A complete characterisation of the orbital shapes of the non-circular Kerr geodesic solutions with circular orbit constants of motion

Abstract

We present analytical solutions describing a family of both inwardly and outwardly spiralling orbits in the Kerr spacetime. The solutions are exact, and remarkable for their simplicity. These orbits all have the angular momentum and energy of a circular orbit at some radius \(r_c\), but are not restricted to remaining on that circular orbit, a property not possible in Newtonian gravity. We demonstrate that there are five distinct orbital solutions which terminate at the black hole singularity, and three solutions which either escape to infinity or remain bound. The different orbital solutions are characterised entirely by the black hole spin \(a\) and the location of \(r_c\). Photon orbits spiralling into or out of their (unstable) circular orbit radii are also analysed. These have properties similar to the hyperbolic class of massive particle orbits discussed herein.