Metric Reconstruction from Timelike Entanglement Entropy

Abstract: Timelike entanglement entropy (TEE) provides a Lorentzian boundary probe of bulk geometry, but its use for metric reconstruction depends on the holographic prescription and on the extremal-surface branch selected by that prescription. We study this inverse problem for strip-shaped TEE data and make these dependencies explicit. In the complex-valued weak extremal surface (CWES) prescription, the time-width dependence of TEE determines an Abel density $H(W)$ on a selected real branch; for Bañados-Teitelboim-Zanelli (BTZ) black holes this gives an analytic reconstruction of the blackening factor once the singularity endpoint fixes the radial origin. After developing a forward numerical method for the complex-coordinate prescription, we formulate it as the main reconstruction scheme for the asymptotically $AdS_{d+1}$ backgrounds with $d\geq 2$. On a chosen complex branch, the time-width dependence of TEE supplies the Abel input that determines the TEE-accessible density, while one additional geometric anchor is required to convert that density into a definite radial metric profile. With UV or horizon-scale anchoring and rational continuation from the reconstructed complex-path samples, the method reproduces the benchmark BTZ, four-dimensional Schwarzschild, and Reissner-Nordström (RN) blackening factors. However, the Gubser-Rocha example shows that a single strip observable with a nontrivial spatial warp factor determines only one functional combination of the metric functions.