A new result on the Klein-Gordon equation in the background of a rotating black hole

TL;DR

This paper proves the essential self-adjointness of the spatial operator for the Klein-Gordon equation in Kerr black hole backgrounds, enabling a rigorous operator-theoretic approach to the wave equation's well-posedness.

Contribution

It introduces a new proof of self-adjointness for the wave operator in Kerr spacetime, extending previous results from Schwarzschild black holes.

Findings

Proved essential self-adjointness of the spatial wave operator in Kerr spacetime.

Established well-posedness of the Klein-Gordon initial value problem in a weighted L^2-space.

Generalized previous Schwarzschild results to Kerr black holes.

Abstract

This short paper should serve as basis for further analysis of a previously found new symmetry of the solutions of the wave equation in the gravitational field of a Kerr black hole. Its main new result is the proof of essential self-adjointness of the spatial part of a reduced normalized wave operator of the Kerr metric in a weighted L^2-space. As a consequence, it leads to a purely operator theoretic proof of the well-posedness of the initial value problem of the reduced Klein-Gordon equation in that field in that L^2-space and in this way generalizes a corresponding result of Kay (1985) in the case of the Schwarzschild black hole. It is believed that the employed methods are applicable to other separable wave equations.