Exploring gravastar-like structures with strongly interacting quark matter shell in the framework of $f(Q)$ gravity under conformal symmetry

Debadri Bhattacharjee; Pradip Kumar Chattopadhyay

REVIEW 4 major objections 4 minor 1 cited by

Exploring gravastar-like structures with strongly interacting quark matter shell in the framework of $f(Q)$ gravity under conformal symmetry

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper constructs a three-layer gravastar in f(Q) gravity whose thin shell is strongly interacting quark matter, and computes shell masses of 1.80, 1.95 and 2.28 solar masses at radii 9.009, 10.009 and 11.009 km.

desk verdict The paper's central 'non-singular' claim is contradicted by its own Eqs. (40)-(42), which give a diverging g_rr and energy density at r=0. read the letter →

arxiv 2505.17583 v1 pith:XZOJGJMF submitted 2025-05-23 gr-qc

Debadri Bhattacharjee , Pradip Kumar Chattopadhyay This is my paper · ORCID

classification gr-qc

keywords gravastarf(Q)gravityconformalsymmetrystronglyinteractingquarkmatterbagconstantthinshellmassjunctionconditionssurfaceredshift

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 establish that a gravastar, a collapsed object with a de Sitter core and a thin shell instead of an event horizon, can be realized in f(Q) gravity when the shell is made of strongly interacting quark matter obeying the maximally causal equation of state $p=\rho-2B_g$. It claims that with a conformal Killing symmetry, the interior and shell field equations admit non-singular, non-vanishing solutions, and that the Israel junction condition makes the shell mass independent of the shell's matter distribution. Working with total mass $2.5\,M_\odot$, bag constant $70\,\text{MeV}/\text{fm}^3$, and radii $9.009$, $10.009$, and $11.009$ km, it obtains shell masses $1.80$, $1.95$, and $2.28\,M_\odot$, all satisfying the compactness bound and the surface-redshift limit $Z_s<2$. If correct, this is a causal and stable black-hole alternative that could be distinguishable by its shell thermodynamics.

What carries the argument

The load-bearing machinery is the combination of a linear f(Q) gravity action, a Conformal Killing Vector ansatz, and the standard thin-shell junction conditions. The conformal symmetry fixes the metric potentials to $e^{2\nu}=c_1^2r^2$ and $e^{2\lambda}=(c_2/\psi)^2$, reducing the field equations to algebraic relations; the requirement that $f_{QQ}=0$ or $Q'=0$ pins $f(Q)$ to the linear form $\alpha_0+\alpha_1 Q$, which is what makes the exterior match the Schwarzschild–(anti) de Sitter spacetime. The strongly interacting quark matter equation of state $p=\rho-2B_g$, taken in the strong-interaction limit of the unified quark-matter equation of state, supplies the shell's causal-limit stiff fluid. The junction conditions, applied at the thin shell, convert the jump in metric derivatives into a surface energy density and hence the shell mass formula $M_{\text{shell}}=R\left(\sqrt{c_4^2R^2}-\sqrt{1-2M/R}\right)$.

What would settle it

Evaluate the Ricci and Kretschmann scalars of the interior metric $ds^2=-c_1^2r^2dt^2+dr^2/(c_4^2r^2)+r^2(d\theta^2+\sin^2\theta\,d\phi^2)$ and take $r\to 0$; if either curvature invariant diverges, the central singularity that gravastars are meant to remove is still present.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the gravastar construction survives the move to f(Q) gravity when the thin shell is switched from the usual stiff fluid $p=\rho$ to strongly interacting quark matter with $p=\rho-2B_g$. With $f(Q)=\alpha_0+\alpha_1 Q$ forced by the field equations, conformal symmetry fixes the interior metric potentials as $e^{2\nu}=c_1^2r^2$ and $e^{-2\lambda}=c_4^2r^2$, the shell solution is non-vanishing with explicit energy density and pressure, and the exterior reduces to the Schwarzschild–(anti) de Sitter form. The junction conditions then yield the thin-shell mass $M_{\text{shell}}=R\left(\sqrt{c_4^2R^2}-\sqrt{1-2M/R}\right)$, which the paper emphasizes is independent of the matter distribution in the shell. For $c_4=0.08$, $M=2.5\,M_\odot$, and $B_g=70\,\text{MeV}/\text{fm}^3$, this formula gives $M_{\text{shell}}=1.80$, $1.95$, and $2.28\,M_\odot$ for $R=9.009$, $10.009$, and $11.009$ km, respectively.

