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REVIEW 3 major objections 4 minor 59 references

Wormhole solutions in generalized Rastall gravity

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

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Pith's one-line read The paper constructs exact wormhole solutions in generalized Rastall gravity whose supporting matter satisfies the weak and null energy conditions at the throat and beyond, so no exotic matter is required.

desk verdict The zero-redshift wormhole family in this paper does not satisfy the paper's own field equations; only the nonzero-redshift class may be worth a second look. read the letter →

arxiv 2412.06863 v2 pith:BWHGBGPI submitted 2024-12-09 gr-qc

Naser Sadeghnezhad This is my paper · ORCID
classification gr-qc MSC 83C1583C1083D05
keywords wormholesgeneralizedRastallgravityenergyconditionsweakconditiongravitationallensingphotonsphereexoticmatterexactsolutions
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 generalized Rastall gravity, where the coupling between matter and curvature varies from point to point, can support traversable wormholes without exotic matter. It constructs two exact families of static spherically symmetric wormhole solutions, one with zero redshift function and one with nonzero redshift, assuming an anisotropic fluid with linear equations of state in the radial and tangential pressures. For both families the author identifies parameter ranges in which the shape function flares out at the throat and the weak and null energy conditions hold at the throat and throughout the spacetime. If these solutions are correct, modified gravity with a running matter–geometry coupling removes the main obstacle to wormhole geometries in general relativity. The paper also argues that the wormhole throat can act as an effective photon sphere, making the light deflection angle diverge there.

What carries the argument

The central mechanism is the spacetime-dependent Rastall coupling $\lambda(r)$, which makes the energy-momentum divergence proportional to $\nabla^\nu(\lambda R)$ instead of zero. This extra freedom closes the underdetermined system when combined with linear equations of state, and the paper treats the throat density $\rho_0$ as a boundary condition that can be adjusted to satisfy $\rho\ge0$, $\rho+p_r\ge0$, and $\rho+p_t\ge0$ while the shape function $b(r)$ keeps the throat flaring out and the metric asymptotically flat. The varying coupling is what is said to absorb what would otherwise be an exotic-matter requirement.

What would settle it

Evaluate the algebraic field equation for the energy density, Eq. (7), directly at the throat $r=r_0$ for the zero-redshift solution (10): with $w_1=0.5$, $w_2=-0.5$, $\kappa=1$, $r_0=1$, the equation gives $\rho_0=-4$, while the paper's condition (18) and Fig. 1 require $\rho_0\ge0$. Checking whether this value is consistent with the imposed boundary condition $\rho(r_0)=\rho_0$ would determine whether the claimed exotic-matter-free parameter regions exist.

Watch

Extended reading notes

Core claim

The central claim is that in generalized Rastall gravity the energy-momentum tensor need not be conserved in the usual sense; instead its divergence is proportional to the gradient of the Ricci scalar through a spacetime-dependent Rastall parameter $\lambda(r)$. With an anisotropic energy-momentum tensor and linear equations of state $p_r(r)=w_1\rho(r)$, $p_t(r)=w_2\rho(r)$, the field equations admit exact wormhole metrics of the Morris–Thorne form and of the nonconstant-redshift form $\Phi(r)=\frac{1}{2}\ln[\alpha+\beta r_0/r]$. The author finds that the flare-out condition, asymptotic flatness, and the inequalities $\rho\ge0$, $\rho+p_r\ge0$, $\rho+p_t\ge0$ can be satisfied simultaneously for suitable parameter ranges, including negative $\kappa$ or particular equation-of-state parameters, so the wormhole is supported by matter that satisfies the standard energy conditions. This is presented as a distinction from ordinary Rastall wormholes previously found, which typically require NEC-violating matter. The lensing section claims that for a Morris–Thorne subclass and for a nonconstant-redshift subclass, the effective potential for null geodesics has a maximum at the throat, so the throat functions as an unstable photon sphere and the deflection angle diverges as the turning point approaches it. In its concluding remarks the paper acknowledges that the proper-distance integral for the general nonzero-redshift family is too complicated to evaluate in elementary functions, so the photon-sphere statement is established for the treated subclass rather than for the full family.

