Replica Trick in Time-Dependent Geometries

We present a unified Lorentzian replica-path-integral framework for computing entanglement entropy in fully time-dependent quantum field theories and gravitational systems. Building on this framework, we apply the real-time replica trick to cosmological spacetimes without spatial boundaries and relate it to existing proposals that identify effective dual QFT descriptions on screen-like surfaces inside the spacetime. We also extend the Lorentzian replica construction to intrinsic gravitational subsystems, focusing on Hawking radiation and including quantum corrections in purely gravitational settings lacking time-reflection symmetry. In both contexts, island-type replica saddles arise naturally. A brief review of the QFT and holographic cases is included to establish the common structure underlying these applications.