Poincare ball embeddings of the optical geometry

Details Poincare ball embeddings of the optical geometry Poincare ball embeddings of the optical geometry

2002-06-10

arxiv.org/abs/gr-qc/0206027v1

Class.Quant.Grav. 19 (2002) 3963-3976

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

16 pages, 8 figures; CQG accepted

Scientific paper

10.1088/0264-9381/19/15/307

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.

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