Impact of Non-metricity and Matter Source on the Geometry of Anisotropic Spheres

Adeeba Arooj; Baiju Dayanandan; M. Sharif; M. Zeeshan Gul

REVIEW 4 major objections 6 minor 111 references

Impact of Non-metricity and Matter Source on the Geometry of Anisotropic Spheres

T0 review · 4 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper argues that four known compact stars can be modeled as stable anisotropic spheres in f(Q,T) gravity, with matter that satisfies every energy condition and stability test.

desk verdict A routine f(Q,T) compact-star paper whose explicit field equations are algebraically impossible, so the central stability claim is unverified. read the letter →

arxiv 2504.21348 v1 pith:7LM3DQEN submitted 2025-04-30 gr-qc

M. Zeeshan Gul , M. Sharif , Adeeba Arooj , Baiju Dayanandan This is my paper · ORCID

classification gr-qc PACS 97.10.Cv04.50.Kd97.60.Jd04.20.Jb

keywords compactstarsanisotropicfluidspheresf(QT)gravitynon-metricityenergyconditionssoundspeedstabilityadiabaticindexstellarequilibrium

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 establish that four observed compact stars—Vela X-1, 4U 1608-52, PSR J1903+327, and PSR J1614-2230—can be described as anisotropic fluid spheres in $f(Q,T)$ gravity, an extended theory in which gravity is carried by non-metricity and coupled to the trace of the energy-momentum tensor. Using a linear model $f(Q,T)=hQ+kT$ and a prescribed metric ansatz, the authors derive explicit density and pressure profiles, fix the free constants by matching the interior metric to the exterior spacetime at the stellar surface, and then check the standard physical criteria. They report that the energy conditions hold, the forces in the stellar equilibrium condition balance, the sound speeds are causal, and the adiabatic index exceeds the stability bound, so the stars are physically viable and stable. A sympathetic reader would care because this is evidence that a non-metricity-based modification of general relativity can accommodate realistic neutron-star masses and radii without exotic matter.

What carries the argument

The central object is the linear model $f(Q,T)=hQ+kT$, where $Q$ is non-metricity (the failure of the connection to preserve the metric tensor) and $T$ is the trace of the energy-momentum tensor. From the variational field equations, the paper writes explicit density and pressure formulas in terms of the metric functions $\xi(r)$ and $\eta(r)$; substituting the ansatz turns those formulas into closed expressions for $\rho$, $p_r$, and $p_t$. Surface matching conditions fix the three constants from each star's observed mass and radius, and the resulting profiles are then fed into the energy conditions, equilibrium, sound-speed, and adiabatic-index tests. The load-bearing device is therefore the linear $f(Q,T)$ model together with this specific metric ansatz: it is what converts a complicated modified-gravity system into testable stellar profiles.

What would settle it

Re-derive the explicit density and pressure formulas from the general field equations by direct substitution of $f=hQ+kT$; the printed equations give density a denominator $2h^2+k-1$ while both pressures carry $2k^2+k-1$, so a single correct substitution should settle whether the system is consistent.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the chosen metric ansatz with the constants fixed by the junction conditions produces anisotropic compact-star solutions in $f(Q,T)$ gravity whose matter variables are regular and decreasing, satisfy the null, weak, strong, and dominant energy conditions, obey the equilibrium condition, and lie within the causality and adiabatic-index stability windows. The density and both pressures peak at the center and fall toward the boundary; tangential pressure exceeds radial pressure, so the anisotropy is positive and repulsive. The mass function rises monotonically, and the compactness and surface redshift stay below the standard bounds. The conclusion is stated as unconditional: the proposed compact stars in the $f(Q,T)$ framework are physically viable and stable.

Load-bearing premise

The entire analysis rests on the unshown algebra that turns the general $f(Q,T)$ field equations into the explicit density and pressure formulas; if those formulas are wrong, every graph and stability conclusion built on them fails.

Editorial extensions

If this is right

Vela X-1, 4U 1608-52, PSR J1903+327, and PSR J1614-2230 can each be fitted by the same two-function ansatz with constants determined from their observed mass and radius.
The matter in these fits is ordinary: all four energy conditions hold throughout the interior.
The configurations are in hydrostatic equilibrium: the gravitational, hydrostatic, and anisotropic forces sum to zero at every radius.
The models are causally stable and resist radial collapse: sound speeds lie in $[0,1]$, their difference obeys the cracking condition, and the adiabatic index stays above $4/3$.
Compactness and surface redshift respect the standard bounds, so the stars are neither too compact nor produce unphysically large redshift.

