Exploring the viability of charged Spheres admitting non-metricity and matter source

Faisal Javed; M. Sharif; M. Zeeshan Gul; Shalan Alkarni

REVIEW 4 major objections 8 minor 106 references

Exploring the viability of charged Spheres admitting non-metricity and matter source

T0 review · 4 major / 8 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper claims that charged, anisotropic pulsar interiors built from a Tolman-type metric ansatz in linear f(Q,T) gravity are physically viable and stable, passing every standard stability and viability test.

desk verdict Routine f(Q,T) compact-star application with a load-bearing matching inconsistency: the tabulated constants cannot satisfy the stated Darmois conditions, so the viability plots do not describe matched pulsar models. read the letter →

arxiv 2412.01411 v1 pith:LS4WT6CR submitted 2024-12-02 gr-qc

M. Zeeshan Gul , Faisal Javed , M. Sharif , Shalan Alkarni This is my paper · ORCID

classification gr-qc PACS 04.50.Kd97.10.Cv97.60.Jd04.20.Jb

keywords f(QT)gravitynon-metricityextendedsymmetricteleparallelchargedanisotropiccompactstarspulsarcandidatesTolman-typemetricansatzReissner-Nordstrommatchingstellarstabilitycriteria

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 sets out to show that charged, anisotropic compact stars — modeled on nine pulsars with observed masses and radii — remain legitimate solutions in $f(Q,T)$ gravity, a modified theory in which gravity is encoded in the non-metricity of spacetime and coupled to the trace of the matter energy-momentum tensor. Working with the linear model $f(Q,T)=\mu Q+\nu T$, a Tolman-type interior metric (Eqs. (19)–(20)), and a Reissner-Nordstrom exterior joined by Darmois junction conditions, the authors reconstruct the density and pressures inside each star. They then run the standard viability battery: energy conditions, equation-of-state parameters, TOV force balance, causality of sound speeds, Herrera cracking, compactness and redshift bounds, and the adiabatic index. Every candidate passes every test, and the paper concludes that the charged spheres in this framework are physically viable and stable. The point, if right, is that non-metricity and matter-coupling terms need not spoil stellar structure; the modified theory accommodates realistic compact objects.

What carries the argument

The load-bearing object is the Tolman-type interior metric of Eqs. (19)–(20), with $\xi(r)=\ln[a(1+r^2/b)]$ and $\eta(r)=\ln[(1+2r^2/b)((1+r^2/b)(1-r^2/c))^{-1}]$, combined with the linear model $f(Q,T)=\mu Q+\nu T$ and the charge profile $q(r)=Q(r/R)^3$. Darmois junction conditions — continuity of the metric and its first derivatives across the stellar surface — fix $a$, $b$, $c$ in terms of each pulsar's observed mass $M$, radius $R$, and charge parameter $Q$, tying the interior geometry to the Reissner-Nordstrom exterior. Substituting the ansatz into the $f(Q,T)$-Maxwell field equations yields explicit closed forms for the density, radial pressure, and tangential pressure (Eqs. (25)–(27)); every subsequent check — energy conditions, TOV forces, sound speeds, cracking, adiabatic index — is computed from these three functions, so the whole viability argument rests on this ansatz-and-matching procedure.

What would settle it

Compare the density–pressure relation reconstructed from Eqs. (25)–(27) with an independently measured interior profile of any candidate — for instance from X-ray pulse-profile modeling of PSR J1614-2230 or from the tidal deformability encoded in gravitational-wave signals of a neutron-star merger. If the observed interior of even one star deviates from the Tolman-type geometry, or requires a charge profile different from $q(r)=Q(r/R)^3$, then that star's matched constants and stability conclusion no longer follow, and the blanket claim of viability would need to be qualified.

