Detecting horizons of symmetric black holes using relative differential invariants

TL;DR

This paper introduces a method using relative differential invariants derived from Lie algebra structures to detect horizons of symmetric black holes in Lorentzian manifolds.

Contribution

It develops a new invariant-based approach for horizon detection that applies to black holes with specific symmetry properties, expanding the toolkit for geometric analysis of horizons.

Findings

Invariant vanishes on Killing horizons of symmetric black holes

Constructs a general order 0 invariant based on Lie algebra structure

Applicable to various well-known black hole solutions

Abstract

Let $\mathfrak{k}$ be a nontrivial finite-dimensional Lie algebra of vector fields on a manifold M, and consider the family of Lorentzian metrics on M whose Killing algebra contains $\mathfrak{k}$. We show that scalar relative differential invariants, with respect to a Lie algebra of vector fields on M preserving $\mathfrak{k}$, can be used to detect the horizons of several well-known black holes. In particular, using the Lie algebra structure of $\mathfrak{k}$, we construct a general relative differential invariant of order 0 that always vanishes on $\mathfrak{k}$-invariant Killing horizons.