Load-bearing premise

The load-bearing premise is that a core whose radial metric component and energy density diverge as $1/r^2$ at the centre still counts as a non-singular gravastar interior.

Editorial extensions

If this is right

Replacing the usual $p=\rho$ shell with the SIQM equation of state $p=\rho-2B_g$ preserves maximal causality, so the gravastar shell can be built from a QCD-motivated matter state.
The thin-shell mass $M_{\text{shell}}=R\left(\sqrt{c_4^2R^2}-\sqrt{1-2M/R}\right)$ depends only on the junction radius and the constants $M$ and $c_4$, giving $1.80$, $1.95$, and $2.28\,M_\odot$ for the three chosen radii.
The model satisfies the compactness bound $2M/r<8/9$ and the surface-redshift bound $Z_s<2$, so it passes the standard static stability criteria.
Proper length decreases while shell energy and entropy increase with shell thickness, characterizing the thermodynamic behavior of the SIQM shell.

Reading between the lines

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

Because the constant $c_4$ was fixed at $0.08$ rather than derived from $B_g$, the quoted shell masses do not test the quark-matter equation of state; varying $B_g$ while redetermining $c_4$ would yield a mass-radius relation that gravitational-wave events could constrain.
The exterior vacuum solution is geometrically the same as a black-hole–de Sitter spacetime, so the model is observationally distinguishable from a black hole only through thin-shell effects such as proper length, surface redshift, and entropy content.
A natural extension is a stability check beyond the static redshift bound, such as radial oscillations or tidal deformability of the two-layer configuration, which could connect the model to gravitational-wave observables.

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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

Quick take: the abstract's "non-singular" claim is broken by the paper's own equations. The interior metric from Eqs. (40)-(41) has g_rr = 1/(c4^2 r^2), which diverges at r=0, and Eq. (42) gives rho = -p = alpha0/(16*pi) + 3 c3^2 alpha1/(8*pi c2^2) - alpha1/(8*pi r^2), which diverges as +1/r^2 for alpha1<0. The paper's Section 4.1 actually says the energy density and pressure "undergo central singularity, which is natural in stellar models with CKVs," while the abstract and conclusion call the same solution "singularity free." That is an internal contradiction, and since removing the central singularity is the defining purpose of a gravastar, the main advertised result fails.

What is genuinely new is the SIQM shell EoS p = rho - 2B_g in a gravastar context, and the numerical shell masses (1.80, 1.95, 2.28 M_sun) are new numbers. The derivation is a standard exercise in the gravastar template: conformal Killing ansatz, linear f(Q) forced by f_QQ=0, Israel junction. That part is competently executed.

Several secondary issues. The shell mass from Eq. (66) depends only on c4, R, and M, not on the shell EoS. The paper says this openly, but it means the advertised "quark matter shell mass" is not actually probing the SIQM shell. The choice c4=0.08 is hand-picked inside the bounds of Table 1; the c4 values computed from the boundary conditions in Table 2 (0.047 to 0.052) would give much smaller or negative shell masses, so the headline numbers are not robust. The stability check is circular: Table 1 bounds c4 with Zs<2, then Section 7 picks c4=0.08, and Section 9.1 uses the resulting M_shell to verify Zs<2. The finite-thickness shell used for proper length, energy, and entropy is also inconsistent with the zero-thickness Israel junction used for M_shell. And because Eq. (27) forces f(Q) linear, there is no genuine modified-gravity effect here—it is GR with a cosmological constant, plus a factor-of-two mismatch in the Lambda identification between Eqs. (57) and (58).

This paper is for readers tracking gravastar models with exotic matter shells. It is not a waste of time, but the central physical claim fails. I would send it to a referee—the math is checkable and the subfield takes these models seriously—but the referee should be pointed at Section 4.1 before anything else. Expect rejection unless the authors either introduce an inner cutoff to excise r=0 or drop the word "non-singular."

Referee Report

4 major / 4 minor

Summary. The paper constructs a three-layer gravastar-like solution in linear f(Q)=alpha0+alpha1 Q gravity with a conformal Killing vector, replacing the usual p=rho shell by a strongly interacting quark matter shell with EoS p=rho-2B_g. It derives interior, shell, and exterior metrics, uses Israel junction conditions to define a thin-shell mass, fixes constants through boundary conditions, and reports shell masses 1.80, 1.95, and 2.28 solar masses for radii 9.009, 10.009, and 11.009 km. It also presents proper length, energy, entropy, and surface redshift as physical validation.

