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REVIEW 3 major objections 5 minor 25 references

Gravitational lensing by Lema\^{\i}tre-Tolman-Bondi wormholes in a Friedmann universe

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper shows that the shadow of a dynamic wormhole in an expanding universe first shrinks and then grows, and that its angular size obeys a specific power law in the wormhole's size.

desk verdict Careful first lensing study of dynamic LTB wormholes, with a time-dependent shadow and a scaling law, but the plotted observation-time range in Fig. 5 appears to be before the wormhole is causally visible, so the headline results need re-scoping. read the letter →

arxiv 2509.09797 v1 pith:XYO6R2JN submitted 2025-09-11 gr-qc

Kirill A. Bronnikov , Valeria A. Ishkaeva , Sergey V. Sushkov This is my paper · ORCID
classification gr-qc
keywords gravitationallensingwormholeshadowLemaitre-Tolman-BondisolutionFriedmannuniversenullgeodesicsdynamiccosmicexpansiondustcosmology
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies gravitational lensing by a wormhole built from the Lemaitre-Tolman-Bondi dust solution and embedded in a closed dust-filled Friedmann universe. It claims that the shadow a distant observer sees has an angular size that is non-monotonic in observation time: at early times the shadow shrinks as photons with smaller angular momentum gradually arrive, and at later times it grows because cosmic expansion dominates. It also derives a power law d_sh ~ chi_*^{1+2k/3} connecting the shadow angular size to the wormhole boundary coordinate and the density-profile exponent. If correct, evolving wormholes would have a distinctive time-dependent lensing signature that static wormhole models do not possess. The authors present this as a first-stage result under a small-eta approximation with tau_0(R)=0, so the detailed conclusions are restricted to wormholes born with the universe and observed in their early evolution.

What carries the argument

The central object is an LTB wormhole matched to a closed dust-filled Friedmann universe through the choices F(R)=2b(1+$R^{2}$)^k and h(R)=1/(1+$R^{2}$), with matching conditions at a boundary chi_* linking the wormhole coordinate R_* to the cosmological coordinate chi_*. The argument runs through the small-eta approximation r ~ (3/2)^{2/3} $F^{{1/3}}$ $tau^{{2/3}}$, which reduces photon motion to turning points at F(R_t)=(500/243)(L/C)^3. The shadow radius is then obtained from the standard coordinate formula for image angles, with L_sh replaced by the minimum angular momentum L_min ~ $\sqrt$(K) chi_*^{1+2k/3}, yielding the power-law angular size.

What would settle it

Integrate null geodesics numerically in the exact LTB metric for the same F(R)=2b(1+$R^{2}$)^k and h(R)=1/(1+$R^{2}$) without expanding in eta, place an observer at chi_obs=1, and measure the shadow angular diameter at several observation times. If the curve is monotone rather than shrink-then-grow, or if the d_sh versus chi_* data at fixed observation time deviate from the chi_*^{1+2k/3} slope, the paper's central claim is refuted.

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Extended reading notes

Core claim

In the small-eta regime, the boundary of the wormhole shadow on the observer's sky is set by the smallest photon angular momentum L_sh that has reached the observer by the observation time tau_obs. Early in the observation, this boundary moves inward because lower-L photons keep arriving, so the shadow shrinks; later, the expansion of the Friedmann universe stretches the image and the shadow grows. Analytically, the shadow diameter obeys d_sh ~ chi_*^{1+2k/3}, where chi_* is the coordinate boundary of the wormhole region and k is the exponent in the mass-distribution function F(R)=2b(1+$R^{2}$)^k, and the paper reports that numerical ray tracing confirms this power law. The authors emphasize that the analysis covers only the beginning of the wormhole's evolution, when it appears together with the universe, and that late-forming wormholes require dropping the small-eta and tau_0=0 assumptions.

Load-bearing premise

The whole shadow calculation uses the small-eta approximation with tau_0(R)=0, so it describes only the very first stage of a wormhole born together with the universe; if that early-birth, small-eta regime is abandoned, the non-monotonic shadow and the power law need not survive.

