Impact of Pulsar SAX J1748.9-2021 Observations on $f(\mathcal{Q}, \mathbb{T})$ Gravity

Iqra Ibrar; M. Sharif

REVIEW 4 major objections 6 minor 1 cited by

Impact of Pulsar SAX J1748.9-2021 Observations on $f(\mathcal{Q}, \mathbb{T})$ Gravity

T0 review · 4 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper argues that the observed mass and radius of pulsar SAX J1748.9-2021 support a stable anisotropic stellar model in f(Q,T) gravity, where the action couples non-metricity to the trace of the energy-momentum tensor.

desk verdict Routine re-scan of the authors' own f(Q,T) star program, with a fatal junction-condition mismatch that invalidates the whole viability claim. read the letter →

arxiv 2509.02641 v1 pith:NOQPHPAT submitted 2025-09-02 gr-qc astro-ph.HE

M. Sharif , Iqra Ibrar This is my paper · ORCID

classification gr-qcastro-ph.HE PACS 97.60.Gb04.50.Kd97.10.-q

keywords pulsarf(QT)gravitynon-metricityanisotropicmatterKrori-BaruaansatzstellarstabilityenergyconditionsSAXJ1748.9-2021

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 tries to show that observations of the pulsar SAX J1748.9-2021 are compatible with a specific modified theory of gravity whose action depends on both non-metricity Q and the trace T of the energy-momentum tensor. It adopts the non-singular Krori-Barua metric for the star's interior, matches it to a Schwarzschild exterior, and inserts the pulsar's observed mass and radius into the resulting anisotropic fluid model. The authors report that density and pressures peak at the core and fall outward, radial pressure vanishes at the surface, and the configuration satisfies energy conditions, causality (0.44 < v_r^2 < 0.50, 0.40 < v_t^2 < 0.46), the adiabatic index, the Zeldovich condition, TOV equilibrium, and the Buchdahl bound. If correct, the result would establish a stable anisotropic stellar configuration in f(Q,T) gravity consistent with this pulsar's data, extending modified-gravity tests to strong-field compact objects.

What carries the argument

The load-bearing object is the Krori-Barua ansatz for the interior metric potentials, α(r)=A1(r/R)^2+a1 and β(r)=a(r/R)^2, inserted into the f(Q,T)=ζQ+ηT field equations. This ansatz gives non-singular analytic expressions for energy density, radial and tangential pressures, and their derivatives, which then feed every stability criterion: sound speeds come from dP/dρ, the adiabatic index from (ρ+P)v^2/P, TOV balance from the gradient forces, and compactness from M(r)/r. The Schwarzschild matching at r=R is presented as the way to fix the constants, and the observed mass-radius of SAX J1748.9-2021 is the empirical anchor for the numerical evaluation.

What would settle it

Compute the total mass from the paper's own density integral M(R)=4π∫_0^R χ^2ρ(χ)dχ at R=13.4 km with ζ=1.08; the observed mass is 1.81±0.3 M⊙, yet the paper never reports this value. If the integrated mass falls outside that band—or if re-running with R=10 km, or with any stated value of η different from the implicit one, makes Pr(R)≠0 or drives v_t^2−v_r^2 out of [−1,0]—the stability claim is falsified for that parameter set.

Watch

Extended reading notes

Core claim

The central claim is that the anisotropic solution built from the Krori-Barua potentials with A1=2, a1=0, a=-2 and R=13.4 km, in the model f(Q,T)=ζQ+ηT with ζ=1, 1.04 and 1.08, is a physically viable and stable representation of the pulsar SAX J1748.9-2021. The solution is obtained by solving the f(Q,T) field equations for an anisotropic fluid and matching to the Schwarzschild exterior; the constants, however, are fixed by hand in Section 4.1 rather than derived from the junction conditions. The paper then checks a standard battery of criteria—energy conditions, sound speeds, cracking, adiabatic index, Zeldovich condition, TOV force balance, equation-of-state parameters, redshift, and compac

Load-bearing premise

The stability conclusion rests on hand-picked values—A1=2, a1=0, a=-2, R=13.4 km, and an unstated η—that are fixed in Section 4.1 rather than derived from the junction conditions or from matching the pulsar's observed mass and radius.

