Null Geodesics and Improved Unique Continuation for Waves in Asymptotically Anti-de Sitter Spacetimes

TL;DR

This paper advances the understanding of unique continuation for Klein-Gordon equations in asymptotically Anti-de Sitter spacetimes by establishing new Carleman estimates, broadening the class of spacetimes considered, and connecting assumptions to null geodesic trajectories.

Contribution

It introduces improved Carleman estimates for Klein-Gordon equations on more general asymptotically AdS spacetimes and develops a covariant formalism for tensorial objects at the boundary.

Findings

New Carleman estimates for broader spacetime classes

Connection between boundary assumptions and null geodesics

Development of a covariant formalism for boundary analysis

Abstract

We consider the question of whether solutions of Klein--Gordon equations on asymptotically Anti-de Sitter spacetimes can be uniquely continued from the conformal boundary. Positive answers were first given by the second author with G. Holzegel, under suitable assumptions on the boundary geometry and with boundary data imposed over a sufficiently long timespan. The key step was to establish Carleman estimates for Klein--Gordon operators near the conformal boundary. In this article, we further improve upon the above-mentioned results. First, we establish new Carleman estimates---and hence new unique continuation results---for Klein--Gordon equations on a larger class of spacetimes, in particular with more general boundary geometries. Second, we argue for the optimality, in many respects, of our assumptions by connecting them to trajectories of null geodesics near the conformal boundary; these geodesics play a crucial role in the construction of counterexamples to unique continuation. Finally, we develop a new covariant formalism that will be useful---both presently and more generally beyond this article---for treating tensorial objects with asymptotic limits at the conformal boundary.