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Entanglement entropy of evolving wormholes

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Ben Kain, Hong Y. Ling

General Relativity and Quantum Cosmology · Submitted 23 Oct 2026 · 20 references

Pith's verdict: Solid, honest first computation of entanglement entropy for evolving wormholes; the double-dip is new and interesting, but the dynamical-generation claim is conditional on the untested initial-vacuum choice.

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arXiv ID2610.32675 (version 1)
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CitationBen Kain, Hong Y. Ling. "Entanglement entropy of evolving wormholes." arXiv:2610.32675 (2026). https://pith.science/paper/TZFDZ5X5
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The paper

The paper claims

This paper claims that the time evolution of a wormhole geometry by itself generates spatial entanglement in a scalar field, in both expanding and contracting models, even after coordinate rescaling and changes in the ultraviolet cutoff…

The reading

This paper tries to establish that the mere time evolution of a wormhole spacetime generates spatial entanglement in a quantized field. The authors compute the entanglement entropy of a discretized scalar field on two evolving wormhole geometries, one expanding and one contracting, and find the entropy grows with time in both cases; they argue the growth survives after coordinate rescaling and the changing ultraviolet cutoff are taken into account. If correct, a field that begins in the static ground state develops growing spatial entanglement purely because its background is changing, with no self-interaction of the field. The paper also claims that far from the wormhole throat the entropy obeys the leading area law with a coefficient that numerical evidence suggests approaches a cutoff-independent value in the continuum limit, while near the throat a double-dip structure appears that the standard leading and logarithmic area-law terms cannot explain.

What carries the argument

The argument runs through a lattice reduction of the field theory to $N$ uncoupled harmonic oscillators with time-dependent frequencies and a common time-dependent mass. The time-dependent ground state is governed by the Ermakov-Milne-Pinney equation, a nonlinear second-order equation whose solution $\rho_j(\tilde{t})$ sets the Gaussian width of each oscillator's ground-state wavefunction; the boundary condition (3.42) chooses that state to be the instantaneous static ground state at $\tilde{t}=0$. Back in the original lattice variables the ground state is a complex Gaussian, and the reduced density matrix after tracing out $n$ oscillators has eigenvalues obtained from a generalized eigenvalue problem with $2(N-n)$ eigenvalues paired under $\lambda \leftrightarrow 1/\lambda$; keeping only the $\lambda<1$ partners gives the per-mode entropy $S_\ell$, and the total is $S = \sum_\ell (2\ell+1) S_\ell$. The identity that makes the area-law test clean is $A_{\mathrm{phys}}/\epsilon_{\mathrm{phys}}^2 = 4\pi\tilde{r}^2$: the conformal factor $B(t)$ cancels, so the physical area in units of the physical cutoff is a time-independent comoving quantity, and this same cancellation removes the coordinate-rescaling and cutoff-rescaling objections to reading the entropy growth as dynamical. The method requires the redshift function to vanish ($\Phi = 0$) and the coefficient of the kinetic term to be independent of $x$.

Extended reading notes

Core claim

The central claim is that wormhole evolution dynamically generates entanglement. In both the expanding model ($A = B = e^{\tilde{\omega}\tilde{t}}$) and the contracting model ($A = B = e^{-\tilde{\omega}\tilde{t}}$), each with constant shape function $h(r) = r_{\mathrm{th}}$ and zero redshift function, the entanglement entropy $S(\tilde{t}, n)$ of a massless, discretized, real scalar field increases with time. The increase is not a coordinate artifact: it is present when the entropy is plotted against the comoving radial coordinate $\tilde{x}$, which the conformal factor does not rescale. Nor is it an artifact of the changing physical cutoff $\epsilon_{\mathrm{phys}} = B(\tilde{t})\epsilon$: in the expanding model the physical cutoff grows, which by itself should suppress the measured entropy, yet the entropy still rises; in the contracting model, normalizing by the area-law combination $4\pi\tilde{r}^2 = A_{\mathrm{phys}}/\epsilon_{\mathrm{phys}}^2$, which accounts for the shrinking cutoff, again leaves growth. Far from the throat the entropy is well described by the leading term of the area law, $S \approx c_1\,4\pi\tilde{r}^2$, and as the UV cutoff is decreased the coefficient $c_1$ changes ever more slowly, which the authors read as numerical evidence for a cutoff-independent continuum value. Near the throat, later-time profiles develop a double-dip structure — a local maximum at the throat flanked by two minima — that neither the leading nor the logarithmic contribution to the area law can reproduce.