Load-bearing premise

The whole zero-redshift family rests on taking the throat energy density $\rho_0$ as a free boundary condition that can be chosen to make the weak energy condition hold; if the field equations already fix $\rho_0$ once $b(r)$ and $\lambda$ are chosen, the displayed solutions and parameter ranges collapse.

Editorial extensions

If this is right

  • If the solutions hold, generalized Rastall gravity produces asymptotically flat, traversable wormholes whose matter satisfies the weak and null energy conditions, so the usual exotic-matter obstruction disappears.
  • Allowed parameter regions restrict the equation-of-state parameters and the gravitational coupling; for a dark-energy tangential pressure the metric reduces to the Morris–Thorne wormhole.
  • Null geodesics see the throat as an unstable photon sphere, so light from an appropriately placed source can be deflected by arbitrarily large angles and produce an infinite sequence of relativistic images.
  • Timelike geodesics allow three behaviors—transit through the throat, reflection back to the same universe, and bound oscillatory motion—depending on the particle's energy and angular momentum.
  • The running Rastall parameter grows near the throat and asymptotes to a constant, localizing the matter–geometry interaction that replaces exotic matter.

Reading between the lines

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

  • Since Eq. (7) at the throat may fix $\rho_0$ rather than leaving it free, the zero-redshift parameter-space plots should be re-derived with $\rho_0$ treated as an output; doing so would decide which subfamilies survive.
  • The throat-as-photon-sphere result suggests these wormholes could masquerade as black holes in strong-lensing surveys; a direct comparison of the deflection-angle coefficients with Schwarzschild's would be a concrete next step, and the paper does not perform that comparison.
  • The paper itself notes that the proper-distance integral for the general nonzero-redshift family is too complicated for elementary functions, so extending the photon-sphere analysis to the full family is an open technical step rather than a closed result.
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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 / 4 minor

Summary. The paper constructs static spherically symmetric wormhole solutions in generalized Rastall gravity with anisotropic matter satisfying the linear equations of state pr=w1 rho and pt=w2 rho. Two families are presented: a zero-redshift family (Phi=0) with a power-law shape function and a two-term energy density, and a nonzero-redshift family with Phi=(1/2) ln[alpha+beta r0/r]. The manuscript claims that parameter windows exist for which the flare-out condition, the weak energy condition, and the null energy condition hold at the throat and throughout the spacetime, so that asymptotically flat wormholes can be built without exotic matter. It then studies timelike and null geodesics and argues that the throat can act as a photon sphere with arbitrarily large deflection angles.

Significance. The question of whether wormholes in modified gravity can be supported without NEC-violating matter is of current interest, and the paper offers explicit closed-form ansatze and a transparent enumeration of parameter regions, which is a strength. If the solutions were valid, the zero- and nonzero-redshift examples would be useful additions to the Rastall-gravity literature, and the lensing analysis would be a nice illustration. Unfortunately, the central zero-redshift solution is internally inconsistent with the paper's own field equations, so the claimed significance is not realized as written.

