Stationary Black Holes as Holographs II

Abstract

Smooth four-dimensional electrovac spacetimes in Einstein's theory are considered each possessing a pair of null hypersurfaces, $H_1$ and $H_2$, generated by expansion and shear free geodesically complete null congruences such that they intersect on a two-dimensional spacelike surface, $Z=H_1\cap H_2$. By making use of a combination of the Newman-Penrose formalism and the null characteristic initial value problem it is shown that both the spacetime geometry and the electromagnetic field are uniquely determined, in the domain of dependence of $H_1\cup H_2$ once a complex vector field $ξ^A$ (determining the metric induced on $Z$), the $τ$ spin coefficient and the $ϕ_1$ electromagnetic potential are specified on $Z$. The existence of a Killing vector field---with respect to which the null hypersurfaces $H_1$ and $H_2$ comprise a bifurcate type Killing horizon---is also justified in the domain of dependence of $H_1\cup H_2$. Since, in general, the freely specifiable data on $Z$ do not have any sort of symmetry the corresponding spacetimes do not possess any symmetry in addition to the horizon Killing vector field. Thereby, they comprise the class of generic `stationary' distorted electrovac black hole spacetimes. It is also shown that there are stationary distorted electrovac black hole configurations such that parallelly propagated curvature blow up occurs both to the future and to the past ends of some of the null generators of their bifurcate Killing horizon, and also that this behavior is universal. In particular, it is shown that, in the space of vacuum solutions to Einstein's equations, in an arbitrarily small neighborhood of the Schwarzschild solution this type of distorted vacuum black hole configurations always exist. A short discussion on the relation of these results and some of the recent claims on the instability of extremal black holes is also given.