Editorial extensions

If this is right

If the claim holds, f(Q,T) gravity with f=ζQ+ηT admits stable anisotropic compact-star solutions at nuclear densities, not just cosmological solutions.
The reported sound-speed ranges 0.44 < v_r^2 < 0.50 and 0.40 < v_t^2 < 0.46 provide concrete, testable equation-of-state slopes for the star's interior.
The mass function increases monotonically and compactness stays below 4/9, so the solution would evade gravitational collapse under the Buchdahl criterion.
The TOV equilibrium with four force contributions (anisotropic, gravitational, hydrostatic, and the f(Q,T) correction) implies that the modified gravity contributes a stabilizing force balance inside the star.

Reading between the lines

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

Editorial inference: A direct test the paper leaves implicit is to compute the total mass M(R)=4π∫χ^2ρ(χ)dχ at R=13.4 km for each ζ and compare with the observed 1.81±0.3 M⊙; the paper never reports this number, so agreement with the pulsar's mass is unverified.
Editorial inference: Because η is never assigned a value in Eq. (27) or in Section 4.1, every plotted quantity is actually a one-parameter family; whether the stability windows survive across a range of η is an open question.
Editorial inference: The statement that lower R=10 km 'led to deviations' suggests the model's viability is radius-sensitive; a systematic scan over the observed 1σ range R=11.7±1.7 km would show whether the stability is robust or a contrived choice.
Editorial inference: The same construction could be applied to other pulsars with measured mass-radius pairs to see whether the chosen constants generalize or need retuning, which would test the theory's predictive power beyond a single object.

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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

Quick take: this is the same f(Q,T)=ζQ+ηT model, Krori-Barua ansatz, and the same stability checklist as the authors' earlier papers, applied to a different pulsar. The only new thing is the input mass and radius, so the scientific increment is small.

Fair credit: the paper does derive the field equations from an action, writes out analytic expressions for density, pressures, anisotropy, sound speeds, and runs the standard battery of checks: energy conditions, causality, adiabatic index, TOV, Zeldovich, Buchdahl. The algebra is heavy and the presentation is clear. If you need a worked example of the mechanics of f(Q,T) stellar models, this is readable.

But the central claim fails on the paper's own equations. Section 3.2 states the matching condition β(R) = -ln(1-u) with u>0, so β(R) must be positive. Section 4.1 sets a=-2, so β(R)=-2 and e^{β(R)}=0.135, while the Schwarzschild g_rr is (1-u)^{-1}>1. The interior and exterior do not match. Every subsequent check is applied to a solution that is not attached to the exterior spacetime. This is a load-bearing flaw, not a typo.

Other problems pile up. The central density is ~10^27-10^28 g/cm^3, about 10^13 times nuclear saturation, so calling this a neutron star is not credible. The mass function plotted to 8 km already exceeds the observed 1.81 M⊙. The Zeldovich ratio Pr(0)/ρ(0) is reported as -0.0842, contradicting the positive central radial pressure in Table 2. The coupling η is never assigned a numerical value, so the plots cannot be reproduced. The claimed uncertainty analysis is mentioned but no error propagation is shown. The paper's own conclusion concedes ζ and η lack direct observational constraints.

The authors deserve credit for laying out the algebra and checking many criteria, but the junction mismatch and the density scale mean the paper does not establish a physical model of SAX J1748.9-2021. It is a parameter-fitting exercise inside their own framework.

I would desk-reject. The internal contradiction is decisive and the paper does not warrant referee time. If the authors fixed the junction conditions and reported η, the exercise might become a useful reference for f(Q,T) stellar modeling, but as it stands it is not citable.

Referee Report

4 major / 6 minor

Summary. The paper constructs an anisotropic stellar model in f(Q,T)=ζQ+ηT gravity using the Krori-Barua metric ansatz, and attempts to validate it against observations of the pulsar SAX J1748.9-2021. The authors derive field equations from the action, match the interior to a Schwarzschild exterior via junction conditions, and then check a battery of physical criteria: energy conditions, causality, adiabatic index, TOV equilibrium, Zeldovich condition, redshift, and compactness. The central claim is that the model configuration is physically viable and stable and that it reproduces the observed pulsar.

