Spherical null geodesics of rotating Kerr black holes

TL;DR

This paper analytically studies non-equatorial spherical null geodesics of Kerr black holes, providing an approximate formula for their radii and revealing complex dependencies on black hole spin and inclination angle, including a critical angle for orbit properties.

Contribution

It introduces an approximate analytical formula for non-equatorial spherical null geodesic radii in Kerr spacetime and uncovers non-monotonic behaviors and a critical inclination angle affecting orbit properties.

Findings

Derived an approximate formula for spherical null geodesic radii.

Discovered non-monotonic dependence of radii on black hole spin for certain orbits.

Identified a critical inclination angle separating different horizon proximity behaviors.

Abstract

The non-equatorial spherical null geodesics of rotating Kerr black holes are studied analytically. Unlike the extensively studied equatorial circular orbits whose radii are known analytically, no closed-form formula exists in the literature for the radii of generic (non-equatorial) spherical geodesics. We provide here an approximate formula for the radii r_{ph}(a/M;cos i) of these spherical null geodesics, where a/M is the dimensionless angular-momentum of the black hole and cos i is an effective inclination angle (with respect to the black-hole equatorial plane) of the orbit. It is well-known that the equatorial circular geodesics of the Kerr spacetime (the prograde and the retrograde orbits with cos i=\pm 1) are characterized by a monotonic dependence of their radii r_{ph}(a/M;cos i=\pm 1) on the dimensionless spin-parameter a/M of the black hole. We use here our novel analytical formula to reveal that this well-known property of the equatorial circular geodesics is actually not a generic property of the Kerr spacetime. In particular, we find that counter-rotating spherical null orbits in the range (3\sqrt{3}-\sqrt{59})/4 \lesssim \cos i<0 are characterized by a non-monotonic dependence of r_{ph}(a/M;cos i=const) on the dimensionless rotation parameter a/M of the black hole. Furthermore, it is shown that spherical photon orbits of rapidly-rotating black holes are characterized by a critical inclination angle, cos i=\sqrt{4/7}, above which the coordinate radii of the orbits approach the black-hole radius in the extremal limit. We prove that this critical inclination angle signals a transition in the physical properties of the spherical null geodesics: in particular, it separates orbits which are characterized by finite proper distances to the black-hole horizon from orbits which are characterized by infinite proper distances to the horizon.