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REVIEW 4 major objections 5 minor 108 references

Implications of Rastall Theory on Stellar Solutions admitting Vanishing Complexity: A New Perspective

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that anisotropic stellar models built from the vanishing complexity condition are physically viable in Rastall gravity, and that a polytropic model unstable in general relativity becomes stable for nonzero values of the…

desk verdict The vanishing-complexity condition is algebraically wrong: Eq. (36) contradicts Eq. (35), so the models are not complexity-free and the Rastall-superiority claim is unsupported. read the letter →

arxiv 2507.07425 v1 pith:GTFYSOGN submitted 2025-07-10 gr-qc

Tayyab Naseer This is my paper · ORCID
classification gr-qc PACS 04.20.-q04.40.Dg97.10.-q
keywords RastallgravityVanishingcomplexityfactorAnisotropicstellarmodelsStructurescalarsPolytropicequationofstateEnergyconditionsStability
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 tries to show that the complexity-factor approach, originally developed for general relativity, can be transplanted into Rastall's non-conservative gravity to build physically realistic anisotropic stellar models. It constructs three families of static spherical solutions by imposing the vanishing complexity condition $\mathcal{Y}_{TF}=0$ together with one of three extra constraints: zero radial pressure, a polytropic equation of state, or a non-local equation of state. It then reports that all three models satisfy the standard viability checks for the chosen values of the Rastall parameter $\alpha$, including energy conditions, redshift bound, Buchdahl limit, and cracking stability. The paper's central comparative claim is that the polytropic model, which suffers cracking at $\alpha=0$ (general relativity), becomes stable for $\alpha=0.1$ and $\alpha=0.2$, which it reads as Rastall theory being superior to Einstein's theory.

What carries the argument

The central object is the structure scalar $\mathcal{Y}_{TF}$, the complexity factor obtained from the orthogonal splitting of the Riemann tensor into Weyl and Ricci parts. It carries the whole construction: setting $\mathcal{Y}_{TF}=0$ produces the vanishing-complexity condition that links the metric potentials with the fluid variables, and combining it with one of three constraints reduces the five-unknown system to solvable fourth-order differential equations in the metric potentials. The Rastall parameter $\alpha$ enters through the effective energy-momentum tensor and controls how far the solutions depart from general relativity.

What would settle it

Take the numerically generated solutions for the polytropic model and map the Rastall energy-momentum tensor to the effective Einstein tensor; if the resulting anisotropic fluid in general relativity satisfies the same energy and cracking conditions, then the stabilization is reproduced inside general relativity, undermining the claim that Rastall theory is distinct.

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Extended reading notes

Core claim

In Rastall gravity, where the energy-momentum tensor has a non-zero divergence proportional to the Ricci scalar gradient, the complexity factor $\mathcal{Y}_{TF}$ obtained from the orthogonal decomposition of the Riemann tensor still encodes the combined effect of density inhomogeneity and pressure anisotropy. Setting $\mathcal{Y}_{TF}=0$ yields a non-local condition (Eq. 36) that, together with each of the three constraints, closes the under-determined field equations. The paper claims the resulting numerical solutions are non-singular, have energy density and pressures peaked at the center, meet all energy conditions, keep gravitational redshift below the observational bound, respect the Rastall-adjusted Buchdahl limit, and pass the cracking criterion except in the cases it identifies as unstable. Its headline result is that the polytropic model is unstable in general relativity but stable for nonzero Rastall parameter, which it presents as evidence that Rastall corrections improve stellar stability.

Load-bearing premise

The load-bearing premise is that Rastall gravity is a genuinely distinct theory from general relativity, with the non-conserved energy-momentum tensor as the true matter content; if the theory is only a relabeling of general relativity through an effective stress tensor, the claimed superiority of the Rastall model reduces to a choice of which anisotropic fluid is being described.

