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Quark Stars in $f(T,\mathcal{T}) $ Gravity: Structure, Stability, and Observational Constraints

T0 review · 2 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Quark stars built on a torsion–matter-trace coupling in f(T,𝒯)=T+β𝒯 gravity can reach 2.021 solar masses at β≈3.10, satisfying the two-solar-mass pulsar bound only for a finite window of positive couplings.

desk verdict The numerics and validation are solid, but the paper's own Eq. (21) implies the exterior mass is not M(R) — the headline 2-solar-mass claim likely fails unless a junction analysis fixes it. read the letter →

arxiv 2607.27365 v1 pith:W5LJZLYB submitted 2026-07-29 gr-qc

Takol Tangphati , Ayan Banerjee , Izzet Sakalli , Sayantan Ghosh , Aseel Smerat This is my paper · ORCID
classification gr-qc PACS 04.50.Kd97.60.Jd26.60.Kp04.40.Dg
keywords quarkstarsf(TT)gravityteleparalleltorsionmodifiedTOVequationsbagmodelequationofstatemaximummasstwo-solar-masspulsarconstraint
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 asks whether quark stars—hypothetical compact stars made entirely of deconfined quark matter—can exist in f(T,𝒯) gravity, a teleparallel theory in which torsion is coupled directly to the trace of the energy–momentum tensor. The authors derive the modified stellar-structure equations for the linear model f=T+β𝒯 with the conformal bag equation of state and scan the full admissible coupling range. They find that the maximum gravitational mass is non-monotonic in β: it rises above the general-relativistic value, peaks at 2.021 solar masses near β≈3.10, and then falls steeply as β approaches the singular value 4π where the pressure equation's denominator vanishes. As a result, the standard two-solar-mass pulsar constraint is satisfied for β in (0,~5.5), establishing moderate positive trace coupling as an observationally viable option. The paper explicitly notes in its stability section that the full radial pulsation equations in this non-conservative theory have not been derived, so the stable-branch classification rests on the turning-point criterion as a diagnostic.

What carries the argument

The load-bearing construction is the modified Tolman–Oppenheimer–Volkoff system for the linear model f(T,𝒯)=T+β𝒯, using the conformal bag equation of state p=(ρ−4B)/3 with B=60 MeV fm⁻³. The system consists of equations for A′, dp/dr, and dM/dr, closed by an algebraic relation for the radial metric function; the pressure equation contains the coupling-dependent coefficient K(β)=(β−8π)/(4(4π−β)), which reduces to the GR value −1/2 at β=0 and diverges at β=4π. A rotated 'good tetrad' is used to keep the pure-tetrad field equations consistent, and the terminal integrated mass is matched to the exterior Schwarzschild solution. Stability along each sequence is classified by the turning-point crit

What would settle it

Derive the radial pulsation (Sturm–Liouville) equations for f(T,𝒯)=T+β𝒯 and compute the fundamental-mode squared frequency along the sequence at β≈3.10: if it crosses zero before the mass–central-density maximum, the claimed 2.021 M_sun configuration is unstable. A second, independent check: measure a quark-star mass above ~2.05 M_sun with the same bag constant and equation of state, which the maximum-mass curve cannot accommodate for any admissible β.

Watch

Extended reading notes

Core claim

The central claim is that the maximum mass of quark stars in f(T,𝒯)=T+β𝒯 gravity is non-monotonic in the coupling: M_max(β) increases from 1.549 M_sun at β=−10, crosses the GR limit near β=0, peaks at 2.021 M_sun at β≈3.10, and drops toward zero as β→4π⁻, the singular surface at which the hydrostatic-equilibrium denominator 4/3(4π−β) vanishes. The paper shows that this turnover cannot be blamed on a sign change of any single factor, such as (16π−7β); it emerges from the combined effect of the trace coupling on the pressure gradient, the metric potential, and the algebraic radial-metric closure, and is established numerically. The authors conclude that quark stars in this model are compatible

Load-bearing premise

The load-bearing assumption is that the general-relativistic turning-point criterion—the first maximum of mass versus central density marks the onset of radial instability—remains valid in this non-conservative, trace-coupled theory, even though the full radial pulsation equations have not been derived.

Editorial extensions

If this is right

  • If the claim is right, a moderate positive trace coupling lets quark stars exceed the two-solar-mass pulsar threshold, with the maximum-mass configuration at 2.021 M_sun and β≈3.10.
  • The allowed coupling window β∈(0,~5.5) means pulsar mass measurements translate directly into constraints on the trace coupling: a >2 M_sun quark star would force β into this interval.
  • Configurations on the candidate stable branch are causal, satisfy the local adiabatic-index criterion Γ>4/3, and stay below the general-relativistic Buchdahl compactness bound and the z_s=0.85 surface-redshift reference.
  • The peak mass falls short of the ~2.35 M_sun black-widow pulsar estimate, so reproducing that object would require a stiffer quark-matter equation of state or an extended coupling model.
  • The singular surface at β=4π bounds the model: no regular static solutions exist at or beyond that coupling, so the admissible parameter space is cut off exactly where the pressure equation's denominator vanishes.

