Universal Entropy and Hawking Radiation of Near-Extremal AdS$_4$ Black Holes

We compute the Bekenstein-Hawking entropy of near-extremal asymptotically AdS$_4$ electrically charged rotating black holes using three different methods: (i) from the gravity solution, (ii) from the near-horizon Kerr/CFT correspondence and (iii) from the boundary conformal field theory. The results from these three different approaches match exactly, giving us a unique and universal expression for the entropy and the microstate counting of near-extremal AdS black holes via the AdS/CFT correspondence. In the second method, we extend the Kerr/CFT correspondence to the near-extremal case to compute the left and right central charges. We also use hidden conformal symmetry of the near-horizon geometry to compute the Frolov-Thorne temperatures. From the results of the near-extremal AdS$_4$ black hole entropy, we provide a microscopic foundation for Hawking radiation.


Introduction
The groundbreaking work of Strominger and Vafa [1] provided the first microscopic explanation for the entropy of certain asymptotically flat black holes, and inspired a considerable amount of work in regards to understanding the microstates of asymptotically flat black holes. In the last few years, this has been expanded on for asymptotically AdS black holes using the AdS/CFT correspondence [2,3]. In particular, for a special class of supersymmetric extremal AdS black holes, which we call BPS black holes, the Bekenstein-Hawking entropies can be reproduced from the corresponding boundary conformal field theories (CFT) [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18].
Combining with the near-horizon Kerr/CFT correspondence [19][20][21], we now have a unifying picture as shown in Fig. 1. More precisely, for BPS black holes, their entropies can be computed in various ways, including (i) the original Bekenstein-Hawking formula on the gravity side, (ii) the near-horizon Kerr/CFT correspondence and (iii) the microstate counting from the boundary CFT via AdS/CFT correspondence. In addition, the useful limit called the gravitational Cardy limit can simplify the geometry near the horizon, by producing an AdS 3 subgeometry, which has been explicitly verified for AdS 4,5,6,7 BPS black holes in [22].
Most of the previous work has assumed supersymmetry, and one natural question that may arise is whether or not the entropy matching is successful when the BPS boundthe intersection of supersymmetry and extremality is relaxed. In order to investigate this question, we study near-extremality in this paper. There has already been progress made in this direction, such as in [23], where AdS 5 black holes have been discussed from gravity, including near-extremal (T H ą 0) and extremal near-BPS (T H " 0) configurations. For the latter case, the boundary field theory was also explored. Later, [24] resolved some technical problems and successfully derived the heat capacity for near-extremal AdS 5 black holes from the boundary N " 4 supersymmetric Yang-Mills theory (SYM). In addition, [24] used the Bardeen-Horowitz near-horizon rescaling [25] and the Kerr/CFT correspondence [19][20][21] to reproduce the near-extremal AdS 5 black hole entropy from the near-horizon CFT 2 . Consequently, it provided a new microscopic formalism of Hawking radiation, similar to the D1-D5 CFT interpretation for asymptotically flat black holes studied by Callan and Maldacena [26]. In this work, we extend the analyses of [22] from the BPS case to the near-extremal case. However, unlike [22], we do not assume the Cardy limit.
More explicitly, we compute the near-extremal electrically charged rotating AdS 4 black hole entropy using three different approaches: 1. expansion of the non-extremal AdS 4 black hole solution around the BPS solution, 2. near-extremal Kerr-Newman-AdS/CFT correspondence from the near-horizon CFT 2 , 3. microstate counting via AdS/CFT correspondence from the boundary 3d superconformal ABJM theory at small temperature.
