A gauge-invariant unique continuation criterion for waves in asymptotically Anti-de Sitter spacetimes

TL;DR

This paper develops a gauge-invariant unique continuation criterion for wave equations in asymptotically Anti-de Sitter spacetimes, extending previous results by broadening geometric assumptions and enabling applications to nonlinear Einstein equations.

Contribution

It introduces a more general, gauge-invariant null convexity criterion and extends unique continuation results to larger boundary domains in asymptotically AdS spacetimes.

Findings

Replaces null convexity criterion with a gauge-invariant version.

Extends unique continuation to broader boundary domains.

Links failure of the criterion to null geodesics, enabling counterexamples.

Abstract

We reconsider the unique continuation property for a general class of tensorial Klein-Gordon equations of the form \begin{align*} \Box_{g} \phi + \sigma \phi = \mathcal{G}(\phi,\nabla \phi) \text{,} \qquad \sigma \in \mathbb{R} \end{align*} on a large class of asymptotically anti-de Sitter spacetimes. In particular, we aim to generalize the previous results of Holzegel, McGill, and the second author [14,15,24] (which established the above-mentioned unique continuation property through novel Carleman estimates near the conformal boundary) in the following ways: (1) We replace the so-called null convexity criterion (the key geometric assumption on the conformal boundary needed in [24] to establish the unique continuation properties) by a more general criterion that is also gauge invariant. (2) Our new unique continuation property can be applied from a larger, more general class of domains on the conformal boundary. (3) Similar to [24], we connect the failure of our generalized null convexity criterion to the existence of certain null geodesics near the conformal boundary. These geodesics can be used to construct counterexamples to unique continuation. Finally, our gauge-invariant criterion and Carleman estimate will constitute a key ingredient in proving unique continuation results for the full nonlinear Einstein-vacuum equations, which will be addressed in a forthcoming paper of Holzegel and the second author [16].