Area Spectrum of Kerr and extremal Kerr Black Holes from Quasinormal Modes

TL;DR

This paper investigates the quantization of the area spectrum of Kerr and extremal Kerr black holes using quasinormal modes, finding a discrete but non-uniform spectrum for Kerr and a uniformly spaced spectrum for extremal Kerr, challenging some previous assumptions.

Contribution

It derives the area spectrum for Kerr and extremal Kerr black holes using quasinormal modes, showing non-uniform spacing for Kerr and uniform spacing for extremal Kerr, and questions previous proposals.

Findings

Kerr black hole has a discrete, non-uniform area spectrum.

Extremal Kerr black hole has a discrete, evenly spaced area spectrum.

The real part of quasinormal frequencies for Kerr may not be the correct function.

Abstract

Motivated by the recent interest in quantization of black hole area spectrum, we consider the area spectrum of Kerr and extremal Kerr black holes. Based on the proposal by Bekenstein and others that the black hole area spectrum is discrete and equally spaced, we implement Kunstatter's method to derive the area spectrum for the Kerr and extremal Kerr black holes. The real part of the quasinormal frequencies of Kerr black hole used for this computation is of the form $m\Omega$ where $\Omega$ is the angular velocity of the black hole horizon. The resulting spectrum is discrete but not as expected uniformly spaced. Thus, we infer that the function describing the real part of quasinormal frequencies of Kerr black hole is not the correct one. This conclusion is in agreement with the numerical results for the highly damped quasinormal modes of Kerr black hole recently presented by Berti, Cardoso and Yoshida. On the contrary, extremal Kerr black hole is shown to have a discrete area spectrum which in addition is evenly spaced. The area spacing derived in our analysis for the extremal Kerr black hole area spectrum is not proportional to $\ln 3$. Therefore, it does not give support to Hod's statement that the area spectrum $A_{n}=(4l^{2}_{p}ln 3)n$ should be valid for a generic Kerr-Newman black hole.