Killing tensor and Carter constant for Painleve-Gullstrand form of Lense-Thirring spacetime

TL;DR

This paper demonstrates that a novel Painleve-Gullstrand form of the Lense-Thirring spacetime admits a nontrivial Killing tensor, enabling the separability of key equations and the explicit integration of geodesics, revealing new insights into its geometric structure.

Contribution

The paper shows that this new spacetime admits a Killing tensor and Carter constant, allowing for separability and explicit integration of geodesics, which was not previously known.

Findings

Existence of a nontrivial Killing tensor in the spacetime

Separability of the Hamilton-Jacobi and Klein-Gordon equations

Explicit integrability of geodesics due to Carter constant

Abstract

Recently, the authors have formulated and explored a novel Painleve-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 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.