$\mathrm{SL}(2,\mathbb{R})$ families of Kerr black holes

TL;DR

This paper explores the $ ext{SL}(2, ext{R})$ symmetry in vacuum general relativity, specifically how it acts on Kerr black holes, revealing a deep connection between symmetries, spacetime parameters, and black hole entropy.

Contribution

It demonstrates the action of the $ ext{SL}(2, ext{R})$ symmetry on a three-parameter Kerr solution and links the Casimir operator to the Bekenstein-Hawking entropy, revealing a new algebraic structure.

Findings

The $ ext{SL}(2, ext{R})$ acts on a three-parameter Kerr family.

The Casimir operator corresponds to the Bekenstein-Hawking entropy.

The Kerr solution's symmetry extends into a Kac-Moody algebra with entropy as the level.

Abstract

The stationary, axisymmetric sector of vacuum general relativity (with zero cosmological constant) enjoys an $\mathrm{SL}(2,\mathbb{R})$ symmetry called the Matzner-Misner group. We study the action of the Matzner-Misner group on the Kerr black hole. We show that the group acts naturally on a three parameter generalization of the usual two parameter Kerr solution. The new parameter represents a large diffeomorphism which gives the spacetime an asymptotic angular velocity. We explain how the $\mathrm{SL}(2,\mathbb{R})$ symmetry organizes the space of three parameter Kerr solutions into the classical analogue of principal series representations. We show that the $\mathrm{SL}(2,\mathbb{R})$ Casimir operator is the Bekenstein-Hawking entropy. The Matzner-Misner group sits inside a much larger Kac-Moody symmetry called the Geroch group. We show that the Kac-Moody level of the Kerr black hole is the Bekenstein-Hawking entropy.