Integrability of Geodesics in Near-Horizon Extremal Vanishing Horizon Myers–Perry Black Holes

Abstract

We study geodesics on the near horizon geometry of odd-dimensional Myers–Perry black hole in the extremal vanishing horizon (EVH) limit. Knowing the fact that geodesics equations on Myers–Perry black hole are integrable in ellipsoidal coordinates, we take the near horizon limit in this coordinate system and obtain the near horizon EVH geometry and its Killing tensors. We explicitly show that geodesics equations are integrable on the near horizon EVH geometry ofMyers–Perry and compute the constants of motion. The resultant Killing tensors of this constants of motion matches with the Killing tensors given by taking the near-horizon limit.