Covariant Holographic Entanglement Entropy Inversion to Reconstruct Bulk Geometry

Abstract: We present an analytic inversion of covariant holographic entanglement entropy beyond the usual one-function equal-time radial inverse problem, in which one effective radial coefficient remains after the radial gauge and transverse density are fixed. The formula reconstructs the stationary radial block seen by the HRT family, including a nontrivial spatial warp factor and the shift responsible for frame dragging. On a smooth HRT branch, the interval entropy (S(Δt,Δx)) is an on-shell Hamilton--Jacobi functional: its endpoint derivatives give conserved HRT charges (E) and (J), and the ratio (κ=E/J) labels characteristic leaves. For each fixed (κ), the entropy defines an Abel reconstruction of a candidate radial block. A single classical radial geometry exists only when the candidates obtained at different (κ)'s agree as functions of the same radial coordinate. This cross-leaf agreement is the hidden integrability condition of the covariant problem. When it holds, the same data determine the projected bulk light cone, making the frame-dragging profile, cone opening, horizon generator, and ergosurface visible from entanglement. Failure of the condition is not a failure of holography; it means that the HEE cannot be interpreted within the assumed radial HRT reduction without further input, such as branch resolution, transverse-density separation, additional region data, or genuinely time-dependent information. We illustrate the construction with pure AdS, rotating BTZ, a static warped metric, a boosted Einstein--scalar black brane, a higher-dimensional strip degeneracy, and a thin-shell obstruction. The analysis uses the classical HRT area term.