
with the $m=2$ unstable superradiant region. $\alpha$ lies at
$(\widetilde{\Omega}_{H},\widetilde{k}_{(0)},\widetilde{L}_{(0)})_{\alpha}\simeq(0.69054,2.95350,2.12737)$,
where to the left of $\alpha$ one has $\widetilde{k}_{(0)}\big{|}_{\hbox{\tiny
SR}}>\widetilde{k}_{(0)}\big{|}_{\hbox{\tiny GL}}$, while to the right of
$\alpha$ one has $\widetilde{k}_{(0)}\big{|}_{\hbox{\tiny
SR}}<\widetilde{k}_{(0)}\big{|}_{\hbox{\tiny GL}}$. $\beta$ lies at
$(\widetilde{\Omega}_{H},\widetilde{k}_{(0)},\widetilde{L}_{(0)})_{\beta}\simeq(0.63478,2.31810,2.71049)$.

We can now show the superradiant onset curve and confining cutoff curves
$\widetilde{k}^{(m)}_{(0)}$ and $\widetilde{k}^{(m)}_{\star}$ also for other
values of $m$. The left panel of Fig. 6 show the results for $m=2,3,4$. (We
also obtained qualitatively similar curves for half-integer modes $m=5/2,7/2$,
but do not show them to avoid clutter.) The unstable regions
$A^{\prime}B^{\prime}c^{\prime}$ and
$A^{\prime\prime}B^{\prime\prime}c^{\prime\prime}$ for $m=3$ and $m=4$,
respectively are qualitatively similar to the unstable region $ABc$ for $m=2$
but transported to higher values of $\widetilde{k}$ as $m$ grows. The cutoff
curves $cB$, $c^{\prime}B^{\prime}$ and $c^{\prime\prime}B^{\prime\prime}$ are
described by (24) with $m=2,3,4$, respectively, and points $c$, $c^{\prime}$
and $c^{\prime\prime}$ are described by (25) with $m=2,3,4$, respectively. The
zero mode curves $cA$, $c^{\prime}A^{\prime}$ and $c^{\prime\prime}A^{\prime}$
for $m=2,3,4$ were obtained numerically solving the eigenvalue problem for
$\widetilde{k}^{(m)}_{(0}(\widetilde{\Omega}_{H})$, which approach the points
$A$, $A^{\prime}$ and $A^{\prime\prime}$ at extremality as given by (27).
Perturbations with adjacent $m$’s also overlap in some small regions of
parameter space close to extremality. So there are Myers-Perry strings that
are superradiant unstable to a few $m$’s (but not an infinite number of them
like in other superradiant black hole systems).

We also see that the unstable regions ($ABc$,
$A^{\prime}B^{\prime}c^{\prime}$,
$A^{\prime\prime}B^{\prime\prime}c^{\prime\prime}$, …) decrease as $m$
increases. Essentially, this occurs because the intersection points
$c,c^{\prime},c^{\prime\prime},\ldots$ described by (25) have an angular
velocity that is increasingly close to $\widetilde{\Omega}_{H}^{\rm ext}$ as
$m$ increases, being exactly $\widetilde{\Omega}_{H}^{\rm ext}$ in the limit
$m\to\infty$. So in the eikonal limit ($m\to\infty$), the unstable region
becomes a measure zero region in parameter space and the eikonal limit of
points $A,B$ and $c$ all collapse to a common single point at
$\widetilde{\Omega}_{H}=\widetilde{\Omega}_{H}^{\rm ext}$ and
$\widetilde{k}\to\infty$, as follows from (25) and (27) in the limit
$m\to\infty$. That is to say, the superradiant instability for Myers-Perry
black strings shuts down as $m\to\infty$, which is in agreement with the fact
that the Myers-Perry black string has no stable circular geodesics
Tangherlini:1963 ; Frolov:2003en . In this precise sense, this is a ‘finite-m’
superradiant instability that occurs in black strings, but is not observed in
(known) superradiant black holes.

In Fig. 6, we also include the onset curve (orange squares) for the Gregory-
Laflamme instability that we have already computed in section 3. We see that
there are regions of parameter space where either, none, or both types of
instabilities are present.

Figure 7: Frequency for $m=3$ unstable superradiant modes, as a function of
parameters of Myers-Perry black strings. The instability is present (i.e.
${\rm Im}\,\widetilde{\omega}>0$) for
$\widetilde{\Omega}_{H}|_{c^{\prime}}\leq\widetilde{\Omega}_{H}\leq
1/\sqrt{2}$, with $\widetilde{\Omega}_{H}|_{c^{\prime}}=\frac{5}{\sqrt{61}}$,
and
$\widetilde{k}_{\star}(\widetilde{\Omega}_{H})\leq\widetilde{k}\leq\widetilde{k}_{(0)}(\widetilde{\Omega}_{H})$.
The zero mode curve $\widetilde{k}_{(0)}(\widetilde{\Omega}_{H})$ (and ${\rm
Im}\,\widetilde{\omega}=0$) was computed independently (see
$c^{\prime}A^{\prime}$) in Fig. 6 and we show it as a continuous brown line in
the ${\rm Im}\,\widetilde{\omega}=0$ plane. On the other hand, the light-brown
curve represents the confining cutoff line
$\widetilde{k}_{\star}(\widetilde{\Omega}_{H})$ as defined parametrically in
(24) for $m=3$ (i.e. the cutoff curve $c^{\prime}B^{\prime}$ in Fig. 6). It
meets the zero mode curve at $\widetilde{\Omega}_{H}|_{c^{\prime}}$.

Figure 8: Frequency for $m=4$ unstable superradiant modes, as a function of
parameters of Myers-Perry black strings. The instability is present (i.e.
${\rm Im}\,\widetilde{\omega}>0$) for
$\widetilde{\Omega}_{H}|_{c^{\prime\prime}}\leq\widetilde{\Omega}_{H}\leq
1/\sqrt{2}$, with
$\widetilde{\Omega}_{H}|_{c^{\prime\prime}}=\frac{7}{\sqrt{113}}$, and
$\widetilde{k}_{\star}(\widetilde{\Omega}_{H})\leq\widetilde{k}\leq\widetilde{k}_{(0)}(\widetilde{\Omega}_{H})$.
The zero mode curve $\widetilde{k}_{(0)}(\widetilde{\Omega}_{H})$ (and ${\rm
Im}\,\widetilde{\omega}=0$) was computed independently (see
$c^{\prime\prime}A^{\prime\prime}$) in Fig. 6 and we show it here as a
continuous black line in the ${\rm Im}\,\widetilde{\omega}=0$ plane. On the
other hand, the gray curve represents the line
$\widetilde{k}_{\star}(\widetilde{\Omega}_{H})$ as defined parametrically in
(24) for $m=4$ (i.e. the cutoff curve $c^{\prime\prime}B^{\prime\prime}$ in
Fig. 6). It meets the zero mode black curve at
$\widetilde{\Omega}_{H}|_{c^{\prime\prime}}$. It is very difficult to obtain
solutions in the neighborhood of the cutoff gray line
$\widetilde{k}_{\star}(\widetilde{\Omega}_{H})$ so we do not attempt to find
them in the present $m=4$ case; hence we find a gap between when the surface
approaches this gray borderline.

For completeness, in Figs. 7 and 8, we show the real and imaginary parts of
the frequency for $m=3$ and $m=4$ superradiant instabilities (in the same
manner as the $m=2$ case in Fig. 4). We find that for each $m$, increasing the
rotation also (typically) increases the growth rate. We also see that
increasing $m$ by one roughly reduces the growth rate by two orders of
magnitude.

Until now, we have parametrised the Myers-Perry black strings using
$\widetilde{\Omega}_{H}$ and $\widetilde{k}$. This was for the convenience of
computation and presentation. However, for the purposes of discussing the time
evolution of the instabilities, as well as the associated novel black string
solutions that branch from the onset of these instabilities, it is more
convenient to use the conserved quantities of energy and angular momenta as
parameters. In time evolution, it is natural to keep the Kaluza-Klein circle
fixed (i.e. we fix its length $L=\widetilde{L}r_{+}$), so it is natural to
express quantities in units of $L$.

