Karmarkar-Tolman Embedded Charged Anisotropic Stars in f(R) Gravity

De-Liang Yao; M.Ilyas; W.U.Rahman; Yi Zhong

REVIEW 4 major objections 5 minor 59 references

Karmarkar-Tolman Embedded Charged Anisotropic Stars in f(R) Gravity

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Three f(R) gravity models yield stable charged anisotropic compact stars.

desk verdict Routine extension of Karmarkar-Tolman charged stars to three f(R) models, but the matching is underdetermined: Q is never specified and the tabulated constants are inconsistent with the stated junction conditions. read the letter →

arxiv 2505.07296 v1 pith:ZJ6YJJSO submitted 2025-05-12 gr-qc hep-th

W.U.Rahman , M.Ilyas , Yi Zhong , De-Liang Yao This is my paper · ORCID

classification gr-qchep-th MSC 83C1583D05 PACS 04.50.Kd04.40.Dg97.60.Jd

keywords f(R)gravitychargedcompactstarsanisotropicmatterdistributionKarmarkar-TolmanspacetimeReissner-Nordströmmatchingenergyconditionssoundspeedcausalitystrangestarmodels

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 sets out to show that charged, anisotropic compact stars can be modeled consistently inside $f(R)$ gravity by building the interior on the Karmarkar-Tolman embedding and fixing the charge profile from a bag-model equation of state. Three curvature-modified gravity models — a quadratic correction $f(R)=R+\alpha R^2$, an exponential correction $f(R)=R+\alpha R(e^{-R/\gamma}-1)$, and a cubic extension $f(R)=R+\alpha R^2(1+\gamma R)$ — are each matched to the Reissner-Nordström exterior for three known compact-star candidates. The authors report that every model passes the standard physical tests: density and pressures fall monotonically from the core, all energy conditions hold, the four TOV forces balance, and both sound speeds remain subluminal. The payoff is a set of exact, observationally anchored stellar solutions in modified gravity that can be compared directly with mass-radius measurements.

What carries the argument

The argument is carried by the Karmarkar-Tolman metric ansatz $e^{a}=A(1+Br^2)^4$, $e^{b}=1+64B^2ACr^2/(1+Br^2)^2$, whose functions satisfy the Karmarkar embedding condition — a differential relation that lets a 4D spherically symmetric spacetime be embedded in 5D flat space. Plugging this ansatz into the $f(R)$ field equations produces the density and pressure expressions; requiring the bag equation of state then determines the interior charge function in Eq. (19). Matching the interior to the Reissner-Nordström exterior fixes the constants $A$, $B$, $C$ from the star's mass and radius. The same machinery is run for three $f(R)$ models, and the output profiles are tested for monotonicity, energy conditions, TOV force balance, causality, and anisotropy.

What would settle it

Evaluate the charge function in Eq. (19) at $r=R$ for each star and check whether the resulting $Q(R)$ equals the total charge used to compute $A$, $B$, $C$ in Table I from Eq. (21). If the two values disagree, or if no $Q$ is specified, then the plotted density, pressure, and charge profiles do not follow from the stated equations.

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Extended reading notes

Core claim

The central claim is that the Karmarkar-Tolman metric, together with a charge profile derived from the bag equation of state $P_r=(\rho-4B_g)/3$, produces physically viable charged anisotropic star interiors in $f(R)$ gravity. The paper derives explicit density, radial-pressure, and tangential-pressure functions from the $f(R)$ field equations, obtains the charge function from the assumed equation of state, fixes the metric constants by matching to the Reissner-Nordström exterior, and then checks the resulting configurations against energy conditions, TOV equilibrium, causality of sound speeds, and the anisotropy sign. For the three $f(R)$ models and three strange-star candidates considered, the paper concludes that the models are stable and physically acceptable, with charge and anisotropy providing outward forces that help balance gravity. The intended result is a constructive demonstration that modified-gravity corrections do not break the viability of charged anisotropic stellar models, but can support them.

