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.
Highlights
Academic Editors: PanayiotisRecently, the current authors have introduced and explored a new variant of theLense–Thirring spacetime [1], specified by the line elementStavrinos and Emmanuel N
This variant of the Lense–Thirring spacetime is rather useful since the metric is recast into the Painlevé–Gullstrand form [2,3,4,5]
Nontrivial Killing tensors are incredibly useful mathematical objects that are present in almost all candidate spacetimes and can be thought of as generalisations of Killing vectors
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. This variant of the Lense–Thirring spacetime is rather useful since the metric is recast into the Painlevé–Gullstrand form [2,3,4,5]. Gullstrand form, while the Kerr metric cannot [30,31,32,33] Given that this variant of the Lense–Thirring metric is amenable to significantly more tractable mathematical analysis and is a valid approximation for the gravitational fields of rotating stars and planets in the same regime as the Kerr solution is appropriate, there is a compelling argument to use the Painlevé–Gullstrand form of Lense–Thirring to model various astrophysically interesting cases [34,35,36]. The existence of this additional constant of motion implies complete separability of the Hamilton–Jacobi equation, which makes the geodesic equations fully integrable, at least in principle
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