Exact Formulas for Spherical Photon Orbits Around Kerr Black Holes

TL;DR

This paper derives exact and approximate formulas for spherical photon orbits around Kerr black holes, extending known results to non-equatorial orbits and analyzing the polynomial equations involved.

Contribution

It provides the first exact formulas for nonequatorial spherical photon orbits around Kerr black holes, including critical inclination angles and explicit radii.

Findings

Exact formulas for non-equatorial photon orbits

Identification of a critical inclination angle with four photon orbits

Accurate approximate formulas for general orbits

Abstract

Exact formulas relating the radii of the spherical photon orbits to the black hole's rotation parameter and the effective inclination angle of the orbit have been known only for equatorial and polar orbits up to now. Here we provide exact formulas for nonequatorial orbits that lie between these extreme limits. For a given rotation parameter of the black hole, there is a critical inclination angle below which there are four null photon orbits two of which are in the exterior region. At the critical angle, the radii of these orbits are explicitly found. We also provide approximate but very accurate formulas for any orbit. To arrive at these analytical solutions, we carefully study the sextic polynomial in the radius that arises for null spherical geodesics.