Towards Bulk Metric Reconstruction from Extremal Area Variations

TL;DR

This work proves that in bulk dimensions $d\ge4$, the metric in any region foliated by boundary-anchored extremal disks is uniquely determined (up to diffeomorphism) by first and second variations of the areas of those disks, i.e. by boundary entanglement-entropy data via HRT. The authors develop a four-step, covariant reconstruction strategy: fix coordinates from the foliation and isothermal charts on each disk; determine the normal-m bundle metric components $g^{ij}$ from the Jacobi operator using boundary data and Alb\'in–Guillarmou–Tzou–Uhlmann results; fix the off-diagonal components $g^{\alpha i}$ by tilting the foliation and solving a linear system; and determine the conformal factor on each disk from the extremality condition as a first-order hyperbolic PDE along a one-parameter foliation. The result does not rely on symmetries and remains applicable to dynamical spacetimes, including black-hole interiors, and it provides a clear route toward an explicit spacetime metric reconstruction, with potential extensions to quantum corrections and deeper bulk probing. This advances the program of bulk emergence from boundary entanglement by establishing a rigorous boundary-data–driven uniqueness framework for bulk metrics in holography.

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.

Figures (7)

• Figure 1: An illustration of the foliation $\Sigma(\lambda^i)$ of extremal surfaces we consider (for clarity we suppress a dimension so the extremal surfaces appear as curves). These surfaces foliate some portion of the bulk, and in an appropriate limit of the $\lambda^i$ they degenerate to a point on $\partial M$.
• Figure 2: In $d = 4$, we sketch the construction of a family of extremal surfaces $\Sigma(\lambda^i)$ which foliate $\mathcal{R}$ under some mild, but certainly nontrivial, assumptions. \ref{['subfig:bndryfoliation']}: Consider some slicing of the boundary into slices of constant $t \in (t_i, t_f)$, and on each such slice introduce a one-parameter family of regions $R_t(s)$ which shrink to a point as $s$ is decreased. \ref{['subfig:bulkfoliation']}: for each $t$, the HRT surfaces $X[R_t(s)]$ will sweep out a three-dimensional surface $\Xi_t$ (shaded) as long as they never jump. If $\Xi_t$ moves everywhere continuously to the future as $t$ is increased, then the $\Xi_t$ foliate some region $\mathcal{R}$, and the family of extremal surfaces $\Sigma(t,s) = X[R_t(s)]$ do as well.
• Figure 3: To show uniqueness of the off-diagonal metric components, we deform the foliation $\Sigma(\lambda^i)$ to a family of foliations $\Sigma(s; \lambda_s^i)$. This deformation is generated by a one-parameter group of diffeomorphisms $\phi_s$ which are fixed by requiring that under the action of $\phi_s$, the isothermal coordinates of each point remain unchanged. In other words, if the point $p$ is labeled by the isothermal coordinate values $(x^1_*, x^2_*)$ on the surface $\Sigma(\lambda^i = \lambda^i_*)$, then the mapped point $\phi_s(p)$ must be labeled by the same values of the isothermal coordinates on the deformed surface $\Sigma(s; \lambda_s^i = \lambda^i_*)$.
• Figure 4: A set of coordinates $\{y^\alpha\}$ on $\Sigma$ corresponds to a map $\psi: \Sigma \to \mathbb{R}^2$. A set of isothermal coordinates $\{x^\alpha\}$ can be obtained by another map $\Phi: \mathbb{R}^2 \to \mathbb{R}^2$. For two different metrics $g_1$, $g_2$ on $\Sigma$, the corresponding maps $\Phi_1$, $\Phi_2$ can be chosen to yield the same image $\Phi_1(\psi(\Sigma)) = \Phi_2(\psi(\Sigma))$, but they need not agree pointwise; in other words, the isothermal coordinates of the point $p$ obtained by the map $\Phi_1$ need not be the same as those obtained by the map $\Phi_2$, as shown. Thus if the (pointwise) boundary data corresponding to $g_1$ and $g_2$ agrees on $\Sigma$, it necessarily agrees in the coordinates $\{y^\alpha\}$, but it need not agree in the coordinates $\{x^\alpha\}$.
• Figure 5: The extension of $\Sigma$ to an asymptotically flat manifold. After choosing a coordinate system $\{y^\alpha\}$ on $\Sigma$ defined by the map $\psi$, the metric components $g_{\alpha\beta}$, connection one-forms ${\omega_{\alpha i}}^j$, and potential ${Q_i}^j$ are extended to the entire $(y^1, y^2)$ plane by requiring that the former two vanish outside of $\psi(\Sigma)$ while $g_{\alpha\beta}$ should be continuous at $\partial\psi(\Sigma)$ and equal to the Euclidean metric $\delta_{\alpha\beta}$ outside of some set containing $\psi(\Sigma)$, denoted by the dotted line.