Load-bearing premise

The load-bearing assumption is that the total surface charge $Q$ has a definite value that makes the matching constants in Eq. (21) consistent with the charge profile produced by Eq. (19); the paper does not state that value, so the entire numerical construction depends on an unspecified input.

Editorial extensions

If this is right

If the models are correct, the three $f(R)$ corrections can each accommodate charged anisotropic strange-star candidates with masses and radii matching Her X-1, SAX J 1808.4-3658, and 4U 1820-30.
The positive anisotropy $\Delta=P_t-P_r>0$ and outward electric force imply that charge is not a small perturbation but a structural element that helps prevent collapse in these $f(R)$ interiors.
Because both sound speeds stay below unity and $0<|v_{st}^2-v_{sr}^2|<1$, the models satisfy the standard causality and stability criteria, so they are candidates for further dynamical or oscillation analysis.
The same construction procedure could be applied to additional $f(R)$ forms, since the charge function and matching conditions are derived before a specific $f(R)$ is chosen.

Reading between the lines

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

The paper never reports the surface charge $Q$ that enters the matching constants via Eq. (21), so the numerical profiles in the figures are not yet reproducible; a natural next step is to solve Eq. (19) at the surface self-consistently and report the resulting charges.
The matching procedure could be turned into a parameter-estimation tool: treating the surface charge and the $f(R)$ parameters as free, one could fit the predicted mass-radius curves to pulsar data and see which $f(R)$ model is favored.
The same Karmarkar-embedding plus equation-of-state-derived charge construction could be tested with more realistic equations of state, such as ones with superfluidity or strong magnetic fields, to see whether the stability conclusions survive.
Because the paper compares with general relativity only through the $\alpha=0$ limit, an explicit GR-limit check of the same stars would isolate how much of the stability comes from the $f(R)$ corrections rather than from the embedding and charge.

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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

The paper adds charge and three f(R) Lagrangians to the familiar Karmarkar-Tolman embedding. The field equations, the quark EoS, and the standard checks (energy conditions, TOV balance, sound speeds) are all handled in the usual way. That part is competently executed, and someone working in compact-star modeling could lift the formulas directly.

But there is a load-bearing problem in the matching. The constants A, B, C are supposed to come from Eq. (21), which depends explicitly on the total charge Q at the surface. Q is never reported anywhere. Worse, inverting Eq. (21) with the tabulated masses, radii, and constants gives Q² ≈ -3.5 km² for Her X-1, and similar negative values for the others. No real charge reproduces the table. Setting Q = 0 gives A ≈ 0.51, B ≈ 0.00115, C ≈ 176, far from the printed values. So Table I is not a solution of the stated junction conditions for any real Q.

This is not cosmetic. Everything downstream—density, pressures, energy conditions, TOV forces, stability—is computed from those constants. The paper also never checks that the charge function Q(r) from Eq. (19) equals the Q used in the matching at the surface. Without that self-consistency, the interior is not matched to the Reissner-Nordström exterior, and the physical-viability claim rests on an unverified foundation.

There are smaller problems in the same spirit. The TOV equation in Eq. (30) is the standard GR expression with an electromagnetic term; in f(R) gravity hydrostatic equilibrium generically picks up extra curvature contributions, and the paper does not justify using the GR form. The summary calls the metric the Krori-Barua solution, but the body uses Karmarkar-Tolman. And the f(R) parameters α and γ are chosen to make the models look good (α=0.03, γ=0.5), so the energy-condition checks are not independent predictions.

What is genuinely useful: the three-model comparison is a reasonable template, and the charge profiles, electric fields, and anisotropy plots could be a starting point if the matching were repaired. The novelty is extension-level, not conceptual.

Who is this for? Someone in the anisotropic-star-in-modified-gravity subfield who wants a worked example. But as it stands, the central construction is underdetermined and the numbers do not support the claims.

Recommendation: I would not publish this in its current form. The Q problem will be caught immediately by any referee; the authors need to either specify Q and produce a consistent table, or show that the matching can be done without Q. If they fix that, the paper could be a minor contribution. Otherwise desk reject.