We find that the entropy computations from the three different approaches lead to one universal result for the entropy, which can be formally written as where S˚denotes the electrically charged rotating AdS 4 black hole entropy in the BPS limit, while pC{T H q˚stands for the heat capacity in the BPS limit. Therefore, our results use a concrete example to support the existence of a near-horizon CFT 2 , which accounts for the black hole entropy, and show that the Kerr/CFT correspondence [19][20][21], which was originally applied only to extremal black holes, can also be extended to the nearextremal black holes. In addition, the near-extremal AdS 5 black hole discussed in [24] and the near-extremal AdS 4 black hole considered in this paper support that the unifying picture shown in Fig. 1 is valid for both BPS black holes and near-extremal black holes. As a byproduct, we clean up some technical issues in previous literature. This includes the computation of the right central charge of the near-horizon CFT 2 and that c L " c R similar to [27]. In addition, we can unambiguously derive the right Frolov-Thorne temperature T R and consequently the black hole entropy from the right sector without any cutoffs, as in [27,28]. Several recent works based on 2d JT gravity coupled to a 2d bath CFT [29][30][31][32] have provided a holographic description of Hawking radiation and black hole evaporation, which can potentially be used to address the long-standing problem of black hole information paradox. This paper together with [24] give an alternative approach of studying black hole evaporation and information paradox via AdS/CFT correspondence. In contrast to [29][30][31][32], which are intrinsically 2d models, both [24] and our work include both higher-dimensional CFTs on the boundary and a near-horizon 2d CFT, which may be more robust.
This paper is organized as follows. In Sec. 2 we carefully discuss how to achieve nearextremal black holes by perturbing the extremal condition for the supersymmetric black hole solutions, and then compute the near-extremal electrically charged rotating AdS 4 black hole entropy in three different ways. In Sec. 3 we connect the near-extremal AdS 4 black hole entropy obtained in the previous section with Hawking radiation, and provide a microscopic formalism similar to [26] for asymptotically flat black holes and [24] for AdS 5 black holes. Some discussions for the future research directions are presented in Sec. 4.
While this work was being completed, we became aware of the related work [33], which has some overlap with Subsection 2.2 in this paper.

AdS 4 Black Hole Solution
In this subsection, we review the asymptotically AdS 4 black holes of interest along with some of their properties. The non-extremal asymptotically AdS 4 electrically charged rotating black hole solution with a gauge group U p1qˆU p1q was constructed in [34,35] as a solution to 4d N " 4 gauged supergravity. This is a special case, where there are two electric charges that are pairwise equal (Q 1 " Q 3 , Q 2 " Q 4 ) along with one angular momentum J. The solution is characterized by four parameters pa, m, δ 1 , δ 2 q. The metric, the scalar fields and the gauge fields are given by where r i " r`2m sinh 2 pδ i q , pi " 1, 2q ∆ r " r 2`a2´2 mr`g 2 r 1 r 2 pr 1 r 2`a 2 q , ∆ θ " 1´g 2 a 2 cos 2 θ , W " r 1 r 2`a 2 cos 2 θ , and g " ´1 4 is the inverse of the AdS 4 radius. We have added two pure gauge terms to the two gauge fields A 1,2 with constants α 41 and α 42 . The metric (2.1) can be written equivalently as with the factors B and f given by The 1 4 -BPS supersymmetric solutions can be obtained by imposing the condition e 2δ 1`2 δ 2 " 1`2 ag , (2.6) or equivalently, a " a 0 , with a 0 " 2 g pe 2δ 1`2 δ 2´1 q . (2.7) The extremal black hole solutions are achieved when the function ∆ r prq has a double root, or equivalently when the discriminant of ∆ r prq vanishes, which can be viewed as an equation for m. We can solve when the discriminant is zero and obtain the extremal value of m as a function of a and δ 1,2 , i.e., m " m ext pa, δ 1 , δ 2 q . (2.8) Since this computation is straightforward, we omit the lengthy expression of m ext pa, δ 1 , δ 2 q. In this case, for m ă m ext the function ∆ r prq has two different real roots corresponding to the outer and the inner horizons of a non-extremal black hole. For m ą m ext the function ∆ r prq does not have real roots, which implies that the solution has a naked singularity instead of a black hole. We would like to emphasize that the asymptotically AdS 4 Kerr-Newman black holes have also been discussed in [36]. However, the supersymmetric solutions considered [36] have 1 2 -BPS supersymmetry instead of 1 4 -BPS supersymmetry discussed in [34,35], which makes some features of the black holes different.
