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

TL;DR

The authors address the problem of determining the transverse expansion of the ambient metric near a general null hypersurface using a covariant, field-equation-free approach based on hypersurface data. They derive general identities that relate Lie derivatives of the Ricci tensor to transverse derivatives of the metric, introduce higher-order expansion data $\{\textup{Y}^{(m)}\}$ and a generalized master equation that leverages symmetry generators, and establish asymptotic uniqueness results for non-degenerate Killing horizons. In particular, for EKH data with $\bm\tau(n)\neq 0$ and hierarchical Ricci dependence (notably $\Lambda$-vacuum), the full transverse expansion is determined by horizon data alone, yielding near-horizon rigidity and a Lambda-vacuum SdS characterization to infinite order. The forthcoming Mio4 will prove existence results given horizon data and further analyze the Lambda-vacuum case, linking geometric data on the horizon to the ambient spacetime geometry.

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 $Λ$-vacuum spacetimes admitting a non-degenerate horizon is uniquely determined in terms of abstract data at the horizon.

Theorems & Definitions (97)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Lemma 3.1
• Proposition 3.2
• proof
• Definition 3.3
• Definition 3.4