Study on physical properties and characteristics of an anisotropic compact star model using Karmarkar Condition in F(Q) gravity

Jitendra Kumar; Sat Paul; S. K. Maurya

Study on physical properties and characteristics of an anisotropic compact star model using Karmarkar Condition in F(Q) gravity

T0 review · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A Karmarkar-based anisotropic star model in linear F(Q) gravity is fitted to Cen X-3, but its own equations and tables disagree.

desk verdict Fails its own central consistency check: the paper's density formula gives a negative central density for its own fitted parameters, so the physical-plausibility claim does not survive. read the letter →

arxiv 2505.15853 v2 pith:35ISOE6J submitted 2025-05-20 gr-qc astro-ph.HEastro-ph.SR

Sat Paul , Jitendra Kumar , S. K. Maurya This is my paper · ORCID

classification gr-qcastro-ph.HEastro-ph.SR

keywords compactstaranisotropicmodelconditionequationfieldphysical

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

The paper tries to build a mathematical model of a compact star, an extremely dense object like a neutron star, using a modified theory of gravity called F(Q) gravity. In this theory, gravity is described by a quantity called non-metricity instead of curvature. However, the authors only use the simplest linear form, F(Q)=δ1 Q+δ2, which their own equations show is mathematically the same as Einstein's general relativity with a re-scaled gravitational constant and a tiny cosmological constant. So the model does not actually test any new physics.

The authors choose a specific form for the star's interior (a hyperbolic function for one of the metric components), impose the Karmarkar condition, and then use the known mass and radius of the compact star Cen X-3 to solve for the remaining constants. They then check that density and pressure are positive, that the energy conditions hold, and that the star is stable against various perturbations. They conclude that the model is physically plausible.

There are serious problems. The numbers in the tables do not seem to come from the equations: using the stated constants on the density formula gives a negative central density, while the table reports positive values. The reported central density is about 4×10^12 g/cm^3, but the abstract and conclusion claim 10^14 g/cm^3. The cosmological constant implied by δ2 is about 2.5×10^-5 km^-2, which is not negligible at all. And because the mass and radius are used as inputs to fix the model, the agreement with Cen X-3 is not a prediction.

Extended reading notes

Core claim

The paper's central assertion is that the solution from Eq (34) and (35) is "a physically plausible representation of a compact star with an energy density of order 10^14 g/cm^3" and that for Cen X-3 the model satisfies all energy conditions, causality, TOV equilibrium, and stability criteria (Table II and conclusion). If true, this would establish a new exact anisotropic stellar model valid in F(Q) gravity and consistent with the observed mass and radius M=1.48-1.49 M_sun, R=11.6 km.

Load-bearing premise

The unverified premise that the fitted parameters in Table I (b=0.107, f=-0.0011, h=1.39, δ2=0.00001, and the boundary-fitted A1, B1) produce a non-negative, monotonically decreasing central density of the order stated. In Section III.C and Table III the authors assert central densities of about 0.04-0.2×10^14 g/cm^3, but substituting Table I into Eq (36) at r=0 gives a negative central density in geometric units. Either the table values or the field equations (or both) contain an error, and without this premise the model is not physical.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Bottom line: the model fails its own central consistency check. Eq (36) with Table I gives a negative central density, and that mismatch sinks the physical-plausibility claim. The paper is a competent template exercise otherwise, but the arithmetic contradiction is load-bearing.

What it does well: it is clearly organized, sets up the F(Q) field equations carefully, notes that F_QQ=0 forces a linear F(Q), matches the interior to a Schwarzschild-(anti-)de Sitter exterior, and runs the standard battery of checks: energy conditions, causality, TOV equilibrium, adiabatic index, stability. The algebra is explicit in an appendix, which is more than many papers in this genre do.

Soft spots, in descending order:

1. Central density mismatch. Evaluating (36) at r=0 with b=0.107, delta_1=0.2, delta_2=1e-5, h=1.39 gives 8*pi*rho(0) = delta_2/2 - 3*b^2*delta_1*csch^2(h) ≈ 5e-6 - 1.9e-3 < 0. In physical units that is about -1e14 g/cm^3, not +0.041e14 as in Table III. This is not an interpretation difference; it is a direct contradiction between the printed formula, the fitted constants, and the reported numbers. Everything downstream—density plots, energy conditions, stability—is drawn from a central density that is not positive.

