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arxiv: 2604.03332 · v1 · submitted 2026-04-03 · 🌀 gr-qc

Analysis of Charged Compact Stars with Bardeen Black Hole in f(mathfrak{Q}, mathcal{T}) Gravity

M. Sharif , Iqra Ibrar This is my paper

Pith reviewed 2026-05-13 19:10 UTC · model grok-4.3

classification 🌀 gr-qc
keywords f(Q,T) gravitycharged compact starsBardeen black holeFinch-Skea metricenergy conditionsstellar stabilitymodified gravityTolman-Oppenheimer-Volkoff equation
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The pith

Charged compact stars remain physically valid when modeled in f(Q, T) gravity with Bardeen exteriors.

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

The paper constructs interior solutions for charged compact stars inside the f(Q, T) framework. It matches a Finch-Skea metric potential for the stellar interior to a Bardeen black hole geometry outside. Physical quantities including energy density, pressure profiles, anisotropy, and energy conditions are examined through graphical plots. Equilibrium is checked with the Tolman-Oppenheimer-Volkoff equation, and stability is tested using causality and the adiabatic index. The resulting configurations are shown to meet all required physical criteria.

Core claim

The central claim is that Finch-Skea interior solutions matched to Bardeen exteriors produce charged compact star models in f(Q, T) gravity that exhibit finite central density and pressure, obey the energy conditions, satisfy the Tolman-Oppenheimer-Volkoff equilibrium equation, and pass stability tests based on the causality condition and adiabatic index greater than 4/3.

What carries the argument

The f(Q, T) gravitational action, with Q the non-metricity scalar and T the trace of the energy-momentum tensor, which alters the field equations and permits new interior-exterior matching for charged stars.

If this is right

  • Density and pressure decrease monotonically from the center to the surface.
  • The equation of state parameters remain within realistic bounds for stellar matter.
  • The mass-radius relation produces configurations consistent with known compact objects.
  • Both the causality condition and adiabatic index criteria are satisfied throughout the interior.

Where Pith is reading between the lines

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

  • The modified field equations may allow higher compactness or different mass limits than in general relativity.
  • These models could be tested against X-ray observations of neutron star radii or gravitational wave signals from mergers.
  • Switching to other regular exteriors or including rotation would provide additional constraints on the theory.

Load-bearing premise

The assumption that the Finch-Skea potential combined with the Bardeen exterior accurately represents the spacetime geometry of a charged compact star and that graphical checks alone confirm physical validity.

What would settle it

Detection of a compact star whose measured adiabatic index falls below 4/3 in the interior or whose energy conditions are violated would falsify the claim that these solutions are physically valid.

Figures

Figures reproduced from arXiv: 2604.03332 by Iqra Ibrar, M. Sharif.

Figure 1
Figure 1. Figure 1: Graphs of ρ, pr and pt as functions of r. 0 2 4 6 8 10 -0.04 -0.03 -0.02 -0.01 0.00 rPkmT dΡ dr Ζ=1.9 Ζ=1.7 Ζ=1.5 Ζ=1.3 Ζ=1.1 0 2 4 6 8 10 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 rPkmT dpr dr Ζ=1.9 Ζ=1.7 Ζ=1.5 Ζ=1.3 Ζ=1.1 0 2 4 6 8 10 -0.14 -0.12 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 rPkmT dpt dr Ζ=1.9 Ζ=1.7 Ζ=1.5 Ζ=1.3 Ζ=1.1 [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Graphs of dρ dr , dpr dr and dpt dr with respect to r. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Graph of ∆ against r. for analyzing these variations: when ∆ = 0, the pressure is uniformly dis￾tributed, indicating isotropy. If ∆ > 0, it corresponds to an outward directed anisotropic force, whereas ∆ < 0 indicates an inward directed force. This characteristic is critical in understanding the stability and behavior of com￾pact stars. As depicted in [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Graphs of energy conditions as functions of [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Graphs of EoS against r. 3.4 Equation of State Parameter The EoS parameters, ωr = pr ρ and ωt = pt ρ , are essential in defining the rela￾tionship between pressure and energy density within compact stars. These parameters provide valuable insights into the star matter distribution and thermodynamic behavior. Ensuring the physical consistency of the stellar model requires that the EoS parameters fall betwee… view at source ↗
Figure 6
Figure 6. Figure 6: Plots of Forces against r. • Anisotropic force: Fa = 2 r (pt − pr), • Gravitational force: Fg = α ′ (r) 2 (ρ + pr), • Electric force: Fe = σ(r)E(r)e β(r) 2 , • Hydrostatic force: Fh = dpr dr [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Plot of M(r) against r. 0 2 4 6 8 10 0.00 0.05 0.10 0.15 0.20 0.25 rPkmT UHrL Ζ=1.9 Ζ=1.7 Ζ=1.5 Ζ=1.3 Ζ=1.1 [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Plot of U(r) against r. As r → 0, the behavior of the mass function reveals that M(r) → 0, indicat￾ing it remains regular and well-defined at the center, even under the Bardeen framework. Additionally, as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Graph of Z(r) against r. at the object surface. Consequently, surface redshift acts as an important parameter in the evaluation of dense stellar remnants. Its mathematical representation is given as Z(r) = 1 − 2U(r) − 1 2 − 1. In the case of an anisotropic configuration, the surface redshift must remain below Z(r) ≤ 5.211 for the compact star to be deemed physically viable [58]. As depicted in [PITH_FULL… view at source ↗
Figure 10
Figure 10. Figure 10: Plots of causality conditions against r. 0 2 4 6 8 10 0.155 0.156 0.157 0.158 0.159 0.160 0.161 rPkmT v 2 r-v 2 t Ζ=1.9 Ζ=1.7 Ζ=1.5 Ζ=1.3 Ζ=1.1 [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Plot of Herrera cracking against r. v 2 t − v 2 r |≤ 1. When this condition is met, it signifies that cosmic structures are stable and can preserve their configuration over time. In contrast, a violation of this criterion indicates instability, which could lead to structural collapse [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Plots of Γr and Γt against r. the formation of compact stars, enabling them to resist gravitational collapse and preserve energy with minimal dissipation. 4 Concluding Remarks Modified theories are pivotal in astrophysics for understanding the complex￾ities of stellar configurations. Among these, f(Q, T ) gravity has attracted considerable interest due to its unique integration of non-metricity and mat￾te… view at source ↗
read the original abstract

