Self-Gravitating Matter in Stationary and Axisymmetric Black Hole Spacetimes

TL;DR

This paper investigates the properties of matter in stationary, axisymmetric black hole spacetimes, revealing conditions on matter at horizons and how matter behavior relates to spacetime features, with implications for various matter types and black hole models.

Contribution

It provides a comprehensive analysis of matter properties in a broad class of black hole spacetimes, establishing conditions at horizons and linking matter flow to spacetime geometry.

Findings

Matter typically flows along timelike Killing orbits outside the black hole.

At the horizon, the matter's energy density and pressure satisfy $p_n = -psilon$, restricting matter types that can thread the horizon.

The framework is verified with various known black hole solutions containing different matter fields.

Abstract

All black holes (BHs) in nature are expected to be described by the Kerr vacuum solution of general relativity (GR). However, the Kerr BH interior contains several problematic features such as a Cauchy horizon, a curvature singularity, and a causality-violating region. Non-Kerr BH models, which are used to examine the genericity of these features, typically contain nontrivial matter content. When such matter is minimally-coupled to Einstein-Hilbert gravity, the Einstein equations can be directly used to investigate its physical properties. We examine properties of the matter in a broad class of stationary and axisymmetric, geodesically-integrable BH spacetimes, and how they are linked to features of the spacetime geometry. In these spacetimes, we find the matter to typically flow along timelike Killing orbits in the BH exterior, usually exhibiting differential rotation but sometimes additionally also non-rigid rotation. At a horizon, the matter rest-frame energy density, $\epsilon$, and principal normal pressure, $p_n$, are shown to necessarily satisfy $p_n = -\epsilon$, implying that only specific types of matter can thread stationary event horizons (e.g., electromagnetic fields but not massless real scalar fields). Furthermore, we show the matter to be comoving with the interior cosmology. We also obtain simple expressions for the expansions of the ingoing and outgoing zero angular momentum null congruences and comment on the light-focussing behavior of the cosmology. Finally, we verify above results explicitly by working with a representative set of well-known BH spacetimes which contain various types of matter -- scalar fields, electromagnetic fields, anisotropic fluids. Some spacetimes have singularities while others have regular interiors. In the exterior, the matter satisfies the weak energy condition. The framework developed here can be extended to cover more general spacetimes.