2. The mass-radius relation is not a prediction. M and R of Cen X-3 are inputs used to fix A1 and B1; Fig. 12 then shows curves that reproduce those inputs. That is fitting, not testing.

3. Physical novelty is thin. With F(Q)=delta_1 Q + delta_2, the theory is GR with a rescaled G and a cosmological constant; the paper itself labels delta_1=1 as the GR case. The metric ansatz comes from the authors' own earlier paper. The new ingredient is only applying a known ansatz to a linear case already present in the cited literature.

None of this requires deep expertise to verify. A referee would need one calculation on the back of an envelope. The stress-test note holds up.

Who this is for: readers who want another example in the Karmarkar-anisotropic-star genre might browse it, but they should not cite the physical conclusions. Recommendation: desk reject. If it is sent to review, the referee should check Eq (36) at r=0 first.

Assumptions & free parameters 7 free parameters · 5 assumptions · 0 invented entities

The central solution depends on six parameters fitted to a single star (δ1, δ2, b, f, h and the boundary-fitted A1/B1). There is no external prediction: the mass and radius are inputs. The Karmarkar condition and the linear F(Q) ansatz are standard assumptions, and the metric ansatz is self-cited from Ref [99].

free parameters (7)

δ1 = 0.2, 0.4, 0.6, 0.8, 1.0
Integration constant in F(Q)=δ1Q+δ2; varied by hand; δ1=1.0 labeled 'GR case'.
δ2 = 0.00001
Integration constant set arbitrarily; contributes a cosmological-constant-like term Λ=δ2/(2δ1).
b = 0.107 km^-1
Parameter in the metric ansatz e^σ chosen to match Cen X-3.
f = -0.0011 km^-1
Parameter in the metric ansatz, chosen by hand.
h = 1.39
Parameter in the metric ansatz, chosen by hand.
A1 = 0.02018 to 0.02698
Integration constant from Karmarkar solution, fixed by boundary matching to measured M and R.
B1 = 0.02633 to 0.02656
Integration constant fixed by boundary matching to measured mass and radius.

assumptions (5)

domain assumption The F(Q) gravity action (9) and field equations (10)-(12) in the coincident gauge are the correct framework for stellar structure.
The whole paper operates inside this theory; no independent justification is given.
domain assumption F(Q) must be linear, F(Q)=δ1Q+δ2, because Eq (25) forces F_{QQ}=0.
This makes the model equivalent to GR with a re-scaled gravitational constant and a cosmological constant, eliminating genuine modified-gravity effects.
domain assumption The Karmarkar condition Eq (31) is imposed, requiring the spacetime to be of embedding class one.
A geometric restriction used to relate metric components; it is a modeling assumption, not a physical requirement.
ad hoc to paper The metric ansatz e^σ = b^2 r^2 csch^2(f r^2+h)+1 is regular and admissible.
Chosen for solvability and taken from the authors' previous paper [99]; no physical derivation is provided.
domain assumption The exterior spacetime is Schwarzschild-(anti-)de Sitter.
Standard matching assumption; the cosmological constant is identified with δ2/(2δ1).

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Cite this review

Pith. "Pith review of Study on physical properties and characteristics of an anisotropic compact star model using Karmarkar Condition in F(Q) gravity." pith.science (2026). https://pith.science/paper/35ISOE6J

@misc{pith2026250515853,
  author       = {Pith},
  title        = {Pith review of: Study on physical properties and characteristics of an anisotropic compact star model using Karmarkar Condition in F(Q) gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/35ISOE6J}},
  note         = {Machine review of arXiv:2505.15853}
}

abstract

The main aim of this study is to examine the behaviour of physical parameters of an anisotropic compact star model demonstrating spherical symmetry in F(Q) modified gravity. To evaluate the behaviour and the stability of an anisotropic compact star model, we utilise the measured mass and radius of an anisotropic compact star model. This study obtained an anisotropic compact star model by solving Einstein field equations. The field equations have been simplified by an appropriate selection of the metric elements and the Karmarkar condition. By solving the field equation to develop a differential equation that establishes a relationship between two essential components of spacetime. A physical analysis of this model reveals that the resulting stellar structure for anisotropic matter distribution is a physically plausible representation of a compact star with an energy density of order $10^14 g/cm^3$. Using the Tolman-Oppenheimer-Volkoff equation, causality condition and Harrison-Zeldovich-Novikov Condition, we investigate the hydrostatic equilibrium and stability of the compact star Cen X-3. We further determined the mass-radius relation of this compact star for different values of delta}1.

