A note on circular geodesics in the equatorial plane of an extreme Kerr-Newman black hole

TL;DR

This paper investigates the behavior of circular geodesics around extreme Kerr-Newman black holes, revealing specific conditions under which key orbits coincide with the event horizon based on the black hole's angular momentum.

Contribution

It provides a detailed analysis of the conditions for the coincidence of photon, marginally bound, and innermost stable orbits with the event horizon in extreme Kerr-Newman black holes.

Findings

Radii of key orbits coincide with the horizon when angular momentum exceeds certain thresholds.

Specific inequalities relate the black hole's angular momentum to orbit radii.

Results extend known properties from extreme Kerr to Kerr-Newman black holes.

Abstract

We examine the behaviour of circular geodesics describing orbits of neutral test particles around an extreme Kerr-Newman black hole. It is well known that the radial Boyer-Lindquist coordinates of the prograde photon orbit $r=r_{\rm ph}$, marginally bound orbit $r=r_{\rm mb}$ and innermost stable orbit $r=r_{\rm ms}$ of the extreme Kerr black hole all coincide with the event horizon's value $r=r_+$. We find that for the extreme Kerr-Newman black hole with mass $M$, angular momentum $J$ and electric charge $Q=\pm\sqrt{M^2-J^2/M^2}$ ($|J|\le M^2$) the coordinate equalities $r_{\rm ph}=r_+$, $r_{\rm mb}=r_+$ and $r_{\rm ms}=r_+$ hold if and only if $|J|$ is greater than or equal to $M^2/2$, $M^2/\sqrt{3}$ and $M^2/\sqrt{2}$, respectively.