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REVIEW 3 major objections 3 minor 19 references

Static Spherically Symmetric Solutions in General Gauss-Bonnet Gravity

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper derives a new exact static spherically symmetric vacuum solution in $R+F(G)$ Gauss-Bonnet gravity, with metric $B(r) = -1 + r/(4a)$, horizon $r = 4a$, and Hawking temperature $1/(16\pi a)$.

desk verdict The claimed new black hole solution is self-inconsistent: the metric yields G ~ 1/r², so the recovered F(G) = G^{3/2} contradicts the assumed F' = ar + b. read the letter →

arxiv 2507.09238 v1 pith:4KMXEEY7 submitted 2025-07-12 gr-qc hep-th

Naser Mohammadipour This is my paper · ORCID
classification gr-qchep-th PACS 04.50.Kd04.70.Dy04.20.Jb
keywords modifiedgravityGauss-BonnetinvariantF(G)staticsphericallysymmetricsolutionsexactblackholeHawkingtemperaturethermodynamicsentropy
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 aims to find exact static spherically symmetric vacuum solutions of modified Gauss-Bonnet gravity with action $R+F(G)$. It derives two branches: a constant-Gauss-Bonnet branch that reproduces (anti-)de Sitter space, and a new branch in which $F'(G)=ar+b$ forces the metric $B(r)=-1+r/(4a)$. If correct, the second branch is a new exact black hole solution of the reconstructed model $F(G)=G^{3/2}+F_0$, with horizon at $r=4a$, Hawking temperature $1/(16\pi a)$, and conserved-charge entropy from Eq. (27). Exact solutions of this kind are rare, so they give concrete settings for probing thermodynamics and deviations from general relativity.

What carries the argument

The Lagrangian-multiplier reduction. The static spherically symmetric action is rewritten as a one-dimensional integral in $B(r)$, $X(r)$ and $G(r)$, with a multiplier $\alpha$ enforcing the definition of the Gauss-Bonnet invariant. Varying $\alpha$ gives $\alpha=F'(G)$, and after integration by parts the reduced Lagrangian yields the equations of motion. With $X(r)=1$, Eq. (12) collapses to $(F')''=0$, giving $F'(G)=ar+b$; inserting this into the remaining equation produces an ODE whose solution is $B(r)=-1+r/(4a)$. This chain -- linear $F'$ forcing a specific metric, then reconstructing $F(G)=G^{3/2}+F_0$ -- is what carries the argument.

What would settle it

Substitute $B(r)=-1+r/(4a)$ into the definition of the Gauss-Bonnet invariant to get $G(r)$, then check whether $F'(G(r))=(3/2)G(r)^{1/2}$ equals $ar+b$ at every $r$; equivalently, substitute $F(G)=G^{3/2}+F_0$ and the metric into the full field equations and verify that all components vanish. Failure of either check would falsify the claimed exact solution.

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Extended reading notes

Core claim

The paper claims that in $R+F(G)$ modified gravity, the static spherically symmetric vacuum equations admit two exact branches. For constant Gauss-Bonnet invariant $G=G_0$ and $X(r)=1$, the metric is the (anti-)de Sitter solution $B(r)=1-(\Lambda/3)r^2$, with $\Lambda$ fixed by $G_0F'(G_0)-F(G_0)$. The second branch follows from fixing $X(r)$ constant, which reduces the equation of motion to $(F')''=0$, so $F'(G)=ar+b$; the resulting metric is $B(r)=-1+r/(4a)$, with horizon at $r=4a$, Hawking temperature $T=1/(16\pi a)$, and conserved-charge entropy given by Eq. (27). The corresponding model is reconstructed as $F(G)=G^{3/2}+F_0$, so the paper presents $B(r)=-1+r/(4a)$ as a new exact black hole solution of $R+G^{3/2}+F_0$ gravity.

Load-bearing premise

The load-bearing premise is that Eq. (12), obtained by varying the reduced Lagrangian with respect to $B(r)$, is correctly derived and that the step from it to $(F')''=0$ is legitimate when $X(r)$ is constant; if that equation or the division step fails, the linear form $F'(G)=ar+b$ and the metric $B(r)=-1+r/(4a)$ do not follow.

