On integrability of geodesics in near-horizon extremal geometries: Case of Myers-Perry black holes in arbitrary dimensions

TL;DR

This paper demonstrates that the geodesic equations in near-horizon extremal Myers-Perry black holes are integrable and separable in arbitrary dimensions, providing explicit solutions and exploring superintegrable cases.

Contribution

It extends previous results by showing integrability and separability of geodesics in higher-dimensional near-horizon extremal Myers-Perry black holes, including special superintegrable cases.

Findings

Geodesic equations are integrable and separable in arbitrary dimensions.

Explicit solutions for Hamilton-Jacobi equations are derived.

Superintegrable cases with additional constants of motion are identified.

Abstract

We investigate dynamics of probe particles moving in the near-horizon limit of extremal Myers-Perry black holes in arbitrary dimensions. Employing ellipsoidal coordinates we show that this problem is integrable and separable, extending the results of the odd dimensional case discussed in arXiv:1703.00713. We find the general solution of the Hamilton-Jacobi equations for these systems and present explicit expressions for the Liouville integrals, discuss Killing tensors and the associated constants of motion. We analyze special cases of the background near-horizon geometry were the system possesses more constants of motion and is hence superintegrable. Finally, we consider near-horizon extremal vanishing horizon case which happens for Myers-Perry black holes in odd dimensions and show that geodesic equations on this geometry are also separable and work out its integrals of motion.