Geroch Group Description of Black Holes

TL;DR

This paper demonstrates the explicit construction and factorization of Geroch group matrices for various black holes, providing a new algebraic approach to understanding their gravitational configurations.

Contribution

It applies the inverse scattering method to construct and factorize Geroch group matrices for multiple black hole solutions, including Myers-Perry, Kaluza-Klein, and Kerr-Newman.

Findings

Explicit Geroch group matrices for five-dimensional black holes

Relations between Geroch group matrices and charge matrices

Demonstration of the inverse scattering method for black hole solutions

Abstract

On one hand the Geroch group allows one to associate spacetime independent matrices with gravitational configurations that effectively only depend on two coordinates. This class includes stationary axisymmetric four- and five-dimensional black holes. On the other hand, a recently developed inverse scattering method allows one to factorize these matrices to explicitly construct the corresponding spacetime configurations. In this work we demonstrate the construction as well as the factorization of Geroch group matrices for a wide class of black hole examples. In particular, we obtain the Geroch group SL(3,R) matrices for the five-dimensional Myers-Perry and Kaluza-Klein black holes and the Geroch group SU(2,1) matrix for the four-dimensional Kerr-Newman black hole. We also present certain non-trivial relations between the Geroch group matrices and charge matrices for these black holes.