Covariant Holographic Entanglement Entropy Inversion to Reconstruct Bulk Geometry

Abstract

We derive an analytic inverse map from the renormalized covariant interval entropy to a symmetry-reduced stationary bulk metric. The main result applies to stationary homogeneous asymptotically AdS${}_3$ geometries with a radial lapse, an independent spatial warp factor, and a stationary shift. On a smooth HRT branch, the interval entropy \(S_{\ren}(Δt,Δx)\) is an on-shell Hamilton--Jacobi functional. Its endpoint derivatives determine two conserved charges, and the ratio \(κ=E/J\) organizes the extremal curves into one-parameter families with varying bulk turning points. For each fixed \(κ\), the length and endpoint-separation data give three Volterra equations with the standard Abel square-root kernel. Their inverse kernels reconstruct the radial coordinate and the metric functions \(f,h,v\). We also state the local consistency conditions and the agreement required when two local reconstructions of the same smooth HRT family cover an overlapping radial interval. The construction is verified in pure AdS, rotating BTZ, a static warped geometry, and an Einstein--scalar black brane, and extended to higher-dimensional strips when the transverse area density is fixed by symmetry.