Abstract
We investigate the separability of the Klein-Gordon equation on near horizon of $d$-dimensional rotating Myers-Perry black hole in two limits: (i) generic extremal case and (ii) extremal vanishing horizon case. In the first case, there is a relation between the mass and rotation parameters so that black hole temperature vanishes. In the latter case, one of the rotation parameters is restricted to zero on top of the extremality condition. We show that the Klein-Gordon equation is separable in both cases. Also, we solved the radial part of that equation and discuss its behavior in small- and large-$r$ regions.
Highlights
The four-dimensional Kerr black hole has been widely studied from various aspects
The integrability of the Klein-Gordon, Maxwell field and gravitational perturbation equations have been studied on the d-dimensional Kerr-(A)dS-NUT space-time [2,3,5,6,7,8,9]
One can construct another solution to Einstein equations in the near horizon extremal limit [10,11,12,13] of Myers-Perry (NHEMP) black hole [14]
Summary
The four-dimensional Kerr black hole has been widely studied from various aspects. From the geometric point of view, Carter showed that it has integrable geodesics [1]. The integrability of the Klein-Gordon, Maxwell field and gravitational perturbation equations have been studied on the d-dimensional Kerr-(A)dS-NUT space-time [2,3,5,6,7,8,9] One can construct another solution to Einstein equations in the near horizon extremal limit [10,11,12,13] of Myers-Perry (NHEMP) black hole [14] (see [15,16,17,18,19,20] for recent studies). The near horizon geometry of Myers-Perry black holes contains integrable and superintegrable systems like Rosochatius and Pöschl-Teller which are interestingly related to the Klein-Gordon equation through a geometrization procedure [36]. The radial part is solved by Bessel’s functions which are restricted to Bessel’s function Jν by demanding the smoothness in the large- regions
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