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Field Sources for Wormholes With Multiple Throats/Anti-throats

T0 review · 4 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper shows that a wormhole metric with multiple throats and anti-throats is an exact solution of general relativity, with explicit matter sources given by a phantom scalar field and nonlinear electrodynamics.

desk verdict The multi-throat construction is algebraically sound, but the advertised well-defined NED source is not established because L(F) is multi-valued for the showcased parameters. read the letter →

arxiv 2412.05236 v2 pith:QCW57ZBX submitted 2024-12-06 gr-qc

T. M. Crispim , Marcos V. de S. Silva , G. Alencar , Celio R. Muniz , Diego Sáez-Chillón Gómez This is my paper · ORCID
classification gr-qc
keywords wormholesmultiplethroatsanti-throatsphantomscalarfieldnonlinearelectrodynamicsenergyconditionsgeodesicsgeneralrelativity
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 aims to show that a wormhole geometry with several throats and anti-throats, described by the areal function $\Sigma^2(r) = (d^2+r^2)e^{b^2/(c_3+r^2)}$, is not merely a constructed line element but an exact solution of Einstein's equations. The matter source is built explicitly as a phantom scalar field, a field with negative kinetic energy, coupled to nonlinear electrodynamics, with the coupling $h(\phi)$, potential $V(\phi)$, and electromagnetic Lagrangian $L(F)$ written out for two different scalar profiles. If the construction holds, multi-throat wormholes can be realized inside general relativity without modifying gravity, and the usual energy-condition violations can be shifted away from the central region for some parameters. The paper also studies geodesics and shows that photon orbits are tied to the positions of the throats and anti-throats.

What carries the argument

The load-bearing object is the areal function $\Sigma^2(r) = (d^2+r^2)e^{b^2/(c_3+r^2)}$. Its extrema control the wormhole structure: minima of the area $A = 4\pi\Sigma^2$ are throats, maxima are anti-throats, and the regularity conditions $\Sigma \neq 0$ with finite $\Sigma'$ and $\Sigma''$ keep the Kretschmann scalar finite. The reconstruction of the sources is carried by the identity $h(\phi)\phi'^2 = -\Sigma''/\Sigma$, together with $V' = -(\Sigma\Sigma''' + 3\Sigma'\Sigma'')/\Sigma^2$ and the resulting expressions for $L$ and $L_F$; the consistency relation $L_F\,dF/dr - dL/dr = 0$ then checks that the electromagnetic pieces are mutually compatible.

What would settle it

Take a parameter set with non-monotonic $F(r)$, such as $b=4$, $c_3=3$, $d=1$: trace $L(F)$ around one full cycle of $r$ through an extremum; if the curve does not close onto a single branch, no standard single-valued electromagnetic Lagrangian exists and the source construction fails as a field theory. A complementary test is to evolve the scalar and electromagnetic field equations from the throat with the proposed $h$, $V$, and $L$ and check that the metric is reproduced and stays regular.

Watch

Extended reading notes

Core claim

The central claim is the exactness of the source construction. With the action $S = \int \sqrt{|g|}[R - 2h(\phi)g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi + 2V(\phi) + L(F)]$, where $F = F_{\mu\nu}F^{\mu\nu}$, the metric $ds^2 = dt^2 - dr^2 - \Sigma^2(r)\,d\Omega^2$ with $\Sigma^2(r) = (d^2+r^2)e^{b^2/(c_3+r^2)}$ satisfies the full set of field equations provided $h$, $V$, $L$, and $L_F$ are obtained from the reconstruction equations. A prominent feature is that $L(r)$, $L_F(r)$, and $V(r)$ are fixed by the metric alone and are identical for both scalar profiles considered, while $h(\phi)$ and $V(\phi)$ depend on the profile. For the arctangent profile the relation between $L$ and $F$ develops cusps whenever $F(r)$ has extrema, and for the tanh profile the scalar coupling grows like $\cosh^4(r/d)$ at large radius. The authors conclude that the combination of a scalar field and nonlinear electrodynamics can generate physically consistent wormhole solutions with multiple throats and anti-throats with well-defined fields and potentials, and that for suitable parameters all energy conditions can be satisfied in the central region.

Load-bearing premise

The whole construction stands on treating the derived $h(\phi)$, $V(\phi)$, and $L(F)$ as genuine field-theory functions, yet for some parameters $L(F)$ is multi-valued and cusped because $F(r)$ is not monotonic, and for the tanh profile the scalar coupling $h(\phi)$ diverges at infinity.

