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

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 $r=r_{\mathrm{ph}}$, marginally bound orbit $r=r_{\mathrm{mb}}$ and innermost stable orbit $r=r_{\mathrm{ms}}$ 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 $r_{\mathrm{ph}}=r_{+}$, $r_{\mathrm{mb}}=r_{+}$ and $r_{\mathrm{ms}}=r_{+}$ 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.

5 pages, 2 figures, v2: reference added

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