Complete Set of Commuting Symmetry Operators for the Klein-Gordon Equation in Generalized Higher-Dimensional Kerr-NUT-(A)dS Spacetimes

TL;DR

This paper proves the complete set of commuting symmetry operators for the Klein-Gordon equation in higher-dimensional Kerr-NUT-(A)dS spacetimes, enabling separation of variables and integrability analysis.

Contribution

It establishes the commutativity and completeness of symmetry operators derived from Killing tensors and vectors in these spacetimes, extending previous results.

Findings

Operators form a complete set of commuting symmetries.

Separated solutions are joint eigenfunctions of all operators.

Explicit zero mode solution for massless Klein-Gordon equation.

Abstract

We consider the Klein-Gordon equation in generalized higher-dimensional Kerr-NUT-(A)dS spacetime without imposing any restrictions on the functional parameters characterizing the metric. We establish commutativity of the second-order operators constructed from the Killing tensors found in arXiv:hep-th/0612029 and show that these operators, along with the first-order operators originating from the Killing vectors, form a complete set of commuting symmetry operators (i.e., integrals of motion) for the Klein-Gordon equation. Moreover, we demonstrate that the separated solutions of the Klein-Gordon equation obtained in arXiv:hep-th/0611245 are joint eigenfunctions for all of these operators. We also present explicit form of the zero mode for the Klein-Gordon equation with zero mass. In the semiclassical approximation we find that the separated solutions of the Hamilton-Jacobi equation for geodesic motion are also solutions for a set of Hamilton-Jacobi-type equations which correspond to the quadratic conserved quantities arising from the above Killing tensors.