Replica Trick in Time-Dependent Geometries

Abstract: We extend the Lorentzian replica framework for computing entanglement entropy to fully time-dependent gravitational settings, with emphasis on cosmological spacetimes without boundaries and gravitational theories lacking a dual quantum field theory description. Our approach constructs the replica path integral directly in real time, avoiding reliance on Euclidean continuation or time-reflection symmetry, and identifies the geometric conditions under which Lorentzian replica saddles appear in dynamical backgrounds. We analyze replica wormholes in both contexts, recovering a generalized version of the island rule. For completeness, we provide a concise review of the existing Lorentzian replica construction for quantum field theories and their holographic duals, which forms the foundation for our generalization.

• The paper introduces a Lorentzian replica trick that computes entanglement entropy in dynamic gravitational and QFT setups without relying on Euclidean continuation.
• It extends the holographic framework by incorporating replica wormholes and the island formula, thereby resolving entropy puzzles in black hole and cosmological contexts.
• Numerical results confirm that the approach recovers known semiclassical behaviors and smoothly transitions between no-island and island-dominated regimes.

Replica Trick in Time-Dependent Geometries: Formalism and Implications

Introduction and Motivation

The study rigorously develops a Lorentzian signature formulation of the replica trick for computing entanglement entropy in generic gravitational and quantum field theoretic setups, extending beyond static or Euclidean backgrounds. This approach is particularly relevant for cosmological spacetimes without spatial boundaries and for gravitational theories lacking a dual boundary QFT. Rather than relying on Euclidean continuation or time-reflection symmetry, the work constructs the replica path integral explicitly in real time, setting a unified framework for analyzing entanglement in time-dependent, dynamical geometries.

Lorentzian Replica Trick in Quantum Field Theory

The paper first reviews the real-time (Lorentzian) formulation of the replica trick in non-gravitational QFTs, closely following the Schwinger-Keldysh contour methodology [Dong:2016hjy]. Entanglement entropy of a spatial subregion $A$ is computed by evaluating the partition function on an $n$-sheeted branched manifold, created by gluing $n$ copies along $A$. The dominant contributions, particularly near $n=1$, are assumed to stem from $\mathbb{Z}_n$-symmetric replica saddles. This path-integral construction remains valid for general time-dependent states and goes beyond previous analyses that required time-reflection symmetry.

Bulk Replica Path Integrals in Holographic Theories

The framework is then extended to QFTs with gravitational duals, where the replica path integral is implemented as a sum over bulk geometries $\mathcal{M}_n$ with boundaries matching the $n$-sheeted QFT geometry. In the semiclassical regime, the dominant saddle-point configurations exhibit $\mathbb{Z}_n$ symmetry, characterized by regular conical singularities at entangling surface endpoints. The gravitational action yields a leading contribution proportional to the surface area of the fixed points, while quantum corrections arise from matter fields integrated out. This reproduces the generalized entropy prescription

$$S_A = \frac{\mathrm{Area}(\tilde{A})}{4 G_N} + S_{\text{matter}}({\Sigma}_A)$$

with $\tilde{A}$ extremized among all codimension-two candidate surfaces, recovering the HRT and QES rules [Faulkner:2013ana, Engelhardt:2014gca, Lewkowycz:2013nqa].

A significant aspect is the inclusion of replica wormhole geometries, which connect different replica sheets through islands disconnected from $A$ in the bulk. These geometries resolve entropy puzzles like the Page curve in evaporating black holes and implement the island rule:

$$S_A = \min \left\{ \frac{\mathrm{Area}(\tilde{A}\cup\partial I)}{4G_N} + S_{\text{matter}}(\Sigma_A \cup I) \right\}$$

where $I$ denotes the island surface [Penington:2019kki, Almheiri:2019qdq].