Watch

Extended reading notes

Core claim

The central discovery claimed is that non-metricity and matter-trace coupling in the action do not undermine stellar viability: for $f(Q,T)=\mu Q+\nu T$, the field equations together with the Tolman-type ansatz $\xi(r)=\ln[a(1+r^2/b)]$ and $\eta(r)=\ln[(1+2r^2/b)((1+r^2/b)(1-r^2/c))^{-1}]$ produce regular, monotone-decreasing density and pressure profiles whose radial pressure vanishes at the boundary and whose anisotropy is positive throughout. With the constants $a$, $b$, $c$ fixed by matching mass and radius to the Reissner-Nordstrom exterior for nine pulsar candidates, and with the charge profile $q(r)=Q(r/R)^3$, the reconstructed interiors satisfy the null, weak, strong, and dominant energy conditions, keep both equation-of-state parameters inside $(0,1)$, obey the TOV equilibrium equation with vanishing net force, keep both sound speeds in the causal range, satisfy the Herrera cracking condition, respect the Buchdahl and surface-redshift bounds, and meet the adiabatic-index stability condition. The paper's conclusion is therefore that charged anisotropic spheres are physically viable and stable in this modified framework.

Load-bearing premise

The load-bearing premise is that the Tolman-type interior metric of Eqs. (19)–(20) is the actual geometry of these pulsars: it is assumed as an ansatz rather than derived from a microphysical equation of state, and every reconstructed density, pressure, and stability result collapses if a real star's interior differs from it.

Editorial extensions

If this is right

If the conclusion holds, $f(Q,T)$ gravity with the linear action admits charged, anisotropic stellar interiors that are regular at the center and matched to Reissner-Nordstrom exteriors, so the theory is not excluded by the existence of compact stars.
The same construction succeeds for all nine pulsar candidates across a wide range of masses (from 0.9 to 1.97 solar masses) and radii, suggesting the viability is not tuned to a single object.
All energy bounds hold with the modification terms active, so the reconstructed matter is compatible with ordinary, nonexotic fluids supporting these stars.
The force balance shown in the TOV analysis has the anisotropic force offsetting the hydrostatic gradient against gravity, which the paper gives as the reason the configurations remain in equilibrium rather than collapsing.

Reading between the lines

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

A reader checking Section IV.C should note that the paper states $\Gamma<4/3$ as the stable side of the adiabatic-index condition, whereas the standard criterion takes $\Gamma>4/3$ as stability; the plotted indices exceed $4/3$ in both components, so whether the same graphs read as stable depends on which direction of the criterion is intended.
The matched constant $c$ is negative for three of the nine candidates (SAX J1808.4-3658, 4U 1820-30, SMC X-4 in Table I), a feature the paper does not discuss; since $c$ enters the metric function $\eta$, checking whether negative $c$ alters the causal structure or stability window for those three stars would tighten the claim.
The construction treats the charge profile $q(r)=Q(r/R)^3$ and the couplings $\mu$, $\nu$ as free inputs; mapping the region of the $\mu$–$\nu$–$Q$ parameter space where viability holds would show how much of the result is structural rather than tuned.
The same ansatz-and-matching machinery transfers to any future mass–radius measurement with tighter errors, so the framework yields concrete predictions for where the next compact-star observation should fall.

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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

Bottom line: the central claim doesn't hold up. The Darmois matching in Section III.A is internally inconsistent with Table I. For EXO 1785-248, imposing the derivative junction condition forces Q^2 = MR - aR^4/b ≈ -8.97 km^2, i.e. imaginary charge, and solving all three junction conditions simultaneously gives c = R^3/M ≈ 536 km, while the table lists c = 7687.56. The printed formulas (22)-(24) mix the uncharged derivative condition a = 1 - 3M/R with Q-dependent expressions for b and c. So the interior metric (19)-(20) with the tabulated constants is not matched to any Reissner-Nordstrom exterior. That means the density, pressure, energy-condition, TOV, and stability plots are not profiles of the observed pulsars; a surface layer or thin shell would be needed, and the paper doesn't include one.

What the paper does well: it is clearly organized and runs the standard battery of checks: energy conditions, TOV equilibrium, causality, Herrera cracking, and adiabatic index. The algebra of the field equations is executed mechanically, and the observational masses and radii are taken from the literature.

Soft spots: the adiabatic index section states the stability threshold backwards (says Gamma < 4/3 stable, Gamma > 4/3 unstable; the references they cite say the opposite), even though Figure 12 shows Gamma > 4/3 and the text concludes stable—so the conclusion is accidentally right while the stated criterion is wrong. The theory parameters mu and nu and the total charge Q are never specified, so no figure is reproducible. The charge profile q(r) = Q(r/R)^3 is posited without justification. The construction is an incremental re-run of a template already published by the same groups (refs 55-59, 67-76, 84), with no independent prediction and no way to discriminate f(Q,T) from GR.

