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arxiv: 2409.04487 · v2 · submitted 2024-09-06 · 🌀 gr-qc

Physical properties and the maximum compactness bound of a class of compact stars in f(Q) gravity

Arpita Ghosh , Abhishek Paul , Ranjan Sharma , Samstuti Chanda This is my paper

Pith reviewed 2026-05-23 21:02 UTC · model grok-4.3

classification 🌀 gr-qc
keywords f(Q) gravitycompact starsVaidya-Tikekar ansatzcompactness boundanisotropic fluidmass-radius relationKarmarkar conditionBuchdahl bound
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The pith

In linear f(Q) gravity the maximum compactness of a compact star depends only on the Vaidya-Tikekar curvature parameter K.

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

The paper constructs closed-form interior solutions for anisotropic relativistic stars in linear f(Q) gravity by combining the Vaidya-Tikekar metric ansatz with the Karmarkar embedding condition and matching the result to the Schwarzschild exterior. Enforcing finite central pressure then yields a maximum compactness bound that is controlled solely by the single parameter K. When K vanishes the bound reduces to the standard Buchdahl limit for an isotropic homogeneous sphere. Numerical integration of the modified Tolman-Oppenheimer-Volkoff equation shows that larger values of the model parameter alpha reduce both maximum mass and radius, shifting stable configurations toward more compact objects and allowing the model to accommodate comparatively low-mass stars whose radii can be tuned to match observed pulsars.

Core claim

By adopting f(Q) = alpha Q + beta together with the Vaidya-Tikekar ansatz and the Karmarkar condition, the authors obtain an exact interior solution whose physical quantities (density, pressures, anisotropy) are fully determined. Imposing regularity at the center produces a compactness upper bound that depends only on the curvature parameter K of the Vaidya-Tikekar metric; the bound recovers the classic Buchdahl value 4/9 precisely when K = 0. Integration of the modified hydrostatic equilibrium equation then yields mass-radius curves whose maximum mass and radius both decrease with increasing alpha, permitting the construction of stable ultra-compact configurations and the fitting of radii (

What carries the argument

Vaidya-Tikekar metric ansatz together with the Karmarkar embedding condition inside linear f(Q) gravity

If this is right

  • The bound collapses exactly to the Buchdahl value 4/9 when K=0, recovering the isotropic homogeneous case.
  • Larger alpha produces smaller maximum mass and radius and shifts the stable branch toward ultra-compact objects.
  • The model supports stable configurations of comparatively low mass whose radii can be adjusted to fit observed pulsars.
  • Mass-radius relations are obtained by direct numerical integration of the modified Tolman-Oppenheimer-Volkoff equation using the derived density and radial pressure profiles.

Where Pith is reading between the lines

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

  • The K-dependent bound supplies a new, parameter-free way to translate future radius measurements of pulsars into constraints on the non-metricity coupling alpha.
  • If the same Vaidya-Tikekar ansatz continues to work in other modified-gravity theories, the same compactness formula may apply beyond f(Q).
  • The reduction in maximum mass with alpha suggests that f(Q) corrections could help reconcile theoretical mass limits with the lightest observed neutron stars.

Load-bearing premise

The interior geometry is exactly described by the Vaidya-Tikekar metric ansatz combined with the Karmarkar condition, which permits a closed-form solution matched to the Schwarzschild exterior.

What would settle it

A single compact star whose measured compactness exceeds the analytic upper bound computed from its independently determined Vaidya-Tikekar parameter K would falsify the central claim.

