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

## TL;DR

This paper characterizes the set of trapped photons in subcritical Kerr spacetimes, proving it forms a smooth 5-dimensional submanifold with specific topology, offering new insights beyond previous analyses.

## Contribution

It provides a new proof that trapped photons at constant Boyer-Lindquist radius are unique and identifies the trapped photon set as a smooth submanifold with explicit topology.

## Key findings

- Trapped photons at constant radius are the only such photons in the exterior region.
- The set of trapped photons forms a smooth 5-dimensional submanifold.
- The topology of this submanifold is $SO(3)\times\mathbb R^2$.

## 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.