major comments (3)
  1. [Section 2.1, Eqs. (7) and (10)] The zero-redshift solution does not satisfy the paper's own field equation for generic allowed parameters. For Phi=0, Eq. (7) reduces to rho(r)=(1-2 kappa lambda) b'(r)/(kappa r^2). With the constant lambda and power-law b(r) from Eq. (10), the right-hand side is a single power of r, proportional to (r/r0)^gamma. The extra rho1 r^eta term in Eq. (10) is therefore incompatible with Eq. (7) unless rho1=0 (or eta=gamma with matching coefficients); imposing rho(r0)=rho0 cannot be an independent boundary condition. Explicitly, w1=0.5, w2=-0.5, kappa=1, r0=1 satisfy condition (18), and Eq. (7) gives rho(r)=-4 r^(-8), hence rho0=-4, contradicting rho0>0 required by Eq. (13). Consequently the parameter windows (16)-(20), the throat energy conditions, and the zero-redshift no-exotic-matter claim are invalid.
  2. [Section 2.2, Eq. (26) and condition V] For the nonzero-redshift family, the 'throughout the spacetime' part of the WEC claim is not demonstrated. Condition V imposes a decay ordering epsilon3+|epsilon2| < epsilon1-1 and positivity at the throat, but this does not by itself preclude rho(r) from becoming negative at intermediate radii, and no inequality or monotonicity argument is given for rho+pr and rho+pt away from the throat. The statement therefore rests on the selected plots in Fig. 2 rather than on a proof.
  3. [Section 3, Eqs. (35)-(43)] The lensing analysis for the zero-redshift family is built on the invalid solution from Eq. (10), so the claim that the throat acts as a photon sphere with divergent deflection angle is not established for that family. For the nonzero-redshift branch the analysis is limited to a single parameter example through Eq. (43), so it does not constitute a general derivation of the abstract's lensing claim.
minor comments (4)
  1. [Eq. (10)] The formula for rho1 is typeset in a way that is hard to parse and appears dimensionally inconsistent as printed; please rewrite it with an explicit normalization and check the powers of r0.
  2. [Throughout] There are several typographical issues, including 'also also satisfies' in Section 2.1, 'theses articles' in the Introduction, and inconsistent notation rim versus r_im in Eqs. (35)-(37).
  3. [Section 2.2, condition III] For alpha != 1, e^{2Phi} tends to alpha rather than 1; the time rescaling that makes the asymptotic form explicitly Minkowskian should be stated.
  4. [Figure 1 caption] The caption appears mismatched with the text: it describes the right panel as an allowed region, while the text refers to energy-density plots; please clarify which panel shows what.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the wormhole solutions are constructed explicitly from stated ansaetze and boundary conditions, and the WEC/NEC checks are imposed restrictions rather than back-fed inputs.

full rationale

The derivation chain is self-contained in the relevant sense. The field equations (6)-(9) are solved under explicitly stated ansaetze: a zero or nonzero redshift function and the linear equations of state pr=w1 rho, pt=w2 rho. The shape function b(r), energy density rho(r), and Rastall coupling lambda(r) are then written down explicitly, with integration constants fixed by the throat conditions b(r0)=r0 and rho(r0)=rho0. The weak and null energy conditions are not inserted into the field equations to force a solution; they are evaluated after the solution is obtained and are used only to restrict the parameter regions (16)-(20) and (I)-(V). The gravitational lensing analysis follows from the standard null-geodesic equations and the photon-sphere condition is taken from independent literature (Shaikh et al.), not from a self-citation. The paper's citation of the author's own earlier Rastall wormhole work is contextual and comparative, not load-bearing: no uniqueness theorem, ansatz, or central premise is imported from that self-citation. The main caveat is an internal-consistency concern rather than circularity: with Phi=0 and constant lambda, Eq. (7) is algebraic and would force rho to be a single power law, whereas Eq. (10) includes an additional rho1 r^eta term unless special parameter relations hold. That is a correctness issue about whether Eq. (10) satisfies the stated field equations, not a reduction of a claimed prediction to its own input, so it does not raise the circularity score.

Assumptions & free parameters 7 free parameters · 4 assumptions · 0 invented entities

The central construction rests on the GRG field equations, the wormhole metric ansatz, and the linear EoS. In addition, the zero-redshift solution implicitly assumes rho0 is a free constant, which conflicts with the algebraic field equation for rho. No new particle, force, or conserved quantity is introduced; the variable Rastall parameter lambda(r) is inherited from Ref. [34].

