Near-extremal Kerr-like ECO in the Kerr/CFT Correspondence

The Kerr/CFT correspondence has been established to explore the quantum theory of gravity in the near-horizon geometry of a extreme Kerr black holes. The horizon can be replaced by partially reflective membrane due to quantum gravitational corrections on the near-horizon region. Because of this modification, black holes now can be seen as a horizonless exotic compact object (ECO). In this work, with Kerr/CFT correspondence we study the properties of Kerr-like ECO in near-extremal condition. We compute the quasinormal modes and absorption cross-section and compare the results with CFT computation. The dual CFT one needs to consider finite size/finite N effects in the dual CFT terminology. We find consistency between properties of the ECOs from gravity side and from CFT side. The quasinormal mode spectrum is in line with non-extreme case with differences in the length of the circle where the dual CFT lives, and phase shift of the incoming perturbation. The absorption cross-section has oscillatory feature that start to vanish in near extremal limit. We also show that the echo time-delay depends on the position of the membrane and extremality of the ECOs.


Introduction
The Anti-de Sitter/conformal field theory (AdS/CFT) correspondence is proposed by 't Hooft [1], which suggests that a higher-dimensional theory can be described by a corresponding lower-dimensional one and leads to significant development in string theory which is based on the idea of the holographic principle.This correspondence provides a useful tool for exploring strongly coupled theories by relating them to weakly one and vice versa.In terms of investigating the thermodynamics of black holes, the well known Kerr/CFT correspondence [2] is successfully study of the microscopic origin of Kerr black hole's entropy.It is shown that its near-horizon geometry exhibits an exact SL(2, R) × U (1) symmetry leading to the precise computation of Bekenstein-Hawking entropy from two-dimensional (2D) Cardy formula.For non-extremal Kerr black hole, unlike extreme case, the conformal symmetry is encoded in the solution space of a probe scalar field in the near-horizon region within the low-frequency limit approximation.One can see the conformal symmetry from the quadratic Casimir operator that satisfies SL(2, R) × SL(2, R) algebra.However, the conformal symmetry is globally broken into U (1) × U (1) symmetry by the periodic identification of the azimuthal ϕ.The spontaneous breaking of the symmetry is caused by the left-and right-moving temperatures T L,R .For more example on extremal Kerr/CFT correspondence, one can find in [3,4,5,6,7] and for nonextremal one in [8,9,10] cases.For a review, one can find in Ref. [11].
After the success of Kerr/CFT correspondence as an alternative to study the properties of classical rotating black holes, one may expect that this correspondence can be extended for quantum black holes.One of the studies that considers quantum black holes in the relation with Kerr/CFT correspondence is Ref. [12].Without losing the generality, we may assume that due to the strong gravitational attraction from the black holes, quantum effect can affect the structure of near-horizon region of the black holes.As shown in Ref. [13,14], the quantum effect appears to influence the apparent horizon of the classical black holes that leads to potentially observable consequences such as the quasinormal modes (QNMs).The structure of the horizon is assumed to be significantly altered so it becomes infinitesimal quantum membrane.Unlike usual horizon, this quantum membrane is reflective and located slightly in front of the wouldbe horizon in the order of Planck length.This novel representation offers another alternative way to solve the black hole information paradox.Beside this alternative, one may read some astrophysical objects that also come to solve the same problem which are fuzzballs [15,16], 2-2 holes [17], gravastars [18,19,20], and Kerr-like wormholes [21].These objects is known as Kerrlike exotic compact object (ECO).Due to the lack of an exact calculation within a theory of quantum gravity, the precise origin of the quantum reflective membrane is still not well grasped.However, the analysis of dual CFT on this ECO is portrayed in terms of Boltzmann factor that matches with the reflectivity of the quantum horizon [22].Some potential observable is expected to emerge because of the existence of the quantum reflective membrane.One of them is the presence of gravitational echoes.The gravitational echoes may be detected in the postmerger ringdown signal of a binary system coalescence such as black hole coalescence [23,24,25,26,14].In particular, the detection of QNMs is major realizations for such observable detection of echoes since the ringdown phase is dominated by this modes.Due to the presence of the reflective membran, the echo signals that bring these modes are separated from the primary ringdown signal by the corresponding time-delay.In several papers, it is claimed that the potential evidence of gravitational echoes has been discovered within LIGO data [27,28,29,30].
It is pointed out in Ref. [26] that the time between the main merger event and the first echo may be affected by non-linear physics.Due to this effect, the magnitude of the time-delay may be shifted around 2%-3%.For an example, this shift appear in Rastall theory of gravity where the covariant derivative of the matter tensor is not null and proportional to an arbitrary constant [31].Not only the time-delay, the analysis of QNMs in the ringdown spectrum can be a proper procedure to verify the natural structure of black holes or ECOs.Furthermore, this also can be another fascinating playground to probe some features beyond general relativity.The calculation of QNMs might probe the extra dimension from the ECOs in such braneworld scenario [32,33].
In near future, LIGO and Event Horizon Telescope or some other experiments will run to improve the precision related to the study of astrophysical objects and phenomena.Specifically, they will run some experiments to advance black hole's observations and related to its properties.Kerr (rotating) black hole is considered as the most physical black hole in the universe.A huge number of observations denotes that the observed black holes are rotating very fast, nearly the extremal limit, especially the supermassive ones in the active galactic nuclei [34,35].As a very specific example, the near-extremal Kerr is found to be the source of X-ray in the binary system GRS 1915+105 [36].Therefore, it is very relevant to study the near-extremal black hole and its quantum counterpart (ECOs).
In this work, we will consider near-extremal Kerr ECO.We will study the scalar field in that background.Instead of imposing purely ingoing boundary condition, we impose reflective 3 boundary conditions at the place slightly near the event horizon of the near-extremal Kerr metric (would-be horizon).Due to the scalar field perturbation, we compute the QNMs by solving the propagation of massless scalar field modes.We then compare the results by using CFT dual computation as has been done for generic Kerr ECO.In the end, we will compute the echo time-delay produced by the propagation of the fields on the near-extremal Kerr ECO background.
The organization of this work is given as follows.In the next section, we consider Kerr metric and its properties as ECO.In Section 3, we consider massless scalar field in the near-extremal Kerr ECO background and assume the reflective boundary condition to compute the QNMs and absorption cross-section.In Section 4, we compare the result from gravity computation with the CFT dual computation.Then in Section 5, the echo time-delay is computed for all fields.Finally, we summarize our work in the last section.