Referee Report

4 major / 5 minor

Summary. The manuscript constructs static, spherically symmetric, anisotropic and charged stellar models in f(R) gravity by imposing the Karmarkar-Tolman embedding condition and using three f(R) functionals: R+αR², R+αR(e^{-R/γ}-1), and R+αR²(1+γR). After deriving the modified field equations, the authors use the MIT bag-model equation of state P_r = (ρ - 4B_g)/3 to determine the charge function, match the interior to the Reissner-Nordström exterior through Eq. (21), and tabulate the metric constants for three candidate stars. They then test energy conditions, TOV force balance, sound speeds, equation-of-state parameters, and anisotropy, concluding that the models are physically acceptable. The central claim is that these charged anisotropic star configurations satisfy the standard physical viability criteria in f(R) gravity.

Significance. If the construction were reliable, the paper would be a useful addition to the f(R) stellar-structure literature: it gives explicit f(R)-modified density and pressure expressions, derives a charge function from a bag-model equation of state, and provides a broad graphical survey of physical conditions for three star candidates under three different f(R) functionals. These derivations are presented in a systematic way, and the attempt to combine the Karmarkar embedding with charge and modified gravity is a relevant line of work. However, the central quantitative output is currently not reproducible because the junction constants depend on an unspecified surface charge, and several of the physical-viability checks are either based on the wrong equilibrium equation for f(R) gravity or are enforced by construction through parameter choices. The significance of the claimed result therefore cannot be assessed without a reworking of the matching and the equilibrium analysis.

major comments (4)

[§II.C, Eq. (21), and Table I] The constants A, B, and C in Table I are claimed to be computed from Eq. (21), but Eq. (21) contains the total charge Q explicitly and the manuscript never specifies Q. For Her X-1 (M = 0.88 M_sun ≈ 1.30 km, R = 7.70 km), setting Q = 0 gives A ≈ 0.51, B ≈ 0.00115, C ≈ 176, which is far from the tabulated (0.385712, 0.00175805, 134.844); direct inversion of Eq. (21) for Q² leads to negative values, so no real charge reproduces the table. Since all subsequent density, pressure, stability, and charge profiles are computed from these constants, the central construction is underdetermined. In addition, the paper never checks that the interior charge function Q(r) from Eq. (19) evaluated at r = R equals the Q used in the junction conditions.
[§II.C, matching conditions] Matching to the Reissner-Nordström exterior via Eqs. (20) and (21) uses continuity of the metric components only. In f(R) gravity, the junction conditions also require continuity of f_R and of its normal derivative across the boundary (or a consistent prescription for a thin shell), because the f(R) field equations contain terms with second derivatives of f_R. No such additional junction conditions are stated or verified, so the exterior matching is not established in this theory.
[§III.C, Eqs. (30)-(32)] The TOV equilibrium condition used is the standard general-relativistic charged anisotropic TOV equation. In f(R) gravity, the matter stress-energy tensor is not separately conserved; the correct equilibrium condition contains extra curvature terms involving f_R and its derivatives, which the paper explicitly includes in the field equations (11)-(13) but omits from Eq. (30). No derivation of Eq. (30) from the f(R) field equations is provided, so the force-balance analysis in Fig. 10 does not, as it stands, demonstrate hydrostatic equilibrium for these f(R) models.
[§III opening, and Table II] The parameters α = 0.03 and γ = 0.5 are chosen to maintain physical consistency, and the bag constant Bg is fixed in Table II by imposing Pr(R) = 0. With these choices, the energy conditions, causality, and stability checks in Figs. 7-12 are tests at a hand-picked point in parameter space, not independent predictions of the model. The paper should either scan the allowed ranges of α and γ and report where the conditions fail, or clearly state that the physical-acceptability claim is conditional on these fitted values.

minor comments (5)

[§IV, first paragraph] The summary refers to the 'Krori-Barua solution' as the stellar metric, but the paper uses the Karmarkar-Tolman spacetime; the terminology should be corrected.
[Eqs. (7) and (20)] The line element in Eq. (7) and the exterior metric in Eq. (20) contain typographical errors ('r2sin2θdφ2' and 'sinθ2dϕ2'); they should read r² sin²θ dφ².
[Figures] Several figures (e.g., Figs. 1 and 13) have garbled or overlapping axis labels and missing units; all panels should have clear, consistent axis labels and complete captions.
[§II, charge notation] The notation Q in Eq. (19) denotes the charge function, while Q in Eqs. (20)-(21) denotes the total surface charge; this dual use is confusing and should be distinguished (for example, q(r) versus Q_surface).
[§III.C, Fig. 10] The text states that the electric force is strong near the center, but the plotted electric force appears small near r = 0; the description and the figure should be reconciled.

