Quasi-cosmological Traversable Wormholes in $f(R)$ Gravity

Hanif Golchin; Mohammad Reza Mehdizadeh

REVIEW 2 major objections 4 minor 41 references

Quasi-cosmological Traversable Wormholes in $f(R)$ Gravity

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper constructs exact traversable wormhole solutions in $f(R)$ gravity and claims that, unlike in general relativity, they can satisfy the null energy condition at the throat.

desk verdict The paper's density and pressure formulas drop the nonzero H term from its own field equations, so the NEC/WEC claims are not statements about the stated f(R) theory. read the letter →

arxiv 1908.04378 v1 pith:6MLVO3XG submitted 2019-08-12 gr-qc

Hanif Golchin , Mohammad Reza Mehdizadeh This is my paper · ORCID

classification gr-qc PACS 04.20.-q04.50.Kd

keywords traversablewormholesf(R)gravitynullenergyconditionweaknonconstantRicciscalarquasi-cosmologicalexactsolutionsshapefunction

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

This paper tries to show that traversable wormholes can be built in $f(R)$ modified gravity without invoking the exotic matter that general relativity demands. The authors construct exact wormhole solutions with a nonconstant Ricci scalar, $R=6c_1+6c_2 r^{-n}$, whose shape functions are chosen so that the geometry matches a flat, spherical, or hyperbolic cosmological background at large radius. They then check the standard energy conditions for six widely used $f(R)$ models. Their central result is that, in parameter regions where $F=df/dR<0$, these wormholes can satisfy the null energy condition at and near the throat, and in some cases the weak energy condition everywhere outside it. If this holds, wormhole throats could be supported by ordinary matter in a modified-gravity setting, removing the main obstruction to traversable wormholes.

What carries the argument

The argument runs on three pieces. First, the $f(R)$ field equations are rewritten as $G_{\mu\nu}=T^c_{\mu\nu}+\tilde T^m_{\mu\nu}$, placing all curvature corrections into an effective stress tensor, and the matter is taken as an anisotropic fluid with density $\rho$, radial pressure $p_r$, and transverse pressure $p_t$. Second, the nonconstant Ricci scalar ansatz $R=6c_1+6c_2 r^{-n}$ fixes the shape function $b(r)$ and makes the wormhole asymptotically match cosmological backgrounds. Third, the paper uses a known no-go result stating that static wormholes cannot satisfy the null energy condition when $F=df/dR>0$ and its second derivative is nonzero, so it searches parameter regions with $F<0$; there the effective gravitational constant is negative. The energy-condition inequalities are then evaluated for six viable $f(R)$ models, with $n$, $r_0$, and the model parameters scanned to identify allowed regions.

What would settle it

Compute $H(r)=\frac{1}{4}(FR+\Box F+T)$ along the claimed solutions, evaluating $F$, $\Box F$, and the matter trace $T$ from the paper's shape function and a concrete model such as $f(R)=R-\mu R_* \tanh(R/R_*)$. If $H$ does not vanish identically, the density and pressure formulas used in the energy-condition plots are not the solutions of the full field equations, and the central claim is not established.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes a family of exact, static, spherically symmetric wormhole metrics in $f(R)$ gravity, with shape function $b(r)=(-r_0^n c_1+r_0^{n-2})r^{3-n}+c_1 r^3$, where $c_1=-1,0,1$ selects hyperbolic, flat, or spherical asymptotics. The Ricci scalar is nonconstant, $R=6c_1+6c_2 r^{-n}$, which is the feature that distinguishes these solutions from earlier constant-$R$ wormhole constructions. Testing the energy conditions using the density and pressures obtained from the field equations, the authors find that, when $F=df/dR$ is negative, the null energy condition can hold at the throat and near it for asymptotically flat and hyperbolic wormholes in most of the models considered, and the weak energy condition can hold through the whole exterior for some flat and hyperbolic solutions. Asymptotically spherical wormholes satisfy the null energy condition only in two of the six models. The paper reads this as evidence that $f(R)$ gravity can replace exotic matter with anti-gravitational regions as the physical support for traversable wormholes.

