Classification of the orthogonal separable webs for the Hamilton-Jacobi and Laplace-Beltrami equations on 3-dimensional hyperbolic and de Sitter spaces

Abstract

We review the theory of orthogonal separation of variables on pseudo-Riemannian manifolds of constant non-zero curvature via concircular tensors and warped products. We then apply this theory simultaneously to both the three-dimensional Hyperbolic and de Sitter spaces, obtaining an invariant classification of the thirty-four orthogonal separable webs on each space, modulo action of the respective isometry groups. The inequivalent coordinate charts adapted to each web are also determined and listed. The results obtained for Hyperbolic 3-space agree with those in the literature, while the results for de Sitter 3-space appear to be new.

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

“…The article is an extension of the work of Rajaratnam, McLenaghan and Valero [28] who solved a similar problem for 2-dimensional Minkowski space E 2 1 and de Sitter space dS 2 . McLenaghan and Valero have also used this method to obtain a classification of the orthogonal separable webs for 3-dimensional hyperbolic and de Sitter spaces [31]. A classification of separable webs in these spacetimes could be used to solve boundary value problems whose geometries are adapted to one of these coordinate systems, or to study the integrability of Hamilton-Jacobi or Klein-Gordon equations when coupled to an external field.…”

“…The approach used in this paper is based on the theory of concircular tensors and warped products developed by Rajaratnam [25] and Rajaratnam and McLenaghan [26,27]. It is a synthesis of the results of Kalnins [17], Crampin [7] and Benenti [1] that has been extended to include general pseudo-Riemannian spaces with application to pseudo-Riemannian spaces of constant curvature [28,31]. This theory is derived from Eisenhart's [9] characterization of orthogonal separability by means of valence-two Killing tensors which have simple eigenvalues and orthogonally integrable eigendirections, called characteristic Killing tensors.…”

“…In Section 3 the theory of warped products is summarized and an algorithm is given for the construction of reducible separable webs. In both Sections 2 and 3, we restrict ourselves to the case of constant zero curvature; for a compact review of the theory in the case of constant non-zero curvature, we direct the reader to [31,Sections 2 and 3]. In Section 4, the theory reviewed in Sections 2 and 3 are applied to obtain a complete list of the separable webs and inequivalent adapted coordinate systems on E 3 1 .…”

“…Here we recall warped product decompositions of E n ν and their importance for constructing separable webs in E n ν . In this section, we quote the relevant results, referring the reader to [31] or [25] for details. Recall that given pseudo-Riemannian manifolds pN i , g i q for 0 ¤ i ¤ k and smooth positive functions ρ j for 1 ¤ j ¤ k, we define the warped product…”

“…ν having mean curvature normal a i at p. A detailed exposition of this construction is given in [31], for example. For the classification in Section 4, we shall only need warped products with 2 factors; we thus recall the relevant results from [31].…”