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Properties of compact objects in quadratic non-metricity gravity

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Pith's one-line read In quadratic non-metricity gravity, a 2-solar-mass pulsar can have a soft equation of state with radial sound speed below the conformal bound and compactness below the Buchdahl limit.

desk verdict Competent but routine KB-ansatz star model in quadratic f(Q) whose headline sound-speed bound fails a continuity check; the algebra is checkable, but the physical claims are oversold. read the letter →

arxiv 2507.14591 v2 pith:W2TC2PQL submitted 2025-07-19 gr-qc astro-ph.HEastro-ph.SRhep-th

G.G.L. Nashed , Kazuharu Bamba This is my paper
classification gr-qcastro-ph.HEastro-ph.SRhep-th PACS 04.40.Dg04.50.Kd
keywords f(Q)gravitynon-metricitycompactstarsPSRJ0740+6620Krori-BaruametricanisotropicfluidequationofstateBuchdahllimit
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 show that a quadratic correction $\xi Q^2$ to the non-metricity scalar $Q$ can account for the observed properties of the massive millisecond pulsar PSR J0740+6620 without exotic matter. For negative $\xi$, the authors claim the radial sound speed inside the star remains below the conformal upper bound $c_s^2 \le c^2/3$ derived from perturbative QCD, and the compactness never exceeds the Buchdahl limit $C=8/9$. The same geometric term pushes the core density to about 2.7 times nuclear saturation density and leaves the surface density above saturation, which the paper presents as a contrast with black-hole compactness limits in general relativity. If correct, this makes quadratic non-metricity gravity a viable explanation for two-solar-mass pulsars with soft equations of state, and it turns NICER/XMM mass-radius data into a direct probe of the parameter $\xi$.

What carries the argument

The central object is the quadratic non-metricity Lagrangian $f(Q)=Q+\xi Q^2$ in symmetric teleparallel gravity, where the non-metricity scalar $Q = Q_{\lambda\mu\nu}P^{\lambda\mu\nu}$ is built from the connection rather than from curvature and $\xi$ is a length-squared coupling. The argument is carried by the Krori-Barua metric ansatz $a(r)=s_0(r/L_s)^2+s_1$, $b(r)=s_2(r/L_s)^2$, which reduces the field equations to algebraic expressions for density, radial pressure, and tangential pressure. Matching those expressions to the Schwarzschild exterior at $r=L_s$ fixes the constants and links $\xi$ to the compactness; expanding near the center yields a linear equation of state whose slopes are the squared radial and tangential sound speeds. The quadratic term also contributes an extra force $F_Q=2\xi_1 Q'$ to the Tolman-Oppenheimer-Volkoff balance, which for negative $\xi$ opposes collapse and lowers the radial sound speed.

What would settle it

A revised, more precise NICER/XMM measurement of PSR J0740+6620 that places its mass-radius combination outside the model's fitted curves, for example a radius below about 11 km at a mass near $2.07M_\odot$, or a measured compactness above $C=8/9$, would settle the central claim by falsifying the model.

Watch

Extended reading notes

Core claim

The central claim is that in the theory $f(Q)=Q+\xi Q^2$, anisotropic stellar models built on the Krori-Barua metric ansatz are stable and observationally consistent with PSR J0740+6620. With $\xi<0$, the model's radial sound speed stays below $c^2/3$, while with $\xi_1=0.03$ (positive) it rises to about $0.47c^2$; the tangential speed remains below $0.3c^2$ in both cases. Compactness $C=2GM/(c^2R)$ remains below the Buchdahl limit $C=8/9$ for both signs of $\xi$, and the derived linear equations of state, $P_r\simeq v_r^2(\epsilon-\epsilon_s)$ and $P_\perp\simeq v_\perp^2(\epsilon-\epsilon_{ii})$, have slopes that the authors identify as the squared sound speeds. The observed NICER+XMM mass and radius, $M=2.07\pm0.11\,M_\odot$ and $R=12.34^{+1.89}_{-1.67}$ km, are used to constrain the coupling to $|\xi_1|<0.03$, i.e. $|\xi|<3$ km$^2$. The paper's discovery is that the geometric quadratic term alone can keep the heavy pulsar's equation of state soft, below the perturbative-QCD conformal sound-speed limit, and simultaneously keep the object below black-hole compactness.

Load-bearing premise

The calculation assumes that the real interior of PSR J0740+6620 is exactly described by a simple quadratic-in-radius metric ansatz chosen for mathematical convenience, and that the equation of state read off from that ansatz remains the physical one throughout the star.

