# Rotating black holes and exotic compact objects in the Kerr/CFT
correspondence within Rastall gravity

Muhammad F. A. R. Sakti  High Energy Physics Theory Group, Department of
Physics, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
Department of Physics, Universitas Indonesia, Depok, Indonesia 16424
Theoretical Physics Laboratory, THEPi Division, Institut Teknologi Bandung,
Jl. Ganesha 10 Bandung, Indonesia 40132  Indonesia Center for Theoretical and
Mathematical Physics (ICTMP), Institut Teknologi Bandung, Jl. Ganesha 10
Bandung, Indonesia 40132<EMAIL_ADDRESS>Agus Suroso
<EMAIL_ADDRESS>Anto Sulaksono<EMAIL_ADDRESS>Freddy P.
Zen<EMAIL_ADDRESS>
###### Abstract

Quantum gravitational effects on the near horizon may alter the black hole’s
horizon drastically to be partially reflective, portrayed by a quantum
membrane. With this modification, the object can be considered as an exotic
compact object (ECO). Quantum effects on the strong gravitational regime may
also lead to a non-conserved matter tensor that can be described
phenomenologically using Rastall gravity. In this work, we study the
properties of black holes and ECOs within Rastall gravity using Kerr/CFT
correspondence. We systematically investigate the properties of the most
general rotating black hole solutions in Rastall gravity, i.e., Kerr-Newman-
NUT-Kiselev, and reveal its hidden conformal symmetry. The Cardy microscopic
entropy formula and absorption cross-sections from 2D CFT are computed and
then matched with gravity calculation. We also extend the dual CFT analysis
for studying the properties of ECOs. The existence of the quantum membrane
leads to the appearance of the gravitational echoes that is manifested as an
oscillatory feature on the absorption cross-section. We compute the absorption
cross-section and quasi-normal modes in the dual CFT picture. We also compare
the absorption cross-section of ECOs to that of black holes. We find that the
Rastall coupling constant plays a significant role for both objects. We also
obtain that the echo time delay depends explicitly on the Rastall coupling
constant. This coupling constant may play a role to recover the correction on
time delay that is believed as a non-linear physics effect. Henceforth, the
signature of the Rastall gravity can be probed from the time-delay
observation.

###### keywords:

black holes, exotic compact objects, Kerr/CFT correspondence, Rastall gravity

††journal: Physics of the Dark Universe

##  1 Introduction

The AdS/CFT correspondence has been discovered as a major advance of a non-
perturbative formulation of string theory. This correspondence has given a
successful realization of the holographic principle proposed by ’t Hooft [1].
It provides a powerful toolkit for investigating strongly coupled theories by
using weakly coupled theories or vice versa. A more practical application of
this correspondence is in the study of black hole’s thermodynamics. A very
notable breakthrough of this correspondence is the study of Kerr black hole’s
entropy, known as Kerr/CFT correspondence [2]. In the Kerr/CFT correspondence
of extremal black holes, it is found that the near-horizon geometry of
extremal Kerr black holes (spin $a$ is equal to mass $M$) exhibits an exact
$SL(2,R)\times U(1)$ symmetry leading to the precise computation of
Bekenstein-Hawking entropy from Cardy formula from two-dimensional (2D) CFT.
This study then has been extended for the Kerr black hole with general value
of spin parameter (non-extremal) [3]. However, in this non-extremal case,
conformal symmetry cannot be found directly from the geometry of the black
holes. The conformal symmetry is stored (hidden) on the solution space of the
probe scalar field in the near-horizon region within the low-frequency
approximation. The probe scalar field equation can be written in quadratic
Casimir operator that satisfies $SL(2,R)\times SL(2,R)$ algebra. The conformal
symmetry obtained from this calculation acts locally on the solution space but
it is globally obstructed by the periodic identification of the azimuthal
angle $\phi$. This identification leads to the broken $SL(2,R)\times SL(2,R)$
symmetry to $U(1)\times U(1)$ symmetry by the left- and right-moving
temperatures $T_{L,R}$. Using the numerological observation, the central
charges of this non-extremal Kerr black hole are identical to the extremal
one. Then the resulting Cardy entropy will exactly match with the Bekenstein-
Hawking entropy as the extremal Kerr black holes. Furthermore, in [4], the
authors gave a more enticing derivation of the central charges using the
covariant phase space formalism for non-extremal Kerr black holes. By assuming
the Wald-Zoupas counterterms, they managed to compute the left and right
central charges. The local conformal symmetry in the Kerr background is also
sufficient to obtain the cross-section to that of the gravity computation. For
a review regarding this whole Kerr/CFT computation can be referred to Ref.
[5], while it has also been applied in several black hole solutions in
extremal [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21] and
generic [25, 26, 27, 28, 32, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41,
42] cases.

Intriguingly, the Kerr/CFT correspondence could also be implemented to the
object beyond classical black holes, namely exotic compact object (ECO) as
discussed in Ref. [43]. It is believed that the quantum gravitational effects
in the near-horizon region may modify the apparent horizon of the classical
black holes leading to potentially observable consequences. Due to the quantum
gravitational effects in the near-horizon region, the structure on the event
horizon may alter drastically to a reflective quantum membrane (an
infinitesimal reflective layer with an arbitrary reflectivity and placed
slightly (in the order of Planck length) in front of the would-be horizon of
the classical black hole). This description is also considered as an
alternative way to solve the black hole information paradox. Somehow, this
reflective membrane does not correspond to the firewall proposal [44]. The
firewall in the black hole is a hypothetical phenomenon where an observer
falling into a black hole encounters high-energy radiation at the event
horizon. This suggests that the entanglement must somehow get broken
immediately between the infalling particle and the outgoing particle leading
to the release of large amounts of energy and creating a firewall at the black
hole event horizon. There are also some proposals addressing the modification
of the black hole’s structure in order to solve that problem such as
gravastars [45, 46, 47], fuzzballs [48, 49], 2-2 holes [50], and Kerr-like
wormholes [51]. The near-horizon modification that results in a quantum
reflective membrane on the ECOs can originate as finite-size effects in CFT
living on a torus with finite length $L$ and inverse of the temperature $1/T$,
in contrary with the CFT dual of black holes where we need to keep one of
torus parameters infinite. Due to the existence of the quantum membrane, it is
expected that the waves traveling to the would-be horizon will get partially
reflected. This feature will lead to the presence of gravitational echoes in
the postmerger ringdown signal of a binary black hole coalescence [52, 53, 54,
55, 56]. Such detection of echoes has made the investigation of quantum
gravitational effects of astrophysical objects more promising. This fact can
also be a well-turned attempt to seek the unification between gravitational
and quantum theories. One may expect that such unification can be probed in
the observational signatures of the ECOs. In particular, the realization of
the observational signatures of the ECOs lies in the detection of the quasi-
normal modes (QNMs) and echo time delay [56, 57, 43, 58, 59]. The QNM spectrum
primarily dominates the final ringdown phase of a binary black hole
coalescence where we need to impose the boundary conditions at the horizon and
asymptotic infinity. For the ECO, the presence of the reflective membrane
modifies the boundary condition at the horizon for which the ingoing waves can
get partially reflected by the membrane and reach back to the asymptotic
observer at infinity with a time delay.

Precise detection of the QNM spectrum can be obtained from the analysis of the
ringdown spectrum, which may give a adequate idea about the nature of the
black holes or ECOs. Moreover, it may provide signatures to test general
relativity and beyond general relativity. For example, the calculation of QNMs
might be used to probe the extra dimension from the ECOs in the braneworld
scenario [58, 57]. Besides the QNM spectrum, one can expect the time delay is
also sufficiently beneficial for probing the theories beyond general
relativity. As noted in [55], the time between the main merger event and the
first echo could be influenced by the non-linear physics effect. Furthermore,
the influence from non-linear physics may cause the change of magnitude of the
echo time delay at 2$\%$-3$\%$.