Editorial extensions

If this is right

  • The complexity-factor program, developed in general relativity, remains workable in a non-conservative gravity theory: the same scalar $\mathcal{Y}_{TF}$ organizes the stellar equations.
  • Polytropic anisotropic stars that develop cracking in general relativity can be stabilized by switching on the Rastall parameter, at least for $\alpha=0.1$ and $alpha=0.2$ within the paper's numerical setup.
  • All three Rastall stellar models reproduce general-relativistic behavior at $\alpha=0$ and remain consistent with the standard physical viability tests for the $\alpha$ values considered.
  • The Buchdahl compactness limit is numerically smaller for larger $\alpha$, meaning the Rastall corrections lower the maximum allowed mass-radius ratio for these solutions.
  • The non-local equation-of-state model is stable only for $\alpha=0$ and $0.1$, while the zero-radial-pressure model is stable for all tested $\alpha$, giving a concrete stability ordering across the three constructions.

Reading between the lines

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

  • If Rastall gravity is equivalent to general relativity through an effective stress-energy redefinition, then the claimed 'superiority' of the polytropic model may describe a particular anisotropic fluid rather than a genuinely different theory; the paper offers no quantitative rebuttal to that equivalence.
  • The numerical scheme fixes the polytropic index and constant, so stability across a broader polytropic parameter space remains an open question that could be tested by repeating the calculation for other values.
  • Matching these interior solutions to observed neutron-star masses and radii would test whether the Rastall-stabilized models are observationally favored over their general-relativistic counterparts.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper extends Herrera's complexity-factor formalism to Rastall gravity for static, spherically symmetric, anisotropic stellar interiors. It derives the field equations and mass function, performs an orthogonal splitting of the Riemann tensor to obtain structure scalars, and identifies Y_TF as the complexity factor. Three stellar models are constructed by imposing Y_TF = 0 together with, respectively, a vanishing radial pressure, a polytropic equation of state, and a non-local equation of state. The models are solved numerically for several values of the Rastall parameter α, and their physical viability is assessed through energy conditions, redshift bounds, Buchdahl limits, and cracking stability. The paper concludes that the three models are consistent with GR and that Rastall theory yields more suitable results than GR for the polytropic model, 'indicating its superiority over Einstein's gravity theory'.

Significance. If the construction were correct, the paper would provide a useful extension of Herrera's complexity method to a non-conservative gravity theory and would offer explicit anisotropic stellar models that satisfy a vanishing-complexity condition. The authors deserve credit for deriving the Rastall field equations, defining the mass function, and checking a battery of viability criteria for three distinct models. However, the central derivation is undermined by an algebraic inconsistency between the expression for Y_TF and the condition (36) used to build all three models. The paper does not include machine-checked proofs, reproducible code, or a derivation of the structure scalars, and the numerical methods are described only vaguely. As written, the results do not support the claim that the models satisfy Y_TF = 0 or the accompanying conclusion of Rastall superiority.

