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

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 $r=r_{\rm ph}$,marginallyboundorbit$r=r_{\rm mb}$andinnermoststableorbit$r=r_{\rm ms}$ 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 $Q=\pm \sqrt{M^2-J^2\mathrm{/}{M}^2}$ ($\mathrm{\mid }J\mathrm{\mid }\le M^2$) the coordinate equalities $r_{\rm ph}=r_+$,$r_{\rm mb}=r_+$and$r_{\rm ms}=r_+$holdifandonlyif$|J|$ is greater than or equal to $M^2\mathrm{/}2$, $M^2\mathrm{/}\sqrt{3}$ and $M^2\mathrm{/}\sqrt{2}$, respectively.