A note on a third order curvature invariant in static spacetimes

TL;DR

This paper investigates a third order curvature invariant in static, conformally flat spacetimes, demonstrating its vanishing on black hole horizons and discussing its limitations in horizon detection.

Contribution

It explicitly evaluates the invariant for multi black-hole solutions and analyzes its effectiveness in identifying event horizons beyond spherical symmetry.

Findings

Invariant $I$ vanishes on black hole horizons.

Vanishing of $I$ alone does not reliably locate horizons in non-spherical spacetimes.

Discusses tidal effects related to the invariant.

Abstract

We consider here the third order curvature invariant $I=R_{\mu\nu\rho\sigma;\delta}R^{\mu\nu\rho\sigma;\delta}$ in static spacetimes ${\cal M}=R\times\Sigma$ for which $\Sigma$ is conformally flat. We evaluate explicitly the invariant for the $N$-dimensional Majumdar-Papapetrou multi black-holes solution, confirming that $I$ does indeed vanish on the event horizons of such black-holes. Our calculations show, however, that solely the vanishing of $I$ is not sufficient to locate an event horizon in non-spherically symmetric spacetimes. We discuss also some tidal effects associated to the invariant $I$.