Editorial extensions

If this is right

  • An observer watching a young LTB wormhole should see its shadow shrink as lower-angular-momentum photons keep arriving, then grow as cosmic expansion dominates.
  • Because the shadow size scales as chi_*^{1+2k/3}, a measured angular size would constrain the combination of wormhole-region size and dust-density exponent.
  • The shadow boundary is time-dependent rather than set by closed photon orbits, so dynamic wormholes have no single static shadow radius and any observational report must specify the observation time.
  • The results apply to wormholes born simultaneously with the universe; later-forming wormholes or late-time observations require an extension beyond the small-eta, tau_0=0 regime, as the paper itself notes.
  • The dust-density deficit around wormhole regions may connect such objects to observed cosmic voids, suggesting a possible observational target beyond direct shadow imaging.

Reading between the lines

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

  • Beyond the paper, a full numerical integration of null geodesics in the exact LTB metric without the small-eta cutoff could test whether the two-phase shrink-then-grow shadow persists for photons that pass near the throat, where eta grows large.
  • As an extension, if the non-monotonic signature survives for wormholes formed at later cosmic times, it would offer a way to distinguish LTB wormholes from static wormholes and ordinary lenses, whose shadows do not shrink then grow.
  • One testable extension is to scan other density profiles F(R): if the shadow size continues to follow a power law in chi_* with an exponent set by the profile, shadow scaling would become a probe of the radial mass distribution rather than a special property of the chosen model.
  • A further editorial inference is that the late-time growth phase might be observable as a population of lensing objects whose shadows enlarge at a rate tracking cosmic expansion, a signature that could be sought in time-domain surveys.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies gravitational lensing by time-dependent Lemaître-Tolman-Bondi (LTB) dust wormholes that are matched to a closed Friedmann dust-filled universe. The authors derive null geodesic equations in the LTB metric within a small-η approximation, match the wormhole region to the Friedmann cosmology via the conditions of Ref. [14], and define a time-dependent shadow as the set of directions from which photons have not yet reached the observer by a given observation time. The central results are a claimed non-monotonic dependence of the shadow angular size on observation time, caused at early times by the gradual arrival of smaller-angular-momentum photons and at later times by the cosmological expansion, and a scaling law d_sh ∼ χ_*^{1+2k/3} for the shadow size as a function of the wormhole boundary coordinate χ_*. The paper is presented as an initial-stage analysis, with the limitations of the small-η approximation and the choice τ_0(R)=0 acknowledged in the conclusion.

Significance. If the results were established, this would be a useful contribution to the observational phenomenology of dynamic wormholes, providing one of the first treatments of a time-dependent wormhole shadow in a cosmological setting. The paper contains a careful derivation of the geodesic equations in the small-η regime, explicit matching conditions, and numerical plots of photon paths and shadow sizes. However, the two headline claims are derived and plotted in regimes where the small-η approximation is not valid, and the paper's own conclusion admits that the analysis covers only the initial stage. Because the non-monotonic shadow and the power-law scaling are consequences of extrapolating the approximation into regimes where the throat has either collapsed or has η significantly larger than 0.1, the significance is currently conditional on a substantial revision that addresses the validity domain.

