Investigation of hidden symmetry in $AdS_{5}$ black hole with Heun equation
Jafar Sadeghi ⋆111Email<EMAIL_ADDRESS>Mohammad Ali S. Afshar ⋆222Email:
<EMAIL_ADDRESS>Mohammad Reza Alipour ⋆333Email:
<EMAIL_ADDRESS>
Saeed Noori Gashti⋆,†444Email<EMAIL_ADDRESS>
⋆Department of Physics, Faculty of Basic Sciences,
University of Mazandaran P. O. Box 47416-95447, Babolsar, Iran
†School of Physics, Damghan University, P. O. Box 3671641167, Damghan, Iran

###### Abstract

Investigating the existence of algebra and finding hidden symmetries in
physical systems is one of the most important aspects for understanding their
behavior and predicting their future. Expanding this unique method of study to
cosmic structures and combining past knowledge with new data can be very
interesting and lead to discovering new ways to analyze these systems.
However, studying black hole symmetries always presents many complications and
sometimes requires computational approximations. For example, checking the
existence of Killing vectors and then calculating them is not always an easy
task. It becomes much more difficult as the structure and geometry of the
system become more complex. In this work, we will show that if the wave
equations with a black hole background can be converted in the form of general
Heun equation, based on its structure and coefficients, the algebra of the
system can be easily studied, and computational and geometrical complications
can be omitted. For this purpose, we selected two $AdS_{5}$ black holes:
Reissner-Nordström (R-N) and Kerr, and analyzed the Klein-Gordon equation with
the background of these black holes. Based on this concept, we observed that
the radial part of the R-N black hole and both the radial and angular parts of
the Kerr black hole could be transformed into the general form of the Heun
equation. As a result, according to the algebraic structure that governs the
Heun equation and its coefficients, one can easily achieve generalized $sl(2)$
algebra.

Keywords:$AdS_{5}$ Black Holes; Klein–Gordon Particle; Generalized $sl(2)$
Algebra.

###### Contents

  1. 1 Introduction
  2. 2 Warming the mind
  3. 3 Why Heun?
  4. 4 Overview of Heun Equation
  5. 5 $RN-AdS_{5}$ black hole
  6. 6 $Kerr-AdS_{5}$ black hole
    1. 6.1 Angular section
    2. 6.2 Radial section
  7. 7 Concluding remarks
  8. 8 Appendix A: Calculations related to generator’s communication

##  1 Introduction

It must be admitted that throughout the history of physics, conservation laws
are the most basic tool of a physicist to study a dynamic and playful
universe. Undoubtedly, the first step in studying such a system is to check
for the existence of constants and stability within it. In this way, the
existence of ”symmetries” and ”periodicity” is the most important data that
these systems provide, hidden from the effects of time and without any
changes. When symmetry and periodicity are raised, the forced relationship of
system elements together (algebra) and the geometric rules of this repetition
(topology) play a more fundamental role.
Sometimes this role even becomes so powerful that it dominates the basic
cognitive tools of a physicist and, as a result, makes decisions about the
physics of the system based only on geometry, topology, and algebra,
regardless of conventional methods. The meaning of these words is more
tangible for particle physicists and cosmologists because we are not dealing
with a crystal where everything can be tested and verified.
This is a fact! All we have is a black hole in infinity, and we are so far
away that all our seemingly advanced measuring tools seem practically unusable
or useless. This is where the value of long-range measurement tools such as
algebra, which can reproduce system generators based on inductive reasoning,
geometry, and comparison of symmetric behaviors, is determined. Then, with the
help of the golden key of Noter’s theorem, the relevant constants and currents
will be easily available to us so that we can safely make decisions about the
behaviors and results of the system.
But there are two ways to use the algebraic method: first of all, like a
mathematician, regardless of whether the results are meaningful or meaningless
from a physical point of view, consider the black hole as a mathematical
entity and be satisfied only with obtaining logical results. Or be ourselves,
a physicist that prioritizes conceptualization and subjectivity and use
algebra only as a study tool. We have chosen the second approach in this
article.
Also the study of hidden symmetries in black holes is not a new concept and is
often based on analyzing the wave field equation with a black hole background
and then identifying the Killing vectors and the Killing-Yano tensor. This
method has been used in the basic work of Castro, Mallory, and Strominger for
the Kerr-Newman black hole [1], as well as in subsequent works such as [2, 3,
4, 5], which were based on Castro and Maloney’s approach for R-N black holes
and 5D black holes. It is well-known that constructing conformal coordinates
and generating the Killing vectors can be a challenging task when dealing with
complex black hole geometries.
We propose that if it is possible to convert the obtained differential
equations into general form of Heun’s equation, perhaps using this method to
identify the algebra of hidden symmetries in field-particle equations with a
black hole background is more appropriate, since it can provide a direct and
faster way to reveal the algebra and hidden symmetries of black holes, which
we will show that for the several different examples.
Prior to concluding this section, it is imperative to underscore that the
transformation of various physical systems’ differential equations into
recognized forms of Heun’s equations has elevated the significance of these
equations in propelling the exploration of physical systems [10, 11, 12, 13,
14, 15, 16, 17, 18, 19]. Various topics, such as the Schrödinger equation with
anharmonic potential, the Teukolsky linear perturbation theory for the
Schwarzschild and Kerr metrics, traversable wormholes, quantum Rabi models,
the perturbations of rotating and non-rotating black holes, the gravitational
lensing by black holes and wormholes, the quantum tunneling, the Hawking
radiation, the quasinormal modes, Klein-Gordon, Dirac, Schrödinger, Maxwell,
and Einstein equations in various space-times, etc., have motivated many
researchers in recent decades to solve, interpret and justify differential
equations arising from physical systems that turn into different forms of
Heun’s equations.[28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42,
43].
Now, based on all the explanations above, we will organize the article in the
following manner.In sections 2 and 3, we provide an explanation of algebra,
Heun’s equation, and work motivation. In section 4, we will give a general
overview of Heun equations. In section 5, we investigate the scalar field
equation in the background of the $RN-AdS_{5}$ black hole and the Heun
equation of its radial part by introducing the Klein–Gordon particle around
it. We identify its operators and the properties of displacement and consider
the generalized $sl(2)$ algebra for the $RN-AdS_{5}$ black hole. In section 6,
we will follow the same process for the $Kerr-AdS_{5}$ black hole. Finally, we
describe the results in section 7.

