Mode stability on the real axis

TL;DR

This paper extends the mode stability results of the Teukolsky equation to real frequencies, proving that purely outgoing and ingoing solutions are linearly independent, which aids in solving the inhomogeneous equation.

Contribution

It generalizes Whiting's 1989 mode stability result to real frequencies for the Teukolsky equation on Kerr spacetime.

Findings

Purely outgoing solutions at infinity and ingoing at the horizon must vanish if separated.

Existence of linearly independent fundamental solutions for real frequencies.

Provides a representation formula for inhomogeneous solutions.

Abstract

A generalization of the mode stability result of Whiting (1989) for the Teukolsky equation is proved for the case of real frequencies. The main result of the paper states that a separated solution of the Teukolsky equation governing massless test fields on the Kerr spacetime, which is purely outgoing at infinity, and purely ingoing at the horizon, must vanish. This has the consequence, that for real frequencies, there are linearly independent fundamental solutions of the radial Teukolsky equation which are purely ingoing at the horizon, and purely outgoing at infinity, respectively. This fact yields a representation formula for solutions of the inhomogenous Teukolsky equation.