# Analog Schwarzschild black holes of Bose-Einstein condensates in a cavity:
Quasinormal modes and quasibound states

H. S. Vieira1,2,3, Kyriakos Destounis4,5 and Kostas D. Kokkotas2 1Department
of Physics, Institute of Natural Sciences, Federal University of Lavras,
37200-000 Lavras, Brazil  2Theoretical Astrophysics, Institute for Astronomy
and Astrophysics, University of Tübingen, 72076 Tübingen, Germany  3São Carlos
Institute of Physics, University of São Paulo, 13560-970 São Carlos, Brazil
4Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 5,
00185, Roma, Italy  5INFN, Sezione di Roma, Piazzale Aldo Moro 2, 00185, Roma,
Italy

###### Abstract

Analog models of black holes have unequivocally proven to be extremely
beneficial in providing critical information regarding black hole
spectroscopy, superradiance, quantum phenomena and most importantly Hawking
radiation and black hole evaporation; topics that have either recently begun
to bloom through gravitational wave observations or have not yet been
investigated in astrophysical setups. Black hole analog experiments have made
astonishing steps toward the aforementioned directions and are paramount in
understanding the quantum nature of the gravitational field. Recently, a
tabletop analog Schwarzschild black hole has been proposed by placing Bose-
Einstein condensates of photons inside a mirror’s cavity, leading to a sink
with a radial vortex that represents a velocity singularity. Here, we provide
an extensive spectral analysis of both the tabletop acoustic black hole and
its higher-dimensional gravitational analog. We find that quasinormal modes
and quasibound states share qualitative similarities in both systems and show
that the eikonal quasinormal modes of the analog acoustic black hole have a
photon-sphere-like interpretation, which points to the existence of a phonon
sphere in the analog black hole. Our results, complemented with the recently
calculated graybody factors and Hawking radiation of the acoustic analog, can
provide a theoretical test bed for future tabletop experiments with
condensates of light in a mirror’s cavity and provide significant insights
regarding classical and quantum phenomena in higher-dimensional black holes.

##  I Introduction

The emission and detection of gravitational waves (GWs) from the binary
coalescence of black holes (BHs) [1, 2, 3, 4, 5, 6, 7] and compact objects [8]
have established the field of experimental gravitation and continue to provide
insights into the strong field regime towards the search for an ultimate
theory of gravitation. General relativity (GR) is currently been tested
meticulously [9, 10, 11, 12] and has proven to be the most successful to date.

GWs carry unparalleled information during all stages of the binary’s evolution
as it undergoes the inspiral, merger and ringdown stage. During these phases,
GWs bear the externally observable properties of the binary’s components and
the eventual compact object’s parameters that forms after ringdown, described
by quasinormal modes (QNMs) [13, 14, 15]. BHs undergo surprising phenomena,
such as Hawking radiation when semiclassical effects are taken into account
[16], they superradiate [17] under the expense of the BH’s spin [18] or charge
[19, 20, 21, 22, 23], and when perturbed, they vibrate according to their
characteristic spectra.

The BH spectroscopy program is currently aiming towards an ultimate
understanding of QNMs, how to detect them properly from GW data [24, 25, 26],
and if nonlinearities [27, 28, 29, 30], as well as spectral instabilities [31,
32, 33, 34, 35, 36, 37, 38] can interfere with QNM extraction. Nevertheless,
superradiance and Hawking radiation have not been detected experimentally. To
this end, different models are being used in an effort to mimic BHs and
recreate such processes in controlled laboratory experiments.

Analog gravity [39, 40] has the potential to provide a better understanding
regarding phenomena that would otherwise elude observation in the strong field
regime and when quantum effects become significant. Unruh pioneered the first
theoretical BH analog [41], which can be envisioned through a fluid which on
its surface, sound excitations are mapped to perturbation equations that are
typically found and utilized in classical and quantum gravity investigations.
The surface fluctuations endure an effective spacetime geometry that is
determined by the propagation speed of these excitations and their relative
speed with respect to the fluid, which enables the construction of an acoustic
BH analog. Tuning the velocity of the fluid leads to the formation of an
acoustic event horizon analog, beyond which sound fluctuations cannot escape
the dumb hole, in analogy to infalling particles into a BH event horizon which
are causally disconnected with spatial infinity.

Currently, there is an abundance in analog gravity experiments that function
with fluids and superfluids [42, 43, 44, 45, 46, 47, 48, 49], Bose-Einstein
condensates (BECs) [50, 51, 52, 53] and optical media [54]. Through these
analog experiments, Hawking radiation [42, 51, 52, 55], superradiance [56, 43,
44, 45, 57], QNMs and quasibound states (QBSs) [46, 48, 49] have been
experimentally observed. These experiments are not only successful scientific
achievements but also can provide unmatched insights into the future of
gravitational physics.

Very recently, a novel tabletop experimental three-dimensional analog of a
Schwarzschild BH was proposed with BECs of light placed in a mirror’s cavity
[58] which leads to a sink with a radial vortex pointing towards the central
velocity singularity, that was missing in other experiments with BECs where
the analog BH horizons had a one-dimensional flow pattern that smoothly
interpolates between a velocity below the speed of sound and thus did not
possess a singular velocity profile. The novel proposal in [58] serves as an
analog of a rotationally-symmetric five-dimentional Schwarzschild BH with a
sonic event horizon and a velocity singularity due to the radial vortex, thus
can shed light on the properties close to the region of the singularity.

In Ref. [58], the authors have calculated the graybody factors and Hawking
radiation of the proposed experimental setup in an attempt to provide data for
future experiments. In our analysis we complete their study by calculating the
QNMs and QBSs for the three-dimensional BEC of light in a cavity and revisit
the spectrum of five-dimensional Schwarzschild BH. We find that QNMs and QBSs
share qualitative similarities in both the gravitational and the BEC analog
system, thus giving more legitimacy to the analogy, and identify that the
photon sphere interpretation of eikonal QNMs of the gravitational BH is shared
in the analog system. Thus, the newly proposed three-dimensional acoustic BH
analog possesses a phonon sphere which properties are intrinsically connected
to the eikonal QNMs of phase fluctuations.