Before moving on, we want to emphasize the parameter space we are exploring. Particularly, we want to focus on the parameters m, a and δ 1 as well as the conditions that characterize supersymmetry and extremality of the black hole. Our goal is to have a black hole solution that is slightly perturbed from these two conditions. Hence, it satisfies both the supersymmetric condition (2.7) and the extremal condition (2.8). Under the supersymmetric condition (2.7), the extremal condition (2.8) is equivalent to which can also be obtained by requiring the black hole solution to have a regular horizon. The two conditions (2.7) and (2.9) in [35] contain typos, which have been corrected in [37,38] and also [39]. With these two constraints, there are only two independent parameters for asymptotically AdS 4 electrically charged rotating BPS black holes. To illustrate the relations of the parameters, we plot in Fig. 2 the codimension-1 supersymmetric surface defined by (2.7) together with the codimension-1 extremal surface defined by (2.8) in the parameter space pm, a, δ 1 , δ 2 q, where for simplicity we set δ 2 " δ 1 and L " 1. The intersection of these codimension-1 surfaces is a codimension-2 surface corresponding to the BPS solutions. We now collect useful properties of the black hole, including the position of the outer horizon in the BPS limit which coincides with the BPS inner horizon. For the thermodynamic quantities of the non-extremal asymptotically AdS 4 black holes, the gravitational angular velocity Ω and the temperature T H are given by Ω " ap1`g 2 r 1 r 2 q r 1 r 2`a 2 , T H " 11) which are evaluated at the outer horizon r`, and the prime ( 1 ) denotes the derivative with respect to r. The other thermodynamic quantities are [35] S " (2.12)

Near-Extremal AdS 4 Black Hole Entropy from Gravity Solution
The asymptotically AdS 4 black hole solutions discussed in the previous subsection are in general non-extremal. Since our focus is on near-extremality, we perturb the BPS black hole solution. More precisely, we expand the non-extremal AdS 4 black hole solutions around the BPS solution by turning on a small temperature. We shall do this by studying the parameter space. Before imposing the constraints (2.7) and (2.9), there are 4 parameters that characterize the black hole solution, and we interchange one of these parameters, a, with the outer horizon r`, where r`is the biggest root of the equation ∆ r pr`q " 0, i.e., Note that we only do this replacement of a with r`when we compute the entropy in this subsection. There are two reasons why we make this change. The first is pragmatic: this simplifies the algebra significantly. The second is that the outer horizon r`plays a clear role in the nAttractor mechanism [40], which will also be relevant for the discussions later in this subsection. Now, we use (2.13) to solve for the parameter a in terms of r`, and the 4 independent parameters for the non-extremal AdS 4 black hole solutions are pr`, m, δ 1 , δ 2 q. Correspondingly, there are 4 independent physical quantities pT H , J, Q 1 , Q 2 q, where we have set Q 1 " Q 3 and Q 2 " Q 4 as in (2.12). Without loss of generality, we further set δ 2 " δ 1 , and therefore Q 1 " Q 2 , to simplify the discussion.
The black hole entropy in (2.12) is valid for any temperature, including small temperature. This is achieved by expanding around the BPS value of the entropy, leading to the expression where S˚denotes the AdS 4 black hole entropy (2.12) in the BPS limit while C is the heat capacity which is linear in T H , and´C T H¯˚i s evaluated in the BPS limit. Computing´C T H¯˚i s straightforward Once the dust settles, the result iŝ We comment that this result can also be obtained by only varying S with respect to r`, i.e.,ˆC This is similar to the AdS 5 case discussed in [23], which is related to the nAttractor mechanism [40]. This hints that the nAttractor mechanism extends to other dimensions.