From (5), the dimensionless energy, $\mathcal{E}\equiv E/L^{3}$, and
dimensionless angular momenta, $\mathcal{J}\equiv J/L^{4}$, are given by

$\mathcal{E}=\frac{1}{G_{6}}\,\frac{3\pi}{8\widetilde{L}^{2}}\frac{1}{1-\widetilde{a}^{2}}=\frac{1}{G_{6}}\,\frac{3\widetilde{k}^{2}}{32\pi(1-\widetilde{\Omega}_{H}^{2})}\,,\qquad\mathcal{J}=\frac{1}{G_{6}}\,\frac{\pi}{4\widetilde{L}^{3}}\,\frac{\widetilde{a}}{1-\widetilde{a}^{2}}=\frac{1}{G_{6}}\,\frac{\widetilde{k}^{3}\widetilde{\Omega}_{H}}{32\pi^{2}(1-\widetilde{\Omega}_{H}^{2})}\,.$
(28)

At extremality, we have

$\mathcal{J}^{\hbox{\tiny
ext}}(\mathcal{E})=\frac{2^{3/2}}{3^{3/2}\pi^{1/2}}G_{6}^{1/2}\mathcal{E}^{3/2}\;.$
(29)

We can now use (28) to translate the onset and cutoff curves of Figs. 3 and 6
into the $\mathcal{E}$-$\mathcal{J}$ phase diagram. For example, the $m=2$
superradiant onset $cA$ curve $\widetilde{k}_{(0)}(\widetilde{\Omega}_{H})$ of
Fig. 3, for $\widetilde{\Omega}_{H}|_{c}\leq\widetilde{\Omega}_{H}\leq
1/\sqrt{2}$, defines via (28) the onset curve for the $m=2$ superradiant
instability of Myers-Perry strings in the $\mathcal{E}$-$\mathcal{J}$ phase
diagram. This curve turns out to be very close to the curve that describes
extremal Myers-Perry strings described by (29). Therefore, to have a plot
where the various unstable regions can be presented clearly, it is convenient
to plot instead $\mathcal{E}$ vs
$\Delta\mathcal{J}\equiv(\mathcal{J}-\mathcal{J}^{\hbox{\tiny
ext}})|_{\hbox{\tiny same}\,\mathcal{E}}$ where the latter describes the
angular momentum difference between a given solution $-$ e.g. the superradiant
onset or cutoff Myers-Perry family $-$ and the extremal Myers-Perry string at
the same $\mathcal{E}$. This phase diagram $\mathcal{E}$-$\Delta\mathcal{J}$
is displayed in Fig. 9.

Figure 9: Onset/cutoff curves for the $m=2,3,4$ superradiant instability and
the zero mode curve of Gregory-Laflamme instability (orange squares on right
panel). The vertical axis describes the difference between a given solution
and the extremal Myers-Perry solution that has the same energy,
$\Delta\mathcal{J}\equiv\left(\mathcal{J}-\mathcal{J}^{\mathrm{ext}}\right)|_{\hbox{\tiny
same}\,\mathcal{E}}$. Equally-spinning Myers-Perry black strings are unstable
to $m=2$, $m=3$ and $m=4$ superradiant perturbations if they lie inside the
regions $ABc$, $A^{\prime}B^{\prime}c^{\prime}$ and
$A^{\prime\prime}B^{\prime\prime}c^{\prime\prime}$, respectively. They are
unstable to the $m=0$ Gregory-Laflamme instability if they are contained above
the orange square curve (and below the horizontal extremal curve with
$\Delta\mathcal{J}=0$) in the right panel. All the points in this figure are
in a 1-to-1 correspondence to the same points in Fig. 6.

In Fig. 9, Myers-Perry black strings exist with $\mathcal{E}>0$ for
arbitrarily large $\mathcal{E}$, and for any angular momenta at or below
extremality $\Delta\mathcal{J}\leq 0$. The triangular region $ABc$
(respectively, $A^{\prime}B^{\prime}c^{\prime}$ and
$A^{\prime\prime}B^{\prime\prime}c^{\prime\prime}$) is unstable to $m=2$
($m=3$ and $m=4$) superradiant modes.121212The horizontal red line with
$\Delta\mathcal{J}=0$ represents the 1-parameter family of extremal Myers-
Perry strings with $\widetilde{\Omega}_{H}=1/\sqrt{2}$. It extends to
arbitrarily large $\mathcal{E}$. Non-extremal Myers-Perry strings exist below
this line. On the other hand, the blue disk curve $Ac$ represents Myers-Perry
strings at the onset of the $m=2$ superradiant instability, as described by
(28) with $\widetilde{k}\equiv\widetilde{k}_{(0)}(\widetilde{\Omega}_{H})$,
for $\widetilde{\Omega}_{H}|_{c}\leq\widetilde{\Omega}_{H}\leq 1/\sqrt{2}$.
The endpoints of this zero mode $cA$ curve are determined by the solutions
(28) with $(\widetilde{\Omega}_{H},\widetilde{k}_{(0)})|_{c}=(3/5,12/5)$ and
$(\widetilde{\Omega}_{H},\widetilde{k}_{(0)})|_{A}=(1/\sqrt{2},\sqrt{15})$, as
follows from (25) and (27). The confining cutoff curve for $m=2$ superradiant
family, namely the curve $cB$ in Fig. 3 or 6 is described by
$\widetilde{k}^{(m)}_{\star}(\widetilde{\Omega}_{H})$ in (24). This
translates, after using (28), into a curve with constant
$G_{6}\mathcal{E}=27/(32\pi)$ and
$G_{6}\mathcal{J}=27\widetilde{\Omega}_{H}\sqrt{1-\widetilde{\Omega}_{H}^{2}}/(32\pi^{2})$
with $3/5\leq\widetilde{\Omega}_{H}\leq 1/\sqrt{2}$ in the
$\mathcal{E}$-$\mathcal{J}$ phase diagram. This gives the vertical line $Bc$
in the phase diagram $\mathcal{E}$ vs $\Delta\mathcal{J}$ of Fig. 9. We can
now repeat this exercise for the Gregory-Laflamme and $m=3,4$ superradiant
curves of Fig. 6 to find the associated onset/cutoff curves for these modes in
the $\mathcal{E}$ vs $\Delta\mathcal{J}$ phase diagram of Fig. 9. The region
above the orange squares line $A\alpha\beta\ldots$ (and below the red
horizontal extremal line) in the right panel of Fig. 9 are Gregory-Laflamme
unstable.

Finally, recall that although we have obtained qualitatively similar curves
for $m=5/2,7/2$ but do not show them to avoid clutter. Also, recall from the
discussions of section 3.1 that while there are Gregory-Laflamme type
instabilities with angular dependence Dias:2010eu , they only exist above a
critical rotation parameter and typically have lower growth rate.
Additionally, Myers-Perry black strings that are unstable to these other
Gregory-Laflamme modes, are also unstable to the one we consider in Fig. 9).

###  4.4 The many faces of the superradiant onset

The superradiant perturbation $\delta\mathrm{d}s^{2}_{\hbox{\tiny SR}}$ in
(4.2) breaks some of the isometries of the Myers-Perry black string, which
recall from (4) is $\mathbb{R}_{t}\times U(1)_{z}\times U(1)_{\psi}\times
SU(2)$, where the direction $z$ along the string is assumed to be Kaluza-Klein
compactified. In (4.2) one can see that $SU(2)$ is manifestly broken if we
consider a perturbation with $m>2$. However, for $m=2$, we have
$\omega=2m\Omega_{H}=4\Omega_{H}$ at the onset of the superradiant instability
and, in these conditions, the gravitational perturbation (4.2) can be written
as

$\displaystyle h_{MN}\mathrm{d}x^{M}\mathrm{d}x^{N}=r^{2}e^{-4\,{\rm
i}\,\Omega_{H}t+{\rm i}\,k\,z}\,Q(r)\sigma_{-}^{2}\ .$ (30)

This is clearly invariant under $SU(2)$ and hence, when we extend the analysis
nonlinearly, one expects that the $m=2$ superradiant onset may be a
bifurcation line for a new family of black string solutions that preserve
$SU(2)$. However, it is less obvious what other isometries are preserved or
broken by the perturbation (30) and this deserves further discussion. In fact,
some linear combinations of the isometries of the original background can be
preserved, as we discuss next.

