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Efficiently Separating the Source of the Teukolsky Equation

Andrew Spiers
Feb 2024
Recent gravitational wave detections from black hole mergers have underscored the critical role black hole perturbation theory and the Teukolsky equation play in understanding the behaviour of black holes. The separable nature of the Teukolsky equation has long been leveraged to study the vacuum linear Teukolsky equation; however, as theory and measurements advance, solving the sourced Teukolsky equation is becoming a frontier of research. In particular, second-order calculations, such as in quasi-normal mode and self-force problems, have extended sources. This paper presents a novel method for efficiently separating the Teukolsky equation's source analytically, a non-trivial problem due to the angular and radial mixing of generic quantities in Kerr spacetime. We provide a proof of concept demonstration of our method's accuracy, separating the Teukolsky source produced by the stress-energy tensor of an ideal gas cloud surrounding a Kerr black hole. The detailed application of our method is provided in an accompanying \textit{Mathematica} notebook. Our approach opens up a new avenue for efficient, accurate black hole perturbation theory calculations with sources in both the time and frequency domain.
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