Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 019, 28 pages      arXiv:1805.12228      https://doi.org/10.3842/SIGMA.2022.019

Classification of the Orthogonal Separable Webs for the Hamilton-Jacobi and Klein-Gordon Equations on 3-Dimensional Minkowski Space

Carlos Valero a and Raymond G. Mclenaghan b
a) Department of Mathematics and Statistics, McGill University, Montréal, Québec, H3A 0G4, Canada
b) Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

Received July 03, 2021, in final form March 02, 2022; Published online March 12, 2022

Abstract
We review a new theory of orthogonal separation of variables on pseudo-Riemannian spaces of constant zero curvature via concircular tensors and warped products. We then apply this theory to three-dimensional Minkowski space, obtaining an invariant classification of the forty-five orthogonal separable webs modulo the action of the isometry group. The eighty-eight inequivalent coordinate charts adapted to the webs are also determined and listed. We find a number of separable webs which do not appear in previous works in the literature. Further, the method used seems to be more efficient and concise than those employed in earlier works.

Key words: Hamilton-Jacobi equation; Laplace-Beltrami equation; separation of variables; Minkowski space; concircular tensors; warped products.

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