Significance. If the central claim were sound, the paper would provide a concrete example of a stable anisotropic pulsar in f(Q,T) gravity constrained by real observational data, which would be a useful contribution to the modified-gravity stellar structure literature. The paper has some strengths: the field equations are derived from a variational principle, the junction-condition formalism is invoked explicitly, and the set of checks (energy conditions, causality, TOV, adiabatic index, Buchdahl) is extensive. However, the numerical implementation contains multiple load-bearing inconsistencies that invalidate the claim, as detailed in the major comments. The manuscript is not suitable for publication in its present form.

major comments (4)

[Section 3.2, Eq. (34) and Section 4.1] The junction condition for the radial metric potential is e^{β(R)}=(1-2GM/R)^{-1}. The paper sets a=-2 in Eq. (31), so e^{β(R)}=e^{-2}≈0.135. For any physical star, 0<2M/R<1, hence (1-2M/R)^{-1}>1. The two sides can never match; solving gives 2M/R=1-e^2<0, which is unphysical. Thus the interior KB spacetime is not matched to the exterior Schwarzschild spacetime at r=R, and every subsequent physical check is performed on an interior solution that has not been embedded in an exterior geometry.
[Section 4.1 and Figure 3] The mass function shown in Figure 3 reaches M≈3 M_⊙ already in the plotted range r≈8 km for ζ=1.08. The observed mass of SAX J1748.9-2021 is M=1.81±0.3 M_⊙ with R=11.7±1.7 km. The paper never quotes M(R) at R=13.4 km, and the plot does not extend to the stellar surface. The model therefore does not reproduce the observed mass-radius point, undermining the headline claim that the model is validated by the pulsar's observations.
[Section 4.1, Table 2] The central density is reported as ρ_core≈8.93×10^27 g/cm³ (for ζ=1.08), which is about 10^13 times the nuclear saturation density (≈2.8×10^14 g/cm³). The boundary density is also quoted as ≈5.69×10^27 g/cm³. An object with these densities is not a neutron star or pulsar in any standard equation-of-state picture. The identification of this configuration as a pulsar is therefore physically unsupported, and the paper's own numbers contradict its interpretation.
[Eq. (27) and Section 4.1] The model function is f(Q,T)=ζQ+ηT, and all field-equation expressions depend on both ζ and η. The paper fixes ζ=1, 1.04, 1.08, but never assigns a numerical value to η anywhere, including in Table 2 and the figures. The quantitative results are therefore not reproducible. In addition, the constants A1=2, a1=0, a=-2 and R=13.4 km are stated by hand; R=13.4 km is explicitly chosen because lower values 'led to deviations from the expected behavior' rather than being derived from the junction conditions or matched to the observed mass and radius. This circular selection of parameters means the checks are performed on a fine-tuned configuration rather than a predictive model.

minor comments (6)

[Section 4.1] The text states that at the boundary ρI is 5.69×10^27 g/cm³, 'which is about 1.8 times the core density.' For ζ=1.08, the ratio is 5.69/8.93≈0.64, not 1.8. In Section 5, the same symbol ρI is used for the surface density in the linear EoS and given as 5.4×10^14 g/cm³, a discrepancy of 13 orders of magnitude with the boundary density quoted in Section 4.1.
[Section 4.4] The Zeldovich condition is quoted for the ratio P(0)/ρ(0), and the paper reports Pr(0)/ρ(0)=-0.0842. A negative central radial pressure is inconsistent with the positive core pressure listed in Table 2; the sign suggests an algebraic error or an inconsistent choice of parameters in the limits.
[Figure 1] The caption of the second panel lists ζ=2, 2.02, 2.04 for P_r, while the text and the other panels use ζ=1, 1.04, 1.08. This is a clear caption inconsistency.
[Eq. (5)] The torsion tensor is written as T^λ_{ξ̺} = 1/2(hatΓ^λ_{ξ̺} - hatΓ^λ_{ξ̺}); the second term should be hatΓ^λ_{̺ξ}. Also, Eq. (32) writes 'dθ + sin^2θ dφ^2' instead of 'dθ^2 + sin^2θ dφ^2'.
[Appendix A] The heading 'Casuality Condition' is a typo for 'Causality Condition'.
[Section 4.6] The text quotes the cracking condition as 0≤|v_t²−v_r²|≤1, but then reports −0.79<v_t²−v_r²<−0.37. This is fine for the absolute value, but the phrasing is confusing and should be clarified.

Circularity Check

1 steps flagged · score 6.0 of 10

Validation loop closes on hand-picked radius and metric constants: the 'confirmation of viability' restates the parameter-selection criterion.

fitted input called prediction [Section 4.1 (Material Component), after Eqs. (37)-(39)]
"The constants are set as A1 = 2 , a1 = 0 , a = −2 and various values of ζ = 1, 1.04 and 1 .08. Throughout the paper these values are applied consistently to derive and evaluate our equations and results. ... Testing lower values, such as R = 10km, led to deviations from the expected behavior, reinforcing the choice of R = 13 .4km for producing meaningful and consistent results."