##  II Higher-dimensional Schwarzschild black holes

The standard four-dimensional Schwarzschild solution was generalized to higher
dimensions by Tangherlini in [59]. The higher-dimensional action, which is a
generalization of the Einstein-Hilbert action, is given by

$S=\frac{1}{16\pi G_{D}}\int d^{D}x\sqrt{-g}R,$ (1)

where $g\equiv\det(g_{\mu\nu})$, and $G_{D}$ is the $D$-dimensional
gravitational constant. In what follows, for simplicity and without loss of
generality, we adopt the natural units where $G_{D}=c=\hbar=1$. By varying the
action with respect to the spacetime metric $g_{\mu\nu}$, we obtain the
Einstein equations in higher dimensions

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=0,$ (2)

where $R_{\mu\nu}$ and $R$ are the Ricci tensor and Ricci scalar,
respectively. A static, asymptotically flat and spherically-symmetric vacuum
solution of Eq. (2), which generalizes the Schwarzschild BH solution in
$D$-dimensions, has the form

$ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}d\Omega_{D-2}^{2},$ (3)

where the metric function, $f(r)$, and the solid angle element,
$d\Omega_{D-2}^{2}$, are given by

$f(r)=1-\frac{\mathcal{M}}{r^{D-3}},$ (4)

and

$\displaystyle d\Omega_{D-2}^{2}$
$\displaystyle=d\theta_{1}^{2}+\sin^{2}\theta_{1}d\theta_{2}^{2}+\sin^{2}\theta_{1}\sin^{2}\theta_{2}d\theta_{2}^{2}+\cdots$
$\displaystyle+\left[\left(\prod_{j=1}^{D-2}\sin^{2}\theta_{j}\right)d\theta_{D-1}^{2}\right].$
(5)

The parameter $\mathcal{M}$ is related to the BH mass $M$ through the relation

$\mathcal{M}=\frac{16\pi
M}{(D-2)\Omega_{D-2}},\,\,\,\Omega_{D-2}=\frac{2\pi^{\frac{D-1}{2}}}{\Gamma(\frac{D-1}{2})},$
(6)

where $\Omega_{D-2}$ is the volume of the $(D-2)$-dimensional unit sphere and
$\Gamma(x)$ is the gamma function.

Note that the solution (3) simplifies to the standard Schwarzschild metric
when $D=4$. In addition, we observe that when the spacetime is higher-
dimensional, the asymptotic fall-off term in the metric function $f(r)$
becomes steeper. This metric is Ricci flat and is called either the
Schwarzschild-Tangherlini solution or simply the $D$-dimensional Schwarzschild
BH spacetime. In this work, we will focus on the five-dimensional
Schwarzschild BH, due to the explicit analogy between BECs of light in a
cavity that form a three-dimensional Schwarzschild analog, as discussed in the
introduction.

The explicit line element (3) for $D=5$ reads

$\displaystyle ds^{2}=$ $\displaystyle-f(r)dt^{2}+f(r)^{-1}dr^{2}$
$\displaystyle+r^{2}\left[d\theta^{2}+\sin^{2}\theta\left(d\phi^{2}+\sin^{2}\phi\
d\chi^{2}\right)\right],$ (7)

with

$f(r)=1-\frac{\mathcal{M}}{r^{2}},\,\,\,\,\mathcal{M}=\frac{8M}{3\pi},\,\,\,\,\Omega_{D-2}=2\pi^{2},$
(8)

where $\theta$, $\phi$ run over the range 0 to $\pi$, and $\chi$ from 0 to
$2\pi$. The causal structure of spacetime can be identified from the equation

$f(r)=0=(r-r_{1})(r-r_{2}),$ (9)

whose solutions are the event horizon, $r_{1}=\sqrt{\mathcal{M}}$, and a
negative non-physical solution, $r_{2}=-\sqrt{\mathcal{M}}$.

###  II.1 Scalar wave equation

Figure 1: _Top panel:_ Real (left) and imaginary (right) part of the
fundamental massless scalar QNMs $\omega_{0\nu}$ of a five-dimensional
Schwarzschild black hole with respect to $\mathcal{M}$ and varying $\nu$.
_Bottom panel:_ Real and imaginary part of the first overtone of massless
scalar QNMs $\omega_{1\nu}$ of a five-dimensional Schwarzschild black hole
with respect to $\mathcal{M}$ and varying $\nu$.

We consider a minimally-coupled massless scalar perturbation
$\Psi=\Psi(t,r,\theta,\phi,\chi)$, whose equation of motion is given by the
Klein-Gordon equation

$\left[\frac{1}{\sqrt{-g}}\partial_{\mu}\left(g^{\mu\nu}\sqrt{-g}\partial_{\nu}\right)\right]\Psi=0.$
(10)

Under symmetry assumptions we choose an ansatz

$\Psi(t,r,\theta,\phi,\chi)=\mbox{e}^{-i\omega t}U(r)P(\theta,\phi,\chi),$
(11)

where $\omega$ is the frequency of the scalar field, $U(r)=R(r)/r^{3/2}$ is
the radial function and $P(\theta,\phi,\chi)$ is an angular function. By
utilizing the spacetime metric (7), Eq. (10) can be separated into an angular
and a radial equation as

$\displaystyle\left(\frac{1}{\sin^{2}\theta}\frac{\partial}{\partial\theta}\biggl{(}\sin^{2}\theta\frac{\partial}{\partial\theta}\biggr{)}+\lambda-\frac{1}{\sin^{2}\theta}\left[\frac{1}{\sin\phi}\frac{\partial}{\partial\phi}\biggl{(}\sin\phi\frac{\partial}{\partial\phi}\biggr{)}\right.\right.$
$\displaystyle\left.\left.+\bar{\lambda}+\frac{1}{\sin^{2}\phi}\frac{\partial^{2}}{\partial\chi^{2}}\right]\right)P(\theta,\phi)=0,$
(12)

and

$\displaystyle R^{\prime\prime}(r)+\frac{f^{\prime}(r)}{f(r)}R^{\prime}(r)$
$\displaystyle+\left[\frac{\omega^{2}}{f^{2}(r)}-\frac{4\lambda+3f(r)+6rf^{\prime}(r)}{4r^{2}f(r)}\right]R(r)=0.$
(13)

Here, $\lambda$ and $\bar{\lambda}$ are the separation constants to be
determined, and the primes generally denote differentiation with respect to
$r$.

The general exact solution of the angular equation (12) is given by
$P(\theta,\phi,\chi)=P_{\nu,4}^{l}(\cos\theta)Y_{lm}(\phi,\chi)$, where
$P_{\nu,4}^{l}(\cos\theta)$ are the associated Legendre functions in four-
dimensions [60, 61, 62] with arbitrary degree $\nu\in\mathbb{C}$ and order $l$
such that $\lambda=\nu(\nu+2)$, and $Y_{lm}(\phi,\chi)$ are the spherical
harmonics with $l$ and $m$ being the angular and magnetic quantum numbers,
respectively, such that $\bar{\lambda}=l(l+1)$.

In what follows, we will revisit the QNMs of five-dimensional Schwarzschild
BHs and calculate analytically the corresponding QBSs.

