# Analogue gravity and its scientific confirmatory role

Mojtaba Shahbazi<EMAIL_ADDRESS>Department of Physics, Faculty
of Science, Tarbiat Modares University, Tehran, Iran

###### Abstract

Empirical confirmation in some areas of physics is obscure; for example in
Hawking radiation. However, the analogue gravity can simulate these phenomena
in condensed matter systems. That is an important question whether the
observation of these phenomena in the condensed matter systems can be
confirmatory of the original phenomenon or not. In this work we answer
affirmatively to this question via structuralism.

##  1 Introduction

Analogue gravity (AG) teaches us how a field theory in a curved background
could be simulated by a sound wave in a fluid, generally a perturbation in a
condensed matter system. Especially, this method provides a practical way to
simulate a physical phenomenon which is not accessible, say, a black hole. One
of the great achievements of the analogue gravity is detection of analogue of
the Hawking radiation in a water tank [Weinfurtner et.al, 2011] and Bose-
Einstein condensation [Steinhauer, 2016]. The detection of the original
Hawking radiation requires two things: one, an evaporating black hole which
the nearest one is very far away from Earth and the second, the technology of
detectors because the Hawking radiation is too weak particularly when it is
compared with the cosmic microwave background, then the required detectors are
beyond our cutting-edge technology. In this vein, the Hawking radiation is
already a far reached physical phenomenon. The other inaccessible example is
string theory. The length of fundamental string is of order $10^{-33}cm$, then
to detect a fundamental string requires an accelerator of energy order
$10^{16}TeV$ , if compared to LHC which is of order $7TeV$, it seems that the
detection of a fundamental string is due to too-distant future. Dealing with
these theories and phenomena prompts speculation that if these theories are
experimentally confirmed or not? How does one experimentally confirm them?
What are the confirmatory criteria in these cases?

As a first instinct, if one can simulate physical phenomena, they can take the
advantage of them, it means that if a black hole acts as a natural quantum
computer, then an analogue system which simulates a black hole can act as an
analogue of a quantum computer, it means the dynamics of the two are the same.
Nonetheless, it is believed that the analogue gravity takes the role of
confirmation in science. In other words, an observation in the analogue
gravity can confirm the observation in the original gravitational phenomenon.
However, this particular stance receives critiques and supports. Supporters
believe that if the physics of the analogue and the original gravitational
phenomenon are the same, then, observation in one inevitably leads to the
other because of the fact that the same mathematical structure governs the two
phenomena. Nevertheless, the critics believe that there are some phenomena,
say the Hawking radiation and the string theory, are not confirmed
empirically, then on what basis we consider their mathematical structure so
perfectly valid, put it differently, although the equations of motion of the
analogue and the original phenomenon are the same, how do we know that the
original phenomenon is described by that mathematical structure? how do we
know that the real string is described by string theory? They have not been
confirmed empirically yet.

In this paper we provide an argument based on the structuralism which states
that the analogue gravity can play the role of empirical confirmation if it
can simulate the original phenomenon. The rest of the paper is organized as
follows: a brief introduction of physical background of analogue gravity is
provided, then arguments in favor of and against the confirmatory role of the
analogue gravity are considered and finally an argument based on the
structuralism is pushed forward.

##  2 Analogue Gravity

There are some well-known correspondences in physics that relate two different
theories, say an AdS gravity to a conformal field theory known as AdS/CFT
[Aharony et.al, 2000], or physical systems, a gravitational phenomenon to a
condensed matter system. These correspondences let us develop our
understanding of the core nature of the physics, let alone they suitably equip
us with some mathematical techniques to solve problems.

Analogue gravity shows that if there is a perturbation in a condensed matter
system, then the equations of motion are exactly the same as the equation of
motion for a field theory in a curved background [Barcelo et.al, 2011].
Suppose a condensed matter system, say a fluid111Electrons in many condensed
matter systems behave as a fluid., Euler and continuity equations for a vortex
free fluid are as follows [Barcelo et.al, 2011]:

$\displaystyle\nabla\times\vec{v}=0$ (1)
$\displaystyle\rho\big{(}\frac{\partial\vec{v}}{\partial
t}+(\vec{v}.\nabla)\vec{v}\big{)}=-\nabla p-\rho\nabla V$ (2)
$\displaystyle\frac{\partial\rho}{\partial t}+\nabla.(\rho\vec{v})=0$ (3)

where $\vec{v}$ is the velocity of the fluid, $\rho$, the density, $V$,
potential and $p$ the pressure. Changing variables:

$\displaystyle\xi:=ln\rho~{}~{}~{}\vec{v}:=\nabla\psi$ (5) $\displaystyle
g(\xi):=\int^{e^{\xi}}\frac{1}{\rho^{\prime}}\frac{dp(\rho^{\prime})}{d\rho^{\prime}}d\rho^{\prime}$
(6)

Finding linearized equations:

$\displaystyle\xi=\xi_{0}+\bar{\xi}~{}~{}~{}\psi=\psi_{0}+\bar{\psi}$ (7)
$\displaystyle\frac{1}{\rho_{0}}\Big{(}\frac{\partial}{\partial
t}\frac{\rho_{0}}{g^{\prime}(\xi_{0})}\frac{\partial\bar{\psi}}{\partial
t}+\frac{\partial}{\partial
t}\frac{\rho_{0}\vec{v_{0}}}{g^{\prime}(\xi_{0})}.\nabla\bar{\psi}+\nabla.\big{(}\frac{\rho_{0}\vec{v}}{g^{\prime}(\xi_{0})}\frac{\partial\bar{\psi}}{\partial
t}\big{)}-\nabla.\rho_{0}\nabla\bar{\psi}+\nabla.\big{(}\frac{\vec{v}\rho_{0}\vec{v}.\nabla\bar{\psi}}{g^{\prime}(\xi_{0})}\big{)}\Big{)}=0$
(8)

then (8) could be rewritten as follows:

$\displaystyle\partial_{\mu}\big{(}\sqrt{-g}g^{\mu\nu}\partial_{\nu}\bar{\psi}\big{)}=0$
(10)

where the metric is given by:

$\displaystyle
ds^{2}=\frac{\rho_{0}}{c(\rho_{0})}\Big{(}\big{(}c^{2}(\rho_{0})-v^{2}\big{)}dt^{2}+2dt\vec{v_{0}}.d\vec{v}-dx^{2}\Big{)}$
(11) $\displaystyle c^{2}(\rho_{0})=g^{\prime}(ln\rho_{0})$ (12)

where $c$ is the speed of sound. In this manner, a fluid in a flat
background222There is a generalization of the flat to curved background [Hu,
2019]. (2) can mimic a field theory333For simplicity, we considered a scalar
field; however, it could be generalized to an arbitrary field theory. in a
curved background (10) but the speed of light is replaced with the speed of
sound and black hole solutions turn into dumb hole solutions where a sound
wave cannot escape. The classical field theory $\psi$ or $v$ is a classical
field on a curved background constructed by $\rho,p$ and $v$ where the role of
the metric $g^{analogue}_{\mu\nu}$ (the line element (11)) is played by
$(\rho,p,v)$. The model of the classical field theory on the curved background
(CFCB) could be demonstrated as:

$\displaystyle M_{S}^{CFCB}=<(\rho,p,v),v>=<g^{analogue}_{\mu\nu},v>$ (13)

Quantization of the sound waves is as the same as the quantization of the
field theory in a curved background or a black hole and as a consequence,
there appears the Hawking radiation-like phenomenon in the analogue gravity
[Unruh, 1981]. In a fluid, this Hawking radiation is the excitations of the
phonon in the fluid that has been detected in laboratories [Weinfurtner, 2011]
[Steinhauer, 2016]. The model of the quantum aspect of the field theory on the
curved background, in short quantum field theory on the curved background
(QFCB) could be demonstrated as:

$\displaystyle
M_{S}^{QFCB}=<(\rho,p,v)_{classical},v_{quantized}>=<g^{analogue}_{\mu\nu},v_{quantized}>$
(14)

where $v_{quantized}$ is the quantization of the classical field theory $v$ or
$\psi$. The main question is that if the detection of this Hawking radiation-
like phenomenon in a fluid can be accounted as an empirical confirmation of
the original Hawking radiation or not.

##  3 Arguments in Favor/Against of Confirmatory Role of AG

In science, many scientists invoke analogies to confirm their claims or at
least guide them to discoveries. In this vein, the analogue gravity by trading
on syntactic isomorphism between the analogue system and the original which
(is inaccessible) is going to be empirically confirmatory [Dardashti, et al
2017]. A rudimentary example is the correspondence between the oscillation of
a pendulum and the oscillation of electric charges in a system of RLC circuit.
In both cases, the equations of motion are the same, like what happens with
(2), (10) and as a consequence, a syntactic isomorphism; nonetheless, the both
systems have quite different ontology, on the one hand, there is a pendulum,
gravity and the resistance of air, on the other, resistor, inductor and
capacitor. This isomorphism lets us to measure the location of the pendulum
and based on that find the the amount of electric charges passing through the
cross section of the wire. The very similar case is brought when one considers
a negative test charge near a uniform spherical configuration of positive
electric charges and a point mass particle near a uniform spherical
configuration of mass. The equations of motion for the test charge and the
point mass are the same.

In [Dardashti ,et al, 2017] there is a line of reasoning that how this
inference is justified in the analogue gravity. Suppose there is a target
system $T$ which is modeled by $M_{T}$ under conditions $D_{T}$ 444As van
Frasen notes $T$ is a data model and $M_{T}$ is the theory model, that means
$M_{T}$ represents $T$ the data model which is the result of experimental
observations, and $T$ is the representation of the natural phenomenon, roughly
speaking when talking of $T$ it means the natural phenomenon [van Frasen,
2006]. and system $S$ simulates the target which is modeled by $M_{S}$ under
the conditions $D_{S}$:

  1. i.

There is a mathematical similarities between $M_{S}$ and $M_{T}$ sufficient
for a syntactic isomorphism under the conditions $D_{S}$ and $D_{T}$.

  2. ii.

System $T$ is inaccessible within the conditions $D_{T}$.

  3. iii.

System $S$ within the conditions $D_{S}$ is accessible and we can form a claim
such as: a phenomenon $P_{S}$ is exhibited under the conditions $D_{S}$ in the
system $S$.

the isomorphism lets us to infer the following statement from the above
assumptions:

  * •

Claim: under conditions $D_{T}$ the system $T$ exhibits a phenomenon $P_{T}$.

To be assured that the model $M_{T}$ describes $T$ accurately enough, it
presupposes that there are some shared implicit assumptions between $M_{S}$
and $M_{T}$ where the phenomena $P_{S}$ and $P_{T}$ are based on these
implicit assumptions. In addition, these implicit assumptions relate the two
systems. What makes these implicit assumptions different are model-external
basis and the empirically grounded. In other words, if a universality of a
phenomenon is established there would be a model-external basis for the
phenomenon and if it is empirically tested then it is not merely a theoretical
curiosity, it is empirically grounded, what Dardashti et.al called MEEGA
(model-external and empirically grounded argument). [Dardashti et.al, 2017]
claims that Hawking radiation is appeared in different analogue systems and
they have been empirically confirmed then they have concluded:

  * •

There are shared implicit assumptions between the analogue gravity and the
original gravitational phenomenon.

[Dardashti et.al ,2017] emphasizes that this argument is different from the
analogical reasoning. In analogical reasoning by finding a correspondence (
similarity) between two objects one develops the correspondence and
generalizes it to the other properties not included in the primer
correspondence[Bartha,2019]. For example, a flower and a telecommunication
tower are similar in their configuration, a tall construction and a big head.
However, in analogical reasoning one develops this correspondence and
concludes that if the flower has a property, say have a leaf then there would
be the analogous property, say an analogous leaf in the tower. Sometimes it
works and the tower has some wave transmitter as analogous leaf and sometimes
it does not work. The point is that the counter examples of the analogical
reasoning arise from the development of the correspondence and it is worth
mentioning that the analogue gravity works within the domain of the
correspondence and does not develop it. Then as long as one works in the
domain of the correspondence and does not develop it, there is no problem with
the analogical reasoning. Put it differently, as long as one takes the
equations of motion of the analogue gravity and the original Hawking radiation
and their quantizations, the correspondence is concrete and legitimate.
However, if one broadens the correspondence, say to the singularity of
solutions, the analogy does not work because of the fact that the analogue
solutions are not singular but the real black hole solutions.

