Hidden Conformal Symmetry of the Kerr-Newman Black Hole

We investigate the hidden conformal symmetry of the 4-dimensional non-extremal Kerr-Newman (KN) black hole with the idea of the near-region Kerr/CFT correspondence proposed by Castro, Maloney and Strominger in arXiv:1004.0996[hep-th]. The near-region KN black hole is dual to a 2D CFT with left and right temperatures $T_L=(2 M^2-Q^2)/(4\pi J)$ and $T_R=\sqrt{M^4-J^2-M^2Q^2}/(2\pi J)$. Furthermore, we reproduce the microscopic entropy of the KN black hole via the Cardy formula, which is in agreement with the macroscopic Bekenstein-Hawking entropy and precisely match the absorption cross section of a near-region scalar field in the KN black hole with the finite-temperature absorption cross section for a 2D CFT.


Introduction
The study of rotating extremal black holes has been an interesting subject since the original work on a new conjecture called the Kerr/CFT correspondence [1], which states that nearhorizon states of a four-dimensional extremal Kerr black hole could be identified with a two-dimensional chiral CFT on the spatially infinite boundary, and the central charge is proportional to the angular momentum of the black hole. Thus, one can derive the microscopic entropy of the four-dimensional extremal Kerr black hole by the Cardy formula. It is a remarkable fact that this correspondence does not rely on supersymmetry or string theory. It was initiated in the context of the 4D Kerr in general relativity, then extended to the higher-dimensional rotating solutions in supergravity and string theories [2]. More works on the Kerr/CFT correspondence and its generalization are listed in [3].
Recently, Castro, Maloney and Strominger (CMS) [4] presented a remarkable result that hidden conformal symmetry of the 4D non-extremal Kerr Black hole can be obtained by the study on the low-frequency wave equation of a massless scalar field in the near region of the Kerr black hole. A CFT with certain central charges and temperatures is dual to the corresponding Kerr Black hole. This evidence can be verified by interpreting the absorption cross section of the scalar in the near region as finite-temperature absorption section for a 2D CFT. Subsequently, with the idea of hidden conformal structure of the non-extremal black hole, Krishnan [5] has extended this method to include five-dimensional black holes in string theory. Furthermore, the non-extremal uplifted 5D Reissner-Nordstrom black holes have been investigated by Chen and Sun [6].
In 4D general relativity theory, the Kerr black hole can be considered as the uncharged Kerr-Newman (KN) black hole. So, we can mention that does the hidden conformal symmetry of the KN black hole exit, and if so, how? It is the aim of this article to further investigate the hidden conformal symmetry of the four-dimensional KN black hole in general relativity theory following the approach of CMS in [4]. We will study the wave equation of the near-region scalar field and the corresponding SL(2, R) Casimir structure, and look for the agreement for absorption cross sections between the CFT and gravity.
The paper is organized as follows. In Sec. 2, we review the KN black hole and derive the near-region wave equation of a massless scalar in the KN background. Furthermore, with the definition of conformal coordinates, we obtain the SL(2, R) Casimir structure of wave equation and reproduce the corresponding microscopic entropy of the KN black hole via the Cardy formula. In Sec. 3, we show that the absorption cross section for the near-region scalar field is dual to the finite temperature absorption cross section in a 2D CFT. The last section is devoted to conclusion.

Wave equation in the Kerr-Newman background
In this section, we study the symmetry of the near-region scalar wave equation in the KN black hole background. The KN solution describes both the stationary axisymmetric asymptotically flat gravitational field outside a massive rotating body and a rotating black hole with mass, charge and angular momentum.
In terms of Boyer-Lindquist coordinates, the KN metric reads where ∆ is the KN horizon function, 2) and the gauge field A is Here a is the angular momentum for unit mass as measured from the infinity, and M and Q are the mass and electric charge of the KN black hole, respectively. The non-extremal KN black hole has the event horizon r + and the Cauchy horizon r − at In what follows, we will see it is convenient to write the entropy S BH , surface gravities κ ± , Hawking temperature T H , angular velocity Ω H as : , 5) here, we use the subscripts "±" to denote the outer and inner horizons of the KN black hole. Now, we consider the Klein-Gordon equation for a massless uncharged scalar field in the background of the KN black hole: The wave function is written in the eigenmodes of the asymptotic energy ω and angular momenta m as Then, the wave equation (2.6) separates into the angular part 8) and the radial part and For aω ≪ 1, the angular equation (2.8) can be written as the spherical harmonic function equation and the eigenvalues K ℓ is determined by Following the argument of CMS in [4], we also consider the same near-region, which is defined by ωM ≪ 1 and rω ≪ 1.
(2.13) Then using the condition M 2 ≥ a 2 + Q 2 and the above equation, we can obtain ωQ ≪ 1, Q/r ≪ 1. (2.14) With the conditions of (2.13) and (2.14), the term G in Eq. (2.9) can be omitted. Thus the radial part of the wave equation can be reduced to the following near region equation: Next, we show that Eq. (2.15) can be reproduced by the use of the SL(2, R) Casimir operator. First, we introduce the conformal coordinates (w ± , y), which is analogue of the definitions in [4][5][6]: where Following the definition of the local vector fields in [4][5][6]:

