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REVIEW 2 major objections 5 minor 61 references

Negative quadratic corrections to nonmetricity raise neutron-star maximum masses while the exterior remains Schwarzschild.

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2026-07-11 16:39 UTC pith:VQ335OQN

load-bearing objection Clean incremental extension of the authors’ own polytropic f(Q) work to four realistic EOSs; negative ξ raises maximum masses while vacuum stays Schwarzschild. the 2 major comments →

arxiv 2607.04596 v1 pith:VQ335OQN submitted 2026-07-06 gr-qc astro-ph.HEhep-th

Neutron stars in f(Q) = Q +xi Q² gravity

J. C. N. de Araujo , H. G. M. Fortes This is my paper
classification gr-qc astro-ph.HEhep-th
keywords f(Q) gravitysymmetric teleparallel gravityneutron starsmass-radius relationnonmetricity scalarrealistic equations of statemodified TOV equations
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper asks whether a simple quadratic extension of symmetric teleparallel gravity can ease the tension between ordinary nuclear equations of state and the heaviest observed neutron stars. Working in the covariant formulation of f(Q) gravity, the authors replace the nonmetricity scalar Q by Q + ξQ² and derive the corresponding stellar-structure equations for four realistic equations of state (FPS, SLy, ENG, MPA1). Numerical sequences show that negative values of the free parameter ξ systematically increase the maximum gravitational mass relative to General Relativity, while positive ξ lowers it; the vacuum exterior remains exactly the Schwarzschild geometry. The nonmetricity itself vanishes at the centre and outside the star, peaking in magnitude only in the high-density interior. If the pattern holds, quadratic f(Q) gravity supplies a concrete geometric mechanism that can accommodate multi-solar-mass compact objects without inventing exotic matter.

Core claim

For the model f(Q) = Q + ξQ² and the realistic equations of state FPS, SLy, ENG and MPA1, negative values of ξ (measured in units of the solar gravitational radius squared) raise the maximum gravitational mass M_S above the General-Relativity value obtained with the same equation of state, while positive ξ lowers it; the exterior geometry is identical to Schwarzschild and all deviations are confined to the stellar interior.

What carries the argument

The modified Tolman–Oppenheimer–Volkoff system obtained from the covariant field equations of f(Q) = Q + ξQ², closed by the first-order relation that determines A′ + B′ and by the algebraic expression for the nonmetricity scalar Q(r) ≤ 0 that vanishes both at the centre and outside the star.

Load-bearing premise

The whole construction stands or falls with the covariant formulation that keeps a non-vanishing connection; if that formulation is not the correct one for stellar interiors, the quadratic term becomes dynamically trivial and the mass shifts disappear.

What would settle it

A precise mass–radius measurement of a neutron star whose equation of state is independently known (for example from simultaneous NICER and gravitational-wave data) that lies outside every mass–radius sequence generated by the model for any |ξ| of order a few r_gs² would rule out the claimed effect.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies static, spherically symmetric neutron stars in the quadratic model f(Q)=Q+ξQ^{2} of covariant symmetric teleparallel gravity. Starting from the covariant field equations, the authors derive a closed first-order system for the metric derivatives A' and B (Eqs. 12–14) together with an algebraic expression for the nonmetricity scalar Q (Eq. 15). They integrate the system for four realistic EOSs (FPS, SLy, ENG, MPA1) and five values of the free parameter ξ (in units of r_gs^{2}), producing mass–radius and mass–central-density sequences. The principal claim is that negative ξ systematically raises the maximum gravitational mass M_S relative to GR while positive ξ lowers it; the exterior geometry remains exactly Schwarzschild. Profiles of Q(r), A(r) and B(r) are also shown for maximum-mass configurations.

