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

TL;DR

This paper establishes a bulk-boundary correspondence for Einstein equations in asymptotically Anti-de Sitter spacetimes, linking boundary data to the spacetime metric near the boundary under a conformally invariant null convexity condition.

Contribution

It proves a unique determination of the bulk spacetime metric from boundary data without analyticity assumptions, using new tensor calculus, transport-wave systems, and Carleman estimates.

Findings

Boundary data uniquely determines the spacetime metric near the boundary.

Conformal symmetries on the boundary extend to spacetime symmetries.

The method does not require analyticity assumptions.

Abstract

In this paper, we consider vacuum asymptotically anti-de Sitter spacetimes $( \mathscr{M}, g )$ with conformal boundary $( \mathscr{I}, \mathfrak{g} )$. We establish a correspondence, near $\mathscr{I}$, between such spacetimes and their conformal boundary data on $\mathscr{I}$. More specifically, given a domain $\mathscr{D} \subset \mathscr{I}$, 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 $\mathscr{D}$, provided $\mathscr{D}$ satisfies a generalised null convexity condition (GNCC). The GNCC is a conformally invariant criterion on $\mathscr{D}$, first identified by Chatzikaleas and the second author, that ensures a foliation of pseudoconvex hypersurfaces in $\mathscr{M}$ near $\mathscr{D}$, and with the pseudoconvexity degenerating in the limit at $\mathscr{D}$. As a corollary of this result, we deduce that conformal symmetries of $( \mathfrak{g}^{(0)}, \mathfrak{g}^{(n)} )$ on domains $\mathscr{D} \subset \mathscr{I}$ satisfying the GNCC extend to spacetimes symmetries near $\mathscr{D}$. 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.