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In f(Q) gravity, where the connection is dynamical, spherical neutron-star solutions with a power-series expansion at center or infinity collapse back to General Relativity; beyond-GR effects must hide in non-analytic structure.

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2026-08-03 18:53 UTC pith:CW4I6BHQ

load-bearing objection A careful regularity analysis that likely kills the easy route to beyond-GR neutron stars in f(Q) gravity, but the headline overstates what is actually proven. the 2 major comments →

arxiv 2512.03037 v1 pith:CW4I6BHQ submitted 2025-12-02 gr-qc astro-ph.HE

Neutron stars in f(mathbb{Q}) gravity

Lavinia Heisenberg , Carlos Pastor-Marcos This is my paper
classification gr-qc astro-ph.HE MSC 83D0583C5583C15 PACS 04.50.Kd04.40.Dg
keywords f(Q) gravitysymmetric teleparallel gravityneutron starsnon-metricityaffine connectionTolman-Oppenheimer-Volkoff equationsboundary value problembeyond-GR solutions
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 is trying to establish a restrictive result about neutron stars in f(Q) gravity, a modified-gravity theory in which the affine connection — not just the metric — is a dynamical field carrying an extra degree of freedom. For the representative models f(Q) = Q + αQ² and f(Q) = Q^β, the authors show that any static, spherically symmetric, perfect-fluid stellar solution admitting a standard power-series expansion at the center (or a 1/r expansion at infinity) necessarily collapses to General Relativity: regularity forces the non-metricity scalar Q to be constant inside the star and zero outside, which silences the connection's equation of motion. The paper is careful to say this is not a no-go theorem; genuinely beyond-GR solutions, if they exist, must be non-analytic in the radial coordinate, such as the logarithmic connection hair already found for black holes in the same theory. It then supplies a practical roadmap — the problem as a boundary value problem, the numerical traps, and continuation strategies — for hunting such non-analytic branches. A sympathetic reader should care because much of the existing compact-star literature in f(Q) gravity relies on connection choices that this paper shows are GR in disguise.

Core claim

On the paper's own terms, the central discovery is a freeze-out mechanism. For a perfect-fluid neutron star in f(Q) = Q + αQ² or Q^β, the equations close for m(r), p(r), Q(r), and the connection component Γ^r_θθ(r). Maclaurin regularity at r = 0 forces the first derivative of Q to vanish, and the equations then force all higher derivatives to vanish, so Q is constant inside the star. The connection's equation of motion — all terms proportional to ∂_r Q — trivializes and the metric reverts to the GR/TOV system. At infinity, a 1/r expansion with asymptotic flatness forces Q = 0, so the exterior is Schwarzschild. This is, the paper stresses, not a no-go theorem: the series ansatz is the filter.

What carries the argument

The load-bearing object is the symmetry-reduced connection: the most general flat, torsion-free, stationary, spherically symmetric affine connection, taken from earlier black-hole work, with exactly one radial dynamical component Γ^r_θθ(r) (plus a free constant Γ^t_θθ that the dynamics ignore). The paper trades the other radial component, Γ^r_rr, for the non-metricity scalar Q, closing a six-equation system: mass and pressure equations, a second-order equation for Q, a first-order constraint for Γ^r_θθ, the conservation law, and an equation of state. The decisive structural property: every term in the connection's equation of motion is proportional to ∂_r Q, so the branch ∂_r Q ≡ 0 is at onc

Load-bearing premise

The result stands or falls on the completeness of the flat, torsion-free, stationary, spherically symmetric connection ansatz adopted from an earlier study — if that reduction missed an allowed radial component of the connection, regular power-series solutions with a genuinely dynamical non-metricity scalar could exist after all.

What would settle it

The cleanest test is numerical: solve the full f(Q) = Q + αQ² system with a polytropic EoS as a global boundary-value problem, treating {Q(0), ∂_r Q(0), Γ^r_θθ(0)} as unknowns, imposing regularity at r = 0, p(R) = 0, asymptotic flatness, and Q → 0 at large r, and seeding Q(r) non-constant. Any solution with ∂_r Q ≢ 0 that satisfies the connection equation refutes the claim that regular branches inevitably converge to GR. A cheaper analytic probe: a center expansion containing r log r, regular but non-Maclaurin; if it works order by order, it exhibits the predicted non-analytic branch. A third

