Relativistic Invariance and the Square‐Root Klein‐Gordon Equation

1963 ◽  
Vol 4 (1) ◽  
pp. 17-23 ◽  
Author(s):  
J. Sucher
Keyword(s):  
Gordon Equation ◽  
Square Root ◽  
Relativistic Invariance ◽  
Klein Gordon ◽  
Klein Gordon Equation

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

Universe ◽  
2021 ◽  
Vol 7 (12) ◽  
pp. 473
Author(s):  
Joshua Baines ◽  
Thomas Berry ◽  
Alex Simpson ◽  
Matt Visser
Keyword(s):  
Jacobi Equation ◽  
Gordon Equation ◽  
Rotation Parameter ◽  
Slow Rotation ◽  
Square Root ◽  
Killing Tensor ◽  
Constant Of Motion ◽  
Klein Gordon ◽  
Hamilton Jacobi Equation ◽  
Klein Gordon Equation

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.

scholarly journals Fractional Klein-Gordon equation with singular mass

2021 ◽  
Vol 143 ◽  
pp. 110579
Author(s):  
Arshyn Altybay ◽  
Michael Ruzhansky ◽  
Mohammed Elamine Sebih ◽  
Niyaz Tokmagambetov
Keyword(s):  
Gordon Equation ◽  
Klein Gordon ◽  
Klein Gordon Equation
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