Load-bearing premise

The load-bearing premise is that the scalar field begins at time zero in the instantaneous static ground state of its Hamiltonian; because the vacuum of a time-dependent spacetime is not unique, the reported entropy growth is established only for this particular starting state.

Editorial extensions

If this is right

  • A scalar field that begins in the static ground state of its instantaneous Hamiltonian develops growing spatial entanglement purely because the wormhole geometry changes in time, with no self-interaction of the field.
  • Far from the throat the entropy follows the leading area law at all times, and the coefficient $c_1$ is a candidate geometric quantity: numerical evidence indicates it approaches a cutoff-independent value in the continuum limit.
  • In the contracting model the entropy briefly dips before growing, a transient the authors connect to behavior seen in gravitational collapse.
  • Near the throat, the leading plus logarithmic area-law terms are insufficient to describe the entropy at later times, and the origin of the double-dip structure is left open.
  • An observer plotting entropy against proper distance in the expanding model would wrongly infer decreasing entropy; the area-normalized quantity $S/4\pi\tilde{r}^2$ is the diagnostic that correctly reveals dynamical generation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension: because the computation needs only the metric form (3.1) and never uses the wormhole constraint on $A(t)$, the same dynamical-generation mechanism should appear in any spherically symmetric, time-dependent background with a conformally rescaled spatial sector, such as an FLRW interior or a collapsing shell.
  • The cleanest probe of robustness: repeat the computation with a different initial vacuum — for instance an adiabatic state instead of the static ground state fixed by (3.42) — and check whether the entropy growth and the near-throat double-dip survive; if they change, the reported phenomenology is state-dependent rather than geometric.
  • If the double-dip persists as the cutoff shrinks, it may encode the throat's embedding geometry through the minimum of the areal radius, and could become a probe of the wormhole's shape function that is invisible in the static case.
Original 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.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-26: 1 request, 4,898 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.80 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.14 · compares with prior results 0.21 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

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.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We compute the entanglement entropy of a discretized scalar field in the fixed spacetime background of an evolving wormhole. contribution 0.96 0.97 0.09 asserted 0.99
s1 Evolving wormholes are a class of time dependent and spherically symmetric wormhole geometries. background 1.00 0.06 0.03 asserted 1.00
s2 In both expanding and contracting models, we find dynamically generated entanglement. result 1.00 0.96 0.48 asserted 1.00
s3 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. result 1.00 0.88 0.13 asserted 0.99
s4 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. result 0.60 0.92 0.12 asserted 0.64
s5 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. result 1.00 0.97 0.17 asserted 1.00

What kind of paper the classifier reports this is: simulation study 0.96 · evidence from simulation or computation 1.00.