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

If this is right

  • Any neutron-star solution in these f(Q) models that admits a Taylor expansion at r = 0, or a 1/r expansion at infinity, is necessarily GR: Q must be constant inside and zero outside, and the connection's extra mode decouples.
  • Common shortcuts — the coincident gauge or the GR-like connection choice Γ^r_θθ = −r — suppress the degree of freedom that distinguishes f(Q) gravity from GR, so published stellar solutions built on them do not explore beyond-GR physics.
  • A well-posed numerical search must be a global boundary-value problem with the central values {Q(0), ∂_r Q(0), Γ^r_θθ(0)} promoted to unknowns selected by outer data (asymptotic flatness, surface matching, possibly observed mass/radius or tidal deformability), with parameter continuation from a small-but-finite deformation away from GR, because no well-defined GR limit exists for the connection eq
  • If beyond-GR neutron-star branches exist in these models, they must be non-analytic in r — e.g. logarithmic tails like those found for black holes in the same theory — and exterior matching must accommodate such terms (working with ξ, ζ rather than m, or subtracting an asymptotic template) to avoid apparent blow-ups.
  • If a fully consistent BVP still finds no beyond-GR branch, that negative result would itself indicate that the connection's extra mode is stealthy or decoupled on static, isotropic, perfect-fluid backgrounds, and that genuinely new physics requires relaxing the assumptions — rotation, anisotropy, time dependence, hypermomentum, or less restrictive ansätze.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Scope flag (editorial): the paper presents no actual stellar models; its numerical observations come from a shooting implementation it describes as delicate, and its one reported attempt to match the perturbative exterior with Q = 0 inside "leads to an apparent blow-up." The global-BVP strategy is a prescription to be executed, not a demonstrated solution.
  • The freeze-out mechanism looks transferable: because the connection equation's terms are all proportional to the gradient of Q, other non-linear extensions of symmetric teleparallel gravity, or scalar-tensor nonmetricity theories, may show the same decoupling under power-series regularity on static spherical backgrounds. Checking one such model would tell whether the GR collapse is specific to the
  • The paper states its results produce "discrepancies with some of the previous results in the literature, in particular those reported in Ref. [29]" (Sec. IV). If the freeze-out is correct, some published stellar mass-radius relations in f(Q) gravity that used GR-like connection choices may be implicitly GR predictions; reconciling them would require re-running those models with the connection left
  • Observational angle: if beyond-GR branches are confined to non-analytic sectors, corrections to the mass-radius relation, surface redshift, and tidal deformability may be parametrically controlled by α (or β − 1) rather than qualitative, so astrophysical constraints would bound the model parameters instead of ruling out the theory.

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-star configurations in f(Q) gravity, treating the affine connection as a dynamical object. Building on the symmetry-reduced flat, torsionless connection of Ref. [12], the authors write down the full structure equations for the metric functions, pressure, density, non-metricity scalar Q, and the connection component Γ^r_θθ (Eqs. (36)–(39)). For the representative models f(Q)=Q+αQ² and f(Q)=Q^β, they analyze regularity at the stellar center via Maclaurin expansions and at infinity via Laurent expansions in 1/r. They find that, under these ansätze, regularity forces Q to be constant (or zero), which trivializes the connection field equation and makes the solutions converge to GR. The paper also discusses the resulting numerical pathologies in solving the system as a boundary-value problem and proposes strategies such as global BVP formulations, continuation in the deformation parameters, and matched asymptotics. The central claim is that genuinely beyond-GR stellar solutions, if they exist within this setup, cannot be represented by ordinary power-series expansions and must involve non-analytic structure or relaxed assumptions.

Significance. If the central claim holds, the paper provides a useful clarification of why many existing neutron-star studies in f(Q) gravity that use simplified connections or perturbative expansions inadvertently obtain GR-like results. The derivation of the structure equations with the connection kept dynamical, the consistency checks (the α=0 and β=1 limits correctly reduce to the GR TOV system, and the Bianchi-identity route to the connection equation reproduces the direct computation in Ref. [12]) are genuine strengths. The paper is explicitly framed as a roadmap rather than a no-go theorem, and it identifies concrete numerical strategies for future searches. The main limitation is that the generality of the conclusion rests on an adopted connection classification and on regularity assumptions that are not fully proved within the manuscript.