To find which isometries are broken by the perturbation (30), it is convenient
to change to a coordinate frame $\\{T,Z,\Psi\\}$ such that131313In this frame
$\\{T,Z,\Psi\\}$, the background Myers-Perry string is rotating at infinity,
and the angular velocity at the horizon vanishes.

$T=t\,,\quad Z=z\,,\quad\Psi=\psi-\Omega_{H}t+\frac{k}{4}z\,.$ (31)

Their dual vectors are

$\partial_{T}=\partial_{t}+\Omega_{H}\partial_{\psi}\,,\quad\partial_{Z}=\partial_{z}-\frac{k}{4}\partial_{\psi}\,,\quad\partial_{\Psi}=\partial_{\psi}\,.$
(32)

In these coordinates, the isometry group of the background Myers-Perry black
string is $\mathbb{R}_{T}\times U(1)_{Z}\times U(1)_{\Psi}\times SU(2)$. But
now we explicitly see that, using also the RHS of (16), the $t$\- and $z$\-
dependence in (30) is totally absorbed in $\Psi$ by the coordinate
transformation. That is to say, let $\Sigma_{i}$ denote the invariant 1-forms
for $(\theta,\phi,\Psi)$, i.e. with $\psi$ in $\sigma_{i}$ of (3) replaced by
$\Psi$. Then, after the coordinate transformation (31), one finds that the
$m=2$ onset superradiant perturbation (30) simply reads

$h_{MN}\mathrm{d}x^{M}\mathrm{d}x^{N}=r^{2}\,Q(r)\Sigma_{-}^{2}\ ,$ (33)

where $\Sigma_{\pm}\equiv(\Sigma_{1}\mp{\rm i}\,\Sigma_{2})/2$. This breaks
$U(1)_{\Psi}$ because $\Sigma_{-}^{2}$ depends explicitly on $\Psi$. Including
also the complex conjugate contribution to make the perturbation real, the
$m=2$ perturbation (33) at the onset of instability finally reads

$h_{MN}\mathrm{d}x^{M}\mathrm{d}x^{N}=\frac{r^{2}}{2}Q(r)\Sigma_{-}^{2}+\textrm{c.c.}=\frac{r^{2}}{4}Q(r)\left(\Sigma_{1}^{2}-\Sigma_{2}^{2}\right)\,,$
(34)

which preserves all the original Myers-Perry symmetries except axisymmetry.
This suggests that the onset of the $m=2$ instability might be a branching
line to a new family of nonlinear solutions with $\mathbb{R}_{T}\times
SU(2)\times U(1)_{Z}$ symmetry. The null generator of the horizon of such
solution would be given by
$K=\partial_{T}=\partial_{t}+\Omega_{H}\partial_{\psi}$, which corresponds to
the “helical” Killing vector generating $U(1)_{T}$.

There is however another important observation. Indeed, note that the
perturbation equation (18) is invariant under $k\to-k$. Thus, left and right
mover modes along the string are absolutely equivalent and thus the onset (30)
of the $m=2$ superradiant instability can be equivalently written as

$h_{MN}\mathrm{d}x^{M}\mathrm{d}x^{N}=r^{2}e^{-4\,{\rm
i}\,\Omega_{H}t}\left(c_{1}\,e^{{\rm i}\,k\,z}+c_{2}\,e^{-{\rm
i}\,k\,z}\right)\,Q(r)\Sigma_{-}^{2}\ ,$ (35)

where $c_{1}$ and $c_{2}$ are arbitrary constants. We can assume
$c_{1},c_{2}\in\mathbb{R}$ without loss of generality by using a symmetry that
allows for constant shifts of $t$ and $z$.

The interesting observation is that by applying the coordinate transformation
(31), one concludes that (35) breaks $U(1)_{Z}$ unless $c_{1}=0$ or $c_{2}=0$.
This means that the $m=2$ superradiant onset $Ac$, that we identify in Figs. 3
and 9, describes the onset of two distinct sectors of perturbations: one
preserves $U(1)_{Z}$ while the other breaks $U(1)_{Z}$.

It follows that each point of the onset line $Ac$ that we found in Figs. 3 and
9 should be, when we extend the analysis beyond linear order, a degenerate
bifurcation line to a two-parameter family of new black string solutions. The
two parameters correspond to $c_{1}$ and $c_{2}$ in (35) at linear order. One
special case of this family is when $c_{1}=0$ or $c_{2}=0$. The resulting one-
parameter family preserves $U(1)_{Z}$. As this $U(1)_{Z}$ describes a
spatially helical symmetry that mixes $\partial_{z}$ and $\partial_{\psi}$, we
call the $U(1)_{Z}$-preserving solutions helical black strings.141414Existence
of black strings with helical symmetry has been first found in Emparan:2009vd
using the blackfold effective worldvolume theory. All other cases break the
$U(1)_{Z}$, while still preserving the $\mathbb{R}_{T}\times SU(2)$ isometry
group. As it preserves a $U(1)_{T}$ temporal helical symmetry that mixes
$\partial_{t}$ and $\partial_{\psi}$, but breaks other spatial symmetries, we
call these black resonator string family by analogy to similar black hole
solutions (without extended directions) Dias:2011at ; Dias:2011tj ;
Dias:2015rxy ; Ishii:2018oms ; Ishii:2020muv ; Ishii:2021xmn .

Of course these novel nonlinear solutions can exist because at the onset of
the instability, and only here, the mode perturbation is regular not only on
the future horizon but also on the past horizon; hence there is room for the
existence of a novel family of nonlinear stationary black strings that are
regular simultaneously at the future and past horizons, and that bifurcate
from the Myers-Perry black string at the onset of the $m=2$ superradiant
instability. To confirm the existence of these two new families of black
strings, we must solve the Einstein equations nonlinearly, i.e. beyond the
linear analysis done in the present paper. We will do so and find the black
resonator strings and the helical black strings in the future companion papers
DIMSW2 ; DIMSW3 , respectively.

Finally, let us comment on the prospect of additional solutions that branch
from the confining cutoff curve $\widetilde{k}^{(m)}_{\star}$. Here, we indeed
have an onset in the sense that $\mathrm{Im}\,\omega=0$, but the boundary
conditions for the linear problem on this curve remain unclear, and it is
possible that they are not regular (recall the discussion about the
logarithmic behaviour below (24)). We therefore do not seek solutions that
branch from this curve.

##  5 Discussion

We have analysed gravitational instabilities on the six-dimensional equal-
spinning Myers-Perry black string, i.e. the cohomogeneity-1 black strings with
equal angular momenta along the two rotation planes of asymptotically Kaluza-
Klein spacetimes ($\mathcal{M}^{1,4}\times S^{1}$). We find instances of both
the Gregory-Laflamme instability and the superradiant instability, determined
their unstable regions in the 2-dimensional Myers-Perry string parameter
space, and computed their growth rates.

Our results on the Gregory-Laflamme instability complement the previous
results in Kleihaus:2007dg ; Dias:2010eu  (see also Dias:2009iu ; Dias:2010maa
and Dias:2011jg ) and supports the broad idea that rotation typically enhances
the Gregory-Laflamme instability.

Our results on the superradiant instability are new. As we have argued, it
would be difficult to anticipate the existence of this instability within this
system without doing the actual computation. On one hand, Myers-Perry black
strings certainly have an ergoregion where superradiant applification can
occur, and nontrivial superradiant Kaluza-Klein modes are effectively massive
and, in some cases, can be confined as first proposed in Marolf:2004fya  (see
also Cardoso:2004zz ; Cardoso:2005vk ; Dias:2006zv ). On the other hand, we
were unable to find scalar field and Maxwell perturbations that are
superradiant unstable (see appendix A), and the scalar field instability is
similarly absent in the single spinning Myers-Perry black string
Cardoso:2005vk . Furthermore, the lack of stable particle orbits about five
dimensional equal-spinning Myers-Perry black holes suggests that the
instability in the associated black string cannot exist in the eikonal
($m\to\infty$) limit. In spite of these less optimistic hints, we did find
what we call ‘finite-$m$’ superradiant instabilities in the tensor harmonic
sector of gravitational perturbations. Consistent with the lack of particle
orbits, this instability shuts-down in the eikonal limit, unlike known
(uncharged) superradiant instabilities in black holes. Here, the scalar field
system is thus not a good toy model for the gravitational system (unlike in
previously studied superradiant systems for black holes).