##  2 Warming the mind

Maloney, Castro, and Strominger [1] showed at the beginning of the last decade
that the physical structure of a Kerr-Newman black hole can be strongly
influenced by the algebra of conformal field theory. They had their reasons
for this concept: for example, the wave equation around a black hole in a
certain limit (”near region”) can be calculated as a quadratic Casimir of an
$sl(2,R)\times sl(2,R)$ algebra. They were able to calculate the entropy of
the black hole using the concept of central charge near the extreme limit
points and showed that the absorption cross section for scalars in the near-
region can be reinterpreted as a standard finite temperature CFT absorption
cross section [3]. This new path, and the checking of its validity, prompted
others to move to other black holes as R-N to study the hidden symmetries of
this black hole [2]. But the main point in all of these works was that they
all used the conventional method during this path. It means Maloney’s method!
The basis of Castro and Maloney’s work was that they considered the Klein-
Gordon wave equation with the background of the Kerr black hole. Then, they
obtained the radial and angular equations by separating the variables.
Although they pointed out that these equations are solvable according to the
Heun form, they went on to construct conformal coordinates and consider
generators for the vector field on these coordinates. Then, they showed that
these generators follow the $sl(2,R)$ algebra. In other words, they rely on
the conventional method of using Killing vectors to identify symmetry.
However, in this work, we demonstrate that it is not necessary to construct a
new vector field to obtain the hidden algebras of a black hole. Instead, it
can be achieved as soon as the wave equation appears in the form of Heun.
Depending on the type of algebraic structure of the equation and its
coefficients, we can directly proceed to generate and reconstruct the algebra,
and then verify its validity by checking the Lie bracket.

##  3 Why Heun?

Bender and Dunne [6] showed that there is a logical correspondence between
orthogonal polynomials and quasi-exactly solvable (QES) models. In other
words, they demonstrated that the wave function in quantum mechanics can act
as a generator for orthogonal polynomials expressed in terms of energy.
Typically, the algebraic approach is used to analyze QES models, where the
Hamiltonian is represented as a non-linear combination of Lie algebra
generators. Based on this concept, Turbiner claimed in [7] that most QES
models are influenced by the $sl(2,Q)$ algebra (although it was later
discovered that some QES models do not necessarily follow the $sl(2)$
algebra). He was able to show that for different models of specific potentials
of this class of Schrödinger equations, these equations can be written using
the structure of polynomials and Hamiltonian reconstruction in terms of
generators of the $sl(2)$ algebra. Therefore, it can be said that due to the
basic and structural similarity of the Schrödinger and Klein-Gordon equations,
differential equations resulting from physical systems whose solutions are
somehow the result of orthogonal polynomials may lead us to the existence of a
hidden algebra in the physical system. Heun’s equation is one such equation.
For this reason, in this article, this feature can be used as a practical,
easy, and short way to get rid of the construction of Killing vectors, which
usually result from long paths with complex computational problems. The
important point is that Castro and Maloney’s method confirms the result
obtained based on this method.

##  4 Overview of Heun Equation

In 1889, Karl M. W. L. Heun introduced a new equation [8], a linear ordinary
differential equation of second-order type ,which has four regular singular
points at zero, one, an arbitrary point between zero and one, and infinity on
the complex plane.In fact, this equation is a generalization of the
hypergeometric equation, which undoubtedly has found many applications in
various branches of science, including in physics, especially in general
relativity and astrophysics. To study about this equation and its properties,
we can refer to the book of Ronveaux [9], which is considered to be
comprehensive in this field.
Based on this, the general form of Heun’s equation will be as follows

$\begin{split}y^{\prime\prime}(x)+\bigg{[}\frac{\eta}{x}+\frac{\delta}{x-1}+\Sigma^{k}_{i=1}\frac{\epsilon_{i}}{x-a_{i}}\bigg{]}y^{\prime}(x)+\frac{\mu\nu
x^{k}+\Sigma^{k}_{i=1}p_{i}x^{i-1}}{x(x-1)\Pi^{k}_{i=1}(x-a_{i})}y(x)=0,\end{split}$
(1)

where each singularity $a_{i}$ has the characteristic
$(0,1-\epsilon_{i}),(0,1-\eta),(0,1-\delta)$ as well as $(\mu,\nu)$ at
$x=0,x=1$ and $x=\infty$. Therefore, considering the Fuchsian relation, we
will have,

$\begin{split}\alpha+\beta+1=\eta+\delta+\Sigma^{k}_{i=1}\epsilon_{i}.\end{split}$
(2)

So, the Heun equation (1) in the case of $k=1$ is as follows,

$\begin{split}y^{\prime\prime}(x)+\bigg{[}\frac{\eta}{x}+\frac{\delta}{x-1}+\frac{\epsilon_{1}}{x-a}\bigg{]}y^{\prime}(x)+\frac{\alpha\beta
x+p_{1}}{x(x-1)(x-a)}y(x)=0,\end{split}$ (3)

furthermore, we can rewrite Heun’s differential equation (3) in a general form
by using the functions $g_{1}(x)$, $g_{2}(x)$, and $g_{3}(x)$,

$g_{1}(x)y^{\prime\prime}(x)+g_{2}(x)y^{\prime}(x)+g_{3}(x)y(x)=0,$ (4)

where

$\begin{split}&g_{1}(x)=a_{0}x^{3}+a_{1}x^{2}+a_{2}x+a_{3},\\\
&g_{2}(x)=a_{4}x^{2}+a_{5}x+a_{6},\\\ &g_{3}(x)=a_{7}x+a_{8}.\end{split}$ (5)