major comments (3)
  1. [§4.3, Figs. 4–5] The small-η approximation is stated in Section 2 (after Eq. (7)) to be valid for η≤0.1, which at the throat R=0 corresponds to τ/b≈η^3/6≤1.7×10^-4, while the exact LTB throat, from Eq. (5) with h=1 and F=2b, recollapses at τ=2πb≈6.28b. However, Figs. 4 and 5 display observation times τ_obs/b≈27, 38, and 40, which are both orders of magnitude beyond the η≤0.1 validity bound and later than the throat recollapse. At those epochs the configuration is no longer a wormhole, so the claimed non-monotonic shadow is not a property of the physical model under study. The authors need to restrict the plots to times before the throat recollapses and to epochs where η≤0.1 along the entire photon path, or to solve the geodesic equations in the exact LTB metric.
  2. [§4.3, Eqs. (36)–(38)] The shadow boundary at late observation times is controlled by photons whose angular momentum tends to L_min, which have a turning point at the throat R_t=0 according to Eq. (23). At R=0 the parameter η is maximal: for L=L_min the integrand in Eq. (20) diverges at the turning point, so the affine parameter and the coordinate time τ grow without bound, and the small-η assumption fails precisely at the location where the scaling d_sh∼χ_*^{1+2k/3} is evaluated. Section 5 acknowledges that η increases near the throat, but this caveat is not applied to Eq. (38) or to Fig. 5. The power law is therefore not established; a full treatment of the geodesics or a rigorous error bound that covers the photon ensemble defining the shadow boundary is required.
  3. [§4.3, Fig. 5 vs. Table 1] Figure 5 reports τ_obs/b ranging from about 25 to 40 for χ_*≈0.015, but using the matching relation b=a0 sin^{3+2k}χ_* and the observer parameters (χ_obs=1, η_obs=1.1) that are used for Table 1 gives τ_obs/b≈1.4×10^5, not the plotted range. The paper does not state the η_obs values corresponding to Fig. 5; if the plotted range actually corresponds to smaller η_obs, then the claimed non-monotonic dependence is not being presented for the same observer configuration as the rest of the paper. This inconsistency must be clarified and reconciled.
minor comments (5)
  1. [§3, heading] The heading contains a typo: 'pohoton motion' should be 'photon motion'.
  2. [§3.1, Eq. (12)] The symbol L is used both for the Lagrangian and for the conserved angular momentum; please use a different symbol for one of them to avoid confusion.
  3. [§4.3, Eq. (34)] The expression for α_sh should be derived step by step; as written the factors (1−cos η_obs)^2 in the numerator and denominator are not transparent and the overall dimensions are unclear.
  4. [§4.3, Fig. 6] The 'black line' in Fig. 6 should specify whether it is the analytic expression (38) with a fixed normalization or a numerical fit, and the adopted proportionality constant should be given.
  5. [§1, Introduction] The description of Ref. [18] is somewhat dismissive; a neutral formulation would be more appropriate for a journal report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the shadow scaling is an algebraic consequence of the chosen LTB model functions, not a fitted or imported result.

full rationale

The analysis is not circular. The central calculation begins from the LTB metric (1), the small-η expansions (7), and the explicit model functions F=2b(1+R^2)^k, h=1/(1+R^2) chosen in (21). The turning-point condition (23), the minimum angular momentum (36), the junction relations (32), and the shadow-radius formula (34) are all derived in this paper from the geodesic equations; the final power law (38) is an algebraic consequence of these equations, not an input. The parameter k is a free model parameter and is not fitted to the shadow size; no quantity that is 'predicted' is also used to determine a parameter. The citations to the authors' earlier papers [13,14] supply the background wormhole-existence conditions and the junction conditions; they are not the source of the shadow prediction, which is computed here. The paper itself in Section 5 limits the validity to small η and τ0(R)=0 and says the results are incomplete; even if the plotted observation times in Figs. 4-5 exceed the strict domain of the expansion, that is a validity/correctness concern about extrapolating an approximation, not a circular reduction of the result to its inputs.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The model adds two genuinely new free choices (density profile exponent k and wormhole boundary chi_*) plus two observer settings; the rest is standard GR and prior matching conditions. No new particles or forces are introduced.