There remains a big question: whether the model $M_{T}$ and $M_{S}$ are
empirically adequate enough or not? Obviously, the model $M_{S}$ which is
accessible is easy to check in experiments; however, the knotty part of the
problem lays in the accuracy and the adequacy of the model $M_{T}$ which is
inaccessible. It seems that there is not a clue for the accuracy of the
original Hawking radiation and actually it is a vicious circle. Put it another
way [Crowther, 2021]:

  * •

The model describing the original Hawking radiation is not confirmed
experimentally and consequently not empirically adequate.

  * •

The analogue gravity is going to provide the experimental confirmation.

  * •

To do that one should make sure that the model of the original Hawking
radiation is adequate enough.

[Dardashti et.al, 2019] tries to invoke a probabilistic analysis to ascertain
that the more phenomena following the Hawking radiation, the more plausibility
of the original Hawking radiation. Having said that, this analysis makes sure
that if the model $M_{T}$ is partially true then more experimental evidences
in the analogue gravity increases the accuracy of $M_{T}$; nevertheless, this
conclusion still tapped in the previous vicious circle. In other words, the
adequacy of the Hawking radiation model is still under question [Crowther
et.al, 2021].

Some philosophers of physics call for the inductive inference [Evans, 2020].
Evans et.al claim that most of modern physics is due to unobservable
phenomena; however, some of them are manipulable such as spin of the electrons
and some of them are unmanipulable such as cosmological phenomena.
Unmanipulable ones can be divided into accessible (indirectly observed) such
as astrophysical phenomena and inaccessible. They claim that in these areas of
research physicists carry out their research via inductive reasoning and if
one is not that skeptic about inductive reasoning. [Evans, 2020] provides an
interesting example of interstellar nucleosynthesis, where scientists talk
about the nuclear reactions in the core of the stars in unmanipulable and
inaccessible position, so scientists find that if they make some assumptions (
the assumptions are: the atomic theory (manipulable and accessible), photonic
spectra of the surface of the star (unmanipulable and accessible) ), they can
successfully describe the energy emitted from stars555Although Evans et.al
using the inductive inference in the nucleosynthesis argument, it seems that
the proper reasoning is IBE (inference to the best explanation). We talk on
this later.. [Evans et.al, 2020] calls this inductive reasoning as inductive
triangulation, combination of two inductive inferences comes to a conclusion.
Then this kind of inference can be applied to the analogue gravity and
conclude that the observation of the Hawking radiation in the analogue gravity
leads to the empirical confirmation of the original Hawking radiation which
are unmanipulable and inaccessible.

It seems that [Evans et.al, 2020] calls for the inductive inference to show
that it is reasonable to suppose a particular nuclear reaction for the deep
interior of stars; however, the main question to answer is that if the two
assumptions, the atomic theory plus the surface photons of the star, imply the
empirical confirmation of the interior nuclear reaction or not. We think that
they do answer the different question. The question they are answering is if
it is reasonable to describe interior of the stars based on other theories?
They do not tell anything about the fact that whether this description is
empirically adequate. In addition, [Evans et.al, 2020] truly states that in
the inductive inference, systems $S$ and $T$ pertain to the same system;
however, in the analogue gravity $S$ and $T$ are in the same universality
class 666Evans et.al emphasize that if we suppose that $T$ is accurately
described by $M_{T}$, $M_{T}$ and $M_{S}$ are isomorphic..

To demonstrate that the analogy in Hawking radiation is not legitimate Field
draws the relation between the universality and empirical constraints in
theories based on distinguishing between two factors: strength and relevance
[Field, 2021]. Strength shows the possibility of the conclusion from the
premises and the relevance concerns how the universality is positively
relevant to the system of interest. Field addresses the relevance to the
microscopic shared structure of some superficially different macroscopic
theories such as renormalization group theory in condensed matter systems with
the shared microscopic mathematical structure but different macroscopic
theories of condensed matter systems. For inaccessible systems we can make a
guess and see how well our guess works in observation if the guess is relevant
or not; however, in Hawking radiation we cannot follow this procedure due to
the inaccessibility. In addition, Field states that Hawking radiation and the
analogue gravity due to the difference of microscopic structure are not
relevant, then we are not allowed to use of Hawking radiation in analogue
models.

We raise an objection to the Field’s exposition. In the correspondence between
RLC circuit and the pendulum, they share no microscopic mathematical structure
and only share the same macroscopy. However, one is hundred percent sure that
if measuring a quantity in the circuit there would be absolutely an analogous
quantity in the pendulum system if measuring it one gets the same magnitude of
its analogous circuit. In this way, it seems that the correspondence between
the original Hawking radiation and its analogue is reasonable due to their
shared macroscopic structure, with different microscopic structures though. In
other words, if there is Hawking radiation in the analogue gravity then the
shared macroscopic mathematical structure makes sure that the original Hawking
radiation occurs unless there is a doubt that if this shared macroscopic
mathematical structure does describe the original black hole correctly.

###  3.1 Structuralism

It appears that if we make sure that $T$ is adequately described by $M_{T}$
and the models $M_{T}$ and $M_{S}$ are isomorphic then the observation $P_{S}$
can provide an empirical confirmation for $P_{T}$; however, the main problem
in here is around the question: on what basis one can make sure that an
unmanipulable and inaccessible system such as the original Hawking radiation,
$T$, is described by the Hawking’s calculations, $M_{T}$ without empirical
confirmation.

Inference to the best explanation (IBE) is spelled out where there are
competing assumptions and the all are empirically adequate, the assumption
that gives the best explanation is true 777There are some critiques to this
kind of inference that we do not consider them here and take IBE for granted.
For a latest review see [Douven, 2021]. [Ladyman, 2001]. Hawking’s
calculations are based on two assumptions: quantum field theory and general
theory of relativity. Both theories are empirically adequate; however, in
Hawking’s calculations a field theory in a curved background is considered, it
means a combination of the both theories. These calculations lead to a
temperature for black holes. Based on IBE, it is reasonable to generalize the
quantum field theory to the curved background as it happened in generalization
of special theory of relativity to general theory of relativity. In this
manner, IBE provides a justification of the Hawking’s calculations due to the
current evidences we have got in hand from general theory of relativity and
quantum field theory similar to argument in [Evans et.al, 2020] about
nucleosynthesis. However, it seems that if the next radical scientific change
predicts the violation of the Hawking’s calculation, then all of the analogue
gravity program of empirical confirmation is jeopardized.

Nevertheless, structuralism seems to be promising particularly when
considering the explicit reconstruction of theories from empirical science,
structuralism is distinguished between the rivals [Balzer et.al, 2000]
[Niebergall, 2002]. This school of thought committees us to the mathematical
structure of the scientific theories and avoids the theory description of the
furniture of the world [Ladyman, 2014]. Particularly the structuralism evades
the known pessimistic meta-induction888Pessimistic meta-induction asserts that
if there is a radical change in scientific theories then the best current
established scientific theories are abandoned [Ladyman, 2001].,
uderdetermination of scientific theories 999It is not guaranteed that the
evidences determine a unique theory which means that there is an
underdetermination problem in scientific theories. [Lee, 2022] and it supports
the Putnam’s no-miracle argument 101010No-miracle argument is mostly one of
the compelling argument in favor of the scientific realism which states that
there would be a miracle if the success of the scientific theories were not at
least approximately true description of the world [Ladyman, 2001]. 111111It is
worth mentioning that there are some criticism on the structural realism such
as the problem of higher structures [Roberts, 2010]; however, these are due to
the metaphysical status of the structures in the structural realism which are
evaded in the structuralism without metaphysical status of the structures. By
structuralism in this paper we mean the latter one..

The structural continuity of the scientific theories in structuralism gives
rise to the fact that the structure of the new theory reduces to the structure
of the old theory under the old theory condition. For example, it has been
shown that the structure of quantum mechanics reduces to the classical
mechanics [Saunders, 1993], special relativity to classical mechanics [Brown,
1993], general theory of relativity to the Newtonian gravitation [Fletcher,
2019] and general theory of relativity to geometrized Newtonian gravitation
[Masoumi, 2021].

What we mean by structure is the meaning of the structure in the semantic
approach. It is believed that the semantic approach to scientific theories is
more appropriate when philosophers of science considering the practice of
scientists. In this sense, a theory is a family of models rather than
axiomatic systems. In addition, the structure is models of the theory in form
of [French et.al, 2006]:

$\displaystyle M=<S,R_{i},\phi_{j},s_{k}>$ (15)

where $S$ is a non-empty set, $R_{i}$ family of relations, $\phi_{j}$ family
of functions and $s_{k}$ family of distinguished individuals of the set $S$.
One can demonstrate an evaporating black hole by the following model:

$\displaystyle M_{T=Hawking~{}radiation}=<\mathcal{M},g_{\mu\nu},\phi>$ (16)

where $\mathcal{M}$ is a manifold, $g_{\mu\nu}$ a metric (the agent of general
theory of relativity) and $\phi$ a field theory (the agent of field theory).
If the field $\phi$ is a classical field theory, such as Maxwell field, there
are a lot of experimental evidences on the classical field theory on the
curved background; for example, Shapiro time delay effect, blue shift and red
shift in astronomical observations, then the structure of the whatever the
next scientific theory should reduce to the structure of CFCB:

$\displaystyle M_{T}^{new}\rightarrow
M_{T}^{CFCB}=<\mathcal{M},g_{\mu\nu},\phi_{classical}>$ (17)

Then it seems that the simulation and detection of the effects caused by CFCB
in the analogue gravity is reasonable. Such experiments for the Hawking
radiation have been conducted and the ratio of the modes of the classical
field has been measured in a water tank experiment [Weinfurtner et.al ,2011].
But here the empirical confirmation of CFCB is independent of the analogue
gravity and it is based on the direct observation. Because of the fact that
CFCB is empirically adequate ( as said before, under the CFCB condition), then
by structuralism these structures will appear in the whatever the next
scientific theories.

We can summarize the points as follows:

  * •

There is an independent empirical confirmation of classical fields on the
curved background (CFCB)

  * •

There exists an isomorphism between the structure of the analogue gravity,
$M_{S}^{CFCB}$ and the structure of the classical Hawking radiation,
$M_{T}^{CFCB}$

  * •

The whatever the next scientific theory will reduce to the structure of the
classical Hawking radiation:

$\displaystyle M_{T}^{new}\rightarrow M_{T}^{CFCB}\cong
M_{S}^{CFCB}=<g_{\mu\nu}^{analogue},v>$ (18)

So far we have emphasized that the structure of the classical aspect of the
original Hawking radiation is empirically adequate and isomorphic to that of
the analogue gravity but there is no experimental evidence on the presence of
a quantum field theory on a curved background (QFCB).

To show that it is reasonable to expect that the quantum aspect of the Hawking
radiation should be in the whatever the next scientific theory, we restrict
our attention to the early times of the black hole evaporation or the times
before the so-called Page time121212After the Page time there would be a
paradox in the Hawking’s calculations known as the information paradox. This
paradox in a nutshell is that a pure state for a closed system evolves to a
mixed state which is in contradiction to the unitarity of the quantum theory.
Although the Hawking’s calculation leads to an inconsistency after the Page
time, AdS/CFT correspondence provides a resolution to the paradox which is
known as the island prescription [Almhri et al., 2020]. The information
paradox is conspicuous when one is going to find the entanglement entropy of
the Hawking radiation. However, AdS/CFT translates the entanglement entropy of
the Hawking radiation ( a calculation in a quantum field theory or the
boundary theory) into the area of a surface in AdS ( a calculation in a
classical theory or the bulk theory). The island prescription in which the
entanglement entropy is computed as a surface in the classical theory is in
agreement with the Hawking’s calculation before the Page time and different
after the Page time. Owing to the fact that this prescription is a computation
in the gravitational theory (the bulk theory), it does not tell us how the
Hawking radiation is described in the field theory. However, one can simulate
the bulk theory of the Hawking radiation by the analogue gravity and
correspond the information paradox and the island role to the momentum loss
over the dumb hole horizon and pump maintenance of the momentum in the dumb
hole, respectively [Parvizi et al., 2023]. Nonetheless, it is controversial
that if the two dual theories are exactly equivalent or not ( see, [Rickles,
2017] as a proponent and [Dardashti et.al., 2018] as an opponent), let alone
the confirmatory role of the two dual theories. In short, the confirmatory
role of the analogue gravity after the Page time is out of the scope of this
work.. The point is that the Hawking’s calculation after the Page time leads
to the information paradox ( for a review see [Manchak et al,2018][Raju,2022])
which means that as long as we are ahead of the Page time this calculation is
an appropriate description based on IBE. Now we argue as follows:

  1. I)

It is believed that the quantum theory is one of the best current fundamental
physical theories.

