Covariant Holographic Entanglement Entropy Inversion to Reconstruct Bulk Geometry

Abstract

We present an analytic inversion of covariant holographic entanglement entropy beyond equal-time inversion, in which only one effective radial coefficient remains after the radial gauge and transverse density are fixed. The formula reconstructs the stationary radial metric block probed by the corresponding HRT geodesics, including a nontrivial spatial warp factor and the the stationary shift associated with frame dragging. In this sector, the renormalized interval entropy \(S(Δt,Δx)\) is an on-shell Hamilton--Jacobi functional. Its endpoint derivatives determine the conserved charges of the corresponding extremal geodesic, and their ratio characterizes the projective class of the endpoint covector associated with the boundary interval, $κ=E/J=-\partial_{Δt} S_{\rm ren}/\partial_{Δx} S_{\rm ren}$. For each fixed \(κ\), the entropy data define an Abel-type reconstruction of a radial metric block. A single classical geometry is obtained only when the reconstructions from different fixed-\(κ\) families agree as functions of one common radial coordinate. This cross-family compatibility condition is the integrability condition of the covariant inverse problem. The analysis is restricted to the classical Hubeny--Rangamani- Takayanagi(HRT) area term.