AdS 4 Black Hole Solution in the Near-Horizon Limit
In this subsection, we consider near-extremal asymptotically AdS 4 black holes close to the 1 4 -BPS solutions by introducing a small positive temperature T , and discuss the corresponding metrics. It was discussed in [36] that for asymptotically AdS 4 black holes from the extremal case to non-extremal configurations corresponds to perturbing the parameter m from its extremal value m ext . As we can see from Fig. 2, when perturbing around the AdS 4 1 4 -BPS black holes, we can deviate from the extremal surface but still stay in the supersymmetric surface by imposing the supersymmetric condition (2.7). Meanwhile, we expand the parameter m around its BPS value given by (2.9) with a small dimensionless parameter λ [27] corresponding to near-extremal AdS 4 black hole solutions, i.e., where which coincides with m ext under the supersymmetric condition (2.7). In general m ext ‰ m 0 . A similar limit was also used in [41] to study the near-BPS black holes and compared with other limits in [5]. To summarize, near-extremal AdS 4 black holes can be achieved by perturbing the parameter m around m ext while keeping the other parameters fixed. This is made explicit in this paper by imposing the near-extremal condition (2.20) with the parameter a fixed by the supersymmetric condition (2.7). Moreover, we perform a near-horizon scaling to the asymptotically AdS 4 black hole metric (2.4), which was first introduced by Bardeen and Horowitz in [25] and extensively studied [22] for the BPS AdS 4 black holes In principle, for near-extremal black holes we should consider the near-horizon scaling r Ñ r``λ r r. However, the near-extremal condition (2.20) implies that r`and r 0 only differ by a constant of order λ. Hence, we can absorb that constant into r r and still take r Ñ r 0`λ r r in the near-horizon scaling. This kind of near-horizon scaling for near-extremal black holes has been used in [28]. To summarize, we impose the near-horizon scaling (2.22) together with the condition conditions (2.20) and (2.7).
Taking the limit λ Ñ 0, the metric (2.4) becomes From the near-horizon, we are now in a position to extract the necessary details to compute the entropy via the Kerr/CFT correspondence. As we shall see, there are several methods to compute the central charges, which requires a rewriting of the near-horizon geometry in different coordinate systems. To make things clearer, we summarize each of these different expressions of the near-horizon metric. The change of coordinates Therefore, it is clear that the near-horizon scaling we applied to the metric leaves us with a circle fibered over AdS 2 , yielding a warped AdS 3 geometry. We now see that the nearhorizon metric in Poincaré coodinates (2.26) is in the standard form with x i P t r φu for the AdS 4 case, and the coefficients f 0 pθq, f θ pθq, k i and γ ij pθq are functions of θ in general. Now, we transform the Poincaré coordinates pτ, ρ, θ, r φq in the metric (2.26) to the global coordinates pt,r, θ,φq using the following relations g ρ "r`a1`r 2 cosptq , g´1 τ " ?
1`r 2 sinptq r`?1`r 2 cosptq , (2.28) which leads to´ρ where κ " log˜1`? 1`r 2 sinptq cosptq`r sinptq¸. (2.30) Consequently, the metric (2.26) can be rewritten as Besides the near-horizon scaling (2.22), we can also apply a light-cone scaling in the near-horizon region [27] x`" ˆφ`e 4 δ 1´3 e 4 δ 1´1 gt˙, x´" φ´e and then consider the following near-horizon scaling in the light-cone coordinates Together with the condition (2.20) and taking the limit Ñ 0, we obtain the near-horizon metric for the AdS 4 near-extremal black holes in the coordinates px`, r r, θ, x´q ds 2 "´`e 4 δ 1´1˘2`e 8 δ 1`1 0 e 4 δ 1´7˘`e 4 δ 1`c osp2θq8 where Λ AdS 4 pθq is the same as (2.24). Introducing some new coordinateŝ x`" g`e 4 δ 1´1˘`e 8 δ 1`1 0 e 4 δ 1´74 we can rewrite the near-horizon metric in the light-cone coordinates (2.35) as We see that the metric (2.37) is in the standard form 38) with x i P tx´u for the AdS 4 case, and k i , f 0 pθq, f θ pθq and γ ij pθq remain the same as (2.27). To summarize, we now have several different expressions for the near-horizon metric in Poincaré and global coordinates. This is useful when we utilize the near-extremal Kerr/CFT correspondence.

