First we describe the Euclidean geometry in the near-horizon limit of the Kerr–Newmann black hole. The Euclidean Kerr–Newman metric can be written in the form

where the Euclidean time is and the rotation and charge parameters have also been transformed , so that the metric (96) is purely real. Here , where , the quantities and take the form with . This spacetime has a pair of orthogonal Killing vectors which are the respective analogs of the vectors and in the (Euclidean) Schwarzschild case. The horizon surface defined by is the stationary surface of the Killing vector . Near this surface the metric (96) is approximately where and is the metric on the horizon surface up to , where variable is defined by the relation and . The angle co-ordinate and is well defined on . The metric is that of a two-dimensional disk attached to at a point (), where is an angle co-ordinate on .Regularity of the metric near the horizon implies the identifications and . For this latter condition to hold, independently of on the horizon, it is also necessary to identify with , where is the (complex) angular velocity and . The identified points have the same coordinate .

Therefore, near we have the following description of the Euclidean Kerr–Newman geometry: attached to every point ) of the horizon is a two-dimensional disk with coordinates (). The periodic identification of points on holds independently for different points on the horizon , even though is not a global coordinate. As in the static case, there is an abelian isometry generated by the Killing vector , whose fixed set is . Locally we have . The periodicity is in the direction of the vector and the resulting Euclidean space is a regular manifold.

Now consider closing the trajectory of with an arbitrary period . This implies the identification , and the metric on becomes

where is a new angular coordinate, with period . This is the metric of a two dimensional cone with angular deficit , . With this new identification the metric (96) now describes the Euclidean conical space with singular surface .The difference of the Kerr–Newman metric from the static case considered above is that the Euclidean space near the bifurcation surface is not a direct product of the surface and two-dimensional cone . Instead, it is a nontrivial foliation of over . However, this foliation shares certain common features with the static case. Namely, the invariants constructed from quadratic combinations of extrinsic curvature of vanish identically.

In the case of a static black hole we have argued that the presence of an abelian isometry with horizon being the stationary point of the isometry guarantees that the extrinsic curvature identically vanishes on the horizon. In fact this is also true in the case of a rotating black hole. The role of the abelian isometry generated by the Killing vector is less evident in this case. That is why, in this subsection, following the analysis of [170], we explicitly evaluate the extrinsic curvature for the Kerr–Newman black hole and demonstrate that quadratic invariants, that can be constructed with the help of the extrinsic curvature, vanish on the horizon.

With respect to the Euclidean metric (96) we may define a pair of orthonormal vectors :

Covariantly, these are The vectors and are normal to the horizon surface (defined as , which is a two-dimensional surface with induced metric . With respect to the normal vectors one defines the extrinsic curvatures of the surface : . The exact expression for the components of extrinsic curvature is given in [170]. The trace of the extrinsic curvature, , vanishes when restricted to the horizon surface defined by condition . Moreover, the quadratic combinations vanish on the horizon . Consequently, we have on the horizon.

Applying the conical singularity method to calculate the entanglement entropy of a rotating black hole we have to verify that i) the curvature singularity at the horizon of a stationary black hole behaves in the same way as in the static case and ii) there are no extra surface terms in the heat kernel expansion for the rotating black hole. The first point was explicitly checked in [170]: the curvature formulas (56) – (59) are still valid in the stationary case. Regarding the second point, it was shown by Dowker [70] that for a generic metric with conical singularity at some surface the only modification of the surface terms in the heat kernel expansion (70) are due to the extrinsic curvature of . For example, the surface coefficient may be modified by integrals over of terms and . Since, as was shown in Section 3.10.2, these terms identically vanish for the Kerr–Newman metric there is no modification of the surface terms in this case. Thus, the expression for the entropy (82) remains unchanged in the case of a rotating black hole. The Ricci scalar for the Kerr–Newmann metric is zero, . The integrals of the projections of Ricci and Riemann tensors over horizon surface are

The analytic continuation of these expressions back to real values of the parameters and requires the substitution With these identities the quantum entropy of the Kerr–Newman black hole reads [170] where is the area of the horizon . In the limit this expression reduces to that of the Reissner–Nordström black hole (87). An interesting and still somewhat puzzling feature of this result is that, in the case of the Kerr black hole, described by the Kerr–Newman metric with vanishing electric charge (), the logarithmic term in the entropy does not depend on the rotation parameter and is the same as in the case of the Schwarzschild black hole. In particular for the extreme Kerr black hole () one has The entropy of the Kerr black hole in the brick-wall model was calculated in [45] and a result different from Eq. (109) was found. However, the subsequent study in [92] has confirmed Eq. (109).
Living Rev. Relativity 14, (2011), 8
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