Curvature tensors on distorted Killing horizons and their algebraic classification

Abstract

We consider generic static spacetimes with Killing horizons and study properties of curvature tensors in the horizon limit. It is determined that the Weyl, Ricci, Riemann and Einstein tensors are algebraically special and mutually aligned on the horizon. It is also pointed out that results obtained in the tetrad adjusted to a static observer in general differ from those obtained in a free-falling frame. This is connected to the fact that a static observer becomes null on the horizon. It is also shown that finiteness of the Kretschmann scalar on the horizon is compatible with the divergence of the Weyl component $Ψ_{3}$ or $Ψ_{4}$ in the freely falling frame. Furthermore finiteness of $Ψ_{4}$ is compatible with divergence of curvature invariants constructed from second derivatives of the Riemann tensor. We call the objects with finite Krestschmann scalar but infinite $Ψ_{4}$ ``truly naked black holes''. In the (ultra)extremal versions of these objects the structure of the Einstein tensor on the horizon changes due to extra terms as compared to the usual horizons, the null energy condition being violated at some portions of the horizon surface. The demand to rule out such divergencies leads to the constancy of the factor that governs the leading term in the asymptotics of the lapse function and in this sense represents a formal analog of the zeroth law of mechanics of non-extremal black holes. In doing so, all extra terms in the Einstein tensor automatically vanish.