  2. II)

The quantum description of a phenomenon where its classical description is
empirically confirmed, is a more complete description of the phenomenon.

  3. III)

Then: it implies that the quantum description of the classical Hawking
radiation is a more complete description and structurally developed one. As a
consequence, the structure of the whatever the next scientific theory should
reduce to the structure of the quantum aspect of the Hawking radiation.

which means that:

$M_{T}^{new}$ | $\rightarrow$ | $M_{T}^{QFCB}$ | $\rightarrow$ | $M_{T}^{CFCB}$
---|---|---|---|---
|  | $\cong$ |  | $\cong$
|  | $M_{S}^{QFCB}$ | $\rightarrow$ | $M_{S}^{CFCB}$

In the previous section it has been shown that the missing parts of the
arguments in favor of the empirical confirmation via the analogue gravity are
the following:

  1. 1)

Whether real black holes are described by the Hawking’s calculation.

  2. 2)

If so, why one should infer the empirical confirmation of the Hawking
radiation based on its analogous one in the analogue gravity.

The reason for the first one is conclusion III. The reason to support the
second one is the fact that the observation of the Hawking radiation in the
analogue gravity means the empirical confirmation of the structure
$M_{S}^{QFCB}$ which is isomorphic to the structure $M_{T}^{QFCB}$.

All in all, if one takes the stance of structuralism, the mathematical
structure of the new theory reduces to the structure of the old theory and
subsequently, the structure of the Hawking’s calculations (before the Page
time) as an old theory is retrieved. In other words, the structure of the
whatever the next scientific theory, $M_{T}^{new}$, will be reduced to the
structure of the Hawking radiation, $M_{T}^{QFCB}\cong M_{S}^{QFCB}$ where the
structure of the Hawking’s calculation is isomorphic to the structure of the
analogue gravity. Resultantly, empirical confirmation of the Hawking radiation
in the analogue gravity leads to the empirical confirmation of the structure
$M_{S}^{QFCB}\cong M_{T}^{QFCB}$. Moreover, the observation of $P_{S}$ prompts
the empirical confirmation of the original Hawking radiation, $P_{T}$.

##  4 Conclusion

String theory, Hawking radiation and some quantum gravitational phenomena are
yet to be confirmed empirically and they are inaccessible and will be so,
perhaps. Then there would be a problem for the empirical confirmation of these
phenomena. Analogue gravity by simulation of these phenomena seems that
provides a fertile ground for the empirical confirmatory role of these areas
of research. Supporters of this idea claim that the mathematical structure of
analogue gravity models and the quantum gravitational phenomena are the same.
Accordingly, if a phenomenon, say, Hawking radiation occurs in the analogue
gravity side, the shared mathematical structure necessitates the occurrence of
the analogous phenomenon in the quantum gravity side. However, critics believe
that on what basis people think that the original quantum gravitational
phenomenon is described by the shared mathematical structure if it is not
confirmed empirically.

Structuralism believes that the continuity and development of science are
demonstrated by the continuity and development of the mathematical structure
of the scientific theories. As a consequence, the mathematical structure of a
new theory should be partially reduced to the mathematical structure of the
old theory. In this manner, the mathematical structure of new theories about
quantum gravity should be reduced to that of the old ones, say, partially to
the Hawking’s calculations on the Hawking radiation. Because of the fact that
Hawking’s calculations share the same mathematical structure with the analogue
gravity then, partially, the occurrence of the Hawking radiation in the
analogue gravity confirms the occurrence of the Hawking radiation in original
black holes.

## Acknowledgment

The author would like to acknowledge his debt to Saeed Masoumi for helpful
discussion and appreciate Radin Dardashti and Paul Bartha for reading the
manuscript and very useful comments.

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# QNMs of scalar fields on small Reissner-Nordström-AdS5 black holes

Julián Barragán Amado<EMAIL_ADDRESS>Department of
Mathematics, University of Sherbrooke, 2500, boul. de l’Université,
Sherbrooke, Quebec, Canada  Departamento de Física, Universidade Federal de
Pernambuco, 50670-901, Recife, Pernambuco, Brazil  Van Swinderen Institute for
Particle Physics and Gravity, University of Groningen, Groningen, The
Netherlands  Bruno Carneiro da Cunha<EMAIL_ADDRESS>Departamento de
Física, Universidade Federal de Pernambuco, 50670-901, Recife, Pernambuco,
Brazil  Elisabetta Pallante<EMAIL_ADDRESS>Van Swinderen Institute for
Particle Physics and Gravity, University of Groningen, Groningen, The
Netherlands  Nikhef, Science Park, Amsterdam, The Netherlands

###### Abstract

We study the quasinormal modes (QNMs) of a charged scalar field on a Reissner-
Nordström-anti-de Sitter (RN-AdS5) black hole in the small radius limit by
using the isomonodromic method. We also derive the low-temperature expansion
of the fundamental QNM frequency. Finally, we provide numerical evidence that
instabilities appear in the small radius limit for large values of the charge
of the scalar field.

Quasinormal modes, Reissner-Nordström-AdS Black Hole, Heun Equation, Painlevé
Transcendents

##  I Introduction

The energy extraction from a rotating uncharged black hole can be realized by
the scattering of a wave of frequency $\omega$ and axial quantum number $m$
that satisfies the superradiance condition $\omega<m\Omega$, where $\Omega$ is
the angular velocity of the black hole. It turns out that the reflected wave
is amplified with respect to the original incoming wave [1]. By imposing
generic boundary conditions at the horizon and infinity, multiple
amplifications and reflections of the scattered wave can occur and lead to
superradiant instabilities.

An analogous superradiance condition for charged fields on charged black holes
can be given by $\omega<e\Phi$, where $e$ is the charge of the field and
$\Phi$ the electrostatic potential difference between the horizon and the
spatial infinity [2, 3, 4]. Then, by imposing an asymptotic AdS spacetime or a
mirrorlike boundary condition, Reissner-Nordström black holes can develop such
instabilities, as it has been shown in four dimensions [5, 6] and $d\geq 4$
dimensions in the small outer horizon radius, $r_{+}$, limit [7].

Black-hole superradiance is of relevance in numerous areas, from high-energy
physics to astrophysics, and to fundamental questions of General Relativity,
such as the stability of a black-hole solution and the existence of hairy
black hole configurations. In [8] it was suggested that a Reissner-Nordström-
AdS4 black hole coupled to a Higgs field Lagrangian can exhibit a spontaneous
Abelian gauge symmetry breaking due to a scalar condensate near the horizon,
thus implying that superradiant instabilities can occur for a certain range of
values of the charges and the masses of the black hole and the scalar field.
This observation may also suggest a holographic description of spontaneous
symmetry breaking and superfluidity through the gauge-gravity duality – for
comprehensive reviews see [9, 10]. In five dimensions, the phase transition
between the RN-AdS5 black holes and hairy black hole solutions at the onset of
superradiant instabilities has been explored in [11, 12].

We study the occurrence of instabilities in RN-AdS5 by computing the QNMs
frequencies of a charged scalar field scattering on a RN-AdS5 black hole. Due
to the boundary conditions at the black hole horizon and at spatial infinity,
the QNMs are generically complex frequencies, where the real part is related
to the oscillation frequency and the imaginary part gives the inverse of the
damping time. A superradiant mode thus corresponds to an eigenfrequency with a
positive imaginary part.

In this context, we apply the isomonodromic method to recast the associated
boundary value problem into an initial value problem for the corresponding
isomonodromic tau function. Since the radial second-order ordinary
differential equation possesses four regular singular points, see (18), one
should expect that the initial conditions are given in terms of the Painlevé
VI (PVI) tau function [13, 14]. Interestingly, in recent years, the Painlevé
transcendents have been employed to solve a variety of problems such as
correlation functions in Ising and Ashkin-Teller model (see [15] and
references therein), conformal maps [16], black-hole physics [17, 18] and the
Rabi model in quantum optics [19]. The Painlevé transcendents implement the
Riemann-Hilbert map relating the accessory parameters of the differential
equations to the monodromy properties of its solutions, and [15, 20] showed
that their general expansion is expressible in terms of $c=1$ conformal
blocks. We should mention at this point that an alternative proposal directly
using four-dimensional supersymmetric gauge theories in the Nekrasov-
Shatashvili phase, corresponding to semi-classical conformal blocks, was
developed in [21, 22]. Also worthy of note is the effort to relate quantities
of physical interest to monodromy data in [23], where the authors considered
greybody factors, QNMs and Love numbers for the four-dimensional Kerr
background.

As argued in [14], in the small $r_{+}$ limit the first correction to the
(fundamental) QNMs frequencies receives contributions from all the
intermediate channels of the $c=1$ Virasoro conformal blocks expansion.
Nevertheless, these contributions can be resummed by using the generating
function for the Catalan numbers, allowing us to obtain an asymptotic
expression for the frequencies [14].

This manuscript is organized as follows. In Sec. II we study charged scalar
perturbations of the RN-AdS5 black hole, and recast the eigenvalue problem
into a set of transcendental equations involving the PVI tau function. Sec.
III is devoted to the analysis of the QNMs in the small $r_{+}$ limit. We
discuss two different regimes for the temperature of the black hole: the high-
temperature and the low-temperature limit for small black holes. Firstly, we
show that at high temperature the QNMs are given by the zeros of the PVI tau
function, and compute an asymptotic expression in the small $r_{+}$ limit for
the $\ell=0$ modes in Sec. III.1. Secondly, in Sec. III.2 we provide an
asymptotic expression for the fundamental QNM frequency in the low-temperature
limit by applying the confluence limit of the PVI tau function. Sec. III.3
compares the asymptotic analytic expressions and the numerical solutions. We
close in Sec. IV with some remarks on the implications of the obtained
results. Finally, in Appendix A we review the Nekrasov expansion of the PVI
tau function.

##  II Scalar Fields in Reissner-Nordström-$\mathbf{AdS_{5}}$

The line element of the Reissner-Nordström-AdS5 black hole is

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

where $d\Omega_{3}^{2}$ is the metric of the unit three-sphere and the
function $f(r)$ is given by

$f(r)=1-\frac{M}{r^{2}}+\frac{Q^{2}}{r^{4}}+r^{2}=\frac{\Delta_{r}}{r^{4}}=\frac{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})(r^{2}-r_{0}^{2})}{r^{4}},$
(2)

with the AdS radius $L=1$ and the parameters $M$, $Q$ are related to the black
hole mass and charge, respectively. There is a correspondence between the
roots of $\Delta_{r}$ and the Killing horizons of the black hole, whose radial
positions will be parametrized by the largest real root $r_{+}$, corresponding
to the outer horizon. The roots corresponding to $r_{-}^{2}$ and $r_{0}^{2}$
can be written as

$r_{-}^{2}=\frac{1}{2}\left(-1-r_{+}^{2}+\sqrt{(1+r_{+}^{2})^{2}+\frac{4Q^{2}}{r_{+}^{2}}}\right),\qquad
r_{0}^{2}=\frac{1}{2}\left(-1-r_{+}^{2}-\sqrt{(1+r_{+}^{2})^{2}+\frac{4Q^{2}}{r_{+}^{2}}}\right).$
(3)

The temperature at each horizon is

$T_{k}=\frac{1}{4\pi}\frac{d}{dr}\left(\frac{\Delta_{r}}{r^{4}}\right)\biggr{|}_{r=r_{k}}=\frac{1}{2\pi}\frac{(r_{k}^{2}-r_{i}^{2})(r_{k}^{2}-r_{j}^{2})}{r^{3}_{k}},\qquad
i,j\neq k,$ (4)

and specifically, at the outer horizon

$T_{+}=\dfrac{1}{2\pi}\dfrac{(r_{+}^{2}-r_{-}^{2})(r_{+}^{2}-r_{0}^{2})}{r_{+}^{3}},$
(5)

which by means of (3) reads

$T_{+}=\dfrac{1}{2\pi}\left[\frac{1}{r_{+}}-\frac{Q^{2}}{r^{5}_{+}}+2r_{+}\right].$
(6)

For black holes satisfying the cosmic censorship conjecture, we require that
$Q\leq Q_{c}$, where the critical charge $Q_{c}$ is the maximal charge at
extremality $(T_{+}=0)$

$Q_{c}=r_{+}^{2}\sqrt{1+2r_{+}^{2}}.$ (7)

We then parametrize the accessible charge values as $Q=q\,Q_{c}$, where $q\leq
1$ is an extremality parameter for fixed $r_{+}>0$, and in terms of $q$ the
roots $r_{-}^{2}$ and $r_{0}^{2}$ in (3) read

$r_{-}^{2}=\frac{1}{2}\left(-1-r_{+}^{2}+\sqrt{1+2(1+2q^{2})r_{+}^{2}+(1+8q^{2})r_{+}^{4}}\right),\quad
r_{0}^{2}=\frac{1}{2}\left(-1-r_{+}^{2}-\sqrt{1+2(1+2q^{2})r_{+}^{2}+(1+8q^{2})r_{+}^{4}}\right).$
(8)

As we will see in section III, one can think of small black holes near the
critical charge $(q\lesssim 1)$ both at low temperature and at high
temperature.

