Asymptotic properties of linear field equations in anti-de Sitter space

TL;DR

This paper investigates the behavior of wave, Maxwell, and linearized Bianchi equations in anti-de Sitter space, establishing decay estimates and energy bounds crucial for understanding the spacetime's stability under dissipative boundary conditions.

Contribution

It provides the first comprehensive proof of uniform energy boundedness and decay estimates for linear fields in AdS space with dissipative boundaries, highlighting a derivative loss linked to trapping phenomena.

Findings

Proved uniform boundedness of energy with dissipative boundary conditions.

Established decay estimates including a derivative loss near the AdS boundary.

Showed that non-degenerate energy decay without loss of derivatives is impossible due to trapping.

Abstract

We study the global dynamics of the wave equation, Maxwell's equation and the linearized Bianchi equations on a fixed anti-de Sitter (AdS) background. Provided dissipative boundary conditions are imposed on the dynamical fields we prove uniform boundedness of the natural energy as well as both degenerate (near the AdS boundary) and non-degenerate integrated decay estimates. Remarkably, the non-degenerate estimates "lose a derivative". We relate this loss to a trapping phenomenon near the AdS boundary, which itself originates from the properties of (approximately) gliding rays near the boundary. Using the Gaussian beam approximation we prove that non-degenerate energy decay without loss of derivatives does not hold. As a consequence of the non-degenerate integrated decay estimates, we also obtain pointwise-in-time decay estimates for the energy. Our paper provides the key estimates for a proof of the non-linear stability of the anti-de Sitter spacetime under dissipative boundary conditions. Finally, we contrast our results with the case of reflecting boundary conditions.