The Petrov type D equation on genus $>0$ sections of isolated horizons

TL;DR

This paper extends the Petrov type D equation analysis to higher genus surfaces, showing solutions have constant curvature and zero rotation, supporting the topological sphericality of black hole horizons.

Contribution

It derives all solutions of the Petrov type D equation on higher genus surfaces under Einstein vacuum conditions, revealing they have constant curvature and zero rotation.

Findings

Solutions have constant Gauss curvature.

Solutions exhibit zero rotation.

Supports spherical topology of black hole horizons.

Abstract

The Petrov type D equation imposed on the 2-metric tensor and the rotation scalar of a cross-section of an isolated horizon can be used to uniquely distinguish the Kerr - (anti) de Sitter spacetime in the case the topology of the cross-section is that of a sphere. In the current paper we study that equation on closed 2-dimensional surfaces that have genus $>0$. We derive all the solutions assuming the embeddability in 4-dimensional spacetime that satisfies the vacuum Einstein equations with (possibly 0) cosmological constant. We prove all of them have constant Gauss curvature and zero rotation. Consequently, we provide a quazi-local argument for a black hole in 4-dimensional spacetime to have a topologically spherical cross-section.