Let us now examine the gravitational superradiant instability in the equal-
spinning Myers-Perry black string in more detail in light of our results and
the above arguments. We have found that this instability is generally bounded
by extremality and two curves with $\mathrm{Im}\,\omega=0$. One of these
curves (the superradiant onset curve) satisfies $\omega=2m\Omega_{H}$, and
follows the standard criterion for the onset of superradiance required for the
amplification of waves (the factor of 2 reflects the fact that we have equal
rotation along the two planes; see footnote 4). However, the second curve with
$\mathrm{Im}\,\omega=0$ (that we now call the confining cutoff curve)
satisfies $\omega=k$, which suppresses the exponential falloff of fields,
$e^{\pm\sqrt{k^{2}-\omega^{2}}\,r}$, and stops the superradiant instability
due to a lack of confinement. That is, at this cutoff curve, the Schröedinger
potential barrier that confines bound states ceases to exist. Ultimately, in
the eikonal limit, both of these mechanisms conspire to eliminate the
superradiant instability.

As far as we are aware, this mechanism of cutting off confinement is absent in
other gravitational superradiant instabilities in black hole systems, such as
in those with a potential barrier sourced by a massive field. The prototypical
example is a massive scalar field in a Kerr black hole Zouros:1979iw ;
PhysRevD.22.2323 ; Dolan:2007mj . In this case, superradiant instabilities
exist as long as the mass of the scalar field perturbation is below an upper
bound. But there is no (non-zero) lower bound for the mass of the scalar field
for existence of superradiant bound states. In our case, the momentum of the
perturbation along the extended black string direction provides an effective
mass term to the system. Like the massive scalar on Kerr, we still find a
maximum effective mass (i.e. momentum $k_{(0)}^{(m)}r_{+}$) for the existence
of an unstable superradiant bound states. But, surprisingly, we also find a
non-zero lower bound for this effective mass (i.e. the momentum
$k_{\star}^{(m)}r_{+}$) below which we no longer have bound states and thus
the system becomes stable to superradiance.

How general is the finite-$m$ superradiant instability that we found? What
circumstances allow this phenomenon to occur? Our superradiant gravitational
instability of the equal-spinning Myers-Perry black string should also be
present in the Kerr black string, though the Kerr case should instead be a
more typical ’infinite-$m$’ superradiant instability as it contains bound
particle orbits. Although a detailed study of perturbations in the Kerr black
string requires solving PDEs, not ODEs. However, we argue around (21) and in
appendix B that it should not be present in $D>6$ equal-spinning Myers-Perry
strings, so $D=5$ and $D=6$ strings seem to be very special cases. What is
also clear is that the gravitational superradiant instability of the six-
dimensional equal-spinning Myers-Perry black string does not extend to scalar
field nor Maxwell perturbations, even in $D=6$ (see appendix A). Moreover, in
the single-spinning case, Cardoso:2005vk  argues that the effective potential
of the scalar field also fails to allow for bound states. Away from the
eikonal limit, the status of superradiant instabilities in the single-spinning
or more generally-spinning case remains unclear, and will again require PDEs
to study in detail.

Our linear instability should have important consequences for the phase
diagram of stationary black strings. Indeed, our linear results suggest that
novel rotating black string solutions, that break the same rotational
$U(1)_{\psi}$ isometry as our linear perturbation, should bifurcate from the
onset of our superradiant instability. That is, new solutions with fewer
isometries (those of our linear perturbation) than the Myers-Perry black
string should exist. Fortunately, for $m=2$, our perturbations still preserve
the $SU(2)$ isometry of the equal-spinning Myers-Perry solution, which
indicates that these new solutions should be feasible to construct at the full
nonlinear level. Actually, in section 4.4, we argued that the onset of our
$m=2$ superradiant instability should be a degenerate bifurcation line to a
new two-parameter family of black string solutions. In general, the $U(1)_{Z}$
isometry of the system is broken, while this isometry is preserved for a
special family. In two companion manuscripts, we will confirm this is the
case. We will solve the full nonlinear problem (perturbatively to higher order
and numerically) and find cohomogeneity-2 black resonator strings with
isometry group $\mathbb{R}_{T}\times SU(2)$ DIMSW2 , and cohomogeneity-1
helical black strings with isometry group $\mathbb{R}_{T}\times SU(2)\times
U(1)_{Z}$ DIMSW3 , where the coordinates $T,Z$ are defined in (31).

Finally, let us comment on the endpoint of equal-spinning Myers-Perry black
strings that are afflicted by one or both (Gregory-Laflamme and superradiant)
instabilities. There have only been a limited number of studies of the
evolution of the Gregory-Laflamme instability in black strings, and all for
the non-rotating case Horowitz:2001cz ; Lehner:2010pn ; Emparan:2015gva ;
Figueras:2022zkg  (see also Figueras:2015hkb  for the evolution of the
Gregory-Laflamme instability of black rings). Thus far, these results are
consistent with entropic expectations. If non-uniform string solutions are
entropically preferred, which is the case for high spacetime dimensions
Gubser:2001ac ; Wiseman:2002zc ; Sorkin:2004qq ; Kudoh:2004hs ; Sorkin:2006wp
; Headrick:2009pv ; Figueras:2012xj , then they provide a natural endpoint
Emparan:2015gva . Otherwise, evolution proceeds to a “localised” black hole
configuration with spherical horizon topology, and the evolution process goes
through a horizon pinch-off phase that necessarily violates the weak cosmic
censorship conjecture Lehner:2010pn ; Figueras:2022zkg . Of course, classical
evolution stops to be valid before reaching the naked singularity, so the
ultimate classical endpoint as predicted by entropy arguments requires that
the naked singularity be resolved in a sufficiently mild way. The evolution of
the Gregory-Laflamme instability for the rotating black string should proceed
in a similar way. That is, that the most entropic solution is the preferred
endpoint, whether this may be some non-uniform string or localized black
hole.151515Note that it is not yet known whether the $D=6$ non-uniform equal-
spinning black strings of Kleihaus:2007dg  are entropically favored over the
uniform strings.

What about the endpoint of equal-spinning Myers-Perry black strings that are
only unstable to superradiance? The time evolution of the gravitational
superradiant instability is best understood for small rotating black holes in
global anti de-Sitter (AdS) space. In this context, it was argued that the
unstable Kerr-AdS black holes develop towards ‘black resonators’, which are
black holes (of pure Einstein-AdS gravity) with gravitational hair that have a
single Killing vector field (the horizon generator). That is, they are not
time independent nor axisymmetric, but are time-periodic Dias:2015rxy  (see
Dias:2011at ; Dias:2011tj ; Dias:2015rxy ; Ishii:2018oms ; Ishii:2020muv ;
Ishii:2021xmn  for more details). However, black resonators are themselves
unstable to superradiant instabilities with arbitrarily higher azimuthal
numbers $m$ (unlike the case for Myers-Perry black strings). The ultimate
endpoint remains an open problem but if, as conjectured in Dias:2011at ;
Cardoso:2013pza ; Dias:2015rxy ; Niehoff:2015oga , the cascade continues to
ever increasing $m$’s, then a significant amount of energy will be pushed
towards microscopic length scales, violating the weak cosmic censorship
conjecture Dias:2015rxy ; Niehoff:2015oga . The numerical time evolution of
this instability is currently consistent with this picture Chesler:2018txn ;
Chesler:2021ehz .

Note that arguments for the existence of this cascade largely comes from the
fact that there are an infinite number of unstable modes with arbitrarily high
azimuthal numbers $m$ affecting the original black hole superradiant system.
The superradiant instability for the equal-spinning Myers-Perry black string
does not have this property. Indeed, a given Myers-Perry black string can be
unstable to only a few modes with different $m$’s (not an infinite tower of
them, unlike Kerr-AdS). This is ultimately related to the fact that the
finite-$m$ superradiant instability shuts down in the eikonal limit. For this
reason, it is expected that no continuous cascade to higher $m$’s should be
present in the time evolution of the superradiant instability of the Myers-
Perry black string. Rather, it seems more plausible that the superradiant
instability will just drive the solution to a particular black resonator
string or helical string with a given $m$, whichever has the most entropy at
the same energy $\mathcal{E}$ and angular momenta $\mathcal{J}$.

