Pith. sign in

REVIEW 3 major objections 6 minor 35 references

Rastall gravity admits exact closed-form solutions for static, spherically symmetric perfect fluids when the Rastall parameter is tuned to four special relations with w, reproducing known general-relativity solutions and adding five new spa

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review

2026-08-01 15:19 UTC pith:R4S7PQJG

load-bearing objection A solid, honest exact-solutions paper whose results are, as the authors admit, mostly known GR spacetimes in a Rastall disguise — the abstract and uniqueness claim need tightening, but the derivation is worth refereeing. the 3 major comments →

arxiv 2607.26463 v1 pith:R4S7PQJG submitted 2026-07-29 gr-qc

Exact solutions for static spherically symmetric spacetime with a perfect fluid in Rastall theory

Yoshimune Tomikawa , Ichika Obara , Takumi Orimo This is my paper
classification gr-qc MSC 83C1583C2083C57 PACS 04.20.Jb04.50.Kd
keywords Rastall theoryexact solutionsperfect fluidstatic spherically symmetric spacetimeequation of stateLiouvillian solutionsblack holescurvature singularities
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tries to establish that Rastall's modified gravity theory, in which the stress-energy tensor is not conserved, is at least as rich as general relativity in exact solutions: for static, spherically symmetric perfect fluids with pressure p=wρ, closed-form (Liouvillian) metrics exist whenever the Rastall parameter a satisfies any of four special relations with w. It shows that the known general-relativity solutions for w=0, -1/5, -1/3, and -1 reappear as special cases of these relations, so the correspondence with Einstein's theory is preserved for those values. It also finds five new solution patterns on a fifth branch (a=6w/(3w-1)), giving black holes with finite curvature at the horizon, black holes with possible curvature singularities, and a closed cyclic universe with infinitely many singularities. A sympathetic reader cares because exact solutions are the primary testing ground for modified gravity, and these constructions show how the Rastall parameter can be chosen to force the field equations to become solvable.

Core claim

The central claim is that the Rastall field equations reduce to integrable form precisely when the parameter a=(6κλ-1)/(4κλ-1) is linked to the barotropic index w through one of the relations w=-1, a=(5w-1)/(3w-1), a=(11w-1)/(2(3w-1)), or a=6w/(3w-1). With these relations, the density equation from the non-conservation law integrates directly, the metric function h satisfies a linear second-order equation, and f is obtained in closed form. The solutions for w=-1,0,-1/5,-1/3 match the known general-relativity spacetimes (Schwarzschild-(anti-)de Sitter, Minkowski, and the Semiz and Chernin-type solutions), while the last branch yields five new metrics whose causal and singularity structure cha

What carries the argument

The load-bearing device is a set of 'coefficient-vanishing' relations a=a(w): tuning the Rastall parameter so that the combination (3w-1)a-(5w-1) or similar coefficients in the field equations vanishes. This collapses the four coupled equations into a linear ODE for the sphere-radius function h (or equivalently the metric function), which then determines f algebraically; the non-conservation law supplies a first-order equation for the density ρ that integrates to a power of f. The final metrics are written in the Schwarzschild-like gauge ds² = -f dt² + f^{-1} dr̃² + h² dΩ², and their properties are read off from the sign and zeros of f and the curvature invariants.

Load-bearing premise

The derivation assumes the Rastall parameter can be tuned independently for each equation-of-state parameter w; if a is instead a fixed constant of the theory, each solution family lives in a different theory and the arbitrary-w result is a statement across a family of theories rather than within one.

What would settle it

Substitute each claimed closed-form metric, together with its density profile and the corresponding a(w) relation, back into the full Rastall field equations; any non-zero residual would disprove the corresponding solution. A complementary check: numerically integrate the general ODE (A17) for w=-1/6 with a tuned parameter to see whether a Liouvillian solution emerges, settling whether the missing GR counterpart is a genuine gap or an artifact of the chosen ansatz.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Share X Bluesky LinkedIn Reddit HN

If this is right

  • For w=-1, the exact spacetime is Schwarzschild-(anti-)de Sitter for any value of the Rastall parameter, so this solution cannot distinguish Rastall theory from general relativity.
  • For w=0, -1/5, -1/3, the general-relativity solutions survive in Rastall theory with appropriately tuned a, extending the exact-solution catalog beyond Einstein's theory.
  • The a=6w/(3w-1) branch provides five new exact metrics, including black holes with no curvature singularity and a closed universe with infinitely many singularities, which have no direct counterpart in the w=-1/3 GR case except for special constants.
  • The absence of a w=-1/6 counterpart means the known GR solution for that value is not obtained by this coefficient-vanishing method, leaving a gap in the correspondence.