Extension to Cosmological Spacetimes

A primary innovation is the extension of the Lorentzian replica trick to cosmologies lacking spatial boundaries, notably de Sitter and bubble geometries. The construction assigns the location of the would-be "dual QFT" to a codimension-one non-spacelike bulk surface, termed a "screen trajectory," and implements the replica procedure within this interior surface [Franken:2023pni, Irakleous:2025trr]. The entanglement entropy is then computed as the generalized entropy over extremal surfaces homologous to the chosen subregion of the screen, with the prescription justified by the bulk path integral. This approach systematizes screen-based proposals such as the bilayer construction and covariant extensions.

Gravitational Systems Without Dual QFTs: Hawking Radiation

The method is further applied to compute Hawking radiation entropy in gravitational settings with asymptotic regions where gravity is weak and no boundary dual exists. The replica path integral in these circumstances is constructed via gluing gravitational "bra-ket" geometries along the radiation region $R$; the dominant contributions again involve conical singularities and replica wormholes, leading to the island formula for black hole information recovery.

A notable technical advance is demonstrating how the entanglement entropy can be derived using only integer numbers of replicas (avoiding analytic continuation in $n$), illustrated by explicit calculations in maximally entangled toy models. Corrections arising from non-replica-symmetric geometries are also included, showing that they yield subleading terms and smooth out phase transitions between no-island and island-dominated regimes.

Boundary Contributions and Non-Subregion Entropy

The analysis incorporates the role of variational constraints and boundary contributions in the extremization procedure defining quantum extremal surfaces: surfaces can approach the boundary, and in cosmological contexts, causal diamonds serve as relevant boundaries for the extremization domain. These boundary saddles are intrinsic to the Lorentzian replica formalism and not merely technical nuances.

The work also discusses entanglement entropy for subsystems not corresponding to spatial regions, such as decoupled field species or algebraic substructures [Casini:2017roe]. While the replica construction formally applies, the lack of a geometric anchor challenges existing holographic prescriptions and suggests the need for new saddle configurations (e.g., bra-ket wormholes [Chen:2021bra]).

Strong Numerical Results and Claims

• The generalized entropy prescription is explicitly derived in Lorentzian signature, not relying on Euclidean continuation.
• The island rule and Page curve behavior are reproduced in time-dependent cases, with corrections quantified for finite replica numbers.
• The formalism is shown to reproduce known semiclassical results for de Sitter and cosmological spacetimes beyond AdS, supporting screen-based holographic proposals.
• Subleading corrections due to non-replica-symmetric geometries smooth the phase transition between "no-island" and "island" regimes.

Implications and Future Directions

The Lorentzian replica trick provides a principled, nonperturbative framework for defining quantum entropy in gravitational systems, especially for time-dependent backgrounds where Euclidean methods are unreliable or ill-defined. Practically, this approach enables robust computation of entanglement in dynamical scenarios such as black hole evaporation, cosmological evolution, and in spacetimes without conventional boundaries.

Theoretically, the framework clarifies the connection between semiclassical gravity, information recovery (e.g., Page curve), entropy bounds, and nonlocal features of effective quantum gravity (e.g., non-factorization of replica partition functions) [Hernandez-Cuenca:2024pey]. It highlights the necessity of including boundary extremal saddles and constraints as intrinsic components of the entanglement prescription. The ability to define entanglement entropy for non-geometric subsystems remains a critical outstanding question, corresponding to a missing link in holographic and algebraic approaches.

Anticipated developments include a systematic classification of Lorentzian replica saddles in generic backgrounds, precise characterizations of boundary and constraint contributions, and direct extensions to gauge-invariant and algebraic subsystems.

Conclusion

By generalizing the replica trick in Lorentzian signature, this work offers a unified approach for entanglement calculations in fully time-dependent and boundary-less gravitational settings, incorporating quantum corrections, islands, and boundary constraints. The framework is essential for computational and theoretical investigations of quantum gravity, black hole information, and holographic entanglement in dynamical or cosmological domains (2601.08756).