This study investigates the behavior of charged compact stars within the $f(\mathfrak{Q}, \mathcal{T})$ gravitational framework, where $\mathfrak{Q}$ denotes the non-metricity scalar and $\mathcal{T}$ represents the trace of the energy-momentum tensor. Recognized as a promising model for explaining the accelerated expansion of the universe, this approach offers a strong theoretical foundation. A central focus of this research is the application of Bardeen's model to describe the exterior spacetime. Additionally, the study examines the internal structure of compact stars using a solution based on the Finch-Skea metric potential. Various physical properties, including energy density, pressure components, anisotropy, energy conditions and equation of state parameters are analyzed through graphical representations. Equilibrium conditions are explored via the Tolman-Oppenheimer-Volkoff equation while key characteristics such as the mass-radius relationship, compactness, surface redshift and stability criteria based on the causality condition and adiabatic index are thoroughly evaluated. The analysis concludes that the proposed solutions for charged compact stars in this framework are both theoretically consistent and physically valid.

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

1 major / 2 minor

Summary. The manuscript analyzes charged compact stars in f(𝔮, 𝒯) gravity by matching a Finch-Skea interior solution to a Bardeen exterior metric. It examines physical properties such as energy density, pressure anisotropy, energy conditions, Tolman-Oppenheimer-Volkoff equilibrium, mass-radius relations, compactness, surface redshift, and stability criteria (causality and adiabatic index) primarily through graphical representations, concluding that the models are theoretically consistent and physically valid.

Significance. If the exterior consistency is established, the work would add to the literature on compact star solutions in modified gravity theories involving non-metricity and matter coupling. The comprehensive graphical analysis of multiple physical quantities and stability conditions provides a detailed viability check, which could be valuable for exploring alternatives to general relativity in stellar astrophysics.

major comments (1)
  1. [Exterior spacetime and junction conditions] The Bardeen metric is adopted for the exterior without deriving or verifying that it satisfies the f(𝔮, 𝒯) field equations in the region where 𝒯 = 0. The modified field equations include non-metricity and trace terms that do not automatically reduce to the Einstein equations with nonlinear electrodynamics; a specific choice of f or explicit substitution is required to confirm matching. This is central to the claim of global consistency and physical validity.
minor comments (2)
  1. [Graphical analysis] The figures lack error bars or sensitivity analysis with respect to the free parameters in f(𝔮, 𝒯); this would help assess robustness of the physical validity conclusions.
  2. [Notation] Ensure consistent use of symbols for the non-metricity scalar throughout the text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive review and recommendation for major revision. We address the single major comment on exterior spacetime and junction conditions below, agreeing that explicit verification strengthens the work. The revised manuscript will incorporate the necessary addition.

read point-by-point responses

  1. Referee: [Exterior spacetime and junction conditions] The Bardeen metric is adopted for the exterior without deriving or verifying that it satisfies the f(𝔮, 𝒯) field equations in the region where 𝒯 = 0. The modified field equations include non-metricity and trace terms that do not automatically reduce to the Einstein equations with nonlinear electrodynamics; a specific choice of f or explicit substitution is required to confirm matching. This is central to the claim of global consistency and physical validity.

    Authors: We agree with the referee that explicit verification of the exterior solution is required for rigorous global consistency. Our adopted functional form is f(𝔮, 𝒯) = 𝔮 + β 𝒯 (β a constant parameter). In the exterior region where 𝒯 = 0 the field equations reduce to the symmetric teleparallel equivalent of Einstein gravity sourced by the Bardeen nonlinear electromagnetic field. We will add a short dedicated subsection deriving the vacuum field equations under this choice and confirming that the Bardeen metric satisfies them identically. This will also make the junction conditions at the stellar surface fully explicit within the modified-gravity framework. The revision will be incorporated in the next version of the manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper adopts the Finch-Skea ansatz for the interior metric and the Bardeen solution for the exterior, selects a specific form for f(Q,T), derives the field equations under these assumptions, and then evaluates physical viability through standard criteria (energy conditions, TOV equation, stability indices) via graphical inspection. No step reduces a claimed prediction or consistency result to a fitted parameter or self-citation by construction; the matching conditions and exterior application are treated as inputs whose consequences are then checked against independent physical requirements. The derivation therefore remains self-contained rather than tautological.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the selection of specific metric potentials and the functional form of f(Q,T), which introduce free parameters not fixed by the underlying theory.

free parameters (1)
  • Coupling constants in f(Q,T)
    The function f(Q,T) is chosen with parameters that are adjusted to meet the physical conditions of the star model.
axioms (2)
  • domain assumption Finch-Skea ansatz for interior metric potential
    Used as the solution for the interior geometry without derivation from the field equations.
  • domain assumption Bardeen black hole solution for exterior spacetime
    Matched at the boundary for the exterior geometry.

pith-pipeline@v0.9.0 · 5491 in / 1276 out tokens · 49548 ms · 2026-05-13T19:10:02.637660+00:00 · methodology

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