This work investigates the black holes and ECOs in such a scenario within
Rastall theory of gravity. Rastall gravity considers the generalization of the
general relativity, especially in the assumption of the conserved covariant
derivative equation of the matter tensor. Originally, Rastall [60, 61]
considered that the covariant divergence of matter tensor is not null, which
given by

$\nabla_{\mu}T^{\mu\nu}=\lambda\nabla^{\nu}R,$ (1)

where $\lambda$ is a constant that quantifies the deviation from general
relativistic theory, and the covariant divergence is proportional to the
covariant derivative of Ricci scalar $R$. Later on, we denote $\kappa\lambda$
as Rastall coupling constant where $\kappa=8\pi$. Due to the non-vanishing
divergence of matter tensor, the Einstein field equation will be modified as

$G_{\mu\nu}+\kappa\lambda{\color[rgb]{0,0,0}R}g_{\mu\nu}=\kappa T_{\mu\nu}.$
(2)

From above equation, one can find $R=\kappa T/(4\kappa\lambda-1)$. Rastall
noted that in this theory, $\kappa\lambda=1/4$ is the exemption because this
results in an irregular Ricci scalar. For the case $\kappa\lambda=1/4$, this
is explained as unimodular gravitational theory [62, 63] and will not be
considered within this paper. Obviously, Rastall gravitational theory
considers that the matter tensor is coupled non-minimally with gravity as
$f(R,T)$ theory, so Rastall gravity can be considered as a specific condition
for $f(R,T)$ theory [64, 65, 66, 67]. In a strong gravitational regime, the
covariant derivative of the matter tensor might not be satisfied. Nonetheless,
Rastall gravity is still valid in the flat space-time and satisfies the
conserved covariant derivative of matter tensor. Hence, this theory is still
feasible as the generalization of general relativity. Rastall theory has also
been applied to several astrophysical objects [68, 69, 70, 71, 72, 73, 74] and
cosmology [75, 76, 77] in different aspects. Recently, Visser [78] claimed
that Rastall gravity is equivalent with Einstein gravity. However, it was
argued by Darabi et al. [79] that Visser’s conclusion is not correct by
comparing these two gravitational theories. Henceforth, whether these two
theories are equivalent or not, Rastall gravity may face the challenges of
cosmological or astrophysical observations as the general relativity.

In a gravitational system, quantum effects may give rise to the violation of
the conservation of the matter tensor. Rastall gravity can be considered as a
phenomenological way to identify the effects of quantum fields in the curved
space-time in a covariant formulation, for example, in the case of
gravitational anomalies [80, 81]. Since the quantum gravitational effects may
modify the apparent horizon of the black holes giving rise to the presence of
a reflective quantum membrane, we may expect that quantum effects are also
manifested as a non-vanishing Rastall coupling constant. Hence, within the
framework of black holes and ECOs, the Rastall coupling constant is expected
to give a distinctive signature on the observables, such as the QNM spectrum
and echo time delay.

In the following, we address the implementation of Kerr/CFT correspondence to
investigate different astrophysical compact objects. Firstly, we investigate
the hidden conformal symmetry of rotating black holes in Rastall gravity. We
choose the Ker-Newman-NUT-Kiselev (KNUTK) black hole solutions in Rastall
gravity obtained in [68]. These solutions portray the charged rotating and
twisting black holes environed by the quintessential field with a non-
vanishing Rastall coupling constant. The entropy and absorption cross-section
of these generic black hole solutions is computed. This computation denotes
that these black hole solutions are holographically dual with 2D CFT.
Secondly, since the black hole solutions can be described in a 2D CFT picture,
we apply a similar duality calculation to the ECO-type of KNUTK black hole
solutions in Rastall gravity where there is a reflective quantum membrane
close to the would-be horizon. We calculate the QNM spectrum, absorption
cross-section, and echo time delay in the 2D CFT picture. Generally, it is
found that the Rastall coupling constant plays a significant role on the
observables along with the other black hole parameters.

The organization of this paper is given as follows. In the next section, we
focus our investigation on KNUTK black hole solutions in Rastall gravity. This
section is to reveal the hidden conformal symmetry on the wave equation. We
also attest the holographic dual of the black holes by computing the
microscopic entropy and absorption cross-section. The subsequent section is
focused on the study of the KNUTK-like ECO in Rastall gravity. We perform the
duality calculation to compute the QNMs. Then the absorption cross-section is
also investigated to see the effects of the quantum membrane. Afterward, we
also delve into the echo time delay of this ECO. Eventually, we summarize the
whole paper in the last section.

##  2 Rotating black holes in Rastall gravity

The KNUTK black holes are the exact solution to Einstein-Maxwell-perfect fluid
equations with non-vanishing NUT charge in Rastall gravity for a non-vanishing
Rastall coupling constant. These black hole solutions are obtained in [68] by
using Demiański-Newman-Janis algorithm to the Reissner-Nordström-Kiselev black
holes in Rastall gravity. The solutions portray the space-time for a rotating
electrically and magnetically charged massive object with a twisting NUT
charge surrounded by a quintessential field. Kiselev first derived the
quintessential field in the black hole solutions [82], so that it is common to
say the quintessential black holes as Kiselev black holes. The line element of
KNUTK blak hole space-time in Rastall gravity reads as [68]

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\frac{\Delta}{\varrho^{2}}\left[dt\\!-\\!\\{a\sin^{2}\theta+2n(1-\cos\theta)\\}d\phi\right]^{2}+\frac{\varrho^{2}}{\Delta}dr^{2}$
(3)
$\displaystyle+\varrho^{2}d\theta^{2}+\frac{\sin^{2}\theta}{\varrho^{2}}\left[adt-\\{r^{2}+(a+n)^{2}\\}\right]d\phi^{2},\
$

where

$\varrho^{2}=r^{2}+(n+a\cos\theta)^{2},~{}~{}\Delta=r^{2}-2Mr+{\color[rgb]{0,0,0}a^{2}}+e^{2}+g^{2}-n^{2}-\alpha
r^{\upsilon},\ $
$\upsilon=\frac{1-3\omega_{q}}{1-3\kappa\lambda(1+\omega_{q})}.\ $

The parameters $a$ and $n$ represent the rotation parameter and NUT charge or
twisting parameter. The mass parameter, electric charge, and magnetic charge
of space-time are given by $M$, $e$, and $g$, respectively. $\alpha$ is the
intensity of the quintessential field, and $\omega_{q}$ is the equation of the
state’s parameter of the quintessential field. The quintessential field in the
black hole solutions is believed to be the hypothetical realization of the
dark energy represented by the scalar field in the local astrophysical objects
for the given equation of state [82]. It is worth pointing out that the space-
time metric (3) is not asymptotically flat due to the presence of NUT charge
and for arbitrary value of $\omega_{q}$ and $\kappa\lambda$.

The space-time metric (3), in the limit of $a=0$ reduces to the Reissner-
Nordstrom-NUT-Kiselev black hole in Rastall gravity. Moreover, it reduces to
Taub-NUT-Kiselev space-time and Schwarzschild-Kiselev space-time in Rastall
gravity, in the limits of $a=0,~{}e=0,~{}g=0$ and $a=0,~{}e=0,~{}g=0,~{}n=0$,
respectively. It is clearly seen that for $a=0,~{}e=0,~{}g=0,~{}n=0,\alpha=0$,
it reduces to Schwarzschild space-time in Rastall gravity. When the
quintessential field is absent in metric (3), the parameter $\omega_{q}$
vanishes and we can treat $\alpha$ just as an integration constant. Basically,
$\alpha$ appears as an integration constant [82] and it appears too when
considering only the Rastall gravity. This integration constant can be
considered as cosmological constant argued in Ref. [78]. However, this is
still debatable as you can see in Ref. [78] that the authors contrarily argued
to the opinion in Ref. [79]. Hence, our analysis within this paper will try to
provide the possibility of probing Rastall gravity in the black hole and ECO.

The electromagnetic potential related to the space-time metric (3) is given by
[68]

$\displaystyle A_{\mu}dx^{\mu}$ $\displaystyle=$
$\displaystyle\frac{-er\left[adt-\left((a+n)^{2}-(n+a\text{cos}\theta)^{2}\right)d\phi\right]}{a\varrho^{2}}$
(4)
$\displaystyle-\frac{g(n+a\text{cos}\theta)\left[adt-\left(r^{2}+(a+n)^{2}\right)d\phi\right]}{a\varrho^{2}}.\
$

The Bekenstein-Hawking entropy, Hawking temperature, and angular velocity on
the event horizon ($r_{+}$) of the classical KNUTK black hole, are given by

$\displaystyle S_{BH}=\frac{A_{BH}}{4}=\pi\left[r_{+}^{2}+(a+n)^{2}\right],$
(5) $\displaystyle T_{H}=\frac{2(r_{+}-M)-\alpha\upsilon
r_{+}^{\upsilon-1}}{4\pi[r_{+}^{2}+(a+n)^{2}]},$ (6)
$\displaystyle\Omega_{H}=\frac{a}{r_{+}^{2}+(a+n)^{2}},\ $ (7)

respectively.