Significance. If the central claims were correct, the paper would offer a concrete gravastar alternative in f(Q) gravity with a quark-matter shell and definite shell-mass estimates. The paper has strengths: it starts from a microscopic IQM free energy and reduces it to the causal SIQM EoS, it gives explicit analytic expressions for all three regions, and the reported shell masses are definite enough to be checked. However, the main physical claim ('non-singular') is contradicted by the paper's own interior solution, and the headline shell masses are not robust model outputs because they are controlled by a free constant chosen by hand. The stability check is circular, and a finite-thickness shell is treated with a zero-thickness junction formula.

major comments (4)

[Section 4.1, Eqs. (40)-(42)] The claimed 'non-singular and non-vanishing' interior is actually singular at r=0. Eq. (40) gives e^{-2lambda}=c4^2 r^2, hence g_rr = e^{2lambda}=1/(c4^2 r^2), which diverges as r->0. Eq. (42) contains the term -alpha1/(8 pi r^2); with the paper's choice alpha1=-0.5, both rho and p diverge as +1/r^2. The text in Section 4.1 itself states that 'the energy density (rho) and isotropic pressure (p) undergo central singularity, which is natural in stellar models with CKVs.' This directly contradicts the Abstract's 'non-singular' wording and the Conclusion's statement that Eqs. (40)-(41) are 'singularity free.' No inner cutoff or additional junction surface excises r=0, so the singularity is part of the spacetime. Since removing the central singularity is the defining purpose of a gravastar, this invalidates the paper's central physical claim.
[Section 5, Eq. (66)] The shell mass M_shell is obtained from the Israel thin-shell junction formula evaluated at a single surface r=R, using the interior metric of Eq. (40) and the exterior metric of Eq. (59). It therefore contains no information about the finite SIQM shell occupying r1 <= r <= r2, its EoS, the bag constant B_g, or the shell integration constant c5. The advertised property that the shell mass is 'independent of the matter distribution in the shell region' is thus true by construction, not a physical prediction of the model. This also creates a consistency problem: the physical features in Section 8 are computed for a shell of finite thickness, while Eq. (66) assumes a zero-thickness shell.
[Sections 6-7, Tables 1-4 and Eq. (66)] The headline shell masses are not robust predictions because c4 is not determined by the matching conditions; it is chosen by hand as c4 = 0.08 within the bounds of Table 1. At R=9.009 km the allowed range 0.047 < c4 < 0.096 spans shell masses from near zero or slightly negative up to roughly 2.7 solar masses, whereas the chosen value gives 1.80 solar masses. Thus the reported masses 1.80, 1.95, and 2.28 solar masses are tuning of a free parameter, not a model output constrained by the physical input. The conclusion that M_shell increases with radius is a consequence of this choice rather than a derived result.
[Section 9.1, Eq. (73) and Figure 5] The stability check is circular. The bounds on c4 in Table 1 are obtained by imposing M_shell > 0 and Z_s < 2 using Eq. (66) and Eq. (73). Section 7 then fixes c4 = 0.08, computes M_shell from Eq. (66), and Section 9.1 uses Eq. (73) to confirm Z_s < 2. Because Eq. (73) is precisely the relation used to select the c4 interval, this confirmation carries no independent information. The plotted values Z_s ~ 0.56 are simply a reflection of the chosen c4, not evidence of stability obtained from the full dynamics of the model.

minor comments (4)

[Section 8.1, Eq. (70)] The integrand of the proper length integral is written with r1 in place of the integration variable r, making the displayed integral a constant divided by the integration range; the expression should use the running radial coordinate throughout.
[Section 6, Table 2] The table header 'c5r1 r2' is garbled; it should list the matched radii r1 and r2 with the determined constants c1, c2, and c5. The values of c5 in Tables 2 and 3 are set to 0.0001 'without loss of generality,' but c5 enters the shell density, pressure, and entropy; this choice should be justified rather than asserted to be generic.
[Section 2, Eq. (7)] The line break in Eq. (7) makes the algebraic form ambiguous; the missing operator between the 4 chi^2/(9 pi^2) term and the square-root bracket should be displayed explicitly.
[Section 4.1] The positivity condition below Eq. (42) should be stated more carefully: the inequality alpha0 > 2 alpha1/r^2 with alpha1 < 0 is automatically satisfied for the values used later (alpha0 = 10^{-46}, alpha1 = -0.5), so it does not meaningfully restrict the parameter space.