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sources read
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  • references and citing sentences
  • relations to other papers
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Entanglement entropy, S, as a function of the scaled comoving radial coordinate, ˜x, for the expanding wormhole model defined in (4.2) and parameter values ˜rth = 10, ˜ω = 1, and ˜µ = 0. From bottom to top, the curves are for t˜= 0, 1, 2, 3, and 4. The circles are the computed values which have been connected by straight lines. where the omitted terms are finite in the limit ϵphys → 0. The first term in (4.3) is the leading order contribution and the second term is the logarithmic correction. Both terms are written in terms of physical quantities. However, Aphys ϵ 2 phys = 4π[B(t˜
Figure 2. The logarithmic correction in the area law, for the area law as written in (4.6), is plotted as a function of scaled comoving radial coordinate ˜x. This plot applies specifically to the wormhole model under study, where the relationship between ˜x and ˜r is given by (4.7). From top to bottom, the curves are for ˜rth = 10, 20, 30, 40, and 50. which follows directly from (2.14). In figure 2, we show ln(4πr˜ 2 )/4πr˜ 2 as a function of ˜x for, from top to bottom, ˜rth = 10, 20, 30, 40, and 50. We find that the logarithmic correction has the best chance of being relevant near the center
Figure 3. (a) The results shown in figure 1 are plotted here as S/4πr˜ 2 . (b) The top curve is the same as the top curve in (a). From top to bottom, the curves are for ˜rth = 10, 15, 25, 35, and 50, while keeping t and ω (without tilde) unchanged. We may interpret these results as studying the same wormhole model while decreasing the value of the UV cutoff. (c) For the curves shown in (b) (as well as additional curves not shown), we display the rate of change of the area law coefficient c1, as defined in (4.6), in terms of the increasing values of ˜rth. defined in (4.6), as we now explain.…
Figure 4. (a) The area law coefficient c1 for the wormhole model with ˜rth = 50 and ˜ω = 0.2. These same parameters and t˜= 20 are used to compute the bottom curve in figure 3(b). (b) This plot zooms into the early time region in (a). structure and the determination of the asymptotic subleading correction for future work. An important question is whether or not the entanglement entropy obeys the area law. At early times, the answer is yes, as can be seen by the flat curves on the bottom of figure 3(a). At later times, the answer may depend on the UV cutoff. More specifically, the answer could…
Figure 5. (a) The same results shown in figure 1 and (b) the same results shown in figure 3(a), but now with both plots in terms of the scaled physical distance from the center of the wormhole, ˜ℓ. All curves in both plots extend indefinitely far out, but for an expanding spacetime it is numerically time consuming to compute the early times curves to the edge of the plot. A consequence of the curves crossing in figure 5(a) is that an observer making such a measurement would mistakenly think that entropy is decreasing with time, at least away from the wormhole’s center. The takeaway is that…
Figure 6. (a) Entanglement entropy, S, as a function of the scaled comoving radial coordinate, ˜x, for the contracting wormhole model defined in (4.11) and parameter values ˜rth = 10, ˜ω = 1, and ˜µ = 0. From bottom to top, the curves are for t˜ = 0, 1, 2, 3, and 4. (b) The solid blue curve displays the entanglement entropy at ˜x = 0 as a function of time for the same model shown in (a). We can see an initial dip before the entropy starts increasing. For comparison, we include the dashed black curve, which displays the entanglement entropy at ˜x = 0 for the expanding model shown in figure 1.…
Figure 7. These plots are analogous to figure 3, but for the contracting wormhole model. (a) The same results shown in figure 6(a) are plotted here as S/4πr˜ 2 . (b) The top curve is the same as the top curve in (a). From top to bottom, the curves are for ˜rth = 10, 15, 25, 35, and 50, while keeping t and ω (without tilde) unchanged. We may interpret these results as studying the same wormhole model while decreasing the value of the UV cutoff. (c) For the curves shown in (b) (as well as additional curves not shown), we display the rate of change of the area law coefficient c1, as defined in…
Figure 8. (a) The area law coefficient c1 for the wormhole model with ˜rth = 50 and ˜ω = 0.2. These same parameters and t˜= 20 are used to compute the bottom curve in figure 7(b). (b) This plot zooms into the early time region in (a). Note the initial dip in c1, which is consistent with the dip seen in the solid blue curve in figure 6(b). is in fact dynamically generated. In figures 9(a) and 9(b), we show S and S/4πr˜ 2 as functions of the physical distance ˜ℓ. Both of these plots show quantities that are measurable by an observer. An observer making such measurements would deduce that entropy
Figure 9. (a) The same results shown in figure 6(a) and (b) the same results shown in figure 7(a), but now with both plots in terms of the scaled physical distance from the center of the wormhole, ˜ℓ. For (b), all curves extend indefinitely far out, but for a contracting spacetime it is numerically time consuming to compute the late times curves to the edge of the plot. 5 Conclusion We computed the entanglement entropy of a discretized scalar field in the fixed spacetime background of an evolving wormhole. We focused on two related models, one describing an expanding wormhole and the other a…

Figures reproduced from arXiv:2610.32675 by the authors.

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