Editorial extensions

If this is right

  • For $\xi_1=-0.03$, the model predicts a radial sound speed of about $0.25c^2$ and a tangential speed of about $0.18c^2$ at the core, implying a soft equation of state for PSR J0740+6620.
  • The allowed coupling range $|\xi_1|<0.03$ is a concrete observational target: any future mass-radius measurement that cannot be fitted within this range would rule out the quadratic model.
  • With negative $\xi$, the model produces maximum masses above 3 solar masses, and with positive $\xi$ above 4 solar masses depending on the surface density, so the theory permits neutron-star-like objects heavier than the roughly 2.1-solar-mass pulsars observed so far.
  • Compactness remains below $C=8/9$ for both signs of $\xi$, in line with higher-order Gauss-Bonnet gravity but in contrast to $f(R)$ gravity, where compactness can exceed the Buchdahl limit.

Reading between the lines

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

  • Because the sound-speed softening appears only for negative $\xi$, a natural next step is to solve the same $f(Q)=Q+\xi Q^2$ field equations with a realistic nuclear equation of state inserted at the center; if the sub-conformal radial speed survives, the effect is a property of the gravity theory, not just of the chosen metric.
  • The same construction could be applied to other precisely measured pulsars such as PSR J1614-2230; a predicted mass-radius curve that matches all such objects would give an independent check that the quadratic non-metricity coupling is negative and of order $|\xi_1|\simeq0.03$.
  • The paper leaves open whether the non-metricity term affects the moment of inertia and tidal deformability in a way that gravitational-wave observations of binary neutron star mergers could distinguish from general relativity; this is a testable extension of the TOV analysis presented here.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper studies static, spherically symmetric, anisotropic compact stars in quadratic non-metricity gravity, f(Q)=Q+ξQ², using the Krori-Barua metric ansatz for the stellar interior. Matching to the observed mass and radius of PSR J0740+6620 fixes the model parameters, and the authors then examine density, pressure, energy conditions, sound speeds, TOV equilibrium, and mass-radius relations. The main advertised results are that the compactness stays below the Buchdahl limit and that, for negative ξ1, the radial sound speed remains below the pQCD conformal bound c_s²≤c²/3, which the authors present as a novel feature of the theory.

Significance. If the central claim were correct, it would be an interesting result: a modified-gravity explanation for a massive pulsar with a soft effective equation of state, without invoking exotic matter. The paper does provide explicit field-equation expressions, matching conditions, stability checks, and a comparison with NICER/XMM data, and the internal algebra appears largely self-consistent. However, the headline claim about the radial sound speed is not a property of the sign of ξ1; it holds only for the particular fitted value ξ1=−0.03. Moreover, the equation of state used for the pQCD comparison is derived from the same Krori-Barua ansatz and fitted parameters used to match the observed star, so the comparison is circular rather than predictive. These issues undermine the paper's main physical conclusions.