Load-bearing premise

The energy-condition results rest on the assumption that $H=\frac{1}{4}(FR+\Box F+T)$ is zero for the nonconstant-Ricci-scalar solutions, because the density and pressure formulas used to evaluate the energy conditions are only the solutions of the field equations in that case; the paper does not impose or verify $H=0$.

Editorial extensions

If this is right

If the central claim is right, the standard obstruction to traversable wormholes—exotic matter near the throat—does not apply in $f(R)$ gravity; the throat can sit in a region with $F<0$ and still satisfy the null energy condition.
For some asymptotically flat and hyperbolic solutions the weak energy condition holds everywhere outside the throat, so these spacetimes pass the stricter energy conditions, not only the null one.
Asymptotically spherical wormholes are far more constrained: only two of the six models admit them with the null energy condition satisfied near the throat, so spherical asymptotics are not generically supported.
The parameter $n$ in the Ricci scalar acts as a tuning dial: the paper identifies intervals, for example $4<n<5.2$ for the flat case of the hyperbolic-tangent model, where the energy conditions hold, making the existence of such wormholes a parameter-selection question.
The same ansatz and energy-condition inequalities can be applied to other viable $f(R)$ models, so the paper provides a template for scanning model space rather than a single isolated solution.

Reading between the lines

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

A direct test of the central claim is to evaluate $H=\frac{1}{4}(FR+\Box F+T)$ for the ansatz; if it does not vanish, the simplified density and pressure formulas used in the energy-condition plots do not solve the full field equations, and the results would need to be re-derived.
Because the solutions require $F<0$, the matter that formally satisfies the energy conditions lives in a region where the effective gravitational constant is negative; whether that counts as ordinary matter depends on the physical interpretation of the modified-gravity frame, which the paper does not settle.
The asymptotic Ricci scalar is a constant $6c_1$, so these wormholes sit in a cosmological background; one natural extension is to promote $c_1$ to a function of time and look for dynamical wormholes that evolve with the expansion of the universe.
The paper tests six models, but the method is a scan over parameter space; a more systematic search over broader $f(R)$ families could map out which functional forms support energy-condition-respecting wormholes, and would likely find both survivors and exclusions.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know upfront: the central energy-condition analysis does not follow from the field equations as written. Equations (10)-(12) come from (9) only by erasing the H terms, and H doesn't vanish for the Ricci scalar (13) with the models considered. The paper never imposes H=0. So the WEC/NEC plots are not about the theory defined by (2). That's load-bearing, not a typo.

What is genuinely new: the shape function (16) extends the constant-R solutions of [28] to nonconstant R, with three asymptotic FRW spatial geometries. The paper is also honest that its wormholes live in F<0 anti-gravitational regions, consistent with the Bronnikov-Starobinsky no-go theorem. The parameter-space survey is systematic and the citations to the relevant literature look right.

The soft spots are serious. First, the H omission: from (2) directly, rho = f/2 - boxF, while (10) gives rho = F b'/r^2, which agree only when H=0. Second, Eq (18) appears to have factor and sign errors for the c1=±1 cases, which affects the asymptotically spherical and hyperbolic claims. Third, the paper leans on parameter values outside the stated viability ranges (e.g., mu>1 for Tsujikawa) while still framing the results as physical—they flag this, but it weakens the conclusion.

None of this kills the underlying idea, that modified gravity can support wormholes with ordinary matter in anti-gravitational regimes. But this paper's specific derivations don't support the abstract's claim.

Who is this for? Wormhole model-builders in f(R) gravity might mine the shape function, but they should not trust the energy-condition results without redoing the algebra. If it crosses a desk, send it to review—referees exist to catch exactly this kind of inconsistency. As it stands, I'd reject and require a careful re-derivation of (10)-(12) with H included, then a fresh look at the WEC inequalities.

Referee Report

2 major / 4 minor

Summary. The manuscript studies traversable wormholes with the static spherically symmetric metric (6) in f(R) gravity, imposing the nonconstant Ricci scalar R=6c1+6c2/r^n (Eq. 13) and deriving the shape function b(r) in Eq. (16). After imposing the throat conditions, the authors set c1=0,±1 to describe asymptotically flat, hyperbolic, and spherical wormholes. Using the matter density and pressures quoted in Eqs. (10)-(12), they derive the WEC/NEC inequalities (18) and scan the parameter space of six f(R) models, searching for regions with F<0 so as to evade the Bronnikov-Starobinsky no-go theorem. They conclude that, unlike in Einstein gravity, NEC holds at the throat and near it for several models, and WEC holds through the whole space for some asymptotically flat and hyperbolic solutions.