The electromagnetic potential of the charged black hole reads

$A_{\mu}dx^{\mu}=\left(-\dfrac{\sqrt{3}}{2}\dfrac{Q}{r^{2}}+C\right)dt,$ (9)

where the choice of the constant $C$ is determined by the gauge choice [24].
We assume $C=0$ for a vanishing potential at the spatial infinity, while it
has also been shown in [6], for the case of asymptotically AdS, that the role
of $C$ is merely a shift in the real part of the quasinormal frequency
$\omega$, resulting in
$\mathrm{Re}(\omega)\rightarrow\mathrm{Re}(\omega)+e\,C$.

We consider massive charged scalar field perturbations of the RN-AdS5 black
hole. The field equation is the Klein-Gordon equation

$\dfrac{1}{\sqrt{-g}}D_{\mu}\left(\sqrt{-g}g^{\mu\nu}D_{\nu}\right)\Phi-\mu^{2}\Phi=0,$
(10)

where $D_{\mu}=\nabla_{\mu}-i\,eA_{\mu}$, and $e$ and $\mu$ are the field
charge and mass, respectively. The scalar field solution of (10) can be
decomposed by the Ansatz

$\Phi_{\omega,\ell}(t,\theta,\varphi,\psi,r)=e^{-i\omega
t}Y_{\ell}^{m_{1},m_{2}}(\theta,\varphi,\psi)R_{\omega,\ell}(r),$ (11)

where $Y_{\ell}^{m_{1},m_{2}}(\theta,\varphi,\psi)$ are the scalar (spinless)
spherical harmonics on the three-sphere, with eigenvalues determined by the
equation

$\Delta
Y_{\ell}^{m_{1},m_{2}}(\theta,\varphi,\psi)=-\ell(\ell+2)Y_{\ell}^{m_{1},m_{2}}(\theta,\varphi,\psi),$
(12)

with $\ell$ the angular momentum quantum number and $m_{1}$ and $m_{2}$ are
integers. By means of (12), equation (10) reduces to a second-order ordinary
differential equation for the radial function $R_{\omega,\ell}(r)$ of the form

$\left[\frac{1}{r}\frac{d}{dr}\left(\frac{\Delta_{r}}{r}\frac{d}{dr}\right)+\frac{r^{6}}{\Delta_{r}}\left(\omega-\frac{\sqrt{3}}{2}\frac{e\,Q}{r^{2}}\right)^{2}-\mu^{2}r^{2}-\ell(\ell+2)\right]R_{\omega,\ell}(r)=0.$
(13)

Equation (13) possesses four regular singular points, located at the roots of
$\Delta_{r}$ in (2) and infinity. The characteristic exponents
$\beta_{k},\beta_{\infty}$ are determined by the asymptotic behavior of the
solution in (13) near the singular points

$\beta_{k}=\pm\frac{1}{2}\theta_{k},\qquad\beta_{\infty}=\frac{1}{2}(2\pm\theta_{\infty}),\qquad\qquad
k=+,-,0,$ (14)

where

$\theta_{k}=\frac{i}{2\pi\,T_{k}}\left(\omega-\frac{\sqrt{3}}{2}\frac{e\,Q}{r_{k}^{2}}\right),\qquad\theta_{\infty}=2-\Delta,\qquad\qquad
k=+,-,0,$ (15)

and $\theta_{+}$ is the variation of the entropy $\delta S$ of the black hole
as it absorbs a quantum of frequency $\omega$ and electric charge $e$ at the
outer horizon

$\theta_{+}=\frac{i}{2\pi}\delta S.$ (16)

When $\delta S<0$, i.e. $\mathrm{Im}\,\theta_{+}<0$, unstable modes can occur
and the wave function grows exponentially with time, which means that the
black hole is unstable. Furthermore, $\theta_{\infty}$ is related to $\Delta$
– the conformal dimension of the CFT primary field associated to the AdS5
scalar, i.e. $\Delta(\Delta-4)=\mu^{2}$, with $\mu$ the mass of the scalar
field.

One can reduce (13) to the canonical Heun form by performing the following
change of variables:

$z=\frac{r^{2}-r^{2}_{-}}{r^{2}-r^{2}_{0}},$ (17a)
$R_{\omega,\ell}(z)=z^{-\theta_{-}/2}(z-z_{0})^{-\theta_{+}/2}(z-1)^{\Delta/2}F(z),$
(17b)

that leads to an equation for $F(z)$

$\frac{d^{2}F}{dz^{2}}+\left[\frac{1-\theta_{-}}{z}+\frac{1-\theta_{+}}{z-z_{0}}+\frac{\Delta-1}{z-1}\right]\frac{dF}{dz}+\left(\frac{\kappa_{1}\kappa_{2}}{z(z-1)}-\frac{z_{0}(z_{0}-1)K_{0}}{z(z-z_{0})(z-1)}\right)F(z)=0,$
(18)

with

$\kappa_{1}=\frac{1}{2}(\theta_{-}+\theta_{+}-\Delta-\theta_{0}),\qquad\kappa_{2}=\frac{1}{2}(\theta_{-}+\theta_{+}-\Delta+\theta_{0}),$
(19)

and the accessory parameters $\\{z_{0},K_{0}\\}$ are

$z_{0}=\frac{r_{+}^{2}-r_{-}^{2}}{r_{+}^{2}-r_{0}^{2}},$ (20a)
$\begin{split}4z_{0}(z_{0}-1)K_{0}=&-\frac{\ell(\ell+2)+\Delta(\Delta-4)r_{-}^{2}-\omega^{2}}{(r_{+}^{2}-r_{0}^{2})}-(z_{0}-1)\left[(\theta_{-}+\theta_{+}-1)^{2}-\theta_{0}^{2}-1\right]\\\
&-z_{0}\left[2(\theta_{+}-1)(1-\Delta)+(2-\Delta)^{2}-2\right].\end{split}$
(20b)

The QNMs are solutions of (18) obeying specific boundary conditions, in the
sense that there is only an incoming wave at the horizon $r_{+}$ $(z=z_{0})$
and regularity at spatial infinity $(z=1)$. For the radial equation in (13)
with $\mu^{2}>0$, the conditions read as follows:

$R_{\omega,\ell}(r)\sim\begin{cases}(r-r_{+})^{-{\tfrac{1}{2}}\theta_{+}},&r\rightarrow
r_{+},\\\ r^{2-\Delta},&r\rightarrow\infty,\end{cases}$ (21)

and $F(z)$ in (18) is a regular function at the boundaries. The conditions in
(21) will enforce the quantization of the (not necessarily real) frequencies
$\omega$, the QNMs frequencies.

Following the seminal works of the Kyoto school [25, 26, 27], one of the
authors has described in [18], by means of the method of isomonodromic
deformations, the Riemann-Hilbert map between invariants of the representation
of the monodromy group of the four-punctured Riemann sphere and Fuchsian
$2\times 2$ linear systems with four regular singular points. The monodromy
group action on the solutions of the Fuchsian system is represented by
matrices $M_{i}$ describing the analytic continuation as one circles once
around a singular point $z_{i}$ of the system. The map gives a procedure to
compute the accessory parameters $\\{z_{0},K_{0}\\}$ of the Heun equation (18)
in terms of the monodromy data
$\\{\vec{\theta},\vec{\sigma}\\}=\\{\theta_{1},\theta_{2},\theta_{3},\theta_{4};\sigma_{12},\sigma_{23},\sigma_{13}\\}$,
where the single monodromy parameters are defined as follows

$2\cos\pi\theta_{i}=\operatorname{Tr}M_{i},\qquad k=1,2,3,4,$ (22)

and the composite monodromy parameters $\sigma_{ij}$ are given by

$2\cos\pi\sigma_{ij}=\operatorname{Tr}M_{i}M_{j},\qquad i,j=1,2,3,$ (23)

where $M_{i}M_{j}$ represents the analytic continuation around two singular
points. Following [26], only two of the three composite monodromy parameters
$\sigma_{ij}=\\{\sigma_{12},\sigma_{23},\sigma_{13}\\}$ are independent, and
we choose the pair $(\sigma_{12},\sigma_{23})$ in the solution of the
differential equation (18). The parameter $\sigma_{12}$ from now on will be
denoted by $\sigma$.

As shown in [18] the Riemann-Hilbert map between the accessory parameters of
the Heun differential equation (18) and the monodromy parameters can be
expressed implicitly via the PVI tau function

$\tau(\vec{\theta};\sigma,s;z_{0})=0,$ (24a) $\left.\frac{\partial}{\partial
t}\log\tau(\vec{\theta}^{-};\sigma-1,s;t)\right|_{t=z_{0}}-\frac{(\theta_{2}-1)\theta_{3}}{2(z_{0}-1)}-\frac{(\theta_{2}-1)\theta_{1}}{2z_{0}}=K_{0}.$
(24b)

where $\vec{\theta}^{-}=\\{\theta_{1},\theta_{2}-1,\theta_{3},\theta_{4}+1\\}$
and $s$ is a function of the monodromy parameters defined by (79).

Repeating the analysis of [14], in order to solve the first condition (24a),
one can invert the series expansion of the PVI tau function (A), for $t=z_{0}$
sufficiently close to zero, to write

$\chi(\sigma;z_{0})=\kappa\,z_{0}^{\sigma},$ (25)

where $\kappa$ is given in terms of the monodromy data by (82). From the
structure of the tau function expansion, we can see that $\chi$ is analytic in
$z_{0}$, and its expansion can be computed recursively from (A), yielding

$\displaystyle\chi(\sigma;z_{0})=\biggl{[}$
$\displaystyle\frac{((\theta_{2}+\sigma)^{2}-\theta_{1}^{2})((\theta_{3}+\sigma)^{2}-\theta^{2}_{4})}{16\sigma^{2}(\sigma-1)^{2}}z_{0}\biggr{]}\biggl{(}1+(1-\sigma)\frac{(\theta_{1}^{2}-\theta^{2}_{2})(\theta_{3}^{2}-\theta^{2}_{4})+\sigma^{2}(\sigma-2)^{2}}{2\sigma^{2}(\sigma-2)^{2}}z_{0}+{\cal
O}(z_{0}^{2})\biggr{)}.$ (26)

We can then determine the logarithmic derivative of the tau function in (24b)
using the expansion (A) and write the expansion of the accessory parameter
$K_{0}$ in terms of the monodromy data and $z_{0}$