major comments (4)
  1. [§IV F, Eq. (C1)] The claim that negative ξ1 guarantees v_r²<c²/3 is false. Setting ξ1=0 in Eq. (C1) gives b1/c²=(4s0−s2)/(5s2); with the paper's own fitted values s0≈0.491 and s2≈0.684 this is ≈0.374, exceeding 1/3, consistent with Fig. 5(b). Since the matching conditions determine s0 and s2 continuously in ξ1, b1(ξ1) is continuous, so there is an open interval of negative ξ1 near zero for which v_r²/c² still exceeds 1/3. Only the hand-picked value ξ1=−0.03 is shown to satisfy the bound. Therefore the abstract's and Sec. I's statement that the radial sound speed remains below the conformal upper limit is not established as a general property of quadratic non-metricity gravity.
  2. [§IV B, Eq. (27); §IV F] The comparison with the pQCD conformal bound is circular. The linear EoS (27) and the sound speeds (33) are derived from the same Krori-Barua metric ansatz (20) and the same fitted parameters ξ1, s0, s2 that were chosen to reproduce the observed mass and radius of PSR J0740+6620. The statement that the theory 'predicts' a soft EoS is therefore a restatement of the fit, not an independent test. A genuine prediction would require a microphysical EoS input or at least a demonstration that the conclusion is insensitive to the metric ansatz.
  3. [§V, Fig. 9] The mass-radius curves and maximum masses in Fig. 9 are obtained by fixing the surface density ǫs to fitted values (e.g., 2.7, 2.9, and 4.5×10^14 g/cm³) and integrating the KB-based density from Eq. (B1). These results are properties of the ad hoc Krori-Barua ansatz, not of quadratic non-metricity gravity as such. The claim that the compactness never exceeds the Buchdahl limit is thus not shown to be a general feature of the theory; it is a feature of this specific fitted model.
  4. [§IV G and §VI] There is a direct contradiction about the effect of negative ξ1. In §IV G and Fig. 6(c) the authors state that negative ξ1 partially counteracts gravitational collapse, while in §VI they state that when ξ1 is negative the force 'aids in gravitational collapse.' The final paragraph of §VI then says negative ξ1 'prevents gravitational collapsing and reduces the sound speed' but immediately reports v_r²≈0.47c², which is the value for ξ1=+0.03 from §IV F. This inconsistency needs to be resolved before the physical interpretation can be assessed.
minor comments (4)
  1. [§IV D] The bullet list giving the fitted parameter sets contains 'ξ1=0.3' where the context and all surrounding values indicate 'ξ1=0.03'; please correct this typo.
  2. [§IV B] The text refers to 'Fig. 1(d)' when describing the anisotropy plot; the relevant panel is Fig. 2(d).
  3. [Throughout] There are several typographical errors, including 'wither' for 'whether' and 'sellstar' for 'stellar,' and duplicated phrases such as 'where where ∇μ signifies.' A careful proofreading pass is needed.
  4. [§IV F] The paper quotes slightly different values for the fitted sound speeds in §IV B (v_r²≈0.50c² for ξ1=0.03) and in §V (v_r²≈0.47c² for the same case); the source of this discrepancy should be clarified.

Circularity Check

0 steps flagged · score 1.0 of 10

No construction-level circularity: the f(Q) stellar model is algebraically self-contained, and the pQCD sound-speed check is an external post-fit benchmark; the overbroad negative-ξ1 claim is a correctness flaw, not circularity.

full rationale

Although the paper contains several self-citations ([48], [49], [56], [57], [59], [60]) and phrases the pQCD sound-speed result as a 'novel feature' of negative ξ1, I find no step in which a derived output is equivalent to an input by construction. The field equations (11) are reduced with f(Q)=Q+ξQ² (17) and the KB ansatz (20); the boundary conditions (23) and observed M,R fix ξ1 and the metric parameters; the EoS (27) and sound speeds (33) are then computed from these parameters and compared with the external pQCD bound [50]. The pQCD bound is not used to fit ξ1, so the comparison is an independent (though post-fit) benchmark. The self-citations to f(R)/GB comparisons are contextual rather than load-bearing, and the Buchdahl claim is checked within the paper (Fig. 9). The statement that any negative ξ1 keeps v_r² below c²/3 is overbroad — continuity from the displayed GR case (v_r²/c² ≈ 0.374 from Eq. C1 with ξ1=0) means nearby negative values would violate the bound — but an over-generalized inference is a correctness risk, not a circularity. No specific reduction of a 'prediction' to its fitted inputs was found.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The model has three families of free inputs: the f(Q) coupling ξ1, the surface density used to build mass-radius curves, and the Krori-Barua metric parameters. No new particles or geometric entities are introduced. The main physical output (EoS, sound speeds) is a function of these fitted inputs.

free parameters (3)
  • ξ1 (dimensionless quadratic coupling) = ±0.03
    Chosen to match the mass and radius of PSR J0740+6620; the paper claims |ξ1|<0.03 in the conclusion but gives 0≤|ξ1|≤300 km² in Sec. IV.D.
  • Surface density ǫs = 2.5, 2.7, 2.9, 4.5 ×10^14 g/cm³
    Hand-picked in Sec. V to generate mass-radius curves; described as 'fixed by the fit of EoS', but the values are chosen to produce the desired maximum masses.
  • KB metric parameters s0, s1, s2 = e.g., {0.491, -1.1756, 0.684} for GR
    Metric ansatz parameters fixed by matching conditions for a chosen ξ1 and compactness; they are inputs to the model and are not derived from nuclear physics.
assumptions (6)
  • domain assumption The f(Q) field equations as given in Eq. (11) are valid.
    Taken from Ref. [55]; not re-derived in this paper.
  • domain assumption The coincident gauge can be adopted, eliminating the connection via coordinate choice.
    Section II.A assumes flat geometry, zero torsion, and the gauge coincident transformation of Eq. (8).
  • ad hoc to paper The Krori-Barua metric ansatz (Eq. 20) describes the stellar interior.
    This geometric assumption completely determines the density and pressure profiles through the field equations.
  • domain assumption The exterior spacetime is Schwarzschild and matches the interior at r=Ls.
    Section III.C relies on Ref. [65], which states that regular f(Q) exterior solutions coincide with general relativity.
  • ad hoc to paper The linearized EoS (Eq. 27) is valid throughout the star.
    Section IV.B expands in ε=r/Ls and keeps only low-order terms; the resulting linear relationships are then treated as the global EoS.
  • domain assumption The pQCD conformal bound c_s²≤c²/3 is a universal upper limit at neutron star densities.
    Section IV.F applies Ref. [50] to core densities near a few times nuclear saturation, which is an extrapolation of perturbative QCD.