Where Pith is reading between the lines

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

  • If a is a fundamental constant of Rastall theory, then each special relation describes a different theory, and the 'arbitrary w' solutions should be read as a family of theories parametrized by a(w), not as multiple fluids in one theory; this is the main interpretive caveat the paper leaves open.
  • The new cyclic spacetime (V) has an upper bound on the radial coordinate and infinitely many curvature singularities, which if taken as a physical model would be a strong phenomenological signature; it could be tested through gravitational-wave or lensing searches for periodic spacetime structure.
  • The coefficient-vanishing strategy is generalizable: the same method might recover exact solutions in other non-conservative gravity theories, or produce the missing w=-1/6 counterpart by choosing a different ansatz for the metric rather than the perfect-fluid sector.
  • A numerical integration of the general ODE (A17) for a value of w not in the special list would test completeness: if Liouvillian solutions appear for generic w, then the special relations are sufficient but not necessary.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper studies static, spherically symmetric perfect-fluid solutions in Rastall gravity with the linear equation of state p=wρ. It identifies four special relations between the Rastall parameter (encoded in a) and w for which the field equations admit elementary (Liouvillian) solutions: w=-1; a=(5w-1)/(3w-1); a=(11w-1)/(2(3w-1)); and a=6w/(3w-1). Explicit metrics are written for each case and checked against the field equations. The paper shows that the w=-1, 0, -1/5, -1/3 reductions reproduce known GR solutions (Minkowski, Schwarzschild-(anti) de Sitter, Semiz, and Chernin et al. forms), and it identifies five solution branches in the a=6w/(3w-1) case with different horizon and singularity properties.

Significance. If taken at face value, the paper provides a systematic extension of the known GR perfect-fluid solutions to Rastall theory under special parameter choices. Its main strength is the explicit, field-equation-checked construction of several metric families. The result is weaker than a full classification in a fixed Rastall theory, because each relation a(w) fixes the Rastall parameter λ to a different value; the paper acknowledges in Sec. V that the relations are chosen to make coefficients vanish, but the abstract and conclusion need to state this conditionality more carefully. The claimed uniqueness is not established. As a technical contribution to the exact-solution literature, the paper is sound and useful, with the caveats below.

major comments (3)
  1. [Abstract and Sec. IV] The sentence 'when w≠1/3,-1, there exist several types of solutions' is established only under the additional relation a=6w/(3w-1), introduced in Sec. IV. As written, the abstract suggests existence for arbitrary Rastall parameter. Please add the condition explicitly, both in the abstract and in the corresponding summary statement in Sec. V.
  2. [Sec. V] The concluding sentence 'our results also have proven the uniqueness of that spacetime' is not supported. The paper constructs solution families in four special cases but does not prove that no other static spherically symmetric solutions exist in those cases, nor that the five branches in Case (iv) exhaust the possibilities. Please remove this claim or replace it with a precise statement about what has actually been shown.
  3. [Sec. II and III, Eq. (18)] Because a=(6κλ−1)/(4κλ−1), each special relation a(w) fixes the Rastall parameter λ. Thus the families for different w are solutions of different Rastall theories, not different fluids within a single theory with fixed λ. The paper should state this scope limitation explicitly in the introduction and abstract, and qualify phrases such as 'arbitrary w' accordingly. This is not a mathematical flaw in the derivation, but it is essential for correct interpretation of the result.
minor comments (6)
  1. [Sec. III.A and Sec. V] The phrase 'the same solution as in Einstein’s theory is obtained independently of the Rastall parameter when w=-1' is ambiguous: the metric (25) still contains (2a−3)ρ0/3. It should say that the functional form is the same, not that a drops out.
  2. [Eq. (18)] The factor 1/2 present in Eqs. (15)-(17) is omitted in Eq. (18). This is harmless but would be clearer if a consistent coefficient c1 were defined explicitly.
  3. [Sec. IV.C] The coordinate transformations leading to Eqs. (62)-(64) are only sketched. Since the claimed horizon/singularity properties are read off after these transformations, please show the transformations explicitly or state that they are straightforward changes of radial coordinate.
  4. [Appendix A] The appendix is titled 'General case of a and w' but assumes the gauge e^{2G}=(1-b/r)^{-1}. This is a restricted metric ansatz, not the fully general static spherical metric. Please state this explicitly in the appendix title or opening sentence.
  5. [Sec. I] The introduction lists w=0 as a known GR Liouvillian solution, but Sec. III.B shows that in the present formulation the only w=0 solution is Minkowski. To avoid an apparent contradiction, clarify that the GR w=0 entry corresponds to the trivial vacuum solution.
  6. [References] Several references are cited only by arXiv numbers (e.g., [13], [30]). Please add publication data where available and check that all in-text citation numbers are correct.

