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Black holes and their shadows in $F(R)$ gravity

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read F(R) gravity can shift black hole shadows into the observed EHT ranges.

desk verdict Novel ODE for F(R) black hole perturbations, but the EHT 'prediction' reduces to tuning a free metric seed; the technical core deserves review, the conclusion does not. read the letter →

arxiv 2412.13775 v1 pith:GMUSRC55 submitted 2024-12-18 gr-qc hep-th

Shin'ichi Nojiri , S. D. Odintsov This is my paper · ORCID
classification gr-qchep-th MSC 83C5783D05 PACS 04.50.Kd04.70.-s
keywords F(R)gravityblackholeshadowphotonsphereEventHorizonTelescopeSchwarzschildperturbationmodifiedM87*SgrA*
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 shows that in F(R) gravity, a modified-gravity theory whose Lagrangian is a general function of the Ricci scalar, spherically symmetric vacuum black holes can be built as small perturbations around the Schwarzschild solution. The construction leaves the metric perturbation $\nu_1(r)$ free, subject only to a finiteness condition at the photon-sphere radius $r=3M$. For any such perturbation the photon sphere moves to $3M(1+\varepsilon M\nu_1'(3M))$ while the shadow radius becomes $3\sqrt{3}M(1-\varepsilon\nu_1(3M))$, so all observational content is carried by the single combination $\varepsilon\nu_1(3M)$. Tuning that combination puts the shadow inside the reported M87* range $r_{\mathrm{sh}}/M\simeq 5.5\pm 0.8$ and the Sgr A* range $4.21\lesssim r_{\mathrm{sh}}/M\lesssim 5.56$. The upshot is that F(R) gravity as a class is not excluded by current shadow measurements, in contrast to the simplest mimetic-gravity model discussed in the paper's introduction.

What carries the argument

The load-bearing object is the third-order differential equation (15) that the F(R) field equations impose on $F_R(r)$ for a given static spherically symmetric metric function $\nu(r)$, together with its perturbative reduction (29) near the photon sphere. The reconstruction pipeline—solve (15) for $F_R(r)$, use (13) to obtain $\lambda(r)$, compute the scalar curvature $R(r)$, invert to $r(R)$, and integrate $F_R(R)$ to recover $F(R)$—converts an assumed geometry into a concrete F(R) theory. The observable side is the shadow formula $r_{\mathrm{sh}}=r e^{-\nu(r)}|_{r=r_{\mathrm{ph}}}$, which in the perturbative branch collapses to $r_{\mathrm{sh}}=3\sqrt{3}M(1-\varepsilon\nu_1(3M))$; the photon-sphere shift itself drops out of this expression.

What would settle it

For a specified F(R) model, solve the linearized field equation (29) (or the full equation (15)) to compute $\nu_1(r)$, then evaluate $-3\sqrt{3}\,\varepsilon\nu_1(3M)$ and check it against the inequalities (55) and (56); a value outside the relevant window shows that model is incompatible with the corresponding EHT shadow observation.

Watch

Extended reading notes

Core claim

The paper's central claim is that a perturbative reconstruction of F(R) from an assumed geometry near the photon sphere is enough to make black hole shadows compatible with observations. Writing $F_R=1+\frac12\varepsilon f_R$ and $\nu=\nu_0+\varepsilon\nu_1$, the vacuum field equations reduce to an inhomogeneous linear third-order equation (29) for $f_R(r)$. In the region $r\sim 3M$, a regular branch of solutions exists whenever $C_{\nu_1}\equiv(4M\nu_1'''-\frac{32}{3}\nu_1''+\frac{8}{3M}\nu_1')|_{r=3M}$ is finite and nonzero; along that branch the shadow radius is $r_{\mathrm{sh}}=3\sqrt{3}M(1-\varepsilon\nu_1(3M))$. Because $\nu_1(3M)$ is not fixed by the reconstruction, the M87* and Sgr A* bounds reduce to simple inequalities on the combination $-3\sqrt{3}\,\varepsilon\nu_1(3M)$, and parameter choices satisfying them exist. The paper therefore claims that F(R)-gravity black holes can easily pass the Event Horizon Telescope constraints while remaining perturbatively close to Schwarzschild.

Load-bearing premise

The load-bearing assumption is that the metric perturbation $\nu_1(r)$ may be chosen freely and then realized by some local F(R) theory; the paper never derives $\nu_1$ from a specified action, so the tuning freedom that fits the shadow data could disappear for a concrete model.