Reading between the lines

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

If the field equations are correct, the linear $f(Q,T)$ model with the chosen parameters becomes a tunable extension of general relativity; scanning $h$ and $k$ against a larger catalog of neutron stars could map which modifications are compatible with observation.
The same ansatz could be inverted to extract an approximate equation of state $\rho(p_r)$ from each fitted star, something the paper does not do; a derived equation of state would connect the model to nuclear-matter predictions.
Treating $k$ as a free parameter rather than fixing it ahead of the plots would test how strongly the viability conclusion depends on the matter coupling.

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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

Here's the bottom line: this is a by-the-numbers f(Q,T) compact-star paper, and the central claim—that the four stars are viable and stable—does not survive contact with the field equations.

What the paper does well: it follows the standard recipe completely. It writes down the f(Q,T) action, chooses the linear model f = hQ + kT, uses a Tolman ansatz, matches to the Schwarzschild exterior, and runs the usual battery of checks: energy conditions, TOV equilibrium, causality via sound speeds, adiabatic index. The figures are clean and the qualitative behavior is plausible. If the equations were right, this would be a serviceable application of the method.

The problem is that the equations are not right. Equations (35)-(37) are presented as the explicit field equations for f = hQ + kT, but they cannot follow from the generic system (31)-(33). With f_QQ = 0 and f_Q = h, the system is linear in ρ, p_r, p_t, and the only fluid coupling is through kT and f_T terms, so h appears only in the geometric source terms. It cannot appear in any denominator. The denominator 2h^2 + k - 1 in (35) is therefore impossible, and the different denominator 2k^2 + k - 1 in (36)-(37) is a red flag that something was copied or typed incorrectly. No derivation is given, and every figure and conclusion depends on these formulas. On top of that, the printed EoS parameters ω_r and ω_t are identical, and the adiabatic indices Γ_r and Γ_t are identical, even though the model is explicitly anisotropic and the paper claims p_t > p_r. That is a copy-paste error, not a physical result. The matching condition has a factor-of-2 typo (derivative of the exterior metric should be 2m/R^2, not m/R^2), though the constants they solve for are actually consistent with the correct condition.

None of this is rescued by the parameter choice: h=2, k=3 are chosen by hand, and no uncertainty or sensitivity analysis is given. The stability analysis is applied to the same constructed solution, so it is not an independent prediction.

For whom is this paper? Someone collecting yet another example in the f(Q,T) stellar catalogue might find it useful, but only after the algebra is corrected. As submitted, the load-bearing derivation is broken, so I would desk-reject rather than send it to referees. The authors can redo the algebra and resubmit.

Referee Report

4 major / 6 minor

Summary. The manuscript constructs anisotropic compact star models in f(Q,T)=hQ+kT extended symmetric teleparallel gravity. Starting from the generic field equations (31)-(33), the authors adopt the linear model (34), the metric ansatz (38), fix the constants a,b,c by matching to the observed masses and radii of four compact stars (Table 1), and then examine energy conditions, EoS parameters, TOV equilibrium, causality, and adiabatic stability (Sections 3-4). The conclusion is that all four proposed stars are physically viable and stable. The paper also includes appendices deriving the non-metricity scalar Q and its variation.

Significance. If the explicit field equations (35)-(37) were correct, the paper would be a standard application of the f(Q,T) framework to neutron-star-like objects, one of many such ansatz-based studies. The use of four observational data sets and the explicit appendices for Q are useful. However, the central technical step—the reduction of the generic field equations to the explicit expressions—is algebraically inconsistent, and the EoS and adiabatic-index formulas are identical for the radial and tangential components despite the claimed anisotropy. As a result, the plots and the conclusion of viability and stability are not supported. The manuscript therefore does not meet the standard for publication in its present form.

major comments (4)