Significance. If the equations were correct, the paper would provide a potentially important counterexample to the usual expectation that traversable wormholes require exotic matter, within observationally motivated f(R) models. The authors deserve credit for being explicit, for treating six viable models, and for correctly recognizing that the F<0 anti-gravitational regime is the only region in which the cited no-go theorem can be evaded; their parameter scan is not circular. However, the central result is not supported because the matter variables used in the energy conditions do not follow from the stated field equations. The stress-test concern is confirmed, and the conclusion as stated is therefore not established by the manuscript.

major comments (2)

[§2, Eqs. (9)-(12)] Equations (10)-(12) do not solve the paper's own field equations. Equation (9) gives b'/r^2=(ρ+H)/F, with analogous H terms in the radial and tangential equations, where H=(FR+□F+T)/4. Solving for the matter variables gives ρ=F b'/r^2−H, not Eq. (10), and the corresponding H terms are missing from Eqs. (11) and (12). The paper never imposes H=0. For the nonconstant Ricci scalar (13) and the nonlinear f(R) models considered, H is generically nonzero; using the trace (3), H=(FR−f)/2+□F, which does not vanish for these models. Equivalently, the exact density from Eq. (2) is ρ=f/2−□F, whereas Eq. (10) gives F R/2; the difference is exactly H. Since every WEC/NEC inequality in Eq. (18) and all figures are built from (10)-(12), the main claim that ordinary matter can support these wormholes is not a statement about the theory defined by Eq. (2). Working in the F<0 regime does not repair this omission.
[§3, Eq. (18)] For c1=±1, Eq. (18) contains algebraic errors that are independent of the H issue. Substituting b(r)=c1 r^3 + A r^{3−n} with A=r0^{n−2}−c1 r0^n into (10)-(12) gives ρ+p_t = F[2c1+(4−n)A r^{−n}/2] + (F'/r)(c1r^2−1+A r^{2−n}) and ρ+p_r = F[2c1+(2−n)A r^{−n}] + F'[c1r+(2−n)A r^{1−n}/2] − F''(1−c1r^2−A r^{2−n}). Equation (18) instead has 24c1r where 4c1r should appear, 12c1 where 2c1 should appear, 6c1r^2 where c1r^2 should appear, and r0^{n−2}−6c1r0^n in place of A throughout. Consequently, the c1=±1 WEC/NEC plots in Figs. 8-13 are not produced by the stated matter expressions (10)-(12), even before the H-term problem is addressed.

minor comments (4)

[§3, energy-condition definitions] The text writes 'the null energy condition WEC2: ρ+pt>0, WEC3: ρ+pr>0' immediately after defining WEC1, WEC2, and WEC3; this is confusing because the same labels are reused for NEC, and the definitions should be clearly separated.
[Eq. (18) and surrounding text] The primes on F in Eq. (18) are never defined; since F=df/dR, the manuscript should state explicitly that F'=dF/dr=f''(R)R'(r), so that F'' is also unambiguous.
[Figure captions] Several captions contain typos such as 'asymptoticly' in Figs. 5, 6, 8, 9, and 11-13, and 'Fig10' without a space; a careful proofreading pass is needed.
[§2, text before Eq. (10)] The phrase 'By solving the above system' is misleading, because Eqs. (10)-(12) are not the general solution of Eq. (9); they are obtained only after dropping the H terms, as noted in the first major comment.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity; Eqs. (10)-(12) silently drop the H term of Eq. (9), a correctness gap, not a circular reduction.