Thus far, we have discussed the Gregory-Laflamme and superradiant
instabilities in isolation, but as we have shown, both instabilities can be
present and thus compete for certain parameters of the Myers-Perry black
string. The typical growth rate for the Gregory-Laflamme instability (in units
of the horizon radius) is $10^{-1}$ while that of the superrradiant
instability is $10^{-5}$. This large separation is not surprising, given the
vastly different origins of the instabilities. One is governed primarily by
horizon dynamics while the other is driven by the amplification and
confinement of gravitons, and these typically have very different timescales.
Given the disparity of growth rates, it should be the case that the Gregory-
Laflamme instability will likely dominate the early dynamics in cases where
both instabilities are present. We stress, though, that our linear analysis
will quickly cease to be valid in mid to late times in the evolution. It is
also unclear if one instability remains present even when the other has
settled down. We do not know, for instance, if rotating localised Kaluza-Klein
black holes have a superradiant instability (although the typical arguments
for superradiance suggest that they should). There will be various interplay
of the Gregory-Laflamme and superradiant instabilities, which would be
interesting to address in future work.

###### Acknowledgements.

The authors would like to thank Roberto Emparan, Takahisa Igata, and Masashi
Kimura for useful discussions. O.J.C.D. acknowledges financial support from
the STFC “Particle Physics Grants Panel (PPGP) 2018” Grant No. ST/T000775/1.
OD acknowledges the Isaac Newton Institute, Cambridge, and the organizers of
its long term programme “Applicable resurgent asymptotics: towards a universal
theory” during which this work was completed. The work of T.I. was supported
in part by JSPS KAKENHI Grant Number 19K03871. This work of K.M. was supported
in part by JSPS KAKENHI Grant Nos. 20K03976, 21H05186 and 22H01217. BW
acknowledges support from ERC Advanced Grant GravBHs-692951 and MEC grant
FPA2016-76005-C2-2-P. J. E. S. has been partially supported by STFC
consolidated grant ST/T000694/1. The authors also acknowledge the use of the
IRIDIS High Performance Computing Facility, and associated support services at
the University of Southampton, in the completion of this work.

##  Appendix A Scalar field and Maxwell perturbations

As described in the main text, we also attempted to find, numerically,
superradiant instabilities in the sector of scalar field and Maxwell
perturbations about the equal-spinning Myers-Perry black string. But, unlike
for the gravitational sector, we have found no superradiant instability in
these sectors of perturbations. To complement this negative search, in this
appendix, we give analytical arguments in favour of the the absence of
superradiant instability for scalar field and Maxwell perturbations. These
arguments are similar to those employed in section 4.3 $-$ in the discussion
between (23)-(25) $-$ to argue in favour of the gravitational instability.
Here we show only the calculations for massless fields; for massive fields the
analysis and conclusion are similar.161616The mass $\mu$ can be introduced
simply by replacing $k^{2}\to k^{2}+\mu^{2}$ in our analysis, while the final
results for the condition of ${\Omega}_{H}^{(m)}/\Omega^{\mathrm{ext}}_{H}$
are not altered. For gravitational perturbations in the main text, a graviton
mass would affect the value of $k$ for the instability, but we are interested
only in Einstein gravity.

###  A.1 Scalar field

Consider the Klein-Gordon equation, $\Box\Pi=0$, for a massless scalar field
$\Pi$ on the equal-spinning Myers-Perry black string background. Introducing
the separation ansatz

$\Pi(t,r,\theta,\phi,\psi,z)=e^{\mathrm{i}kz}e^{-\mathrm{i}\omega
t}\Pi(r)D^{j}_{\ell^{\prime}\ell=-m}(\theta,\phi,\psi)\,,$ (36)

where $D^{j}_{\ell^{\prime}\ell=-m}$ are the Wigner D-matrices introduced in
section 4.1, the Klein-Gordon equation decouples for each $m$ and the radial
equation one needs to solve reads

$\left(r^{3}F\Pi^{\prime}\right)^{\prime}+r\left[\frac{Hr^{2}}{F}\left(\omega-2m\frac{\Omega}{H}\right)^{2}-\frac{4m^{2}}{H}-4(j^{2}+j-m^{2})-k^{2}r^{2}\right]\Pi=0\,.$
(37)

Defining $\Phi=(rH)^{1/4}\Pi$, we can rewrite this equation in the Schrödinger
form (19) where in the present case the effective potential is

$V=\frac{F}{H}k^{2}-\left(\omega-2m\frac{\Omega}{H}\right)^{2}+\frac{4F}{r^{2}H}\left[j(j+1)-m^{2}\left(1-\frac{1}{H}\right)\right]+\frac{1}{r^{3/2}H^{1/4}}\frac{\mathrm{d}^{2}}{\mathrm{d}r_{\ast}^{2}}(r^{3/2}H^{1/4}).$
(38)

Near infinity, this Schrödinger potential reads

$V|_{r\to\infty}\sim
k^{2}-\omega^{2}+\left[4j(j+1)+\frac{3}{4}-\frac{4k^{2}r_{+}^{2}}{4-\Omega_{H}^{2}r_{+}^{2}}\right]\frac{1}{r^{2}}+\mathcal{O}(r^{-4})\
.$ (39)

Alike for the gravitational perturbation (21), the scalar field perturbation
can be confined near the horizon for a sufficiently large $k$. Yet, unlike the
gravitational case, a simple analytical analysis of our radial equation
suggests absence of scalar field superradiant instabilities as described next.

At infinity, the asymptotic behaviour of the field is given in general by
$\Pi|_{r\to\infty}\sim r^{-3/2}e^{\pm\sqrt{k^{2}-\omega^{2}}\,r}$. But for
$\omega=k$, the exponential decay is absent and we have to follow an analysis
similar to the one described in (23)-(25) for the gravitational sector.
Namely, for $\omega=k$, the asymptotic behaviour of the scalar field is
instead $\Pi|_{r\to\infty}\sim r^{-1\pm\gamma}$ with
$\gamma=\sqrt{(2j+1)^{2}-4\widetilde{k}^{2}/(4-\widetilde{\Omega}_{H}^{2})}$.
As for the gravitational case, the special $\gamma=0$ degenerate case should
select the confining cutoff condition
$\widetilde{k}^{(m)}_{\star}(\widetilde{\Omega}_{H})$ for the scalar
superradiant instability. So setting $\gamma=0$, we obtain

$\widetilde{k}^{(m)}_{\star}=\frac{1}{2}(2j+1)\sqrt{4-\widetilde{\Omega}_{H}^{2}}\,.$
(40)

So this should define the scalar field counterpart of the superradiant cutoff
line $cB$ of the gravitational case observed in Fig. 3. Actually, we can also
proceed and apply the same argument leading to (25) to get the scalar field
counterpart of the critical point $c$ where the onset curve for
$\widetilde{k}^{(m)}_{(0)}(\widetilde{\Omega}_{H})$ and cutoff curve
$\widetilde{k}^{(m)}_{\star}(\widetilde{\Omega}_{H})$ should intersect. Recall
from Fig. 3, that $c$ is a vertex point for the superradiant unstable region,
i.e. instability is present only for rotation above the one of this point. In
this context, the onset of the scalar superradiant instability must certainly
satisfy the condition $\omega=2m\Omega_{H}$ (this follows from the equation of
motion at the horizon). Combining this with $\omega=k^{(m)}_{\star}$ given in
(40), we find that the scalar field superradiant instability should only exist
for horizon angular velocities higher than the critical value

$\frac{{\Omega}_{H}^{(m)}}{\Omega^{\mathrm{ext}}_{H}}=\frac{\sqrt{2}(2j+1)}{\sqrt{(2j+1)^{2}+4m^{2}}}>1\,,$
(41)

where the last inequality follows from $|m|\leq j$, and the ratio approaches
$1$ from above as $m\to\infty$. So the minimum angular velocity required for
the scalar field instability is larger than the velocity at extremality.
Effectively, this argument rules out the existence of a superradiant
instability in the scalar field sector. Our arguments did rely on some
assumptions that are only known to be valid empirically for the gravitational
sector. But, as stated above, we have also searched directly for instabilities
of the scalar field by solving the scalar field eigenvalue problem numerically
but found no instability.

