Null geodesics and improved unique continuation for waves in asymptotically anti-de Sitter spacetimes

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 in [15,16], 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 than in [15,16], 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.

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

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We reconsider the unique continuation property for a general class of tensorial Klein-Gordon equations of the formon 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.

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“…The most current results in this direction, which further improved upon [14,15], are given by McGill and the second author in [24]. Its main result can be roughly stated as follows:…”

“…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.…”

“…(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.…”