###  2.1 Hidden conformal symmetry

Unlike the extremal black holes, the conformal symmetries are hidden on the
scalar wave equation of the black hole’s background. To investigate and
explore the hidden conformal symmetry, we consider a massless neutral scalar
probe in the background of the KNUTK black holes in Rastall gravity (3). Note
that one can also assume a (electrically and magnetically) charged scalar
probe to explore the conformal symmetries in different pictures. Nonetheless,
within this paper, we focus our analysis only for the angular momentum
picture. The massless scalar field equation for the scalar probe, is given by

$\nabla_{\alpha}\nabla^{\alpha}\Phi=0.$ (8)

We notice the KNUTK black hole (3) have two translational Killing vectors
i.e., $\partial_{t}$ and $\partial_{\phi}$. Hence, we can separate the
coordinates in the scalar field as

$\Phi(t,r,\theta,\phi)=\mathrm{e}^{-i\omega t+im\phi}R(r)S(\theta).$ (9)

Plugging Eq. (9) into Eq. (8), leads to two differential equations i.e., the
angular $S(\theta)$ and radial $R(r)$ wave functions,

$\partial_{r}(\Delta\partial_{r})R(r)+\bigg{[}\frac{[(r^{2}+(a+n)^{2})\omega-
ma]^{2}}{\Delta}+2am\omega-K_{l}\bigg{]}R(r)=0,$ (11)

where $K_{l}$ is the separation constant.

The hidden conformal symmetries can only be revealed when we look at the near-
horizon region of the scalar wave equation. Hence, we consider the radial
equation (LABEL:angular1) at the near-horizon region, which is defined by
$\omega{\color[rgb]{0,0,0}r_{+}}\ll 1$. Moreover, we consider the low-
frequency limit for the scalar field $\omega M\ll 1$. Consequently, we also
impose $\omega a\ll 1,\,\omega e\ll 1,\omega g\ll 1,\omega\sqrt{\alpha}\ll 1,$
and $\omega n\ll 1$. As the study of the hidden conformal symmetries on Kerr-
AdS black hole’s family, the function $\Delta$ should be approximated into two
horizons. It is necessary to approximate $\Delta$ by a quadratic polynomial of
$(r-r_{+})$ to set up the radial equation in a suitable form for exploring its
hidden conformal symmetry. We can approximate $\Delta$ by a quadratic
polynomial in $(r-r_{+})$ as,

$\Delta\simeq K(r-r_{+})(r-r_{*}),$ (12)

where

$K=1-\frac{\alpha\upsilon(\upsilon-1)r_{+}^{\upsilon-2}}{2},\ $
$r_{*}=r_{+}-\frac{1}{Kr_{+}}\left[r_{+}^{2}-a^{2}-e^{2}-g^{2}+n^{2}+\alpha(1-\upsilon)r_{+}^{\upsilon}\right].\
$

It is worth noting that $r_{*}$ is not the inner horizon. In terms of $r_{+}$
and $r_{*}$, we can write the Hawking temperature as

$T_{H}=\frac{K(r_{+}-r_{*})}{4\pi[r_{+}^{2}+(a+n)^{2}]}.$ (13)

Such an approximation was used before for the quartic metric function of the
Kerr-Newman-AdS [32, 33], Kerr-NUT-AdS [34], and Kerr-Newman-NUT-AdS [41, 42]
black holes. Using (12), the radial equation (11) reduces to

$\left[\partial_{r}\left[(r-r_{+})(r-r_{*})\partial_{r}\right]+\frac{r_{+}-r_{*}}{r-r_{+}}A+\frac{r_{+}-r_{*}}{r-r_{*}}B+C\right]R(r)=0,\
$ (14)

where

$A=\frac{\left[(r_{+}^{2}\\!+\\!(a+n)^{2})\omega-
am\right]^{2}}{K^{2}(r_{+}-r_{*})^{2}},\ $
$B=\frac{\left[(r_{*}^{2}\\!+\\!(a+n)^{2})\omega-
am\right]^{2}}{K^{2}(r_{+}-r_{*})^{2}},~{}~{}~{}C=\frac{-K_{l}}{K}.\ $

We should note that not all generic rotating black holes are holographically
dual to two different CFTs. Specifically, the four-dimensional charged
rotating magnetized Kerr black holes cannot be written in the above form to
reveal the hidden conformal symmetries [15]. It is important to note also that
we do not explore the angular wave equation because this does not have
$SL(2,R)\times SL(2,R)$ isometry, yet $SU(2)\times SU(2)$ isometry [83].

In order to reveal the hidden symmetry, we perform the following coordinate
transformations for the generic black holes [3, 5]

$\displaystyle\omega^{+}=\sqrt{\frac{r-r_{+}}{r-r_{*}}}e^{2\pi
T_{R}\phi+2n_{R}t},$ (15)
$\displaystyle\omega^{-}=\sqrt{\frac{r-r_{+}}{r-r_{*}}}e^{2\pi
T_{L}\phi+2n_{L}t},$ (16) $\displaystyle
y=\sqrt{\frac{r_{+}-r_{*}}{r-r_{*}}}e^{\pi(T_{L}+T_{R})\phi+(n_{L}+n_{R})t}.\
$ (17)

In terms of the new conformal coordinates $\omega^{+},\,\omega^{-}$ and $y$,
we may define three locally conformal operators

$\displaystyle H_{1}=i\partial_{+},$ (18) $\displaystyle
H_{-1}=i\left(\omega^{+2}\partial_{+}+\omega^{+}y\partial_{y}-y^{2}\partial_{-}\right),$
(19) $\displaystyle
H_{0}=i\left(\omega^{+}\partial_{+}+\frac{1}{2}y\partial_{y}\right),$ (20)

as well as

$\displaystyle\bar{H}_{1}=i\partial_{-},$ (21)
$\displaystyle\bar{H}_{-1}=i\left(\omega^{-2}\partial_{-}+\omega^{-}y\partial_{y}-y^{2}\partial_{+}\right),$
(22)
$\displaystyle\bar{H}_{0}=i\left(\omega^{-}\partial_{-}+\frac{1}{2}y\partial_{y}\right),$
(23)

where $\partial_{+}=\partial/\partial\omega^{+}$ and
$\partial_{-}=\partial/\partial\omega^{-}$. The set of operators (18)-(20)
satisfy the $SL(2,R)$ Lie algebra

$\displaystyle\left[H_{0},H_{\pm 1}\right]=\mp iH_{\pm
1},~{}~{}~{}\left[H_{-1},H_{1}\right]=-2iH_{0},$ (24)

while a similar $SL(2,R)$ algebra exists for the set of operators (21)-(23).
From any of two sets of operators, we can obtain the quadratic Casimir
operator that reads as

$\displaystyle\mathcal{H}^{2}$ $\displaystyle=$
$\displaystyle\bar{\mathcal{H}}^{2}=-H_{0}^{2}+\frac{1}{2}(H_{1}H_{-1}+H_{-1}H_{1})$
(25) $\displaystyle=$
$\displaystyle\frac{1}{4}(y^{2}\partial_{y}^{2}-y\partial_{y})+y^{2}\partial_{+}\partial_{-}.$

It will be easier to notice the relation between the quadratic Casimir
operator (25) and the radial Eq. (14) in old coordinates $(t,r,\phi)$. In old
coordinates, the quadratic Casimir operator is given by

$\displaystyle\mathcal{H}^{2}$ $\displaystyle=$
$\displaystyle(r-r_{+})(r-r_{*})\partial_{r}^{2}+(2r-r_{+}-r_{*})\partial_{r}$
(26) $\displaystyle+$
$\displaystyle\frac{r_{+}-r_{*}}{r-r_{*}}\left(\frac{n_{L}-n_{R}}{4\pi
G}\partial_{\phi}-\frac{T_{L}-T_{R}}{4G}\partial_{t}\right)^{2}$
$\displaystyle-$
$\displaystyle\frac{r_{+}-r_{*}}{r-r_{+}}\left(\frac{n_{L}+n_{R}}{4\pi
G}\partial_{\phi}-\frac{T_{L}+T_{R}}{4G}\partial_{t}\right)^{2},$

where $G=n_{L}T_{R}-n_{R}T_{L}$ and $n_{L},T_{R},n_{R},T_{L}$ are constants.

