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arXiv · 2110.01814

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

Joshua Baines·Thomas Berry·Alex Simpson·Matt Visser
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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.

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Joshua Baines, Thomas Berry, Alex Simpson, Matt Visser. 2021-10-05. Killing tensor and Carter constant for Painleve-Gullstrand form of Lense-Thirring spacetime. https://doi.org/10.3390/universe7120473

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