Editorial extensions

If this is right

  • The multi-throat metric is an exact solution of general relativity once the reconstructed scalar and nonlinear-electrodynamic sources are included, not merely a hand-picked line element.
  • The same spacetime can be generated by different scalar field profiles, because $L(r)$, $L_F(r)$, and $V(r)$ do not depend on which profile is chosen.
  • Radial null and massive geodesics are free, while non-radial null motion feels the effective potential $1/\Sigma^2$; each throat carries an unstable photon sphere and each anti-throat can carry a stable photon orbit.
  • For parameter choices such as $b=4$, $c_3=3$, $d=0.8$, all energy conditions can hold in a central region around $r=0$, although violations remain at large distances where the scalar field is phantom.
  • Setting $b=0$ recovers the Ellis-Bronnikov wormhole, whose energy conditions are globally violated, showing that the new parameters are what allow partial restoration of the energy conditions.

Reading between the lines

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

  • Editorial extension: If the cusps in $L(F)$ cannot be smoothed by a field redefinition, then for multi-throat parameter sets the source may be position-dependent bookkeeping rather than a genuine nonlinear electrodynamics; one could test whether restricting to parameters with monotonic $F(r)$ removes the multi-valuedness while preserving multiple throats.
  • Editorial extension: The diverging $h(\phi)$ for the tanh profile means the scalar sector is not asymptotically standard; computing quasinormal modes or the ringdown of this wormhole would show whether that divergence leaves observable traces.
  • Editorial extension: Because all energy conditions can hold near the center, the strong-field optical appearance of this wormhole may mimic that of a regular black hole; lensing and shadow calculations would be needed to distinguish the two.
  • Editorial extension: The two scalar profiles producing identical $L(r)$, $L_F(r)$, $V(r)$ indicates an underdetermination: the same geometry and electromagnetic sector admit many scalar sectors, so astrophysical matching would require additional input to fix the scalar profile.
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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

4 major / 3 minor

Summary. The paper constructs a family of static, spherically symmetric traversable wormhole spacetimes with multiple throats and anti-throats, based on the areal function Sigma^2(r) = (d^2 + r^2) exp(b^2/(c3 + r^2)) in Eq. (4). It studies embedding diagrams, null geodesics and effective potentials, and then asks which matter sources in the action (24), consisting of GR plus a scalar field with coupling h(phi) and potential V(phi) plus nonlinear electrodynamics, can produce the geometry. For prescribed scalar profiles phi = arctan(r/d) and phi = tanh(r/d), the authors solve algebraically for h, V, L and L_F as functions of r and present explicit closed forms, including the b = 0 limit that recovers the Ellis-Bronnikov source. They further analyze energy conditions, finding regions where all conditions can hold near r = 0 and violations at large r. The central claim is that the multi-throat geometry is an exact GR solution with well-defined scalar and NED sources.

Significance. If established, the construction would be a useful explicit example of multi-throat wormholes in GR with a two-field source, and the energy-condition analysis provides a concrete map of NEC/WEC/SEC/DEC behavior. The paper is carefully reverse-engineered rather than predictive: h is defined by Eq. (39), V by Eq. (41), and L, L_F by Eq. (40), so the Einstein equations are satisfied by construction. Strengths include the systematic reduction to quadratures, the explicit b = 0 consistency check, and the demonstration that h changes sign, so the scalar alternates between standard and phantom behavior. The main weakness is that the assembled matter sector is not always a well-defined Lagrangian: the printed definition of F is inconsistent with Eq. (49), L(F) becomes multi-valued for the showcased multi-throat parameters, and the tanh scalar profile makes h diverge asymptotically. These issues directly affect the claim of 'well-defined fields and potentials' and need to be addressed before the solution can be regarded as a fully consistent field-theoretic source.