Editorial extensions

If this is right

  • F(R) gravity remains viable at horizon scales: the current M87* and Sgr A* shadow measurements do not exclude the theory class.
  • Any metric perturbation $\nu_1$ with $C_{\nu_1}$ finite and nonzero at $r=3M$ determines a locally reconstructed F(R) whose shadow radius is controlled by $\varepsilon\nu_1(3M)$ alone.
  • The constraints become explicit numerical targets for model building: $-0.5<-3\sqrt{3}\,\varepsilon\nu_1(3M)<1.1$ for M87* and $-1.01\lesssim -3\sqrt{3}\,\varepsilon\nu_1(3M)\lesssim 0.36$ for Sgr A*.
  • A shift in the photon-sphere radius at first order does not feed into the shadow radius, because $d(r/\sqrt{1-2M/r})/dr$ vanishes at $r=3M$; the resulting first-order shadow formula depends only on $\nu_1(3M)$.
  • The non-perturbative $\nu=0$ power-law branch yields no finite photon sphere, so not every F(R) vacuum solution can describe observed shadows.

Reading between the lines

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

  • The paper leaves open which explicit F(R) action realizes a given $\nu_1$; committing to a concrete model fixes $\nu_1$ and may remove the tuning freedom used to satisfy the inequalities.
  • The same perturbative scheme should transfer to other metric-modified gravity theories: whenever the field equations linearize to a scalar equation for a metric perturbation near the photon sphere, EHT constraints reduce to a one-parameter test.
  • The 'easily pass' conclusion is an existence statement about parameter choices, not a prediction for generic F(R) models; the method supplies a diagnostic for classifying individual models by their value of $\varepsilon\nu_1(3M)$.
  • A single shadow measurement constrains only the combination $\varepsilon\nu_1(3M)$, not the full radial dependence of the deviation; distinguishing among F(R) models will require additional observations or a theoretical prior on $\nu_1$.
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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 / 4 minor

Summary. The paper investigates photon-sphere and black-hole-shadow radii in F(R) gravity. For a general static spherically symmetric metric (5) the authors derive, in Eq. (15), a third-order ODE for F_R(r); they then specialize to a perturbative setup in which the metric is a small deviation from Schwarzschild and F_R=1+(ε/2)f_R. Solving the linearized Eq. (29) locally near r=3M, they obtain f_R and λ_1 as expansions around the photon-sphere radius. Using the standard null-geodesic potential W(r), they find the shadow radius shift r_sh=3√3 M(1-ε ν_1(3M)), Eq. (54). The EHT constraints on M87* and Sgr A* are then translated into bounds on the combination δ=-3√3 ε ν_1(3M), Eqs. (55)-(56), and the paper concludes that F(R)-gravity black-hole shadows may easily pass EHT constraints.

Significance. If the central claim were established, the paper would show that a wide class of F(R) gravity models can evade EHT shadow-radius constraints, and the perturbative framework for deriving the third-order F_R equation could be a useful technical tool for modified-gravity shadow calculations. The leading-order shadow formula is a straightforward consequence of the null-geodesic calculation once the perturbative ansatz is accepted. However, the main conclusion is not supported by the presented derivation: the quantity constrained by EHT is an arbitrary seed function ν_1, not a parameter of a specified F(R) action, and no global asymptotically flat solution is constructed. The paper is therefore better viewed as a consistency relation for a free metric perturbation than as a prediction of F(R) gravity.

