Detecting Event Horizons and Stationary Surfaces

Abstract

We have investigated the behavior of three curvature invariants for Schwarzschild, Reissner-Nordstr{\o}m, Kerr, and Kerr-Newman black holes. We have also studied these invariants for a Schwarzschild-de Sitter space-time, the $\gamma$ metric, and for a 2+1 charged dimensional black hole. The invariants are $I_{1}=R_{\alpha\beta\mu\nu;\lambda}R^{\alpha\beta\mu\nu;\lambda}$, $I_{2}=R_{\mu\nu;\lambda} R^{\mu\nu;\lambda}$, and $I_{3}=C_{\alpha\beta\mu\nu;\lambda}C^{\alpha\beta\mu\nu;\lambda}$. For all but the Kerr-Newman case these invariants serve as either horizon or stationary surface detectors. The Kerr-Newman case is more complicated. We show that $I_{1}$ vanishs on the horizon in any space-time with a Schwarzschild like metric.