Section 3.2 says the KB constants are to be evaluated from junction conditions, but they are simply assigned here, and R is not derived from the observed mass-radius data but explicitly chosen because lower values 'led to deviations from the expected behavior.' Every subsequent quantity—density, pressures, M(r), redshift, sound speeds, adiabatic index, TOV forces, compactness—is a function of these hand-set inputs. The abstract's conclusion that the findings 'confirm the viability and stability' is therefore a restatement of the selection criterion used to pick R and the constants, not an independent confirmation; the observed mass M = 1.81 ± 0.3 M⊙ is never imposed, so the validation loop closes on the fitted inputs.

full rationale

The main circularity is the hand-picking of R = 13.4 km and the KB constants to produce 'meaningful and consistent results,' followed by reporting those results as confirmation of viability. This is a fitted-input-called-prediction pattern. The derivation is otherwise not self-citation-dependent: the f(Q,T) model is taken from Xu et al. [51], the KB ansatz from Krori-Barua [53], and the stability criteria from independent references; the authors' self-citations [30]-[37] are background only. Two further problems are correctness/reproducibility issues rather than circularities: the junction conditions in Eq. (35) are incompatible with the assigned value a = -2 (for physical u > 0, -ln(1-u) > 0, while β(R) = -2), and the value of η in Eq. (27) is never stated, making numbers such as Pr(0)/ρ(0) = -0.0842 non-reproducible. Because the stability inequalities are nontrivial functions of the chosen parameters and could in principle have failed, the paper is only partially circular, not fully forced by definition; hence a score of 6.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The paper introduces no new particles or fields beyond the f(Q,T) gravity action already present in the cited literature. The load-bearing elements are the free parameters ζ, η, A1, a1, a, and R; among these, η is never specified numerically, and R is explicitly tuned to produce the expected behavior. The KB ansatz and the modified TOV force F(Q,T) are additional modeling assumptions that the central claim depends on.

free parameters (6)

ζ = 1, 1.04, 1.08
Coupling constant in f(Q,T)=ζQ+ηT; varied by hand to produce the three sets of curves. No derivation or observational constraint is given.
η = not stated anywhere in the paper
Coupling constant in f(Q,T)=ζQ+ηT; appears in all field equations, densities, pressures, sound speeds, and stability criteria, yet its numerical value is never given. This is a load-bearing free parameter.
A1 = 2
Constant in the Krori-Barua metric potential α(r)=A1(r/R)^2+a1; assigned in Section 4.1 rather than derived from junction conditions.
a1 = 0
Constant in the Krori-Barua metric potential; assigned by hand in Section 4.1.
a = -2
Constant in the Krori-Barua metric potential β(r)=a(r/R)^2; assigned by hand in Section 4.1.
R (stellar radius) = 13.4 km
Chosen so that the graphs behave as expected; the text says testing lower values such as R=10 km led to deviations. The observed radius is 11.7±1.7 km, so 13.4 km sits at the 1σ upper edge.

assumptions (7)

domain assumption The functional form f(Q,T)=ζQ+ηT is assumed (Eq. 27).
The paper adopts this model from ref [51] without deriving it from a more fundamental principle. The stability conclusion is conditional on this functional form.
domain assumption The interior metric takes the Krori-Barua form α(r)=A1(r/R)^2+a1, β(r)=a(r/R)^2 (Eq. 31).
The KB ansatz is imposed, not derived from the field equations. The paper asserts it is non-singular and useful, but does not show that the true stellar solution in f(Q,T) has this form.
domain assumption The matter is an anisotropic perfect-like fluid with energy-momentum tensor (Eq. 21).
Anisotropy is assumed as the only deviation from isotropy; the results apply only to this matter model.
domain assumption The exterior is the Schwarzschild vacuum and the simple matching conditions g_tt, g_rr continuity plus P_r(R)=0 suffice (Eqs. 32-35).
The paper does not derive the full junction conditions for f(Q,T) gravity, including possible boundary terms from the non-metricity scalar and trace; it assumes the GR-style matching is valid.
ad hoc to paper The TOV equilibrium requires an extra force F(Q,T) to balance (Eq. 47).
In f(Q,T) gravity the energy-momentum tensor is generally not conserved; the paper adds a new force term F(Q,T) to restore equilibrium without deriving it from a conservation law or matching procedure.
standard math Stability is judged by the standard battery of criteria: Zeldovich, causality, Abreu cracking, adiabatic index, energy conditions, and Buchdahl bound.
These criteria are standard in the literature and are used as benchmarks; however, the paper does not prove that satisfying these local inequalities guarantees global stability of the f(Q,T) solution.
standard math Geometrized units with G=c=1 are used for the derivations, followed by a conversion to physical units for the plots.
Standard practice, but the conversion appears to yield central densities near 10^28 g/cm^3, which are unphysical for a neutron star, suggesting either a conversion error or a units inconsistency.

