The Dirichlet-to-Neumann map on asymptotically anti-de Sitter spaces and holography

TL;DR

This paper studies the Dirichlet-to-Neumann map for Klein-Gordon equations on asymptotically anti-de Sitter spaces, revealing its spectral properties and its role in recovering boundary metrics and Einstein metrics.

Contribution

It establishes the fractional power relationship of the Dirichlet-to-Neumann map to boundary wave operators and demonstrates its ability to determine boundary metric Taylor series and Einstein metrics.

Findings

Dirichlet-to-Neumann map is a fractional power of boundary wave operator

Map determines boundary metric Taylor series outside a countable set of masses

Lorentzian Graham-Zworski theorem relates poles to conformally invariant operators

Abstract

We consider the Klein-Gordon equation on asymptotically anti-de Sitter spacetimes, and show that the forward Dirichlet-to-Neumann map (or scattering matrix) is a fractional power of the boundary wave operator modulo lower order terms in the sense of paired Lagrangian distributions. We use it to show that, outside of a countable set of mass parameters, the Dirichlet-to-Neumann map determines the Taylor series of the bulk metric at the boundary, and hence allows the recovery of a real analytic metric or Einstein metric modulo isometries. Furthermore, we prove a Lorentzian version of the Graham-Zworski theorem relating poles of the Dirichlet-to-Neumann map to conformally invariant powers of the boundary wave operator.