Dynamical Wormhole Solutions in $f(R, T)$ Gravity

Maryam Ranjbar; Yaghoub Heydarzade

REVIEW 2 major objections 5 minor 131 references

Dynamical Wormhole Solutions in $f(R, T)$ Gravity

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper claims that in f(R,T)=R+βT² gravity, exact dynamical wormhole solutions can be supported by ordinary matter satisfying the weak energy condition, without exotic matter.

desk verdict New explicit dynamical wormhole solutions in f(R,T) gravity, but the reconstructed matter violates the EoS used to simplify the field equations. read the letter →

arxiv 2504.17910 v2 pith:6ERATVRE submitted 2025-04-24 gr-qc

Yaghoub Heydarzade , Maryam Ranjbar This is my paper · ORCID

classification gr-qc

keywords dynamicalwormholesf(RT)gravitytrace-squaredweakenergyconditionMorris-Thornemetricexactsolutionsenergy-momentumtensorsquaredmodified

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 establish that in the modified gravity theory $f(R,T)=R+\beta T^2$, where the action contains a term quadratic in the trace of the energy-momentum tensor, wormholes can be supported without exotic matter. It constructs exact time-dependent wormhole solutions, with a scale factor $a(t)$ and shape function $B(r)$, for which the ordinary matter density and pressures satisfy the weak energy condition even though the effective energy-momentum tensor that sources the curvature violates it. If correct, this would be a departure from general relativity, where traversable wormholes require negative-energy matter. The construction is carried out for two different matter equations of state, and the allowed parameter ranges are classified.

What carries the argument

The load-bearing object is the effective energy-momentum tensor of trace-squared gravity, $T^{\mathrm{eff}}_{\mu\nu}=T_{\mu\nu}+\frac{1}{\kappa}\,2\beta(-\rho+3P)\left(T_{\mu\nu}+\frac14(-\rho-P)g_{\mu\nu}\right)$, which rewrites the modified field equations as $G_{\mu\nu}=\kappa T^{\mathrm{eff}}_{\mu\nu}$. The argument proceeds by forcing the redshift function to be independent of $r$ through Eq. (19), absorbing it into the time coordinate; simplifying the effective tensor with the assumed ordinary-matter equation of state $2P_l+P_r=\zeta\rho$; and then using an effective equation of state—either $\rho^{\mathrm{eff}}=\frac{\omega}{1+2\gamma}(P_r^{\mathrm{eff}}+2\gamma P_l^{\mathrm{eff}})$ or $P_r^{\mathrm{eff}}=\alpha P_l^{\mathrm{eff}}$—to separate the field equations into an ODE for $B(r)$ and an ODE for $a(t)$. 'Trace-squared gravity' is the name for the particular model $f(R,T)=R+\beta T^2$ studied here.

What would settle it

Substitute the paper's formulas for $\rho$, $P_r$, and $P_l$ (Eqs. (41)-(43) in Case I or (65)-(67) in Case II) into the assumed equation of state $2P_l+P_r=\zeta\rho$ for a parameter set such as the one used in Figure 1 ($\beta=2$, $\zeta=2$, $\gamma=1$, $\omega=3$, $a_0=3$, $a_1=3$, $r_0=2$, $\kappa=8\pi$) at $t=0$, $r=2$. If the equality fails, the simplification used to derive the solutions is invalid and the metric does not satisfy the true field equations.

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Extended reading notes

Core claim

The central claim is that the field equations of $f(R,T)=R+\beta T^2$ gravity, recast with an effective energy-momentum tensor, admit exact dynamical Morris-Thorne wormhole solutions whose ordinary matter obeys the weak energy condition. With the linear equation of state $2P_l+P_r=\zeta\rho$ for the ordinary matter, the effective density and pressures take simple algebraic forms, and imposing one of two effective equations of state separates the field equations into ordinary differential equations for $B(r)$ and $a(t)$. The paper obtains explicit scale factors and shape functions, lists the ranges of the coupling $\beta$ and the equation-of-state parameters for which $\rho\ge 0$, $\rho+P_r\ge 0$, and $\rho+P_l\ge 0$, and notes that the effective tensor violates the weak energy condition while the ordinary matter does not.

