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

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.

6 pages, no figures; typos in eq.(6) fixed; one reference added

References in corpus (5)

• Complete Integrability of Geodesic Motion in General Kerr-NUT-AdS Spacetimes
• Closed conformal Killing-Yano tensor and Kerr-NUT-de Sitter spacetime uniqueness
• Separability of Dirac equation in higher dimensional Kerr-NUT-de Sitter spacetime
• Constants of Geodesic Motion in Higher-Dimensional Black-Hole Spacetimes
• Kerr-NUT-de Sitter Curvature in All Dimensions

Cited by in corpus (5)

• Higher-Dimensional Black Holes: Hidden Symmetries and Separation of Variables
• Closed conformal Killing-Yano tensor and uniqueness of generalized Kerr-NUT-de Sitter spacetime
• Generalized Kerr-NUT-de Sitter metrics in all dimensions
• Separability of the massive Dirac's equation in 5-dimensional Myers-Perry black hole geometry and its relation to a rank-three Killing-Yano tensor
• Symmetry operators and separability of massive Klein-Gordon and Dirac equations in the general 5-dimensional Kerr-(anti-)de Sitter black hole background