###  II.2 Quasinormal modes

Equation (13) can be recast into a Schrödinger-like equation by multiplying it
with $f^{2}(r)$ and utilizing the tortoise coordinate $r_{*}$, with
$dr/dr_{*}=f(r)$, to obtain

$\frac{d^{2}R(r)}{dr_{*}^{2}}+\left(\omega^{2}-V_{\text{BH}}\right)R(r)=0,$
(14)

where the effective potential $V_{\text{BH}}$ is

$V_{\text{BH}}\equiv
f(r)\left(\frac{\nu(\nu+2)}{r^{2}}+\frac{3f(r)}{4r^{2}}+\frac{3f^{\prime}(r)}{2r}\right).$
(15)

QNMs $\omega_{n\nu}$ are a discrete set of mode solutions of the radial
equation (14) with purely ingoing (outgoing) boundary conditions at the event
horizon (infinity), such that

$R(r)\sim e^{-i\omega r_{*}},\,\,r\rightarrow r_{1},\,\,\,\,\,R(r)\sim
e^{i\omega r_{*}},\,\,r\rightarrow\infty,$ (16)

where $n$ is the overtone number of the frequencies. In what follows, we
present some typical scalar QNMs of a five-dimensional Schwarzschild BH. We
have employed three different methods, namely a pseudospectral method [63], a
matrix method [64, 65] and the Wentzel-Kramers-Brillouin (WKB) method [66,
67]. We have found convergence between the pseudospectral and matrix method
QNMs up to at least six digits (convergence can reach even to 20 digits when
more grid points are utilized) and confirmed the validity of eikonal QNMs with
the WKB approximation finding similar precision. Our results also are in
perfect agreement with those reported in [68].

Figure 1 demonstrates the typical behavior of five-dimensional Schwarzschild
scalar QNMs, i.e. the increment of the BH’s mass term decreases the
oscillation frequency and increases the perturbation’s lifetime. Furthermore,
the increment of the angular number $\nu$ increases the real part of QNMs
while the imaginary part is also increased (in absolute value) as expected
from the eikonal limit which probes the photon sphere of the BH, where photons
occupy unstable circular geodesics [69]. There, the angular frequency of null
geodesics is proportional to the real part of the eikonal QNM and their
instability timescale is proportional to the imaginary part of QNMs with large
$\nu$. Therefore, the BH QNMs in study have a typical photon sphere
interpretation [69] which we will further discuss in the upcoming sections.
For a numerical presentation of some QNMs we refer the reader to Table 1 and
Ref. [68].

5D Scwarzschild BH
---
$\nu$ | $n=0$ | $n=1$
1 | 1.0160 - 0.3623 i | 0.8564 - 1.1576 i
2 | 1.5106 - 0.3575 i | 1.3927 - 1.1046 i
10 | 5.5028 - 0.3539 i | 5.4689 - 1.0639 i

Table 1: Massless scalar QNMs of the five-dimensional (5D) Schwarzschild BH
with $\mathcal{M}=1$.

###  II.3 Quasibound states

In what follows, we will analytically solve the radial equation (13) for a
five-dimensional Schwarzschild BH. QBSs are solutions to (13) with purely
ingoing boundary conditions at the event horizon (same as QNMs) and decaying
boundary conditions at infinity. Since both QNMs and QBSs are mode solutions
to the same eigenvalue problem, though with different asymptotic behavior, we
will also refer to QBSs as $\omega_{n\nu}$. We will see that the fact that at
infinity QBSs decay exponentially enables an analytic evaluation. To find
their analytic expression, we apply the VBK approach [70, 71] in order to
write the radial equation (13) as a Heun-type differential equation [72].

We define a new radial coordinate, $x$, and a new parameter, $x_{1}$, as

$x=\frac{r-r_{2}}{-r_{2}},$ (17)

and

$x_{1}=\frac{r_{1}-r_{2}}{-r_{2}},$ (18)

such that the three original singularities $(r_{2},0,r_{1})$ are moved to the
points $(0,1,x_{1})$, while maintaining a regular singularity at spatial
infinity. In addition, the regular singular point at $x=x_{1}$ is always
located outside the unit circle $|x_{1}|>1$. The final step is to perform an
F-homotopic transformation $R(x)\mapsto y(x)$ given by

$R(x)=x^{A_{0}}(x-1)^{A_{1}}(x-x_{1})^{A_{x_{1}}}y(x),$ (19)

where the coefficients $A_{0}$, $A_{1}$, and $A_{x_{1}}$ are given by

$\displaystyle A_{0}$ $\displaystyle=$
$\displaystyle-\frac{i\omega}{r_{1}-r_{2}},$ (20) $\displaystyle A_{1}$
$\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{1}{r_{1}}\sqrt{r_{1}\biggl{(}r_{1}+\frac{\lambda}{r_{2}}\biggr{)}},$
(21) $\displaystyle A_{x_{1}}$ $\displaystyle=$
$\displaystyle-\frac{i\omega}{r_{1}-r_{2}}.$ (22)

Thus, by substituting Eqs. (17)-(22) into Eq. (13), we obtain

$\displaystyle
y^{\prime\prime}(x)+\left(\frac{1+2A_{0}}{x}+\frac{2A_{1}}{x-1}+\frac{1+2A_{x_{1}}}{x-x_{1}}\right)y^{\prime}(x)$
$\displaystyle+\frac{A_{3}x+A_{4}}{x(x-1)(x-x_{1})}y(x)=0,$ (23)

where the coefficients $A_{3}$, and $A_{4}$ are given by

$\displaystyle A_{3}$
$\displaystyle=-3+A_{x_{1}}+2A_{1}(1+A_{x_{1}})+A_{0}(1+2A_{1}+2A_{x_{1}})$
$\displaystyle-\frac{(x_{1}-1)(x_{1}^{2}\lambda-2\omega^{2}+2x_{1}\omega^{2})}{r_{1}^{2}x_{1}^{2}},$
(24) $\displaystyle A_{4}$
$\displaystyle=-A_{0}-A_{x_{1}}-2A_{0}A_{x_{1}}+\frac{3x_{1}}{2}-A_{1}x_{1}-2A_{0}A_{1}x_{1}$
$\displaystyle+\frac{(x_{1}-1)^{2}(x_{1}^{2}\lambda+2\omega^{2})}{r_{1}^{2}x_{1}^{2}}.$
(25)

The radial equation (23) has the form of a general Heun equation [72].
Therefore, the exact solution for the radial part of the Klein-Gordon equation
can be written as

$\displaystyle R_{j}(x)=$ $\displaystyle
x^{\frac{1}{2}(\gamma-1)}(x-1)^{\frac{1}{2}\delta}(x-x_{1})^{\frac{1}{2}(\epsilon-1)}$
$\displaystyle[C_{1,j}\ y_{1,j}(x)+C_{2,j}\ y_{2,j}(x)],$ (26)

where $C_{1,j}$ and $C_{2,j}$ are constants to be determined, and
$j=\\{0,1,x_{1},\infty\\}$ labels the solution at each singular point, which
are given as follows. The pair of linearly independent solutions at $x=0$
($r=r_{2}$) is given by

