We study the order ten polynomial equation satisfied by the radius of the spherical timelike orbits for a massive particle with a generic energy around a Kerr black hole. Even though the radii of the prograde and retrograde orbits at the equatorial or polar plane for particles with zero or unit energy have a monotonic dependence on the rotation parameter of the black hole, we show that there is a critical inclination angle above which the retrograde orbits have a nonmonotonic dependency on the rotation of the black hole. Thus the capture cross-section of these retrograde orbits decrease with increasing black hole spin. Hence their efficiency to reduce the black hole's spin is decreased. We also provide an equation for the critical inclination angle that shows exactly at which point the nonmonotonicity starts. In addition, we employ the Lagrange-Bürmann method to find approximate analytic solutions from the known exact solutions.

Some judiciously chosen local curvature scalars can be used to invariantly characterize event horizons of black holes in D > 3 dimensions, but they fail for the three dimensional Bañados-Teitelboim-Zanelli (BTZ) black hole since all curvature invariants are constant. Here we provide an invariant characterization of the event horizon of the BTZ black hole using the curvature invariants of codimension one hypersurfaces instead of the full spacetime. Our method is also applicable to black holes in generic dimensions but is most efficient in three, four, and five dimensions. We give four dimensional Kerr, five dimensional Myers-Perry and three dimensional warped-anti-de Sitter, and the three dimensional asymptotically flat black holes as examples.PACS numbers:

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We study null and timelike constant radii geodesics in the environment of an over-spinning Kerrtype naked singularity. We are particularly interested in two topics: first, the differences of the shadows of the naked rotating singularity and the Kerr black hole; and second, the spinning down effect of the particles falling from the accretion disk. Our findings are as follows: around the naked singularity, the non-equatorial prograde orbits in the in Kerr black hole remain intact up to a critical rotation parameter (α = 4 √ 2 3 √ 3 ) and cease to exist above this value. This has an important consequence in the shadow of the naked singularity if the shadow is registered by an observer on the rotation plane or close to it as the shadow cannot be distinguished from that of a Kerr black hole viewed from the same angle. We also show that the timelike retrograde orbits in the equatorial plane immediately (after about an 8% increase in mass) reduce the spin parameter of the naked singularity from larger values to α = 1 at which an event horizon appears. This happens because the retrograde orbits have a larger capture cross-section than the prograde ones. So if a naked singularity happens to have an accretion disk, it will not remain naked for long, an event horizon forms.