With $a_{i(i=0,1,...,8)}\in{R}$, we can express the coefficients of $a_{i}$ by
comparing equations (3) and (4) in terms of the parameters of Heun’s equation.
Additionally, as it shown in [20, 21, 22], we can rewrite the Heun equation
using equations (5) and (4) in terms of operators $P^{0}$, $P^{-}$, $P^{+}$,
and $F(x\frac{d}{dx})$, such that they follow the generalized $sl(2)$ algebra
and the following commutation relation holds for them,

$\begin{split}[P^{0},P^{\pm}]=\pm\alpha
P^{\pm}\qquad[P^{+},P^{-}]=FP^{0},\end{split}$ (6)

where

$\begin{split}&P^{0}=\alpha x\frac{d}{dx}+\beta\\\
&P^{+}=a_{0}x^{3}\frac{d^{2}}{dx^{2}}+a_{4}x^{2}\frac{d}{dx}+a_{7}x\\\
&P^{-}=a_{2}x\frac{d^{2}}{dx^{2}}+a_{6}\frac{d}{dx}\\\
&F(x\frac{d}{dx})=a_{1}x^{2}\frac{d^{2}}{dx^{2}}+a_{5}x\frac{d}{dx}+a_{8}.\\\
\end{split}$ (7)

Also note that we have the following conditions to establish equations (6) and
(7),

$\begin{split}&\alpha=-\frac{4a_{0}a_{2}}{a_{1}},\hskip
56.9055pt\beta=-\frac{a_{6}a_{7}}{a_{8}}\\\ &2\alpha a_{1}+\alpha
a_{5}+a_{1}\beta=-\left(6a_{0}a_{2}+3a_{4}a_{2}+3a_{0}a_{6}\right)\\\ &\alpha
a_{5}+\alpha
a_{8}+a_{5}\beta=-\left(2a_{2}a_{4}+2a_{6}a_{4}+2a_{2}a_{7}\right).\\\
\end{split}$ (8)

The calculations related to equation (8) are in Appendix A. Note that the
$P^{0}$ form is significant because it exhibits the symmetry algebra of the
mode. By considering the $P^{0}$ form in relation (7), we can establish the
generalized $sl(2)$ algebra. Additionally, $\alpha$ and $\beta$ are special
parameters that establish the relation (6). In the following, we will discuss
the existence of scalar field symmetry in the background of $RN-AdS_{5}$ and
$Kerr-AdS_{5}$ black holes.

##  5 $RN-AdS_{5}$ black hole

First, we introduce the $RN-AdS_{5}$ black hole. The line element of the $RN-
AdS_{5}$ black hole is as follows[23],

$\begin{split}ds^{2}=-f(r)dt+\frac{1}{f(r)}dr^{2}+r^{2}d\Omega_{3}^{2},\end{split}$
(9)

where $d\Omega_{3}^{2}$ represents the metric of the unit three-sphere.
Additionally, the function $f(r)$ is defined as follows,

$\begin{split}f(r)=1-\frac{M}{r^{2}}+\frac{Q^{2}}{r^{4}}+r^{2}=\frac{\Delta_{r}}{r^{4}}=\frac{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})(r^{2}-r_{0}^{2})}{r^{4}},\end{split}$
(10)

where $M$ and $Q$ are the black hole’s mass and charge, respectively, and the
AdS radius is $L=1$. There is a correspondence between the roots $\Delta_{r}$
and the black hole’s Killing horizons, whose radial positions will be
parameterized by the largest real root of $r_{+}$, corresponding to the outer
horizon. The roots are as follows,

$\begin{split}&r_{-}^{2}=\frac{1}{2}\bigg{[}-1-r_{+}^{2}+\sqrt{(1+r_{+}^{2})^{2}+\frac{4Q^{2}}{r_{+}^{2}}}\bigg{]},\\\
&r_{0}^{2}=\frac{1}{2}\bigg{[}-1-r_{+}^{2}-\sqrt{(1+r_{+}^{2})^{2}+\frac{4Q^{2}}{r_{+}^{2}}}\bigg{]}.\end{split}$
(11)

One can use the Klein-Gordon equation in curved space to derive the field
equation near the black hole [23],

$\begin{split}\frac{1}{\sqrt{-g}}D_{\mu}(\sqrt{-g}g^{\mu\nu}D_{\nu})\Psi-m^{2}\Psi=0,\end{split}$
(12)

where $D_{\mu}=\nabla_{\mu}-iqA_{\mu}$, $A_{\mu}=(-\sqrt{3}Q/2r^{2},0,0,0,0)$,
and $m$ and $q$ are the electromagnetic potential, mass, and charge of the
scalar field, respectively. Also, we can relate the mass of the scalar field
in equation (12) to the CFT dimension, such that $m^{2}=\Delta(\Delta-4)$. In
[23], the differential equation for the radial part of equation (12) for the
scalar field near the $RN-AdS_{5}$ black hole was obtained by defining
$z=\frac{r^{2}-r_{-}^{2}}{r^{2}-r_{0}^{2}}$,

$\begin{split}F^{\prime\prime}(z)+\left[\frac{1-\theta_{-}}{z}+\frac{1-\theta_{+}}{z-z_{0}}+\frac{\Delta-1}{z-1}\right]F^{\prime}(z)+\left[\frac{k_{1}k_{2}}{z(z-1)}-\frac{z_{0}(z_{0}-1)K_{0}}{z(z-z_{0})(z-1)}\right]F(z)=0,\end{split}$
(13)

where

$\begin{split}&k_{1}=\frac{1}{2}(\theta_{+}+\theta_{-}-\theta_{0}-\Delta)\\\
&k_{2}=\frac{1}{2}(\theta_{+}+\theta_{-}+\theta_{0}-\Delta)\\\
&z_{0}=\frac{r_{+}^{2}-r_{-}^{2}}{r_{+}^{2}-r_{0}^{2}}\\\
&K_{0}=-\frac{[(\theta_{+}+\theta_{-}+1)^{2}-1-\theta_{0}^{2}]}{4z_{0}}-\frac{[(2-\Delta)^{2}+2(\theta_{+}-1)(1-\Delta)-2]}{4(z_{0}-1)}\\\
&-\frac{\ell(\ell+2)+\Delta(\Delta-4)r_{-}^{2}-\omega^{2}}{4z_{0}(z_{0}-1)(r_{+}^{2}-r_{0}^{2})},\end{split}$
(14)

where $\ell$ and $\omega$ are the angular momentum quantum number and the
quasinormal frequency, respectively. Also, parameters
$\\{\theta_{n},n=+,-,0\\}$ are $\theta_{n}=\frac{i}{2\pi
T_{n}}\left(\omega-\sqrt{3}qQ/2r_{n}^{2}\right)$. We rewrite equation (13) as
equation (4),

