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

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

2007-11-29

arxiv.org/abs/0711.4623v2

Phys.Rev.D77:044033,2008

Physics

High Energy Physics

High Energy Physics - Theory

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

Scientific paper

10.1103/PhysRevD.77.044033

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.

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