###  A.2 Maxwell field

For the Maxwell field, we focus on the perturbation given by

$a_{M}\mathrm{d}x^{M}=e^{\mathrm{i}kz}e^{-\mathrm{i}\omega
t}A(r)D^{j}_{\ell^{\prime}\ell=-j}(\theta,\phi,\psi)\sigma_{-}\,.$ (42)

where $D^{j}_{\ell^{\prime}\ell=-m}$ are again the Wigner D-matrices
introduced in section 4.1. The azimuthal mode number for this perturbation is
$m=j+1$ and it is a half-integer multiple satisfying $m\geq 1$. This mode
decouples from other Maxwell perturbations and it is expected to be the most
superradiant unstable mode for a given $j$. The gauge transformation of the
Maxwell field is $a_{M}\to a_{M}+\partial_{M}\lambda$. The gauge parameter
$\lambda$ is also expanded by the Wigner D-matrices as
$\lambda=e^{\mathrm{i}kz}e^{-\mathrm{i}\omega
t}\Lambda(r)D^{j}_{\ell^{\prime}\ell}$. The maximal azimuthal mode number of
the gauge parameter $\lambda$ is $m=j$ for a given $j$. It follows that the
Maxwell perturbation (42) which has $m=j+1$ is invariant under the gauge
transformation.

The Maxwell equation $\Box a_{M}=0$ takes the form

$\left(rFA^{\prime}\right)^{\prime}+\frac{1}{r}\left[\frac{Hr^{2}}{F}\left(\omega-2m\frac{\Omega}{H}\right)^{2}-\frac{4m^{2}}{H}-k^{2}r^{2}\right]A=0\,.$
(43)

Defining $\Phi=r^{1/2}H^{1/4}A$, we recast this as a Schrödinger equation (19)
with the effective potential

$V=\frac{F}{H}k^{2}-\left(\omega-2m\frac{\Omega}{H}\right)^{2}+\frac{F}{r^{2}H}\left[\frac{4m^{2}}{H}-(rF)^{\prime}\right]+\frac{1}{r^{3/2}H^{1/4}}\frac{\mathrm{d}^{2}}{\mathrm{d}r_{\ast}^{2}}(r^{3/2}H^{1/4}).$
(44)

Its asymptotic behaviour is

$V|_{r=\infty}\sim
k^{2}-\omega^{2}+\left(4m^{2}-\frac{1}{4}-\frac{4k^{2}r_{+}^{2}}{4-\Omega_{H}^{2}r_{+}^{2}}\right)\frac{1}{r^{2}}+\mathcal{O}(r^{-4})\,,$
(45)

which indicates that Maxwell perturbations can be confined for a sufficiently
large $k$. However, there is no room for the existence of a superradiant
instability as argued next (following again a line of reasoning similar to the
one employed in the gravitational discussion of (23)-(25)).

Typically, Maxwell perturbations decay exponentially as $A|_{r=\infty}\sim
r^{-1/2}e^{\pm\sqrt{k^{2}-\omega^{2}}\,r}$. But for $\omega=k$, the
exponential decay is absent and the asymptotic behaviour of the Maxwell field
is instead $A|_{r\to\infty}\sim r^{\pm\gamma}$ with
$\gamma=\sqrt{4m^{2}-4\widetilde{k}^{2}/(4-\widetilde{\Omega}_{H}^{2})}$. As
for the gravitational case, the special $\gamma=0$ degenerate case should
select the confining cutoff condition
$\widetilde{k}^{(m)}_{\star}(\widetilde{\Omega}_{H})$ for the Maxwell
superradiant instability. This $\gamma=0$ condition yields

$\widetilde{k}^{(m)}_{\star}=m\sqrt{4-\widetilde{\Omega}_{H}^{2}}\,$ (46)

which should define the Maxwell field counterpart of the superradiant cutoff
line $cB$ of the gravitational case observed in Fig. 3. Very much like for the
gravitational discussion leading into (25), the endpoint $c$ of this cutoff
curve can be obtained from the condition that the cutoff curve (46) intersects
the onset curve $\widetilde{k}^{(m)}_{(0)}(\widetilde{\Omega}_{H})$ with
$\omega=m\Omega$. This condition concludes that the superradiant instability
for Maxwell fields should only exist for horizon angular velocities higher
than the critical value

$\frac{{\Omega}_{H}^{(m)}}{\Omega^{\mathrm{ext}}_{H}}=1\,,$ (47)

i.e. for equal-spinning Myers-Perry strings rotating with a velocity above the
extremal value (which are not regular). Thus non-extreme Myers-Perry black
strings do not develop superradiant instabilities for the Maxwell field. This
conclusion is supported by the fact that we directly searched numerically for
instabilities but found none.

We can repeat the same analysis for massive scalar and Proca fields and obtain
the same conclusion that these fields do not induce superradiant instability.
This suggests that the mass of the fields is not essential, and spin
dependence is important for the existence of the superradiant instability for
Myers-Perry black strings.

##  Appendix B Absence of superradiant instability in equal-spinning Myers-
Perry strings in $D>6$

In the main text we found that $D=6$ equal angular momenta Myers-Perry black
strings are unstable to the finite-$m$ superradiant instability (in addition
to being unstable to the Gregory-Laflamme instability). In particular, we also
found a criterion $-$ detailed in the discussion of (23)-(25) of section 4.3
$-$ to find, almost only resorting to analytical computations, the
superradiant unstable region $ABc$ in the phase diagrams of Fig. 3 and Fig. 9.
We also argued that $D=5$ Kerr black strings should also be superradiant
unstable since it has superradiant bound states. However, in this appendix we
employ a line of reasoning that follows the discussion of (23)-(25) of section
4.3 to argue that superradiant instabilities should not be present for $D>6$
equal angular momenta Myers-Perry black strings. In short, although there are
superradiant modes, there is no potential that can confine them as bound
states in a region of the parameter space that is contained inside the
boundary set by the extremal (zero temperature) Myers-Perry black string
configuration. So the $D=6$ case studied in the main text is very special.

Consider $D=2N+4$ dimensional equal-spinning Myers-Perry black strings, i.e.
the product of an equal-spinning $(2N+3)$-dimensional Myers-Perry black hole
and a circle. Unlike for the $N=1$ case, we argue that the gravitational
superradiant instability is absent for $N\geq 2$. The metric of the equal-
spinning Myers-Perry black string in $D=2N+4$ dimensions is

${\mathrm{d}}s^{2}_{\rm
MP\,string}=-\frac{F}{H}\,{\mathrm{d}}t^{2}+\frac{{\mathrm{d}}r^{2}}{F}+r^{2}\left[H\left(d\psi+A_{a}dx^{a}-\frac{\Omega}{H}\,{\mathrm{d}}t\right)^{2}+\hat{g}_{ab}dx^{a}dx^{b}\right]+\mathrm{d}z^{2}\,,$
(48)

where

$F(r)=1-\frac{r_{0}^{2N}}{r^{2N}}+\frac{a^{2}r_{0}^{2N}}{r^{2N+2}}\,,\qquad
H(r)=1+\frac{a^{2}r_{0}^{2N}}{r^{2N+2}}\,,\qquad\Omega(r)=\frac{a\,r_{0}^{2N}}{r^{2N+2}}\,,$
(49)

and $\hat{g}_{ab}$ is the Fubini-Study metric on $CP^{N}$ and $A_{a}$ is the
Kähler potential. In Kunduri:2006qa , “doubly transverse” gravitational
perturbations of the equal-spinning Myers-Perry black hole have been
considered for $D=2N+3$ dimensions with $N\geq 2$. A unified form of the
perturbation equation was given for decoupled gravitational and scalar field
perturbations. While their study was limited to $N\geq 2$, the corresponding
perturbation was examined in detail for $N=1$ in Murata:2008yx . Below we will
check that the unified form of Kunduri:2006qa  also applies to $N=1$.