free parameters (7)
  • w1 (radial EoS parameter) = restricted to allowed regions, e.g. 0<w1<=1
    Chosen by hand to satisfy flare-out and energy conditions; not determined by the theory.
  • w2 (tangential EoS parameter) = e.g. -1<=w2<=-1/2(w1+1) depending on branch
    Chosen by hand; in the nonzero class the special case w2=-1 is used for lensing plots.
  • rho0 (energy density at throat) = treated as free non-negative; e.g. rho0=1 in plots
    Imposed as a boundary condition, but in the zero-redshift class this conflicts with Eq. (7).
  • kappa (Rastall gravitational constant) = allowed negative or positive depending on branch
    Sign and magnitude control whether the WEC can hold; not fixed by observations in this paper.
  • r0 (throat radius) = set to 1 in plots
    Radial scale of the wormhole; not predicted by the theory.
  • alpha, beta (redshift function parameters) = alpha=3, beta=-1 in the lensing example
    Parameters of the nonzero redshift function; chosen to satisfy metric positivity and energy conditions.
  • C1, C2 / rho3 integration constants = fixed by rho(r0)=rho0 and b(r0)=r0
    Constants of integration; in the zero-redshift class the resulting extra term appears inconsistent with the field equations.
assumptions (4)
  • domain assumption Generalized Rastall field equations with variable lambda(r) are accepted as the gravitational theory.
    Adopted from Ref. [34]; the paper does not derive or test this theory and does not discuss its mapping to Einstein gravity with a modified source.
  • domain assumption Static spherically symmetric wormhole metric Eq. (3) with redshift function Phi and shape function b.
    Standard Morris-Thorne ansatz, plus traversability conditions listed in Sec. 2.
  • ad hoc to paper Anisotropic EMT with linear equations of state pr=w1*rho and pt=w2*rho.
    Chosen to close the underdetermined system; the constants are then restricted to satisfy energy conditions.
  • ad hoc to paper In the zero-redshift class, rho0 can be imposed independently as an integration constant.
    Used in Eq. (10) to set rho1; this conflicts with the algebraic field equation Eq. (7), which already fixes rho0.

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Cite this review

Pith. "Pith review of Wormhole solutions in generalized Rastall gravity." pith.science (2026). https://pith.science/paper/BWHGBGPI

@misc{pith2026241206863,
  author       = {Pith},
  title        = {Pith review of: Wormhole solutions in generalized Rastall gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BWHGBGPI}},
  note         = {Machine review of arXiv:2412.06863}
}
read the original abstract

In the present work, we seek for static spherically symmetric solutions representing wormhole configurations in generalized Rastall gravity (GRG). In this theory, a varying coupling parameter could act as dark energy (DE) and thus, it can be considered as responsible for the current accelerated expansion of the universe. We consider an anisotropic energy momentum tensor (EMT) as the supporting source for wormhole structure and further assume that there exists a linear relation between radial and tangential pressures and energy density. We therefore obtain two classes of solutions to the field equations of GRG, including the solutions with zero and nonzero redshift functions. For these solutions we find that the matter distribution obeys the physical reasonability conditions, i.e., the flare-out and the weak (WEC) and null (NEC) energy conditions either at the throat and throughout the spacetime. The conditions on physical reasonability of the wormhole solutions put restrictions on model parameters. Hence, in the framework of GRG, asymptotically flat wormhole configurations can be built without the need of exotic matter. Gravitational lensing effects of the obtained solutions are also discussed and it is found that the throat of wormhole can effectively act as a photon sphere near which the light deflection angle takes arbitrarily large values.

Figures

Figures reproduced from arXiv: 2412.06863 by the authors.

Figure 1
Figure 1. Plot of energy density for ρ0 = r0 = 1. Right panel: The allowed region for the pair of parameters (w1, w2) for r0 = κ = 1, α = 1, β = −0.3 (blue region), β = −0.2 (red region) and β = −0.1 (gray region). 2.2 Solutions with Φ(r) = 1 2 ln[α + β [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Left panel: The allowed values of EoS parameters fo [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. The behavior of inverse of the radial metric compon [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: The behavior of Rastall parameter for the same valu [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Left panel: The behavior of effective potential for [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: Left panel: The behavior of effective potential for [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

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