Who it's for: readers who want another existence example in the f(Q,T) compact-star literature. As written, the viability claim fails because the matching is broken. It deserves a serious referee, because the matching inconsistency is exactly what referee time should catch; but my verdict would be reject unless the matching is redone and the parameters are supplied.

Referee Report

4 major / 8 minor

Summary. The manuscript studies static, spherically symmetric, charged, anisotropic fluid spheres in extended symmetric teleparallel gravity, using the linear model f(Q,T) = mu*Q + nu*T (Eq. (15)). The authors adopt a Tolman-type interior metric, Eqs. (19)-(20), impose the three Darmois junction conditions with the Reissner-Nordstrom exterior (21), and use the observed masses and radii of nine pulsar candidates (Table I) to fix the metric constants a, b, c. With the charge profile q(r) = Q(r/R)^3, they obtain closed-form expressions for the density and pressure components (Eqs. (25)-(27)) and then check, graphically, the metric regularity, energy conditions, equation-of-state parameters, mass, compactness, redshift, Herrera cracking condition, TOV equilibrium, causality, and the adiabatic index. They conclude that the charged spheres are physically viable and stable in this theoretical framework.

Significance. The paper belongs to a very active program of constructing anisotropic stellar models in modified gravity, and it is organized around the standard battery of viability tests. The electromagnetic extension of the f(Q,T) field equations and the closed-form fluid variables (25)-(27) are derived explicitly, which is a strength of presentation. If the construction were sound, the paper would add a further example of charged stars in f(Q,T) theory. In my assessment, however, the central claim is not supported as written. The analysis is an inverse reconstruction: the observed masses and radii are inputs that fix the metric constants, and the same fitted solution is then used to demonstrate the energy conditions and stability; the paper presents no independent, falsifiable prediction. More decisively, the matching on which the identification with the pulsars rests is internally inconsistent (Major comment 1), and the parameters mu, nu, and Q that enter every plotted quantity are never specified (Major comment 2). The paper is therefore not reproducible as it stands, and its astrophysical conclusion does not follow.

major comments (4)

[III.A, Eqs. (21)-(24), Table I] The three Darmois conditions written in Section III.A are not satisfied by the constants reported in Table I, so the constructed interiors are not matched models of the listed pulsars. For the EXO 1785-248 row (M = 1.30 M_sun ~ 1.92 km, R = 10.10 km, a = 0.430832, b = 158.119), the left side of the derivative condition is 2aR/b ~ 0.055 km^-1, whereas the right side, 2(MR - Q^2)/R^3, is at most 2M/R^2 ~ 0.038 km^-1 for any real charge Q; the condition therefore cannot be met. Solving the three junction equations algebraically gives c = R^3/M independently of Q (~ 537 km^2 for this row), while Table I lists c = 7687.56. The tabulated a and b likewise correspond to Q^2 ~ +9.0 km^2 through Eq. (23) but to Q^2 ~ -9.0 km^2 through the derivative condition. Moreover, Eqs. (22)-(24) are not the solution of the junction equations even in the uncharged limit: for Q = 0 the conditions require a = 1 - 3M/R, b = R^3(R - 3M)/(MR), and c = +R^3/M, whereas Eq. (24) gives c = -R^3/M; substituting (22)-(23) into the derivative condition reproduces that condition only when Q = 0. The density, pressure, energy-condition, and stability plots in Sections III-V are therefore not properties of matched interior/exterior solutions for these stars.
[III.B, Eqs. (25)-(27), Figures 2-12] The paper never assigns numerical values to the free parameters that determine every plotted quantity. mu and nu are introduced in Eq. (15) as arbitrary constants, and the total charge Q entering q(r) = Q(r/R)^3 is introduced in Section III.A without a value; the fluid variables (25)-(27), the sound speeds, and the adiabatic indices all depend on mu, nu, and Q. As a consequence, none of Figures 2-12 can be reproduced from the information given, and the claims that the energy conditions, causality bounds, cracking condition, and adiabatic-index criteria are satisfied cannot be checked. The authors should state the parameter values used for each figure and should verify, or scan, the parameter space, including the restrictions needed for the expressions to be well defined (e.g., (1 + nu)(2 nu - 1) != 0) and for the density to be positive.
[IV.C, Adiabatic Index] The stability criterion is stated in reverse. The paper reads: 'If the value of Gamma is less than 4/3 then the compact star is stable. If the value of Gamma is greater than 4/3, the compact stars is unstable and will collapse.' The standard Chandrasekhar criterion for radial stability is Gamma > 4/3 (stable) and Gamma < 4/3 (unstable). As written, the sentence would imply that the large values of Gamma_r shown in Figure 12 (up to about 14) are a sign of instability, directly contradicting the following sentence claiming that the system is stable. The discussion must be corrected. In addition, the bare 4/3 threshold is the isotropic-fluid criterion; the anisotropic corrections discussed in the cited Chan et al. references are not applied, so the threshold should be used with the appropriate generalization or its approximate status acknowledged.
[III.A and V] The logical structure of the viability claim is not a test of the theory against observation. The observed masses and radii of the pulsars are used as inputs to fix the metric constants by matching, and the same inputs are then used to demonstrate the energy conditions, TOV equilibrium, and stability; for instance, the mass function (28) is integrated from the reconstructed density, and the compactness and surface redshift (Figure 8) are functions of the same fitted constants. No quantity is predicted that could fail against independent data. The concluding assertion that the charged spheres are viable and stable should therefore be stated as a consistency check of the chosen ansatz, conditional on the presently unspecified parameters, rather than as an observational validation of the framework.

