Geometric Horizons

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.