Circularity Check

1 steps flagged · score 6.0 of 10

Energy-condition and stability checks are backfitted: the f(R) parameters α and γ are explicitly chosen and adjusted to ensure those very checks, so the viability claim is not an independent first-principles prediction.

fitted input called prediction [Section III (Physical Aspects), introductory paragraph on the f(R) parameters α and γ.]
"In our work, we consider the values of these parameters are α=0.03, and γ=0.5 for the f (R) gravity models [55]. ... The parameters are adjusted carefully to ensure stability, satisfy energy conditions, and avoid ghost instabilities across all the models."

The free f(R) parameters α and γ are selected explicitly to produce physically consistent models, and the text states they were adjusted to ensure stability and energy conditions. The later sections then report, as successful validations, that energy conditions are thoroughly satisfied (Section III B) and that sound speeds remain within stability limits (Section III D). These checks are therefore not independent predictions of the construction; they are restatements of the criteria used to tune the inputs. The viability claim is partially manufactured by parameter choice, even though the field-equation algebra and the model comparisons retain independent mathematical content.

full rationale

The formal derivation chain — Karmarkar condition, metric ansatz (10), f(R) field equations (11)–(17), the assumed MIT-bag-like EoS (18), the charge expression (19), and the Reissner-Nordström junction conditions (20)–(21) — is algebraic and not circular by itself. The circularity arises in the validation layer. At the start of Section III, the authors state that α and γ were chosen to maintain physical consistency and were adjusted to ensure stability and satisfaction of the energy conditions. The same properties are then reported as successful tests in Sections III B, III C, and III D. Thus the central claim that the constructed stars are stable and physically acceptable is, in part, an output of the parameter-tuning procedure rather than a result derived from the model alone. The bag constant Bg is also fitted by imposing Pr(R)=0 (Table II), so the pressure profile used in the subsequent physical checks is adjusted to have the desired surface behavior. This is transparent, but it further weakens the claim that the energy conditions and equilibrium are independent predictions. Separately, the matching constants in Table I are problematic: Eq. (21) determines A, B, and C in terms of M, R, and the total charge Q, yet Q is never reported, and the paper never verifies that the charge function from Eq. (19) evaluated at the surface equals the Q used in Eq. (21). Inversion of Eq. (21) with the tabulated values suggests no real positive Q^2 is compatible with the stated constants. I treat this as a correctness and reproducibility defect, not as a circularity, because it does not reduce the claimed result to its inputs by definition. Overall, the model construction is not vacuous, but the headline viability validation is backfitted through the explicit tuning of α, γ, and Bg. This is a partial circularity, so the score is 6 rather than a lower value.

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

The model rests on a hand-picked metric ansatz, hand-picked f(R) parameters, a fitted bag constant, and an unspecified matching charge. The only external inputs are the masses and radii of three stars.

free parameters (4)

alpha (f(R) coupling) = 0.03
Chosen by hand to maintain physical consistency and satisfy solar-system constraints, as stated in Section III.
gamma (f(R) exponential or cubic parameter) = 0.5
Chosen by hand together with alpha to ensure stability and energy conditions, as stated in Section III.
Bag constant Bg = Values in Table II, e.g., 0.00427539 for Her X-1 in Model 1
Determined per star and per model by imposing that radial pressure vanishes at the surface, Pr(R)=0.
Surface charge Q used in matching = Not specified
Eq. (21) for A, B, and C depends on Q, but the paper never gives Q, leaving Table I underdetermined.

