Geometric Horizons

TL;DR

This paper explores the algebraic properties of black hole horizons, proposing that such horizons are more algebraically special than other spacetime regions, and discusses methods to identify them using scalar curvature invariants.

Contribution

It introduces a conjecture that black hole horizons are more algebraically special and presents invariant-based criteria for their identification in four-dimensional spacetimes.

Findings

Black hole horizons are likely more algebraically special than other regions.

Scalar curvature invariants can identify horizons without foliation dependence.

The conjecture extends to various horizon types and dimensions.

Abstract

We discuss black hole spacetimes with a geometrically defined quasi-local horizon on which the curvature tensor is algebraically special relative to the alignment classification. Based on many examples and analytical results, we conjecture that a spacetime horizon is always more algebraically special (in all of the orders of specialization) than other regions of spacetime. Using recent results in invariant theory, such geometric black hole horizons can be identified by the alignment type $\textbf{II}$ or $\textbf{D}$ discriminant conditions in terms of scalar curvature invariants, which are not dependent on spacetime foliations. The above conjecture is, in fact, a suite of conjectures (isolated vs dynamical horizon; four vs higher dimensions; zeroth order invariants vs higher order differential invariants). However, we are particularly interested in applications in four dimensions and especially the location of a black hole in numerical computations.