Load-bearing premise

The derivation assumes that the wormhole's ordinary matter obeys the linear relation $2P_l+P_r=\zeta\rho$, using it to rewrite the effective energy-momentum tensor, but it never checks whether the matter components computed later from the field equations still satisfy that same relation.

Editorial extensions

If this is right

Traversable wormholes can be embedded in an expanding cosmological background without negative-energy matter when the coupling $\beta$ and equation-of-state parameters lie in the allowed ranges.
The weak energy condition can be satisfied by the ordinary matter tensor even while the effective tensor that curves spacetime violates it, so energy-condition tests in $f(R,T)$ gravity must target the physical matter tensor.
For the effective equation of state $P_r^{\mathrm{eff}}=\alpha P_l^{\mathrm{eff}}$, the wormhole throat size is not fixed by the theory's parameters, allowing wormholes of arbitrary size in principle.
The explicit scale factors give concrete expanding or contracting wormhole cosmologies, with density and pressures decaying to zero at late times for $a_0>0$, so the wormhole becomes asymptotically indistinguishable from a standard cosmological background.

Reading between the lines

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

A direct consistency check of the derived matter components against the assumed equation of state is a natural extension; the parameter set used in the figures provides a concrete place to test it.
The solution procedure—recasting the theory as an effective fluid and separating variables through an effective equation of state—could carry over to other $f(R,T)$ forms or higher-order matter couplings, potentially yielding black-hole or stellar solutions by the same route.
Since the ordinary matter becomes isotropic and homogeneous at large radius, such wormhole models could be probed by looking for small local deviations from a cosmological background, for instance in galactic-halo or galaxy-cluster settings.
The split into allowed parameter ranges with $\beta<0$, $-15<\zeta<1$ versus $\beta>0$, $\zeta>1$ suggests the sign of the matter–geometry coupling determines which ordinary fluids can support a wormhole, a feature that might be tied to observable bounds on such theories.

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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

Read this one so you know the main result is not to be trusted: the paper claims exact dynamical wormhole solutions in f(R,T)=R+βT² gravity supported by ordinary matter satisfying the weak energy condition. That claim fails at a simple consistency check.

The genuinely new part is the construction itself. The authors adapt the solution-generating method from Rastall theory and from Bhattacharya–Bandyopadhyay to the trace-squared f(R,T) model, producing explicit scale-factor/shape-function pairs for two different effective EoS. The algebra is detailed, the classification is systematic, and the distinction between ordinary and effective energy conditions is drawn clearly. I'm not aware of these particular dynamical solutions elsewhere.

The problem is load-bearing and sits right at the center of the derivation. The paper assumes the ordinary matter obeys 2P_l+P_r=ζρ (Eq. 11), uses that to reduce the effective energy-momentum components (8)–(10) to the simpler (12)–(14), then solves the simplified system and reconstructs ρ, P_r, P_l via Eqs. (41)–(43). The reconstructed components are never tested against Eq. (11). For the paper's own Fig. 1 parameters (β=2, ζ=2, γ=1, ω=3, a0=a1=3, r0=2, κ=8π) at t=0, r=2, the left side of the EoS is about 2×10⁻³ while ζρ is about 6×10⁻². That's two orders of magnitude, not a rounding error. The same issue recurs in Case II with Eqs. (65)–(67). Since the simplified effective source was used to determine the metric, a metric whose reconstructed matter violates Eq. (11) is not a solution of the original f(R,T) field equations—it is a solution of a different system.

A secondary concern: the WEC-satisfying parameter ranges are obtained by imposing the inequalities on the derived expressions, so the existence claim is partially baked into the parameter choice. That would be acceptable if the underlying equations were consistent, but they are not.

The paper is not a silly paper; it is a serious technical attempt with visible effort. But the central claim does not survive contact with the equations, and the missing consistency check is exactly the kind of thing a careful referee should catch. As it stands, I would reject. If the authors can demonstrate a parameter set where Eq. (11) holds for the reconstructed components, or recast the analysis without the simplifying EoS, it could be worth a fresh look.