major comments (4)
  1. [II B 1 and III D, Eqs. (29)-(35), (55)-(56)] The EHT constraints are imposed on ε ν_1(3M), where ν_1 is a freely chosen seed function introduced in Eq. (27). The paper explicitly states that 'we did not specify ν_1 as long as Cν_1 is finite and does not vanish.' Thus Eqs. (55) and (56) restrict the choice of ν_1, not any parameter of a concrete F(R) action. The conclusion 'by tuning the parameters, the models can satisfy the constraints' is therefore circular with respect to the claim that F(R) gravity predicts shadow sizes consistent with EHT; a genuine test would require fixing an action F(R), deriving ν_1 from the resulting field equations, and then comparing the predicted ε ν_1(3M) with the bounds.
  2. [II, Eq. (15)] The derivation of Eq. (15) is not shown. The text states that substituting Eq. (13) into Eq. (14) yields Eq. (15), but the algebra is nontrivial and no intermediate steps or independent check are provided. Since Eq. (29) and all subsequent shadow results depend on Eq. (15), the derivation should be supplied in an appendix or at least sketched in sufficient detail for the reader to verify the reduction.
  3. [II B 1, Eqs. (32)-(35)] The solution for f_R and λ_1 is only a local Taylor expansion around r=3M. No argument establishes that these local expressions extend to a global, horizon-regular, asymptotically flat solution. The shadow formula Eq. (41) is evaluated at the photon sphere but refers the radius to an asymptotic observer; without asymptotic flatness the interpretation of r_sh as the observed shadow size is not justified. The paper neither constructs the global metric nor invokes any existence or asymptotic-flatness result that would warrant the local-to-global step.
  4. [IV, Summary and Conclusion] The final sentence, 'Hence, it is proved that BH shadows in F(R) gravity may easily pass the Event Horizon Telescope constraints,' overstates the result. What is proved, given the assumptions, is that for any sufficiently regular seed ν_1 one can locally solve for f_R near r=3M and obtain a shadow shift proportional to ν_1(3M). Since ν_1 is not derived from a specified F(R) model and no global solution is shown, the statement about F(R) gravity as a theory is not established by the manuscript.
minor comments (4)
  1. [II, Eq. (13)] The displayed formula for N has ambiguous fraction formatting; the denominator should be written explicitly, e.g. N = exp(-∫ dr1 [F_R''/(F_R/r1 + (1/2)F_R')]), to avoid confusion.
  2. [III A, Eqs. (43)-(44)] The symbol E is used for both the conserved energy and the Lagrangian, and the sentence 'E = L vanishes identically E = L = 0' is confusing; a distinct symbol for the Lagrangian would make the null-geodesic argument clearer.
  3. [III C, after Eq. (54)] The statement that 'the expression in (41) is general' is too strong; Eq. (54) is derived under the specific local perturbative ansatz of Section II B 1 and the assumption Cν_1 ≠ 0.
  4. [II A, Eqs. (21)-(22)] The notation α_±^3 = -10/27 ± 90/27 and the subsequent numerical values appear correct but are displayed in a compressed way; writing the cube roots explicitly would improve readability.

Circularity Check

2 steps flagged · score 7.0 of 10

The claimed EHT compatibility is a fit to a free seed function ε ν1(3M), not a prediction of any specified F(R) gravity theory.

  1. fitted input called prediction [Section III C-D, Eqs. (50)-(56)]
    "rsh = re−ν(r)| r=rph = 3√3M (1 − ǫν1 (r = 3M)) . ... In the model of Subsubsection II B 1, we did not specify ν1 as long as Cν1 ≡ (4M ν'''1 − 32/3 ν''1 + 8/(3M) ν'1) | r=3M is finite and does not vanish. Therefore, the expression in (41) is general."

    The shadow radius in Eq. (54) is expressed entirely in terms of the free first-order metric seed ν1, which Assumption (27) introduces and Section II B 1 explicitly leaves unspecified. The observational constraints (55) and (56) are inequalities on the same combination −3√3 ε ν1(3M); they do not test F(R) gravity but merely pick out allowed values of the free input. A concrete F(R) action would determine ν1 through the field equations, so the EHT agreement is achieved by construction, not derived.

  2. fitted input called prediction [Section III D, Eqs. (55)-(56) and conclusion]
    "−0.5 < −3√3ǫν1 (r = 3M) < 1.1 , (55) for M87∗ and −1.01 ≲ −3√3ǫν1 (r = 3M) ≲ 0.36 , (56) for Sgr A∗. Therefore by tuning the parameters, the models can satisfy the constraints."