Kerr-like exotic compact object
In this work, we consider horizonless Kerr-like ECO.The horizon is replaced by partially reflective membrane.For Kerr-like ECO with mass M and angular momentum J = aM , the metric in the Boyer-Lindquist coordinate can be written as where ρ2 = r2 + a 2 cos 2 θ, ∆ = r2 + a 2 − 2M r. ( The position of the would-be horizons and the angular velocity on the horizon are given by The Hawking temperature and the Bekenstein-Hawking entropy are given by The partially reflective membrane with reflectivity R can emerges as a correction from quantum gravitational effects in the near-horizon geometry [37,38].The membrane is located outside the would-be horizon (r + + δr), where δr is assumed to be the order of Planck length to avoid instability [37].In black hole case, an incident perturbation with some modes is reflected by the angular momentum barrier while some others will traverse the barrier and absorbed by the black hole through the event horizon.However, because of the presence of the reflective membrane, the perturbations are trapped between membrane and the angular momentum barrier until eventually leak through the barrier as a repeating echoes with some time-delay.
There are various models for the membrane that we can use, such as constant reflectivity and Boltzmann frequency-dependent reflectivity [13,14] with R c is a constant and T QH is defined as "quantum horizon temperature" [22] and expected to be comparable to the Hawking temperature with an arbitrary proportional constant (T QH = 10th Asian Physics Symposium (APS 2023) Journal of Physics: Conference Series 2734 (2024) 012072 γT H ). The proportionality γ depends on the dispersion and dissipation effects in graviton propagation [39].In this work, we only consider the Boltzmann reflectivity.We want to see how this modification of boundary condition at the near-horizon region affects the absorption cross-section and QNMs of Kerr-like ECO in near-extremal condition.Later we will compare the results from gravitation calculation with a dual CFT analysis.