Referee Report

2 major / 5 minor

Summary. The manuscript studies dynamical Morris–Thorne wormholes in the modified gravity theory f(R,T)=R+βT^2. The strategy is to rewrite the field equations in terms of an effective energy-momentum tensor, assume that the ordinary anisotropic fluid satisfies the linear equation of state 2P_l+P_r=ζρ, solve the effective field equations for the scale factor and shape function under two different effective equations of state, and then algebraically reconstruct the ordinary fluid components ρ, P_r, and P_l. The authors claim that these reconstructed components satisfy the weak energy condition, so that wormholes can be supported by ordinary matter, in contrast to general relativity. The paper provides explicit solutions, parameter restrictions, and several figures illustrating the WEC behavior.

Significance. If correct, the paper would provide explicit time-dependent wormhole solutions in a specific modified gravity theory whose ordinary matter content satisfies the weak energy condition, and the solution method could plausibly be adapted to other matter-coupled gravity theories. The manuscript is systematic, the parameter constraints are stated explicitly, and the presentation of the exact expressions is a useful feature. However, the central result is undermined by a consistency gap in the reconstruction of the ordinary matter components, as detailed below. Because the claimed solutions do not satisfy the original field equations, the significance of the paper as written cannot be assessed as claimed.

major comments (2)

[§IV.C, Eqs. (41)–(43) and Eq. (11)] The ordinary matter equation of state (11) is an input used to simplify the effective energy-momentum tensor to the forms (12)–(14), but the ordinary components reconstructed in Eqs. (41)–(43) are never checked against this equation of state. For the paper's own Fig. 1 parameters (β=2, ζ=2, γ=1, ω=3, a0=a1=3, r0=2, κ=8π), at t=0 and r=r0, direct substitution of (26)–(27) into (41)–(43) gives ρ≈0.0296, P_r≈5.8×10^{-5}, and P_l≈1.02×10^{-3}. Hence 2P_l+P_r≈2.1×10^{-3}, whereas ζρ≈5.93×10^{-2}; the assumed EoS (11) is violated by more than an order of magnitude. Consequently, the simplified effective components (12)–(14) are not the actual effective source for the metric defined by (15) and (26)–(27), and the metric does not satisfy the original f(R,T) field equations (6)–(7). This invalidates the central claim that these are wormhole solutions supported by ordinary matter satisfying the WEC.
[§V.B, Eqs. (65)–(67)] The same missing consistency check occurs in Case II. The ordinary components (65)–(67) are obtained by inverting the simplified relations (12)–(14), which were derived under the ordinary EoS (11), but no condition is imposed to ensure that the reconstructed P_r and P_l satisfy 2P_l+P_r=ζρ. The algebraic inversion of (12)–(14) does not automatically preserve (11), so the Case II solutions inherit the same defect. At minimum, the authors would need to impose (11) as an additional constraint on the final ordinary components and verify that the displayed parameter ranges admit solutions compatible with it.

minor comments (5)

[Introduction] The phrase 'to the eader’s attention' should read 'to the reader’s attention'; there are several similar typographical slips throughout the manuscript (e.g., 'for wich' and 'gnerally').
[§IV.B, Eq. (39)] The sentence 'Note that R1 is an integration constant' should refer to a1, not R1.
[Eq. (5)] The matter Lagrangian is set to L_m=P=(P_r+2P_l)/3; this is a standard choice in the f(R,T) literature, but it should be stated more explicitly as an assumption, since the final equations depend on it.
[Table I] Table I is very dense and hard to parse because of the nested inequalities and exponents; reformatting it with clearer brackets or separate entries per parameter regime would improve readability.
[Figures 2 and 7] The captions of Figs. 2 and 7 state that the plots show behavior for 0<t<∞, but the horizontal axis is finite; the time range used in the plots should be stated precisely.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is a self-contained ansatz-based construction; the EoS-consistency gap found in the solutions is a correctness issue, not a circular reduction.