free parameters (4)
  • k = 0.1 in examples
    Exponent in F(R)=2b(1+R^2)^k, controls the wormhole density profile and sets the shadow power-law exponent 1+2k/3. Chosen by hand, not fit to data.
  • chi_* (wormhole boundary coordinate) = e.g., 0.015 (R_*=65.5) in Fig. 5
    Determines the wormhole size via b=a0(sin chi_*)^{3+2k}. Varies over orders of magnitude in Table 1; effectively a free parameter of the model.
  • observer conformal time eta_obs = 1.1
    The time at which the shadow is evaluated; chosen to be early in the universe's history. The shadow size and its time dependence are sensitive to this choice.
  • observer position chi_obs = 1
    The comoving angular position of the observer in the closed Friedmann universe; chosen as a representative location.
assumptions (6)
  • standard math LTB dust solution with elliptic branch and Lambda=0 is a valid GR model.
    Used throughout; standard solution of Einstein equations.
  • domain assumption Wormhole throat conditions hold, as in Eq. (6).
    Taken from prior work by the authors [13,14]; ensures a traversable wormhole throat.
  • domain assumption Matching conditions to the closed Friedmann universe, Eqs. (31) and (32).
    Borrowed from [14]; defines the wormhole as a region inside the Friedmann universe with continuous F and h.
  • ad hoc to paper Small-eta approximation with relative errors around 10^{-3}.
    Restricts the analysis to the early stage of wormhole evolution; used in Eq. (7) and throughout.
  • ad hoc to paper tau0(R)=0 for all R.
    Assumes all dust spheres start at the same time, i.e., the wormhole is born with the universe. Flagged as a limitation in Section 5.
  • domain assumption No light sources inside the wormhole and no photons from the other universe.
    Assumed in Section 4.3 to define the shadow; needed because the small-eta approximation breaks near the throat for the other side.

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Cite this review

Pith. "Pith review of Gravitational lensing by Lema\^{\i}tre-Tolman-Bondi wormholes in a Friedmann universe." pith.science (2026). https://pith.science/paper/XYO6R2JN

@misc{pith2026250909797,
  author       = {Pith},
  title        = {Pith review of: Gravitational lensing by Lema\^\itre-Tolman-Bondi wormholes in a Friedmann universe},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XYO6R2JN}},
  note         = {Machine review of arXiv:2509.09797}
}
abstract

The Lema\^{\i}tre-Tolman-Bondi (LTB) solution to the Einstein equations describes the dynamics of a self-gravitating spherically symmetric dust cloud with an arbitrary density profile and any distribution of initial velocities, encoded in three arbitrary functions $f(R)$, $F(R)$, and $\tau_0(R)$, where $R$ is a radial coordinate in the comoving reference frame. A particular choice of these functions corresponds to a wormhole geometry with a throat defined as a sphere of minimum radius at a fixed time instant. In this paper we explore LTB wormholes and discuss their possible observable appearance studying in detail the effects of gravitational lensing by such objects. For this aim, we study photon motion in wormhole space-time inscribed in a closed Friedmann dustfilled universe and find the wormhole shadow as it could be seen by a distant observer. Since the LTB wormhole is a dynamic object, we analyze the dependence of its shadow size on the observation time and on the initial size of the wormhole region. We reveal that the angular size of the shadow exhibits a non-monotonic dependence on the observation time. At early times, the shadow size decreases as photons with smaller angular momentum gradually reach the observer. At later times, the expansion of the Friedmann Universe becomes a dominant factor that leads to an increase in the shadow size.

Figures

Figures reproduced from arXiv: 2509.09797 by the authors.

