Dynamical Timelike Entanglement Entropy in an Evaporating Schwarzschild--AdS Black Hole

Abstract

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 ($r_h(t)$) and surface gravity $\kappa(t)$. 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 $\Phi(t)=\int_0^t\kappa(t')\,dt'$ as the natural replacement for the static phase $\kappa t$. 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 $r_h=l/\sqrt3$ identified independently from the static thermodynamics. Using $\Phi(t)$, 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 $\kappa(t)$ 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.