Boundary Rigidity and Holography

TL;DR

This paper reviews boundary rigidity theorems and explores their application to reconstructing negative curvature space-times from boundary data in holography, highlighting limitations and alternative methods within AdS/CFT.

Contribution

It applies boundary rigidity theorems to holographic reconstruction, identifies conditions where they fail, and surveys alternative procedures in the context of AdS/CFT.

Findings

Boundary rigidity theorems can reconstruct certain space-times from boundary geodesic spectra.

Examples show that negative-curvature spaces can violate assumptions, hindering reconstruction.

Alternative holographic reconstruction methods are discussed and evaluated.

Abstract

We review boundary rigidity theorems assessing that, under appropriate conditions, Riemannian manifolds with the same spectrum of boundary geodesics are isometric. We show how to apply these theorems to the problem of reconstructing a $d+1$ dimensional, negative curvature space-time from boundary data associated to two-point functions of high-dimension local operators in a conformal field theory. We also show simple, physically relevant examples of negative-curvature spaces that fail to satisfy in a subtle way some of the assumptions of rigidity theorems. In those examples, we explicitly show that the spectrum of boundary geodesics is not sufficient to reconstruct the metric in the bulk. We also survey other reconstruction procedures and comment on their possible implementation in the context of the holographic AdS/CFT duality.