$\begin{split}&(z^{3}-z^{2}(z_{0}+1)+zz_{0})F^{\prime\prime}(z)\\\
&+[(1-\theta_{-}-\theta_{+}+\Delta)z^{2}+(\theta_{+}+\theta_{-}+z_{0}\theta_{-}-z_{0}\Delta-2)z+z_{0}(1-\theta_{-})]F^{\prime}(z)\\\
&+[k_{1}k_{2}z-k_{1}k_{2}z_{0}-z_{0}k_{0}(z_{0}-1)]F(z)=0.\end{split}$ (15)

Now, by comparing relations (15), (5), (4), and (3), we can obtain values for
$a_{i}$, $\alpha$, and $\beta$,

$\begin{split}&a_{0}=1,\hskip 14.22636pta_{1}=-(1+z_{0}),\hskip
14.22636pta_{2}=z_{0},\hskip 14.22636pta_{3}=0\\\
&a_{4}=1+\Delta-\theta_{+}-\theta_{-},\hskip
14.22636pta_{5}=-2+\theta_{+}-z_{0}\Delta+\theta_{-}+z_{0}\theta_{-},\hskip
14.22636pta_{6}=z_{0}(1-\theta_{-})\\\ &a_{7}=k_{1}k_{2},\hskip
14.22636pta_{8}=-z_{0}[k_{1}k_{2}+(z_{0}-1)K_{0}],\hskip
14.22636pt\alpha=-k_{1},\hskip 14.22636pt\beta=-k_{2}.\end{split}$ (16)

Using relations (16) and (7), we obtain the operators $P^{+}$, $P^{-}$,
$P^{0}$, and $F(z\frac{d}{dz})$,

$\begin{split}&P^{0}=-k_{1}z\frac{d}{dz}-k_{2}\\\
&P^{+}=z^{3}\frac{d^{2}}{dz^{2}}+(1+\Delta-\theta_{+}-\theta_{-})z^{2}\frac{d}{dz}+k_{1}k_{2}z\\\
&P^{-}=z_{0}x\frac{d^{2}}{dz^{2}}+z_{0}(1-\theta_{-})\frac{d}{dz}\\\
&F(z\frac{d}{dz})=-(1+z_{0})z^{2}\frac{d^{2}}{dz^{2}}+(-2+\theta_{+}-z_{0}\Delta+\theta_{-}+z_{0}\theta_{-})z\frac{d}{dz}+-z_{0}[k_{1}k_{2}+(z_{0}-1)K_{0}].\end{split}$
(17)

According to equation (8), we obtain the conditions for establishing algebra
for equation (17) as follows,

$\begin{split}&\alpha=\frac{4z_{0}}{z_{0}+1},\hskip
56.9055pt\beta=\frac{\left(1-\theta_{-}\right)k_{1}k_{2}}{k_{1}k_{2}+K_{0}\left(z_{0}-1\right)}\\\
&\alpha\theta_{+}-4\alpha-\beta-
z_{0}\left(\alpha(\Delta+2)+\beta-3(\Delta+4)+3\theta_{+}\right)+\theta_{-}\left(\alpha+(\alpha-6)z_{0}\right)=0\\\
&\left(\theta_{-}+\theta_{+}-2\right)(\alpha+\beta)-\alpha K_{0}z_{0}^{2}\\\
&+z_{0}\left(\theta_{-}\left(\alpha+\beta-2\Delta+2\theta_{+}-6\right)-\Delta(\alpha+\beta-4)+2\theta_{-}^{2}-4\theta_{+}-(\alpha-2)k_{1}k_{2}+\alpha
K_{0}+4\right)=0.\\\ \end{split}$ (18)

Since the operators mentioned above satisfy the relation (6), the radial part
of the scalar field near the $RN-AdS_{5}$ black hole can exhibit generalized
$sl(2)$ symmetry. Furthermore, the quadratic Casimir operator $C$, which is
associated with all current operators of the model, can be obtained using the
formula $C=\frac{1}{2}(P^{-}P^{+}+P^{+}P^{-})+P^{0}P^{0}$.

##  6 $Kerr-AdS_{5}$ black hole

The metric of the $Kerr-AdS_{5}$ spacetime is given by [24, 25, 26],

$\begin{split}&ds^{2}=-\frac{\Delta}{\rho^{2}}\bigg{(}dt-\frac{a_{1}\sin^{2}\theta}{\Xi_{1}}d\phi_{1}-\frac{a_{2}\cos^{2}\theta}{\Xi_{2}}d\phi_{2}\bigg{)}^{2}+\frac{\rho^{2}}{\Delta}dr^{2}\\\
&+\frac{\rho^{2}}{\Delta_{\theta}}d\theta^{2}+\frac{\Delta_{\theta}\sin^{2}\theta}{\rho^{2}}\bigg{(}a_{1}dt+\frac{r^{2}+a_{1}^{2}}{\Xi_{1}^{2}}d\phi_{1}\bigg{)}^{2}+\frac{\Delta_{\theta}\cos^{2}\theta}{\rho^{2}}\bigg{(}a_{2}dt+\frac{r^{2}+a_{2}^{2}}{\Xi_{2}^{2}}d\phi_{2}\bigg{)}^{2}\\\
&+\frac{1+r^{2}/L^{2}}{r^{2}\rho^{2}}\bigg{(}a_{1}a_{2}dt-\frac{a_{2}(r^{2}+a_{1}^{2})\sin^{2}\theta}{\Xi_{1}}d\phi_{1}-\frac{a_{1}(r^{2}+a_{2}^{2})\cos^{2}\theta}{\Xi_{2}}d\phi_{2}\bigg{)}^{2},\end{split}$
(19)

