Marginally bound (critical) geodesics of rapidly rotating black holes

One of the most important geodesics in a black-hole spacetime is the marginally bound spherical orbit. This critical geodesic represents the innermost spherical orbit which is bound to the central black hole. The radii $r_{\text{mb}}(\bar{a})$ of the marginally bound {\it equatorial} circular geodesics of rotating Kerr black holes were found analytically by Bardeen {\it et. al.} more than four decades ago (here $\bar{a}\equiv J\mathrm{/}{M}^2$ is the dimensionless angular-momentum of the black hole). On the other hand, no closed-form formula exists in the literature for the radii of generic ({\it non}-equatorial) marginally bound geodesics of the rotating Kerr spacetime. In the present study we analyze the critical (marginally bound) orbits of rapidly rotating Kerr black holes. In particular, we derive a simple {\it analytical} formula for the radii $r_{\text{mb}}(\bar{a}\simeq 1;\cos i)$ of the marginally bound spherical orbits, where $\cos i$ is an effective inclination angle (with respect to the black-hole equatorial plane) of the geodesic. We find that the marginally bound spherical orbits of rapidly-rotating black holes are characterized by a critical inclination angle, $\cos i=\sqrt{2\mathrm{/}3}$, above which the coordinate radii of the geodesics approach the black-hole radius in the extremal $\bar{a}\to 1$ limit. It is shown that this critical inclination angle signals a transition in the physical properties of the orbits: in particular, it separates marginally bound spherical geodesics which lie a finite proper distance from the black-hole horizon from marginally bound geodesics which lie an infinite proper distance from the horizon.