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Cite this review

Pith. "Pith review of Impact of Pulsar SAX J1748.9-2021 Observations on $f(\mathcal{Q}, \mathbb{T})$ Gravity." pith.science (2026). https://pith.science/paper/NOQPHPAT

@misc{pith2026250902641,
  author       = {Pith},
  title        = {Pith review of: Impact of Pulsar SAX J1748.9-2021 Observations on $f(\mathcalQ, \mathbbT)$ Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NOQPHPAT}},
  note         = {Machine review of arXiv:2509.02641}
}

abstract

The main objective of this study is to investigate the viability and stability of a pulsar filled with anisotropic matter in $f(\mathcal{Q}, \mathbb{T})$ gravity, where $\mathcal{Q}$ represents non-metricity and $\mathbb{T}$ is the trace of the energy-momentum tensor. In this context, we employ non-singular solution and a particular model of this gravity. We use junction conditions to evaluate unknown constants in the metric coefficients. Observations from the pulsar SAX J1748.9-2021 star are employed to validate the model, producing stable configurations that address both geometric and physical characteristics. This framework establishes relationships between various physical quantities, including fluid parameters, anisotropy, mass-radius relation, redshift, the Zeldovich condition, energy conditions, causality conditions, adiabatic index, Tolman-Oppenheimer-Volkoff equation, the equation of state parameter, and compactness. Our findings confirm the viability and stability of the proposed pulsar star in this theoretical framework.

Figures

Figures reproduced from arXiv: 2509.02641 by the authors.

**Figure 1.** Plots of ρ, Pr, Pt versus r. We have analyzed the impact of these uncertainties on key physical parameters, such as ρ, Pr, Pt and compactness. The results confirmed that these parameters remain stable and within physically expected bounds, validating the robustness of our model. Additionally, we have conducted a sensitivity analysis by varying the parameter ζ and studying its effect on the results. We have observed… view at source ↗

**Figure 2.** Plot of σ against r. physical systems, such as stars, where anisotropic pressure can affect their structural integrity and evolution [56]. The anisotropy (σ = Pt − Pr) leads to σ = 1 r 2R4 [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** Plot of mass-radius against r. 4.3 The Geometric Component The gravitational redshift, denoted by Zs(r), describes the change in wavelength (or frequency) of light or other electromagnetic radiation as it escapes from the gravitational field. According to GR, this effect intensifies in stronger gravitational field, particularly around massive objects like stars or black holes. This is defined as follows Zs(r) = 1 … view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Plot of redshift versus r. the EoS remains causal. This condition is derived from relativistic principles, emphasizing that information or perturbations cannot propagate faster than the speed of light. By imposing this restriction, the Zeldovich condition plays a cruc…

**Figure 5.** Figure 5: Plots of energy conditions versus r. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]

**Figure 6.** Figure 6: Plots of causality conditions versus r. 1 [62]. Meeting this criterion indicates that cosmic structures are stable and capable of sustaining their configurations over time. Failure to satisfy this condition suggests instability, potentially leading to the collapse of t…

**Figure 7.** Figure 7: Plots of adiabatic index against r. Γr = ρ + Pr Pr v 2 r , Γt = ρ + Pt Pt v 2 t , where the symbols Γr and Γt represent the radial and tangential components of the adiabatic index, respectively. Clearly, in the case of isotropy (σ = 0), we get 4 3 . For slightly anisot…

**Figure 8.** Figure 8: Plots of the TOV equation versus r. × e 2A1r 2 R2 r 4 − 44e A1r 2 R2 r 2 + 31 η 2 + [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]

**Figure 9.** Figure 9: Plots of EoS versus r [PITH_FULL_IMAGE:figures/full_fig_p029_9.png]

**Figure 10.** Figure 10: Plot of compactness versus r. • We have shown that ρ, Pr and Pt for various values of ζ are highest at the center and decrease towards the surface. The radial pressure Pr reaches zero at the surface of the star ( [PITH_FULL_IMAGE:figures/full_fig_p030_10.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Neutron stars in $f(\mathbb{Q})$ gravity
gr-qc 2025-12 conditional novelty 6.0 of 10

For f(Q)=Q+αQ² and f(Q)=Q^β, neutron-star solutions that admit power-series expansions at the center or at infinity collapse to General Relativity; genuine beyond-GR effects must be non-analytic.

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