[Section 2, Eqs. (35)-(37)] The explicit field equations cannot be obtained from the generic system (31)-(33) with f(Q,T)=hQ+kT. Substituting f_Q=h, f_T=k, f_QQ=0 into (31)-(33) gives a system that is linear in (rho,p_r,p_t), with the fluid-coupling matrix depending only on k; for the signs shown, the determinant is (1+k)^2, so the denominators of the solved expressions can contain only k (e.g., factors (1+k)^2), not h or h^2. The displayed denominator 2h^2+k-1 in (35) and 2k^2+k-1 in (36)-(37) is therefore algebraically impossible, and the manuscript gives no derivation of these equations. Because (35)-(37) are the basis for Figures 2-10 and all subsequent physical conclusions, the central claim is unsupported.
[Section 3.3 and Section 4.2] The radial and tangential EoS parameters omega_r and omega_t are printed with identical expressions, and the adiabatic indices Gamma_r and Gamma_t in (46)-(47) are also identical. This is inconsistent with the anisotropic matter model (30) and with the reported finding p_t > p_r. Either the anisotropic reduction is incorrect, or the printed stability formulas are wrong; in either case the causality and adiabatic-index stability analysis (Figures 9-10) and the anisotropy-based TOV analysis (Figure 8) are not valid.
[Section 3.5] The TOV equilibrium condition is not stated as an equation: the display reads 'MG(r)/r^2 (rho+pr)e^{(xi-eta)/2} + p'_r - 2Delta/r ,' with no '=0', and the text then refers to 'Equation (??)'. The missing equation and broken cross-reference prevent verification of the force-balance calculation. This is a load-bearing part of the equilibrium claim and must be corrected.
[Section 2 (Table 1, Eq. (40)) and Section 5] The constants a,b,c are fixed by matching to the observed mass and radius of each star, and the subsequent checks are performed on the same fitted solution. Describing these checks as 'predictions' (abstract, Section 5) overstates their status; they are internal consistency tests of the ansatz. The conclusions should be rephrased accordingly.

minor comments (6)

[Section 2] The line 'The variation of Eq.(45) yields' refers to an equation number that does not exist; the intended reference is likely Eq. (21).
[Section 3.5] The TOV expression is missing a right-hand side ('=0') and the cross-reference 'Equation (??)' is broken.
[Section 4.1] The sentence 'the difference between ust and ust' should read 'the difference |u_t^2 - u_r^2|' or similar.
[Throughout] The notation f(Q,T) and f (Q, T) is used inconsistently; please standardize.
[Figure 1] The colors mentioned in the text (gray, pink, green, brown) are not visible in a grayscale rendering; the caption should identify the curves explicitly.
[References] The bibliographic details of entries [66] and [67] appear to overlap (both Eur. Phys. J. C 83 (2023) 1088); please verify these references.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ansatz and matching produce a solution whose viability criteria are evaluated inequalities, not quantities forced by the inputs.

full rationale

The paper's workflow is an exact-solution consistency analysis common in modified-gravity astrophysics: choose f(Q,T)=hQ+kT and the Tolman-type metric ansatz (38), fix the constants (a,b,c) via Darmois matching to the observed mass and radius of four compact stars, then compute the fluid variables and test standard physical criteria. None of the paper's central viability claims reduces to a fitted input by construction. The energy conditions, causality bounds, Herrera cracking condition, and adiabatic-index inequality are genuine inequalities evaluated on the constructed solution; they can fail for arbitrary choices of the model parameters or metric ansatz, so their satisfaction is not a renaming of the input. The matching constants are indeed fixed by the observed M and R, and the resulting mass function and compactness at the boundary reflect those inputs, but the paper does not present these as independent predictions; they are consistency checks, and the interior behavior used in the plots is not determined by the boundary data alone. The TOV check in Section 3.5 is the closest point to inspect, since equilibrium equations can be identities for exact solutions of covariant field equations; however, the paper uses the standard TOV combination without deriving it from the f(Q,T) field equations, and whether that combination is automatic in this theory is a correctness question rather than a definitional circularity. The larger concern is algebraic rather than circular: Eqs. (35)-(37) are stated with no derivation from (31)-(34), and the denominators '2h^2+k-1' and '2k^2+k-1' appear inconsistent for f=hQ+kT, since f_QQ=0 leaves h linear and the 3x3 solve would give denominators depending on k only. That is a serious verification risk, but a suspected algebra error is not evidence that a result is equivalent to its inputs. The paper also cites prior work by the same authors extensively, but those citations are background and comparison material, not load-bearing justifications for the ansatz or the field equations; the model and metric are attributed to external references (Xu et al., Tolman, Jimenez et al.). Accordingly, no circular step can be exhibited with the required specificity, and the appropriate finding is no significant circularity.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central claim depends on the hand-picked model parameters h=2 and k=3, the chosen matter Lagrangian, and the Tolman ansatz. No new entities are introduced. The parameters are set a priori; the paper does not scan or justify them. The constants a,b,c are fixed by the junction conditions from observed masses and radii, so they are not free parameters in the usual sense.