$\displaystyle y_{1,0}=\,$
$\displaystyle\mbox{HeunG}(x_{1},q;\alpha,\beta,\gamma,\delta;x),$ (27)
$\displaystyle y_{2,0}=\,$ $\displaystyle
x^{1-\gamma}\mbox{HeunG}(x_{1},(x_{1}\delta+\epsilon)(1-\gamma)+q;$
$\displaystyle\alpha+1-\gamma,\beta+1-\gamma,2-\gamma,\delta;x),$ (28)

where $\mbox{HeunG}(x_{1},q;\alpha,\beta,\gamma,\delta;x)$ denotes a general
Heun function, which is analytic in the disk $|x|<1$, and has a Maclaurin
expansion given by

$\mbox{HeunG}(x_{1},q;\alpha,\beta,\gamma,\delta;x)=\sum_{n=0}^{\infty}c_{n}x^{n},$
(29)

with

$\displaystyle-qc_{0}+x_{1}\gamma c_{1}$ $\displaystyle=$ $\displaystyle
0\quad(c_{0}=1),$ (30) $\displaystyle
P_{n}c_{n-1}-(Q_{n}+q)c_{n}+X_{n}c_{n+1}$ $\displaystyle=$ $\displaystyle
0\quad(n\geq 1),$ (31)

and

$\displaystyle P_{n}$ $\displaystyle=$ $\displaystyle(n-1+\alpha)(n-1+\beta),$
$\displaystyle Q_{n}$ $\displaystyle=$ $\displaystyle
n[(n-1+\gamma)(1+x_{1})+x_{1}\delta+\epsilon],$ (32) $\displaystyle X_{n}$
$\displaystyle=$ $\displaystyle(n+1)(n+\gamma)x_{1}.$

The pair of linearly independent solutions at $x=1$ ($r=0$) is given by

$\displaystyle y_{1,1}$
$\displaystyle=\mbox{HeunG}(1-x_{1},\alpha\beta-q;\alpha,\beta,\delta,\gamma;1-x),$
(33) $\displaystyle y_{2,1}$
$\displaystyle=(1-x)^{1-\delta}\mbox{HeunG}(1-x_{1},((1-x_{1})\gamma+\epsilon)(1-\delta)$
$\displaystyle+\alpha\beta-q;\alpha+1-\delta,\beta+1-\delta,2-\delta,\gamma;1-x).$
(34)

The pair of linearly independent solutions at $x=x_{1}$ ($r=r_{1}$) is given
by

$\displaystyle y_{1,x_{1}}$
$\displaystyle=\mbox{HeunG}\biggl{(}\frac{x_{1}}{x_{1}-1},\frac{\alpha\beta
x_{1}-q}{x_{1}-1};\alpha,\beta,\epsilon,\delta;\frac{x_{1}-x}{x_{1}-1}\biggl{)},$
(35) $\displaystyle y_{2,x_{1}}$
$\displaystyle=\left(\frac{x_{1}-x}{x_{1}-1}\right)^{1-\epsilon}\mbox{HeunG}\left(\frac{x_{1}}{x_{1}-1},\right.$
$\displaystyle\left.\frac{(x_{1}(\delta+\gamma)-\gamma)(1-\epsilon)}{x_{1}-1}+\frac{\alpha\beta
x_{1}-q}{x_{1}-1};\alpha+1-\epsilon,\right.$
$\displaystyle\left.\beta+1-\epsilon,2-\epsilon,\delta;\frac{x_{1}-x}{x_{1}-1}\right).$
(36)

The pair of linearly independent solutions at $x=\infty$ ($r=\infty$) is given
by

$\displaystyle y_{1,\infty}$
$\displaystyle=x^{-\alpha}\mbox{HeunG}\left(\frac{1}{x_{1}},\alpha(\beta-\epsilon)+\frac{\alpha}{x_{1}}(\beta-\delta)\right.$
$\displaystyle\left.-\frac{q}{x_{1}};\alpha,\alpha-\gamma+1,\alpha-\beta+1,\delta;\frac{1}{x}\right),$
(37) $\displaystyle y_{2,\infty}$
$\displaystyle=x^{-\beta}\mbox{HeunG}\left(\frac{1}{x_{1}},\beta(\alpha-\epsilon)+\frac{\beta}{x_{1}}(\alpha-\delta)\right.$
$\displaystyle\left.-\frac{q}{x_{1}};\beta,\beta-\gamma+1,\beta-\alpha+1,\delta;\frac{1}{x}\right).$
(38)

In these solutions, the parameters $\alpha$, $\beta$, $\gamma$, $\delta$,
$\epsilon$, and $q$ are given by

$\displaystyle\alpha$
$\displaystyle=\frac{1}{r_{1}-r_{2}}\left[3r_{1}-\sqrt{r_{1}\biggl{(}r_{1}+\frac{\lambda}{r_{2}}\biggr{)}}\right.$
$\displaystyle\left.+r_{2}\biggl{(}\sqrt{1+\frac{\lambda}{r_{1}r_{2}}}-3\biggr{)}-2i\omega\right],$
(39) $\displaystyle\beta$
$\displaystyle=-\frac{1}{r_{1}-r_{2}}\left[r_{1}+\sqrt{r_{1}\biggl{(}r_{1}+\frac{\lambda}{r_{2}}\biggr{)}}\right.$
$\displaystyle\left.-r_{2}\biggl{(}\sqrt{1+\frac{\lambda}{r_{1}r_{2}}}+1\biggr{)}+2i\omega\right],$
(40) $\displaystyle\gamma$ $\displaystyle=1-\frac{2i\omega}{r_{1}-r_{2}},$
(41) $\displaystyle\delta$
$\displaystyle=1-\frac{2}{r_{1}}\sqrt{r_{1}^{2}+\frac{\lambda r_{1}}{r_{2}}},$
(42) $\displaystyle\epsilon$ $\displaystyle=1-\frac{2i\omega}{r_{1}-r_{2}},$
(43) $\displaystyle q$
$\displaystyle=\frac{r_{1}}{r_{2}}-\frac{\lambda}{r_{2}^{2}}-1+\frac{1}{r_{2}}\biggl{[}\sqrt{r_{1}^{2}+\frac{\lambda
r_{1}}{r_{2}}}+i\omega\biggr{]}$
$\displaystyle-\frac{(r_{2}+2i\omega)}{r_{1}r_{2}}\sqrt{r_{1}^{2}+\frac{\lambda
r_{1}}{r_{2}}}-\frac{2i(r_{1}-r_{2}-2i\omega)\omega}{(r_{1}-r_{2})^{2}}.$ (44)

The first boundary condition related to QBSs (and QNMs) requires that the
radial solution is purely ingoing at the event horizon, i.e. to impose
$r\rightarrow r_{1}$ (or $x\rightarrow x_{1}$) on the radial solution given by
Eq. (26). To do this, we use Eqs. (35) and (36) to get the following
asymptotic behavior

$\lim_{r\rightarrow r_{1}}R_{x_{1}}(r)\sim C_{1,x_{1}}\
(r-r_{1})^{\frac{1}{2}(\epsilon-1)}+C_{2,x_{1}}\
(r-r_{1})^{-\frac{1}{2}(\epsilon-1)},$ (45)

where all the remaining constants were included in $C_{1,x_{1}}$ and
$C_{2,x_{1}}$, which will be determined afterwards. Thus, from Eq. (43), we
can write