We have found that the radial Eq. (14) could be re-written in terms of the
$SL(2,R)$ quadratic Casimir operator as
$\mathcal{H}^{2}R(r)=\bar{\mathcal{H}}^{2}R(r)=-CR(r)$ where we should
identify the constants as

$n_{L}=-\frac{K}{2(r_{+}+r_{*})},~{}~{}~{}n_{R}=0,$ (27)
$T_{L}=\frac{K[r_{+}^{2}+r_{*}^{2}+2(a+n)^{2}]}{4\pi
a(r_{+}+r_{*})},~{}~{}~{}T_{R}=\frac{K(r_{+}-r_{*})}{4\pi a}.$ (28)

We may identify $T_{R,L}$ as the CFT temperatures that arise as a result of
the spontaneously symmetry breaking of the partition function on
$SL(2,R)\times SL(2,R)$ theory to the partition function of $U(1)\times U(1)$
CFT. By periodic identification of the azimuthal coordinate
$\phi\sim\phi+2\pi$, the $SL(2,R)\times SL(2,R)$ symmetry breaks down to
$U(1)\times U(1)$ symmetry.

###  2.2 Microscopic entropy description

In the previous subsection, we have shown the hidden conformal symmetry where
the dual CFTs could describe the KNUTK black holes in Rastall gravity. For the
extremal rotating black holes in Rastall gravity [21], the CFTs have only a
non-zero left temperature. Nevertheless, the dual CFTs have finite and non-
zero left- and right-moving temperatures for the generic black holes. Another
essential piece of information for any CFT is the central charge of the
underlying theory. For the extremal black holes, the central charges of the
dual CFT could be derived by using analysis of the asymptotic symmetry group
[2, 5].

In order to use the analysis, we need to work on the near-horizon region. To
find the near-horizon geometry of the extremal KNUTK black holes in Rastall
gravity (3), we employ the following coordinate transformations

$\displaystyle r=r_{+}+\epsilon
r_{0}y,~{}~{}t=\frac{r_{0}}{\epsilon}\tau,~{}~{}\phi=\varphi+\frac{\Omega_{H}r_{0}}{\epsilon}\tau,\
$ (29)

where $r_{0}^{2}={r_{+}^{2}+(a+n)^{2}}$ and $\epsilon\rightarrow 0$ denotes
the near-horizon limit. In the near-horizon coordinates, the metric (3)
becomes

$ds^{2}=\Gamma(\theta)\left(-y^{2}d\tau^{2}+\frac{dy^{2}}{y^{2}}+Kd\theta^{2}\right)+\gamma(\theta)\left(d\varphi+pyd\tau\right)^{2},\
$ (30)

where

$\Gamma(\theta)=\frac{\varrho_{+}^{2}}{K},~{}~{}~{}\gamma(\theta)=\frac{r_{0}^{4}\sin^{2}\theta}{\varrho_{+}^{2}},\
$
$\varrho_{+}^{2}=r_{+}^{2}+(n+a\cos\theta)^{2},~{}~{}~{}p=\frac{2ar_{+}}{Kr_{0}^{2}},\
$

and we have used the scaling $d\tau\rightarrow d\tau/K$. In the near-horizon
limit, the gauge field (4) is given by

$A_{\mu}dx^{\mu}=f(\theta)\left(d\varphi+\hat{p}yd\tau\right)-\frac{e[r_{+}^{2}-(a+n)^{2}]}{[r_{+}^{2}+(a+n)^{2}]}d\varphi,\
$ (31)

where

$\displaystyle
f(\theta)=\frac{er_{0}^{2}\left[(r_{+}^{2}-n^{2})-a(2n+a\text{cos}\,\theta)\text{cos}\,\theta\right]}{2ar_{+}\rho_{+}^{2}}.\
$ (32)

As denoted in [6], the central charge can be calculated directly from the
near-horizon geometry of the extremal black holes. Hence, the central charge
of the CFT is given by

$c_{L}=3p\int^{\pi}_{0}d\theta\sqrt{\Gamma(\theta)\gamma(\theta)K}=\frac{12ar_{+}}{K}.$
(33)

For generic rotating black holes, CFTs are represented by left- and right-
moving central charges. So, from (33), we can propose the following left- and
right-moving central charges for the CFT dual to the generic black holes as

$c_{L}=c_{R}=\frac{6a(r_{+}+r_{*})}{K}.\ $ (34)

Note that there is a restriction on the central charge in order to be finite
in which $K\neq 0$ or $\alpha\upsilon(\upsilon-1)r_{+}^{\upsilon-2}\neq 2$.
So, besides $\kappa\lambda=1/4$ [60, 61], we can also obtain the bound of
Rastall coupling constant and $\omega_{q}$ in this correspondence.

Another way to compute the central charges given in [4] where they extend to
calculate the Wald-Zoupas counterterms within the covariant formalism that
leads to the well-defined central charges. Our proposal for the central
charges (34) together with the CFT temperatures (28) lead to the Cardy entropy
for the dual CFT to the KNUTK black holes in Rastall gravity, as

$S_{CFT}=\frac{\pi^{2}}{3}(c_{L}T_{L}+c_{R}T_{R})=\pi[r_{+}^{2}+(a+n)^{2}].$
(35)

It is in a perfect agreement with the Bekenstein-Hawking entropy.

###  2.3 Absorption cross-section

Another realization of the correspondence between rotating black holes in
Rastall gravity and the CFTs is the equivalence of the absorption cross-
section. We consider the absorption cross-section of scalar probes in the
background of the generic black holes. In investigating the absorption cross-
section, we consider the near-horizon and asymptotic regions. In the near-
horizon region, the approximation of $\Delta$ in the radial Eq. (14) is still
acceptable. Nevertheless, in the asymptotic region, this approximation breaks
down, except for certain conditions. Accordingly, we cannot exploit the usual
radial equation to discuss the scattering issue in general. However, for near-
extremal black holes, we may pose a well-defined scattering problem. So, in
the calculation of the absorption cross-section, we focus on the near-extremal
black holes [32, 33, 34, 41, 42]. In order to do so, we consider the following
near-extremal coordinate transformations, as

$r=\frac{r_{+}+r_{*}}{2}+\epsilon r_{0}y,~{}r_{+}-r_{*}=\mu\epsilon
r_{0},~{}t=\frac{r_{0}}{\epsilon}\tau,~{}\phi=\varphi+\frac{\Omega_{H}r_{0}}{\epsilon}\tau,\
$ (36)

where $\mu$ is the parameter which quantifies the extremality. We also
consider the scalar probe with frequencies around the super-radiant bound

$\omega=m\Omega_{H}+\hat{\omega}\frac{\epsilon}{r_{0}},$ (37)

where $\Omega_{H}$ are given by (7). We can re-write the radial Eq. (14) by

$\left[\partial_{y}\left(y-\frac{\mu}{2}\right)\left(y+\frac{\mu}{2}\right)\partial_{y}+\frac{A_{s}}{y-\frac{\mu}{2}}+\frac{B_{s}}{y+\frac{\mu}{2}}+C_{s}\right]R(y)=0,$
(38)

where

$\displaystyle
A_{s}=\frac{\hat{\omega}^{2}}{\mu},~{}~{}~{}B_{s}=-\mu\left(\frac{\hat{\omega}}{\mu}-\frac{2m\Omega_{H}r_{+}}{K}\right)^{2},\
$

and $C_{s}$ is the new separation constant. We also use the coordinate
transformation $z=(y-\mu/2)/(y+\mu/2)$. Now the radial Eq. (38) becomes

$\left[z(1-z)\partial_{z}^{2}+(1-z)\partial_{z}+\frac{\hat{A_{s}}}{z}+\hat{B_{s}}+\frac{C_{s}}{1-z}\right]R(z)=0,$
(39)

where

$\displaystyle\hat{A_{s}}=\frac{\hat{\omega}^{2}}{\mu^{2}},~{}~{}~{}\hat{B_{s}}=-\left(\frac{\hat{\omega}}{\mu}-\frac{2m\Omega_{H}r_{+}}{K}\right)^{2}.\
$

The differential Eq. (39) possesses the solutions as given by

$R(z)=z^{-i\alpha_{s}}(1-z)^{\beta_{s}}F(a_{s},b_{s},c_{s};z),$ (40)

where $F(a_{s},b_{s},c_{s};z)$ is the hypergeometric function where

$a_{s}=\beta_{s}+i(\gamma_{s}-\alpha_{s}),~{}b_{s}=\beta_{s}-i(\gamma_{s}+\alpha_{s}),~{}c_{s}=1-2i\alpha_{s},\
$

and

$\alpha_{s}=\sqrt{\hat{A_{s}}},~{}\beta_{s}=\frac{1}{2}\left(1-\sqrt{1-4C_{s}}\right),~{}\gamma_{s}=\sqrt{-\hat{B}_{s}}.\
$