Near-Extremal AdS 4 Black Hole Entropy from Cardy Formula
After obtaining the various expressions of the near-horizon metric of the asymptotically AdS 4 black holes, we are now ready to compute the central charges and the Frolov-Thorne temperatures using the near-extremal Kerr/CFT correspondence as well as hidden conformal symmetry of the near-horizon geometry to find the AdS 4 black hole entropy in the near-extremal limit. For the left central charge c L and the right central charge c R , there are two different ways for computing each of them, depending on which coordinate system we choose. We summarize each of these diverse approaches as a consistency check on our computation as well as to keep things self-contained.
The Kerr/CFT correspondence was originally posed for asymptotically flat extremal Kerr black holes [19] and was later shown to also be valid for asymptotically AdS black holes [20,21]. For the near-extremal case, [27,28] initiated some progress and we extend those results here by computing the entropies of near-extremal AdS 4 black holes via the Cardy formula. Before further exploring the near-extremal case, let us take a step back and recall how the Kerr/CFT correspondence works. The basic idea is the following. Taking the Bardeen-Horowitz near-horizon scaling [25], the near-horizon geometry of an asymptotically flat or asymptotically AdS extremal black hole contains U p1q cycles fibered on AdS 2 . The near-horizon asymptotic symmetries are characterized by diffeomorphims generated by the vectors The mode expansion of a diffeomorphism generating vector ζ is We can define a 2-form k ζ for a general perturbation h µν around the background metric g µν as We also define the Lie derivative with respect to ζ, denoted by L ζ , as The left central charge c L of the near-horizon Virasoro algebra can be computed using the Kerr/CFT correspondence in two slightly different ways. For the first method, the central charge can be computed using the following integral [19][20][21] 1 8πG ż BΣ k ζ pmq rL ζ pnq g, gs "´i 12 c L pm 3`α mq δ m`n, 0 , (2.43) where g denotes the near-horizon metric of the near-extremal AdS 4 black hole in global coordinates (2.31). An explicit evaluation of (2.43) shows that The other way of computing c L is to evaluate the following integral [27] 1 8πG N ż BΣ k ξn rL ξmḡ ,ḡs " δ n`m, 0 n 3 c L 12 , which matches exactly the result of c L (2.44) from the first approach. The right central charge c R can also be obtained in two different ways. Although we describe the two methods, we prefer one method over the other because of its robustness.
The first approach is to compute the quasi-local charge [27,42,43] using the standard form of the near-horizon metric of the near-extremal AdS 4 black hole in Poincaré coordinates (2.27), which is given by the integral where f 0 pθq, f θ pθq, γ ij pθq and k i are defined in (2.27), and the parameter Λ 0 denotes a UV cutoff in r. This approach has been used to compute the right central charge c R for near-extremal AdS 5 black holes [24]. For the four-dimensional case, the integral (2.47) can be applied to the near-horizon metric (2.26) in Poincaré coordinates to compute c R . However, the result is not very illuminating due to the unfixed cutoff Λ 0 .