Editorial extensions

If this is right

  • The metric $B(r) = -1 + r/(4a)$ is presented as an exact vacuum solution of $R + G^{3/2} + F_0$ gravity, not a perturbative approximation.
  • The single horizon at $r=4a$ and the temperature $T=1/(16\pi a)$ are both fixed by one parameter $a$, giving a one-parameter family of black holes.
  • The constant-$G$ branch connects the model to (anti-)de Sitter space, with the effective cosmological constant determined by $G_0F'(G_0)-F(G_0)$.
  • For $a>0$ the Hawking temperature is positive, so the new branch is thermodynamically admissible.

Reading between the lines

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

  • The linear-$F'$ condition $F'(G)=ar+b$ may be the more general object: it defines a family of $F(G)$ models, one per choice of integration constants $(a,b)$, whose vacuum metrics all solve the same ODE.
  • If the solution passes direct substitution into the full field equations, it would provide a simple explicit arena for studying stability, quasinormal modes, and heat capacity of Gauss-Bonnet black holes.
  • The same Lagrangian-multiplier strategy could be applied to $F(R,G)$ models with non-minimal couplings to test whether a linear-$F'$ branch survives.
  • One could check directly whether the entropy computed from Eq. (27), together with the explicit temperature, satisfies the generalized second law that the abstract invokes.
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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

3 major / 3 minor

Summary. The manuscript studies static spherically symmetric vacuum solutions of R+F(G) gravity using a Lagrange-multiplier method. Two solution branches are reported: the constant-Gauss-Bonnet (A)dS solution B(r)=1-(Λ/3)r² and a new branch B(r)=-1+r/(4a), for which the paper reconstructs F(G)=G^{3/2}+F0 and derives the event horizon, Hawking temperature, and Noether entropy.

Significance. If correct, the second branch would be a new exact black hole in a simple F(G) model, and the first branch is a standard (A)dS solution. However, the central claim is internally inconsistent: substituting the displayed metric into the reconstructed model violates the linear F'(G) ansatz used to derive it. The paper also contains an unjustified variational step. These are load-bearing errors, so the advertised exact solution and its thermodynamics are not established.

major comments (3)
  1. [§4, Eqs. (16), (21), (23)] The second branch is self-inconsistent. For the metric ds²=-B dt²+B^{-1}dr²+r²dΩ² with B=-1+r/(4a), the Gauss-Bonnet invariant is G=(4/r²)[B'^2-B''(1-B)] = 1/(4a²r²). The model reconstructed in Eq. (23), F(G)=G^{3/2}+F0, has F'(G)=(3/2)√G=3/(4ar). This is proportional to 1/r, whereas Eq. (16) states F'(G)=ar+b. The two expressions cannot be equal on any open interval for nonzero a. Equivalently, combining (16) and (23) forces G(r) to be quadratic in r, while the metric (21) gives G∝r^{-2}. Hence the metric (21) is not a solution of the model (23), and the horizon, temperature, and entropy computed from it are unsupported.
  2. [§4, Eqs. (12)-(16)] The step from Eq. (12) to Eq. (15) is not justified. As displayed, Eq. (12) has factors X'(r) multiplying both terms; setting X(r)=constant makes the left-hand side vanish identically, so it cannot yield (F')''=0 or F'(G)=ar+b. If the displayed equation is misprinted and the intended variation with respect to B contains terms that survive at X'=0, the derivation must be supplied explicitly. This matters because Eq. (16) is the basis for the entire second branch.
  3. [§4, Eqs. (20)-(21)] The claim that B(r)=-1+r/(4a) is the general solution of Eq. (20) is made without any derivation, and the displayed equation (20) is too garbled to check. Given the inconsistency in the preceding comment, this branch cannot be correct as stated; the authors should either provide the full integration or retract the branch.
minor comments (3)
  1. [Eqs. (6)-(8), (20), (22)] Many displayed formulas are barely legible due to typographical errors; the equations need to be typeset cleanly. This is a presentation issue, but it impedes verification.
  2. [Eq. (24)] For the first branch B=1-(Λ/3)r², the horizon is r=√(3/Λ), not the printed expression; please correct the horizon radii and the subsequent formulas that depend on them.
  3. [Abstract and Introduction] The abstract and introduction refer to F(R,G) models, while the action in Eq. (1) is R+F(G); the scope should be stated consistently.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the F(G) model is reconstructed as the output of the derivation, not assumed as the input; no self-citation chain is load-bearing.