Circularity Check

1 steps flagged · score 6.0 of 10

The surface-redshift stability check is a closed loop: c4 is bounded using Zs < 2, then M_shell from that c4 is fed back into Zs to 'validate' the model.

fitted input called prediction [Section 6 (bounds on c4) and Section 9.1 (surface redshift validation); Eqs. (66) and (73); Table 1]
"while the stipulation that the surface redshift (Zs) of thin shell of an isotropic configuration is less than 2, i.e., Zs < 2 [131], produces, c4 < ... The upper and lower bounds of c4 are tabulated in Table 1. ... within the bounds described in Table 1, we have set c4 = 0.08. ... Using Eq. (66) and Table 2, we explore the surface redshift related to this model ... From Figure 5, it is evident that the our model satisfies the surface redshift bound for an isotropic gravastar."

The upper bound on c4 in Table 1 is derived from the same inequality that Section 9.1 claims to verify. With Eq. (73), Zs < 2 is equivalent to M_shell/R < 4/9, and inserting Eq. (66) gives the stated upper bound on c4. The paper then chooses c4 = 0.08 inside that bound, computes M_shell from Eq. (66), and uses Eq. (73) to confirm Zs < 2. The confirmation is the inverse of the constraint that selected c4, so it cannot fail and provides no independent test of stability.

full rationale

The core construction is largely self-contained: the f(Q) field equations with conformal symmetry are solved for the interior (p = -rho), the SIQM shell (p = rho - 2B_g), and the exterior, and the Israel junction condition yields the thin-shell mass formula Eq. (66). The quoted shell masses (1.80, 1.95, 2.28 M_sun) are honest outputs of that formula for the chosen inputs R, M, and c4, and the 'independence' of M_shell from the shell matter distribution is a direct consequence of the junction formalism, not a circular step. The significant circularity is the stability validation: the c4 bound in Table 1 is imposed by requiring Zs < 2, c4 = 0.08 is then chosen within that bound, and Section 9.1 uses the resulting M_shell to re-derive Zs < 2; this is a fitted-input-called-prediction loop, not an independent check. Separately, the abstract's claim of 'non-singular' interior solutions is contradicted by the paper's own Eq. (42), where rho diverges as -alpha1/(8 pi r^2), and by Section 4.1's admission that 'the energy density (rho) and isotropic pressure (p) undergo central singularity'; this is a scientific correctness problem rather than a circularity, so it does not by itself raise the circularity score but it does undermine the central advertised result. The self-citation [132] for alpha0 and alpha1 is not load-bearing for the mass or stability loop, since those parameters do not enter Eq. (66) or Eq. (73).

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The model rests on the three-layer gravastar ansatz, the conformal Killing solution, the linearized f(Q) theory, and several hand-set constants (c4, c5, Bg, M). The most consequential choices are c4, which directly sets the advertised shell masses, and the acceptance of a divergent interior density, which carries the singularity-free claim. No new particles or fields are introduced.

free parameters (6)

c4 (conformal constant) = 0.08, chosen inside bounds that depend on radius; Table 2 gives 0.047-0.052
Eq. (66) makes the shell mass a direct function of c4: M_shell = R (c4 R - sqrt(1 - 2M/R)). The paper selects c4 = 0.08 after imposing positivity and redshift bounds, so the headline masses are controlled by this hand-set number.
Bg (bag constant) = 70 MeV/fm^3, with scans at 57.55 and 95.11 MeV/fm^3
Sets the shell EoS p = rho - 2 Bg. Chosen from the stable SQM range in Section 6; it affects the shell density and the derived constants c1, c2, c5, though not M_shell after c4 is fixed.
c5 (shell integration constant) = 0.0001
Declared 'without the loss of any generality' in Section 6. It appears in the shell metric Eq. (45) and in the shell density and pressure, Eqs. (47)-(48).
alpha0, alpha1 (linear f(Q) coefficients) = alpha0 = 10^-46 km^-2, alpha1 = -0.5
Taken from the authors' earlier Ref. [132]. They fix the linear f(Q) action and enter the exterior metric Eq. (57) as a cosmological-constant-like term.
Total mass M = 2.5 solar masses
Chosen so the selected radii obey the Buchdahl limit (Section 6). M sets the exterior Schwarzschild mass and appears in Eq. (66) through sqrt(1 - 2M/R).
Shell radii r1, r2 = r1 = 9, 10, 11 km; r2 = 9.009, 10.009, 11.009 km
The three radius choices come from the authors' earlier Ref. [25] rather than being derived in this paper; the shell mass values in Table 4 vary with r2.

