Transverse expansion of the metric at null hypersurfaces I. Uniqueness and application to Killing horizons

TL;DR

This paper develops covariant geometric identities for the transverse expansion of metrics at null hypersurfaces, proving the uniqueness of this expansion at Killing horizons based on horizon data and Ricci tensor derivatives.

Contribution

It introduces general geometric identities relating transverse derivatives of the Ricci tensor and metric expansion, and proves the uniqueness of the transverse metric expansion at Killing horizons.

Findings

Transverse expansion at Killing horizons is uniquely determined by horizon data.

The approach is covariant and independent of field equations.

Results apply to $ ext{Lambda}$-vacuum spacetimes with non-degenerate horizons.

Abstract

This is the first in a series of two papers with sequel [arXiv:2501.03983] where we analyze the transverse expansion of the metric on a general null hypersurface. In this paper we obtain general geometric identities relating the transverse derivatives of the ambient Ricci tensor and the transverse expansion of the metric at the null hypersurface. We also explore the case where the hypersurface exhibits a generalized symmetry generator, namely a privileged vector field in the ambient space which, at the hypersurface, is null and tangent (including the possibility of zeroes). This covers the Killing, homothetic, or conformal horizon cases, and, more generally, any situation where detailed information on the deformation tensor of the symmetry generator is available. Our approach is entirely covariant, independent on any field equations, and does not make any assumptions regarding the topology or dimension of the null hypersurface. As an application we prove that the full transverse expansion of the spacetime metric at a non-degenerate Killing horizon (also allowing for bifurcation surfaces) is uniquely determined in terms of abstract data on the horizon and the tower of derivatives of the ambient Ricci tensor at the horizon. In particular, the transverse expansion of the metric in $\Lambda$-vacuum spacetimes admitting a non-degenerate horizon is uniquely determined in terms of abstract data at the horizon.