major comments (4)
  1. [Section 3, Eqs. (35)-(36)] The condition called 'vanishing complexity' in Eq. (36) is not equivalent to setting Y_TF = 0 in Eq. (35). In the GR limit α = 0, Eq. (35) yields Π = ρ/2 − (3/(2r³))∫₀ʳ ρ r̄² d r̄, which is Herrera's standard condition after integration by parts, whereas Eq. (36) reduces to Π = ρ − (2/r³)∫₀ʳ ρ r̄² d r̄. These differ in both the local term and the integral coefficient. Since all three model-building equations in Section 5, including Eqs. (40), (41), (43), (44), and (51), are stated as consequences of Eq. (36), the numerical solutions are not shown to satisfy vanishing complexity. The paper's central claim about Rastall's superiority in model 2 therefore rests on an unvalidated constraint.
  2. [Section 3, Eqs. (28)-(31)] The four structure scalars are introduced with the comment 'simple but detailed calculations (which are not presented here)'. Because the identification of Y_TF as the complexity factor is foundational to the entire paper, the derivation should either be included in full or a direct reference to the Rastall-specific computation should be provided.
  3. [Section 4, Table 1] The numerical Buchdahl limits are presented without stating the method used to obtain them. The text only says that explicit analytical expressions cannot be derived because the equations are highly nonlinear; it does not specify the differential equations solved, the boundary conditions, or the numerical scheme. Without this information, the values 0.889, 0.772, 0.714, 0.652, and 0.636 cannot be reproduced or checked.
  4. [Sections 2, 5.2, and 6] The 'superiority over GR' conclusion is based on the instability of the α = 0 case versus stability for α = 0.1 and α = 0.2 in model 2. Because the models are constructed from the incorrect Eq. (36), this conclusion is unsupported. Moreover, the dismissal of Visser's equivalence argument is an assertion rather than a quantitative rebuttal; if Rastall solutions are simply GR solutions with a redefined energy-momentum tensor, then the stability difference reflects a relabeled effective fluid rather than a new physical theory. The paper should either demonstrate the physical distinctness of the Rastall fluid explicitly or soften the superiority claim.
minor comments (5)
  1. [Section 5.2, Eq. (42)] The notation 'η3 being a polytropic exponent' is unclear; the polytropic exponent should be defined consistently with the subsequent dimensionless variables in Eq. (45).
  2. [Section 5 (general)] The numerical integration is described as relying on 'carefully chosen initial conditions' without giving the actual values or solver details; please provide the initial conditions and numerical method used for each model.
  3. [Figures 1-13] The figure captions contain placeholder symbols such as '/ScriptR' and '/ScriptE'; these should be replaced with standard mathematical notation (e.g., r, e^δ1, e^−δ2).
  4. [Abstract and Section 1] The abstract calls the vanishing complexity condition 'well-known', but Eq. (36) differs from the standard Herrera condition in the GR limit; this wording is misleading and should be revised.
  5. [Throughout] There are numerous typographical and language errors, such as 'heavily systems' in Section 1, 'deriv[e]' in Section 5.2, and 'orthogonally to' in Section 6; a careful proofread is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stellar models are constructed by imposing Y_TF=0 and chosen equations of state, with parameters scanned by hand rather than fitted, so no prediction reduces to its input by construction.

full rationale

The paper makes no empirical prediction: it constructs three static, anisotropic stellar models by imposing the vanishing complexity condition Y_TF=0 together with three different constraints (pr=0, a polytropic equation of state, and a non-local equation of state). The construction is a self-contained derivation from the Rastall field equations (10)-(12), the mass function (18), and the orthogonal decomposition of the Riemann tensor (23)-(31). No parameter is fitted to observational data: alpha, I, U, and tau are chosen or scanned by hand, and the physical viability checks in Section 4 are evaluated against the inequalities (37)-(39), not against any measured stellar dataset. There is thus no fitted-input-called-prediction pattern. The paper's rejection of Visser's GR-equivalence argument is an interpretive claim supported by an argument about other matter-geometry coupled theories, not by a self-citation chain, and it does not enter the algebraic construction of the models. The self-citations in the reference list are background references to the author's prior work on Rastall gravity and complexity; none is load-bearing for the three solutions presented here. A separate correctness concern should be noted: Eq. (36), as printed, does not follow from Eq. (35). In the GR limit alpha=0, setting Y_TF=0 in Eq. (35) gives Pi = rho/2 - (3/(2r^3)) * integral rho rbar^2 drbar (Herrera's condition), whereas Eq. (36) gives Pi = rho - (2/r^3) * integral rho rbar^2 drbar. This is an algebraic inconsistency in the vanishing-complexity constraint used for the models, and it is a serious correctness risk, but it is not circularity: a false derivation is not an input-output equivalence by construction. Because the circularity pass targets definitional or fit-based reductions rather than mathematical validity, the appropriate finding is no significant circularity.