full rationale

The paper's central chain is: assume f(R,T)=R+βT^2, take an anisotropic fluid (4), impose the linear EoS (11) to simplify the effective tensor (8)-(10) into (12)-(14), impose one of two effective EoSs (22)/(50), separate variables to solve for B(r) and a(t), then algebraically recover ordinary-matter components (41)-(43)/(65)-(67) and check WEC inequalities. Each step is a standard construction from explicit assumptions. The WEC parameter ranges are obtained by imposing ρ≥0, ρ+P_r≥0, and ρ+P_l≥0 on the derived expressions; this is an existence-region calculation, not a fit of data or a prediction forced by a self-citation. Self-citations to [68] are used as a methodological antecedent and as a comparison with Rastall theory, but the ODEs and inequalities are re-derived in the present paper, so no load-bearing claim rests on unverified prior work. A serious internal-consistency problem does exist: the recovered components are never checked against the assumed EoS (11), and direct substitution for the Fig. 1 parameters gives 2P_l+P_r≈2.1×10^-3 while ζρ≈5.9×10^-2, so the simplified effective components (12)-(14) are not the actual effective source for the derived metric. This threatens the correctness of the exact-solution claim, but it is an algebraic consistency failure rather than a circular reduction: the EoS is an input assumption, not a predicted output, and the WEC statement is not equivalent to that input. The admitted restriction of the Case II WEC analysis to C=0 (Sec. V) is an omitted case, not a circular step. Therefore the circularity score is 0.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The construction relies on several hand-chosen parameters and two EoS closure relations; the ordinary EoS (11) is load-bearing but not verified against the final matter solutions, which is the main source of incorrectness.

free parameters (8)

β = 2 (figures); β<0 or β>0 ranges
Coupling constant of the T^2 term; chosen by hand to make square roots real and WEC inequalities hold.
ζ = 2 (figures); ζ>1 or -15<ζ<1 ranges
EoS parameter for ordinary matter; selected to satisfy WEC.
ω = 3 (Fig. 1); ω<-3, -3<ω<-2, -2≤ω<-1, ω>0 ranges
EoS parameter in Case I effective EoS; chosen from WEC-compatible ranges.
γ = 1 (Fig. 1)
EoS parameter in Case I; selected from ranges satisfying WEC and flatness.
a0, a1 = a0=3, a1=3 (Fig. 1); or a1<0 in Case II
Integration constants in scale factors; restricted by WEC constraints.
r0 = 2 (Fig. 1), 0.2-2 in Case II
Throat radius; lower bounds imposed by WEC inequalities.
α, n, ε = α=-3, n=-0.2, ε=-2 (figures)
Shape function parameters in Case II; chosen to satisfy flaring-out and WEC.
C = 0 for WEC analysis
Separation constant; C≠0 solutions excluded because WEC fails in f(R,T).

assumptions (6)

domain assumption f(R,T) gravity field equations of Harko et al. with Lm=P
The theory and matter Lagrangian are taken from Ref. [20] without independent derivation.
ad hoc to paper Ordinary matter EoS: 2P_l+P_r=ζρ (Eq. 11)
This linear EoS is assumed to simplify the effective energy-momentum tensor; it is never verified for the derived solutions.
ad hoc to paper Effective EoS: ρ_eff=ω/(1+2γ)(P_r^eff+2γP_l^eff) (Eq. 22) or P_r^eff=αP_l^eff (Eq. 50)
Closure relations chosen to make the field equations separable; many parameter choices are possible.
ad hoc to paper Separation constant C=0 restriction for WEC analysis
The authors exclude C≠0 because those solutions fail the WEC in f(R,T), limiting the classification.
domain assumption Redshift function φ'=0 from Eq. (19) and absorbing φ(t) into time
Standard for this metric ansatz; requires no horizons.
standard math Flaring-out and asymptotic flatness conditions for the shape function
Morris-Thorne wormhole conditions; applied as constraints on parameters.

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Cite this review

Pith. "Pith review of Dynamical Wormhole Solutions in $f(R, T)$ Gravity." pith.science (2026). https://pith.science/paper/6ERATVRE

@misc{pith2026250417910,
  author       = {Pith},
  title        = {Pith review of: Dynamical Wormhole Solutions in $f(R, T)$ Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6ERATVRE}},
  note         = {Machine review of arXiv:2504.17910}
}

abstract

A class of $f(R, T)$ theories extends the Einstein-Hilbert action by incorporating a general function of $R$ and $T$, the Ricci scalar and the trace of the ordinary energy-momentum tensor $T_{\mu\nu}$, respectively, thereby introducing a specific modification to the Einstein's field equations based on matter fields. Given that this modification is intrinsically tied to an energy-momentum tensor $T_{\mu\nu}$ that a priori respects energy conditions, we explore the potential of $f(R, T)$ theories admitting wormhole configurations satisfying energy conditions, unlike General Relativity, which typically necessitates exotic matter sources. Consequently, we investigate the existence of dynamical wormhole geometries that either uphold energy conditions or minimize their violations within the framework of trace of energy-momentum squared gravity. To ensure the generality of our study, we consider two distinct equations of state for the matter content and systematically classify possible solutions based on constraints related to the wormhole's throat size, the coupling parameter of the theory, and the equation of state parameters.