major comments (4)
  1. [III.A, Eqs. (31) and (49)] There is an inconsistency in the definition of the electromagnetic invariant. Eq. (31) defines F = 2q^2/Sigma^2, but Eq. (49) gives F = 2q^2 exp(-2b^2/(c3+r^2))/(d^2+r^2)^2 = 2q^2/Sigma^4, and the latter is the standard invariant for F_{theta phi} = q sin(theta) in the metric (2). As printed, Eq. (50) does not follow from Eqs. (40) and (41) with F = 2q^2/Sigma^2: one obtains L_F F' = 2 Sigma' B / Sigma instead of L' = 4 Sigma' B / Sigma^3, where B = Sigma Sigma'' + Sigma'^2 - 1. With F = 2q^2/Sigma^4 the relation does follow. Please correct Eq. (31) and make all subsequent definitions and plots consistent with the corrected invariant.
  2. [III.A, Fig. 7 and paragraph after Eq. (49)] The paper acknowledges that for parameter choices with extrema of F(r), the relation between F and r cannot be inverted and L(F) has cusps. For the cases shown in Fig. 7 (e.g., b = 4, c3 = 3, d = 1 and b = 4, c3 = 4.5, d = 1), the same value of F is attained at several radii with different values of L, so L is not a single-valued function of the invariant F. This means the NED sector of action (24) is not a well-defined local Lagrangian for those parameters, which contradicts the Conclusion that the sources have 'well-defined fields and potentials.' Please either restrict the multi-throat parameter claims to ranges where L(F) can be made single-valued, provide an explicit branch prescription and verify the field equations on that prescription, or reformulate the claim as a parametric reconstruction rather than a Lagrangian field theory. If cusped Lagrangians are considered admissible in the NED literature, the paper should cite the precise sense and justify that variations of the action remain well-defined.
  3. [III.B, Eq. (51) and Conclusions] For the tanh profile, Eq. (51) contains a factor cosh^4(r/d); since phi = tanh(r/d), this behaves as (1 - phi^2)^{-2} as r -> +/- infinity. The corresponding h(phi) in Eq. (53) has denominators that vanish at phi = +/- 1, so h diverges as phi -> +/- 1. The statement in the Conclusions that 'the coupling h(phi) approaches a constant' is therefore not correct for this model. Please correct the asymptotic description and discuss whether the divergent coupling is physically acceptable.
  4. [III.A, after Eq. (49)] The consistency relation (50) is asserted without a shown verification for the lengthy expressions (47)-(49), and the long forms of L and L_F are otherwise unverified in the text. Since Eq. (50) is the principal check that the reconstructed L and L_F represent a single NED source, please include the derivation or provide a supplementary file with a symbolic verification of Eqs. (47)-(50).
minor comments (3)
  1. [II.A, condition (12)] The paper states that condition (12) is not satisfied for all parameter values, but it never gives the allowed parameter ranges; a brief statement of the admissible ranges would help readers reproduce the embedding diagrams.
  2. [III.A, paragraph after Eq. (39)] The statement that different choices of phi yield the same L(r), L_F(r), and V(r) is a consequence of the reconstruction procedure, not an independent dynamical result; please phrase it as such.
  3. [Abstract and Section IV] The abstract and conclusions say that 'all energy conditions can be partially satisfied in certain regions of spacetime,' but Section IV also shows that violations remain at large radii; the phrase 'partially satisfied' should be clarified to mean that each condition holds only in restricted radial intervals, not that the energy conditions are globally relaxed.

Circularity Check

2 steps flagged · score 6.0 of 10

The sources are reverse-engineered by definition: h is defined from the metric and chosen scalar profile via Eq. (39), and V, L, LF follow from the same metric by Eqs.

  1. self definitional [Section III, Eqs. (38)-(41), text after Eq. (39)]
    ""From the equations of motion, we can also write: h(ϕ)ϕ′(r)2 = −Σ′′/Σ. ... Written h(ϕ) = −Σ′′/(Σϕ′2), we can simplify the relations ... In this way, once we know the form of the function Σ, we can obtain the functions L, LF, and V, regardless of the scalar field model we choose.""

    The source functions are not independent inputs: h(ϕ) is defined by Eq. (39) as a functional of the metric Σ and the chosen scalar profile ϕ; V is obtained by integrating (41), which is metric-determined; L and LF are then given by (40), also metric-determined. Therefore any smooth profile ϕ placed in (39) automatically satisfies the scalar-field equation, and the metric (4) satisfies the Einstein equations with these algebraically constructed sources. The abstract's claim that 'distinct scalar field profiles can generate the same spacetime geometry' is true by construction, because h absorbs the profile, and the Conclusion's claim that the scalar-plus-NED action 'is capable of generating' the wormhole reduces to the definitional relations (39)-(41).

