Forms of all spacetime metrics which admit [(11) (11)] Killing tensors with nonconstant eigenvalues

Abstract

Forms are obtained for all of the spacetime metrics which admit or are conformal to those which admit Killing tensors whose Segre characteristics are [(11) (11)] and whose eigenvalues λ(1) and λ(2) are not constants. No a priori assumptions are made concerning separability, isometries, invertibility, Petrov type, or the matter tensor. Some of our metrical forms, viz., those for which the Hamilton–Jacobi equation is separable or partially separable after multiplication by an integrating factor, are already known; in particular, they include all of Carter’s Hamilton–Jacobi separable spacetimes. Also, some of our metrical forms are new and may be applicable to finding interesting nonvacuum metrics which admit [(11) (11)] Killing tensors, but which do not necessarily have any Killing vectors. Though our results include a class of metrics for which λ(1) and/or λ(2) are constant, they do not include all such metrics; in an appendix, we prove that the set of all metrics which admit [(11) (11)] Killing tensors with constant λ(1) and λ(2) is identical except for conformal factors with the set of all metrics which admit [(11) (11)] conformal Killing tensors.