Figures

Figures reproduced from arXiv: 2505.15853 by the authors.

**Figure 2.** FIG. 2: The behaviour of energy density( [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** FIG. 3: The behaviour of gradient of energy density( [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: FIG. 4: The behaviour [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: FIG. 5: The behaviour of NEC, WEC, DEC, SEC and TEC for a fixed value of [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: FIG. 6: Graphical representation of Adiabatic index Γ for a [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: FIG. 7: The behaviour of gravitational force ( [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

**Figure 8.** Figure 8: FIG. 8: The behaviour of radial velocity ( [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

**Figure 9.** Figure 9: FIG. 9: The behaviour of [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

**Figure 10.** Figure 10: FIG. 10: The behaviour of Mass and [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]

**Figure 11.** Figure 11: FIG. 11: The behaviour of Gravitational Redshift ( [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]

**Figure 12.** Figure 12: FIG. 12: The behaviour of [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Neutron stars in $f(\mathbb{Q})$ gravity
gr-qc 2025-12 conditional novelty 6.0 of 10

For f(Q)=Q+αQ² and f(Q)=Q^β, neutron-star solutions that admit power-series expansions at the center or at infinity collapse to General Relativity; genuine beyond-GR effects must be non-analytic.

Reference graph

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Q+ 1 r2 + e−σ r (Υ′ +σ ′) # ,(21) pr = F 2 − FQ

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The causality condi- tions provide boundaries on the radial and tangential ve- locities, denoted asV 2 r andV 2 t , respectively

Causality and Stability requirement This study examines the validity of the causality crite- rion for the proposed stellar model. The causality condi- tions provide boundaries on the radial and tangential ve- locities, denoted asV 2 r andV 2 t , respectively. The bound- aries are specified as 0<|V 2 i |<1, wherei=r, t. The 10 0.00010 0.00015 0.00020 0.000...
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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Equilibrium requirement 9

[2] [2]

Causality and Stability requirement 9

[3] [3]

Stability Versus Convection 10 ∗Email:satpaul232039cuh@gmail.com †Email:jitendark@gmail.com ‡Email:sunil@unizwa.edu.om

[4] [4]

Q+ 1 r2 + e−σ r (Υ′ +σ ′) # ,(21) pr = F 2 − FQ

Harrison-Zeldovich-Novikov Stability Condition 13 J. Redshift Profile 13 K. Mass and compactness of the model 13 IV. Conclusion14 Acknowledgments17 Data access statement18 Conflict of interest18 Appendix19 References23 I. INTRODUCTION Various universal events [1–3] have demonstrated that improvements to general relativity (GR) are essential at a geometric...

arXiv 1916

[5] [5]

Einstein, Albert, Albert Einstein: Akademie-Vortr¨ age: Sitzungsberichte der Preußischen Akademie der Wis- senschaften 1914–1932, 316–321, 2005,

1914

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Equilibrium requirement The Tolman-Oppenheimer-Volkoff (TOV) equation has been described as an equilibrium condition for a compact star, combining gravitational, hydrostatic and anisotropic forces [107]. The standard representation of the TOV equation is expressed as dpr dr + Υ′(ρ+p r) 2 − 2(pt −p r) r = 0 (58) On the other hand, it can be stated as Fg +F...

[7] [7]

The causality condi- tions provide boundaries on the radial and tangential ve- locities, denoted asV 2 r andV 2 t , respectively

Causality and Stability requirement This study examines the validity of the causality crite- rion for the proposed stellar model. The causality condi- tions provide boundaries on the radial and tangential ve- locities, denoted asV 2 r andV 2 t , respectively. The bound- aries are specified as 0<|V 2 i |<1, wherei=r, t. The 10 0.00010 0.00015 0.00020 0.000...

[8] [8]

It was illus- trated in [109], such that ρ′′(r)≤0 (60) We demonstrate the graphical representation ofρ ′′ as a function of r in Fig

Stability Versus Convection The buoyancy principle inside a fluid demonstrates that every displaced fluid component will revert to its original position, as evidenced by the stability of a self- gravitating sphere in contrast to convection. It was illus- trated in [109], such that ρ′′(r)≤0 (60) We demonstrate the graphical representation ofρ ′′ as a funct...