assumptions (4)

domain assumption Karmarkar class-one embedding condition (Eq. 9) holds for the interior spacetime.
The metric ansatz is restricted by this geometric condition, which is imposed rather than derived from the field equations.
domain assumption The MIT bag model equation of state Pr = (rho - 4Bg)/3 applies.
Invoked in Eq. (18) to derive the charge function; no microphysical justification is given beyond quark confinement.
ad hoc to paper The standard GR TOV equation Eq. (30) describes equilibrium in f(R) gravity.
The equilibrium test uses the unchanged GR TOV equation even though the model is built from f(R) field equations.
ad hoc to paper Matching to Reissner-Nordstrom requires only continuity of metric functions.
Junction conditions in f(R) gravity can require additional constraints beyond the first fundamental form; these are not discussed.

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Cite this review

Pith. "Pith review of Karmarkar-Tolman Embedded Charged Anisotropic Stars in f(R) Gravity." pith.science (2026). https://pith.science/paper/ZJ6YJJSO

@misc{pith2026250507296,
  author       = {Pith},
  title        = {Pith review of: Karmarkar-Tolman Embedded Charged Anisotropic Stars in f(R) Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZJ6YJJSO}},
  note         = {Machine review of arXiv:2505.07296}
}

read the original abstract

We investigate various anisotropic spherical distributions of charged celestial bodies within the context of f(R) gravity, where R represents the Ricci scalar. The properties of specific charged compact objects are analyzed by using the Karmarkar-Tolman spacetime and three distinct gravitational models. The behavior of the structural parameters is examined via graphical methods. Energy constraints are applied to assess how well the results align with the Karmarkar-Tolman spacetime model. The physical acceptability of the stellar models is evaluated by checking the energy conditions and the equation of state parameter. Additionally, we explore the influence of anisotropy on the stability and internal structure of the models. Our findings are compared with predictions from general relativity to highlight the effects of f(R) gravity on charged compact stars. The obtained results are useful to enhance our understanding of how modified gravity theories affect the properties of compact astrophysical objects.

Figures

Figures reproduced from arXiv: 2505.07296 by the authors.

**Figure 2.** FIG. 2. Graph depicting the evolution of radial pressure for the st [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** FIG. 3. Graph depicting the evolution of transverse pressure for [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

Figures from the paper (15 more)

**Figure 4.** Figure 4: FIG. 4. Plot of [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: FIG. 5. A graph of [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: FIG. 6. The graph of [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: FIG. 7. Different Energy conditions for strange stars, Model 1. [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

**Figure 8.** Figure 8: FIG. 8. Different Energy conditions for strange stars, Model 2. [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

**Figure 9.** Figure 9: FIG. 9. Different Energy conditions for strange stars, Model 3. [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]

**Figure 10.** Figure 10: FIG. 10. A graph showing the gravitational force ( [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]

**Figure 11.** Figure 11: FIG. 11. Variations of [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]

**Figure 12.** Figure 12: FIG. 12. Variations of [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]

**Figure 13.** Figure 13: FIG. 13. Variations of the radial EoS parameter with respect to th [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]

**Figure 14.** Figure 14: FIG. 14. Changes in the transverse equation of state (EoS) para [PITH_FULL_IMAGE:figures/full_fig_p019_14.png]

**Figure 15.** Figure 15: FIG. 15. Variations of anisotropic measure ∆ with respect to the ra [PITH_FULL_IMAGE:figures/full_fig_p020_15.png]

**Figure 16.** Figure 16: FIG. 16. Radial distribution of the charge function [PITH_FULL_IMAGE:figures/full_fig_p020_16.png]

**Figure 17.** Figure 17: , is obtained from the derivative of q(r) and reveals the manner in which charge is localized within the star. These profiles provide valuable insights into how charge accumulates and varies across different regions of the star. The presence of charge introduces an el…

**Figure 18.** Figure 18: FIG. 18. Squared electric field [PITH_FULL_IMAGE:figures/full_fig_p021_18.png]

Discussion (0). Continue with ORCID to comment.

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