Conformal rigidity from focusing

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 paper, 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 anti-de Sitter/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.

The holographic entropy cone from marginal independence

“…e.g. [10,11]) are visible directly in the singularities of boundary correlators. This can be generalized to the so-called 'bulk-point singularities' of Landau diagrams corresponding to n-particle scattering [12], used for the construction of the 'light-cone cuts' [13][14][15], which offer a much more elegant extraction from much more complicated and harder-to-access CFT data.…”

“…For a fixed number of parties N, the collection of all entropy vectors for all possible Hilbert spaces and density matrices was shown by [30] to be a convex cone 11 known as the N-party quantum entropy cone (QEC N ). 12 By construction, the QEC N is clearly symmetric under an arbitrary permutation of the N parties, as can be seen by just permuting Hilbert space factors.…”

Light-cone cuts and hole-ography: explicit reconstruction of bulk metrics

“…We write the one-dimensional bifurcation surface as Ξ A . The minimal surface 9) of A, E A , lies on Σ, and let R A be the surface on Σ with ∂R A = A ∪ E A . The entanglement wedge of A is the bulk domain of dependence of R A (Fig.…”

“…, both of the top and bottom vertex of ♦ Ip(θ) are null-related to p, since p is by definition on E Ip(θ) , i.e., on 9) For example in a geometry with a black hole, we focus on the case where I p (θ) is not so large, hence the minimal surface becomes a geodesic. This causes no problem to our reconstruction method (see footnote 14)).…”

“…In the former, one component of Einstein equation is assumed, and in the latter, boundary entanglement entropy is used. Other than that, constraints on conformal factors were discussed recently in [9], by using curvature conditions. As another concern, the bulk-point singularity is not suitable for finding light-cone cuts corresponding to points near the horizon of an eternal black hole [4,8], and finding light-cone cuts by the singularity is a tough computation.…”