Significance. If the numerical results hold under the covariant formulation, the work supplies a concrete, observationally relevant extension of the authors’ earlier polytropic study: realistic EOSs can support maximum masses above the GR limit for negative ξ, offering a possible geometric interpretation of the heaviest known pulsars and of the secondary component of GW190814. The analytic recovery of the Schwarzschild exterior (Eq. 18) and the explicit algebraic form of Q (Eq. 15) are clean technical strengths that make the model falsifiable by future mass–radius measurements. The paper therefore contributes a usable benchmark for strong-field tests of f(Q) gravity.

major comments (2)
  1. Section 4 and Figs. 2–5: the sequences are plotted only for M_S; the companion mass M obtained by integrating the energy density is never shown, even though the text repeatedly notes that M_S eq M in general and that regions with M_S > M_0 appear. Without a quantitative comparison of the two mass definitions (or a clear statement that only M_S is observationally relevant), the claim that negative ξ produces “more massive configurations” remains ambiguous for observers who measure gravitational mass via orbital dynamics or gravitational waves.
  2. Section 4.1: stability is assessed solely by the inequality M_S < M_0. No radial-perturbation analysis, no turning-point criterion applied to the M_S(ρ_c) curves, and no discussion of the adiabatic index are provided. Because the paper’s central phenomenological claim concerns the maximum mass, a minimal stability check (e.g., the sign of dM_S/dρ_c along each sequence) is required before the higher-mass configurations can be regarded as viable neutron-star models.
minor comments (5)
  1. Abstract and Introduction: the phrase “a family of f(Q) gravity models” is used, yet only the single quadratic model is studied; rephrase for precision.
  2. Equation (12): the lengthy expression for A'' would benefit from a short intermediate derivation or a reference to the corresponding equation in the polytropic companion paper, so that readers can verify the algebra without reconstructing every term.
  3. Figures 2–5: the shaded or labelled regions where M_S > M_0 are mentioned in the captions but are not visually distinct in the text description; a clearer legend or hatching would help.
  4. Section 4.2: the statement that “the more negative (positive) ξ is, the greater (smaller) the magnitude of Q” is illustrated only for SLy; a single additional panel for a stiffer EOS (e.g., MPA1) would strengthen the claim that the Q-profile behaviour is universal.
  5. References: a few recent observational papers on massive pulsars (e.g., the latest NICER radius measurements) are missing and would better anchor the comparison with current constraints.

Circularity Check

0 steps flagged

No significant circularity: modified TOV system is derived from covariant field equations and numerically integrated for scanned ξ; self-citations are background, not load-bearing for the mass-radius claim.

full rationale

The paper starts from the covariant f(Q) action and field equations (Eq. 4, citing Zhao 2022 for the covariant formulation), specializes to f(Q)=Q+ξ Q^{2}, and obtains the modified TOV system (Eqs. 12–14) together with the algebraic expression for Q (Eq. 15). Vacuum recovery of the Schwarzschild exterior (Eq. 18) follows directly from those equations when ρ=P=0. The free parameter ξ is scanned over a discrete set of values rather than fitted to any mass-radius data; the resulting sequences (Figs. 2–5) are therefore genuine numerical outputs of the differential system for the chosen realistic EOSs. Self-citations to the authors’ prior polytropic study [51] supply only the same differential system and the mass-definition discussion; they do not insert the target M_S(R) relation or force the sign of the ξ effect. The single minor self-reference is therefore non-load-bearing. The derivation chain is self-contained against external benchmarks and exhibits no definitional reduction of prediction to input.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

The central claim rests on the covariant f(Q) field equations, the perfect-fluid stress-energy tensor, spherical static symmetry, the quadratic ansatz with free ξ, and four named nuclear EOSs treated as external input. No new particles or forces are introduced; the only free parameter is ξ, which is scanned rather than fitted.