where,

$\begin{split}&\Delta:=\frac{1}{r^{2}}(r^{2}+a_{1}^{2})(r^{2}+a_{2}^{2})(1+r^{2}/L^{2})-2M=\frac{1}{r^{2}L^{2}}(r^{2}-r_{0}^{2})(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2}),\\\
&\Delta_{\theta}:=1-\frac{a_{1}^{2}}{L^{2}}\cos^{2}\theta-\frac{a_{2}^{2}}{L^{2}}\sin^{2}\theta,\\\
&\rho^{2}:=r^{2}+a_{1}^{2}\cos^{2}\theta+a_{2}^{2}\sin^{2}\theta,\\\
&\Xi_{1}:=1-\frac{a_{1}^{2}}{L^{2}},\\\
&\Xi_{2}:=1-\frac{a_{2}^{2}}{L^{2}},\end{split}$ (20)

where $0\leq\phi_{1},\phi_{2}\leq 2\pi,0\leq\theta\leq\pi/2$. The rotational
parameters a and b are constrained such that $a^{2},b^{2}\leq\ell^{-2}$. Also,
$M$ is the mass parameter, $a_{1}$ and $a_{2}$ are spin parameters, and
$r_{k}$ (where $k=+,-,0$) describe the horizon radii. We assume that they are
distinct and that $r_{0}^{2}<r_{-}^{2}<r_{+}^{2}$. Based on the above
explanations, $r_{0}$ is purely imaginary, while $r_{-}$ and $r_{+}$ are real
and satisfy $0<r_{-}<r_{+}$. Therefore, $r_{-}$ represents the radius of the
inner horizon, and $r_{+}$ represents the radius of the outer horizon. As
shown in [27] we will have,

$\begin{split}&r_{0}^{2}+r_{-}^{2}+r_{+}^{2}+a_{1}^{2}+a_{2}^{2}+L^{2}=0,\\\
&r_{0}^{2}r_{-}^{2}+r_{0}^{2}r_{+}^{2}+r_{-}^{2}r_{+}^{2}-a_{1}^{2}a_{2}^{2}+\big{(}2M-a_{1}^{2}-a_{2}^{2}\big{)}L^{2}=0,\\\
&r_{0}^{2}r_{-}^{2}r_{+}^{2}+a_{1}^{2}a_{2}^{2}L^{2}=0.\end{split}$ (21)

The angular velocities and Hawking temperatures at each event horizon, denoted
by $r_{n}$ where $n=+,-,0$, are obtained according to [27],

$\begin{split}\Omega_{n,1}\equiv\frac{a_{1}\Xi_{1}}{r_{n}^{2}+a_{1}^{2}},\hskip
8.5359pt\Omega_{n,2}\equiv\frac{a_{2}\Xi_{1}}{r_{n}^{2}+a_{2}^{2}},\hskip
8.5359ptT_{n}=\frac{r_{n}(r_{n}^{2}-r_{i}^{2})(r_{n}^{2}-r_{j}^{2})}{2\pi
L^{2}(r_{n}^{2}+a_{1}^{2})(r_{n}^{2}+a_{2}^{2})},\hskip 8.5359pti,j\neq
n.\end{split}$ (22)

We solve the Klein-Gordon equation (12) in the background of the $Kerr-
AdS_{5}$ black hole. Also, considering $\Psi=\psi(r)\psi(\theta)e^{-i\omega
t+im_{1}\phi_{1}+im_{2}\phi_{2}}$ (where $m_{1}$ and $m_{2}$ are integers) and
using the separation of variables [27], we convert the differential equation
into two radial and angular parts.

###  6.1 Angular section

In the angular section, when $|a_{1}|\neq|a_{2}|$, and by changing the
variables, as shown in [27] we will have,

$\begin{split}&x=\sin^{2}\theta,\hskip
14.22636pt\psi(\theta)=(x-1)^{m_{2}/2}x^{m_{1}/2}(x-x_{0})^{\tau/2}S(x)\\\
&x_{0}=\frac{\Xi_{1}}{\Xi_{1}-\Xi_{2}},\hskip
14.22636pt\tau=\frac{m_{1}a_{1}}{L}+\frac{m_{2}a_{2}}{L}+\omega L.\end{split}$
(23)

Then, as shown in [27] for the Heun equation we will have,

$\begin{split}\frac{d^{2}S(x)}{dx^{2}}+\bigg{[}\frac{\gamma_{a}}{x}+\frac{\delta_{a}}{x-1}+\frac{\epsilon_{a}}{x-x_{0}}+\bigg{]}\frac{dS(x)}{dx}+\frac{\alpha_{a}\beta_{a}x+p_{a}}{x(x-1)(x-x_{0})}S(x)=0,\end{split}$
(24)

where

$\begin{split}&p_{a}=-\frac{1}{2}\left[(m_{1}+m_{2}+m_{1}m_{2})x_{0}+(m_{1}+1)(\tau+m_{1})\right]\\\
&-\frac{x_{0}}{4\Xi_{1}}\left[\omega^{2}L^{2}+m^{2}a_{1}^{2}-\lambda+\Xi_{2}(m_{2}^{2}+m_{1}^{2}-\tau^{2})\right],\\\
&\alpha_{a}=\frac{1}{2}(\tau+m_{1}+m_{2}+\sqrt{m^{2}L^{2}+4}+2),\hskip
8.5359pt\beta_{a}=\frac{1}{2}(\tau+m_{1}+m_{2}-\sqrt{m^{2}L^{2}+4}+2),\\\
&\epsilon_{a}=1+\tau,\hskip 11.38092pt\delta_{a}=1+m_{2},\hskip
11.38092pt\gamma_{a}=1+m_{1}.\end{split}$ (25)

Now, we can rewrite equation (24) as follows,

$\begin{split}&(x^{3}-x^{2}(x_{0}+1)+xx_{0})S^{\prime\prime}(x)\\\
&+[(\gamma_{a}+\delta_{a}+\epsilon_{a})x^{2}-(\gamma_{a}+x_{0}\gamma_{a}+x_{0}\delta_{a}+\epsilon_{a})x+x_{0}\gamma_{a}]S^{\prime}(x)\\\
&+[\alpha_{a}\beta_{a}+p_{a}]S(x)=0.\end{split}$ (26)

Therefore, by comparing relations (24), (5), (4), and (3), we obtain values
for $\alpha$, $\beta$, and $a_{i}$.

