Probing bulk geometry via pole skipping: from static to rotating spacetimes

A bstract We investigate an analytical framework for reconstructing bulk geometries from pole-skipping data. Previously, this method enabled the recursive recovery of near-horizon metric derivatives in static, planar-symmetric black holes. Building on this framework, we systematically extend it to more intricate geometries, specifically static topological black holes and rotating black holes. For three-dimensional rotating black holes, we demonstrate that the metric can be fully reconstructed from boundary pole-skipping data. For four-dimensional rotating spacetimes admitting a separable coordinate system (such as the Kerr family), standard near-horizon pole-skipping successfully reconstructs the purely radial metric functions. To recover the remaining angular metric functions, we introduce a mathematical counterpart termed “angular pole-skipping,” defined via a near-axis analysis. Although its precise holographic dictionary remains an open question, this bulk-side formalism completes the geometric reconstruction algorithm. Furthermore, we demonstrate that the vacuum Einstein equations can be recast as a set of algebraic equations governing the pole-skipping data, and that the Null Energy Condition imposes algebraic inequalities on this boundary data. Finally, we establish general polynomial constraints dictated by the overdetermined nature of the metric reconstruction, highlighting the highly redundant encoding of bulk geometry in boundary data.