To compute c R , we choose a more concrete approach using light-cone coordinates as introduced in [27]. More precisely, a scale-covariant right central charge c pcovq R can be computed from the near-horizon metric (2.37) by using where the factors γ ij pθq, f θ pθq and k´are defined in (2.38). Like in [27], we can define a scale-invariant right central charge c R " c pcovq R { , which in this case is We see that the result is exactly the same as the left central charge computed in (2.44) and (2.46). To summarize, the explicit expression for the integral changes slightly depending on the coordinate system, and we have shown that all the results do indeed lead to the same central charge. Now that we have taken care of the central charges, and have consistently gotten that c L " c R , the final ingredient is the Frolov-Thorne temperatures T L and T R . We have seen in [22] that for the BPS case T R " 0. For the near-extremal case, T L can still be computed in the same way discussed in [22], and its value remains the same as the BPS case, as it is unaffected by whether we impose the condition (2.20). Therefore, we find On the other hand, T R is proportional to the physical Hawking temperature T H . To find the exact expression of T R , we apply the technique of hidden conformal symmetry. This method was first introduced in [44], and later generalized to many different cases. The basic idea is to define a set of near-horizon conformal coordinates and corresponding locally-defined vector fields with SU p2, Rq Lie algebra, such that the wave equation of an uncharged massless scalar field becomes the quadratic Casimir of the SU p2, Rq Lie algebra.
In this way, we can fix the Frolov-Thorne temperatures T L,R and the mode numbers N L,R for non-extremal black holes.
In particular, [28] has considered the hidden conformal symmetry of an AdS 4 black hole close to the solutions discussed in this paper. We can apply the same technique by first expanding ∆ r defined in (2.3) ∆ r " kpr´r`qpr´r s q`O`pr´r`q 3˘, (2.51) where k and r s can be read off from the Taylor expansion to quadratic order in pr´r`q.
Based on hidden conformal symmetry [28], the right temperature is We also find that the expression obtained using hidden conformal symmetry for T L is (2.50) as expected. Using the Cardy formula, we obtain the near-extremal AdS 4 black hole entropy where the BPS entropy S˚is S˚" π 2 3 c L T L , (2.56) while the near-extremal correction to the black hole entropy is We see that this result from the near-horizon CFT 2 and the Cardy formula is exactly the same as the results from the gravity side ((2.14), (2.15) and (2.18)).

Near-Extremal AdS 4 Black Hole Entropy from Boundary CFT
What remains is the computation of the near-extremal entropy from the boundary CFT.
In the BPS limit, the AdS 4 black hole entropy can be obtained by extremizing an entropy function, which was derived by the superconformal index or supersymmetric localization of the 3d ABJM theory on the boundary of electrically charged rotating AdS 4 BPS black holes [11,12]. More precisely, the BPS entropy function is where r ∆ I are chemical potentials corresponding to the electric charges Q I , and r ω is the angular velocity. To extremize the entropy function (2.58), we solve the equations which can be expressed explicitly as (2.61) Substituting these equations back into the entropy function (2.58), we obtain Moreover, the equations (2.60) and (2.61) can be combined into one equation: which can be written more compactly as with the real-valued coefficients A " (2.65) In order to obtain a real-valued black hole entropy, the expression (2.62) implies that Λ should have a purely imaginary root. Since (2.64) is a quartic equation of Λ with real coefficients, the imaginary roots should come in pairs. Consequently, (2.64) can be factorized as Comparing (2.66) with (2.64), we find A " β , B " α`µ , C " αβ , D " αµ , (2.67) or equivalently, According to (2.62), the imaginary root Λ " i ?
A leads to the real-valued AdS 4 BPS black hole entropy For the special case Q 1 " Q 3 , Q 2 " Q 4 , the expression above becomes After imposing the identifications of parameters introduced in [11,12,38] Q BH,I " g 2 Q I , J BH " J , I P t1,¨¨¨, 4u (2.71) and using an entry from the AdS/CFT dictionary we can rewrite the BPS black hole entropy (2.70) as which can be subsequently written in terms of the free parameters pδ 1 , δ 2 q on the gravity side in the BPS limit. For the special case δ 1 " δ 2 , the BPS black hole entropy obtained from the boundary CFT is which is exactly the same as the BPS result from the gravity side (2.15) and the one from the near-horizon Kerr/CFT correspondence (2.56).