$\begin{split}&a_{0}=1,\hskip 14.22636pta_{1}=-(1+x_{0}),\hskip
14.22636pta_{2}=x_{0},\hskip 14.22636pta_{3}=0\\\
&a_{4}=\gamma_{a}+\delta_{a}+\epsilon_{a},\hskip
14.22636pta_{5}=-(\gamma_{a}+x_{0}\gamma_{a}+x_{0}\delta_{a}+\epsilon_{a}),\hskip
14.22636pta_{6}=\gamma_{a}x_{0}\\\ &a_{7}=\alpha_{a}\beta_{a},\hskip
14.22636pta_{8}=p_{a},\hskip 14.22636pt\alpha=\alpha_{a},\hskip
14.22636pt\beta=\beta_{a}.\end{split}$ (27)

Also, by using relations (27) and (7), we obtain the operators $P^{+}$,
$P^{-}$, $P^{0}$, and $F(x\frac{d}{dx})$

$\begin{split}&P^{0}=\alpha_{a}x\frac{d}{dx}+\beta_{a}\\\
&P^{+}=x^{3}\frac{d^{2}}{dx^{2}}+(\gamma_{a}+\delta_{a}+\epsilon_{a})x^{2}\frac{d}{dx}+\alpha_{a}\beta_{a}x\\\
&P^{-}=x_{0}x\frac{d^{2}}{dx^{2}}+\gamma_{a}x_{0}\frac{d}{dx}\\\
&F(x\frac{d}{dx})=-(1+x_{0})x^{2}\frac{d^{2}}{dx^{2}}-(\gamma_{a}+x_{0}\gamma_{a}+x_{0}\delta_{a}+\epsilon_{a})x\frac{d}{dx}+p_{a}.\end{split}$
(28)

So, we will have according to equation (8),

$\begin{split}&\alpha=\frac{4x_{0}}{x_{0}+1},\hskip
56.9055pt\beta=-\frac{x_{0}\alpha_{a}\beta_{a}\gamma_{a}}{p_{a}}\\\
&-\alpha\gamma_{a}-\alpha\epsilon_{a}-x_{0}\left((\alpha-6)\gamma_{a}+(\alpha-3)\delta_{a}-3\epsilon_{a}+2\alpha+\beta-6\right)-2\alpha-\beta=0\\\
&\alpha p_{a}-(\alpha+\beta)\left(\gamma_{a}+\epsilon_{a}\right)\\\
&+x_{0}\left(-\gamma_{a}\left(-2\delta_{a}-2\epsilon_{a}+\alpha+\beta-2\right)+2\alpha_{a}\beta_{a}-\alpha\delta_{a}-\beta\delta_{a}+2\gamma_{a}^{2}+2\delta_{a}+2\epsilon_{a}\right)=0\\\
.\end{split}$ (29)

Since the above operators obey relation (6), we can say that there is
generalized $sl(2)$ symmetry for the angular part of the scalar field near the
$ker-AdS_{5}$ black hole.

###  6.2 Radial section

In the radial section, the following transformation is applied [27],

$\begin{split}&z=\frac{r^{2}-r_{+}^{2}}{r^{2}-r_{0}^{2}},\hskip
14.22636pt\psi(r)=(z-1)^{\sigma/2}(z-z_{0})^{\theta_{-}/2}z^{\theta_{+}/2}R(z)\\\
&z_{0}=\frac{r_{-}^{2}-r_{+}^{2}}{r_{-}^{2}-r_{0}^{2}},\hskip
14.22636pt\theta_{n}=\frac{i}{2\pi T_{n}}(\omega-
m_{1}\Omega_{k,1}-m_{2}\Omega_{k,2}),\hskip
14.22636pt\sigma=2+\sqrt{4+m^{2}L^{2}},\end{split}$ (30)

using the above transformations, the radial part of equation will become into
the form of Heun equation [27],

$\begin{split}\frac{d^{2}R(z)}{dz^{2}}+\bigg{[}\frac{\gamma_{r}}{z}+\frac{\delta_{r}}{z-1}+\frac{\epsilon_{r}}{z-z_{0}}+\bigg{]}\frac{dR(z)}{dz}+\frac{\alpha_{r}\beta_{r}x+p_{r}}{z(z-1)(z-z_{0})}R(z)=0,\end{split}$
(31)

where

$\begin{split}&p_{r}=\frac{1}{4}\left[\frac{(\lambda+m^{2}r_{+}^{2}-\omega^{2}L^{2})L^{2}}{r_{-}^{2}-r_{0}^{2}}+2z_{0}(\sigma+\sigma\theta_{+}-\theta_{+})+(\theta_{-}+\theta_{+}+2)(\theta_{-}+\theta_{+})-\theta_{0}^{2}\right]\\\
&\alpha_{r}=\frac{1}{2}(\sigma+\theta_{0}+\theta_{-}+\theta_{+}),\hskip
8.5359pt\beta_{r}=\frac{1}{2}(\sigma-\theta_{0}+\theta_{-}+\theta_{+}),\\\
&\epsilon_{r}=\theta_{-}+1,\hskip 11.38092pt\delta_{r}=\sigma-1,\hskip
11.38092pt\gamma_{r}=\theta_{+}+1.\end{split}$ (32)

Note that equation (2) holds. We rewrite equation (31) as equation (2)