In the asymptotic infinity of the radial coordinate $r$ ($y\gg\mu/2$), the
coordinate $z$ becomes 1 that leads to the solutions

$\displaystyle R(y)\sim D_{1}y^{-\beta_{s}}+D_{2}y^{\beta_{s}-1},$ (41)

where

$D_{1}=\frac{\Gamma(c_{s})\Gamma(2h-1)}{\Gamma(c_{s}-a_{s})\Gamma(c_{s}-b_{S})},~{}~{}~{}D_{2}=\frac{\Gamma(c_{s})\Gamma(1-2h)}{\Gamma(a_{s})\Gamma(b_{s})}.~{}~{}~{}$
(42)

The conformal weight is given by

$h=1-\beta_{s}=\frac{1}{2}\left(1+\sqrt{1-4C_{s}}\right).$ (43)

Hence, the absorption cross-section of the scalar fields is given by

$P^{\text{BH}}_{\text{abs}}\sim\left|D_{1}\right|^{-2}=\frac{\sinh\left({2\pi\alpha_{s}}\right)}{2\pi\alpha_{s}}\frac{{\left|{\Gamma\left(c_{s}-a_{s}\right)}\right|^{2}\left|{\Gamma\left(c_{s}-b_{s}\right)}\right|^{2}}}{{|{\Gamma\left({2h-1}\right)}|^{2}}}.$
(44)

In [22], it is shown that the real-time correlator could be computed
holographically in the bulk. So, we can also find the real-time correlation
function $G_{R}$ for our black hole solutions. By considering $D_{1}$ as the
source and $D_{2}$ as the response, we can obtain

$\displaystyle
G_{R}\sim\frac{D_{2}}{D_{1}}=\frac{\Gamma(1-2h)}{\Gamma(2h-1)}\frac{\Gamma(c_{s}-a_{s})\Gamma(c_{s}-b_{s})}{\Gamma(a_{s})\Gamma(b_{s})}.$
(45)

The absorption cross-section can also be obtained from the correlation
function where $P^{\text{BH}}_{\text{abs}}\sim\text{Im}(G_{R})$, the imaginary
part of the correlation function. To further support the correspondence
between the rotating black hole (3) and 2D CFT, we show that the absorption
cross-section for the scalar fields (44) can be obtained from the absorption
cross-section in a 2D dual CFT. In a 2D CFT with conformal weights $h_{L,R}$
and temperatures $T_{L,R}$, the absorption cross-section for the scalar fields
with frequencies $\tilde{\omega}_{L,R}$, is given by [3, 22]

$\displaystyle P^{\text{BH}}_{\text{abs}}$ $\displaystyle\sim$
$\displaystyle{T_{L}}^{2h_{L}-1}{T_{R}}^{2h_{R}-1}\sinh\left({\frac{{{\tilde{\omega}}_{L}}}{{2{T_{L}}}}+\frac{{{\tilde{\omega}}_{R}}}{{2{T_{R}}}}}\right)$
(46) $\displaystyle\times$
$\displaystyle\left|{\Gamma\left({h_{L}+i\frac{{{\tilde{\omega}}_{L}}}{{2\pi{T_{L}}}}}\right)}\right|^{2}\left|{\Gamma\left({h_{R}+i\frac{{{\tilde{\omega}}_{R}}}{{2\pi{T_{R}}}}}\right)}\right|^{2}.$

The matching between absorption cross-sections (44) and (46) can be obtained
by choosing proper left and right frequencies
${\tilde{\omega}}_{L},{\tilde{\omega}}_{R}$. In order to do so, we consider
the first law of black hole’s thermodynamics which is given by

$T_{H}\delta S_{BH}=\delta M-\Omega_{H}\delta J.$ (47)

By varying Cardy entropy, we obtain

$\delta S_{CFT}=\frac{{\delta E_{L}}}{{T_{L}}}+\frac{{\delta
E_{R}}}{{T_{R}}}.$ (48)

We can identify $\delta M$ as $\omega$ and $\delta J$ as $m$ which leads to
the identification of $\delta E_{R,L}$ as $\tilde{\omega}_{R,L}$. From the
variations of entropy in (47) and (48), we can find the left and right
frequencies as

$\displaystyle\tilde{\omega}_{L}=\frac{r_{+}^{2}+r_{*}^{2}+2(a+n)^{2}}{2a}\omega,~{}~{}~{}\tilde{\omega}_{R}=\tilde{\omega}_{L}-m.\
$ (49)

We note that the conformal weights $h_{L,R}$ in (46) are equal to $h$, as in
(43). So, we have found the holographic dual of the KNUTK black holes in
Rastall gravity.

##  3 Exotic compact object in Rastall gravity

In this section, we assume a model of an exotic compact object (ECO) where the
exterior space-time metric of this ECO is given by Eq. (3). The near-horizon
geometry of the classical black hole given by Eq. (3) is modified due to the
existence of quantum membrane structure originating from near-horizon quantum
gravitational effects. These quantum gravitational effects lead to the
formation of ECO. This ECO does not possess an event horizon, but a partially
reflective membrane located slightly outside the common position of the event
horizon. Due to quantum gravitational effects, the reflective membrane can be
grasped as an effective description of the quantum corrections on the horizon.
This work is inspired by the paper [43] where they studied the Kerr-like ECO.
Furthermore, in Rastall theory of gravity, quantum gravitational effects can
be manifested as non-conserved divergence of matter tensor, giving rise
phenomenologically to non-vanishing Rastall coupling constant.

Figure 1: Schematic diagram of the near-horizon region from a black hole and
an ECO.

The reflective membrane is assumed to be located at  $\bar{r}_{0}$ where
$\bar{r}_{0}=r_{+}+\delta r_{+}$ with reflectivity $\mathcal{R}$. The constant
$\delta r_{+}$ is very small and in the order of Planck length. Due to quantum
corrections, the apparent horizon gets shifted as given by [57, 58]

$\delta r_{+}\sim\frac{N^{2}l_{p}^{2}}{M},\ $ (50)

where $N$ is the degree of freedom of the dual CFT living on the torus and
$l_{p}$ is the Planck length. $N$ corresponds to length of the torus $L$ as
$N^{2}\sim L^{2}/l^{2}_{p}$. We can calculate $\delta r_{+}$ from

$\int^{r_{+}+\delta r_{+}}_{r_{+}}\sqrt{g_{rr}}|_{\theta=0}~{}dr\sim\eta
l_{p},\ $ (51)

where $\eta$ is just a constant proportional to CFT degree of freedom.

In the black hole space-time, the low- (high-)frequency modes scalar wave will
be reflected (transmitted) by the angular momentum barrier, while the higher
(lower) frequencies modes will cross the barrier and go to infinity (see Fig.
1). However, due to a quantum membrane for ECO, the low-frequency modes will
be trapped between the membrane and the angular momentum barrier. The trapped
waves will slowly leak out as repeating echoes. For a scalar perturbation in
this ECO background, the asymptotic behavior of the waveform would be given as

$R\sim\left\\{\begin{array}[]{lr}\frac{C}{r}e^{-i\omega
r^{*}}+\frac{D}{r}e^{i\omega
r^{*}}~{}~{}~{}\text{for}~{}~{}~{}r^{*}\rightarrow\infty,\\\
Ae^{-i\tilde{\omega}r^{*}}+Be^{i\tilde{\omega}r^{*}}~{}~{}~{}\text{for}~{}~{}~{}r^{*}\rightarrow-\infty,\end{array}\right.$
(52)

where $\tilde{\omega}$ is the horizon-frame frequency, $r^{*}$ is the tortoise
coordinate and $A,B,C,D$ are the asymptotic amplitudes as shown in Fig. 1. The
quantum membrane possesses a reflectivity which can be defined in terms of the
given asymptotic amplitudes as

$\mathcal{R}e^{i\pi\delta}\equiv\frac{B}{A}z_{0}^{2i\sigma},$ (53)

where $\delta$ is a phase identified by the quantum properties of the ECO,
$z_{0}$ is the position of the quantum membrane in the coordinate $z$, and
$\sigma=(\omega-m\Omega_{H})/4\pi T_{H}$.