Circularity Check

0 steps flagged

No significant circularity: the a(w) relations are an explicit ansatz for solving the field equations, and the GR correspondence follows from a=1 at the matching values of w.

full rationale

The derivation is self-contained and non-circular. The authors begin from the Rastall field equations (4)-(5), impose the general static spherically symmetric metric (6)/(7), and reduce the system to the ODEs (15)-(18). The special relations between a and w are introduced explicitly as a device to make coefficients vanish ('it is necessary to consider ... the case a=(5w-1)/(3w-1)'; 'one can see that the cases a=(11w-1)/(2(3w-1)) and a=6w/(3w-1) are also specific relations by observing the transformation of the equations'). These choices set the coefficient ((3w-1)a-(5w-1)) in Eq. (18) to convenient values, which is a legitimate parameter/ansatz reduction, not a hidden fit. The solutions are then obtained by integrating the resulting ODEs (e.g., Eqs. (23)-(24), (26)-(32), (33)-(55)) and are verified to satisfy the remaining field equations; the target spacetimes are not assumed at the start. The match to the known GR solutions of Semiz and Chernin et al. is not an input but follows because at w=0,-1/5,-1/3 the chosen a(w) equals 1, at which Rastall's Eq. (4) reduces exactly to Einstein's equation. Recovering known GR solutions in that limit is a consistency check, not circularity. There are no load-bearing self-citations: the cited GR classification [25,26] and Rastall definitions [19,20] are external. The conditional nature of the result—each a(w) corresponds to a different Rastall parameter λ—is a scope limitation, not a circular reduction. The paper's 'uniqueness of that spacetime' sentence in Sec. V is an overclaim about exhaustiveness, but that is a completeness issue, not a circular step. The explicit omission of w=-1/6 is likewise a method limitation and is acknowledged by the authors. Overall, no circular step can be identified, and the central derivation has independent mathematical content.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

The central derivation rests on the Rastall field equations, the perfect-fluid EoS, the static spherically symmetric ansatz, and the ad hoc imposition of special a(w) relations. No data or independently motivated constraints are used; integration constants are free but not fitted.

free parameters (1)
  • Special Rastall-parameter relations a(w) = a=(5w-1)/(3w-1); a=(11w-1)/(2(3w-1)); a=6w/(3w-1)
    Imposed ad hoc to make coefficients in Eqs. (15)-(18) vanish; these choices enable Liouvillian integration but are not derived from the theory. Each relation selects a different Rastall theory.
axioms (5)
  • domain assumption Rastall field equations (1)-(5) with a := (6κλ-1)/(4κλ-1) are the governing equations.
    The paper builds on Rastall's 1972 theory and the rewritten form (4)-(5); no derivation of the field equations is given.
  • domain assumption Perfect fluid stress-energy tensor (11) with p=wρ and constant w.
    Assumed from the outset; the equation of state is the focus.
  • domain assumption Static, spherically symmetric metric ansatz (6) and the equivalent form (7).
    The paper restricts to this symmetry class; the form (7) with h(ř) is an ansatz.
  • standard math λ ≠ 1/(4κ) and a ≠ 3/2 to avoid singular cases.
    Assumed to rewrite the field equations; the excluded cases are handled in Ref. [19] for a=3/2, but w=-1/6 is not addressed.
  • ad hoc to paper Special relations a(w) are admissible physical choices.
    The core method: these are chosen so coefficients vanish; the paper provides no physical justification for why the Rastall parameter should depend on the fluid EoS.

reviewed 2026-08-01 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Exact solutions for static spherically symmetric spacetime with a perfect fluid in Rastall theory." pith.science (2026). https://pith.science/paper/R4S7PQJG

@misc{pith2026260726463,
  author       = {Pith},
  title        = {Pith review of: Exact solutions for static spherically symmetric spacetime with a perfect fluid in Rastall theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R4S7PQJG}},
  note         = {Machine review of arXiv:2607.26463}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