Reading between the lines

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

  • A decisive extension would be to derive the radial pulsation equations in this non-conservative theory and test whether the fundamental-mode frequency squared changes sign exactly at the M(ρ_c) maximum; the stability of the 2.021 M_sun star hinges on that match.
  • The paper's maximum-mass enhancement over GR is only 2.9 percent, so mass measurements alone will struggle to distinguish this theory; the surface-redshift difference Δz_s≈0.026 that the paper notes could be a more discriminating observable.
  • Computing the tidal deformability Λ within this framework would let gravitational-wave data bound β directly, effectively converting the theoretical window (0,~5.5) into an observationally measurable parameter.
  • The sharp decline near β→4π hints that the theory has a finite coupling ceiling for static stars; asking whether rotating or time-dependent configurations can exist beyond that ceiling would probe the nature of the singularity.
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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

2 major / 3 minor

Summary. This paper studies static, spherically symmetric quark stars in the linear f(T,T)=T+βT model with matter Lagrangian L_m=p and the conformal MIT bag equation of state (B=60 MeV fm^-3, ω=1/3). The authors derive modified TOV-like equations, integrate them numerically using the rotated good tetrad, and scan the coupling over β∈[−10,12.25]. They report a non-monotonic maximum-mass curve M_max(β) peaking at 2.021 M_⊙ near β≈3.10, and state that the 2 M_⊙ pulsar constraint is satisfied for β∈(0,~5.5). Stability is classified with the GR turning-point criterion, which the paper itself labels as only a diagnostic because the full pulsation equations in f(T,T) have not been derived.

Significance. If the mass identification and stability assumptions were valid, this would be a useful numerical survey of quark-star structure in a trace-coupled teleparallel theory, with a concrete prediction for the allowed coupling interval. The paper has clear strengths: a careful validation protocol (GR-limit gate ΔM=1.8×10^-13 M_⊙, benchmark within 1%, tolerance/grid convergence, independent computer-algebra checks) and a transparent statement of the equation of state and parameter choices. However, the central observable claim is undermined by the surface-matching issue detailed below, and the stability conclusion is explicitly conditional. As written, the headline quantitative result does not stand.

major comments (2)
  1. [Sec. III B, Eq. (21), Table I] The mass quoted as M_max is not the exterior gravitational mass. Eq. (21) gives e^{-B_m}=1+(M/r)Σ+r^2ξ/3; the exterior is Schwarzschild, e^{-B}=1-2M_s/r. Continuity at r=R forces M_s=-(Σ/2)M(R)-R^3ξ/6, not M(R). For β=3, Σ=-2+3/(6π)≈-1.841 and, with B=60 MeV fm^-3 and ω=1/3, R^3ξ/6≈0.11 M_⊙, so M_s≈0.920·2.021-0.11≈1.75 M_⊙. Even neglecting the R^3 term, M_s≈1.86 M_⊙<2 M_⊙. Thus the claimed β≈3.10 peak satisfying the 2 M_⊙ constraint is an artifact of the operational prescription in Sec. III B. Because Eq. (21) itself fixes g_{rr}, this is an internal inconsistency of the solution, not merely an unexamined junction condition as suggested in Sec. VII.
  2. [Sec. IV; Sec. VII] The stability classification is load-bearing but unsupported. Sec. IV states that the full radial pulsation equations in f(T,T) have not been derived and that neither those equations nor the GR turning-point theorem can be assumed to carry over. The paper's central claim—that the 2.021 M_⊙ peak configuration is on the stable branch and satisfies the 2 M_⊙ pulsar constraint—depends on this criterion. The authors disclose the limitation, but the title and abstract still assert stability. Either derive the pulsation equations or explicitly restrict all claims to equilibrium sequences and remove the 'stable branch' language from the abstract and conclusions.
minor comments (3)
  1. [Sec. II, Eqs. (5), (8)] Notation: T is used both for the torsion scalar and for the trace of the energy-momentum tensor. Please use a calligraphic \mathcal{T} for the trace throughout to avoid ambiguity.
  2. [Eq. (23)] Define B_km explicitly, including the conversion from MeV fm^-3 to km^-2 in geometrized units; this is needed to reproduce the numerical results.
  3. [Abstract] The phrase 'full admissible range' should be qualified: the paper treats only the linear model T+βT over β∈[−10,12.25], and the word 'candidate' should appear in the abstract if stability is not established.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: beta is scanned rather than fitted, the MIT bag EOS is external, and the claimed peak mass is a numerical output of the modified TOV system rather than an input.