One can see the existence of the echoes coming from the massless scalar waves
trapped between the quantum membrane and the angular momentum barrier from the
calculation of the absorption cross-section. Since the quantum membrane will
reflect the massless scalar waves, the absorption cross-section of this ECO
will differ from that of a black hole (44), resulting in echoes. As given in
[43], we can write the relation between absorption cross-section of the black
holes and ECOs as

$P^{\text{ECO}}_{\text{abs}}=P^{\text{BH}}_{\text{abs}}\frac{1-|\mathcal{R}|^{2}}{|1-\mathcal{R}e^{-2ir_{0}^{*}(\omega-m\Omega_{H})+i\delta}|^{2}},\
$ (54)

where $r_{0}^{*}$ is the position of the membrane in terms of tortoise
coordinate. It is obvious that for $\mathcal{R}=0$, it will reduce to that of
classical black holes. It has been derived in [56, 84] that the reflectivity
for a quantum black hole (ECO) is given by

$\mathcal{R}\sim e^{-\frac{|\omega-m\Omega_{H}|}{2T_{QH}}}.\ $ (55)

$\mathcal{R}$ depends on the frequency and angular momentum of the scalar
waves. This finding is the standard Boltzmann reflectivity. Note that $T_{QH}$
is identified as the quantum horizon temperature and comparable to the Hawking
temperature for black holes $(T_{QH}=\gamma T_{H})$. As noted in [53, 59],
$\gamma$ is the proportionality constant depending on the dispersion and
dissipation effects in graviton propagation.

###  3.1 QNMs

Gravitational radiation from ECO oscillation exhibits a specific
characteristic spectrum independent of the processes that give rise to these
oscillations. The QNM spectrum is directly related to the parameters contained
by the ECOs. Hence, QNM may carry important information about the near-horizon
quantum structure of the ECOs, which is dependent on those parameters. In
investigating the black holes, especially using AdS/CFT duality, QNM can be
obtained from the poles of the retarded correlation function from the CFTs. In
this duality calculation, the quantum gravity in the near-horizon region of
the rotating black holes is dual to the 2D CFT. It has been shown in  [23, 24,
25, 29, 30, 32, 33, 34, 41, 42] by employing this duality that QNM computation
is consistent with the gravity result, even for non-asymptotically flat
solutions, for example for black holes with NUT charge and cosmological
constant. For the ECOs, it is crucial to note that the quantum effects that
modify the near-horizon region can originate from the finite-size effects in
CFT living on a torus with finite periodicities on its length $L$ and inverse
of the temperature $1/T$. The finite-size effects on the CFT side are also
believed to produce discreet QNMs spectrum. For ECOs, the QNM spectrum appears
from the poles of the exponential parts of the CFT two-point function (see A).
The two-point function is distinct from the CFT two-point function of the dual
CFT for the black holes. The difference that appears is caused by the
implementation of the method of images. By using this method, there will be an
infinite sum in front of the two-point function. This infinite sum induces the
emergence of the exponential parts.

In this paper, we basically compute QNM and other quantities from the CFT
analysis. Moreover, for the ECOs with the exterior metric given by (3) and
from CFT analysis, the QNM is given by other poles of the exponential parts of
the CFT two-point function. From Eq. (71) and the frequencies given in Eq.
(49), we obtain the following QNM spectrum

The Eq. (LABEL:eq:QNMRastall) is obtained by expanding the factor containing a
complex number to the first order. Note that for electromagnetic perturbation,
the factor $(2R_{n}+1+\delta)$ becomes $(2R_{n}+\delta)$. The QNM spectrum
(LABEL:eq:QNMRastall) will reduce to the QNM spectrum of Kerr-like ECO [43]
when all parameters vanish, except $M$ and $a$. Interestingly, we can also
find the QNM spectrums for various ECOs. Simply from QNM spectrum given in Eq.
(LABEL:eq:QNMRastall), we can obtain the QNM of Kerr-Newman-NUT-like ECO when
$\alpha=0$, QNM of Kerr-Newman-like ECO when $n,\alpha=0$, and QNM of Kerr-
NUT-like ECO when $e,g,\alpha=0$.

The reflectivity of the near-horizon quantum membrane of the ECO can be
obtained from the imaginary part of the QNM spectrum [59]. From the imaginary
part of the QNM spectrum for Boltzmann ECOs (LABEL:eq:QNMRastall), we can find
that the reflectivity of the membrane is given by

$\mathcal{R}=e^{-\frac{|\omega-m\Omega_{H}|\mathbf{a}}{T_{H}}},\ $ (57)

where we also need the modular parameter $\mathbf{a}$ to be $1/2\gamma$ and
the following relation for the length of the torus

$L=\frac{a|r_{0}^{*}|}{\pi[r_{+}^{2}+r_{*}^{2}+2(a+n)^{2}]}.\ $ (58)

However, we can generalize the modular parameter $\mathbf{a}$. Subsequently,
it is important to see the role of every parameter to the QNM spectrum because
it can be the plausible observational signatures of the quantum gravitational
effects on this ECO. To see the role of the parameters, in Fig. 2, we give the
numerical value of the (real and imaginary) QNMs for different values of
parameters to see the role of the parameters. From top panels, we can see that
higher $\upsilon$ and higher $\alpha$ produce lower $\omega$. The bottom
panels denote that higher $a,e$ and $g$ corresponds to higher $\omega$ while
higher $n$ corresponds to lower $\omega$. It is worth noting that the
information of $\omega_{q}$ and $\kappa\lambda$ analytically stored in the
function $\upsilon$ as we can see in its definition after Eq. (3).
$\omega_{q}$ and $\kappa\lambda$ determine the number of the horizons (as
explained in Ref. [68]). For several values of $\omega_{q}$ and
$\kappa\lambda$, we can obtain a similar $\upsilon$. Hence, there will be a
degeneracy on $\upsilon$, bot not on the values of $\omega_{q}$ and
$\kappa\lambda$.

Figure 2: Real and imaginary parts of the QNMs as a function of integer
$R_{n}$. with $v$ and $\alpha$ variations in upper panels and with $a$, $n$,
and $e,g$ varitations in lower panels. We set
$M=1,r_{0}^{*}=100,m=2,\delta=0,\gamma=1$ for all panels.

###  3.2 Absorption cross-section

In the two-point function of the ECOs, the quantum membrane has contributed to
produce new poles beside the poles coming from the gamma functions. In Eq.
(54), it is seen that the presence of the quantum membrane very close to the
would-be horizon gives an intriguing contribution to the absorption cross-
section of the ECO. In addition, in the absorption cross-section, the presence
of the quantum membrane contributes to the emergence of the echoes portrayed
by the oscillatory feature. In the CFT description, the absorption cross-
section of the ECO can be obtained by considering the dual field theory on a
circle of length $L$ and delve into the finite-size effects on the boundary.
The detailed derivation of the absorption cross-section of the ECO is given in
[43].

For the two-point function of 2D CFT, the absorption cross-section is given by
the following expression

$\displaystyle P^{\text{ECO}}_{\text{abs}}$ $\displaystyle\sim$
$\displaystyle\omega^{2l-1}{T_{L}}^{2h_{L}-1}{T_{R}}^{2h_{R}-1}\sinh\left({\frac{{{\tilde{\omega}}_{L}}}{{2{T_{L}}}}+\frac{{{\tilde{\omega}}_{R}}}{{2{T_{R}}}}}\right)$
(59) $\displaystyle\times$
$\displaystyle\left|{\Gamma\left({h_{L}+i\frac{{{\tilde{\omega}}_{L}}}{{2\pi{T_{L}}}}}\right)}\right|^{2}\left|{\Gamma\left({h_{R}+i\frac{{{\tilde{\omega}}_{R}}}{{2\pi{T_{R}}}}}\right)}\right|^{2}.$
$\displaystyle\times$
$\displaystyle\frac{1-e^{-2\mathbf{a}\big{|}\frac{\tilde{\omega}_{L}}{T_{L}}+\frac{\tilde{\omega}_{R}}{T_{R}}\big{|}}}{\bigg{|}1-e^{2\pi
iL(\tilde{\omega}_{L}+\tilde{\omega}_{R})-\mathbf{a}\big{|}\frac{\tilde{\omega}_{L}}{T_{L}}+\frac{\tilde{\omega}_{R}}{T_{R}}\big{|}}\bigg{|}^{2}}$

The last factor is the source of the echoes where for the black holes, the
last factor will vanish. The general left- and right-moving temperatures
$(T_{L},T_{R})$ are given by Eq. (28). The CFT frequencies
$(\tilde{\omega}_{L},\tilde{\omega}_{R})$ are also identical with those of
black holes which are given by Eq. (49). For our ECO, in the low-frequency
limit, the absorption cross-section reads as

Note that we have taken $\mathbf{a}=1/2$.

