The Petrov type D isolated null surfaces

Abstract

Generic black holes in vacuum-de Sitter/anti-de Sitter spacetimes are studied in quasi-local framework, where the relevant properties are captured in the intrinsic geometry of the null surface (the horizon). Imposing the quasi-local notion of stationarity (null symmetry of the metric up to second order at the horizon only) we perform the complete classification of all the so-called special Petrov types of these surfaces defined by the properties (structure of principal null direction) of the Weyl tensor at the surface. The only possible types are: II, D and O. In particular, all the geometries of type O are identified. The condition distinguishing type D horizons, taking the form of a second order differential equation on certain complex invariant constructed from the Gaussian curvature and the rotation scalar, is shown to be an integrability condition for the so-called near horizon geometry equation. The emergence of the near horizon geometry in this context is equivalent to the hyper-suface orthogonality of both double principal null directions. We further formulate a no-hair theorem for the Petrov type D axisymmetric null surfaces of topologically spherical sections, showing that the space of solutions is uniquely parametrized by the horizon area and angular momentum.