Linear Stability of Schwarzschild-Anti-de Sitter spacetimes II: Logarithmic decay of solutions to the Teukolsky system

TL;DR

This paper establishes boundedness and inverse logarithmic decay of solutions to the Teukolsky equations on Schwarzschild-Anti-de Sitter backgrounds, advancing understanding of stability in these spacetimes.

Contribution

It introduces a novel physical space transformation and new energy and Carleman estimates to analyze the Teukolsky system with boundary conditions.

Findings

Proves boundedness of solutions.

Demonstrates inverse logarithmic decay over time.

Develops new Carleman estimates for coupled equations.

Abstract

We prove boundedness and inverse logarithmic decay in time of solutions to the Teukolsky equations on Schwarzschild-Anti-de Sitter backgrounds with standard boundary conditions originating from fixing the conformal class of the non-linear metric on the boundary. The proofs rely on (1) a physical space transformation theory between the Teukolsky equations and the Regge-Wheeler equations on Schwarzschild-Anti de Sitter backgrounds and (2) novel energy and Carleman estimates handling the coupling of the two Teukolsky equations through the boundary conditions thereby generalising earlier work of \cite{Hol.Smu13} for the covariant wave equation. Specifically, we also produce purely physical space Carleman estimates.