2026/07/20 by Digen Das, Prabwal Phukon
#hep-th
Timelike entanglement entropy (tEE) has recently emerged as a novel probe of temporal quantum correlations in gravitational systems. Existing studies are largely restricted to static backgrounds. In this work we extend the construction of tEE to an evaporating Schwarzschild--AdS black hole. The evaporation is modeled through an effective Stefan--Boltzmann description of Hawking radiation coupled to an external absorptive bath, yielding a time-dependent horizon radius (rh(t)) and surface gravity κ(t). We derive, from the near-horizon Rindler structure of the evolving horizon, an adiabatic generalization of the static Kruskal construction, obtaining the accumulated thermal phase Φ(t)=∫0tκ(t') dt' as the natural replacement for the static phase κt. The validity of this construction is governed by an explicit adiabatic parameter \mathcal A(t), which we verify numerically remains small (\lesssim0.012) throughout the regime of interest. It vanishes exactly where the horizon crosses the critical radius rh=l/√3 identified independently from the static thermodynamics. Using Φ(t), we construct a dynamical timelike entanglement entropy that continuously tracks the evaporation process and derive the corresponding dynamical Page-like times. Unlike the uniformly spaced Page-like times of the static geometry, evaporation induces non-uniform temporal spacing, together with a progressive phase delay and amplitude modulation of the oscillatory tEE. Because the dynamical entropy depends on the full accumulated history of κ(t) rather than its instantaneous value alone, it retains a memory of the entire evaporation process. These results establish a first-principles dynamical framework for investigating temporal quantum correlations in evaporating black holes.