full rationale

The paper's central chain is: field equations (2)/(4) -> (9) with H defined -> (10)-(12) -> (18) -> parameter scans. I find no circular reduction in the rubric's sense. The self-citations (refs [5], [9], [14]) are background references in the introduction and are not load-bearing. The Bronnikov-Starobinsky theorem [29,30] is external, not this group's own theorem, and the paper's anti-gravitational F<0 search is a legitimate use of that theorem. The serious defect is that (10)-(12) do not solve (9); they are what remains after deleting every H term, and H=1/4(FR+boxF+T) is generically nonzero for the nonconstant Ricci scalar (13). Thus the energy-condition plots are not statements about the theory defined by (2). But deleting a term is not the same as assuming the conclusion: the omitted H does not encode the NEC/WEC result, and the paper does not fit data or rename its inputs. Accordingly the circularity score is low (2), while the correctness risk from the dropped H term is high.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The solution family depends on free parameters c1, c2, n and r0, plus hand-chosen f(R) model parameters. No new particles, forces, or dimensions are introduced. The central calculation rests on the metric f(R) field equations, an imposed Ricci scalar ansatz, and the Bronnikov-Starobinsky theorem.

free parameters (5)

c1 = 0, -1, +1 (chosen by hand)
Sets the asymptotic spatial curvature in the Ricci scalar ansatz R=6c1+6c2/r^n, Eq (13).
c2 = eliminated by throat condition b(r0)=r0
c2 is fixed in terms of c1, r0 and n by the throat condition leading to Eq (16).
n = varied, e.g., 2.02 to 5
Exponent in the Ricci scalar ansatz; constrained by flaring-out and selected to make the energy condition inequalities hold.
r0 = 1, 15, 25, etc.
Wormhole throat radius; arbitrary and varied across figures to satisfy the required inequalities.
f(R) model parameters = e.g., mu=1.02, 5; lambda=0.98, 1; R*=0.001 to 1; a,b,c=2,20,20
Model parameters are chosen by hand, with several values outside the stated viability ranges, to make F<0 and satisfy energy conditions.

assumptions (4)

domain assumption The metric f(R) field equations (2) with matter minimally coupled describe the theory.
Standard starting point; the paper's entire analysis uses Eq (2) and its trace.
domain assumption The Morris-Thorne wormhole metric (6) with zero redshift function is traversable and static.
Standard wormhole ansatz; no horizon or singularity is assumed.
ad hoc to paper The Ricci scalar can be imposed as R=6c1+6c2/r^n.
This ansatz defines the solution family and is not derived from a specific f(R) or matter model.
domain assumption The Bronnikov-Starobinsky nonexistence theorem (20) applies, and F<0 evades it.
The search strategy explicitly requires F<0 to violate the theorem, which limits the physical interpretation.

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Cite this review

Pith. "Pith review of Quasi-cosmological Traversable Wormholes in $f(R)$ Gravity." pith.science (2026). https://pith.science/paper/6MLVO3XG

@misc{pith2026190804378,
  author       = {Pith},
  title        = {Pith review of: Quasi-cosmological Traversable Wormholes in $f(R)$ Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6MLVO3XG}},
  note         = {Machine review of arXiv:1908.04378}
}

abstract

In this paper we study traversable wormholes in the context of $f(R)$ gravity. Exact solutions of traversable wormholes are found by imposing the nonconstant Ricci scalar. These solutions asymptotically match spherical, flat and hyperbolic FRW metric. By choosing some static $f(R)$ gravity models, we verify the standard energy conditions for the asymptotically spherical, flat and hyperbolic wormhole solutions. Unlike the Einstein gravity, we find that in the context of $f(R)$ modified gravity, the asymptotically spherical, flat and hyperbolic wormhole solutions can respect the null energy condition (NEC) at the wormhole throat and near that. We find that in some static $f(R)$ models, asymptotically flat and hyperbolic wormholes respect the weak energy condition (WEC) through the whole space.

Figures

Figures reproduced from arXiv: 1908.04378 by the authors.

