Complete Integrability of Geodesic Motion in General Kerr-NUT-AdS Spacetimes

TL;DR

This paper demonstrates that geodesic motion in general Kerr-NUT-AdS spacetimes in any dimension is completely integrable by explicitly constructing D independent, commuting constants of motion.

Contribution

It provides explicit construction of a complete set of independent constants of motion for geodesics in general Kerr-NUT-AdS spacetimes, proving their complete integrability.

Findings

Constructed D independent constants of motion.

Proved all constants commute under Poisson brackets.

Established complete integrability of geodesic motion.

Abstract

We explicitly exhibit n-1 constants of motion for geodesics in the general D-dimensional Kerr-NUT-AdS rotating black hole spacetime, arising from contractions of even powers of the 2-form obtained by contracting the geodesic velocity with the dual of the contraction of the velocity with the (D-2)-dimensional Killing-Yano tensor. These constants of motion are functionally independent of each other and of the D-n+1 constants of motion that arise from the metric and the D-n = [(D+1)/2] Killing vectors, making a total of D independent constants of motion in all dimensions D. The Poisson brackets of all pairs of these D constants are zero, so geodesic motion in these spacetimes is completely integrable.