full rationale

The paper is not circular. The modified TOV system (18)-(21) is obtained by varying the action (6), and the coupling beta is scanned independently (Sec. VI C, Delta beta = 0.25) rather than tuned to reproduce the 2 solar mass threshold. The conformal MIT bag EOS (25) with B = 60 MeV/fm^3 is an external input from refs. [10,13], and the GR limit and published MIT-bag maximum mass are used only as validation benchmarks (Sec. V). The observational 2-solar-mass bound is applied after the fact as a filter selecting beta in (0, ~5.5), not as an inverse-fitting target. Stability statements are explicitly hedged: Sec. IV concedes that the full radial pulsation equations in f(T,T) have not been derived and that neither the pulsation equations nor the GR turning-point theorem can be assumed to carry over, so the turning-point criterion is used only as a diagnostic. Similarly, the identification of M(R) with the exterior gravitational mass is an acknowledged prescription from Pace and Levi Said [67], and Sec. VII flags the absence of an explicit junction-condition analysis. These are omitted proofs or validity risks, not circular reductions: no equation or fitted parameter in the derivation is equivalent by construction to the reported M_max(beta) curve. The self-citations present ([37], [39], [40]) are contextual background, not load-bearing; no uniqueness theorem or unverified prior result by the present authors is invoked to force the central result.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

Central results rest on the modified TOV equations of the linear f(T,T) model (unverified by shipped artifact), the MIT bag EOS with fixed B, an adopted surface-mass prescription, and an unproven stability criterion. No free parameters are fitted to the target observations; β is scanned and B is a literature value.

free parameters (1)
  • Bag constant B = 60 MeV fm^-3
    Chosen from the MIT bag literature and held fixed. M_max scales approximately with B^{1/2}, so the absolute peak mass and hence the β-window depend on this value; a scan over B is left to future work (Sec. VII).
assumptions (4)
  • domain assumption The f(T,T) field equations (14) and reduced TOV system (18)–(21) are correct for f=T+βT with matter Lagrangian L_m=p and the rotated good tetrad.
    The derivation is sketched and said to be cross-checked by computer algebra, but no verification artifact is provided; all numerical results depend on these equations.
  • domain assumption The conformal MIT bag equation of state p=(ρ−4B)/3 with B=60 MeV/fm^3 describes quark matter.
    Standard model input from prior literature; not varied in this paper, although the results are sensitive to B.
  • domain assumption The exterior is exactly Schwarzschild and the terminal value M(R) of the integrated mass function equals the gravitational mass of the star.
    Explicitly an operational prescription adopted from Pace and Levi Said [67]; the paper states that an explicit surface junction analysis is left for future work.
  • ad hoc to paper The GR turning-point criterion (dM/dρ_c=0) marks the onset of instability in f(T,T) gravity.
    Used to identify the candidate stable branch, while the paper admits the full radial pulsation equations in f(T,T) have not been derived and the GR theorem cannot be assumed to carry over in this non-conservative theory.

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Pith. "Pith review of Quark Stars in $f(T,\mathcal{T}) $ Gravity: Structure, Stability, and Observational Constraints." pith.science (2026). https://pith.science/paper/W5LJZLYB

@misc{pith2026260727365,
  author       = {Pith},
  title        = {Pith review of: Quark Stars in $f(T,\mathcalT) $ Gravity: Structure, Stability, and Observational Constraints},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W5LJZLYB}},
  note         = {Machine review of arXiv:2607.27365}
}
abstract

Quark stars-hypothetical compact stars made entirely of deconfined quark matter-offer a clean testing ground for gravity beyond general relativity. We study their structure in $f(T,\mathcal{T})$ gravity, a teleparallel theory in which torsion is coupled directly to the trace of the energy-momentum tensor through a single constant coupling. Using the standard MIT bag description of quark matter, we solve the modified stellar structure equations and follow how the mass, radius, compactness, and surface redshift respond as the coupling is varied across its full admissible range. The maximum mass turns out to depend on the coupling in a non-monotonic way: it rises above the general relativity value, peaks near 2.02 solar masses at a moderate positive coupling, and then falls steeply as the coupling approaches a critical value at which the structure equations become singular. The two-solar-mass pulsar constraint is satisfied within a finite window of positive couplings. All configurations on the candidate stable branch satisfy causality and remain below the standard general-relativistic compactness and surface-redshift benchmarks.

Figures

Figures reproduced from arXiv: 2607.27365 by the authors.

Figure 2
Figure 2. FIG. 2. Gravitational mass [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 1
Figure 1. FIG. 1. Mass–radius relation for quark stars in [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Compactness [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Surface gravitational redshift [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.