minor comments (8)

[III.A, Figure 1] The metric functions are defined as xi and eta in Eqs. (19)-(20), but Figure 1 labels the plotted components e^nu and e^lambda; the notation should be made consistent.
[Table I] The table lists masses in solar masses while the matching formulas (22)-(24) require M in geometric length units; the conversion used should be stated explicitly.
[III.E, Figure 6] The second panel of Figure 6 carries the same axis label omega_r as the first; if it shows omega_t, it should be relabeled.
[IV.A, Eq. (33)] The expression for the anisotropic force F_a = q^2/(2 pi r^5) + 2 pi r^5 nu is dimensionally inconsistent (the second term has different units from the first) and no derivation from Eqs. (26)-(27) is shown; it should be re-derived and corrected.
[IV.A, Eq. (30)] The standard TOV equation (30) is used without comment, although in f(Q,T) gravity the matter stress-energy tensor is not generally covariantly conserved; the authors should justify that Eq. (30) follows from the field equations (12)-(14) or state that it is used as an approximation.
[V] The statement that all parameters attain their maximum levels in comparison to both GR and other modified gravity theories is not substantiated by any comparison calculation in the paper and should either be supported or removed.
[References] Reference [88] cites Phys. Rev. D 55 (1939), a journal and volume combination that did not exist; this should be Phys. Rev. 55 (1939), and several other references contain typographical errors in journal titles and page numbers.
[Figures 2-12] The multiple curves in Figures 2-12 are not identified within the figures, although the text refers to line colors (black line, blue line, etc.) keyed only to Table I; legend entries or consistent labeling should be added.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the viability checks are genuine inequalities on the reconstructed profiles, not identities forced by the fitted matching constants.

full rationale

After walking the derivation chain, I find no circular reduction. The paper starts from observed masses and radii, fixes the metric constants via the printed Darmois conditions (Eqs. 22-24), and then substitutes the chosen Tolman-type ansatz (19)-(20) into the f(Q,T) field equations to obtain the density and pressure profiles (25)-(27). The subsequent energy-condition, causality, cracking, TOV, and adiabatic-index checks are genuine inequalities evaluated on those profiles; they are not identities imposed by the matching, since whether they hold depends on the free parameters mu, nu, Q and on the chosen ansatz. The mass-function plot is an internal consistency check rather than an independent prediction of the input masses. Self-citations, including the comparison to f(R,T2) in the conclusion, are contextual and are not load-bearing for the central claim. The serious defects in this manuscript, such as the apparent inconsistency between the tabulated constants and the stated junction equations and the inverted adiabatic-index stability criterion, are correctness and consistency problems rather than circularity. Therefore the circularity score is 0.