$\lim_{r\rightarrow r_{1}}R_{x_{1}}(r)\sim C_{1,x_{1}}\ \Psi_{{\rm
in},x_{1}}+C_{2,x_{1}}\ \Psi_{{\rm out},x_{1}},$ (46)

where $\Psi_{{\rm in},x_{1}}$ and $\Psi_{{\rm out},x_{1}}$ are the ingoing and
outgoing scalar wave solutions, respectively, given by

$\displaystyle\Psi_{{\rm in},x_{1}}(r>r_{1})$ $\displaystyle=$
$\displaystyle\mbox{e}^{-i\omega t}(r-r_{1})^{-\frac{i\omega}{2\kappa_{1}}},$
(47) $\displaystyle\Psi_{{\rm out},x_{1}}(r>r_{1})$ $\displaystyle=$
$\displaystyle\mbox{e}^{-i\omega t}(r-r_{1})^{+\frac{i\omega}{2\kappa_{1}}},$
(48)

with the surface gravity of the event horizon, $\kappa_{1}$, being defined as

$\kappa_{1}=\frac{1}{2}\left.\frac{df(r)}{dr}\right|_{r=r_{1}}=\frac{r_{1}-r_{2}}{2}.$
(49)

The ingoing boundary condition is satisfied when $C_{2,x_{1}}=0$ in Eq. (46)
and Eq. (26). Thus, we have

$\lim_{r\rightarrow r_{1}}R_{x_{1}}(r)\sim C_{1,x_{1}}\ \Psi_{{\rm
in},x_{1}}.$ (50)

Figure 2: _Top panel:_ Real (left) and imaginary (right) part of the
fundamental massless scalar QBSs $\omega_{0\nu}$ of a five-dimensional
Schwarzschild black hole with respect to $\mathcal{M}$ and varying $\nu$.
_Bottom panel:_ Real and imaginary part of the first overtone of massless
scalar QBSs $\omega_{1\nu}$ of a five-dimensional Schwarzschild black hole
with respect to $\mathcal{M}$ and varying $\nu$.

The second boundary condition related to the QBSs (that differs from that of
QNMs) is to require that the radial solution vanishes at asymptotic infinity,
i.e. to impose a decaying boundary condition at the limit $r\rightarrow\infty$
(or $x\rightarrow\infty$) on the radial solution given by Eq. (26). Thus, we
use Eqs. (37) and (38) to obtain the following asymptotic behavior

$\lim_{r\rightarrow\infty}R_{\infty}(r)\sim C_{1,\infty}\
\frac{1}{r^{\sigma}},$ (51)

where we have imposed that $C_{2,\infty}=0$. The parameter $\sigma$ is given
by

$\sigma=\alpha-A_{0}-A_{1}-A_{x_{1}}.$ (52)

The sign of the real part of $\sigma$ determines the asymptotic behavior of
the radial scalar wave function as $r\rightarrow\infty$: if
$\mbox{Re}[\sigma]>0$, the radial solution tends to zero at spatial infinity
and then it fully satisfies the second boundary condition for QBSs; if
$\mbox{Re}[\sigma]<0$, the radial solution diverges at spatial infinity.

The final asymptotic behavior of the radial scalar wave solution at spatial
infinity will be determined when we know the values of the coefficients
$A_{j}$, as well as the frequencies $\omega$, and the parameter $\alpha$.
Those will be obtained by the VBK approach.

The general Heun functions become (class I) polynomials of degree $n$ $(\geq
0)$ if and only if they satisfy two conditions [72], namely,

$\displaystyle\alpha+n$ $\displaystyle=$ $\displaystyle 0,$ (53)
$\displaystyle c_{n+1}(q)$ $\displaystyle=$ $\displaystyle 0,$ (54)

where the accessory parameter $q$ is given by Eq. (44). The first condition,
given by Eq. (53), is called the $\alpha$-condition, which is used to find the
frequency eigenvalues. The second condition, given by Eq. (54), determines the
value of the separation constant $\lambda$ for each value of $n$, which is
used to find the eigenvalues $\nu$ and both the radial and angular wave
eigenfunctions. It is worth emphasizing that the two polynomial conditions
could be applied in any order we wish, that is, the first condition could give
the values of the separation constant, and the second condition could give the
frequency eigenvalues. Thus, by imposing Eq. (53), we obtain the exact
spectrum of QBSs given by

$\omega_{n\nu}=-i(\sqrt{\mathcal{M}}(n+3)-\sqrt{\mathcal{M}-\lambda}),$ (55)

where we have used the explicit expressions of the event horizon. Here, the
separation constant is, again, $\lambda=\nu(\nu+2)$, and $n$ is the overtone
number, which can be, without loss of generality, called the principal quantum
number. On the other hand, by substituting Eqs. (20), (21), (22), (53) and
(55) into Eq. (52), we get

$\sigma=4,$ (56)

which means that the parameter $\sigma$ is a real, positive number for any
value of the principal quantum number $n$, as well as for any value of the
separation constant $\lambda$, and hence the frequency eigenvalues given by
Eq. (55) represent QBSs for massless scalar fields in the five-dimensional
Schwarzschild spacetime. In addition, by substituting these equations into Eq.
(44), it is possible to obtain a nonlinear equation for $q=q(\lambda)$ and
then impose the second polynomial condition given by Eq. (54) to obtain the
angular eigenvalues $\lambda$.

Now, let us discuss a very important case of these QBSs: the fundamental mode,
where $n=0$. In this case, we have

$\omega_{0\nu}=-i(3\sqrt{\mathcal{M}}-\sqrt{\mathcal{M}-\lambda}),$ (57)

where the second polynomial condition, given by Eq. (54), is automatically
satisfied, since it leads to

$c_{n+1}(q)\biggl{|}_{n=0}=q=n(4+n)\biggl{|}_{n=0}=0.$ (58)

This means that there is no restriction on the value of the separation
constant $\lambda=\nu(\nu+2)$, with $\nu=0,1,2,\dots$. In particular, the
ground state $(n,\nu)=(0,0)$ of QBSs is given by
$\omega_{00}=-2i\sqrt{\mathcal{M}}$. This is a very important result because
it designates the absence of bound states when the angular number is null.
These solution are not oscillatory but only decay in time therefore no bound
state can be formed in this particular case, and the BH only oscillates in
accord to QNMs.

In Fig. 2 we show the behavior of massless scalar QBSs in the the BH under
study. The real part for the QBS $\omega_{00}$ is always zero since it does
not depend on $\mathcal{M}$ and corroborates the aforementioned discussion.
When $\nu\neq 0$ QBSs exist; the mass parameter acts in the opposite way with
respect to that of QNMs, namely their real parts increase with $\mathcal{M}$
while their lifetime decreases. The angular number $\nu$ has a somewhat
similar effect to that of QNMs, that is its increment increases (in absolute
value) the oscillation frequency of QBSs and quickly drives the imaginary
parts to their asymptotic, large $\nu$, value. Of course, since these modes
have different boundary conditions they have no connection to null geodesics
in the photon sphere at the eikonal limit.