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

The central derivation rests on three domain assumptions (distinctness of Rastall theory, transferability of Herrera's complexity factor, validity of the cracking criterion) and six hand-chosen parameters. The Visser equivalence question is the most consequential: if the effective EMT mapping is invertible, the new models are formally GR solutions.

free parameters (6)
  • Rastall parameter α = 0, 0.1, 0.2 (models); 0.3, 0.4 (Table 1)
    Theory coupling parameter scanned by hand; stability and compactness conclusions depend on its chosen values.
  • Polytropic constant I = 0.9
    Chosen ad hoc for model 2; no data or physical constraint.
  • Polytropic index U = 0.05 (first run); 0.06, 0.07, 0.08 (second run)
    Chosen ad hoc; the stability analysis in Figure 8 uses U=0.05.
  • Pressure ratio τ = 0.1
    Dimensionless analysis in Section 5.2; chosen by hand.
  • Integration constants c1, c2, c4 = not specified
    Initial values of e^{δ1} at center; the paper says 'carefully chosen' but gives no values, preventing exact replication.
  • Non-local EoS constant c3 = 0
    Set to zero to avoid central singularity (Section 5.3).
assumptions (6)
  • domain assumption Rastall gravity is physically distinct from GR; the physical EMT T is not equivalent to the effective conserved T̃.
    Section 2 defines T̃ in Eq (6) and dismisses Visser's equivalence argument without quantitative rebuttal. If false, the models reduce to GR with a redefined fluid.
  • domain assumption Herrera's complexity factor Y_TF retains its physical interpretation in Rastall gravity.
    Section 3 claims Y_TF is the complexity factor by analogy with GR; no independent derivation is provided.
  • domain assumption The vanishing complexity condition Y_TF=0 is a valid closure condition for stellar models.
    Used in all three models (Eq 36); the paper cites a non-local equation of state argument but does not prove it selects physical solutions.
  • domain assumption The cracking criterion 0 ≤ v_r^2 - v_t^2 ≤ 1 determines stability.
    Section 4 uses this inequality to declare stability or instability; it is a heuristic condition from Herrera.
  • domain assumption Pointwise energy conditions (38) on the physical fluid T are the correct viability conditions.
    The paper checks energy conditions on the non-conserved physical fluid rather than the conserved effective fluid; this choice is non-trivial.
  • standard math Misner-Sharp mass function m(r)=r(1-e^{-δ2})/2 applies in Rastall gravity.
    Standard in spherical symmetry; used to derive Eq (19) and the boundary conditions.

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Pith. "Pith review of Implications of Rastall Theory on Stellar Solutions admitting Vanishing Complexity: A New Perspective." pith.science (2026). https://pith.science/paper/GTFYSOGN

@misc{pith2026250707425,
  author       = {Pith},
  title        = {Pith review of: Implications of Rastall Theory on Stellar Solutions admitting Vanishing Complexity: A New Perspective},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GTFYSOGN}},
  note         = {Machine review of arXiv:2507.07425}
}
abstract

In this paper, the notion of complexity factor and its implication is extended to the framework of non-conserved Rastall theory of gravity. First of all, the field equations governing a static spherical geometry associated with the anisotropic fluid are formulated. The mass function corresponding to the considered geometry is defined in terms of both matter and geometric quantities. The orthogonal decomposition of the Riemann tensor is then performed through which a family of scalar quantities, known as structure scalars, is obtained. Using the Herrera's recent definition, one of the scalars among them is claimed as the complexity factor, \emph{i.e.}, $\mathcal{Y}_{TF}$. Since there are extra degrees of freedom in the gravitational equations, some constraints are needed to make their solution possible to obtain. In this regard, a well-known vanishing complexity condition is introduced along with three different constraints which ultimately lead to distinct stellar models. In order to check their physical feasibility, a detailed graphical interpretation is provided using multiple values of the Rastall parameter. It is concluded that the obtained results in all three cases are consistent with those of general relativity. Further, the Rastall theory provides more suitable results in the case of model 2, indicating its superiority over Einstein's gravity theory.

