Global embedding of the Kerr black hole event horizon into hyperbolic 3-space

TL;DR

This paper presents a method for explicitly embedding the event horizon of Kerr black holes into hyperbolic 3-space, enabling visualization of complex geometries for various angular momenta, including cases not embeddable in Euclidean space.

Contribution

It provides the first explicit global isometric embedding of Kerr-Newman event horizons into hyperbolic 3-space for arbitrary angular momentum values.

Findings

Embedding exists for Kerr-Newman horizons with any angular momentum.

Embedding fits within a fundamental domain of the Picard group.

Some horizons, like double-Kerr, cannot be globally embedded.

Abstract

An explicit global and unique isometric embedding into hyperbolic 3-space, H^3, of an axi-symmetric 2-surface with Gaussian curvature bounded below is given. In particular, this allows the embedding into H^3 of surfaces of revolution having negative, but finite, Gaussian curvature at smooth fixed points of the U(1) isometry. As an example, we exhibit the global embedding of the Kerr-Newman event horizon into H^3, for arbitrary values of the angular momentum. For this example, considering a quotient of H^3 by the Picard group, we show that the hyperbolic embedding fits in a fundamental domain of the group up to a slightly larger value of the angular momentum than the limit for which a global embedding into Euclidean 3-space is possible. An embedding of the double-Kerr event horizon is also presented, as an example of an embedding which cannot be made global.