Horizon Data: Existence Results and a Near-Horizon Equation on General Null Hypersurfaces
Miguel Manzano, Marc Mars
TL;DR
This work develops a general framework for horizon geometry that does not presuppose a specific ambient spacetime. By introducing metric $\mathcal{K}$-tuples and the non-isolation tensor $\boldsymbol{\Pi}^{\overline{\eta}}$, it encodes the zeroth and first transverse derivatives of the deformation tensor on detached null hypersurfaces and derives a covariant generalized near-horizon equation valid for arbitrary topology and horizon symmetry. It then proves two existence theorems ensuring the embeddability of totally geodesic horizons with prescribed horizon data into spacetimes obeying generic field equations, including cross-section cases and special reductions to known Killing/homothetic scenarios. Overall, the paper broadens horizon data analysis beyond classical Killing or isolated horizons and provides a toolkit for constructing spacetimes with prescribed horizon properties under general dynamics.
Abstract
In a spacetime $(\mathcal{M},g)$, a horizon is a null hypersurface where the deformation tensor $\mathcal{K}:=£_ηg$ of a null and tangent vector $η$ satisfies certain restrictions. In this work, we develop a formalism to study the geometry of \textit{general} horizons (i.e. characterized by any $\mathcal{K}$), based on encoding the zeroth and first transverse derivatives of $\mathcal{K}$ on null hypersurfaces detached from any ambient spacetime. We introduce the notions of \textit{$\mathcal{K}$-tuple} and \textit{non-isolation tensor}. The former encodes the order zero of $\mathcal{K}$, while the latter is a symmetric $2$-covariant tensor that codifies the ``degree of isolation" of a horizon. In particular, the non-isolation tensor vanishes for homothetic, Killing and isolated horizons. As an application we derive a \textit{generalized near-horizon equation}, i.e., an identity that holds on any horizon (regardless of its topology or whether it contains fixed points), which relates the non-isolation tensor, a certain torsion one-form, and curvature terms. By restricting this equation to a cross-section one can recover the near-horizon equation of isolated horizons and the master equation of multiple Killing horizons. Our formalism allows us to prove two existence theorems for horizons. Specifically, we establish the necessary and sufficient conditions for a non-degenerate totally geodesic horizon with any prescribed non-isolation tensor to be embeddable in a spacetime satisfying any (non-necessarily $Λ$-vacuum) field equations. We treat first the case of arbitrary topology, and then show how the result can be strengthened when the horizon admits a cross-section.