The scalar-gravitational unified treatment of Kunduri:2006qa  can be
generalized straightforwardly to the $D=2N+4$ dimensional Myers-Perry black
string with $N\geq 1$ since one just needs to add an extra dimension to the
formalism which introduces a Kaluza-Klein mass term to the perturbation
equations. The unified perturbation equation can be written in the
Schröedinger form

$-\frac{\mathrm{d}^{2}}{\mathrm{d}r_{\ast}^{2}}\Phi+V\Psi=0,$ (50)

where $dr_{\ast}=\sqrt{H}/Fdr$ denotes the tortoise coordinate. The unified
Schröedinger potential for scalar field and anti-hermitian gravitational
perturbations is given by

$\displaystyle V=$
$\displaystyle\frac{F}{H}k^{2}-\left(\omega-2m\frac{\Omega}{H}\right)^{2}+\frac{4F}{r^{2}H}\left[l(l+N)-m^{2}\left(1-\frac{1}{H}\right)\right]$
(51)
$\displaystyle+\frac{1}{r^{N+1/2}H^{1/4}}\frac{\mathrm{d}^{2}}{\mathrm{d}r_{\ast}^{2}}\left(r^{N+1/2}H^{1/4}\right),$
(52)

where $l=n+|m|\ (n=0,1,2,\dots)$ for the scalar field, while $l$ is restricted
to $l\geq 1$ for the gravitational perturbation, and $m$ takes either integer
or half-integer values.171717The quantum numbers in Kunduri:2006qa  are
related to ours as $m_{\mathrm{theirs}}=2m_{\mathrm{ours}}$ and
$l_{\mathrm{theirs}}=2l_{\mathrm{ours}}$. Also, the perturbations we focus on
corresponds to “$\sigma=1$” in Kunduri:2006qa . We have checked that (50)
reproduces the perturbation equation we have been using for $N=1$. Indeed,
setting $l=j$, the scalar field perturbation (37) is reproduced. Moreover, for
$l=m-1$ one recovers (18).181818Also, the range of $l$ is consistent. In
Kunduri:2006qa , the range of $l_{\mathrm{theirs}}$ was restricted to
$l_{\mathrm{theirs}}\geq 2$, which would also applied to the case of $N=1$.
This condition is translated to $l\geq 1$ for our $l$. Meanwhile, we have
$m\geq 2$, which gives $l=m-1\geq 1$.

For $N\geq 2$, the asymptotic behaviour of the potential $V$ becomes

$V|_{r\to\infty}\simeq
k^{2}-\omega^{2}+\frac{4l(l+N)+N^{2}-1/4}{r^{2}}+\mathcal{O}(r^{-4})\ .$ (53)

Unlike for the $N=1$ case of (21), the $1/r^{2}$ term is always positive.
Moreover, because $F/H|_{r\to\infty}\simeq 1-r_{0}^{2N}/r^{2N}$ in (52),
$k^{2}$ does not appear in the coefficient of the $1/r^{2}$. Altogether, this
suggests that for $N\geq 2$, the scalar field and gravitational perturbations
cannot be confined inside the ergoregion and thus we should not expect the
existence of a superradiant instability. We actually searched numerically for
the onset of the instability for $N=2,3,4$, but we did not find evidence of
its existence, consistent with the above analysis.

We can also repeat the same line of arguments detailed in the discussion of
(23)-(25) of section 4.3 (and further explored in appendix A) to arrive to the
same conclusion. For $k\neq\omega$, the asymptotic solutions of (50) decay
exponentially as $\Psi|_{r\to\infty}\sim e^{-\sqrt{k^{2}-\omega^{2}}\,r}$.
However, for $k=\omega$, the two possible decays are instead polynomial.
Concretely, for $N\geq 2$, they read

$\Psi\sim r^{\frac{1}{2}\pm(l+N)}.$ (54)

Unlike the $N=1$ counterpart case discussed around (23), there is no room for
the two solutions of (54) to become degenerate since $l+N>0$ (and more
importantly the exponents are not a function of $k$ and $\Omega_{H}$). Hence,
this analysis also gives no evidence for the existence of a superradiant
instability (be it in the scalar or gravitational sector) for $N\geq 2$
($D\geq 8$) equal-spinning Myers-Perry black strings, unlike for the $N=1$
($D=6$) case discussed in the main text.