$\begin{gathered}4z_{0}K_{0}=\left(\sigma-1\right)^{2}-\left(\theta_{1}+\theta_{2}-1\right)^{2}+\bigg{[}2(\theta_{3}-1)(\theta_{2}-1)+\frac{\left((\sigma-1)^{2}+\theta_{2}^{2}-\theta_{1}^{2}-1\right)\left((\sigma-1)^{2}+\theta_{3}^{2}-\theta_{4}^{2}-1\right)}{2\sigma(\sigma-2)}\bigg{]}z_{0}\\\
+\bigg{[}\frac{13}{32}\sigma(\sigma-2)+2\left(\theta_{3}-1\right)\left(\theta_{2}-1\right)-\frac{1}{32}\left(5+14(\theta^{2}_{1}+\theta^{2}_{4})-18(\theta^{2}_{2}+\theta^{2}_{3})\right)+\frac{\left(\theta^{2}_{1}-\theta^{2}_{2}\right)^{2}\left(\theta^{2}_{3}-\theta^{2}_{4}\right)^{2}}{64}\left(\frac{1}{\sigma^{3}}-\frac{1}{(\sigma-2)^{3}}\right)\\\
-\frac{\left((\theta^{2}_{1}-\theta^{2}_{2})(\theta^{2}_{3}-\theta^{2}_{4})+8\right)^{2}-2(\theta^{2}_{1}+\theta^{2}_{2})(\theta^{2}_{3}-\theta^{2}_{4})^{2}-2(\theta^{2}_{1}-\theta^{2}_{2})^{2}(\theta^{2}_{3}+\theta^{2}_{4})-64}{32\sigma(\sigma-2)}\\\
+\frac{\left((\theta_{1}-1)^{2}-\theta^{2}_{2}\right)\left((\theta_{1}+1)^{2}-\theta^{2}_{2}\right)\left((\theta_{3}-1)^{2}-\theta^{2}_{4}\right)\left((\theta_{3}+1)^{2}-\theta^{2}_{4}\right)}{32(\sigma+1)(\sigma-3)}\bigg{]}z^{2}_{0}+\mathcal{O}(z_{0}^{3}).\end{gathered}$
(27)

Equations (26) and (27) were derived under the assumption that $\sigma$ lies
in the strip $0<\mathrm{Re}\,\sigma<1$. Formulas for ${\rm Re}\,\sigma<0$ can
be obtained by the replacement $\sigma\rightarrow-\sigma$. The expansion for
$K_{0}$ is symmetric under the replacement $\sigma\rightarrow 2-\sigma$ and
has the generic structure

$4z_{0}K_{0}=k_{0}+k_{1}z_{0}+k_{2}z_{0}^{2}+\ldots+k_{n}z_{0}^{n}+\ldots,$
(28)

where $k_{n}$ is a rational function of the monodromy parameters, the
numerator is a polynomial in the monodromy parameters $\theta_{i}$ and
$\sigma$, and the denominator is a polynomial of $\sigma$ alone. The
coefficient $k_{n}$ has single poles at integer $\sigma=3,\ldots,n+1$, as well
as poles of order up to $2n-1$ at $\sigma=2$. The leading behavior of $k_{n}$
near $\sigma=2$ is

$k_{n}=(-1)^{n-1}\mathbf{C}_{n-1}\frac{(\theta_{2}^{2}-\theta_{1}^{2})^{n}(\theta_{3}^{2}-\theta_{4}^{2})^{n}}{4^{2n-1}(\sigma-2)^{2n-1}}+\ldots,\qquad
n\geq 1,$ (29)

where $\mathbf{C}_{n}$ is the $n$-th Catalan number. A structure analogous to
(28) exists for $\chi(\sigma;z_{0})$ in (26), and the reader is referred to
[14] for further details. Recently, similar analytic expansions involving
irregular conformal blocks have been analyzed in [28, 29].

For our particular problem of the radial system, the single monodromy
parameters $\theta_{i}$ can be read directly from (18) as follows

$\theta_{1}=\theta_{-},\quad\theta_{2}=\theta_{+},\quad\theta_{3}=2-\Delta,\quad\theta_{4}=\theta_{0},$
(30)

and then the conditions (24a) and (24b) define for us the monodromy parameters
$\sigma$ and $\kappa$ (or, rather, $s$) as functions of the modulus $z_{0}$
and the accessory parameter $K_{0}$. As it was shown in [17], physically
relevant quantities such as scattering coefficients can be expressed in terms
of $\sigma$ and $s$.

For the QNM problem at hand, we have to enforce the boundary conditions in
(21). As shown in [14], these can be encoded with the requirement that the
composite monodromy matrix around the two singular points $r_{+}$ and $\infty$
is triangular. In turn, this implies that the composite monodromy parameter
$\sigma_{23}$ satisfies

$\sigma_{23}=\theta_{2}+\theta_{3}+2n,\qquad n\in\mathbb{Z},$ (31)

which will be referred to as the quantization condition for the radial
function. With this restriction, we can solve for the $s$ parameter in (79) to
yield

$s=\frac{\sin\tfrac{\pi}{2}(\theta_{2}+\theta_{1}+\sigma)\sin\tfrac{\pi}{2}(\theta_{2}-\theta_{1}+\sigma)\sin\tfrac{\pi}{2}(\theta_{3}+\theta_{4}+\sigma)\sin\tfrac{\pi}{2}(\theta_{3}-\theta_{4}+\sigma)}{\sin\tfrac{\pi}{2}(\theta_{2}+\theta_{1}-\sigma)\sin\tfrac{\pi}{2}(\theta_{2}-\theta_{1}-\sigma)\sin\tfrac{\pi}{2}(\theta_{3}+\theta_{4}-\sigma)\sin\tfrac{\pi}{2}(\theta_{3}-\theta_{4}-\sigma)},$
(32)

which, given the relation between $\kappa$ and $s$ in (82), can then be
substituted in (25) to give

$\frac{\Gamma^{2}(1-\sigma)}{\Gamma^{2}(1+\sigma)}\frac{\Gamma(1+\tfrac{1}{2}(\theta_{2}+\theta_{1}+\sigma))}{\Gamma(1+\tfrac{1}{2}(\theta_{2}+\theta_{1}-\sigma))}\frac{\Gamma(1+\tfrac{1}{2}(\theta_{2}-\theta_{1}+\sigma))}{\Gamma(1+\tfrac{1}{2}(\theta_{2}-\theta_{1}-\sigma))}\frac{\Gamma(1+\tfrac{1}{2}(\theta_{3}+\theta_{4}+\sigma))}{\Gamma(1+\tfrac{1}{2}(\theta_{3}+\theta_{4}-\sigma))}\frac{\Gamma(1+\tfrac{1}{2}(\theta_{3}-\theta_{4}+\sigma))}{\Gamma(1+\tfrac{1}{2}(\theta_{3}-\theta_{4}-\sigma))}s\,z_{0}^{\sigma}=\chi(\sigma;z_{0}).$
(33)

Equations (27) and (33) can be interpreted as an overdetermined system for
$\sigma$ in terms of the parameters of the differential equation, namely
$z_{0}$ in (20a), $K_{0}$ in (20b) and $\vec{\theta}$ in (30), which will only
admit solutions for specific (discrete) values of $\omega$. As we will see in
the following, these equations can be solved perturbatively in the small
$r_{+}$ limit to provide us with the QNMs frequencies.

Similar methods have been employed to study scalar and vector perturbations in
Kerr-AdS5 backgrounds [14, 30], and more recently in the near-extremal limit
[31]. In four dimensions, rotating black holes in asymptotically flat and de-
Sitter spacetimes were analyzed in [32, 28, 29].

##  III Small RN-AdS5 black holes via isomonodromy

We proceed to study the small $r_{+}$ limit of RN-AdS5 black holes. By means
of (8), the accessory parameter $z_{0}$ in (34) can be rewritten in terms of
$r_{+}$ and $q$ only

$z_{0}=\frac{r_{+}^{2}-r_{-}^{2}}{r_{+}^{2}-r_{0}^{2}}=\frac{1+3r_{+}^{2}-\sqrt{1+2(1+2q^{2})r_{+}^{2}+(1+8q^{2})r_{+}^{4}}}{1+3r_{+}^{2}+\sqrt{1+2(1+2q^{2})r_{+}^{2}+(1+8q^{2})r_{+}^{4}}}.$
(34)

For small $r_{+}$ and arbitrary $0\leq q\leq 1$

$z_{0}=(1-q^{2})r_{+}^{2}+\mathcal{O}(r_{+}^{4}),$ (35)

so that $r_{+}\rightarrow 0$ implies $z_{0}\rightarrow 0$ in (35). Hence, the
analytic expansions for $\chi(\sigma;z_{0})$ in (26) and the accessory
parameter $K_{0}$ in (27) for small $z_{0}$ apply in the small $r_{+}$ limit.

By means of (8) and $q=Q/Q_{c}$, the temperature (5) reduces to

$2\pi
T_{+}=\frac{1-q^{2}}{r_{+}}+2(1-q^{2})r_{+}=\frac{\epsilon(2-\epsilon)}{r_{+}}+2\epsilon(2-\epsilon)r_{+},$
(36)

where in the second equality we introduced the extremality parameter
$\epsilon=1-q$. For small black holes $r_{+}\ll 1$ and $\epsilon$ fixed, the
terms of order $1/r_{+}$ in the last equality of (36) dominate as
$r_{+}\rightarrow 0$.

Equation (36) also suggests that we can identify two different regimes,
depending on the behavior of $\epsilon$:

  1. 1.

The first regime corresponds to $0<\epsilon<1$ fixed and small black hole,
$r_{+}\ll 1$, hence high temperature $T_{+}\sim 1/r_{+}$. Specifically, we
should set $r_{+}\ll\epsilon<1$ in our expansions. This regime is investigated
in Sec. III.1.

  2. 2.

The second regime describes the scaling limit for $\epsilon\ll r_{+}\ll 1$, as
both $\epsilon$ and $r_{+}$ vary. This is the low temperature regime and it is
explored in Sec. III.2.

###  III.1 QNMs frequencies at high temperature

The single monodromy parameters $\theta_{\pm}$ and $\theta_{0}$ can be
expanded for small $r_{+}$ as follows

$\displaystyle\theta_{-}$
$\displaystyle=-i\frac{q^{2}\left(2q\omega-\sqrt{3}e\right)}{2(1-q^{2})}r_{+}+\mathcal{O}(r_{+}^{3}),$
$\displaystyle\theta_{+}$
$\displaystyle=i\frac{\left(2\omega-\sqrt{3}eq\right)}{2(1-q^{2})}r_{+}+\mathcal{O}(r_{+}^{3}),$
(37) $\displaystyle\theta_{0}$
$\displaystyle=\omega+\frac{1}{2}\left(\sqrt{3}eq-3(1+q^{2})\omega\right)r_{+}^{2}+\mathcal{O}(r_{+}^{4}),$

with

$\theta_{-}=-i\vartheta_{-}r_{+},\qquad\theta_{+}=i\vartheta_{+}r_{+},$ (38)

and the coefficients $\vartheta_{\pm}$ are finite for $q<1$ fixed, such that
$\theta_{\pm}$ approach zero as $r_{+}\rightarrow 0$. The expansion (27) for
$z_{0}K_{0}$ allows us to compute the composite monodromy parameter
$(\sigma=\sigma_{\ell})$ by inverting the series in (27). For $\ell\geq 2$ the
result is

$\displaystyle\sigma_{\ell}$ $\displaystyle=\sum_{n=0}^{\infty}c_{n}r_{+}^{n}$
$\displaystyle=\ell+2-\frac{((1+q^{2})(3\omega^{2}+3\ell(\ell+2)-\Delta(\Delta-4))-2\sqrt{3}eq\omega)}{4(\ell+1)}r_{+}^{2}+\mathcal{O}(r_{+}^{4}),\qquad\ell\geq
2.$ (39)

For $\ell=0$ and $\ell=1$, the pole structure of $k_{n}$ described in (28)
renders the derivation of the coefficients $c_{n}$ more complicated, and we
will deal separately with the $\ell=0$ case below. At any rate,
$\sigma_{\ell}$ always admits the generic structure

$\sigma_{\ell}=\ell+2-\nu_{\ell}r_{+}^{2}+\mathcal{O}(r_{+}^{4}).$ (40)

By substituting (III.1) in (26), one obtains an expansion for small $r_{+}$
for $\chi(\sigma_{\ell};z_{0})$

$\chi_{\ell}\equiv\chi(\sigma_{\ell};z_{0})=-(1-q^{2})\frac{\omega^{2}-(\Delta-\ell-4)^{2}}{16(\ell+1)^{2}}\left(1+i\frac{2\vartheta_{+}}{(\ell+2)}r_{+}\right)r_{+}^{2}+\mathcal{O}(r_{+}^{4}),\qquad\ell\geq
2,$ (41)

where $\ell\geq 2$ avoids the poles in (26).