In general relativity, exact Liouvillian solutions for a static and spherically symmetric spacetime with a perfect fluid and the equation of state $p(r)=w\rho (r)$ are known only for $w=0,-\frac{1}{6}, -\frac{1}{5}, -\frac{1}{3}, -1$. We extend this setup to Rastall's theory, presenting the relation between the Rastall parameter and the constant $w$, and deriving exact solutions that correspond to the known counterparts in general relativity, except for $w=-\frac{1}{6}$. Furthermore, we find that, when $w\neq \frac{1}{3}, -1$, there exist several types of solutions whose behavior changes depending on the choice of constants.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

35 extracted references · 4 linked inside Pith

  1. [1]

    k5 tanh r − w+ 1 2 k0˜r+α ! − 1 2k1k2 ·(−(w+ 1)k 0) # dt2 +

    Therefore,ρ=ρ 0 is obtained, whereρ 0 is a constant. In this case, Eqs. (15) and (16) yield 2f h−1h′′ = 0,(22) 1 Whenw=−1, the relationa= 5w−1 3w−1 impliesa= 3 2 , hencea̸= 5w−1 3w−1 holds. 4 and then we can derive h=ρ 1˜r+ρ2,(23) whereρ 1, ρ2 are constants. Hence, Eq. (16) yields f= 1 ρ2 1 1− ρ2 −ρ 2 1ρ3 ρ1˜r+ρ2 + (2a−3)ρ 0 3 (ρ1˜r+ρ2)2 ,(24) whereρ 3 is...

  2. [2]

    12 Equation (A11) yields ρ=a 1e w+1 c1 F ,(A15) wherea 1 is a constant

    Casea̸= 5w−1 3w−1 withw̸= 1 3 ,−1 We definec 1 = 1 2 ((3w−1)a−(5w−1)) andc 2 = 1 2 ((3w−1)a−3(w−1)). 12 Equation (A11) yields ρ=a 1e w+1 c1 F ,(A15) wherea 1 is a constant. Therefore, Eq. (A9) leads to b= r3 1 + 2rF′ 2 r F ′ +c 1ρ .(A16) From the above two equations and Eqs. (A8) and (A10), we can see thatF(r) satisfies F ′′ + 2F ′2 + 2 r F ′ −a 1c1r2 F ′...

  3. [3]

    L. L. Smalley, J. Phys. A: Math. Gen.16, no.10, 2179-2185 (1983)

    1983

  4. [4]

    Therefore, Eq

    Casew= 1 3 Equation (A11) yields ρ=a 0e−4F ,(A12) wherea 0 is a constant. Therefore, Eq. (A9) leads to b= r3 1 + 2rF′ 2 r F ′ − 1 3 ρ .(A13) From the above two equations and Eqs. (A8) and (A10), we can see thatF(r) satisfies F ′′ + 2F ′2 + 2 r F ′ + a0 3 r2 F ′′ −2F ′2 − 6 r F ′ − 3 r2 e−4F = 0.(A14)

  5. [5]

    Darabi, H

    F. Darabi, H. Moradpour, I. Licata, Y. Heydarzade and C. Corda, Eur. Phys. J. C78, 25 (2018)

    2018

  6. [6]

    Rastall, Phys

    P. Rastall, Phys. Rev. D6, 3357-3359 (1972)

    1972

  7. [7]

    Lindblom and W

    L. Lindblom and W. A. Hiscock, J. Phys. A: Math. Gen.15, no.6, 1827-1830 (1982)

    1982

  8. [8]

    sin2 r w+ 1 2 k0˜r+γ ! (dθ2 + sin2 θdφ2).(61) For these comparisons, we apply an appropriate coordinate transformation to each metric. The metrics of (I) and (II) are written as ds2 =− − w+ 1 2 k0 k3 − 1 2R2 dT 2 + 1 R2 − w+ 1 2 k0 k3 − 1 2R2 −1 dR2 +R2(dθ2 + sin2 θdφ2) (62) via an appropriate coordinate transformation, respectively. This metric has a hor...