Near-extremal Kerr ECO background
From the correspondence between Kerr black holes and the CFTs, it is shown that absorption cross section is equivalent to two-point function in 2D CFT [40].Turns out that this correspondence also appear in Kerr-like ECO as shown in [12].They show that additional features from the new boundary condition can be interpreted as an effect from 2D CFT living on a finite circle.Another physical properties that we can investigate is QNM, where the reflectivity influences imaginary part of the QNM.As both quantities are affected by the reflectivity, we cannot take extremal limit right away to study the scattering problem because the Boltzmann reflectivity will vanish.However, for near-extremal case, we may be able to explore the scattering problem.So in this section, we focus on near-extremal condition of the Kerr-like ECO.
In order to study the scattering of a massless scalar field, we choose ansatz From Klein-Gordon equation, the field equations for a Kerr metric (1) are and where K l is separation constant.We then define a dimensionless coordinate x and dimensionless Hawking temperature τ H [41] as where the near-extremal condition corresponds to τ H ≪ 1.In this new coordinate, the radial equation ( 8) becomes where the prime denotes ∂ x and In the near-extremal limit, n is held fixed when T H → 0. That means only modes with energy near superradiant bound ω ≃ Ω H will cross the angular momentum barrier.We will solve the radial equation in near-horizon region x ≪ 1, then impose the new boundary condition.The solution derivation is similar to the near-extremal black holes case [41,50] as the ECO has Kerr spacetime as its exterior background.The difference becomes apparent when we impose reflective boundary condition to include the reflective membrane.
In the near-horizon region x ≪ 1, the radial equation becomes The solution can be written as F (a, b, c; z) where and z = −x/τ H .The solution behaves near the membrane x → 0 as where r * is the tortoise coordinate defined as The first part on the right hand side of Eq. ( 15) can be seen as the ingoing wave and the second part as the outgoing wave.Thus the new boundary condition at the membrane can be defined as where x 0 is the position of the membrane and δ is phase shift determined by the properties of the ECO.After we impose this new boundary condition, the solution behaves asymptotically as where and 10th

Absorption cross-section
The absorption cross-section is the ratio between absorbed flux near the would-be horizon and incoming flux from infinity There is actually another factor comes from asymptotic region contribution [41], but in this calculation we only consider near-horizon contribution to match with CFT.Since we want to study near-extremal condition τ H ≪ 1, |C| is more dominant than |D|.Using |C| given by ( 19), we find where r * 0 = r * (r 0 ) is the position of membrane in tortoise coordinate, ln(x 0 /τ H ) ∼ r * 0 (r + − r − )/(r 2 + + a 2 ).In addition to the classical features where the cross-section is negative, there is an oscillatory feature unique for ECO that depends on R. Obviously for R = 0, it is associated with the classical black hole.

Quasinormal modes
The oscillation of radiation from ECO corresponds to the resonance at the ECO's QNM frequencies.Parameters contained by the ECO contribute to the QNM spectrum.Moreover, since those parameters also determine the near-horizon quantum structure of the ECO, QNM may contain significant information about it, as shown in Ref. [31].To obtain the QNM spectrum of black holes or ECOs, we impose a purely outgoing boundary condition at the asymptotic infinity.However, it is difficult to solve the wave equation analytically, so it is impractical to get the QNMs of ECOs without make some assumptions.In this work, we assume low-frequency limit and near-superradiant bound to solve approximately the QNM spectrum.Another method is using AdS/CFT duality where QNM is obtained from poles of retarded Green's function [42].With no incoming waves from infinity and the modified near-horizon boundary condition, the QNM spectrum is given by where q is a positive integer.If we use the Boltzmann reflectivity, we get This result has same explicit form with the QNM spectrum of Kerr-like ECO for non-extreme case [12].The difference are in the definition of r * 0 where it is taken to the near-extremal condition here, and in the value of δ which we will see in the next section.

Hidden conformal symmetry
It has been established that Kerr black hole is dual to 2D CFT [2] through Kerr/CFT correspondence.However, the conformal symmetry is hidden for non-extremal Kerr black holes [40].It is hidden locally on the radial scalar wave equation and globally broken under periodic identification of ϕ.To unveil the symmetry we need to investigate the scalar wave equation at the near-horizon region.As we see in Eq. ( 14), the solution of scalar wave equation in this region is given by the hypergeometric functions, which fall into SL(2, R) representation.
We introduce the following set of conformal coordinates to understand the hidden symmetry [40,43] where we define right-and left-moving temperatures as In terms of these new coordinates, we can define two sets of vector fields, and where ∂ ± = ∂/∂w ± .Each set of the vector fields can form a quadratic Casimir operator that reads as It is shown in Ref. [40] that if we transform this back to the original coordinates ( t, r, φ), we can write the scalar field equation at the near-horizon region for Kerr background (1) as the SL(2, R) Casimir operator.As we have mentioned earlier, the SL(2, R) × SL(2, R) symmetry breaks into U (1) × U (1) under periodic identification of azimuthal coordinate, φ → φ + 2π.
For fixed r, we can write the relation between the conformal coordinates and Boyer-Lindquist coordinates as where we define In order to match between gravitation and CFT computation, we need to determine left and right frequencies ωL , ωR .We can derive the relation between these frequencies and (ω, m) of Kerr spacetime through the first law of black hole's thermodynamics We identify δM as ω and δJ as m, and consider conjugate charges (δE L , δE R ), which leads to identification of δE L,R as ωL,R .Thus, relations between (ω, m) and δE L,R are