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Cite this review

Pith. "Pith review of Properties of compact objects in quadratic non-metricity gravity." pith.science (2026). https://pith.science/paper/W2TC2PQL

@misc{pith2026250714591,
  author       = {Pith},
  title        = {Pith review of: Properties of compact objects in quadratic non-metricity gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W2TC2PQL}},
  note         = {Machine review of arXiv:2507.14591}
}
read the original abstract

Astrophysical compact objects are studied in the context of quadratic non-metricity gravity. The solutions to the gravitational field equations, which include fluid components, are analyzed to investigate the density and pressure properties of radio pulsars. It is explicitly demonstrated that the theoretically stable models are consistent with astronomical data, due to the geometric features of the quadratic component. Furthermore, it is shown that, in contrast to the compactness limits of black holes in general relativity, the core density can significantly exceed the density at which nuclear saturation occurs, and the surface density can also surpass the value of nuclear saturation. Additionally, it is found that the radial sound speed remains below the conformal upper bound for sound velocity established by perturbative quantum chromodynamics.

Figures

Figures reproduced from arXiv: 2507.14591 by the authors.

Figure 1
Figure 1. The visual representation of pulsar J 0740 + 6620 in (a) displays gtt and grr inside the star, calculated using KB form, and the outside the star, obtained from the external vacuum solution in the Schwarzschild spacetime. It is seen that the potentials of the metric remain finite values within the interior of pulsars and have a smooth transition to the exterior region; (b) shows the pattern of the redshift in the ca… view at source ↗
Figure 2
Figure 2. The properties of ǫ, Pr and P⊥ for the star J 0740 + 6620, are shown in (a) to (c). These are the case for: ξ1 = 0, ±0.03. It is shown how the density and pressures stay in limits of the inner part of a pulsar, reducing as we get closer to the surface [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. The function of the mass given by Eq. (30) for the pulsar J 0740 + 6620 is shown in the limitations imposed by the observations in terms of the radius and mass (Ls = 12.35 ± 0.11 km and M = 2.07 ± 0.11M⊙) in Ref. [69]. If ξ1 = −0.03, the parameter set of KB as [ s0 ≈ 0.457, s1 ≈ −1.142, s2 = 0.684] is used, and when ξ1 = 0.03, that as [s0 ≈ 0.525, s1 ≈ −1.21, s2 ≈ 0.684] is taken. For ξ1 = 0 general relativity, we e… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Visual representations that validate the pulsar m [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: The velocity of sound in the pulsar J 0740 + 6620 is shown for ξ1 = 0, ±0.03 in Figs. 4(a) and 4(b), illustrating the propagation of the sound wave in tangential and radial directions based on Eq. (33). The images in (c) show that the model meets the stability requirem…
Figure 6
Figure 6. Figure 6: The TOV constraint represented by Eq. (34): Various forces acquired from Eq. (35), on the pulsar J 0740 + 6620 are shown when ξ1 = 0, ±0.03. If ξ1 = 0.03, the additional negative force introduced by the quadratic correction increases the force of gravitational collapse…
Figure 7
Figure 7. Figure 7: The plots of (a), (b), and (c) verify the stability of pulsar J 0740 + 6620, due to γ > 4/3, and show that both Γr and Γt are greater than γ across the entire of the star. This is a vital condition for a fluid with its strong anisotropy. In Newtonian gravity, if the ad…
Figure 8
Figure 8. Figure 8: The set of density and radial pressure data are obta [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: The compactness is shown in Fig. (a). The dashed horizontal line represents the Buchdahl constraint if the compact￾ness is C = 8/9. The behavior of the Compactness-radius relation linked to the fits of the EoS is presented in [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]

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