Figures

Figures reproduced from arXiv: 2507.07425 by the authors.

Figure 1
Figure 1. Metric potentials e δ1 (⋆) and e −δ2 (⋆) for α = 0 (upper left), 0.1 (upper right) and 0.2 (lower) analogous to model I. 0.05 0.10 0.15 0.1 0.2 0.3 0.4 0.5 r Ρ 0.00 0.05 0.10 0.15 0.00 0.05 0.10 0.15 r pt 0.00 0.05 0.10 0.15 -0.15 -0.10 -0.05 0.00 r P [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Governing parameters for α = 0 (⋆), 0.1 (⋆) and 0.2 (⋆) corre￾sponding to model I. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Energy bounds for α = 0 (⋆), 0.1 (⋆) and 0.2 (⋆) corresponding to model I. 0.05 0.10 0.15 0.5 1.0 1.5 r z 0.05 0.10 0.15 0.0 0.1 0.2 0.3 0.4 0.5 0.6 r vr 2-vt 2 [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Redshift and cracking for α = 0 (⋆), 0.1 (⋆) and 0.2 (⋆) corre￾sponding to model I. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: Metric potentials e δ1 (⋆) and e −δ2 (⋆) for α = 0 (upper left), 0.1 (upper right) and 0.2 (lower) analogous to model II. explicit expressions due to numerically solving the above two equations. The findings show that e δ1(0) = c2 (a positive constant), and e −δ2(0) = …
Figure 6
Figure 6. Figure 6: Governing parameters for α = 0 (⋆), 0.1 (⋆) and 0.2 (⋆) corre￾sponding to model II. occurs at the core, and the minimum at the boundary, following a mono￾tonically decreasing trend. The upper left plot shows that pr disappears at the interface for every value of α [PI…
Figure 7
Figure 7. Figure 7: Energy bounds for α = 0 (⋆), 0.1 (⋆) and 0.2 (⋆) corresponding to model II. 0.02 0.04 0.06 0.08 0.10 0.12 1.0 1.5 2.0 2.5 r z 0.02 0.04 0.06 0.08 0.10 0.12 -0.5 0.0 0.5 r vr 2-vt 2 [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]
Figure 8
Figure 8. Figure 8: Redshift and cracking for α = 0 (⋆), 0.1 (⋆) and 0.2 (⋆) corre￾sponding to model II. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]
Figure 9
Figure 9. Figure 9: Profile of Γ, λ and Π for U = 0.06 (⋆), 0.07 (⋆) and 0.08 (⋆) corresponding to model II. 5.3 Model admitting Non-local Equation of State and YT F = 0 Hernandez and Nunez [87] proposed a constraint that relates the radial pres￾sure to both the energy density and an inte…
Figure 10
Figure 10. Figure 10: Metric potentials e δ1 (⋆) and e −δ2 (⋆) for α = 0 (upper left), 0.1 (upper right) and 0.2 (lower) analogous to model III. 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.1 0.2 0.3 0.4 0.5 r Ρ 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.000 0.002 0.004 0.006 0.008 0.010 0.012…
Figure 11
Figure 11. Figure 11: Governing parameters for α = 0 (⋆), 0.1 (⋆) and 0.2 (⋆) corre￾sponding to model III. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_11.png]
Figure 12
Figure 12. Figure 12: Energy bounds for α = 0 (⋆), 0.1 (⋆) and 0.2 (⋆) corresponding to model III. 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.6 0.8 1.0 1.2 1.4 1.6 r z 0.04 0.06 0.08 0.10 0.12 0.14 0.16 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 r vr 2-vt 2 [PITH_FULL_IMAGE:figures/full_fig_p027_12.png]
Figure 13
Figure 13. Figure 13: Redshift and cracking for α = 0 (⋆), 0.1 (⋆) and 0.2 (⋆) corre￾sponding to model III. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_13.png]

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