In addition to the black hole entropy, the electric charges Q I 's and the angular momentum J should also satisfy a constraint, which originates from the consistency of two expressions of µ in (2.68), i.e., More explicitly, for the special case Q 1 " Q 3 , Q 2 " Q 4 the constraint is We emphasize that the constraint is not unique. A constraint multiplied by a constant or some regular function of Q I and J can produce new constraints. For later convenience, we define Hence, only the imaginary part of δΛ will contribute to the real part of δS. If we assume that δh is purely imaginary, then where δT H " T H´TH " T H . Now, we apply (2.86) to the explicit choice of h given by (2.77). In the unit G N " 1, the near-extremal correction to the BPS entropy for the special case δ 1 " δ 2 becomes Combining the BPS black hole entropy from the boundary CFT (2.74) and the nearextremal correction (2.89), we obtain the near-extremal AdS 4 black hole entropy from the boundary CFT

Hawking Radiation and Near-Extremal AdS 4 Black Hole
In Sec. 2, we have derived the near-extremal AdS 4 black hole entropy using three different approaches and obtained one universal result. In particular, the approach of the nearhorizon Kerr/CFT correspondence shows that there exists a near-horizon CFT 2 , which accounts for the low-energy spectrum of the black hole microstates.
As we have seen in Subsection 2.4, the near-extremal black hole entropy can be decomposed into the contributions from the left and the right sectors of the near-horizon CFT 2 . The expression from the canonical ensemble is which has been discussed extensively for the asymptotically flat black holes [45][46][47][48][49][50][51][52], while the expression from the microcanonical ensemble is [53] S BH " 2π where N L and N R are the left and the right mode numbers, respectively. The relation (3.2) can be applied to asymptotically AdS black holes as well (see e.g. [54]). Comparing the expressions (3.1) and (3.2), we find that the temperatures T L,R can be related to the mode numbers N L,R The explicit expressions of N L and N R for near-extremal AdS 4 black holes considered in this paper with δ 1 " δ 2 are Suppose that the left and the right mode numbers in the BPS limit are NL and NR respectively, where NR " 0 . As discussed in [24,26], for the near-extremal case the left-moving and the right-moving modes become with δN L " δN R ! NL, which can be seen from (3.4). Because T H {g ! 1 for near-extremal AdS 4 black holes, the right mode number N R " pT H {gq 2 is much smaller than the left mode number N L " pT H {gq 0 . If we assume that the right modes obey a microcanonical ensemble, then the partition function of the right sector can be written as where q " e´1 {T R . We evaluate this partition function using a saddle-point approximation with respect to N R , and the result is Bq logZ R « c R π 2 6 plogpqqq 2 , with logpqq ă 0 . (3.8) The inequality logpqq ă 0 can be justified by the definition q " e´1 {T R . For near-extremal AdS 4 black holes 0 ă T R ! 1, which implies that 0 ă q ! 1 and consequently logpqq ă 0. The occupation number in the right sector is given by Bose-Einstein statistics ρ R pk 0 q " q n 1´q n " where n is the momentum quantum number of the mode moving in the time circle for the near-horizon region of AdS 4 black holes, and k 0 is the typical energy of the right-moving modes. From (3.8) we can solve for q in terms of δN R " N R , and then combining it with (3.9) we obtain where we used k 0 " n. A similar expression holds for T L , i.e., We see that (3.10) and (3.11) are completely consistent with (3.3). Since the right sector obeys the Bose-Einstein statistics, the typical energy k 0 of the right modes can be characterized by the right temperature T R . In the limit k 0 " T R ! T L , the occupation number in the left sector can be approximated as According to [24,26], Hawking radiation can be formulated as a scattering process of left and right modes in the near-horizon CFT 2 . Therefore, we can evaluate the Hawking radiation rate for near-extremal AdS 4 black holes based on the analyses above where the central charge c L provides the degrees of freedom for a given momentum quantum number n, and A is the disc amplitude of strings depending on details of the near-horizon CFT 2 . From (3.12) we see that c L ρ L pk 0 q " S L 9 phorizon areaq . (3.14) Consequently, the Hawking radiation rate becomes which is a consequence of Bose-Einstein statistics, and implies that the radiation spectrum is thermal and governed by a temperature T R proportional to the Hawking temperature T H . Therefore, we have found a microscopic formalism of Hawking radiation in the nearhorizon CFT 2 . According to this picture, the scattering of modes is unitary; hence there is no information loss during the Hawking radiation process. Since the boundary CFT can exactly reproduce the near-extremal black hole entropy (3.1), this microscopic formalism of Hawking radiation can in principle be embedded in higher-dimensional boundary CFT, which is the 3d superconformal ABJM theory for AdS 4 black holes.