Figures

Figures reproduced from arXiv: 2409.04487 by Abhishek Paul, Arpita Ghosh, Ranjan Sharma, Samstuti Chanda.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Metric potentials [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Plot of radial pressure [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Anisotropic factor ∆ plotted against radial distance [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Compactness bound [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Compactness bound [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Mass-radius relationship for different values of [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Maximum mass vs radius curve for a pulsar [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Compactness ( [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
read the original abstract

Motivation: Motivated by the growing interest in understanding the role of non-metricity in describing dense stellar systems, in this paper, we study compact stellar configurations within the framework of linear $f(Q)$ gravity. Methodology: By adopting a linear modification of the form $f(Q) = \alpha Q+\beta$, we analyze the internal structure and physical properties of an anisotropic relativistic star within the framework of $f(Q)$ gravity. We employ the Karmarkar's condition together with the Vaidya-Tikekar metric ansatz to obtain a closed-form interior solution of the star. The interior solution is then matched to the Schwarzschild exterior solution across the boundary of the star. By varying the model parameters, we analyze physical features of the resultant stellar configuration. Results: We note distinctive features in the density, pressure, anisotropy and total mass of the star under a such modification. By enforcing the condition that the central pressure remains finite, we obtain the maximum compactness bound which is shown to depend solely on the Vaidya-Tikekar curvature parameter $K$. We recover the Buchdahl bound for the curvature parameter $K=0$, which corresponds to the solution for an isotropic and homogeneous fluid sphere. Utilizing the energy density and radial pressure profiles, we numerically integrate the modified Tolman-Oppenheimer-Volkoff equations and obtain the mass-radius ($M-R$) relationships for different values of the model parameter $\alpha$. We note that for higher values of $\alpha$, the maximum mass and radius decrease, shifting the stable branch towards ultra-compact configurations. An interesting observation in our analysis is that a linearly modified $f(Q)$ gravity model can support comparatively low mass stars. Utilizing the observed mass of some known pulsars, we demonstrate how our model can be used to fine-tune the radius of the star.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript studies anisotropic compact stars in linear f(Q) gravity with f(Q)=αQ+β. Adopting the Vaidya-Tikekar ansatz together with the Karmarkar condition yields a closed-form interior solution that is matched to the Schwarzschild exterior. Enforcing finite central pressure produces a maximum compactness bound that depends only on the ansatz parameter K (recovering the Buchdahl bound at K=0). Physical quantities are examined by varying α and β; the modified TOV equation is integrated numerically to generate M-R curves, which are then compared with observed pulsar masses to illustrate radius tuning.

Significance. If the exterior matching and field-equation verification hold, the work supplies an explicit K-dependent compactness bound in f(Q) gravity and demonstrates that linear non-metricity modifications can accommodate low-mass compact objects with tunable radii. The closed-form interior solution is a technical strength, though the bound remains tied to the chosen metric ansatz and the M-R analysis incorporates a fitting step.

major comments (2)
  1. [Matching to exterior] Matching to exterior (§ on junction conditions): The interior solution is matched to the Schwarzschild metric without an explicit demonstration that the Schwarzschild line element satisfies the vacuum f(Q) field equations for arbitrary α and β. Because the junction conditions fix the integration constants that enter the compactness bound (obtained from the finite-central-pressure requirement), this consistency check is load-bearing for the central claim.
  2. [Compactness bound derivation] Compactness bound derivation (section deriving the bound from central pressure): The bound is expressed solely in terms of the free parameter K inserted by the Vaidya-Tikekar ansatz. While the K=0 case recovers Buchdahl, the subsequent M-R curves are generated by varying α and then using observed pulsar masses to tune radii; this fitting step reduces the predictive content of the model relative to the claimed bound.
minor comments (1)
  1. The abstract states that the modified TOV equation is integrated numerically, yet no error estimates, convergence tests, or verification that the numerical solution satisfies the field equations pointwise are supplied.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and will incorporate clarifications and verifications in a revised manuscript.

read point-by-point responses

  1. Referee: Matching to exterior (§ on junction conditions): The interior solution is matched to the Schwarzschild metric without an explicit demonstration that the Schwarzschild line element satisfies the vacuum f(Q) field equations for arbitrary α and β. Because the junction conditions fix the integration constants that enter the compactness bound (obtained from the finite-central-pressure requirement), this consistency check is load-bearing for the central claim.

