The bulk-boundary correspondence for the Einstein equations in asymptotically Anti-de Sitter spacetimes

Abstract

In this paper, we consider vacuum asymptotically anti-de Sitter spacetimes $(\mathcal{M},g)$ with conformal boundary $(\mathcal{I},\mathfrak{g})$. We establish a correspondence, near $\mathcal{I}$, between such spacetimes and their conformal boundary data on $\mathcal{I}$. More specifically, given a domain $\mathcal{D}\subset \mathcal{I}$, we prove that the coefficients ${\mathfrak{g}}^{(0)}=\mathfrak{g}$ and ${\mathfrak{g}}^{(n)}$ (the undetermined term or stress energy tensor) in a Fefferman-Graham expansion of the metric $g$ from the boundary uniquely determine $g$ near $\mathcal{D}$, provided $\mathcal{D}$ satisfies a generalised null convexity condition (GNCC). The GNCC is a conformally invariant criterion on $\mathcal{D}$, first identified by Chatzikaleas and the second author, that ensures a foliation of pseudoconvex hypersurfaces in $\mathcal{M}$ near $\mathcal{D}$, and with the pseudoconvexity degenerating in the limit at $\mathcal{D}$. As a corollary of this result, we deduce that conformal symmetries of $({\mathfrak{g}}^{(0)},{\mathfrak{g}}^{(n)})$ on domains $\mathcal{D}\subset \mathcal{I}$ satisfying the GNCC extend to spacetimes symmetries near $\mathcal{D}$. The proof, which does not require any analyticity assumptions, relies on three key ingredients: (1) a calculus of vertical tensor-fields developed for this setting; (2) a novel system of transport and wave equations for differences of metric and curvature quantities; and (3) recently established Carleman estimates for tensorial wave equations near the conformal boundary.

60 pages, 1 figure. Version accepted at ARMA

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• On the uniqueness and global dynamics of AdS spacetimes
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• A gauge-invariant unique continuation criterion for waves in asymptotically Anti-de Sitter spacetimes

Cited by in corpus (1)

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