On space–time Killing tensors with a Segre characteristic [(11)(11)]

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Abstract
We consider the set of all space–times which admit a Killing tensor Kαβ whose Segre characteristic is [(11)(11)] and whose eigenvalues λ1 and λ2 satisfy the condition that dλ1 ‐ dλ2 is not null. We prove that there locally exist scalar fields φ3, φ4 such that λ1, λ2, φ3, φ4 constitute a chart, and dλ1⋅dλ2 =dλi⋅dφr=0 for i=1, 2 and r=3, 4. Also, relative to this coordinate system, the metric has the same general form as Carter’s Hamilton–Jacobi separable metric except that his arbitrary functions of λi are replaced by functions of λi, φ3, φ4. If Rαβ(∇αλ1)(∇βλ2) =0, there exists a two parameter Abelian group of isometries which commute with Kαβ; also (φ3,φ4) can be chosen so that ∂/∂φ3 and ∂/∂φ4 are the Killing vectors. Rαβ(∇αλ1)(∇βλ2) =0 is necessary and sufficient for Schrödinger equation separability in case (a) of Carter. A Newtonian analog of our results is discussed.

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