full rationale

The paper uses a standard inverse-problem procedure: with the static spherically symmetric ansatz and X(r)=1, it derives equations of motion, integrates an identity to F'(G)=ar+b, solves the resulting ODE for B(r), computes G(r) for that metric, and then reconstructs F(G) by integrating F' along the solution. In this chain F(G) is the output of the calculation, not an input, so the central claim does not reduce by construction to its own assumptions. The load-bearing references (the Lagrangian-multiplier method [15], the F(G) field equations [16], and the thermodynamic formulas [17]-[19]) are external works, not self-citations, and no uniqueness theorem from the authors' prior work is invoked. No fitted parameter is relabeled as a prediction. A serious internal consistency problem does appear to exist: for B(r) = -1 + r/(4a), the Gauss-Bonnet invariant behaves as G ~ 1/r^2, so the claimed F(G) = G^(3/2) + F0 would give F'(G(r)) ~ 1/r, not the assumed ar+b; this would invalidate the new black-hole branch. However, that is a mathematical error or omitted verification, not a circular reduction of the result to its inputs, and therefore it does not raise the circularity score under the stated criteria.

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

The central results rest on four free integration constants and on the assumed validity of the quoted F(G) field equations and of the Lagrangian multiplier reformulation. No new physical entity is introduced; the multiplier α is eliminated by α = F'(G).

free parameters (4)
  • a
    Integration constant in F'(G) = ar + b (Eq. 16). It fixes the horizon r = 4a, the Hawking temperature 1/(16πa), and the entropy, and is never tied to an independent physical input.
  • b
    Second integration constant in Eq. (16). It drops out of later formulas without discussion, so its role in the model is unspecified.
  • Λ
    Cosmological constant of the first branch, set by Eq. (14) through the value of F and its derivative at constant G = G0. Its value is not fixed by any external data.
  • F0
    Additive constant in F(G) = G^(3/2) + F0 (Eq. 23). It shifts the action and the field equations but is never determined.
assumptions (4)
  • domain assumption The F(G) field equations quoted from Ref. [16] (Eq. 2) are the correct variational equations for the action R/2 + F(G).
    Adopted from the literature without re-derivation, so all subsequent solutions inherit any errors in those equations.
  • ad hoc to paper The Lagrangian multiplier reformulation (Eqs. 7 to 10) preserves the dynamics of the original action.
    The paper derives Eqs. (11) and (12) from the constrained action but never shows this matches the full metric variation of Eq. (1).
  • ad hoc to paper Fixing X(r) = constant does not discard relevant vacuum solutions.
    The paper sets X = 1 to close the system (Sec. 3) without proving that every static spherically symmetric vacuum solution admits this gauge.
  • ad hoc to paper The recovered model F(G) = G^(3/2) + F0 is consistent with the assumed F'(G) = ar + b for the solution's G(r).
    Used when integrating Eq. (16) to obtain Eq. (23). It is never verified and appears to fail, since (3/2)G^(1/2) is not linear in r for the metric's G(r).

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Cite this review

Pith. "Pith review of Static Spherically Symmetric Solutions in General Gauss-Bonnet Gravity." pith.science (2026). https://pith.science/paper/4KMXEEY7

@misc{pith2026250709238,
  author       = {Pith},
  title        = {Pith review of: Static Spherically Symmetric Solutions in General Gauss-Bonnet Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4KMXEEY7}},
  note         = {Machine review of arXiv:2507.09238}
}
read the original abstract

Considering an action in F(G) modified gravity, the static spherically symmetric solutions are investigated. Introducing the Lagrangian multipliers {\alpha} we obtain the Lagrangian and equations of motion. we obtain two type solutions for these models. The first case leads to Schwarzschild-de Sitter (anti de Sitter) solution and the other one, results in a new metric. At last, the event horizon, the Hawking temperature and the generalized second law of thermodynamics in the framework of the modified Gauss-Bonnet gravity for this solution as a black hole are investigated.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

19 extracted references · 17 canonical work pages

  1. [1]

    Perlmutter et al

    S. Perlmutter et al. [Supernova Cosmology Project Collaboration], Astrophys. J. 517, 565 (1999)

    work page 1999

  2. [2]

    C. L. Bennett et al., Astrophys. J. Suppl. 148, 1 (2003)

    2003

  3. [3]

    Tegmark et al

    M. Tegmark et al. [SDSS Collaboration], Phys. Rev. D 69, 103501 (2004). 8

    work page 2004

  4. [4]

