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arXiv:2610.32675

Entanglement entropy of evolving wormholes

Ben Kain, Hong Y. Ling

Published Oct 23, 2026• 0 citations
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Abstract

We compute the entanglement entropy of a discretized scalar field in the fixed spacetime background of an evolving wormhole. Evolving wormholes are a class of time dependent and spherically symmetric wormhole geometries. In both expanding and contracting models, we find dynamically generated entanglement. Asymptotically far from the wormhole throat, the entanglement entropy is well described by the leading term in the area law, although its coefficient can depend sensitively on the value of the UV cutoff. As the UV cutoff is decreased, the coefficient becomes progressively less sensitive to the cutoff, providing numerical evidence that it approaches a cutoff-independent value in the continuum limit. Near the throat, we find a double-dip structure in the entanglement entropy which cannot be described by the leading and logarithmic contributions to the area law.

DOI
10.48550/arxiv.2610.32675

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We compute the entanglement entropy of a discretized scalar field in the fixed spacetime background of an evolving wormhole. Evolving wormholes are a class of time dependent and spherically symmetric wormhole geometries. In both expanding and contracting models, we find dynamically generated entanglement. Asymptotically far from the wormhole throat, the entanglement entropy is well described by the leading term in the area law, although its coefficient can depend sensitively on the value of the UV cutoff. As the UV cutoff is decreased, the coefficient becomes progressively less sensitive to the cutoff, providing numerical evidence that it approaches a cutoff-independent value in the continuum limit. Near the throat, we find a double-dip structure in the entanglement entropy which cannot be described by the leading and logarithmic contributions to the area law.
We investigate the dynamics of the ground state entanglement entropy for a discretized scalar field propagating within the Oppenheimer-Snyder collapse metric. Starting from a well-controlled initial configuration, we follow the system as it evolves toward the formation of a horizon and, eventually, a singularity. Our approach employs an Ermakov-like equation to determine the time-dependent ground state of the field and calculates the resulting entanglement entropy by tracing out the degrees of freedom inside a spherical region within the matter sphere. We find that the entanglement entropy exhibits nontrivial scaling and time dependence during collapse. Close to the horizon, the entropy can deviate from the simple area law, reflecting the rapid changes in geometry and field configuration. Although the model is idealized, these results provide insights into the generation and scaling of entanglement in the presence of realistic, dynamically evolving gravitational fields.
Subdominant contributions to the entanglement entropy of quantum fields include logarithmic corrections to the area law characterized by universal coefficients that are independent of the ultraviolet regulator and capture detailed information on the geometry around the entangling surface. We determine two universal coefficients of the entanglement entropy for a massive scalar field in a static closed universe $\mathbb{R}\ifmmode\times\else\texttimes\fi{}{\mathbb{S}}^{3}$ perturbatively and verify the results numerically. The first coefficient describes a well-known generic correction to the area law independent of the geometry of the entangling surface and background. The second coefficient describes a curvature-dependent universal term with a nontrivial dependence on the intrinsic and extrinsic geometries of the entangling surface and curvature of the background. The numerical calculations confirm the analytical results to a high accuracy. The first and second universal coefficients are determined numerically with a relative error with respect to the analytical values of the orders ${10}^{\ensuremath{-}4}$ and ${10}^{\ensuremath{-}2}$, respectively.
A scalar field in the ground state, when partially hidden from observation by a spherical boundary, acquires entanglement entropy SSS proportional to the area of the surface. This area law is well established in flat space, where it follows almost directly from dimensional arguments. We study its validity in an Einstein universe, whose curvature provides an additional physical parameter on which the entropy could, in principle, depend. The surprisingly simple result is that the entanglement entropy still scales linearly with the area. This is supported by other observations to the effect that the entanglement entropy arises mostly from degrees of freedom near the boundary, making it insensitive to the large-scale geometry of the background space.
We use holographic duality to study the entanglement entropy (EE) of Conformal Field Theories (CFTs) in various spacetime dimensions d, in the presence of various deformations: a relevant Lorentz scalar operator with constant source, a temperature T , a chemical potential μ, a marginal Lorentz scalar operator with source linear in a spatial coordinate, and a circle-compactified spatial direction. We consider EE between a strip or sphere sub-region and the rest of the system, and define the “entanglement density” (ED) as the change in EE due to the deformation, divided by the sub-region’s volume. Using the deformed CFTs above, we show how the ED’s dependence on the strip width or sphere radius, L, is useful for characterizing states of matter. For example, the ED’s small-L behavior is determined either by the dimension of the perturbing operator or by the first law of EE. For Lorentz-invariant renormalization group (RG) flows between CFTs, the “area theorem” states that the coefficient of the EE’s area law term must be larger in the UV than in the IR. In these cases the ED must therefore approach zero from below as L→∞. However, when Lorentz symmetry is broken and the IR fixed point has different scaling from the UV, we find that the ED often approaches the thermal entropy density from above, indicating area theorem violation.
Abstract Entanglement entropy is now widely accepted as having deep connections with quantum gravity. It is therefore desirable to understand it in the context of causal sets, especially since they provide in a natural manner the UV cutoff needed to render entanglement entropy finite. Formulating a notion of entanglement entropy in a causal set is not straightforward because the type of canonical hypersurface-data on which its definition typically relies is not available. Instead, we appeal to the more global expression given in Sorkin (2012 (arXiv:1205.2953)) which, for a Gaussian scalar field, expresses the entropy of a spacetime region in terms of the field’s correlation function within that region (its ‘Wightman function’ W ( x , x ′ ) ). Carrying this formula over to the causal set, one obtains an entropy which is both finite and of a Lorentz invariant nature. We evaluate this global entropy-expression numerically for certain regions (primarily order-intervals or ‘causal diamonds’) within causal sets of 1 + 1 dimensions. For the causal-set counterpart of the entanglement entropy, we obtain, in the first instance, a result that follows a (spacetime) volume law instead of the expected (spatial) area law. We find, however, that one obtains an area law if one truncates the commutator function (‘Pauli–Jordan operator’) and the Wightman function by projecting out the eigenmodes of the Pauli–Jordan operator whose eigenvalues are too close to zero according to a geometrical criterion which we describe more fully below. In connection with these results and the questions they raise, we also study the ‘entropy of coarse-graining’ generated by thinning out the causal set, and we compare it with what one obtains by similarly thinning out a chain of harmonic oscillators, finding the same, ‘universal’ behaviour in both cases.