major comments (2)
  1. [Sec. III.A (Table I)] The paper adopts, without re-derivation, the symmetry-reduced connection from Ref. [12] and describes it as 'the most general flat, torsionless, stationary and spherically symmetric connection' (Sec. III.A). This classification is load-bearing: the structure equations (36)–(39), and in particular the homogeneous-in-∂Q form of Eq. (38), are specific to this ansatz. If additional r-dependent connection components compatible with flatness and spherical symmetry exist (e.g., Γ^t_tr(r) or Γ^r_tt(r)), the conclusion that Maclaurin/Laurent regularity forces Q=const would hold only for a subclass of connections. Please either provide a self-contained proof of the classification, state it as a theorem with a precise reference to Ref. [12], or explicitly restrict the headline conclusions to this ansatz. As written, the claim in Sec. V.B that 'any attempt' to find a solution deviating from GR using
  2. [Sec. V.B, Tables III–IV] The argument that regularity of the interior solutions forces Q(1)=0 and hence Q=const is not fully demonstrated. The tables list Q(1)=0 as a boundary condition in the 'different possibilities' for the quadratic model and the power-law model, but they do not show the order-by-order elimination that excludes branches with Q(1)≠0. It is true from Eq. (38) that if Q'(0)=0, and the equation is of the form Q'' = Q' × F, then uniqueness of the ODE implies Q'≡0; however, the harder step is proving that Q'(0)=0 is necessary for regularity. Please supply the explicit recurrence relations obtained by substituting the expansions (57) into (36)–(39), and show that any regular solution necessarily has Q(1)=0, or qualify the conclusion as conditional on this additional assumption.
minor comments (5)
  1. [Sec. V.B] Typo: 'analiticity' should be 'analyticity'.
  2. [Eq. (59)] The expression with '±' is ambiguous; please clarify which sign is meant and how the branch is chosen consistently with the α→0 limit.
  3. [Sec. V.A] The phrase 'guillotining the additional degree of freedom' is informal; consider replacing with 'eliminating' or 'freezing'.
  4. [Tables II–IV] The formatting of Table II and III has malformed entries (X's and overstrikes); please clean up so the conditions and expansions are legible.
  5. [General] Since the connection classification of Ref. [12] is central to the argument, consider stating the theorem explicitly in an appendix rather than only citing it, to make the paper more self-contained.

Circularity Check

0 steps flagged

No significant circularity: the central GR-convergence result is derived internally from the stated EoMs; the imported connection classification is a non-circular geometric prior.

full rationale

The paper's main claim is conditional and derived, not assumed: substituting the Maclaurin ansatz (57) or the Laurent ansatz (58) into the equations of structure (36)–(39) and solving order by order leads to Q = const and hence to GR-like dynamics. This is an internal recurrence calculation; no parameter is fitted to a subset of data and then 'predicted', and 'Q = const' is not an input but an output of the regularity analysis. The exterior log r/r solution from [12] is used only as a heuristic counterexample to illustrate that non-analytic terms are missed by Laurent expansions, not as evidence for the no-go itself. The one load-bearing imported element is the symmetry-reduced connection classification in Table I, taken from [12] ('we refer to this study for more details', Sec. III.A). This is a parameter-free, purely geometrical classification (flat, torsionless, stationary, spherically symmetric connections) whose stated assumptions do not include the target GR-convergence result; under the rubric it counts as independent support rather than circularity. The no-hypermomentum condition is explicitly stated and relaxable, and the paper repeatedly frames the result as holding 'within our setup' and 'under stated assumptions'. Any concern about the completeness of Table I is a correctness/robustness issue, not a circularity of the derivation chain.

Axiom & Free-Parameter Ledger

5 free parameters · 7 axioms · 0 invented entities

The central claim rests on: (i) the teleparallel postulates and the symmetry-reduced connection ansatz from [12]; (ii) the no-hypermomentum, perfect-fluid matter assumptions; (iii) the regularity/series ansatz (57)–(58), which is the assumption being tested. No data are fitted; the free constants are model couplings and BVP boundary data, and the conclusions are robust to their values. The series ansatz is where the conditional no-go lives: the paper cannot and does not claim a full no-go, and explicitly identifies rotation, anisotropy, time dependence, hypermomentum, and non-analytic radial profiles as escape routes.

