Darkness cannot bind them: a no-bound theorem for $d=5$ Myers-Perry null & timelike geodesics

In Newtonian gravity, it is well known that Kepler's problem admits no bound solutions in more than three spatial dimensions. This limitation extends naturally to General Relativity, where Tangherlini demonstrated that Schwarzschild black holes in higher dimensions admit no bound timelike geodesics. However, an analogous result for the rotating counterpart of the five-dimensional Tangherlini spacetime - the $d=5$ Myers-Perry black hole - has not yet been established. This work addresses this gap by proving that no bound timelike geodesics exist outside the event horizon of a $d=5$ Myers-Perry black hole, for any choice of spin parameters that avoid naked singularities. With this result in place, we further generalize to null geodesics. It is shown that radially bound null geodesics, which are absent in the four-dimensional Kerr spacetime as established by Wilkins, also cannot exist in the $d=5$ Myers-Perry spacetime. These results complete the geodesic analysis of this spacetime and provide a direct generalization of Wilkins' classical result to higher dimensions. Specifically, we establish the following theorem: no radially bound timelike or null geodesics are possible outside the event horizon of a $d=5$ Myers-Perry black hole, regardless of the spin configuration.