Exponentially growing finite energy solutions for the Klein-Gordon equation on sub-extremal Kerr spacetimes

TL;DR

This paper demonstrates the existence of exponentially growing solutions to the Klein-Gordon equation on sub-extremal Kerr spacetimes, revealing a superradiant instability that contrasts with stability results for the wave equation.

Contribution

It provides the first rigorous construction of a superradiant instability for the Klein-Gordon equation on Kerr backgrounds.

Findings

Existence of exponentially growing solutions for certain masses.

Construction of solutions with arbitrary non-zero integer m.

Contrasts with recent linear stability results for the wave equation.

Abstract

For any sub-extremal Kerr spacetime with non-zero angular momentum, we find an open family of non-zero masses for which there exist smooth, finite energy, and exponentially growing solutions to the corresponding Klein-Gordon equation. If desired, for any non-zero integer m, an exponentially growing solution can be found with mass arbitrarily close to |am|/2Mr_+. In addition to its direct relevance for the stability of Kerr as a solution to the Einstein-Klein-Gordon system, our result provides the first rigorous construction of a superradiant instability. Finally, we note that this linear instability for the Klein-Gordon equation contrasts strongly with recent work establishing linear stability for the wave equation.