The Petrov type D isolated null surfaces

TL;DR

This paper classifies Petrov type D null surfaces in vacuum-de Sitter/Anti-de Sitter spacetimes, revealing their geometric properties, conditions for type D horizons, and a no-hair theorem linking solutions to horizon area and angular momentum.

Contribution

It provides a complete classification of special Petrov types of null surfaces under quasi-local stationarity and establishes a no-hair theorem for type D horizons.

Findings

Only Petrov types II, D, and O are possible.

Type D horizons satisfy a specific second order differential equation.

Solutions are uniquely determined by horizon area and angular momentum.

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.