Figure 1
Figure 1. Paths R(τ ) of photons with different L. Dashed lines mark the photon motion at R < 0. The wormhole parameters are b = 1, k = 0.1. The photons are launched at R0 = −10, τ0 ≈ 0.005. All photons have the same initial energy E0 ≈ 0.3. 4 A Friedmann universe with a dynamic wormhole In what follows we are going to study photon paths in a Friedmann universe containing a dynamic wormhole, therefore, let us first consider n… view at source ↗
Figure 2
Figure 2. Paths R(ϕ) of photons with different L. Dashed lines mark the photon motion at R < 0. The wormhole parameters are b = 1, k = 0.1. The photons are launched at R0 = −10, τ0 ≈ 0.005. All photons have the same initial energy E0 ≈ 0.3. where a(η) is the cosmological scale factor, and the Friedmann metric can be written as ds2 = a 2 (η) [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. shows a schematic representation of photon motion from a light source to the observer through the wormhole. All photons are emitted simultaneously. Photons 3 and 4 (red trajectories) have higher angular momentum L and reach the observer by the observation time. Photons 1 and 2 (brown trajectories) have lower angular momenta L and do not reach the observer by that time. Thus, at this moment, the shadow boundary is fo… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Photon paths r(ϕ) in a Friedmann universe with a dynamic wormhole, with R∗ = 5, at different observation times τobs . We will also obtain the dependence of the shadow size dsh on the boundary value χ∗ . To do that, it is sufficient to find the minimum value |Lmin| of p…
Figure 5
Figure 5. Figure 5: The wormhole shadow angular size dsh for a wormhole with χ∗ ≈ 0.015(R∗ ≈ 65.5), k = 0.1 vs. observation time τobs/b. observer is at the point χobs = 1. From the expression (23) we get |Lmin| = 3 C 5  9 b 2 1/3 . (36) From the conditions (32) we obtain that the parame…
Figure 6
Figure 6. Figure 6: shows the dependence of the angular size of the wormhole shadow dsh on the boundary value χ∗ at the observation time ηobs = 1.1. The observer is located at the point χobs = 1. The orange dots represent the values of χ∗ and dsh taken from [PITH_FULL_IMAGE:figures/full_…

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Works this paper leans on

25 extracted references · 12 canonical work pages

  1. [14]

    Possible wormholes in a Friedmann universe

    K.A. Bronnikov, P.E. Kashargin, S.V. Sushkov, Possible wormholes in a Friedmann universe, Universe 9, 465 (2023). doi:10.3390/universe9110465 [arXiv:2309.03166 [gr-qc]]

    work page Pith review arXiv 2023

  2. [1]

    Morris, K.S

    M.S. Morris, K.S. Thorne, Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity. Am. J. Phys.56(5), 395–412 (1988).https://doi.org/10.1119/1.15620

    doi:10.1119/1.15620 1988

  3. [2]

    Hochberg, M

    D. Hochberg, M. Visser, Geometric structure of the generic static traversable wormhole throat. Phys. Rev. D56, 4745 (1997)

    work page 1997

  4. [3]

    Will, The confrontation between General Relativity and experiment, Living Rev

    Clifford M. Will, The confrontation between General Relativity and experiment, Living Rev. Relativity 17, 4 (2014); arXiv: 1403.7377

    arXiv 2014

  5. [4]

    Jose Luis Bl´ azquez-Salcedo, Christian Knoll, Eugen Radu, Traversable wormholes in Einstein-Dirac- Maxwell theory, Phys. Rev. Lett.126, 101102 (2021). 16

    work page 2021

  6. [5]

    R. A. Konoplya, A. Zhidenko, Traversable wormholes in General Relativity without exotic matter, Phys. Rev. Lett.128, 091104 (2022); arXiv: 2106.05034

    arXiv 2022

  7. [6]

    Traversable wormholes in Einstein-Dirac-Maxwell theory

    Kirill Bronnikov, Sergey Bolokhov, Serguey Krasnikov, Milena Skvortsova, A note on “Traversable wormholes in Einstein-Dirac-Maxwell theory”, Grav. Cosmol.27, 401–402 (2021); arXiv: 2104.10933

    arXiv 2021

  8. [7]

    Bronnikov, V.G

    K.A. Bronnikov, V.G. Krechet,. Potentially observable cylindrical wormholes without exotic matter in GR. Phys. Rev. D99, 084051 (2019); arXiv: 1807.03641

    arXiv 2019

Show all 25 references
  1. [8]

    Bronnikov, V.G

    K.A. Bronnikov, V.G. Krechet, V.B. Oshurko, Rotating Melvin-like universes and wormholes in general relativity, Symmetry2020(12), 1306 (2020); arXiv: 2007.01145

    2020 arXiv

  2. [9]