  9. [9]

    Visser, Phys

    M. Visser, Phys. Lett. B782, 83-86 (2018)

    2018

  10. [10]

    W. A. G. De Moraes and A. F. Santos, Gen. Rel. Grav.51, no.12, 167 (2019)

    2019

  11. [11]

    Chagoya, J

    J. Chagoya, J. C. L´ opez-Dom ´ ınguez and C. Ortiz, Class. Quant. Grav.40, no.7, 075005 (2023)

    2023

  12. [12]

    Golovnev, Annals Phys.461, 169580 (2024)

    A. Golovnev, Annals Phys.461, 169580 (2024)

    2024

  13. [13]

    L. L. Smalley, Nuovo Cim. B80, no.1, 42-48 (1984)

    1984

  14. [14]

    R. V. d. Santos and J. A. C. Nogales, [arXiv:1701.08203 [gr-qc]]

    Pith/arXiv arXiv

  15. [15]

    C. E. M. Batista, M. H. Daouda, J. C. Fabris, O. F. Piattella and D. C. Rodrigues, Phys. Rev. D85, 084008 (2012)

    2012

  16. [16]

    Shabani and A

    H. Shabani and A. Hadi Ziaie, EPL129, no.2, 20004 (2020)

    2020

  17. [17]

    J. C. Fabris, O. F. Piattella and D. C. Rodrigues, Eur. Phys. J. Plus138, no.3, 232 (2023)

    2023

  18. [18]

    J. A. P. Paiva, J. C. Fabris and V. T. Zanchin, [arXiv:2606.27761 [gr-qc]]

    Pith/arXiv arXiv

  19. [19]

    A. S. Al-Rawaf and M. O. Taha, Gen. Rel. Grav.28, no.8, 935-952 (1996)

    1996

  20. [20]

    obtained the exact solutions for a static and spherically symmetric spacetime with a scalar field. References [21, 22] derived static and spherically symmetric black hole solutions with a perfect fluid under an additional assumption for the metric (namely,g ttgrr =−1 andg AB =r 2σAB). Furthermore, wormhole solutions were investigated in Ref. [24]. In this...

    Pith/arXiv arXiv 2026

  21. [21]

    A. M. Oliveira, H. E. S. Velten, J. C. Fabris and L. Casarini, Phys. Rev. D92, no.4, 044020 (2015)

    2015

  22. [22]

    Khyllep and J

    W. Khyllep and J. Dutta, Phys. Lett. B797, 134796 (2019)

    2019

  23. [23]

    Astorga-Moreno, K

    J. Astorga-Moreno, K. Jacobo, S. Arteaga, M. ´A. Garc ´ ıa-Aspeitia and A. Hern´ andez-Almada, Class. Quant. Grav.41, no.6, 065003 (2024)

    2024

  24. [24]

    A. M. Oliveira, H. E. S. Velten and J. C. Fabris, Phys. Rev. D93, no.12, 124020 (2016)

    2016

  25. [25]

    K. A. Bronnikov, J. C. Fabris, O. F. Piattella and E. C. Santos, Gen. Rel. Grav.48, no.12, 162 (2016)

    2016

  26. [26]

    Heydarzade, H

    Y. Heydarzade, H. Moradpour and F. Darabi, Can. J. Phys.95, no.12, 1253-1256 (2017)

    2017

  27. [27]

    Heydarzade and F

    Y. Heydarzade and F. Darabi, Phys. Lett. B771, 365-373 (2017)

    2017

  28. [28]

    Hansraj, A

    S. Hansraj, A. Banerjee and P. Channuie, Annals Phys.400, 320-345 (2019)

    2019

  29. [29]

    Halder, S

    S. Halder, S. Bhattacharya and S. Chakraborty, Mod. Phys. Lett. A34, no.12, 1950095 (2019)

    2019

  30. [30]

    B. V. Ivanov, J. Math. Phys.43, 1029 (2002)

    2002

  31. [31]

    Iacono and J

    R. Iacono and J. Llibre, J. Math. Phys.55, no.10, 102501 (2014)

    2014

  32. [32]

    Semiz, Class

    ˙I. Semiz, Class. Quant. Grav.39, no.21, 215002 (2022)

    2022

  33. [33]

    A. D. Chernin, D. I. Santiago and A. S. Silbergleit, Phys. Lett. A294, 79-83 (2002)

    2002

  34. [34]

    Simpson and M

    A. Simpson and M. Visser, JCAP02, 042 (2019)

    2019

  35. [35]

    Semiz, [arXiv:2210.16648 [gr-qc]]

    ˙I. Semiz, [arXiv:2210.16648 [gr-qc]]

    Pith/arXiv arXiv

This paper was first reviewed by deepseek-v4-flash on August 1, 2026.