Conformal Rigidity from Focusing

TL;DR

This paper demonstrates that the null curvature condition and causal structure impose strong constraints on the metric, affecting conformal rescalings and enabling metric reconstruction in AdS/CFT contexts.

Contribution

It proves that conformal rescaling in vacuum spacetimes leads to geodesic incompleteness or negative null curvature, and applies these results to metric reconstruction in AdS/CFT.

Findings

Conformal rescaling in vacuum spacetimes causes geodesic incompleteness or negative null curvature.

The results enable metric reconstruction in regions probed by null geodesics in AdS/CFT.

Constraints on the conformal factor in non-vacuum spacetimes facilitate approximate metric reconstruction.

Abstract

The null curvature condition (NCC) is the requirement that the Ricci curvature of a Lorentzian manifold be nonnegative along null directions, which ensures the focusing of null geodesic congruences. In this note, we show that the NCC together with the causal structure significantly constrain the metric. In particular, we prove that any conformal rescaling of a vacuum spacetime introduces either geodesic incompleteness or negative null curvature, provided the conformal factor is non-constant on at least one complete null geodesic. In the context of bulk reconstruction in AdS/CFT, our results combined with the technique of light-cone cuts can be used in vacuum spacetimes to reconstruct the full metric in regions probed by complete null geodesics reaching the boundary. For non-vacuum spacetimes, our results constrain the conformal factor, giving an approximate reconstruction of the metric.