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
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
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 , 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 :. The trace of the extrinsic curvature, ,
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 : the curvature formulas (56) – (59) are still valid in the stationary case. Regarding the second point, it was shown by Dowker  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  and a result different from Eq. (109) was found. However, the subsequent study in  has confirmed Eq. (109).
Living Rev. Relativity 14, (2011), 8
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