    The paper itself states that the constraints are satisfied by tuning parameters. But ε and ν1(3M) are not parameters of a fixed F(R) model; ν1 was declared arbitrary in Subsubsection II B 1. Thus the conclusion 'Hence, it is proved that BH shadows in F(R) gravity may easily pass the Event Horizon Telescope constraints' reduces to the statement that a free function can be chosen to match the data.

full rationale

The central claim is circular in the fitted-input sense: the constrained quantity is a free input of the perturbative construction. The paper never constructs a concrete F(R) action for which ν1 is forced; instead, Eq. (29) solves locally for f_R given an arbitrary ν1, and Eq. (54) then gives the shadow shift as −3√3 M ε ν1(3M). The EHT inequalities (55)-(56) bound this same free combination, so satisfying them is a choice of ν1, not a prediction of F(R) gravity. There is no self-citation chain or uniqueness theorem forcing the result, so this is not pattern 3 or 4; the circularity is of the fitted-input-called-prediction type. The separate absence of a global asymptotically flat extension is a correctness gap but does not affect the circularity scoring. Overall score 7: one or more 'predictions' reduce by construction to choosing a free seed function, and the paper explicitly says the parameters are tuned to satisfy the constraints.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The derivation rests on a perturbative ansatz with a free metric function nu1, integration constants, and the assumption that local expansions extend globally. No new particles, forces, or dimensions are introduced. The observational consistency is achieved by fitting the free combination epsilon nu1(3M), so the theory itself is not constrained independently.

free parameters (5)
  • epsilon (perturbative expansion parameter) = unspecified small value
    Introduced in Eqs. (26) and (27) to control the deviation from general relativity and from Schwarzschild. No fixed value is derived from the theory.
  • delta = -3*sqrt(3)*epsilon*nu1(3M) = M87*: between -0.5 and 1.1; Sgr A*: between -1.01 and 0.36
    This is the quantity entering the shadow shift in Eq. (54) and is constrained by observations in Eqs. (55) and (56). The paper states that tuning this parameter allows the models to satisfy the constraints.
  • f_R^0 and f_R^1 = integration constants
    Appear in the local solution for f_R in Eqs. (32) and (33). Their values are not fixed by the theory.
  • lambda_1^0 = integration constant
    Integration constant in Eq. (35) for the perturbed metric function lambda_1.
  • r0 = integration constant
    Integration constant in Eq. (24) for the non-perturbative nu=0 example.
assumptions (6)
  • standard math The F(R) field equations in vacuum, Eq. (2), are the correct equations of motion.
    Standard result from varying the action (1); the paper uses it without proof.
  • domain assumption The metric can be written in the static spherically symmetric form (5).
    Restricts the analysis to static, spherically symmetric configurations, which is central to the derivation.
  • ad hoc to paper The deviation from Schwarzschild and from Einstein gravity is small, with F_R = 1 + (epsilon/2) f_R and nu = nu0 + epsilon nu1.
    Perturbative ansatz in Eqs. (26) and (27); the paper does not justify that the actual F(R) solutions of interest lie in this regime.
  • ad hoc to paper A local solution of the approximate equation near r=3M extends to a global, physically acceptable solution.
    The paper solves only the leading behavior around r=3M; global existence is not established.
  • standard math The photon sphere is determined by W(r)=W'(r)=0 and the shadow radius by r_sh = r e^{-nu(r)} at r=r_ph.
    Standard geodesic and shadow results, Eqs. (41) and (46).
  • domain assumption The observational constraints quoted from the literature apply to the asymptotic shadow radius.
    Used in Section III D; requires the spacetime to be asymptotically flat at the observer, which is assumed but not proven for the constructed solutions.

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Cite this review

Pith. "Pith review of Black holes and their shadows in $F(R)$ gravity." pith.science (2026). https://pith.science/paper/GMUSRC55

@misc{pith2026241213775,
  author       = {Pith},
  title        = {Pith review of: Black holes and their shadows in $F(R)$ gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GMUSRC55}},
  note         = {Machine review of arXiv:2412.13775}
}
abstract

We investigate the radii of the photon sphere and the black hole shadow in the framework of $F(R)$ gravity. For this purpose, we derive the field equation for the corresponding theory when the general spherically symmetric and static configuration is considered. This equation is the third-order differential equation with respect to $F_R(r)\equiv \left. \frac{dF(R)}{dR}\right|_{R=R(r)}$, where $r$ is the radial coordinate. Solving the equation, we find $F(R)$ as a function of $r$, $F_R=F_R(r)$. By using the assumed and obtained geometry, one can calculate the scalar curvature $R$ as a function of $r$, $R=R(r)$, which could be solved with respect to $r$ as $r=r(R)$. Then one finds the functional form of $F_R$ as a function of the scalar curvature $R$, $F_R=F_R(R)=F_R\left( r=r\left(R\right)\right)$.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.