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

The central claim rests on the chosen linear f(Q,T) action, the Tolman metric ansatz, the Reissner-Nordstrom exterior, and the anisotropic fluid model. The theory parameters mu, nu and the total charge Q are free inputs that are never specified, and the metric constants a, b, c are fitted to observed masses and radii. No new entities are introduced.

free parameters (5)

mu (f(Q,T) coupling) = not specified
Appears in every fluid expression (Eqs. 16-18 and 25-27); no numeric value or unit is given, yet all plots depend on it.
nu (f(Q,T) coupling) = not specified
Enters alongside mu in all plotted profiles and stability quantities; no value is reported.
Total charge Q = not specified
Sets the charge profile q(r)=Q(r/R)^3 and the Reissner-Nordstrom exterior; no value is listed in Table 1 or the text.
Charge profile exponent = 3 (chosen by hand)
q(r)=Q(r/R)^3 is assumed without derivation; this choice controls the electromagnetic terms in the field equations.
Metric constants a, b, c = values in Table 1 for each star
Determined by Darmois matching to observed masses and radii; they are inputs to the analysis, but their derivation assumes the exterior Reissner-Nordstrom solution and the chosen interior ansatz.

assumptions (5)

ad hoc to paper Linear f(Q,T) = mu Q + nu T
Assumed in Eq. (15) to simplify the field equations; no derivation from an underlying theory or observational requirement is given.
ad hoc to paper Tolman metric ansatz, Eqs. (19) and (20)
The interior metric is posited, not derived from a microphysical equation of state; all fluid profiles follow from it.
domain assumption Reissner-Nordstrom exterior and Darmois matching
The exterior of a charged pulsar is assumed to be exactly Reissner-Nordstrom, and the Darmois junction conditions are assumed sufficient to fix the interior constants.
domain assumption Anisotropic fluid energy-momentum tensor, Eq. (8)
Matter is assumed to be a charged anisotropic fluid with radial and tangential pressures differing, standard in this literature.
domain assumption Energy conditions and stability criteria from cited literature
The paper uses NEC, WEC, SEC, DEC, TOV equilibrium, causality, cracking, and adiabatic index criteria; the adiabatic index criterion is stated incorrectly in the text.

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Cite this review

Pith. "Pith review of Exploring the viability of charged Spheres admitting non-metricity and matter source." pith.science (2026). https://pith.science/paper/LS4WT6CR

@misc{pith2026241201411,
  author       = {Pith},
  title        = {Pith review of: Exploring the viability of charged Spheres admitting non-metricity and matter source},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LS4WT6CR}},
  note         = {Machine review of arXiv:2412.01411}
}

read the original abstract

This research paper investigates the impact of non-metricity and matter source on the geometry of charged spheres in the presence of anisotropic matter configuration. We use a specific model of extended symmetric teleparallel theory to minimize the complexity of the field equations. Moreover, the feasible non-singular solutions are used to examine the interior composition of the charged spheres. The Darmois junction conditions are used to determine the unknown constants in the metric coefficients. We explore some significant properties in the interior of compact stars under consideration to check their viable existence in this modified framework. The equilibrium state of the charged spheres is discussed using the Tolman-Oppenheimer-Volkoff equation and stability is analyzed by sound speed and Herrera cracking approach. We find that the charged spheres in this theoretical framework are physically viable and stable.

Figures

Figures reproduced from arXiv: 2412.01411 by the authors.

**Figure 2.** FIG. 2: Graphical analysis of matter contents for different c [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FIG. 3: Graphs of rate of change of fluid parameters for variou [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: FIG. 4: Investigation of anisotropic pressure for different [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: FIG. 5: Examination of energy bounds for different charged sp [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6: Graphical study of EoS parameters. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: FIG. 7: Investigation the behavior of mass function for vari [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: FIG. 8: Examination of graphical behavior of compactness an [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

**Figure 9.** Figure 9: FIG. 9: Investigation of cracking for different charged sphe [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

**Figure 10.** Figure 10: FIG. 10: Investigation of different forces [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

**Figure 11.** Figure 11: FIG. 11: Study of behavior of sound speed components for diffe [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]

**Figure 12.** Figure 12: FIG. 12: Graphical behavior of adiabatic index for different [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]

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