    David Hochberg, Matt Visser, Dynamic wormholes, anti-trapped surfaces, and energy conditions, Phys. Rev. D58, 044021 (1998); arXiv: gr-qc/9802046

    1998 arXiv

  3. [10]

    Aaron V. B. Arellano, Francisco S. N. Lobo, Evolving wormhole geometries within nonlinear electro- dynamics, Class. Quantum Grav.23, 5811 (2006); arXiv: grqc/0608003

    2006

  4. [11]

    Bronnikov

    K.A. Bronnikov. Nonlinear electrodynamics, regular black holes and wormholes. Int. J. Mod. Phys. D 27, 1841005 (2018); arXiv: 1711.00087

    2018 arXiv

  5. [12]

    Kashargin, S

    P. Kashargin, S. Sushkov, Collapsing wormholes sustained by dustlike matter. Universe6(10), 186 (2020)

    2020

  6. [13]

    Bronnikov, P.E

    K.A. Bronnikov, P.E. Kashargin, S.V. Sushkov, Magnetized dusty black holes and wormholes, Universe 7(11), 419 (2021). doi:10.3390/universe7110419 [arXiv:2109.12670 [gr-qc]]

    2021 arXiv

  7. [15]

    Lemaˆ ıtre, The expanding universe

    A.G. Lemaˆ ıtre, The expanding universe. Gen. Rel. Grav.29, 641–680 (1997).https://doi.org/10. 1023/A:1018855621348

    1997

  8. [16]

    Tolman, Effect of inhomogeneity on cosmological models

    R.C. Tolman, Effect of inhomogeneity on cosmological models. Proc. Nat. Acad. Sci.20, 169–176 (1934).https://doi.org/10.1073/pnas.20.3.169

    1934 doi

  9. [17]

    Bondi, Spherically symmetrical models in general relativity

    H. Bondi, Spherically symmetrical models in general relativity. Mon. Not. Roy. Astron. Soc.107, 410–425 (1947).https://doi.org/10.1093/mnras/107.5-6.410

    1947 doi

  10. [18]

    Subenoy Chakraborty, Madhukrishna Chakraborty, A study on existence of wormhole in Lemaitre- Tolman-Bondi model, arXiv: 2409.01331

    arXiv

  11. [19]

    Sajadi, N

    S.N. Sajadi, N. Riazi, Gravitational lensing by polytropic wormholes, arXiv: 1611.04343

    arXiv

  12. [20]

    Bugaev, Igor D

    Mikhail A. Bugaev, Igor D. Novikov, Serge V. Repin, Polina S. Samorodskaya, Uniform sky glow (CMB) observed through the throat of a wormhole, arXiv: 2305.18041

    arXiv

  13. [21]

    Mikhail Bugaev, Igor Novikov, Serge Repin, Polina Samorodskaya, Igor Novikov Jr., Observing of back- ground electromagnetic radiation of the real sky through the throat of a wormhole, arXiv: 2412.06872. 17

    arXiv

  14. [22]

    Bronnikov, K.A

    K.A. Bronnikov, K.A. Baleevskikh, On gravitational lensing by symmetric and asymmetric wormholes, Grav. Cosmol.25, 44-49 (2019); arXiv: 1812.05704

    2019 arXiv

  15. [23]

    Ishkaeva, S.V

    V.A. Ishkaeva, S.V. Sushkov, Image of an accreting general Ellis-Bronnikov wormhole, Phys. Rev. D 108(8), 084054 (2023).https://doi.org/10.1103/PhysRevD.108.084054

    2023 doi

  16. [24]

    Landau, E.M

    L.D. Landau, E.M. Lifshitz,The Classical Theory of Fields. Oxford: Pergamon Press, 1975, Volume 2, 402 p

    1975

  17. [25]

    Vazquez, E.P

    S.E. Vazquez, E.P. Esteban, Strong field gravitational lensing by a Kerr black hole. Nuovo Cimento B.119(5), 489–519 (2004)

    2004

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