## References

  * (1) G. T. Horowitz and A. Strominger, _Black strings and P-branes_ , _Nucl. Phys. B_ 360 (1991) 197–209.
  * (2) R. Gregory and R. Laflamme, _Black strings and p-branes are unstable_ , _Phys.Rev.Lett._ 70 (1993) 2837–2840, [hep-th/9301052].
  * (3) R. Gregory and R. Laflamme, _The Instability of charged black strings and p-branes_ , _Nucl. Phys. B_ 428 (1994) 399–434, [hep-th/9404071].
  * (4) G. T. Horowitz and K. Maeda, _Fate of the black string instability_ , _Phys. Rev. Lett._ 87 (2001) 131301, [hep-th/0105111].
  * (5) L. Lehner and F. Pretorius, _Black Strings, Low Viscosity Fluids, and Violation of Cosmic Censorship_ , _Phys.Rev.Lett._ 105 (2010) 101102, [1006.5960].
  * (6) P. Figueras, T. França, C. Gu and T. Andrade, _The endpoint of the Gregory-Laflamme instability of black strings revisited_ , 2210.13501.
  * (7) S. S. Gubser, _On nonuniform black branes_ , _Class. Quant. Grav._ 19 (2002) 4825–4844, [hep-th/0110193].
  * (8) T. Wiseman, _Static axisymmetric vacuum solutions and nonuniform black strings_ , _Class. Quant. Grav._ 20 (2003) 1137–1176, [hep-th/0209051].
  * (9) E. Sorkin, _A Critical dimension in the black string phase transition_ , _Phys. Rev. Lett._ 93 (2004) 031601, [hep-th/0402216].
  * (10) H. Kudoh and T. Wiseman, _Connecting black holes and black strings_ , _Phys. Rev. Lett._ 94 (2005) 161102, [hep-th/0409111].
  * (11) E. Sorkin, _Non-uniform black strings in various dimensions_ , _Phys. Rev. D_ 74 (2006) 104027, [gr-qc/0608115].
  * (12) M. Headrick, S. Kitchen and T. Wiseman, _A New approach to static numerical relativity, and its application to Kaluza-Klein black holes_ , _Class.Quant.Grav._ 27 (2010) 035002, [0905.1822].
  * (13) P. Figueras, K. Murata and H. S. Reall, _Stable non-uniform black strings below the critical dimension_ , _JHEP_ 11 (2012) 071, [1209.1981].
  * (14) M. Hanada and T. Nishioka, _Cascade of Gregory-Laflamme Transitions and U(1) Breakdown in Super Yang-Mills_ , _JHEP_ 09 (2007) 012, [0706.0188].
  * (15) N. Kawahara, J. Nishimura and S. Takeuchi, _Phase structure of matrix quantum mechanics at finite temperature_ , _JHEP_ 10 (2007) 097, [0706.3517].
  * (16) O. J. C. Dias, G. T. Horowitz and J. E. Santos, _Black holes with only one Killing field_ , _JHEP_ 07 (2011) 115, [1105.4167].
  * (17) O. J. C. Dias, J. E. Santos and B. Way, _Black holes with a single Killing vector field: black resonators_ , _JHEP_ 12 (2015) 171, [1505.04793].
  * (18) T. Ishii and K. Murata, _Black resonators and geons in AdS5_ , _Class. Quant. Grav._ 36 (2019) 125011, [1810.11089].
  * (19) T. Ishii and K. Murata, _Photonic black resonators and photon stars in AdS 5_, _Class. Quant. Grav._ 37 (2020) 075009, [1910.03234].
  * (20) T. Ishii, K. Murata, J. E. Santos and B. Way, _Multioscillating black holes_ , _JHEP_ 05 (2021) 011, [2101.06325].
  * (21) B. E. Niehoff, J. E. Santos and B. Way, _Towards a violation of cosmic censorship_ , _Class. Quant. Grav._ 33 (2016) 185012, [1510.00709].
  * (22) P. M. Chesler and D. A. Lowe, _Nonlinear Evolution of the AdS 4 Superradiant Instability_, _Phys. Rev. Lett._ 122 (2019) 181101, [1801.09711].
  * (23) P. M. Chesler, _Hairy black resonators and the AdS4 superradiant instability_ , _Phys. Rev. D_ 105 (2022) 024026, [2109.06901].
  * (24) R. Brito, V. Cardoso and P. Pani, _Superradiance: New Frontiers in Black Hole Physics_ , _Lect. Notes Phys._ 906 (2015) pp.1–237, [1501.06570].
  * (25) D. Marolf and B. Cabrera Palmer, _Gyrating strings: A New instability of black strings?_ , _Phys. Rev. D_ 70 (2004) 084045, [hep-th/0404139].
  * (26) V. Cardoso and J. P. Lemos, _New instability for rotating black branes and strings_ , _Phys. Lett. B_ 621 (2005) 219–223, [hep-th/0412078].
  * (27) V. Cardoso and S. Yoshida, _Superradiant instabilities of rotating black branes and strings_ , _JHEP_ 07 (2005) 009, [hep-th/0502206].
  * (28) O. J. Dias, _Superradiant instability of large radius doubly spinning black rings_ , _Phys. Rev. D_ 73 (2006) 124035, [hep-th/0602064].
  * (29) R. C. Myers and M. Perry, _Black Holes in Higher Dimensional Space-Times_ , _Annals Phys._ 172 (1986) 304.
  * (30) S. W. Hawking, C. J. Hunter and M. M. Taylor-Robinson, _Rotation and the ads-cft correspondence_ , _Physical Review D_ 59 (Feb, 1999) .
  * (31) F. Tangherlini, _Schwarzschild field in n dimensions and the dimensionality of space problem_ , _Nuovo Cim._ 27 (1963) 636.
  * (32) V. P. Frolov and D. Stojkovic, _Particle and light motion in a space-time of a five-dimensional rotating black hole_ , _Phys. Rev. D_ 68 (2003) 064011, [gr-qc/0301016].
  * (33) T. Igata, _Stable Bound Orbits in Six-dimensional Myers-Perry Black Holes_ , _Phys. Rev. D_ 92 (2015) 024002, [1411.6102].
  * (34) T. Igata, H. Ishihara and Y. Takamori, _Stable Bound Orbits around Black Rings_ , _Phys. Rev. D_ 82 (2010) 101501, [1006.3129].
  * (35) G. Gibbons, H. Lu, D. N. Page and C. Pope, _Rotating black holes in higher dimensions with a cosmological constant_ , _Phys. Rev. Lett._ 93 (2004) 171102, [hep-th/0409155].
  * (36) G. Gibbons, H. Lü, D. N. Page and C. Pope, _The general kerr–de sitter metrics in all dimensions_ , _Journal of Geometry and Physics_ 53 (Jan, 2005) 49–73.
  * (37) V. Kagramanova and S. Reimers, _Analytic treatment of geodesics in five-dimensional Myers-Perry space–times_ , _Phys. Rev. D_ 86 (2012) 084029, [1208.3686].
  * (38) P. Basu, J. Bhattacharya, S. Bhattacharyya, R. Loganayagam, S. Minwalla and V. Umesh, _Small Hairy Black Holes in Global AdS Spacetime_ , _JHEP_ 10 (2010) 045, [1003.3232].
  * (39) S. Bhattacharyya, S. Minwalla and K. Papadodimas, _Small Hairy Black Holes in $AdS_{5}xS^{5}$_, _JHEP_ 11 (2011) 035, [1005.1287].
  * (40) O. J. C. Dias, P. Figueras, S. Minwalla, P. Mitra, R. Monteiro and J. E. Santos, _Hairy black holes and solitons in global $AdS_{5}$_, _JHEP_ 08 (2012) 117, [1112.4447].
  * (41) Dias, Oscar J.C. and Santos, Jorge E., _Boundary Conditions for Kerr-AdS Perturbations_ , _JHEP_ 1310 (2013) 156, [1302.1580].
  * (42) V. Cardoso, O. J. Dias, G. S. Hartnett, L. Lehner and J. E. Santos, _Holographic thermalization, quasinormal modes and superradiance in Kerr-AdS_ , _JHEP_ 1404 (2014) 183, [1312.5323].
  * (43) C. A. R. Herdeiro and E. Radu, _Kerr black holes with scalar hair_ , _Phys. Rev. Lett._ 112 (2014) 221101, [1403.2757].
  * (44) T. Ishii, K. Murata, J. E. Santos and B. Way, _Superradiant instability of black resonators and geons_ , _JHEP_ 07 (2020) 206, [2005.01201].
  * (45) T. Ishii, O. J. Dias, K. Murata, J. E. Santos and B. Way, _Gregory-Laflamme and Superradiance encounter black resonator strings_ , .
  * (46) T. Ishii, O. J. Dias, K. Murata, J. E. Santos and B. Way, _Superradiance and resonators encounter helical black strings_ , .
  * (47) B. Kleihaus, J. Kunz and E. Radu, _Rotating nonuniform black string solutions_ , _JHEP_ 05 (2007) 058, [hep-th/0702053].
  * (48) O. J. Dias, P. Figueras, R. Monteiro, J. E. Santos and R. Emparan, _Instability and new phases of higher-dimensional rotating black holes_ , _Phys.Rev._ D80 (2009) 111701, [0907.2248].
  * (49) O. J. Dias, P. Figueras, R. Monteiro, H. S. Reall and J. E. Santos, _An instability of higher-dimensional rotating black holes_ , _JHEP_ 05 (2010) 076, [1001.4527].
  * (50) O. J. Dias, P. Figueras, R. Monteiro and J. E. Santos, _Ultraspinning instability of rotating black holes_ , _Phys.Rev._ D82 (2010) 104025, [1006.1904].
  * (51) O. J. Dias, R. Monteiro and J. E. Santos, _Ultraspinning instability: the missing link_ , _JHEP_ 08 (2011) 139, [1106.4554].
  * (52) P. Hoxha, R. Martinez-Acosta and C. Pope, _Kaluza-Klein consistency, Killing vectors, and Kahler spaces_ , _Class. Quant. Grav._ 17 (2000) 4207–4240, [hep-th/0005172].
  * (53) H. K. Kunduri, J. Lucietti and H. S. Reall, _Gravitational perturbations of higher dimensional rotating black holes: Tensor perturbations_ , _Phys. Rev. D_ 74 (2006) 084021, [hep-th/0606076].
  * (54) J. E. Martin and H. S. Reall, _On the stability and spectrum of non-supersymmetric AdS(5) solutions of M-theory compactified on Kahler-Einstein spaces_ , _JHEP_ 03 (2009) 002, [0810.2707].
  * (55) O. J. Dias, J. E. Santos and B. Way, _Numerical Methods for Finding Stationary Gravitational Solutions_ , _Class. Quant. Grav._ 33 (2016) 133001, [1510.02804].
  * (56) B. L. Hu, _Separation of tensor equations in a homogeneous space by group theoretical methods_ , _J. Math. Phys._ 15 (1974) 1748–1755.
  * (57) R. Emparan, T. Harmark, V. Niarchos and N. A. Obers, _New Horizons for Black Holes and Branes_ , _JHEP_ 04 (2010) 046, [0912.2352].
  * (58) T. J. M. Zouros and D. M. Eardley, _Instabilities of massive scalar perturbations of a rotating black hole_ , _Annals Phys._ 118 (1979) 139–155.
  * (59) S. Detweiler, _Klein-gordon equation and rotating black holes_ , _Phys. Rev. D_ 22 (Nov, 1980) 2323–2326.
  * (60) S. R. Dolan, _Instability of the massive Klein-Gordon field on the Kerr spacetime_ , _Phys. Rev. D_ 76 (2007) 084001, [0705.2880].
  * (61) R. Emparan, R. Suzuki and K. Tanabe, _Evolution and End Point of the Black String Instability: Large D Solution_ , _Phys. Rev. Lett._ 115 (2015) 091102, [1506.06772].
  * (62) P. Figueras, M. Kunesch and S. Tunyasuvunakool, _End Point of Black Ring Instabilities and the Weak Cosmic Censorship Conjecture_ , _Phys. Rev. Lett._ 116 (2016) 071102, [1512.04532].
  * (63) K. Murata and J. Soda, _Stability of Five-dimensional Myers-Perry Black Holes with Equal Angular Momenta_ , _Prog. Theor. Phys._ 120 (2008) 561–579, [0803.1371].