assumptions (6)

domain assumption The gravastar has three layers with EoS p = -rho (interior), p = rho - 2 Bg (shell), p = rho = 0 (exterior).
The construction is defined by these imposed equations of state, following Mazur-Mottola [1-3] and Zhang-Mann [105].
domain assumption The conformal Killing vector ansatz sets the metric via e^{2 nu} = c1^2 r^2 and e^{2 lambda} = (c2/psi)^2, Eqs. (33)-(34).
This ansatz restricts all interior and shell solutions; the interior branch psi = r c3 is selected and psi = 0 is discarded in Eq. (39).
domain assumption f(Q) is linear: f(Q) = alpha0 + alpha1 Q, from Eq. (27).
The authors choose the solution f_QQ = 0 of Eq. (26), excluding the Q' = 0 branch. This reduces the theory to STEGR/GR with a cosmological constant, so the modified gravity content is not active.
ad hoc to paper The central divergence of the interior density and metric is acceptable in a gravastar core.
Section 4.1 states that the 1/r^2 divergence in rho and p is 'natural in stellar models with CKVs'. The non-singular claim depends on this judgment, which is not supported by any alternative regularization.
standard math The Israel/Darmois-Israel junction conditions apply at a single radius R to define the shell surface energy density and mass.
Section 5 uses the Lanczos equation to obtain Eqs. (63)-(66). This is a standard tool for zero-thickness shells, though the paper later treats the shell as having finite thickness.
domain assumption The Buchdahl surface redshift bound Zs < 2 is valid for the shell and can be used to constrain c4.
Section 6 converts Zs < 2 into an upper bound on c4 and Section 9.1 then verifies Zs < 2 using the resulting M_shell, which makes the test circular.

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Pith. "Pith review of Exploring gravastar-like structures with strongly interacting quark matter shell in the framework of $f(Q)$ gravity under conformal symmetry." pith.science (2026). https://pith.science/paper/XZOJGJMF

@misc{pith2026250517583,
  author       = {Pith},
  title        = {Pith review of: Exploring gravastar-like structures with strongly interacting quark matter shell in the framework of $f(Q)$ gravity under conformal symmetry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XZOJGJMF}},
  note         = {Machine review of arXiv:2505.17583}
}

abstract

In this work, we investigate gravastar-like structures in static and spherically symmetric space-time within the framework of $f(Q)$ gravity coupled with conformal symmetry. We have modified the conventional gravastar model by introducing a strongly interacting quark matter shell which maintains the apex of causal limit through the EoS, $p=\rho-2B_{g}$, where, $B_{g}$ is the bag constant. Non-singular and non-vanishing solutions for the interior and shell regions are obtained, respectively. We have used the Israel junction condition to evaluate the mass of the thin shell for different choices of characteristic radii. Interestingly, the mass of the shell is independent of the matter distribution in the shell region. We found that for radii 9.009, 10.009 and 11.009, the mass increases as $1.80,~1.95$ and $2.28~M_{\odot}$. The physical features, such as, proper length, energy and entropy of the shell region are studied within the parameter space. Surface redshift calculations were used to validate the proposed model.

Figures

Figures reproduced from arXiv: 2505.17583 by the authors.

**Figure 2.** Variation of proper length (ℓ) with shell thickness (ϵ) for (a) different characteristic radii and Bg = 70 MeV /fm3 . Here, the solid, dashed and dotdashed lines represent 9-9.009, 10-10.009 and 11-11.009 Km, respectively, (b) for different bag constant (Bg) and radius 10-10.009 Km. Here, the solid, dashed and dotdashed lines represent Bg = 57.55, 70 and 95.11 MeV /fm3 , respectively. that for a particular radius, t… view at source ↗

**Figure 3.** Variation of shell energy (E) with shell thickness (ϵ) for (a) different characteristic radii and Bg = 70 MeV /fm3 . Here, the solid, dashed and dotdashed lines represent 9-9.009, 10-10.009 and 11-11.009 Km, respectively, (b) for different bag constant (Bg) and radius 10-10.009 Km. Here, the solid, dashed and dotdashed lines represent Bg = 57.55, 70 and 95.11 MeV /fm3 , respectively. illustrates that with increasing… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Variation of shell entropy (S) with shell thickness (ϵ) for (a) different characteristic radii and Bg = 70 MeV /fm3 . Here, the solid, dashed and dotdashed lines represent 9-9.009, 10-10.009 and 11-11.009 Km, respectively, (b) for different bag constant (Bg) and radius…

**Figure 5.** Figure 5: Radial variation of surface redshift for different characteristic radii and [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

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