##  III Analog black holes: Bose-Einstein condensates of light in a cavity

In this section, we begin by considering the general acoustic BH solution in
Minkowski spacetime obtained by Unruh [41] (see also Refs. [39, 40]), and then
show how to properly choose the sound velocity in order to obtain a
Schwarzschild BH in BECs of photons trapped in a mirror cavity [58, 73].

The fundamental equations of motion for an irrotational fluid are given by

$\displaystyle\nabla\times\mathbf{v}$ $\displaystyle=$ $\displaystyle 0,$ (59)
$\displaystyle\partial_{t}\rho+\nabla\cdot(\rho\mathbf{v})$ $\displaystyle=$
$\displaystyle 0,$ (60)
$\displaystyle\rho\frac{d\mathbf{v}}{dt}\equiv\rho[\partial_{t}\mathbf{v}+(\mathbf{v}\cdot\nabla)\mathbf{v}]$
$\displaystyle=$ $\displaystyle-\nabla p,$ (61)

where $\mathbf{v}$, $\rho$, and $p$ are the velocity, density, and pressure of
the fluid, respectively. Next, we introduce the velocity potential $\Psi$,
such that $\mathbf{v}=-\nabla\Psi$, and assume the fluid as barotropic, which
means that $\rho=\rho(p)$. Then, by linearizing these equations of motion
around some background $(\rho_{0},p_{0},\Psi_{0})$, namely,

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\rho_{0}+\epsilon\rho_{1},$
(62) $\displaystyle p$ $\displaystyle=$ $\displaystyle p_{0}+\epsilon p_{1},$
(63) $\displaystyle\Psi$ $\displaystyle=$
$\displaystyle\Psi_{0}+\epsilon\Psi_{1},$ (64)

we obtain the following equation

$\displaystyle-\partial_{t}\biggl{[}\frac{\partial\rho}{\partial
p}\rho_{0}(\partial_{t}\Psi_{1}+\mathbf{v}_{0}\cdot\nabla\Psi_{1})\biggr{]}+$
$\displaystyle\nabla\cdot\biggl{[}\rho_{0}\nabla\Psi_{1}-\frac{\partial\rho}{\partial
p}\rho_{0}\mathbf{v}_{0}(\partial_{t}\Psi_{1}+\mathbf{v}_{0}\cdot\nabla\Psi_{1})\biggr{]}=0,$
(65)

where the local speed of sound, $c_{s}$, is defined by

$c_{s}^{-2}\equiv\frac{\partial\rho}{\partial p}.$ (66)

Equation (65) describes the propagation of the linearized scalar potential
$\Psi_{1}$, that is, it governs the propagation of the phase fluctuations as
weak excitations in a homogeneous stationary condensate, which can be
rewritten as a wave equation in an analog curved spacetime as

$\frac{1}{\sqrt{-g}}\partial_{\mu}(g^{\mu\nu}\sqrt{-g}\partial_{\nu}\Psi_{1})=0.$
(67)

Note that Eq. (67) is similar to the covariant Klein-Gordon equation (10),
where the acoustic line element can be written as

$ds^{2}=\frac{\rho_{0}}{c_{s}}[-c_{s}^{2}dt^{2}+(dx^{i}-v_{0}^{i}dt)\delta_{ij}(dx^{j}-v_{0}^{j}dt)].$
(68)

Now, let us revisit how the $(1+2)$-dimensional acoustic BH was set up in Ref.
[58]. Firstly, we can assume that the fluid is a static spherically-symmetric
conservation flow, with a polar angle $\theta=\pi/2$, which implies that
$d\theta=0$. Next, we define the speed of sound as
$c_{s}\equiv\sqrt{kn_{0}/\mu}$, where $k$ is the strength of the effective
contact interaction, $n_{0}$ is the condensate density, and $\mu$ is the
effective mass of the photon gas. Then, the velocity of the fluid $\mathbf{v}$
is the flow rate along the radial direction (representing the radial vortex),
that is it points to the center of the acoustic BH towards the velocity
singularity, given by

$\mathbf{v}_{0}=-\frac{\hbar c_{0}}{\mu r}\hat{\mathbf{r}}=-\xi
c_{s}\frac{c_{0}}{r}\hat{\mathbf{r}},$ (69)

where $\xi(=\hbar/\mu c_{s})$ is the correlation length, and $c_{0}(>0)$ is a
constant related to the acoustic horizon’s radius. Thus, the acoustic line
element resulting by Eq. (68) can be written as

$ds^{2}=-\biggl{(}1-\frac{c_{0}^{2}}{r^{2}}\biggr{)}dt^{2}+2\frac{c_{0}}{r}drdt+dr^{2}+r^{2}d\phi^{2}.$
(70)

With the coordinate transformation

$dt\rightarrow dt-\frac{c_{0}}{r(1-c_{0}^{2}/r^{2})}dr,$ (71)

we can write the line element of an acoustic $(1+2)$-dimensional BEC of light
in a cavity as

$ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}d\phi^{2},$ (72)

where the warp factor $f(r)$ has the form

$f(r)=1-\frac{c_{0}^{2}}{r^{2}}.$ (73)

The radial vortex exists in a spatially $(1+2)$-dimensional BEC, therefore
since the warp factor (73) is identical to the metric function (8), under the
identification $\mathcal{M}=c_{0}^{2}$, then this model is an analog of a
$(1+4)$-dimensional Schwarzschild BH. This is a consequence of the fact that
the metric of the BEC does not obey Einstein’s field equations, but rather it
is determined by pure hydrodynamics [58].

The causal structure of the particular analog BH is strikingly similar to that
presented in Sec. II. The warp factor equation

$f(r)=0=(r-r_{1})(r-r_{2}),$ (74)

has two solutions, an acoustic horizon $r_{1}=c_{0}$, which is the outermost
marginally trapped surface for outgoing phonons, and an unphysical negative
solution $r_{2}=-c_{0}$. The exterior acoustic event horizon is where the
velocity of the fluid reaches the sound velocity. The central point of the
acoustic BH can be regarded as a sink that leads to the higher-dimensional
space from which the fluid flows to the three-dimensional space.

Figure 3: _Top panel:_ Real (left) and imaginary (right) part of the
fundamental QNMs $\tilde{\omega}_{0m}$ of a three-dimensional acoustic
Schwarzschild black hole with respect to $c_{0}$ and varying $m$. _Bottom
panel:_ Real and imaginary part of the first overtone of QNMs
$\tilde{\omega}_{1m}$ of the same system with respect to $c_{0}$ and varying
$m$.

