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

Abstract

Recently, the authors have formulated and explored a novel Painlevé–Gullstrand variant of the Lense–Thirring spacetime, which has some particularly elegant features, including unit-lapse, intrinsically flat spatial 3-slices, and some particularly simple geodesics—the “rain” geodesics. At the linear level in the rotation parameter, this spacetime is indistinguishable from the usual slow-rotation expansion of Kerr. Herein, we shall show that this spacetime possesses a nontrivial Killing tensor, implying separability of the Hamilton–Jacobi equation. Furthermore, we shall show that the Klein–Gordon equation is also separable on this spacetime. However, while the Killing tensor has a 2-form square root, we shall see that this 2-form square root of the Killing tensor is not a Killing–Yano tensor. Finally, the Killing-tensor-induced Carter constant is easily extracted, and now, with a fourth constant of motion, the geodesics become (in principle) explicitly integrable.

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

“…Based on the algorithm presented in two recent papers by Papadopoulos and Kokkotas [75,76], which are, in turn, based on considerably older results by Benenti and Francaviglia [77], in reference [2] we found the non-trivial Killing tensor:…”

“…Recently, the current authors have introduced and extensively explored a specific new variant of the slow-rotation Lense-Thirring spacetime [1,2], described by the explicit line element…”

“…We shall now extend the physical and mathematical analysis of this spacetime, paying particular attention to the geodesics. We first unavoidably need to provide a brief summary of the key results derived in references [1,2].…”

“…In this variant of the Lense-Thirring spacetime the metric possesses both unit lapse [3], and also exhibits a flat spatial 3-metric. That is, the spacetime metric is presented in socalled Painlevé-Gullstrand form [4][5][6][7], (sometimes called Gullstrand-Painlevé form [8]), with a relatively simple globally defined tetrad [1,2]. (For a textbook-level physically motivated discussion of the tetrad formalism see, for instance, reference [9].)…”

“…From a purely theoretical perspective, the Lense-Thirring metric is algebraically much simpler than the Kerr metric, making most calculations significantly easier to conduct, and the Lense-Thirring metric can be recast into Painlevé-Gullstrand form, while the Kerr metric cannot [46][47][48][49]. This spacetime exhibits a separable Klein-Gordon equation (wave operator) [2] and also possesses a non-trivial Killing tensor, thereby implying separability and complete (formal) integrability of the Hamilton-Jacobi equations for geodesic motion [2]. Below, we shall discuss two particularly interesting classes of geodesics; the generic case involving ultra-elliptic integrals, and the case of vanishing Carter constant where the analysis can be completely performed in terms of elementary functions.…”