Poincare ball embeddings of the optical geometry

TL;DR

This paper demonstrates that the optical geometry of Reissner-Nordstrom black holes can be fully embedded in hyperbolic space using Poincare ball models, capturing the horizon and distinguishing extremal black holes.

Contribution

It introduces a novel embedding method of optical geometry into hyperbolic space that covers the entire spacetime extent, including the horizon, unlike previous flat-space embeddings.

Findings

Embedding extends to the horizon in hyperbolic space

Poincare ball embedding distinguishes extremal black holes

Embedding remains valid below the Buchdahl-Bondi limit

Abstract

It is shown that optical geometry of the Reissner-Nordstrom exterior metric can be embedded in a hyperbolic space all the way down to its outer horizon. The adopted embedding procedure removes a breakdown of flat-space embeddings which occurs outside the horizon, at and below the Buchdahl-Bondi limit (R/M=9/4 in the Schwarzschild case). In particular, the horizon can be captured in the optical geometry embedding diagram. Moreover, by using the compact Poincare ball representation of the hyperbolic space, the embedding diagram can cover the whole extent of radius from spatial infinity down to the horizon. Attention is drawn to advantages of such embeddings in an appropriately curved space: this approach gives compact embeddings and it distinguishes clearly the case of an extremal black hole from a non-extremal one in terms of topology of the embedded horizon.