It is worth noting that one can also obtain this effective metric, given by
Eq. (72), by using the approach developed in [74], which is based on the
Gross-Pitaevskii theory [75, 76] that describes a nonlinear complex scalar
field propagating in a curved spacetime background. In this approach, we
simply require a fluid flow with velocity given by Eq. (69) in a flat
Minkowski spacetime.

In what follows, we will calculate, for the first time, the QNMs and QBSs of
the analog BH proposed in Ref. [58], by using the same methods used for the
five-dimensional Schwarzschild BH analysis in Sec. II.

###  III.1 Scalar wave equation

In order to solve the equation of motion given by Eq. (67), for the acoustic
metric Eq. (72), we will use a similar separation ansatz for the phase
fluctuation wave function

$\Psi_{1}(t,r,\phi)=\mbox{e}^{-i\tilde{\omega}t}U(r)\mbox{e}^{im\phi},$ (75)

where $\tilde{\omega}$ is the frequency of phase fluctuations,
$U(r)=R(r)/r^{1/2}$ is the radial function, and $m$ is the magnetic quantum
number. Thus, by substituting Eqs. (72) and (75) into Eq. (67), we obtain

$\displaystyle R^{\prime\prime}(r)+\frac{f^{\prime}(r)}{f(r)}R^{\prime}(r)$
$\displaystyle+\biggl{[}\frac{\tilde{\omega}^{2}}{f^{2}(r)}-\frac{4m^{2}-f(r)+2rf^{\prime}(r)}{4r^{2}f(r)}\biggr{]}R(r)=0.$
(76)

###  III.2 Quasinormal modes

Equation (76) can be recast into a Schrödinger-like equation by multiplying
with $f^{2}(r)$ and utilizing the tortoise coordinate $r_{*}$, with
$dr/dr_{*}=f(r)$, to obtain

$\frac{d^{2}R(r)}{dr_{*}^{2}}+\left(\tilde{\omega}^{2}-V_{\text{BEC}}\right)R(r)=0,$
(77)

where the effective potential $V_{\text{BEC}}$ is

$V_{\text{BEC}}\equiv
f(r)\left(\frac{m^{2}}{r^{2}}-\frac{f(r)}{4r^{2}}+\frac{f^{\prime}(r)}{2r}\right).$
(78)

QNMs $\tilde{\omega}_{nm}$, similarly, form a discrete set of mode solutions
to the radial equation (77) with purely ingoing (outgoing) boundary conditions
at the acoustic horizon (infinity), such that

$R(r)\sim e^{-i\tilde{\omega}r_{*}},\,\,r\rightarrow r_{1},\,\,\,\,\,R(r)\sim
e^{i\tilde{\omega}r_{*}},\,\,r\rightarrow\infty.$ (79)

In what follows, we present the QNMs of the three-dimensional BH analog of
BECs in a cavity, where we have employed the same methods to numerically
obtain and benchmark our results.

3D BEC analog
---
$m$ | $n=0$ | $n=1$
1 | 0.4068 - 0.3412 i | 0.1968 - 1.2341 i
2 | 0.9527 - 0.3507 i | 0.7856 - 0.2234 i
10 | 4.9906 - 0.3535 i | 4.9532 - 1.1248 i

Table 2: QNMs of the three-dimensional (3D) BEC acoustic BH with $c_{0}=1$.

Figure 3 demonstrates the behavior of the BEC BH analog’s QNMs. The increment
of the vortex radius $c_{0}$ decreases the oscillation frequency and increases
the perturbation’s lifetime. The increment of $m$ increases the real part of
QNMs while the imaginary part is also increased (in absolute value) as
expected from the eikonal limit which probes, as we will show in the following
section, the instability timescale of phase fluctuations at a critical radius
where phonons are trapped in unstable curcular orbits around the acoustic BH.
For some typical QNMs we refer the reader to Table 2.

####  III.2.1 Eikonal quasinormal mode interpretation

We are already aware that the real and imaginary part of eikonal QNMs are
intrinsically connected to the angular frequency and instability timescale of
null geodesics, respectively, at the photon sphere of a vast variety of BH
geometries [69, 77, 78, 79, 80, 81, 82]. Our calculations demonstrate that the
same behavior occurs for the acoustic BEC BH, which means that the system,
though hydrodynamic in nature, has an analog photon sphere, i.e. a phonon
sphere. By calculating the QNMs for large $m$ and comparing with those of the
five-dimensional BH for large $\nu$ we find that both asymptote to the same
value.

Specifically, the QNMs of the acoustic BH in the eikonal limit are related to
the Lyapunov exponent $\lambda_{0}$ of phase fluctuations at a critical radius
where phonons are trapped in unstable circular orbits. The Lyapunov exponent
is inversely-proportional to the instability timescale of phonon orbits there,
i.e. [69]

$\displaystyle\tilde{\omega}_{\text{WKB}}=m\Omega_{\text{ph}}-i\left(n+\frac{1}{2}\right)|\lambda_{0}|,$
(80)

where

$\Omega_{\text{ph}}=\sqrt{\frac{f(r_{\text{ph}})}{r^{2}_{\text{ph}}}},$ (81)

is the angular frequency of phonons at the phonon sphere $r=r_{\text{ph}}$ and

$|\lambda_{0}|=\sqrt{-\frac{1}{2}\frac{r^{2}_{\text{ph}}}{f(r_{\text{ph}})}\left(\frac{d^{2}}{dr^{2}_{*}}\frac{f(r)}{r^{2}}\right)}\Bigg{|}_{r=r_{\text{ph}}}$
(82)

is the Lyapunov exponent or instability timescale of phonons.

Figure 4 shows that both the BH and the acoustic BEC analog reach the same
decay timescale of QNMs rapidly as $\nu$ and $m$ become arbitrarily large. The
asymptotic values of eikonal QNMs extracted with the numerical methods
utilized above agree perfectly with those obtained by Eq. (80) for large $m$.
The fact that both the gravitational and acoustic BH asymptote to the same
eikonal QNMs is routed to the form of their effective potentials. Equations
(15) and (78) are both dominated by $\nu$ and $m$ when those constants acquire
large values, hence the effective potentials practically coincide and lead to
the same mode solutions.

A peculiar, though not necessarily important, aspect of the convergence of the
imaginary part of QNMs is that when $m$ increases the trend of convergence to
the WKB prediction (shown with a dashed horizontal black line in Fig. 4) is
opposite to that of the BH when $\nu$ asymptotes to infinity.

Figure 4: Imaginary parts of fundamental QNMs of a five-dimensional
Schwarzschild black hole, $\text{Im}(\omega_{0\nu})$ (purple dots), with
$\mathcal{M}=1$ and increasing $\nu$, and of the acoustic analog Schwarzschild
black hole in a cavity, $\text{Im}(\tilde{\omega}_{0m})$ (orange dots), with
$c_{0}=1$ and increasing $m$. The dashed horizontal line designates the WKB
prediction at the eikonal limit.