**Figure 1.** Fig(1.a), Fig(1.b) and Fig(1.c) show the asymptotically flat (c1 = 0), asymptotically hyperbolic (c1 = −1) and asymptotically spherical (c1 = 1) wormhole solutions respectively. Wormhole throat is chosen at r0 = 1, we also set n = 4 in these figures. It is worth to mention that in this model, there is a 4-dimensional parameter space for the wormhole solutions which is made of µ, R∗, r0 and n. Among these parameters,… view at source ↗

**Figure 2.** b shows n − r/r0 diagram. The blue region in this figure corresponds to asymptotically flat traversable wormhole solutions in the background of (21), which respect the WEC. It is obvious that by choosing 4 < n < 5.2, the WEC is respected at the throat r0 and through the whole space outside that. Note that unlike the Einstein gravity, considering f(R) gravity model in the form (21), it is possible to find traversabl… view at source ↗

**Figure 2.** F = df(R)/dR and weak energy condition requiements for a wormhole solution of the Tsujikawa model with n = 5 are depicted in Fig(2.a). It is obvious that F < 0, so the condition of wormhole nonexistence is violated. It is also clear that WEC is respected at the throat r0 and beyond it. In the blue region of Fig(2.b), F < 0 and (18) are simultaneously satisfied, so this region shows traversable asymptotically flat wo… view at source ↗

Figures from the paper (11 more)

**Figure 3.** Figure 3: Setting n = 2.8 in Fig(3.a), we can see that df(R)/dR < 0 around the throat and the obtained wormhole solution of Hu-Sawicki model respect NEC at r0. The blue region in Fig(3.b) corresponds to traversable asymptotically flat wormholes of the Hu-Sawicki model, which res…

**Figure 4.** Figure 4: It is clear that df(R)/dR < 0 at the throat, but the wormhole solutions in Starobinsky model do not respect NEC at the throat r0. In this figures we set q = 2, λ = 0.98 , R∗ = 0.01 and r0 = 15. Fig (5.a) Fig (5.b) [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: Fig(5.a) shows that df(R)/dR < 0 and the NEC is satisfied at the throat for the wormhole solution of the NojiriOdintsov model with n = 2.8. According to Fig(5.b) asymptoticly flat traversable wormholes in this model, respect NEC by choosing adequate values for n. In t…

**Figure 6.** Figure 6: The NEC for a wormhole solution in the case of Amendola-Gannouji-Polarski-Tsujikawa model is respected ( [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: Setting n = 2.65 in Fig(7.a) we found a wormhole solution in the exponential gravity model that respect NEC. The blue region in Fig(7.b) also shows that it is possible to find wormholes solutions in this model that respect NEC around the throat. In these figures we set…

**Figure 8.** Figure 8: The blue region in Fig8.a (Fig8.c) shows the asymptotically spherical (hyperbolic) wormhole solutions of the Tsujikawa model that respect the NEC. Fig(8.b) which is depicted by setting n = 4.5, r0 = 1, shows that F = df(R)/dR < 0 and the asymptotically spherical wormh…

**Figure 9.** Figure 9: Similar to the previous model, there are asymptotically spherical (Fig(9.a)) and hyperbolic (Fig(9.c)) wormholes in Hu-Sawicki model that respect the NEC. In Fig(9.b) we draw F and WEC2,3 for an asymptoticly spherical solution with n = 2.8. Fig(9.d) also shows that an …

**Figure 10.** Figure 10: In the case of Starobinsky model, Fig(10.a) and Fig(10.b) show F, WEC2,3 for the asymptotically hyperbolic and spherical wormhole solutions respectively. In both cases by fixing λ = 1, R∗ = 1 the F < 0 is satisfied around r0, however the NEC are violated. We set r0 = …

**Figure 11.** Figure 11: Fig(11.a) shows that the asymptoticly hyperbolic wormholes with n > 2 in the Nojiri-Odintsov model, respect the WEC almost through the whole space outside the throat r0. We have depicted WEC1,2,3 and F for an asymptotically hyperbolic wormhole with n = 3.2 in Fig(11.b…

**Figure 12.** Figure 12: The blue region in Fig(12.a) corresponds to asymptoticly hyperbolic wormhole solutions that respect the NEC in Amendola-Gannouji-Polarski-Tsujikawa model. It is clear from Fig(12.b) that F < 0 and NEC is respected around r0, for an asymptoticly hyperbolic wormhole wit…

**Figure 13.** Figure 13: In the case of c1 = −1, Fig(13.a) shows the region that F < 0 and (18) are satisfied in blue. This means that the WEC is respected by the asymptotically hyperbolic wormhole solutions of the exponential gravity model, almost through the whole space outside the throat. …

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