  2. other [Section III.A, after Eq. (49) and Fig. 7]
    ""For each non-zero maximum/minimum of the function F (r), there will be a cusp present in the function L(F ) [28, 66]. ... we cannot analytically invert r(F ) to explicitly express the function L(F ).""

    This is a self-acknowledged breakdown of the 'well-defined field' claim, not a circular reduction, but it is load-bearing for the Conclusion. For the showcased multi-throat parameters (e.g., b=4, c3=3, d=1; b=4, c3=4.5, d=1 in Fig. 7), F(r)=2q^2 exp(-2b^2/(c3+r^2))/(d^2+r^2)^2 has extrema away from r=0, so the parametric curve (F(r), L(r)) folds: the same F value corresponds to multiple L values. Then L(F) is multi-valued with cusps and is not a function of the local invariant F, so the action (24) does not define a local NED Lagrangian for those parameters. The authors' own admission undercuts the Conclusion's assertion of 'well-defined fields and potentials' for the very parameter sets displayed.

full rationale

The paper's derivation is an explicit inverse-problem construction: starting from the fixed metric (4), the authors define the scalar coupling h by Eq. (39), obtain V by integrating Eq. (41), and then solve for L and LF via Eq. (40). This makes the central existence claim true by construction for any smooth scalar profile, because the sources are functionals of the chosen metric. The statement that different scalar profiles generate the same geometry is therefore self-definitional rather than a physical prediction. However, the algebraic consistency checks, the geodesic analysis, and the energy-condition plots are independent content and are not circular. The largest weakness is not circularity but well-definedness: the paper admits that for several displayed parameter choices, F(r) is non-monotonic so L(F) is multi-valued with cusps, meaning the NED Lagrangian is not a single-valued function of the field invariant; the tanh profile also gives an h(phi) that diverges as phi approaches ±1, contradicting the Conclusion's asymptotic claim. These issues reduce the scope of the 'physically consistent' claim but do not by themselves constitute circular reasoning. Weighing the by-construction nature of the source construction and the admitted breakdown of L(F), the partial circularity score is 6.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

We place in the ledger the hand-chosen shape parameters (b, c3, d), the magnetic charge q, and the implicit choice of integration constant for V (set by V(infinity)=0). The reverse-engineering axioms are the ones that guarantee the reconstructed objects define a field theory rather than mere bookkeeping: the descent of h and V to functions of phi, the consistency relation for L and LF, and the admissibility (single-valuedness/analyticity) of L(F). The only external benchmark is the b=0 Ellis-Bronnikov limit.

free parameters (5)
  • b = chosen by hand (e.g., 3.3, 3.5, 4 in figures)
    Controls the number and depth of throats/anti-throats; not predicted by the theory, selected to display the desired area extrema.
  • c3 = chosen by hand (e.g., 3, 4.5, 7)
    Regulates the exponential factor; required non-zero for regularity; selected freely.
  • d = chosen by hand (e.g., 0.8, 1, 2, 2.5)
    Sets the radial scale; in the b=0 limit it recovers the Ellis-Bronnikov throat parameter.
  • q = not specified (drops out of metric and energy conditions)
    Magnetic charge scaling the NED Lagrangian; free parameter of the solution, affects L(F) and F but not the geometry.
  • V integration constant = set so V(infinity)=0
    A constant shift between V and L is a gauge-like freedom; the paper implicitly fixes it via the asymptotic condition, but does not state this explicitly.
assumptions (5)
  • domain assumption The line element (2) with Sigma given by (4) is a valid wormhole geometry admitting minima/maxima of the areal function (imported from [53]).
    The whole construction starts from this ansatz; no derivation from a more fundamental principle is given.
  • ad hoc to paper Given h=-Sigma''/(Sigma phi'^2) (eq 39), the scalar field equation reduces to eq (41), so V(r) is profile-independent.
    This is the reverse-engineering step: h and V are solved from the geometry, not independently motivated.
  • ad hoc to paper The functions L and LF defined in eq (40) satisfy the consistency relation LF F' = L' (eq 50).
    Needed for a well-defined NED; asserted but not proved in the text.
  • ad hoc to paper The reconstructed matter theory is physically admissible, i.e., h(phi), V(phi) and L(F) are well-defined (single-valued or suitably branched) functions of their arguments.
    The paper notes L(F) develops cusps (multi-valuedness) when F(r) has extrema, and h(phi) diverges for the tanh model, so admissibility is not automatic.
  • standard math Standard Einstein equations with the action (24) are the correct gravitational dynamics.
    GR is assumed; the paper constructs sources within this theory.