free parameters (2)

h = 2
Coefficient of Q in f(Q,T)=hQ+kT. Set to 2 with no derivation or observational fit; all plots use this value (Section 2, after Eq. (34)).
k = 3
Coefficient of T in f(Q,T)=hQ+kT. Set to 3 without justification; controls the coupling to matter trace and hence all density/pressure profiles (Section 2, after Eq. (34)).

assumptions (3)

domain assumption The interior of the star is a static, spherically symmetric, anisotropic perfect fluid with zero heat flux and the exterior is Schwarzschild (Eqs. (29), (39)).
Standard compact star modeling assumptions; the junction conditions then fix the metric constants.
ad hoc to paper The matter Lagrangian is L_m = -(p_r + 2p_t)/3 (stated in Section 2 before Eq. (31)).
This particular choice is needed to close the f(Q,T) field equations and is not derived from microphysics.
ad hoc to paper The metric ansatz (38), e^xi = a(1+r^2/b) and e^eta = (2r^2/b+1)/((r^2/b+1)(1-r^2/c)), is a valid interior geometry.
The ansatz is taken from Tolman (ref [95]) and not derived from the field equations; it is the backbone of the model.

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Cite this review

Pith. "Pith review of Impact of Non-metricity and Matter Source on the Geometry of Anisotropic Spheres." pith.science (2026). https://pith.science/paper/7LM3DQEN

@misc{pith2026250421348,
  author       = {Pith},
  title        = {Pith review of: Impact of Non-metricity and Matter Source on the Geometry of Anisotropic Spheres},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7LM3DQEN}},
  note         = {Machine review of arXiv:2504.21348}
}

abstract

This paper delves into the impact of extended symmetric teleparallel theory on anisotropic compact stellar structures. The explicit field equations were formulated by considering a minimum model of this extended gravity. Basically, the Darmois junction conditions are used to determine the unknown constants of metric coefficients. We explore some significant properties of the compact stars under consideration to check their viable existence in this modified framework. The Tolman-Oppenheimer-Volkoff equation assess the equilibrium state of the compact stars. Moreover, the stability analysis is defined by using methods based on sound speed (related to how disturbances propagate in the star) and adiabatic index (related to the thermodynamic behavior of the star). We find that the proposed compact stars in the $f(\mathbb{Q}, \mathbb{T})$ gravity are physically viable and stable.

Figures

Figures reproduced from arXiv: 2504.21348 by the authors.

**Figure 2.** Evolution of fluid parameters. 3.1 Behavior of Matter Contents The study of matter contents is significant to comprehend the inner characteristics of CSOs. These material factors are expected to reach their peak at a core because of their high density, countering gravity and preserving the CSOs from collapse. Figures 2 and 3 show the graphical representation of matter contents and their rates of change for each CSO… view at source ↗

**Figure 3.** Evolution of gradient of fluid parameters. [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Graphs of energy bounds for different compact stars. [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

**Figure 5.** Figure 5: Graphs of EoS parameter for different compact stars. [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]

**Figure 6.** Figure 6: Plot of mass function. 0 2 4 6 8 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 r u 0 2 4 6 8 0.00 0.05 0.10 0.15 0.20 r Z [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

**Figure 7.** Figure 7: Graph of compactness and redshift functions. [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]

**Figure 8.** Figure 8: Plot of TOV equation. 3.5 Analysis of Different Forces The TOV equation refers an equilibrium state of static spherical symmetric objects [102], defined as MG(r) r 2 (̺ + pr)e ξ−η 2 + p ′ r − 2∆ r , (43) where MG(r) = 4π Z (T 0 0 − T 1 1 − T 2 2 − T 3 3 )r 2 e ξ+η 2 dr…

**Figure 9.** Figure 9: Plots of causality condition. employ the sound speed method and adiabatic index, offering valuable insights into the structural soundness and resilience of celestial objects. 4.1 Sound Speed To analyze the stability of CSOs, the causality principle can be considered w…

**Figure 10.** Figure 10: Stability analysis by adiabatic index. Using Eqs.(35)-(37), the above equation becomes Γr = − 2(b + 2c)(b + r 2 )(2k − 1)(b + 2r 2 − 2bk) ((b + 2r 2 )(b − c + 3r 2 ) + (b(2b + c) + 6br2 + 6r 4 )k)(5b(2k − 1) + 2r 2 (4k − 1))−1 , (46) Γt = − 2(b + 2c)(b + r 2 )(…

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