Improved Analytic Solution of Black Hole Superradiance
Abstract
The approximate solution of the Klein-Gordon equation for a real scalar field of mass μ in the geometry of a Kerr black hole obtained by Detweiler Detweiler:1980uk is widely used in the analysis of the stability of black holes as well as the search of axion-like particles. In this work, we confirm a missing factor 1/2 in this solution, which was first identified in Ref.~Pani:2012bp. The corrected result has strange features that put questions on the power-counting strategy. We solve this problem by adding the next-to-leading order (NLO) contribution. Compared to the numerical results, the NLO solution reduces the percentage error of the LO solution by a factor of 2 for all important values of rg μ. Especially the percentage error is 10\% in the region of rgμ 0.35. The NLO solution also has a compact form and could be used straightforwardly.