As the QNM spectrum, the absorption cross-section (LABEL:eq:abscrossRastall)
will reduce to the absorption cross-section of Kerr-like ECO [43] when only
$M$ and $a$ which do not vanish and by changing $h\rightarrow l$. It is worth
to note that we can also find the absorption cross-section for various ECOs.
From the absorption cross-section (LABEL:eq:abscrossRastall), we can obtain
the absorption cross-section of Kerr-Newman-NUT-like ECO when $\alpha=0$,
absorption cross-section of Kerr-Newman-like ECO when $n,\alpha=0$ and
absorption cross-section of Kerr-NUT-like ECO when $e,g,\alpha=0$.

Figure 3: Absorption cross-section as a function of $\omega$ with variations
of $v$, $\alpha$,$\omega_{q}$, and $\kappa\lambda$. We set
$m=2,C_{s}=-2,r_{+}=1,r_{0}^{*}=100,\delta=0,a=0.6,e,g=0.3,n=0.5$ and
$\alpha=0.03$.

To show the membrane reflectivity’s effect of the ECO and the parameters on
the absorption cross-section, we need to plot the absorption cross-section by
varying all parameters. We give the plot in Fig. 3. The absorption cross-
section in the low-frequency limit is negative due to superradiance. In the
top panels and bottom left panel of Fig. 3, we vary $\kappa\lambda$ and
$\omega_{q}$ for the ECO and compare it to the black hole $(\mathcal{R}=0)$.
The absorption cross-section for the ECO possesses an oscillatory feature that
corresponds with the reflectivity of the quantum membrane. For positive
$\kappa\lambda$, orange dashed line, the absorption cross-section is quite
distinctive compared to vanishing and negative $\kappa\lambda$ where for
$\omega_{q}=-2/3$, the reflectivity is higher than those of when
$\omega_{q}=0,-1/3$. The bottom right panel shows the plot of the variations
of $\upsilon$ and quintessential intensity $\alpha$. The higher $\upsilon$
results in the lower reflectivity. Moreover, we can see that for vanishing
quintessential intensity, the reflectivity goes higher than the non-vanishing
ones. In Fig. 4, we vary $a,\alpha,n,e$ and $g$. From the top left panel, we
can see that higher $a$ makes $P_{abs}$ more negative and has higher
reflectivity. On the top right panel, it can be seen that higher $\alpha$
corresponds to the lower $P_{abs}$ and lower reflectivity. From the bottom
panels, we can see that NUT charge $n$ possesses similar feature as $\alpha$
while $e,q$ have similar feature as $a$.

Figure 4: Absorption cross-section as the variation of $a,\alpha,n,e$ and $g$.
We set $\upsilon=1,m=2,l=2,r_{0}^{*}=100,\delta=0,M=1$.

From Figs. 3 and 4, we can also see the role of the parameters regarding the
superradiant instability (also called ergoregion instability because it occurs
near the ergoregion). For ECOs, the transmitted wave from the effective
potential is reflected by the ECO’s surface. This wave then passes through the
ergoregion and extracts the rotational energy from the ECO (Penrose’s
process). A part of these waves which has gained energy from the ECO will have
a higher amplitude and experience similar processes. Since these processes are
repeating, the amplitude will grow exponentially, leading to instability. The
instability always occurs when the real part of the frequency satisfies the
superradiant condition $\omega<m\Omega_{H}$. From the absorption cross-section
analysis, this instability is depicted in the region below the superradiant
bound. From Fig. 3, we can analyze that $\kappa\lambda$ affects the
instability by increasing or reducing the absorption cross-section. For
$\omega_{q}=0,-1/3$, the coupling constant may reduce the instability, while
for $\omega_{q}=-2/3$, the instability increases for the positive coupling
constant and decreases for the negative one. In Fig. 4, we can see that the
increasing value of the spin parameter and electromagnetic charges may
increase the instability while the increasing value of quintessential
intensity and NUT charge may reduce the instability.

###  3.3 Echo time delay

Since there is a reflective membrane near the would-be horizon, the ingoing
wave is partially reflected by the membrane while copies of the reflected wave
reach infinity after some time delay. This time delay is said to be the time
for the wave to travel from the angular momentum barrier to the quantum
membrane and travel back to the angular momentum barrier again. The time gap
between two consecutive echoes can be determined in terms of the black hole
parameters and the location of the reflective membrane. The echoes time delay
can be computed as

$\Delta\tau=2|r_{0}^{*}|,$ (61)

where $r_{0}^{*}$ is the position of the reflective membrane in tortoise
coordinate. The tortoise coordinate is given by
$r^{*}=\int[r^{2}+(a+n)^{2}]dr/\Delta$. In the near-horizon region, the
integral will become

$r^{*}\simeq\int\frac{r^{2}+(a+n)^{2}}{K(r-r_{+})(r-r_{*})}dr.$ (62)

By integrating above equation in the near-horizon region and using (61), we
find the echo time delay as

$\Delta\tau=\frac{2[r_{+}^{2}+(a+n)^{2}]}{K(r_{+}-r_{*})}\ln\left(\frac{\delta
r_{+}}{r_{+}-r_{*}}\right).\ $ (63)

The constant $\delta r_{+}$ is very small that can be calculated from Eq.
(51). Hence, we can obtain

$\delta
r_{+}=\frac{\eta^{2}l_{p}^{2}K(r_{+}-r_{*})}{4M[r_{+}^{2}+(a+n)^{2}]}.\ $ (64)

From Eq. (63), we can see that the echo time delay is sensitive to the value
of the position of the membrane in front of the horizon. Moreover, the
membrane position is determined by the constant $\eta$, which is related to
the length of the torus $L$ and proportional to the CFT degree of freedom $N$.
It is evident also that the echo time delay depends on all parameters,
including the Rastall coupling constant.

Figure 5: Echo time delay as the variation of $\kappa\lambda,\omega_{q}$
(left) and $a,\alpha,n,e,g$ (right). For the left panel, we set
$r_{+}=1,a=0.6,e,g=0.3,n=0.5;\alpha=0.03,\eta^{2}=10^{30}$ and
$\Delta\tau_{0}$ is the time delay for Einstein gravity. For the right panel,
we set $\upsilon=1$ for the variation of $a,n,e,g$ and $M=1$. $\Delta\tau_{0}$
is the time delay for the zero value of the parameters that we vary on the
right panel.

The black hole parameters, including the Rastall coupling constant, can modify
the time delay between two successive echoes. Hence, by observing the time
delay of the ringdown waveform of the ECOs, one can observe the parameters,
especially to determine the Rastall coupling constant. When we can quantify
the deviation from the Einstein gravity, we can expect that there exists
gravitational theory beyond Einstein theory, which is, in this case, Rastall
theory. In Fig. 5, we show the plot of the echo time delay normalized by the
time delay for vanishing parameters. For example, when we vary $n$, it means
that $\Delta\tau_{0}$ is the time delay when $n=0$. From the left panel, we
can see that for every arbitrary $\omega_{q}$, an irregular time delay
corresponds to specific $\kappa\lambda$. The irregularity occurs because
$\upsilon$ can be infinite for arbitrary $\omega_{q}$ and $\kappa\lambda$.
Despite that, generally, we can observe for the ECOs in Rastall gravity that
the presence of the Rastall coupling constant hastens the echoes compared to
the ECO in general relativity, except when $\kappa\lambda$ approaches the
irregularity point from the left side. The positive coupling constant hastens
the echoes more than the negative one, except near the irregular point. In
[55], they note that the first echo in the time delay could be affected by
nonlinear physics, so there can be a magnitude change on the time delay at
$2\%$-$3\%$ level, and it might be detected in the future observations. In
Rastall gravity, for example, we can find the change of magnitude at that
level when $\sim 0.32\leq\kappa\lambda\leq 0.41$ and $\omega_{q}=0$. These
results provide fair information on the existence of a non-conserved energy-
momentum tensor in a strong gravity regime that can be probed from echo time
delay when we have the real observational data of echo time delay. We also
argue that the quantum effects on the near horizon region can be manifested as
a non-zero Rastall coupling constant and can be observed from the echo time
delay.