19)
we can obtain the two sets of SL(2, R) Lie algebra with respect to (H 0 , H ±1 ) and (H,H ±1 ), respectively: The corresponding quadratic Casimir operator of the SL(2, R) Lie algebra reads as Thus, we rewrite the vector fields in terms of the coordinators (t, r, φ) as

23)
So, we obtain the corresponding Casimir operator In the background of a KN black hole, the near region wave equation (2.15) can be rewritten asH 2 Φ = H 2 Φ = l(l + 1)Φ. (2.26) From the above procedure, we know that, in the near region, the scalar field can have SL(2, R) L × SL(2, R) R weight to obtain the microscopical entropy of the non-extremal KN black hole by the use of the Cardy Formula. However, we don't know how to compute the corresponding central charges c L and c R . Here, as did in [4], we also assume that the central charges of the non-extremal Kerr-Newman still keep same as the extremal case, i.e., c L = c R = 12J. Then, using the Cardy formula for the microstate, we have From the 2D dual CFT, we reproduce the KN black hole entropy in four dimensions which reaches agreement with the macroscopic entropy S BH .

Scalar absorption
In this section we will calculate the absorption cross section for the KN black hole in the near region. The absorption cross section of a low-frequency massless scalar in the nearextremal limit of the KN black hole has been calculated and analyzed in [7][8][9]. In the near region, the solutions of the radial wave equation (2.15) with respect to the ingoing and outgoing boundary conditions at the horizon are and where F (a, b; c; z) is the hypergeometric function. At the outer boundary of the matching region r ≫ M (but still r ≪ 1 ω ) , the ingoing wave behaves as .

(3.4)
The absorption cross section can be written as ( 3.5) In order to match the absorption cross section of a near-region scalar field in the KN black hole background with the finite-temperature absorption cross section for the corresponding 2D CFT, we need study the first law of black hole thermodynamics. Since the existence of non-zero charge of the KN black hole, the first law of black hole thermodynamics is where ϕ = Q r + r 2 + +a 2 is the electric potential of the KN black hole. It is obvious that δQ is equal to zero with respect to the uncharged scalar field. With the conjugate charges δE R and δE L , we can suppose the dual CFT entropy is The solution is With the variations δM = m and δJ = ω, the left and right moving frequencies are turned out to be By using of the above formula, we can rewrite the absorption cross section as which agrees precisely with finite-temperature absorption cross section for a 2D CFT.

Conclusion
In this paper, the method of CMS is extended to the four-dimensional KN Black Hole in general relativity theory. We investigate the hidden conformal structure of the KN black hole by the low-frequency wave equation of a massless scalar in the near region. The dual CFT with the left and right temperatures T L = (2M 2 − Q 2 )/(4πJ) and T R = M 4 − J 2 − M 2 Q 2 /(2πJ) is obtained. Furthermore, under the assumption that the central charges of the non-extremal Kerr-Newman still keep same as the extremal case, one can reproduce the macroscopic Bekenstein-Hawking entropy via the Cardy formula. At last, by rewriting the absorption cross section of a near-region scalar field as the finite-temperature absorption cross section for the 2D CFT, one can verify that the near-region KN black hole is dual to a 2D CFT.
From the result of this paper, we can see that the method of CMS is valid for the nonextremal Kerr-Newman black hole in 4D general relativity theory background. It would be interesting to investigate other charged and rotating black holes in diverse dimensions and in various gravity theories.