$\begin{split}&(z^{3}-z^{2}(z_{0}+1)+zz_{0})R^{\prime\prime}(z)\\\
&+[(\gamma_{r}+\delta_{r}+\epsilon_{r})z^{2}-(\gamma_{r}+z_{0}\gamma_{r}+z_{0}\delta_{r}+\epsilon_{r})z+z_{0}\gamma_{r}]R^{\prime}(z)\\\
&+[\alpha_{r}\beta_{r}+p_{r}]R(r)=0.\end{split}$ (33)

Now, by comparing relations (33), (5), (4) and (3) we get

$\begin{split}&a_{0}=1,\hskip 14.22636pta_{1}=-(1+z_{0}),\hskip
14.22636pta_{2}=z_{0},\hskip 14.22636pta_{3}=0\\\
&a_{4}=\gamma_{r}+\delta_{r}+\epsilon_{r},\hskip
14.22636pta_{5}=-(\gamma_{r}+z_{0}\gamma_{r}+z_{0}\delta_{r}+\epsilon_{r}),\hskip
14.22636pta_{6}=\gamma_{r}z_{0}\\\ &a_{7}=\alpha_{r}\beta_{r},\hskip
14.22636pta_{8}=p_{r},\hskip 14.22636pt\alpha=\alpha_{r},\hskip
14.22636pt\beta=\beta_{r}.\end{split}$ (34)

Therefore, using relations (34) and (7), we obtain the following operators,

$\begin{split}&P^{0}=\alpha_{r}z\frac{d}{dz}+\beta_{r}\\\
&P^{+}=z^{3}\frac{d^{2}}{dz^{2}}+(\gamma_{r}+\delta_{r}+\epsilon_{r})z^{2}\frac{d}{dz}+\alpha_{r}\beta_{r}z\\\
&P^{-}=z_{0}z\frac{d^{2}}{dz^{2}}+\gamma_{r}z_{0}\frac{d}{dz}\\\
&F(z\frac{d}{dz})=-(1+z_{0})z^{2}\frac{d^{2}}{dz^{2}}-(\gamma_{r}+z_{0}\gamma_{r}+z_{0}\delta_{r}+\epsilon_{r})z\frac{d}{dz}+p_{r}.\end{split}$
(35)

Here, we can obtain,

$\begin{split}&\alpha=\frac{4z_{0}}{z_{0}+1},\hskip
56.9055pt\beta=-\frac{z_{0}\alpha_{r}\beta_{r}\gamma_{r}}{p_{r}}\\\
&-2\alpha-\beta-\gamma_{r}\alpha-\alpha\epsilon_{r}-z_{0}\left(2\alpha+\beta+(\alpha-6)\gamma_{r}+(\alpha-3)\delta_{r}-3\epsilon_{r}-6\right)=0\\\
&\alpha p_{r}-(\alpha+\beta)\left(\gamma_{r}+\epsilon_{r}\right)\\\
&+z_{0}\left(-\gamma_{r}\left(\alpha+\beta-2\delta_{r}-2\epsilon_{r}-2\right)+2\alpha_{r}\beta_{r}-\alpha\delta_{r}-\beta\delta_{r}+2\gamma_{r}^{2}+2\delta_{r}+2\epsilon_{r}\right)=0.\\\
\end{split}$ (36)

Note that the operators mentioned above are valid with respect to relation
(6). Therefore, we can conclude that the radial part of the scalar field near
the Kerr black hole follows the generalized $sl(2)$ algebra. Our research
reveals that the scalar field near the $RN-AdS_{5}$ black hole only adheres to
the generalized $sl(2)$ algebra in the radial part, whereas near the $Kerr-
AdS_{5}$ black hole, it follows the generalized $sl(2)$ algebra in both the
radial and angular parts.

##  7 Concluding remarks

In the investigation of hidden symmetries in black holes, conventional methods
are either based on a purely algebraic and geometric approach or on the method
of finding symmetries based on the Killing vector, which has a more dominant
physical approach [1, 2, 3, 4, 5]. However, each of these methods has its
complexities. As the metric function and geometric structure of the system
become more complex, these methods may lose their effectiveness due to the
complexity or difficulty of the problem. By studying the method of Castro and
Maloney in the search for hidden symmetries of Kerr’s black hole and other
works based on this method, as well as articles that did not seek to find
symmetry in cosmic structures, we realized that Klein Gordon’s equations with
a black hole background often become differential equations that can be solved
with orthogonal polynomials. We also know that studies of Schrödinger
equations in quasi-exact solvable models have shown a logical relationship
between orthogonal polynomials and the method of solving these models. The
similarity between the Klein-Gordon and Schrödinger equations on one hand, and
the properties of orthogonal polynomials on the other, led us to consider
whether the differential equation resulting from the system, which has been
transformed into a special and well-known form (Heun’s equation) according to
the orthogonal polynomials, can be a good candidate for directly finding the
hidden algebra of the system. To this end, we selected two black holes in
$AdS_{5}$, namely R-N and Kerr, and analyzed the Klein-Gordon equation with
the background of these black holes. As we have seen that the Klein-Gordon
differential equations in the background of Reissner-Nordström and Kerr black
holes can be transformed into the general form of Heun’s equation by using a
suitable change of variable, both in the radial and angular parts. As a
result, we can easily achieve the algebraic structure based on the algebraic
structure that governs the Heun equation and its coefficients. We selected the
general form of Heun and demonstrated that the same results can be obtained
more simply and efficiently based on the structure of Heun’s coefficients,
which is a shorter method than the alternative. Our work is accurate as it
matches the previous works precisely. Another interesting point is that in the
work of Castro and Maloney, they were forced to impose restrictions on the
omega function to remove powers higher than 2 and divide the space into two
parts - near and far [1]. Unlike any other application of this approximation
for the Castro and Maloney method, it seems that no preconditions need to be
applied on the way to the algebra in our method. This method, in addition to
being more concise and practical, is also more general and comprehensive
across the entire space.
Finally, it may be worthwhile to ask what the purpose and benefit of finding
hidden symmetries, such as sl(2), for the studied systems are and what new
perspectives they offer for further research? To answer this question briefly,
we can say that: The sl(2) symmetry implies that the wave equations with a
black hole background can be related to a two-dimensional conformal field
theory (CFT) via the AdS/CFT correspondence. This means that the black hole
physics can be described by a dual CFT that lives on the boundary of the AdS
space. The sl(2) generators act as the Virasoro operators of the CFT and the
Casimir operator corresponds to the central charge. The sl(2) symmetry also
enables us to find the eigenvalues and eigenfunctions of the Heun equation
using the representation theory of the algebra [44, 45]. However, In this
article, our goal is not to explore these properties, but to demonstrate that
we can use the wave equations with a black hole background to reveal the
hidden symmetries in a system, without resorting to the common methods that
may sometimes be complicated.
In addition to all the mentioned concepts,we also suggest some questions that
can serve as motivations for further research.
Could renowned variants of Heun’s equation, such as the Trigonometric,
Jacobi’s Elliptic,Confluent, or Weierstrass’s Form [46], be employed by
reconfiguring the coefficients to unveil concealed algebras and symmetries
inherent in the system?
Finally, this fundamental and historical question: if it is possible to solve
the differential equations of a physical system using orthogonal polynomials,
is it possible to point out the existence of a hidden algebra in the system
based on the algebraic structure of these polynomials?