free parameters (5)
  • α (f(Q) = Q + αQ² coupling) = not fitted; α→0 is the GR limit
    Model parameter controlling deviation from GR; the regularity analysis holds for any real α, so the GR-convergence conclusion does not depend on a tuned value.
  • β (f(Q) = Q^β exponent) = not fitted; β=1 is the GR limit
    Second model's deviation parameter; same status as α. Note Q^β requires care with sign/domain of Q for non-integer β, which the paper does not address.
  • Γ^t_θθ (constant connection component) = arbitrary constant (drops out of all equations)
    Sec. III.A shows no equation depends on it; dynamically inert, so it does not affect the central result.
  • Central BVP data {Q(0), ∂Q(0), Γ^r_θθ(0)} = to be fixed by global boundary conditions / observables (not fixed here)
    Sec. V.C: the free data of the shooting/BVP problem. The paper argues regularity alone cannot fix them; they must be solved for globally.
  • Polytropic EoS constants k, γ = chosen by hand (γ ∈ [2,3] for NS)
    Sec. III.C: deliberately simplified EoS; realistic or piecewise-polytropic EoS are left to future work.
axioms (7)
  • standard math The affine connection is flat and torsion-free; the connection is purely inertial (Γ^α_μν = (∂x^α/∂ξ^λ) ∂_μ∂_νξ^λ, Eq. (11)); the coincident gauge exists but is not spherically symmetric in the adopted chart.
    Foundational postulates of symmetric teleparallel geometry (Sec. II.A). The non-spherical-symmetry of the coincident gauge is asserted following [12] (Sec. III.A).
  • domain assumption The most general flat, torsionless, stationary, spherically symmetric connection is exhausted by Table I (from [12]): 10 non-zero components, one dynamical function Γ^r_θθ(r), one free constant Γ^t_θθ.
    Load-bearing structural input, cited from [12] and not re-derived here (Sec. III.A). If further r-dependent connection components exist, the GR-convergence conclusion could fail; the paper's own conclusions list less restrictive connection ansätze as a relaxation.
  • domain assumption Matter is minimally coupled to the metric only; hypermomentum vanishes, H^μν_α = 0 (Eq. (16) area).
    Sec. II.B: "we always consider H^μν_α = 0." With hypermomentum the connection EoM would acquire matter sources and Q=const would not necessarily trivialize it.
  • domain assumption Matter is a perfect fluid (Eq. (31)) with polytropic EoS p = k ρ^γ (Eq. (41)).
    Sec. III.B/C; standard for NS modeling. The conclusions flag rotation, anisotropy, and time dependence as open directions.
  • domain assumption Regularity conditions 1–3 (regular center, asymptotic flatness, Maclaurin expansion at r=0 and Laurent in 1/r at infinity), Eqs. (57)–(58).
    Sec. V.B. The central no-go is conditional on these ansätze; the paper is explicit that non-analytic forms (r log r, etc.) could escape it.
  • standard math The connection EoM is equivalent to the covariant divergence of the metric EoMs: ˚∇_μM^μ_ν + C_ν = 0 (Eq. (21)), with matter conservation ˚∇_μT^μν = 0 imposed.
    Used to derive Eq. (28) in Sec. III.A; verified against the direct computation in [12]. The route is a shortcut, not an independent axiom, but it is load-bearing for the form of (38).
  • domain assumption Non-degeneracy throughout the domain: f′(Q) ≠ 0 and denominators such as (Γ^r_θθ)² + 2rm − r² and r − 2m do not vanish.
    The structure equations (36)–(39) contain 1/f′(Q), 1/[(Γ^r_θθ)² + 2rm − r²], and 1/(r − 2m); the paper does not analyze solutions crossing these loci.

pith-pipeline@v1.3.0-alltime-deepseek · 28463 in / 26113 out tokens · 230626 ms · 2026-08-03T18:53:03.780333+00:00 · methodology

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We investigate the challenges of constructing neutron star (NS) solutions in $f(\mathbb{Q})$ gravity, highlighting the importance of treating the affine connection as an active, dynamical component of the theory. We begin by clarifying under what conditions standard simplifications -- such as the coincident gauge or General Relativity (GR)-like connections -- inadvertently lead to GR behavior, even in non-trivial $f(\mathbb{Q})$ models. Building on previous work in black hole (BH) spacetimes, we adapt the formalism to NS and extend it to non-vacuum configurations. Focusing on two representative models, $f(\mathbb{Q}) = \mathbb{Q} + \alpha \mathbb{Q}^2$ and $f(\mathbb{Q}) = \mathbb{Q}^\beta$, our analysis suggests that, under standard regularity assumptions, solutions with Maclaurin/Laurent-type series recover GR dynamics, pointing to more intricate structures as the likely seat of beyond-GR effects, and reflecting the constraints imposed by the connection's dynamics on the asymptotic behavior of genuinely beyond-GR solutions. We then formulate the problem as a boundary value problem (BVP) and highlight the numerical pathologies that may arise, together with possible strategies to prevent them. This work aims to provide a concrete framework for future numerical studies and outlines the theoretical consistency conditions required to construct physically meaningful beyond-GR NS solutions in $f(\mathbb{Q})$ gravity.

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