QNMs from CFT
As mention earlier, another way to find QNM spectrum is from poles of retarded CFT correlation function based on AdS/CFT duality perspective.By using this correspondence, the QNM computation is agree with the gravity side as shown in Refs.[8,9,44,45,46].As for the ECOs, modification of near-horizon region as a result of quantum gravitational effect may come from finite size/finite N effects in the usual AdS/CFT terminology [47,48,49].Furthermore, discrete QNMs spectrum is believed to be the result this effects on the CFT side.
In order to obtain QNM spectrum of ECO and later its absorption cross-section, we start with two-point function of 2D CFT living on a torus with finite spatial cycle of length L and temporal cycle of length 1/T .The detailed derivation of the two-point function is given in [12].We keep both spatial and temporal coordinates to be finite, unlike for black hole case where cylindrical approximation of torus (L ≫ 1/T ) is used.We then add additional thermal periodicity of imaginary time to the azimuthal one With this new identification, the CFT coordinates (39) become The CFT two-point function based on torus coordinate (39) and their periodicity ( 44) is given as where a is general value of modular parameter and we defined new torus coordinate tR,L as There are two poles lying in the upper and lower half of the ω plane that come from exponential part of the two-point function.These poles correspond to QNM spectrum unique for ECOs.Other poles from the singularities of the Gamma function correspond to the usual Kerr black hole QNM spectrum.In this case, we need poles from the retarded correlation function, so the poles of (45) in the lower half plane is the relevant one.Based on the definition of CFT temperatures (30) and CFT frequencies (42), consider near-extremal and near-superradiant limit, the QNM spectrum is Both real and imaginary part of QNM for ECOs would match with the CFT result if we take Compared with non-extreme case, this QNM spectrum dependency of spin of the ECO (a) is different, since a and M is interchangeable in this case.In fact, if we consider a ≃ M condition of Eq. ( 40) in [12], it reproduces the same QNM spectrum with (47).Because of this, The definition of L and δ is also different which then will contribute to the cross-section and echo time-delay.We can find from the imaginary part of QNM (47) that reflectivity of the membrane is which match exactly with Boltzmann reflectivity.
From QNMs, we can study the condition of ergoregion instability.It happens because of infinite amplification of trapped waves inside the ergoregion, between reflective membrane and potential barrier.When the wave travel across the ergoregion, it will gain energy from the ECO through Penrose's process.Since some waves are trapped because of the reflection of the membrane and potential barrier, the amplification is exponentially increase, create an instability.In the QNMs, it is described as Im(ω) > 0. To quench this ergoregion instability, some of the incoming waves need to be absorbed by the ECO, in other words the membrane should be partially reflective [37,38].From Fig. (1), we can see in our case the instability is avoided for all modes by choosing the modular parameter a = 1/2 (γ = 1) or positive in general.As for the real part, modular parameter does not contribute into it.We can also see that nearer extremal limit (higher a/M ) produces higher ω and smaller L (lower r 0 ) also produces higher ω.

Absorption cross-section from CFT
In the previous section, the contribution of reflective membrane is marked by new poles in the two-point function.The identification (48) proves QNM from CFT consistent with gravity calculation.In Eq. ( 24), the presence of the reflective membrane appears as an oscillatory features.We want to show if the identification (48) can produce the same features to the absorption cross-section on CFT computation.
In the CFT computation, we again use CFT living on a circle with finite length L and look into the finite size effects on the boundary.The absorption cross-section can be defined using Fermi's golden rule given by The ±iϵ determine the poles that contribute while performing the integration.Thus we obtain We get identification of CFT parameter so the result agrees with gravity calculation ( 24) along with (48).If we take (30) and (42) in near-extremal near-superradiant limit, we will get the same CFT temperatures and frequencies with our results here.From Fig.
(2), the absorption cross-section is negative in low frequency similar to classical black hole because the superradiant condition.The oscillatory feature comes from the last factor in (51) which corresponds to reflective membrane contribution.If this factor vanishes or the reflectivity is zero, the crosssection is reduced to that of classical black holes.We can also see that when a/M → 1, the oscillatory feature is damped and starts to vanish.We can interpret this as for near-extremal case a ≃ M the Boltzmann reflectivity is suppressed by the spin of the ECO.