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Cite this review

Pith. "Pith review of Field Sources for Wormholes With Multiple Throats/Anti-throats." pith.science (2026). https://pith.science/paper/QCW57ZBX

@misc{pith2026241205236,
  author       = {Pith},
  title        = {Pith review of: Field Sources for Wormholes With Multiple Throats/Anti-throats},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QCW57ZBX}},
  note         = {Machine review of arXiv:2412.05236}
}
read the original abstract

In this work, we investigate wormhole geometries with multiple throats and anti-throats in general relativity. The existence of these structures is identified through the analysis of minima and maxima in the area of the solution. Using embedding diagrams, we visualize the geometry and demonstrate that these objects exhibit a complex structure, distinct from standard single-throat wormholes. We further analyze the geodesic motion in such spacetimes. The solutions are derived from Einstein's equations by coupling a phantom scalar field to nonlinear electrodynamics, and we show that distinct scalar field profiles can generate the same spacetime geometry. Additionally, we examine the energy conditions and demonstrate that, for specific parameter choices, all energy conditions can be partially satisfied in certain regions of spacetime.

Figures

Figures reproduced from arXiv: 2412.05236 by the authors.

Figure 1
Figure 1. Parametric plot of z(r) vs ρ(r) showing the so-called isometric embedding diagrams for different choices of parameters. These curves stand out significantly compared to the typical catenary curve of the Ellis-Bronnikov wormhole. The plot in [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Embedded surface diagrams for different choices of parameters, where it is clearly [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Effective potential for null geodesics on the background of the wormhole solution [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Graphical behavior of the functions V (ϕ(r)) (top panel) and h(ϕ(r)) (bottom panel) in terms of the coordinate r for different combinations of the wormhole parameters d, b and c3 and the model ϕ = arctan(r/d). Explicitly, in terms of the radial coordinate, the function…
Figure 5
Figure 5. Figure 5: Graphical behavior of the functions V (ϕ) (top panel) and h(ϕ) (bottom panel) in terms of the scalar field ϕ for different combinations of the wormhole parameters b, d and c3. This consistency relation shows that, even though they are obtained independently through Max…
Figure 6
Figure 6. Figure 6: Behavior of the electromagnetic Lagrangian [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Behavior of the electromagnetic Lagrangian [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: Behavior of the function h(ϕ(r)) in terms of the coordinate r for different combinations of the wormhole parameters b, d and c3 and the model ϕ = tanh (r/d). h(ϕ) = −  c3 + d 2 tanh−1 (ϕ) 2 −4 (ϕ2 − 1)2  tanh−1 (ϕ) 2 + 12 n d 2 tanh−1 (ϕ) 2 h b 4 d 2 + 2b 2 c3  d …
Figure 9
Figure 9. Figure 9: Graphical behavior of the functions V (ϕ) (top panel) and h(ϕ) (bottom panel) in terms of the scalar field ϕ for different combinations of the wormhole parameters b, d and c3. (SEC), weak energy condition (WEC), and dominant energy condition (DEC). These con￾ditions ar…
Figure 10
Figure 10. Figure 10: Behavior of the energy density ρ (bottom panel) and the combinations of the pressures ρ + pr (top panel) and ρ + pt (middle panel) for different combinations of the wormhole parameters b, d and c3. LF (r), and V (r). We tested two widely studied models for ϕ in the li…
Figure 11
Figure 11. Figure 11: Behavior of the combinations of the pressures [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On regular black string spacetimes in nonlinear electrodynamics

    gr-qc 2026-03 unverdicted novelty 6.0 of 10

    No regular purely electric black strings exist in NED recovering the Maxwell limit, but regular cylindrical Bardeen and Hayward analogues are constructed with finite curvature.

  2. Embedding Wormholes and Dyonic Black Strings in Warped Braneworlds via Local Sum Rules

    gr-qc 2026-01 reject novelty 4.0 of 10

    Embedding of Ellis-Bronnikov wormhole and NED-sourced magnetic/dyonic black strings into RS braneworlds using Local Sum Rules; the dyonic q→0 limit is inconsistent as written.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.