Besides the Rastall coupling constant, other parameters can also modify the
time delay leading to the delayed echoes. As we can see from the right panel
of Fig. 5, the increasing value of the parameters leads to the increasing of
time delay, yet with different curves. The red solid, dotted, and dashed lines
are the curves for the $\alpha$ variation for different $\upsilon$. For the
same $\alpha$, higher $\upsilon$ will produce higher time delay. It is also
seen that for higher $a,n,e,g$, the produced time delay will be higher,
denoting that the echoes will be delayed.

##  4 Summary

In this work, we performed the Kerr/CFT correspondence computation to delve
into the physical properties of rotating black holes and ECOs in Rastall
theory of gravity in which the KNUTK black hole solutions in Rastall gravity
have been chosen. Firstly, it has been shown that there exists a hidden
conformal invariance in the low-frequency scattering off KNUTK black hole in
Rastall gravity. Although the conformal symmetry is broken by periodic
identification in the configuration space, it acts on the solution space of
the scalar probe and associates a dual CFT description to the black hole. To
be specific, the KNUTK black holes in Rastall gravity are dual to 2D CFTs with
temperatures (28) and central charges (34). Nevertheless, since we wanted to
have regular central charges and non-vanishing temperatures, we required that
$K\neq 0$ which corresponds to
$\alpha\upsilon(\upsilon-1)r_{+}^{\upsilon-2}\neq 2$. From the Cardy formula
counting the microstate degeneracy in the dual CFT, we recovered the
Bekenstein-Hawking entropy. For the massless neutral scalar scattering, we
identified the dual operators with conformal dimension, left and right
frequencies, which allow us to find the perfect match with the CFT prediction.
Our findings on this black hole support that the conformal symmetry in the
solution space of the scalar probe is essential to set up a CFT description.
Henceforth, not only on the extremal condition [21], yet the non-extremal
rotating black holes in Rastall gravity are also holographically dual with
CFTs.

We also analyzed the holographic CFT description of an ECO for which the
exterior space-time resembles the KNUTK black hole in Rastall gravity. There
exists a reflective quantum membrane very close to the would-be horizon of the
ECO, which produces distinct physical properties, especially the oscillatory
feature on the absorption cross-section compared to those of black holes. We
have implemented the dual CFT description of this ECO to compute the QNM
spectrum where the spectrum comes from the poles on the exponential parts of
the two-point function in the upper and lower half of $\omega$-plane. The QNM
spectrum is very sensitive to the change of the black hole’s parameters,
including the Rastall coupling constant, resulting in several consequences.
For example, higher $\upsilon$ produces a lower QNM spectrum.

From the calculation of the absorption cross-section, the Rastall coupling
constant and other parameters possess significant contributions to the
superradiant instability of the ECO where the sign of the Rastall coupling
constant is notably essential to increase or decrease the instability. We also
computed the echo time delay as the necessary observable from the
gravitational echoes observation in the postmerger ringdown signal. Besides
the quantum gravitational effects that could modify the near-horizon region,
it could also be manifested as a non-conserved matter tensor giving rise to a
non-vanishing Rastall coupling constant. It was shown that the Rastall
coupling constant played a significant role in time delay. For instance, near
the irregular point from the negative sign side of the coupling constant, the
time delay gets higher, which denotes the change of magnitude in time delay
coming from the correction of non-linear physics. In a sense, the non-
vanishing Rastall coupling constant could be grasped as the phenomenological
realization of that correction. Black hole and ECO considered in this paper
contain many parameters which are $M,a,e,g,n,\alpha,\omega_{q}$, and
$\kappa\lambda$. It is very important to observe the effect of each parameter.
We believe that from our calculation on the QNM spectrum, absorption cross-
section and time delay, the effects of each parameter can be studied
effectively. Hence, our analysis within this paper tries to provide the
possibility of probing Rastall gravity in the black hole and ECO. We believe
that the future observations from interferometric detectors at higher
sensitivity will confirm these findings of which the time delay can probe the
existence of the Rastall gravity.

## Acknowledgments

M. F. A. R. Sakti is partially supported by C2F Fellowship, Chulalongkorn
University. A. Sulaksono is partially supported by DRPM UI’s (Skema PPI Q1
2021) Grants No. NKB-586/UN2.RST/HKP.05.00/2021. A. Suroso and F. P. Zen are
supported in part by Riset ITB. F. P. Zen is also supported by the Ministry of
Education, Culture, Research and Technology of Indonesia.

##  Appendix A CFT Two-point Function

It has been conjectured that the quantum gravity living in the near-horizon
region of the Kerr-like ECO is dual to a 2D CFT living on a circle of a finite
length $L$ [43]. Similarly with those in the black holes, for ECOs, the CFT
coordinates are defined by

$t_{L}=-2\pi T_{L}\phi-2n_{L}t,~{}~{}~{}t_{R}=2\pi T_{R}\phi+2n_{R}t.$ (65)

We can also apply the following azimuthal periodicity with the addition of
thermal periodicity of imaginary time

$\phi\rightarrow\phi+2\pi
L+i\frac{\Omega_{H}}{T_{H}},~{}~{}~{}t=t+\frac{i}{T_{H}}.$ (66)

Under the above identifications, the CFT coordinates now become

$\displaystyle(t_{L},t_{R})$ $\displaystyle\sim$
$\displaystyle(t_{L},t_{R})+\left(-4\pi^{2}T_{L}L-2\pi
iT_{L}\frac{\Omega_{H}}{T_{H}}\right.$ (67) $\displaystyle\left.-2\pi
i\frac{n_{L}}{T_{H}},4\pi^{2}T_{R}L+2\pi
iT_{R}\frac{\Omega_{H}}{T_{H}}\right).\ $

The QNM spectrum can be found from the poles of the retarded CFT correlation
function. However, the correlation function that will be used is generalized
for general value of modular parameters ($\mathbf{a,b,c,d}$). It is worth
performing Fourier transformation of the retarded CFT correlation function to
the momentum space. The Fourier transform of the two-point function for the
general value of modular parameters is given as

$\displaystyle\tilde{G}(\omega_{L},\omega_{R})$ $\displaystyle=$
$\displaystyle\int dt_{L}dt_{R}e^{-i\omega_{L}T_{L}}e^{-i\omega_{R}T_{R}}$
(68) $\displaystyle\times$
$\displaystyle\sum_{\bar{p}\in\mathcal{Z}}\frac{(\pi
T_{L})^{2h_{L}}}{\left[\sinh\left\\{\pi T_{L}\left(\frac{t_{L}}{2\pi
T_{L}}+2\pi\bar{p}L+\frac{i\mathbf{a}\bar{p}}{T_{L}}\right)\right\\}\right]^{2h_{L}}}$
$\displaystyle\times$ $\displaystyle\frac{(\pi
T_{R})^{2h_{R}}}{\left[\sinh\left\\{\pi T_{R}\left(\frac{t_{R}}{2\pi
T_{R}}+2\pi\bar{p}L+\frac{i\mathbf{a}\bar{p}}{T_{R}}\right)\right\\}\right]^{2h_{R}}}.\
$

Then by defining the following new coordinates

$\displaystyle\tilde{t}_{R}=\frac{t_{R}}{2\pi
T_{R}}+2\pi\bar{p}L+\frac{i\mathbf{a}\bar{p}}{T_{R}},$
$\displaystyle\tilde{t}_{L}=\frac{t_{L}}{2\pi
T_{L}}+2\pi\bar{p}L+\frac{i\mathbf{a}\bar{p}}{T_{L}},\ $ (69)

one can find

In above two-point function, there are two types of poles emerging from the
gamma functions and exponential parts lying in the upper and lower half of the
$\omega$-plane. It has been shown in [22, 24] that the QNM from the poles
emerging from the gamma function agree with the QNM spectrum of the black
holes. In addition, the poles emerging from the exponential parts will result
in the QNM spectrum for ECOs. In order to produce the QNM spectrum for ECOs,
we choose the pole in the lower half plane which gives

$2\pi
iL(\tilde{\omega}_{L}+\tilde{\omega}_{R})-\mathbf{a}\big{|}\frac{\tilde{\omega}_{L}}{T_{L}}+\frac{\tilde{\omega}_{R}}{T_{R}}\big{|}=2\pi
iR_{n},$ (71)

where $R_{n}$ is positive integer. From above equation, we can approximate the
value of QNM spectrum.

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