We now focus on the $\ell=0$ case, and defer the analysis of higher $\ell$
modes to future work. The $\ell=0$ case is peculiar, since the leading
behavior of $\sigma_{0}-2$ in (40) is of order $r_{+}^{2}$ and the scaling
limit of $\theta_{\pm}$ in (38) suggests that the accessory parameter $K_{0}$
receives contributions from all orders in $z_{0}$ in the small $r_{+}$ limit.
The left hand side of (27) is given by (20b) for $\ell=0$, and for small
$r_{+}$ we have

$4z_{0}K_{0}=-2i(\vartheta_{-}-\vartheta_{+})r_{+}+\left((\vartheta_{-}-\vartheta_{+})^{2}+\Delta^{2}-2(1+q^{2})\Delta+\omega(\sqrt{3}eq-3(1+q^{2})\omega)+(1+2q^{2})\omega^{2}\right)r_{+}^{2}+\mathcal{O}(r_{+}^{3}),$
(42)

whereas the right hand side of (27) for $\ell=0$ is

$\displaystyle 4z_{0}K_{0}$
$\displaystyle=-2i(\vartheta_{-}-\vartheta_{+})r_{+}+\biggl{[}(\vartheta_{-}-\vartheta_{+})^{2}-2\nu_{0}-\frac{1}{2}(1-q^{2})(\omega^{2}-\Delta^{2})$
$\displaystyle+\frac{(1-q^{2})(\omega^{2}-(\Delta-2)^{2})(\vartheta_{-}^{2}-\vartheta_{+}^{2})}{4\nu_{0}}+\frac{(1-q^{2})^{2}(\omega^{2}-(\Delta-2)^{2})^{2}(\vartheta_{-}^{2}-\vartheta_{+}^{2})^{2}}{64\nu_{0}^{3}}$
$\displaystyle+\frac{(1-q^{2})^{3}(\omega^{2}-(\Delta-2)^{2})^{3}(\vartheta_{-}^{2}-\vartheta_{+}^{2})^{3}}{512\nu_{0}^{5}}+\frac{5(1-q^{2})^{4}(\omega^{2}-(\Delta-2)^{2})^{4}(\vartheta_{-}^{2}-\vartheta_{+}^{2})^{4}}{16384\nu_{0}^{7}}+\ldots\biggr{]}r_{+}^{2}+\mathcal{O}(r_{+}^{3}),$
(43)

where $\nu_{0}=\nu_{\ell}(\ell=0)$ in (40) and the dots are higher orders in
$z_{0}$ that contribute to the $r_{+}^{2}$-term. It was shown in [14] that all
the contributions from the conformal blocks expansion can be resummed using
the generating function for the Catalan numbers to compute the first
correction to $\sigma_{0}$ of order $r_{+}^{2}$. By defining

$X=\frac{(1-q^{2})(\omega^{2}-(\Delta-2)^{2})(\vartheta_{-}^{2}-\vartheta_{+}^{2})}{16\nu_{0}^{2}},$
(44)

and resumming all the terms in the series will lead to an equation for the
coefficient $\nu_{0}$

$\nu_{0}\sqrt{1-4X}=\frac{1}{2}\omega(3(1+q^{2})\omega-\sqrt{3}eq)-\frac{1}{4}(1+q^{2})(3\omega^{2}-\Delta(\Delta-4))+\mathcal{O}(r_{+}^{2}).$
(45)

Then, $\nu_{0}$ in (45) admits the following small $r_{+}$ expansion

$\begin{split}&\nu_{0}=\frac{1}{4}\biggl{[}(1+q^{2})^{2}((3\omega^{2}-\Delta(\Delta-4))^{2}-4\omega^{2}(\omega^{2}-(\Delta-2)^{2}))+4q^{2}\omega^{2}(\omega^{2}-(\Delta-2)^{2})\\\
&\qquad\quad+3e^{2}q^{2}(3\omega^{2}+(\Delta-2)^{2})-8\sqrt{3}(1+q^{2})eq\omega(\omega^{2}+2)\biggr{]}^{\\!1/2}+\mathcal{O}(r_{+}^{2}).\end{split}$
(46)

Hence, by means of (44) and resumming all the contributions of order
$r_{+}^{2}$, we get for $\chi_{0}$ in (41)

$\chi_{0}=-(1-q^{2})\frac{(\omega^{2}-(\Delta-4)^{2})}{64}r^{2}_{+}(1+i\vartheta_{+}r_{+})\left(1+\sqrt{1-\frac{(1-q^{2})(\omega^{2}-(\Delta-2)^{2})(\vartheta_{-}^{2}-\vartheta_{+}^{2})}{4\nu_{0}^{2}}}\right)^{2}+\mathcal{O}(r_{+}^{4}),$
(47)

where

$\vartheta_{-}=\frac{q^{2}\left(2q\omega-\sqrt{3}e\right)}{2(1-q^{2})},\qquad\vartheta_{+}=\frac{\left(2\omega-\sqrt{3}eq\right)}{2(1-q^{2})}.$
(48)

The knowledge of the parameter $s$ in (33) is now sufficient to determine the
QNMs frequencies. By expanding for small $r_{+}$ and employing (47) in (33),
we obtain

$s=\Sigma_{0}\left(1+\frac{i2\nu_{0}\vartheta_{+}r_{+}}{(\vartheta_{+}^{2}-\vartheta_{-}^{2})}\right)+\mathcal{O}(r_{+}^{2},r_{+}^{2}\log
r_{+}),$ (49)

with

$\Sigma_{0}=\nu_{0}^{2}\frac{\Gamma(\tfrac{1}{2}(2-\omega-\Delta))\Gamma(\tfrac{1}{2}(2+\omega-\Delta))}{\Gamma(\tfrac{1}{2}(6-\omega-\Delta))\Gamma(\tfrac{1}{2}(6+\omega-\Delta))}\frac{(\omega^{2}-(\Delta-4)^{2})}{4(1-q^{2})(\vartheta_{+}^{2}-\vartheta_{-}^{2})}\left(1+\sqrt{1+\frac{(1-q^{2})(\omega^{2}-(\Delta-2)^{2})(\vartheta_{+}^{2}-\vartheta_{-}^{2})}{4\nu_{0}^{2}}}\right)^{2}.$
(50)

Equation (49) is a non-analytic expansion in $r_{+}$ due to terms
$\mathcal{O}(r_{+}^{2}\log r_{+})$, which are originated by the term
$z_{0}^{\sigma_{0}}$ in (33). However, the limit as $r_{+}\rightarrow 0$ is
finite.

We now parametrize the QNMs frequencies $\omega_{n,\ell}$ with $n\geq 0$ and
$\ell=0$ in terms of a perturbation to the normal modes in empty AdS5

$\omega_{n,0}=2n+\Delta+\eta_{n,0}r_{+}^{2},$ (51)

under the assumption that $\eta_{n,0}$ has a finite limit as $r_{+}\rightarrow
0$. Then, by replacing (51) in (III.1) for $\omega$, one obtains

$\begin{split}\theta_{-}&=-i\frac{q^{2}\left(2q(2n+\Delta)-\sqrt{3}e\right)}{2(1-q^{2})}r_{+}+\mathcal{O}(r_{+}^{3}),\\\
\theta_{+}&=i\frac{\left(2(2n+\Delta)-\sqrt{3}eq\right)}{2(1-q^{2})}r_{+}+\mathcal{O}(r_{+}^{3}),\\\
\theta_{0}&=2n+\Delta+\beta_{n,0}r_{+}^{2}+\mathcal{O}(r_{+}^{4}),\end{split}$
(52)

where $\beta_{n,0}$ encodes the correction $\eta_{n,0}$ to the QNMs as follows

$\beta_{n,0}=\eta_{n,0}+\frac{1}{2}\left(\sqrt{3}eq-3(1+q^{2})(2n+\Delta)\right).$
(53)

On the other hand, $\beta_{n,0}$ can also be computed by equating (32) to
(49), where $\Sigma_{n,0}\equiv\Sigma_{0}(\omega=\omega_{n,0})$ and the radial
monodromies are given by (52), and by expanding for small $r_{+}$ we have

$\beta_{n,0}=\nu_{n,0}\frac{1+\Sigma_{n,0}}{1-\Sigma_{n,0}}+4i\frac{\vartheta_{+}\nu_{n,0}^{2}}{(\vartheta_{-}^{2}-\vartheta_{+}^{2})}\frac{\Sigma_{n,0}}{(\Sigma_{n,0}-1)^{2}}r_{+}+\mathcal{O}(r_{+}^{2},r_{+}^{2}\log
r_{+}),$ (54)

with $\nu_{n,0}\equiv\nu_{0}(\omega=\omega_{n,0})$ in (46). By means of (54),
(53) yields for $\eta_{n,0}$

$\eta_{n,0}=\frac{1}{2}\left(3(1+q^{2})(2n+\Delta)-\sqrt{3}eq-
Z_{n,0}^{1/2}-i(n+1)(\Delta+n-1)(2(2n+\Delta)-\sqrt{3}eq)r_{+}\right)+\mathcal{O}(r_{+}^{2},r_{+}^{2}\log
r_{+}),$ (55)

where

$Z_{n,0}=\left(\frac{1}{2}(1+q^{2})(3(2n+\Delta)^{2}-\Delta(\Delta-4))-\sqrt{3}eq(2n+\Delta)\right)^{2}.$
(56)

Finally, by employing (55), (51) admits the small $r_{+}$ expansion

$\begin{split}&\omega_{n,0}=2n+\Delta-\frac{1}{2}\left(Z_{n,0}^{1/2}-3(1+q^{2})(2n+\Delta)+\sqrt{3}eq\right)r_{+}^{2}-i\frac{1}{2}(n+1)(\Delta+n-1)(2(2n+\Delta)-\sqrt{3}eq)r_{+}^{3}\\\
&\qquad\quad+\mathcal{O}(r_{+}^{4},r_{+}^{4}\log r_{+}).\end{split}$ (57)

In fact, equation (57) is different from the analytical result in [7] due to
the correction to the real part of $\omega_{n,0}$ of order $r_{+}^{2}$. The
asymptotic expansion (57) for $n=0$ reproduces the numerical results for the
real part of $\omega_{0,0}$ of Table $\mathrm{I}$ in [7]. In addition, the
imaginary part of $\omega_{n,0}$ for $n=0$ in (57) for specific values of $e$,
$\Delta$, $q$ and $r_{+}$ falls in between the numerical and analytical values
given in Tables $\mathrm{II,IV,V}$ for the five-dimensional case in [7].
Importantly, for $q>\tfrac{2(2n+\Delta)}{\sqrt{3}e}$, the imaginary part in
(57) becomes positive, and the superradiance condition is satisfied.

Figure 1 shows our numerical solutions for the imaginary part of
$\omega_{0,0}$ and illustrates explicitly that for
$q=\\{0.5775,0.57775,0.578,0.57825\\}$ the superradiance condition is
satisfied in a finite interval of $r_{+}$, and the stability is eventually
restored as the black hole radius increases. As $q\rightarrow 1$, the
superradiance condition is always satisfied in the small $r_{+}$ limit.

Figure 1: Numerical results for the imaginary part of the fundamental QNM
$\omega_{0,0}$ as a function of $r_{+}$ for increasing values of $q$ in the
interval $\left[0.1,0.9\right]$. The charge and the scaling dimension of the
scalar field are $e=8$ and $\Delta=4$, respectively.

The QNMs frequencies shown in Fig. 1 have been obtained through the numerical
implementation of the initial conditions for the PVI tau function using an
arbitrary-precision library in Julia programming language [33]. For the fast
convergence of the numerical results, we have introduced the Fredholm
determinant formulation of the PVI tau function, instead of its combinatorial
description. We refer to previous works [13, 14, 16] for a more detailed
discussion of the numerical methods.

Finally, in terms of $\epsilon=1-q$ and for $n=0$, (57) reads

$\omega_{0,0}=\Delta-\frac{1}{2}(\Delta-1)(\Delta(\epsilon^{2}-2\epsilon+2)-\sqrt{3}e(1-\epsilon))r_{+}^{2}-i\frac{1}{2}(\Delta-1)(2\Delta-\sqrt{3}e(1-\epsilon))r_{+}^{3}+\mathcal{O}(r_{+}^{4},r_{+}^{4}\log
r_{+}),$ (58)

where $e<2(\Delta+2)/\sqrt{3}$. Equation (58) should provide a good
description of the high-temperature regime $r_{+}\ll\epsilon<1$. We perform a
detailed comparison of (58) with the numerical solution in Sec. III.3 (Fig.
3).

