Dirty rotating black holes: regularity conditions on stationary horizons

TL;DR

This paper investigates the regularity conditions of stationary rotating black holes surrounded by matter, establishing constraints on metric behavior near horizons and implications for horizon rotation and curvature invariants.

Contribution

It derives new regularity conditions for 'dirty' black holes, linking curvature boundedness to horizon properties and providing a detailed analysis of metric coefficients near horizons.

Findings

Bounded Ricci scalar implies horizon rotation rigidity.

Finiteness of quadratic invariants leads to constant surface gravity.

Curvature tensor components become diagonal in ZAMO frame at the horizon.

Abstract

We consider generic, or "dirty" (surrounded by matter), stationary rotating black holes with axial symmetry. The restrictions are found on the asymptotic form of metric in the vicinity of non-extremal, extremal and ultra-extremal horizons, imposed by the conditions of regularity of increasing strength: boundedness on the horizon of the Ricci scalar, of scalar quadratic curvature invariants, and of the components of the curvature tensor in the tetrad attached to a falling observer. We show, in particular, that boundedness of the Ricci scalar implies the "rigidity" of the horizon's rotation in all cases, while the finiteness of quadratic invariants leads to the constancy of the surface gravity. We discuss the role of quasiglobal coordinate r that is emphasized by the conditions of regularity. Further restrictions on the metric are formulated in terms of subsequent coefficients of expansion of metric functions by r. The boundedness of the tetrad components of curvature tensor for an observer crossing the horizon is shown to lead in the horizon limit to diagonalization of Einstein tensor in the frame of zero angular momentum observer on a circular orbit (ZAMO frame) for horizons of all degrees of extremality.