Killing Tensor and Carter Constant for Painlevé–Gullstrand Form of Lense–Thirring Spacetime

Geodesics for the Painlevé–Gullstrand form of Lense–Thirring Spacetime

Recently, the current authors have formulated and extensively explored a rather novel Painlevé–Gullstrand variant of the slow-rotation Lense–Thirring spacetime, a variant which has particularly elegant features—including unit lapse, intrinsically flat spatial 3-slices, and a separable Klein–Gordon equation (wave operator). This spacetime also possesses a non-trivial Killing tensor, implying separability of the Hamilton–Jacobi equation, the existence of a Carter constant, and complete formal integrability of the geodesic equations. Herein, we investigate the geodesics in some detail, in the general situation demonstrating the occurrence of “ultra-elliptic” integrals. Only in certain special cases can the complete geodesic integrability be explicitly cast in terms of elementary functions. The model is potentially of astrophysical interest both in the asymptotic large-distance limit and as an example of a “black hole mimic”, a controlled deformation of the Kerr spacetime that can be contrasted with ongoing astronomical observations.