# Towards Bulk Metric Reconstruction from Extremal Area Variations

## TL;DR

This paper demonstrates that in four or more bulk dimensions, the entanglement entropy variations of boundary regions uniquely determine the bulk metric in a covariant manner, even in dynamical spacetimes like black hole interiors.

## Contribution

It extends metric reconstruction from boundary entanglement data to higher dimensions without symmetry assumptions, providing a covariant and explicit reconstruction approach.

## Key findings

- Boundary entanglement entropies fix the bulk metric in 4D and higher.
- The method applies to dynamical regions like black hole interiors.
- The approach offers a pathway to explicit spacetime metric reconstruction.

## 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 $d \geq 4$, 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.