Towards bulk metric reconstruction from extremal area variations

Abstract

The Ryu-Takayanagi and Hubeny-Rangamani-Takayanagi formulae suggest that bulk geometry emerges from the entanglement structure of the boundary theory. Using these formulae, we build on a result of Alexakis, Balehowsky, and Nachman to show that in four bulk dimensions, the entanglement entropies of boundary regions of disk topology uniquely fix the bulk metric in any region foliated by the corresponding HRT surfaces. More generally, for a bulk of any dimension d ≥ 4, knowledge of the (variations of the) areas of two-dimensional boundary-anchored extremal surfaces of disk topology uniquely fixes the bulk metric wherever these surfaces reach. This result is covariant and not reliant on any symmetry assumptions; its applicability thus includes regions of strong dynamical gravity such as the early-time interior of black holes formed from collapse. While we only show uniqueness of the metric, the approach we present provides a clear path towards an explicit spacetime metric reconstruction.

Bulk reconstruction beyond the entanglement wedge

“…In this letter, we restricted ourselves to twodimensional surfaces to exploit the BCFK metric reconstruction results [15]. Metric reconstruction may also be possible using the area data of minimal k-surfaces, which would lead to similar conclusions for CFT operators whose expectation values correspond to such areas.…”

“…This conjecture has been tested in several cases, and has since become the de facto standard for computing such Wilson lines in holography. Second, a recent result shows that knowing the areas of these surfaces is sufficient to uniquely fix the space-time metric in the region that they probe [15]. We therefore arrive at an alternative: Either 1. a boundary subregion contains information about much more of the bulk than just the entanglement wedge, or 2. a boundary subregion fails to contain information about operators whose expectation values give the areas of extremal 2-surfaces anchored within that subregion, for example, smooth, non-self-intersecting Wilson lines.…”

“…where A[m(B)] is the regularized area of the minimal 2surface anchored to B, and ∆ is a prefactor whose contribution is subleading at large λ (except in the presence of kinks or self-intersections, which we will not need below in the context of the BCFK result [15]). Therefore, the areas of minimal 2-surfaces anchored within A can be deduced from the reduced density matrix ρ A by extracting the expectation values of Wilson loops.…”

“…We focused on smooth boundary Wilson loops in particular because knowing the areas of all minimal 2surfaces whose anchors lie on smooth, closed curves within A is sufficient to uniquely fix the full bulk metric in W 2 (A) for a large class of boundary subregions. This is essentially the content of the BCFK theorem [15]: Let d ≥ 3, and let A be a (d − 1)-dimensional boundary subregion that is topologically a ball. Suppose that every point in W 2 (A) lies on a minimal 2-surface m(B), which can be retracted via a family of minimal surfaces to a single point in A.…”

Surface theory: the classical, the quantum, and the holographic

“…First, the applications we present in Section 6 are just an example of how the formalism that we present, which relates perturbations of extremal surfaces to elliptic operators, can be used to deduce new information about the bulk; they are far from an exhaustive study of the applications of elliptic operator theory in this context. For instance, the first application (that we are aware of) of elliptic operator theory to classical extremal surfaces via subregion/subregion duality may be found in [33], which gave a holographic account of dynamical black hole entropy; a more recent application of elliptic operator theory via the equation of classical extremal deviation to bulk reconstruction may be found in [34]. Since this article is the first presentation of the equation of quantum extremal deviation, the results discussed in Section 6 constitute the first applications of elliptic operator theory to subregion/subregion duality in the semiclassical regime.…”