Mode stability for the Teukolsky equations on Kerr-anti-de Sitter spacetimes

TL;DR

This paper proves mode stability for the Teukolsky equations on Kerr-anti-de Sitter spacetimes under certain conditions, ruling out non-stationary solutions and stationary solutions within specific parameter bounds, thus laying groundwork for decay estimates.

Contribution

It establishes the absence of non-stationary and certain stationary mode solutions for the Teukolsky equations on Kerr-anti-de Sitter spacetimes, advancing understanding of their stability.

Findings

No non-stationary real mode solutions exist.

Stationary solutions do not exist under specific parameter bounds.

Mode stability is established, supporting future decay estimates.

Abstract

We prove that there are no non-stationary (with respect to the Hawking vectorfield) real mode solutions to the Teukolsky equations on all $(3+1)$-dimensional subextremal Kerr-anti-de Sitter spacetimes. We further prove that stationary solutions do not exist if the black hole parameters satisfy the Hawking-Reall bound and $\left|a\sqrt{-\Lambda}\right|<\frac{\sqrt{3}}{20}$. We conclude with the statement of mode stability which preludes boundedness and decay estimates for general solutions which will be proven in a separate paper. Our boundary conditions are the standard ones which follow from fixing the conformal class of the metric at infinity and lead to a coupling of the two Teukolsky equations. The proof relies on combining the Teukolsky-Starobinsky identities with the coupled boundary conditions. In the stationary case the proof exploits elliptic estimates which fail if the Hawking-Reall bound is violated. This is consistent with the superradiant instabilities expected in that regime.