Geometry and topology of the Kerr photon region in the phase space

Abstract

We study the set of trapped photons of a subcritical (a<M) Kerr spacetime as a subset of the phase space. First, we present an explicit proof that the photons of constant Boyer--Lindquist coordinate radius are the only photons in the Kerr exterior region that are trapped in the sense that they stay away both from the horizon and from spacelike infinity. We then proceed to identify the set of trapped photons as a subset of the (co-)tangent bundle of the subcritical Kerr spacetime. We give a new proof showing that this set is a smooth 5-dimensional submanifold of the (co-)tangent bundle with topology $SO(3)\times {\mathbb{R}}^2$ using results about the classification of 3-manifolds and of Seifert fiber spaces. Both results are covered by the rigorous analysis of Dyatlov [5]; however, the methods we use are very different and shed new light on the results and possible applications.

arXiv comment2 figures; comments very welcome; much simplified proof of Theorem 10 (thanks to Gregory J. Galloway for pointing this out to us)

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• Uniqueness of photon sphere for Reissner-Nordström electric-magnetic system
• Synthetic gravitational lens image of the Sagittarius A${}^{\ast }$ black hole with a thin disk model
• Higher-dimensional electric-magnetic photon sphere uniqueness
• Uniqueness of electric-magnetic spacetimes with massive particle