Imprints of phase transitions on Kasner singularities

Under the AdS/CFT correspondence, asymptotically AdS geometries with backreaction can be viewed as CFT states subject to a renormalization group (RG) flow from an ultraviolet (UV) description towards an infrared (IR) sector. For black holes however, the IR point is the horizon, so one way to interpret the interior is as an analytic continuation to a "trans-IR" imaginary-energy regime. In this paper, we demonstrate that this analytic continuation preserves some imprints of the UV physics, particularly near its "endpoint" at the classical singularity. We focus on holographic phase transitions of geometric objects in round black holes. We first assert the consistency of interpreting such black holes, including their interiors, as RG flows by constructing a monotonic $a$-function. We then explore how UV phase transitions of entanglement entropy and scalar two-point functions, each of which are encoded by bulk geometry under the holographic mapping, are connected to the structure of the near-singularity geometry, which is characterized by Kasner exponents. Using 2d holographic flows triggered by relevant scalar deformations as test beds, we find that the 3d bulk's near-singularity Kasner exponents can be viewed as functions of the UV physics precisely when the deformation is nonzero.