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

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## Abstract

**:**

## 1. Introduction

## 2. Killing Tensor

## 3. Two-Form Square Root of the Killing Tensor

## 4. Separability of the Klein–Gordon Equation

## 5. Carter Constant and Other Conserved Quantities

- For ${L}^{2}=\mathcal{C}$, we have $\theta =\pi /2$; the motion is restricted to the equatorial plane.
- For $L=0$ with $\mathcal{C}>0$, the range of $\theta $ is a priori unconstrained; $\theta \in [0,\pi ]$.
- For $L=0$ with $\mathcal{C}=0$, the declination is fixed $\theta \left(\lambda \right)={\theta}_{0}$, and the motion is restricted to a constant declination conical surface.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Wave Operators

## Note

1 |

## References

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**MDPI and ACS Style**

Baines, J.; Berry, T.; Simpson, A.; Visser, M.
Killing Tensor and Carter Constant for Painlevé–Gullstrand Form of Lense–Thirring Spacetime. *Universe* **2021**, *7*, 473.
https://doi.org/10.3390/universe7120473

**AMA Style**

Baines J, Berry T, Simpson A, Visser M.
Killing Tensor and Carter Constant for Painlevé–Gullstrand Form of Lense–Thirring Spacetime. *Universe*. 2021; 7(12):473.
https://doi.org/10.3390/universe7120473

**Chicago/Turabian Style**

Baines, Joshua, Thomas Berry, Alex Simpson, and Matt Visser.
2021. "Killing Tensor and Carter Constant for Painlevé–Gullstrand Form of Lense–Thirring Spacetime" *Universe* 7, no. 12: 473.
https://doi.org/10.3390/universe7120473