free parameters (1)
  • ξ = scanned: −2,−1,0,1,2 (units r_gs² ≃ 2.95 km²)
    Dimensionful coupling of the quadratic nonmetricity term; scanned over the discrete set {−2,−1,0,1,2} r_gs². Controls the sign and magnitude of the mass shift relative to GR.
axioms (5)
  • domain assumption Covariant formulation of f(Q) gravity is required; coincident gauge at the action level collapses the theory to GR.
    Stated in §2 and justified by reference to Zhao (2022). Without it the quadratic term is inert.
  • domain assumption Matter is a perfect fluid with isotropic pressure; energy-momentum tensor is diagonal.
    Used from §2 onward to close the field equations into the modified TOV system.
  • domain assumption Spacetime is static and spherically symmetric with metric ansatz (5).
    Standard stellar-structure assumption; enters every subsequent equation.
  • domain assumption The four nuclear EOSs (FPS, SLy, ENG, MPA1) correctly describe cold dense matter.
    Taken as external input; no microphysical derivation is attempted.
  • standard math Vacuum solutions of the modified theory coincide with Schwarzschild (A′+B′=0 outside matter).
    Derived from the field equations in §2–3; used to define the geometric mass M_S.

pith-pipeline@v1.1.0-grok45 · 17134 in / 2767 out tokens · 21510 ms · 2026-07-11T16:39:36.974957+00:00 · methodology

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read the original abstract

Modified theories of gravity based on the symmetric teleparallel framework have recently attracted considerable attention as viable alternatives to General Relativity. In this context, f(Q) gravity, in which the gravitational interaction is encoded in the nonmetricity scalar Q, provides a consistent geometrical formulation that differs from the standard curvature-based description. In this work, we investigate the structure of neutron stars within a family of f(Q) gravity models by employing realistic equations of state, namely FPS, SLy, ENG and MPA1. Using the covariant formulation of f(Q) gravity, we derive the corresponding Tolman-Oppenheimer-Volkoff equations and apply them to model compact stellar configurations. Numerical integration of the field equations provides the mass-radius relations and the maximum masses supported by each equation of state, enabling a direct comparison with current observational constraints. Furthermore, we analyze the behavior of the nonmetricity scalar both inside and outside the stellar object, providing additional insight into the gravitational structure of compact stars in this framework.

Figures

Figures reproduced from arXiv: 2607.04596 by H. G. M. Fortes, J. C. N. de Araujo.

Figure 1
Figure 1. Figure 1: Left (Right): General Relativity sequences of M [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left (Right): Sequences of MS vs. radius R (central energy density ρc) for the FPS EOS, and different values of ξ, which is given in units of the square of the gravitational radius of the Sun (rgs = 2GM⊙/c2 ≃ 2.95 km). 4.2 Nonmetricity It is useful to analyze, for the realistic equations of state considered in this work, the behavior of the nonmetricity scalar Q(r), as well as the metric functions A(r) and… view at source ↗
Figure 3
Figure 3. Figure 3: The same as in Figure 2 for SLy EOS [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The same as in Figure 2 for ENG EOS [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The same as in Figure 2 for MPA1 EOS [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Q(r) for maximum MS of ξ = −2, 0 and 2 for SLy EOS. Data availability statement In this study, no new data was created or analyzed. Acknowledgment J.C.N.A. thanks CNPq (307803/2022-8) for partial financial support. References [1] C.M. Will, The Confrontation between General Relativity and Experiment, Living Rev. Relativity 17, 4 (2014). [2] S. Capozziello et al., Comparing equivalent gravities: common feat… view at source ↗
Figure 7
Figure 7. Figure 7: A(r) and B(r) for maximum MS of ξ = −2, 0 and 2 for SLy EOS. [5] T.P. Sotiriou and V. Faraoni, f(R) theories of gravity, Rev. Mod. Phys. 82, 451 (2010). [6] J.B. Jim´enez et al., The Geometrical Trinity of Gravity, Universe 5, 173 (2019). [7] R. Aldrovandi and J.G. Pereira, Teleparallel Gravity: An Introduction, Springer, Dordrecht - Heidelberg - New York - London (2013). [8] S. Bahamonde et al., Teleparal… view at source ↗

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