Figures

Figures reproduced from arXiv: 2504.17910 by the authors.

**Figure 2.** FIG. 2. This figure indicates the behavior of [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** FIG. 3. The behavior of [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4. The behavior of [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

**Figure 5.** Figure 5: FIG. 5. The behavior of [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

**Figure 6.** Figure 6: FIG. 6. The behavior of [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

**Figure 7.** Figure 7: FIG. 7. The behavior of [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]

Discussion (0). Continue with ORCID to comment.

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This also helps to study how wormholes might behave or survive in an evolving spacetime, specially if they were formed in the early universe

Cosmological Context: Embedding wormhole solutions within a cosmological background allows one to explore their evolution in time, potentially making them more realistic in the context of the expanding universe. This also helps to study how wormholes might behave or survive in an evolving spacetime, specially if they were formed in the early universe
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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

This also helps to study how wormholes might behave or survive in an evolving spacetime, specially if they were formed in the early universe

Cosmological Context: Embedding wormhole solutions within a cosmological background allows one to explore their evolution in time, potentially making them more realistic in the context of the expanding universe. This also helps to study how wormholes might behave or survive in an evolving spacetime, specially if they were formed in the early universe

[2] [2]

This extra interplay of the geometry and matter content leads to dynamic matter distributions and, consequently, time-dependent geometries

Modified Gravity Framework: In f(R,T ) gravity, the coupling between matter and curvature naturally allows for non-conservation of the energy-momentum tensor. This extra interplay of the geometry and matter content leads to dynamic matter distributions and, consequently, time-dependent geometries. Thus, the structure of the theory itself supports and just...

[3] [3]

For instance, a time-varying pressure could stem from cosmological evolution or particle production effects in f(R,T ) theory

Choice of Stress-Energy tensor: If the matter source is modeled as a time-dependent perfect fluid or an anisotropic fluid with ρ(t,r ) and p(t,r ), the field equations naturally admit dynamical solutions. For instance, a time-varying pressure could stem from cosmological evolution or particle production effects in f(R,T ) theory

[4] [4]

Phenomenological Interest: Time-dependent wormholes are of considerable interest in the literature, as they may exhibit rich physical phenomena such as horizon formation/disappearance, energy condition violation/evolution, and interaction with cosmological fluids

[5] [5]

on the other side

Rahaman et al. recently proposed that galactic halos may exhibit properties conducive to supporting traversable wormholes [97], prompting subsequent investigations into their potential detectability [98–100]. If such galactic 3 wormholes do exist, it is reasonable to assume they were formed in the early Universe, thereby raising important questions regard...

[6] [6]

f(R,T ) =f1(R) +f2(T ),

[7] [7]

Key features of f(R,T ) theory can be noted as follows

f(R,T ) =f1(R) +f2(R)f3(T ), where α is an arbitrary constant and f1,f 2,f 3 are arbitrary functions of their variables [20]. Key features of f(R,T ) theory can be noted as follows. • While GR and f(R) gravity both feature minimal coupling between geometry and matter, the f(R,T ) theory generally introduces a non-minimal matter-geometry coupling directly ...

[8] [8]

(34) Integrating the above equation (34) yields the following first-order nonlinear differential equation ˙a2(t) = C −(3 +ω) 1−a0a− 3+ω ω , (35) where a0 is an integration constant

Solution for the scale factor a(t) The equation (25) can be rewritten in a different but equivalent form as follows: −2ωa(t)¨a(t)− (3 +ω)˙a2 =C. (34) Integrating the above equation (34) yields the following first-order nonlinear differential equation ˙a2(t) = C −(3 +ω) 1−a0a− 3+ω ω , (35) where a0 is an integration constant. Consequently, the next integra...