On the existence of Kundt's metrics with compact sections of null hypersurfaces

TL;DR

This paper proves that vacuum Kundt's metrics with compact, boundaryless null hypersurface sections are limited to tori and spheres, excluding higher genus surfaces, based on analysis of the fundamental PDE derived from Einstein's equations.

Contribution

It demonstrates the non-existence of vacuum Kundt's metrics with compact sections of null hypersurfaces of higher genus, refining the understanding of possible horizon geometries in general relativity.

Findings

Higher genus compact sections are excluded for vacuum Kundt's metrics.

The basic PDE constrains the geometry of null hypersurface sections.

Results apply also to degenerate Killing and isolated horizons.

Abstract

It is shown that Kundt's metric for vacuum cannot be constructed when two-dimensional space-like sections of null hypersurfaces are compact, connected manifolds with no boundary unless they are tori or spheres, i.e. higher genus $\mathbf{g} \geq 2$ is excluded by vacuum Einstein equations. The so-called {\em basic equation} (resulting from Einstein equations) is examined. This is a non-linear PDE for unknown covector field and unknown Riemannian structure on the two-dimensional manifold. It implies several important results derived in this paper. It arises not only for Kundt's class but also for degenerate Killing horizons and vacuum degenerate isolated horizons.