Unique continuation from infinity in asymptotically Anti-de Sitter spacetimes

TL;DR

This paper proves unique continuation properties for Klein-Gordon equations in asymptotically Anti-de Sitter spacetimes, showing that high-order vanishing on the boundary implies vanishing nearby, with implications for boundary conditions and gravitational perturbations.

Contribution

It establishes new Carleman estimates and demonstrates boundary-to-bulk unique continuation for Klein-Gordon equations in asymptotically Anti-de Sitter spacetimes, including applications to gravitational perturbations.

Findings

Unique continuation from boundary to interior in AdS spacetimes.

Conditions under which Dirichlet and Neumann boundary data imply solution uniqueness.

Application to gravitational perturbations and global spacetime properties.

Abstract

We consider the unique continuation properties of asymptotically Anti-de Sitter spacetimes by studying Klein-Gordon-type equations $\Box_g \phi + \sigma \phi = \mathcal{G} ( \phi, \partial \phi )$, $\sigma \in \mathbb{R}$, on a large class of such spacetimes. Our main result establishes that if $\phi$ vanishes to sufficiently high order (depending on $\sigma$) on a sufficiently long time interval along the conformal boundary $\mathcal{I}$, then the solution necessarily vanishes in a neighborhood of $\mathcal{I}$. In particular, in the $\sigma$-range where Dirichlet and Neumann conditions are possible on $\mathcal{I}$ for the forward problem, we prove uniqueness if both these conditions are imposed. The length of the time interval can be related to the refocusing time of null geodesics on these backgrounds and is expected to be sharp. Some global applications as well a uniqueness result for gravitational perturbations are also discussed. The proof is based on novel Carleman estimates established in this setting.