Echo Time-delay
The trapped waves eventually will leak as repeating echoes.The time gap between two consecutive echoes is the time to travel from the angular momentum barrier to the membrane and travel back again.This time-delay can be described in terms of the ECO parameters and the position of the reflective membrane.The echoes time-delay is defined as We can see that ∆τ is sensitive to the position of the membrane.In terms of distance of the membrane from the position of would-be horizon δr = r 0 − r + , the time-delay becomes where plus sign is for δr > r + −r − and minus sign is for δr < r + −r − .However, we only consider the later case because as we see in Fig. (3) as the distance between the angular momentum barrier and the reflective membrane decreases, larger δr should have shorter time-delay.Compared to non-extreme case [12] the difference is on the logarithmic factor.This is because in this case we use near-extremal limit (9) when converting the position of the membrane in tortoise coordinate to the proper distance.Nonetheless, the sensitivity of the echoes time-delay to the position of the membrane is still exist.In addition to that, there is also sensitivity to the extremality of the ECOs.The length of the torus in terms of echo time-delay is Thus, we obtain From this relation we can see that when δr → 0, we have L → ∞ which corresponds to classical case.If we expand (56) and take up to the first order, we have relation between δr and dimensionless Hawking temperature Since δr is assumed in order of Planck length for stability, this relation fits with near-extremal condition τ H ≪ 1.In Fig.
(3) we show the plot of echo time-delay as a function of a/M near extremality with variation of δr.When a/M → 1, echo time-delay will approach infinity, in other words there is no echoes.This can be interpreted as for extremal case, the reflectivity is suppressed by the spin of the ECO thus ECO absorbs the incoming wave like classical black hole and does not produce echoes.

Conclusions
In this work, we studied the CFT dual of Kerr-like ECO in near-extremal condition.Quantum gravitational effects allow us to replace horizon with partially reflective membrane that placed slightly outside the would-be horizon.To keep the reflectivity in the near-extremal condition nonzero, we only considered fields with energies near-superradiant bound.With this new reflective boundary conditions, we obtained the QNM spectrum and absorption cross-section of massless scalar field.We compared the results by using CFT dual computation with generic Kerr ECO.
The CFT computation of modified near-horizon region is done by considering dual field theory that lives on a finite toroidal two-manifold, where we kept the spatial and temporal periodicity finite.This method of modification can be interpreted as finite size/finite N effects in AdS/CFT.We showed the consistency between QNMs and absorption cross-section computed from the gravity side and similar quantities from the dual CFT.The differences of QNMs with nonextreme case are in the identification of torus length L and phase shift δ.We reproduced Boltzmann reflectivity from imaginary parts of QNMs from CFT, which in line with nonextreme case.The imaginary part of QNMs is also associated with instability of the ECO.In near-extremal condition, it is suppressed by the partially reflective membrane as long as we choose the modular parameter a to be positive.Other than that, the absorption cross-section contains oscillatory features that start to vanish when a/M → 1.It shows that near extremality the reflectivity of ECO is suppressed by the spin of the ECO.The CFT side reproduced this features by choosing CFT quantities that is consistent with near-extremal condition.
In the end, we calculated the echo time-delay produces by the perturbation on the nearextremal Kerr ECO background.Echo time-delay is one of the observable from the gravitational echoes observation in the postmerger ringdown signal.The modification of near-horizon region is manifested in this observable.We showed that the echo time-delay has sensitivity to the position of the membrane and the extremality of the ECOs.It is very important to explore furthermore the quantum corrections to the near-horizon geometry of near-extremal Kerr from the CFT side.It may lead to a better understanding of the microscopic theory.We believe that our calculation on the QNMs, absorption cross-section, and echo time-delay in near-extremal condition can be used to advance the study of ECO, especially with the relation with experiment since in the near future more experiments will run to improve the study of astrophysical objects.In the future, extending our analysis for higher spin fields may provide the possibility of observing these physical properties of ECO such as through gravitational waves.

10thFigure 1 .
Figure 1.Real and imaginary part of QNM as a function of q with a/M and r 0 variations.We set m = 1 and a = 1/2.

10thFigure 3 .
Figure 3. Echo time-delay as a function of a/M near extremality with variation of δr.

10th
Asian Physics Symposium (APS 2023) Journal of Physics: Conference Series 2734 (2024) 012072 This relation reminds us to the relation between Minkowski coordinates (w ± ) and Rindler coordinates (t R,L ).The frequencies (ω L , ω R ) related with the Killing vectors (i∂ t L , i∂ t R ) are conjugate to (t L , t R ). 8