Bulk Spacetime Encoding via Boundary Ambiguities

Abstract

We propose a method to reconstruct the metric and its arbitrary-order derivatives at the horizon for any static, planar-symmetric black hole, using an infinite set of discrete pole-skipping points in momentum space where the boundary Green's function becomes ambiguous. This method is fully analytical and involves solving only linear equations. The near-horizon reconstruction can extend either inside or outside the horizon until reaching the nearest singularity in the complex radial plane. It further enables a reinterpretation of any pure gravitational field equation in pole-skipping data. Moreover, our method reveals that the pole-skipping points are redundant: only a subset is independent, while the rest are fixed by an equal number of homogeneous polynomial constraints. These identities are universal, independent of the details of the bulk geometry, including its dimensionality, asymptotic behavior, or the existence of a holographic duality.

Figures (2)

• Figure 1: Heatmaps of the Green's function in deformed CFTs dual to BTZ black holes with a finite radial cutoff. Red and blue lines denote the poles and zeros respectively, while black dots mark the pole-skipping points. From left to right, the location of the radial cutoff surface $r_c$ approaches the horizon, making higher-order poles and zeros increasingly difficult to distinguish.
• Figure 2: The black hole geometry can be reconstructed from different subsets of boundary pole-skipping points (represented by black dots) by solving the linear system $\text{Det}(\vec{\mathcal{M}}^{(n)}(\textbf{g})) = 0$. The green and red quadrilaterals enclose portions of two such subsets.