Photon escape in the extremal Kerr black hole spacetime

TL;DR

This paper characterizes the conditions under which photons emitted from any position around an extremal Kerr black hole can escape to infinity, providing a complete visualization of the escapable photon region.

Contribution

It introduces a comprehensive method to identify and visualize the escapable photon region in the entire extremal Kerr black hole spacetime, extending previous near-horizon results.

Findings

Complete identification of the escapable photon region.

A method to visualize the two-dimensional impact parameter space.

Generalization of previous near-horizon results to the entire spacetime.

Abstract

We consider necessary and sufficient conditions for photons emitted from an arbitrary spacetime position of the extremal Kerr black hole to escape to infinity. The radial equation of motion determines the necessary conditions for photons emitted from $r=r_*$ to escape to infinity, and the polar angle equation of motion further restricts the allowed region of photon motion. From these two conditions, we provide a method to visualize a two-dimensional photon impact parameter space that allows photons to escape to infinity, i.e., the escapable region. Finally, we completely identify the escapable region for the extremal Kerr black hole spacetime. This study has generalized our previous result [K.~Ogasawara and T.~Igata, Phys. Rev. D \textbf{103}, 044029 (2021)], which focused only on light sources near the horizon, to the classification covering light sources in the entire region.