A note on a third order curvature invariant in static spacetimes

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 $\mathcal{M}=R\times \Sigma$ 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 $I$.

5 pages, 3 figures. Extra material available at http://vigo.ime.unicamp.br/inv