On the shape of rotating black-holes

TL;DR

This paper thoroughly describes the shape of rotating black-hole horizons, showing how rotation influences their geometry, limits their shape, and makes them resemble the extreme Kerr horizon at high spins, independent of surrounding matter.

Contribution

It provides a detailed geometric analysis of rotating black-hole horizons, establishing bounds and relations based solely on area and angular momentum, and demonstrating their shape's dependence on rotation.

Findings

Rotation causes horizons to widen centrally (rotational thickening).

Stable black holes with fixed area and angular momentum form a precompact family.

Horizon geometry approaches that of the extreme Kerr black hole at near-maximal rotation.

Abstract

We give a thorough description of the shape of rotating axisymmetric stable black-hole (apparent) horizons applicable in dynamical or stationary regimes. It is found that rotation manifests in the widening of their central regions (rotational thickening), limits their global shapes to the extent that stable holes of a given area A and angular momentum J (non zero) form a precompact family (rotational stabilization) and enforces their whole geometry to be close to the extreme-Kerr horizon geometry at almost maximal rotational speed (enforced shaping). The results, which are based on the stability inequality, depend only on A and J. In particular they are entirely independent of the surrounding geometry of the space-time and of the presence of matter satisfying the strong energy condition. A complete set of relations between A, J, the length L of the meridians and the length R of the greatest axisymmetric circle, is given. We also provide concrete estimations for the distance between the geometry of horizons and that of the extreme Kerr, in terms only of A and J. Besides it own interest, the work has applications to the Hoop conjecture as formulated by Gibbons in terms of the Birkhoff invariant, to the Bekenstein-Hod entropy bounds and to study the compactness of classes of stationary black-hole space-times.