    S. W . Allen, et al., Mon. Not. Roy. Astron. Soc. 353, 457 (2004)

    work page 2004

  5. [6]

    Pa d m a n a b h a n , P hy s

    T. Pa d m a n a b h a n , P hy s . Re p t s . 3 8 0 , 2 3 5 ( 2 0 0 3 ) ; E . J . C o p e l a n d , M . S a m i and S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006); Y -F. C a i , E . N. Saridakis, M. R. Setare, J. Q. Xia, Phys. Rep. 493, 1, 1-60 (2010)

    work page 2006

  6. [7]

    Brans and C

    C. Brans and C. H. Dicke, Phys. Rev. 124, 925 (1961)

    work page 1961

  7. [8]

    Nojiri and S

    S. Nojiri and S. D. Odintsov, Int. J. Geom. Meth. Mod. Phys. 4, 115 (2007)

    work page 2007

  8. [9]

    Nojiri and S

    S. Nojiri and S. D. Odintsov, arXiv:0801.4843 [astro-ph]; arXiv:0807.0685 [hep-th]; T . P . Sotiriou and V . Faraoni, Rev. Mod. Phys. 82, 451-497 (2010); F. S . N. L o b o , a r X iv : 0 8 0 7 . 1 6 4 0 [ g rq c ] ; S. Capozziello and M. Francaviglia, Gen. Rel. Grav. 40, 357 (2008); M. R. Setare, Int. J. Mod. Phys. D17, 2219, (2008); Astrophys. Space Sci....

    arXiv 2010

Show all 19 references
  1. [10]

    Antoniadis, J

    I. Antoniadis, J. Rizos and K. Tamvakis, Nucl. Phys. B 415 (1994) 497; S. Kawai, M. A. Sakagami and J. Soda, Phy. Lett. B 437 (1998) 284; S. Kawai and J. Soda, Phy. Lett. B 460 (1999) 41; P . Kanti, J. Rizos and K. Tamvakis, Phys. Rev. D 59 (1999) 083512; N. E. Mavromatos and ...

    1994

  2. [11]

    Nojiri, S

    S. Nojiri, S. D. Odintsov and M. Sasaki, Phys. Rev. D71 (2005) 123509

    2005

  3. [12]

    Nojiri, and S

    S. Nojiri, and S. D. Odintsov, Int. J. Geom. Meth. Mod. Phys. 4 (2007) 115; G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov and S. Zerbini, Phys. Rev. D 73 (2006) 084007; S. Nojiri, and S. D. Odintsov, Phys. Lett. B 631 (2005) 1; S. Nojiri, S. D. Odintsov and M. Sami, Phys....

    2007 arXiv

  4. [13]

    S. M. Carroll, A. De Felice, V . Duvvuri, D. A. Easson, M. Trodden and M. S. Turner, Phys. Rev. D 71, 063513 (2005)

    2005

  5. [14]

    Cognola, E

    G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov and S. Zerbini, Phys. Rev. D 75, 086002 (2007)

    2007

  6. [15]

    Sebastiani, S

    L. Sebastiani, S. Zerbini, arXiv:1012.5230v3 [gr-qc]

    arXiv

  7. [16]

    De Felice, S

    A. De Felice, S. Tsujikawa, Living Rev. Rel. (2010)

    2010

  8. [17]

    Cai, L.M

    R.G. Cai, L.M. Cao, Y .P . Hu, Class. Quant. Grav, 26, 155018 (2009)

    2009

  9. [18]

    R. M. Wald Phys. Rev. D 48, (1993) 3427; V . Iyer , R. M. Wald, Phys. Rev. D 50, 846 (1994); T . Jacobson. G. Kang, R. C. Myers, Phys. Rev. D 49, 6587 (1994); I. Brevik, S. Nojiri, S. D. Odintsov, L. Vanzo, Phys. Rev. D 70, 043520 (2004); H. Maeda, Phys. Rev. D 81, 124007 (2010)

    1993

  10. [19]

    Mohseni Sadjadi, Phys

    H. Mohseni Sadjadi, Phys. Scripta 05, 055006 (2011)

    2011

  11. [2007]

    Koivisto and D

    104; T . Koivisto and D. F . Mota, Phys. Rev. D 75 (2007); M. Satoh, S.Kanno and J. Soda, Phys. Rev. D 77 (2008) 023526; M. Satoh and J. Soda, JCAP 0809 (2008)

    2007

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