    Authors: We agree that an explicit check was omitted. In linear f(Q) gravity the vacuum equations outside the star reduce to the Einstein equations with an effective cosmological term set by β; the Schwarzschild metric satisfies these equations when the non-metricity scalar vanishes and β is absorbed into the integration constants fixed by junction conditions. We will add a short appendix deriving the vacuum field equations for f(Q)=αQ+β and confirming that the Schwarzschild exterior is an exact solution, thereby validating the matching used to obtain the compactness bound. revision: yes

  2. Referee: Compactness bound derivation (section deriving the bound from central pressure): The bound is expressed solely in terms of the free parameter K inserted by the Vaidya-Tikekar ansatz. While the K=0 case recovers Buchdahl, the subsequent M-R curves are generated by varying α and then using observed pulsar masses to tune radii; this fitting step reduces the predictive content of the model relative to the claimed bound.

    Authors: The compactness bound is obtained solely from the finiteness of central pressure in the closed-form interior solution and depends only on K, recovering the Buchdahl limit at K=0. This is a theoretical upper limit within the chosen ansatz and is independent of α and β. The M-R curves are generated by integrating the modified TOV equation with the α-dependent density and pressure profiles; the subsequent comparison with observed pulsar masses is presented only to illustrate that the model can accommodate known objects by radius tuning via α. We do not claim parameter-free predictions. We will revise the text to emphasize the distinction between the K-dependent bound and the illustrative role of the M-R analysis. revision: partial

Circularity Check

2 steps flagged

Compactness bound depends on Vaidya-Tikekar parameter K by construction from the adopted ansatz; M-R relations incorporate observational tuning

specific steps
  1. self definitional [Abstract]
    "By enforcing the condition that the central pressure remains finite, we obtain the maximum compactness bound which is shown to depend solely on the Vaidya-Tikekar curvature parameter K. We recover the Buchdahl bound for the curvature parameter K=0, which corresponds to the solution for an isotropic and homogeneous fluid sphere."

    The interior geometry is constructed from the Vaidya-Tikekar metric ansatz (introduced as an input to obtain a closed-form solution), which defines K as a free curvature parameter. The bound depending solely on K is therefore a direct algebraic consequence of that ansatz choice rather than an independent result derived from the f(Q) field equations alone.

  2. fitted input called prediction [Abstract]
    "Utilizing the observed mass of some known pulsars, we demonstrate how our model can be used to fine-tune the radius of the star."

    After obtaining M-R relations by numerical integration of the modified TOV equations for varying α, the paper uses observed pulsar masses to adjust/tune the radii. This makes the reported stellar configurations a result of fitting to external data rather than a pure prediction from the theory.

full rationale

The paper adopts the Vaidya-Tikekar ansatz plus Karmarkar condition as the starting point for a closed-form interior solution in linear f(Q) gravity, then derives a compactness bound that depends only on the ansatz parameter K. This reduces the reported bound to a property of the chosen metric family. The M-R curves are generated by varying α and integrating the modified TOV equations, followed by using observed pulsar masses to tune radii, introducing a fitting element that limits independent predictive content. No self-citation chains or uniqueness theorems from the same authors are invoked as load-bearing. The exterior matching and field-equation consistency issues raised by the skeptic are correctness concerns rather than circularity. Overall moderate circularity because the central bound result is tied directly to the input ansatz.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central results rest on the choice of linear f(Q), the Vaidya-Tikekar ansatz, and the Karmarkar condition; α and K function as free parameters that are varied rather than derived.

free parameters (2)
  • α
    Coefficient of the linear term in f(Q) = αQ + β; varied to generate families of M-R curves and to match observed pulsar masses.
  • K
    Curvature parameter of the Vaidya-Tikekar ansatz; directly sets the value of the maximum compactness bound.
axioms (2)
  • domain assumption Karmarkar's condition is imposed to obtain a closed-form interior solution
    Invoked in the methodology paragraph to reduce the field equations to an integrable system.
  • ad hoc to paper The linear form f(Q) = αQ + β is an adequate modification for stellar interiors
    Chosen at the outset without derivation from a more fundamental principle.

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    A controlled two-parameter deformation in linear f(Q) gravity with gravitational decoupling enlarges the stellar mass window for compact objects while satisfying causality and regularity.

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