##  8 Appendix A: Calculations related to generator’s communication

In this section, we want to calculate the equation (7), for a general function
$\psi(x)$, to obtain the necessary conditions for establishing this
relationship[20, 21, 22].

$\begin{split}[P^{+},P^{-}]\psi(x)=FP^{0}\psi(x).\end{split}$ (37)

First from the left side we will have

$\begin{split}&p^{+}p^{-}\psi(x)=a_{7}a_{2}x^{2}\psi^{\prime\prime}(x)+a_{7}a_{6}x\psi^{\prime}(x)\\\
&+a_{4}a_{2}x^{3}\psi^{(3)}(x)+a_{4}a_{2}x^{2}\psi^{\prime\prime}(x)+a_{4}a_{6}x^{2}\psi^{\prime\prime}(x)\\\
&+a_{0}a_{2}x^{4}\psi^{(4)}(x)+2a_{0}a_{2}x^{3}\psi^{(3)}(x)+a_{0}a_{6}x^{3}\psi^{(3)}(x),\\\
\end{split}$ (38)

and

$\begin{split}&p^{-}p^{+}\psi(x)=2a_{2}a_{7}x\psi^{\prime}(x)+a_{6}a_{7}x\psi^{\prime}(x)+2a_{2}a_{4}x\psi^{\prime}(x)+2a_{4}a_{6}x\psi^{\prime}(x)+a_{6}a_{7}\psi(x)\\\
&+6a_{0}a_{2}x^{2}\psi^{\prime\prime}(x)+4a_{2}a_{4}x^{2}\psi^{\prime\prime}(x)+3a_{0}a_{6}x^{2}\psi^{\prime\prime}(x)+a_{4}a_{6}x^{2}\psi^{\prime\prime}(x)+a_{2}a_{7}x^{2}\psi^{\prime\prime}(x)\\\
&+a_{0}a_{2}x^{4}\psi^{(4)}(x)+6a_{0}a_{2}x^{3}\psi^{(3)}(x)+a_{2}a_{4}x^{3}\psi^{(3)}(x)+a_{0}a_{6}x^{3}\psi^{(3)}(x).\\\
\end{split}$ (39)

Using the equations (38) and (39), we have

$\begin{split}&(p^{+}p^{-}-p^{-}p^{+})\psi(x)=(-4a_{0}a_{2}x^{3}\psi^{(3)}(x)-\left(6a_{0}a_{2}+3a_{2}a_{4}+3a_{0}a_{6}\right)x^{2}\psi^{\prime\prime}(x)\\\
&-\left(2a_{2}a_{4}+2a_{6}a_{4}+2a_{2}a_{7}\right)x\psi^{\prime}(x)-a_{6}a_{7}\psi(x).)\end{split}$
(40)

Now, we rewrite the relation (40) as below

$\begin{split}&(p^{+}p^{-}-p^{-}p^{+})\psi(x)=\big{(}-4a_{0}a_{2}x^{3}\frac{d^{3}}{dx^{3}}\\\
&-\left(6a_{0}a_{2}+3a_{2}a_{4}+3a_{0}a_{6}\right)x^{2}\frac{d^{2}}{dx^{2}}\\\
&-\left(2a_{2}a_{4}+2a_{6}a_{4}+2a_{2}a_{7}\right)x\frac{d}{dx}\\\
&-a_{6}a_{7}\big{)}\psi(x).\\\ \end{split}$ (41)

From the righthand side of equation (37) we will also have

$\begin{split}&Fp^{0}\psi(x)=\big{(}\alpha a_{1}x^{3}\frac{d^{3}}{dx^{3}}\\\
&+\left(2\alpha a_{1}+\alpha
a_{5}+a_{1}\beta\right)x^{2}\frac{d^{2}}{dx^{2}}\\\ &+\left(\alpha
a_{5}+\alpha a_{8}+a_{5}\beta\right)x\frac{d}{dx}\\\
&+a_{8}\beta\big{)}\psi(x),\\\ \end{split}$ (42)

with the comparison of equations(41) and (42) it will be concluded that

$\begin{split}&\alpha=-\frac{4a_{0}a_{2}}{a_{1}},\hskip
56.9055pt\beta=-\frac{a_{6}a_{7}}{a_{8}}\\\ &2\alpha a_{1}+\alpha
a_{5}+a_{1}\beta=-\left(6a_{0}a_{2}+3a_{4}a_{2}+3a_{0}a_{6}\right)\\\ &\alpha
a_{5}+\alpha
a_{8}+a_{5}\beta=-\left(2a_{2}a_{4}+2a_{6}a_{4}+2a_{2}a_{7}\right).\\\
\end{split}$ (43)

Now, by using the above relations , the necessary conditions for establishing
the equation (7) will be obtained.

## Acknowledgments

The authors would like to express their appreciation to Professor Adel Rezaei-
Aghdam for his valuable feedback and advice, which has improved the quality
and academic accuracy of this manuscript.

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