###  III.2 The low-temperature limit of the fundamental QNM

The description of the low-temperature regime requires a more careful approach
since the single monodromy parameters (III.1) may diverge a priori. This issue
is solved by means of a double series expansion for small $r_{+}$ and
$\epsilon$, and by considering the scaling regime $\epsilon\ll r_{+}\ll 1$,
which indeed corresponds to the low-temperature regime of small black holes,
i.e. near-extremal small black holes. In the low-temperature limit (36) is
given by

$2\pi
T_{+}=\frac{2\epsilon}{r_{+}}+\mathcal{O}\left(\frac{\epsilon^{2}}{r_{+}},\epsilon
r_{+}\right),$ (59)

and the single monodromies in terms of $\epsilon$ and $r_{+}$ are

$\begin{split}&\theta_{+}=i\dfrac{(2\omega-\sqrt{3}e)}{4}\dfrac{r_{+}}{\epsilon}+i\dfrac{1}{16}(2\omega+\sqrt{3}e)(2+\epsilon)r_{+}+\mathcal{O}\left(\epsilon^{2}r_{+},\frac{r_{+}^{3}}{\epsilon},r_{+}^{3},\epsilon
r_{+}^{3}\right),\\\
&\theta_{-}=-i\dfrac{(2\omega-\sqrt{3}e)}{4}\dfrac{r_{+}}{\epsilon}+i\dfrac{1}{8}(10\omega-3\sqrt{3}e)r_{+}-i\dfrac{1}{16}(14\omega-\sqrt{3}e)\epsilon
r_{+}+\mathcal{O}\left(\epsilon^{2}r_{+},\frac{r_{+}^{3}}{\epsilon},r_{+}^{3},\epsilon
r_{+}^{3}\right),\\\
&\theta_{0}=\omega-\dfrac{1}{2}(6\omega-\sqrt{3}e)(1-\epsilon)r_{+}^{2}+\mathcal{O}(\epsilon^{2}r_{+}^{2},r_{+}^{4},\epsilon
r_{+}^{4}),\end{split}$ (60)

where $\theta_{\pm}$ diverge as $\epsilon\rightarrow 0$, as it is also evident
from the pole at $q^{2}=1$ in (III.1).

The initial conditions for the confluent limit of the PVI tau function were
derived in [31]. The procedure is further analogous to the rotating case in
[31]. We need to compute the correction of order $r_{+}^{2}$, as well as the
first non-trivial correction in $\epsilon$ for
$\omega_{0,0}=\Delta+\eta_{0,0}r_{+}^{2}$ in (51) and

$\sigma_{0,0}\equiv\sigma_{0}(\omega=\omega_{0,0})=2-\nu_{0,0}r_{+}^{2}.$ (61)

For this purpose, we expand the left hand side of (27) in the small $\epsilon$
and $r_{+}$ limit to obtain

$\begin{split}4z_{0}K_{0}&=\frac{1}{4}i(6\Delta-\sqrt{3}e)(2-\epsilon)r_{+}+\frac{1}{8}i(2\Delta+\sqrt{3}e)\epsilon^{2}r_{+}+\frac{1}{16}(3e^{2}+4\sqrt{3}e\Delta+4\Delta(\Delta-16))(1-\epsilon)r_{+}^{2}\\\
&+\mathcal{O}(\epsilon^{3}r_{+},\epsilon^{2}r_{+}^{2},r_{+}^{3},\epsilon
r_{+}^{3})\end{split}$ (62)

while the right hand side of (27) gives

$\begin{split}4z_{0}K_{0}&=\frac{1}{4}i(6\Delta-\sqrt{3}e)(2-\epsilon)r_{+}+\frac{1}{8}i(2\Delta+\sqrt{3}e)\epsilon^{2}r_{+}\\\
&-\biggl{[}2\nu_{0,0}-\frac{1}{16}(\sqrt{3}e-6\Delta)^{2}-4\nu_{0,0}x\left(1+x+2x^{2}+5x^{3}+\ldots\right)\biggr{]}r_{+}^{2}\\\
&-\biggl{[}\frac{1}{16}(\sqrt{3}e-6\Delta)^{2}+8\nu_{0,0}x\left(1+2x+6x^{2}+20x^{3}+\ldots\right)\biggr{]}\epsilon
r_{+}^{2}+\mathcal{O}(\epsilon^{3}r_{+},\epsilon^{2}r_{+}^{2},r_{+}^{3},\epsilon
r_{+}^{3}),\end{split}$ (63)

where the dots are the higher order poles in $\sigma_{0,0}$ that contribute to
the $r_{+}^{2}$ and $\epsilon r_{+}^{2}$ terms, and we have defined

$x=\frac{(2\Delta-\sqrt{3}e)(\sqrt{3}e-6\Delta)(\Delta-1)}{16\nu_{0,0}^{2}}.$
(64)

Equating (62) and (63) leads to the following equation for $\nu_{0,0}$

$\frac{2\nu_{0,0}}{\sqrt{1-4x}}-\frac{8\nu_{0,0}x(1-\epsilon)}{\sqrt{1-4x}}=\Delta(2(\Delta+2)-\sqrt{3}e)(1-\epsilon)+\mathcal{O}(\epsilon^{2}),$
(65)

that can be solved to yield

$\nu_{0,0}=\frac{1}{2}(1-\epsilon)\sqrt{4\Delta^{2}(\Delta^{2}+\Delta+7)-4\sqrt{3}e\Delta(\Delta^{2}+2)+3e^{2}(\Delta^{2}-\Delta+1)}+\mathcal{O}(\epsilon^{2}).$
(66)

Hence, $\sigma_{0,0}$ in (67) reads

$\begin{split}\sigma_{0,0}&=2-\nu_{0,0}r_{+}^{2}\\\
&=2-\tilde{\nu}_{0,0}(1-\epsilon)r_{+}^{2}+\mathcal{O}(\epsilon^{2}r_{+}^{2}),\end{split}$
(67)

where

$\tilde{\nu}_{0,0}=\frac{1}{2}\sqrt{4\Delta^{2}(\Delta^{2}+\Delta+7)-4\sqrt{3}e\Delta(\Delta^{2}+2)+3e^{2}(\Delta^{2}-\Delta+1)}.$
(68)

The asymptotic expansion for $\chi_{0,0}\equiv\chi_{0}(\omega=\omega_{0,0})$
now receives contributions from all orders in $z_{0}$ due to the pole
structure at $\sigma=2$ in (26). Nevertheless, one can again use the
generating function for the Catalan numbers to obtain an analytic expansion
for $\epsilon\ll r_{+}\ll 1$ given by

$\chi_{0,0}=\frac{1}{2}+\frac{(\Delta-1)(2\Delta-\sqrt{3}e)(6\Delta-\sqrt{3}e)}{16\tilde{\nu}_{0,0}^{2}}+\frac{1}{2}\sqrt{1+\frac{(\Delta-1)(2\Delta-\sqrt{3}e)(6\Delta-\sqrt{3}e)}{4\tilde{\nu}_{0,0}^{2}}}+\mathcal{O}(\epsilon^{2},r_{+}^{2},\epsilon
r_{+}^{2}).$ (69)

By means of the expansion for $\theta_{0}$ in (60), with $\omega=\omega_{0,0}$
in (51), we obtain

$\begin{split}\theta_{0}&=\Delta-\beta_{0,0}r_{+}^{2}\\\
&=\Delta-\left(\frac{1}{2}(6\Delta-\sqrt{3}e)(1-\epsilon)-\eta_{0,0}\right)r_{+}^{2},\end{split}$
(70)

where

$\eta_{0,0}=\frac{1}{2}(1-\epsilon)(6\Delta-\sqrt{3}e)-\beta_{0,0}.$ (71)

Inserting (69) in the right hand side of (33) for $n=\ell=0$ and expanding the
left hand side of (33) for $\epsilon\ll r_{+}\ll 1$, we obtain

$\begin{split}&\frac{(\Delta-1)(6\Delta-\sqrt{3}e)(2\Delta-\sqrt{3}e)}{16\tilde{\nu}_{0,0}^{2}}\frac{\beta_{0,0}+\tilde{\nu}_{0,0}}{\beta_{0,0}-\tilde{\nu}_{0,0}}\biggl{[}1-\frac{2\tilde{\nu}_{0,0}\beta_{0,0}\epsilon}{\beta_{0,0}^{2}-\tilde{\nu}_{0,0}^{2}}+\frac{i4\tilde{\nu}_{0,0}r_{+}}{(6\Delta-\sqrt{3}e)}-\frac{i2\tilde{\nu}_{0,0}(\beta_{0,0}^{2}+4\beta_{0,0}\tilde{\nu}_{0,0}-\tilde{\nu}_{0,0}^{2})\epsilon
r_{+}}{(6\Delta-\sqrt{3}e)(\beta_{0,0}^{2}-\tilde{\nu}_{0,0}^{2})}\\\
&+\mathcal{O}(\epsilon^{2},\epsilon^{2}r_{+},r_{+}^{2},r_{+}^{2}\log
r_{+},\epsilon r_{+}^{2},\epsilon r_{+}^{2}\log r_{+})\biggr{]}\\\
&=\frac{1}{2}+\frac{(\Delta-1)(2\Delta-\sqrt{3}e)(6\Delta-\sqrt{3}e)}{16\tilde{\nu}_{0,0}^{2}}+\frac{1}{2}\sqrt{1+\frac{(\Delta-1)(2\Delta-\sqrt{3}e)(6\Delta-\sqrt{3}e)}{4\tilde{\nu}_{0,0}^{2}}}+\mathcal{O}(\epsilon^{2},r_{+}^{2},\epsilon
r_{+}^{2})\end{split}$ (72)

whose perturbative solution for $\beta_{0,0}$ reads

$\beta_{0,0}=\frac{1}{2}\Delta(2(\Delta+2)-\sqrt{3}e)(1-\epsilon)+i\frac{1}{4}(\Delta-1)(2\Delta-\sqrt{3}e)(2-3\epsilon)r_{+}+\mathcal{O}(\epsilon^{2},\epsilon^{2}r_{+},r_{+}^{2},r_{+}^{2}\log
r_{+},\epsilon r_{+}^{2},\epsilon r_{+}^{2}\log r_{+}).$ (73)

Therefore, by means of (71) and (73), the fundamental QNM frequency is

$\begin{gathered}\omega_{0,0}=\Delta-\frac{1}{2}(\Delta-1)(2\Delta-\sqrt{3}e)(1-\epsilon)r_{+}^{2}-i\frac{1}{4}(\Delta-1)(2\Delta-\sqrt{3}e)(2-3\epsilon)r_{+}^{3}+\mathcal{O}(\epsilon^{2}r_{+}^{2},r_{+}^{4},r_{+}^{4}\log
r_{+},\epsilon r_{+}^{4},\epsilon r_{+}^{4}\log r_{+}),\end{gathered}$ (74)

which is valid for small black holes in the low-temperature limit, i.e.
$\epsilon\ll r_{+}\ll 1$. The imaginary part of $\omega_{0,0}$ in (74), at
this order of perturbation theory, suggests that the black hole is stable
$(\mathrm{Im}\,\omega_{0,0}<0)$ under perturbations with charge
$e<2\Delta/\sqrt{3}$ and $\Delta>1$. The imaginary part has a leading
correction in $\epsilon$ of order $\epsilon r_{+}^{3}$, indicating that
contributions from the black hole temperature are not sufficiently enhanced to
trigger superradiant instabilities for $e<2\Delta/\sqrt{3}$. Such
instabilities can instead be triggered by the charge of the scalar field
provided that $\Delta>1$. Incidentally, we notice that although equations (57)
and (74) were derived in two different regimes, they coincide in the extremal
limit $\epsilon=0$ $(q=1)$ at least at the given order in $r_{+}$.

###  III.3 Comparison with numerical results

It is interesting to compare the analytical expression (74) for small black
holes at low temperature with the numerical prediction for a given charge of
the scalar field $e$ as a function of the radius of the outer horizon $r_{+}$,
as well as the $\epsilon$ parameter. We recall that the numerical results have
been obtained by solving the transcendental equations (24) for the Fredholm
determinant of the PVI tau function [20]. The initial conditions have been
implemented in Julia, using an arbitrary-precision library. Then, the QNM
frequencies have been found by applying a root-finding algorithm which employs
the Newton-Raphson method.