Like the AdS 5 case discussed in [24], we have not taken into account the global structure of AdS space. Particularly, due to the conformal boundary of AdS space, once the radiation reaches the boundary, it will bounce back and head towards the black hole. Therefore, our current model provides a microscopic description for the Hawking radiation immediately after creation. We leave the full evolution of Hawking radiation for future work.

Discussion
We have studied the electrically charged rotating AdS 4 black holes in the near-extremal limit. Moreover, by studying the parameter space we have successfully defined a way to approach near-extremal supersymmetric black holes. We have then computed the entropy using three different approaches: (i) from the gravity solution, (ii) from the nearhorizon CFT 2 via the Kerr/CFT correspondence and (iii) from the boundary CFT via the AdS/CFT correspondence. Remarkably, these three results match precisely, giving us a universal and unique expression for the entropy in the near-extremal limit. This supports the near-extremal microstate counting in the boundary CFT and in the near-horizon CFT 2 . We also have shown that the extension of the Kerr/CFT correspondence, originally posed for extremality, to near-extremal black holes is valid. Using the results of near-extremal black hole entropy, we provide a microscopic description of Hawking radiation, and qualitatively show that unitarity and information are preserved during the Hawking radiation process.
The success of this work provides motivation to further study near-extremality in other dimensions and indeed show that the three diverse entropy computations lead to one universal entropy. Besides the near-extremal AdS 5 black holes discussed in [24] and the AdS 4 case discussed in this paper, we can also consider the known AdS 6 and AdS 7 [33] black hole solutions. Similar results from different approaches listed in Fig. 1 are expected. Moreover, the unifying picture Fig. 1 can potentially be valid beyond the Bekenstein-Hawking entropy. Hence, it would be interesting to study the subleading corrections to the Bekenstein-Hawking entropy and see if the different approaches still provide a unique expression for the entropy, in the same spirit of [55][56][57][58][59][60][61][62][63][64][65]. A recent work [66] shows that Sen's classical entropy function formalism [67] can be applied to asymptotically AdS 4 black holes to capture higher derivative corrections to the Bekenstein-Hawking entropy, which complements the methods in Fig. 1.
Besides the microstate counting of black holes, a more interesting question is how to use field theory techniques to study dynamical process in black hole physics. For instance, Hawking radiation on asymptotically AdS black holes has been studied within the frame-work of AdS/CFT correspondence previously in [68,69]. Some recent progress has been made for microscopic description of Hawking-Page transition [70]. Another related problem is to reproduce the Page curve in the black hole evaporation process [71][72][73], which has been studied in the framework of 2d JT gravity coupled to a 2d bath CFT [29][30][31][32]. Our approach in [24] and in this paper provides another powerful framework of studying these problems. In order to do that, however, we have to first carefully study the Hawking radiation at a later time in the dual boundary field theory and in the near-horizon CFT 2 to resolve the issues from the global property of AdS space. We hope to refine our microscopic models and study these more physical problems in the near future.