Transverse deformations of extreme horizons

TL;DR

This paper studies infinitesimal transverse deformations of extreme black hole horizons, establishing a finite-dimensional moduli space and proving uniqueness for the extreme Kerr horizon, with potential implications for discovering new black hole solutions.

Contribution

It characterizes the space of transverse deformations of extreme horizons and proves uniqueness for the Kerr case, also identifying a family of deformations in five dimensions that may lead to new black holes.

Findings

Finite-dimensional moduli space of deformations

Uniqueness of Kerr horizon deformations

Discovery of a three-parameter family in five dimensions

Abstract

We consider the inverse problem of determining all extreme black hole solutions to the Einstein equations with a prescribed near-horizon geometry. We investigate this problem by considering infinitesimal deformations of the near-horizon geometry along transverse null geodesics. We show that, up to a gauge transformation, the linearised Einstein equations reduce to an elliptic PDE for the extrinsic curvature of a cross-section of the horizon. We deduce that for a given near-horizon geometry there exists a finite dimensional moduli space of infinitesimal transverse deformations. We then establish a uniqueness theorem for transverse deformations of the extreme Kerr horizon. In particular, we prove that the only smooth axisymmetric transverse deformation of the near-horizon geometry of the extreme Kerr black hole, such that cross-sections of the horizon are marginally trapped surfaces, corresponds to that of the extreme Kerr black hole. Furthermore, we determine all smooth and biaxisymmetric transverse deformations of the near-horizon geometry of the five-dimensional extreme Myers-Perry black hole with equal angular momenta. We find a three parameter family of solutions such that cross-sections of the horizon are marginally trapped, which is more general than the known black hole solutions. We discuss the possibility that they correspond to new five dimensional vacuum black holes.