Figure 2 shows the comparison between the numerical results (in red, solid
line) and the low-temperature asymptotic formula (74) for the fundamental QNM
frequency as a function of $r_{+}$ for fixed small $\epsilon=5\times 10^{-6}$
and the charge of the scalar field $e=0.005$. We observe a small discrepancy
between the two curves, both for the real and imaginary part of
$\omega_{0,0}$, increasing to about $2\,\%$ for $r_{+}\simeq 0.01$. In Figure
3 we display the transition from low to high temperatures — small and large
$\epsilon$, respectively — by varying $\epsilon$ while keeping $r_{+}=0.001$
fixed. The numerical results (again in red, solid line) are compared with the
asymptotic formulas (in blue, dashed line) for high and low temperature, given
by (58) and (74), respectively. For low-temperatures, we have $\epsilon\ll
r_{+}\ll 1$ and thus $r_{+}^{3}$ corrections to (74) are more relevant for the
approximation than the $\epsilon r_{+}^{2}$ terms. In other words, one should
resum the series in $r_{+}$ at each given order in $\epsilon$ for a full
comparison. Nevertheless, the $\epsilon$ corrections encode the temperature
dependence, and can be checked to agree with the numerical prediction by
comparing the slope of the two curves for small $\epsilon$ in the imaginary
part of $\omega_{0,0}$ in Fig. 3.

Figure 2: The analytical result (blue dashed line) in (74) for the real (left)
and imaginary (right) part of the fundamental QNM frequency $\omega_{0,0}$
compared with the numerical result (red solid line) as function of $r_{+}$ for
fixed $\epsilon=5\times 10^{-6}$ and charge of the scalar field $e=0.005$.
Figure 3: The analytical result (blue dashed line) in (74) for the real (left)
and imaginary (right) part of the fundamental QNM frequency $\omega_{0,0}$
compared with the numerical result (red solid line) as function of $\epsilon$
for $e=2$ and $r_{+}=0.001$. The ascending and descending dashed lines for the
$\mathrm{Im}\,\omega_{0,0}$ in the right panel reproduce (74) and (58),
respectively. The ascending line describes the same slope of the numerical
result, however, the shift w.r.t. the numerical result is due to higher orders
in $r_{+}$, for $\epsilon=0$ in (74).

##  IV Discussion

In this paper, we have investigated the quasinormal modes of a charged scalar
field scattering on small RN-AdS5 black holes, where the outer horizon radius
$r_{+}$ is much smaller than the AdS radius, via the isomonodromy method. We
analytically explored two regimes in terms of $r_{+}$ and the parameter
$\epsilon=1-q$, with $0\leq q\leq 1$ the charge of the black hole in units of
the critical charge $Q_{c}$. In section III.1, we have considered the small
black hole approximation at high temperatures $(r_{+}\ll\epsilon<1)$ and
provided an asymptotic expression for the QNMs frequencies for $n$ generic and
$\ell=0$ that generalizes previous results found in the literature [7]. These
asymptotic expressions display instabilities for the fundamental QNM mode for
large enough $q$, and we verified numerically (for $e=8$ and $\Delta=4$) that
there is a superradiance window whose width in $r_{+}$ increases with $q$. We
leave open the question whether, in parallel to the Hawking-Page transition
for uncharged black holes [34], stability might be restored as the radius of
the black hole increases.

A different approach is required in the low-temperature limit $\epsilon\ll
r_{+}\ll 1$ and we have presented the analysis of this case in section III.2.
By considering a double series expansion for small $\epsilon$ and $r_{+}$, we
have computed the first non-trivial correction in $\epsilon$ to the
fundamental QNM frequency. We found that small black holes at low temperature
are stable under scalar perturbations with $\Delta$ satisfying the unitarity
bound $\Delta>1$, and charge $e<2\Delta/\sqrt{3}$. When the latter conditions
are not met, the imaginary part of (74) becomes positive, meaning that the
black hole is unstable.

In [7], the correction to the empty AdS frequency is derived from the matching
condition between the analytical solutions constructed in terms of
hypergeometric functions. In contrast, we have shown via the isomonodromy
method that the asymptotic expansion for the fundamental QNM frequency
receives contributions from all descendants of the CFT primaries in the vacuum
module. These contributions can be resummed in terms of the generating
functions of the Catalan numbers and the result suggests that in the matching
approach one should consider all levels in the Frobenius expansion. It is
worth mentioning that for numerical purposes the Fredholm determinant
representation of the PVI tau function can reach a faster convergence time
than direct instanton counting methods based on Nekrasov functions.

This work can be seen as an application to RN-AdS5 of the ideas exposed in
[31], where the authors have written the initial conditions for the Painlevé V
tau function to compute an asymptotic expansion for the fundamental QNM
frequency in the low-temperature regime.

Finally, in [5] it was argued that large RN-AdS4 black holes, i.e $r_{+}\gg 1$
in units of the AdS curvature radius, become unstable below a critical
temperature. Since one can control $\epsilon$ and $r_{+}$ in the accessory
parameter expansion, it would be interesting to explore this regime and the
nature of these instabilities in terms of the PVI tau function and its
confluent limits. It is also interesting to study the hydrodynamical modes in
RN-AdS5 black holes, since scalar perturbations in this background have been
introduced as dual CFT states of strongly correlated holographic models with
finite density, for which the bound on the speed of sound can be violated
[35]. The method presented here and in [31] works for higher spin
perturbations on rotating black holes [30] and can be used to study those
perturbations in RN-AdS5.

## Acknowledgments

J.B.A. is grateful for the support from the University of Sherbrooke as well
as from the Mathematical Physics Laboratory at the Centre de Recherches
Mathématiques in Montréal.

##  Appendix A Nekrasov expansion for the PVI tau function

The series representation of the PVI tau function, written in [36, 15], around
the critical point $t=0$ is given by

$\tau(t)=\sum_{n\in\mathbb{Z}}C(\vec{\theta},\sigma+2n)s^{n}t^{\tfrac{1}{4}((\sigma+2n)^{2}-\theta_{1}^{2}-\theta_{2}^{2})}\mathcal{B}(\vec{\theta},\sigma+2n;t),$
(75)

where $\vec{\theta}=\\{\theta_{1},\theta_{2},\theta_{3},\theta_{4}\\}$ are the
single monodromy parameters, and the parameters $\sigma$, $s$ are two
integration constants. The structure constants $C(\vec{\theta},\sigma)$ are
given in terms of Barnes’ functions

$C(\vec{\theta},\sigma)=\frac{\prod_{\alpha,\beta=\pm}G(1+\tfrac{1}{2}(\theta_{3}+\alpha\theta_{4}+\beta\sigma))G(1+\tfrac{1}{2}(\theta_{2}+\alpha\theta_{1}+\beta\sigma))}{G(1+\sigma)G(1-\sigma)},$
(76)

and $\mathcal{B}(\vec{\theta},\sigma;t)$ in (75) coincides with the $c=1$
Virasoro conformal blocks, which are explicitly given by a combinatorial
series in terms of Nekrasov functions

$\mathcal{B}(\vec{\theta},\sigma+2n;t)=(1-t)^{\tfrac{1}{2}\theta_{2}\theta_{3}}\sum_{\lambda,\mu\in\mathbb{Y}}{\cal
B}_{\lambda,\mu}(\vec{\theta},\sigma+2n)t^{|\lambda|+|\mu|},$ (77)

where $\mathbb{Y}$ denotes the space of Young diagrams, $\lambda$ and $\mu$
are two of its elements, with number of boxes $|\lambda|$ and $|\mu|$, and the
coefficients in (77) are

$\begin{split}{\cal
B}_{\lambda,\mu}(\vec{\theta},\sigma)&=\prod_{(i,j)\in\lambda}\frac{((\theta_{t}+\sigma+2(i-j))^{2}-\theta_{0}^{2})((\theta_{1}+\sigma+2(i-j))^{2}-\theta_{\infty}^{2})}{16h_{\lambda}^{2}(i,j)(\lambda^{\prime}_{j}-i+\mu_{i}-j+1+\sigma)^{2}}\\\
&\times\prod_{(i,j)\in\mu}\frac{((\theta_{t}-\sigma+2(i-j))^{2}-\theta_{0}^{2})((\theta_{1}-\sigma+2(i-j))^{2}-\theta_{\infty}^{2})}{16h_{\lambda}^{2}(i,j)(\mu^{\prime}_{j}-i+\lambda_{i}-j+1-\sigma)^{2}}.\end{split}$
(78)

For each box situated at $(i,j)$ in $\lambda$, $\lambda_{i}$ is the number of
boxes at row $i$ of $\lambda$ and $\lambda^{\prime}_{j}$ is the number of
boxes at column $j$ of $\lambda$;
$h(i,j)=\lambda_{i}+\lambda^{\prime}_{j}-i-j+1$ is the hook length of the box
at $(i,j)$. The parameter $s$ can be determined in terms of the monodromy
matrices $\\{\sigma,\sigma_{23}\\}$ from the formula $(3.48a)$ in [37]:

$\displaystyle\sin^{2}\pi\sigma\cos\pi\sigma_{23}$
$\displaystyle=\cos\pi\theta_{1}\cos\pi\theta_{4}+\cos\pi\theta_{2}\cos\pi\theta_{3}$
$\displaystyle-\cos\pi\sigma(\cos\pi\theta_{1}\cos\pi\theta_{3}+\cos\pi\theta_{2}\cos\pi\theta_{4})$
$\displaystyle-\frac{1}{2}(\cos\pi\theta_{4}-\cos\pi(\theta_{3}-\sigma))(\cos\pi\theta_{1}-\cos\pi(\theta_{2}-\sigma))s$
$\displaystyle-\frac{1}{2}(\cos\pi\theta_{4}-\cos\pi(\theta_{3}+\sigma))(\cos\pi\theta_{1}-\cos\pi(\theta_{2}+\sigma))s^{-1}.$
(79)

For $t$ sufficiently close to zero111One can choose different series
expansions around the other critical points of the PVI tau function at $t=1$
and $t=\infty$ [27]., and generic monodromy parameters in the sense that

$\sigma\notin\mathbb{Z},\qquad\sigma\pm\theta_{1}\pm\theta_{2}\notin\mathbb{Z},\qquad\sigma\pm\theta_{3}\pm\theta_{4}\notin\mathbb{Z},$
(80)

we have

$\displaystyle\tau(t)=C_{0}\,$ $\displaystyle
t^{\tfrac{1}{4}(\sigma^{2}-\theta^{2}_{1}-\theta^{2}_{2})}(1-t)^{\tfrac{1}{2}\theta_{3}\theta_{2}}\biggl{\\{}1+\bigg{[}\frac{\theta_{3}\theta_{2}}{2}+\frac{(\theta^{2}_{1}-\theta^{2}_{2}-\sigma^{2})(\theta^{2}_{4}-\theta^{2}_{3}-\sigma^{2})}{8\sigma^{2}}$
$\displaystyle-\frac{(\theta^{2}_{1}-(\theta_{2}-\sigma)^{2})(\theta^{2}_{4}-(\theta_{3}-\sigma)^{2})}{16\sigma^{2}(1+\sigma)^{2}}\kappa\,t^{\sigma}-\frac{(\theta^{2}_{1}-(\theta_{2}+\sigma)^{2})(\theta^{2}_{4}-(\theta_{3}+\sigma)^{2})}{16\sigma^{2}(1-\sigma)^{2}}\frac{1}{\kappa\,t^{\sigma}}\bigg{]}t+\cdots\biggr{\\}},$
(81)

where $0<\mathrm{Re}\,\sigma<1$, $C_{0}$ is a constant independent of $t$, and
$\kappa$ is a known function of the monodromy parameters:

$\kappa=s\frac{\Gamma^{2}(1-\sigma)}{\Gamma^{2}(1+\sigma)}\frac{\Gamma(1+\tfrac{1}{2}(\theta_{2}+\theta_{1}+\sigma))\Gamma(1+\tfrac{1}{2}(\theta_{2}-\theta_{1}+\sigma))}{\Gamma(1+\tfrac{1}{2}(\theta_{2}+\theta_{1}-\sigma))\Gamma(1+\tfrac{1}{2}(\theta_{2}-\theta_{1}-\sigma))}\frac{\Gamma(1+\tfrac{1}{2}(\theta_{3}+\theta_{4}+\sigma))\Gamma(1+\tfrac{1}{2}(\theta_{3}-\theta_{4}+\sigma))}{\Gamma(1+\tfrac{1}{2}(\theta_{3}+\theta_{4}-\sigma))\Gamma(1+\tfrac{1}{2}(\theta_{3}-\theta_{4}-\sigma))}.$
(82)

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