###  III.3 Quasibound states

Figure 5: _Top panel:_ Real (left) and imaginary (right) part of the
fundamental QBSs $\tilde{\omega}_{0m}$ of a three-dimensional acoustic
Schwarzschild black hole with respect to the vortex radius $c_{0}$ and varying
$m$. _Bottom panel:_ Real and imaginary part of the first overtone of QBSs
$\tilde{\omega}_{1m}$ of the same system with respect to $c_{0}$ and varying
$m$.

The exact solution for the radial part of the massless Klein-Gordon equation,
in the acoustic BH system, can be written through Eqs. (26)-(38), with the new
radial coordinate, $x$, the new parameter, $x_{1}$, and the F-homotopic
transformation, $R(x)\mapsto y(x)$, defined by Eqs. (17), (18), and (19),
respectively. However, in the present case, the coefficients $A_{0}$, $A_{1}$,
$A_{x_{1}}$, $A_{3}$, and $A_{4}$ are given by

$\displaystyle A_{0}$ $\displaystyle=$
$\displaystyle-\frac{i\tilde{\omega}}{2c_{0}},$ (83) $\displaystyle A_{1}$
$\displaystyle=$ $\displaystyle\frac{1}{2}-\frac{im}{c_{0}},$ (84)
$\displaystyle A_{x_{1}}$ $\displaystyle=$
$\displaystyle-\frac{i\tilde{\omega}}{2c_{0}},$ (85) $\displaystyle A_{3}$
$\displaystyle=$
$\displaystyle-\frac{(m+\tilde{\omega})(2ic_{0}+m+\tilde{\omega})}{c_{0}^{2}},$
(86) $\displaystyle A_{4}$ $\displaystyle=$
$\displaystyle\frac{(m+\tilde{\omega})(2ic_{0}+m+\tilde{\omega})}{c_{0}^{2}},$
(87)

such that the parameters $\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$,
and $q$ are given by

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle
2-\frac{i(m+\tilde{\omega})}{c_{0}},$ (88) $\displaystyle\beta$
$\displaystyle=$ $\displaystyle-\frac{i(m+\tilde{\omega})}{c_{0}},$ (89)
$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle
1-\frac{i\tilde{\omega}}{c_{0}},$ (90) $\displaystyle\delta$ $\displaystyle=$
$\displaystyle 1-\frac{2im}{c_{0}},$ (91) $\displaystyle\epsilon$
$\displaystyle=$ $\displaystyle 1-\frac{i\tilde{\omega}}{c_{0}},$ (92)
$\displaystyle q$ $\displaystyle=$
$\displaystyle-\frac{(m+\tilde{\omega})(2ic_{0}+m+\tilde{\omega})}{c_{0}^{2}}.$
(93)

Similarly to Sec. II.3, by imposing ingoing boundary conditions at the
acoustic horizon, decay at infinity, and the resulting $\alpha$-condition
given by Eq. (53), we obtain the exact spectrum of QBSs $\tilde{\omega}_{nm}$
in the BEC Schwarzschild analog

$\tilde{\omega}_{nm}=-m-ic_{0}(2+n).$ (94)

Furthermore, from Eqs. (83)-(94), we get

$\sigma=\frac{i(m+ic_{0}n+\tilde{\omega}_{nm})}{c_{0}}=2,$ (95)

which means that the parameter $\sigma$ is a real, positive (constant) number
for any value of the overtone number $n$, as well as for any value of the
magnetic quantum number $m$, and hence the frequency eigenvalues given by Eq.
(94) represent QBSs for phase fluctuations in the acoustic BEC BH studied
here.

By focusing on the fundamental mode ($n=0$) we obtain

$\tilde{\omega}_{0m}=-m-2ic_{0},$ (96)

where the second polynomial condition, given by Eq. (54), is automatically
satisfied, since it leads to

$c_{n+1}(q)\biggl{|}_{n=0}=q=n(2+n)\biggl{|}_{n=0}=0.$ (97)

This means that there is no restriction on the value of the magnetic quantum
number $m$; we can consider it as $m=-\infty,\dots,0,\dots,+\infty$. It is
noteworthy to point that as in the BH case, the QBSs of the acoustic BH do not
exist when $m=0$ since the real part is zero and the modes only decay in time,
i.e. $\tilde{\omega}_{00}=-2ic_{0}$.

Fig. 5 demonstrates the behavior of QBSs in the analog BEC BH. Surprisingly,
they have almost identical tendencies as those found for five-dimensional
Schwarzschild BHs which further supports the analogy between them. Most
surprisingly, the imaginary parts depend linearly on the vortex radius
$c_{0}$, which is a much more simplified version of what occurs in the BH
case.

##  IV Conclusions

In this study, we have calculated the QNMs and QBSs of five-dimensional
Schwarzschild BHs which are analog models of BECs of light in a mirror cavity;
an experiment recently proposed in [58]. We found that the spectra for both
systems share striking qualitative similarities with respect to their tuning
parameters, i.e. the BH mass and the cavity’s radius. Most importantly, our
investigation provides proof of a phonon-sphere interpretation of the acoustic
BEC BH since the imaginary part of the eikonal QNMs asymptotes to the
instability timescale of phonons at the phonon sphere in accord to the WKB
prediction. Our results are timely, and together with the recent investigation
of the graybody factors and Hawking radiation of the analog proposed in [58],
can prescribe numerical data for actual experimental devices that may be
constructed based on such proposal.

In fact, although a multitude of BH analog experiments have taken place and
made significant breakthroughs so far, the proposal in [58] is quite simple
and elegant, includes a velocity singularity at the center of the vortex
(aspect that other BEC-based experiments lack), has tabletop dimensions and
can provide intuition regarding classical and quantum aspects of the
singularity, as well as a better understanding of higher-dimensional BH
geometries and their properties.

An interesting extension of the analog [58] can be the manipulation of the
nonlinear coupling of the BECs of light in order to obtain a massive Klein-
Gordon-like equation for phase fluctuations through the underlying
hydrodynamic equations of motion [57] and study classical and quantum
phenomena. Most importantly, a generalization of the model in [58] onto a
rotating BEC acoustic BH analog should be of utmost importance in order to
simulate phenomena that astrophysical BHs experience. We leave these
directions of research for the future.

###### Acknowledgements.

H.S.V. is partially supported by the Alexander von Humboldt-
Stiftung/Foundation (Grant No. 1209836). This study was financed in part by
the Conselho Nacional de Desenvolvimento Científico e Tecnológico – Brasil
(CNPq) – Research Project No. 150410/2022-0. It is a great pleasure to thank
the Theoretical Astrophysics at Tübingen (TAT Group) for its hospitality and
technical support. K.D. acknowledges financial support provided under the
European Union’s H2020 ERC, Starting Grant agreement no. DarkGRA–757480 and
the MIUR PRIN and FARE programmes (GW-NEXT, CUP: B84I20000100001).

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