Accelerating electrically charged ModMax black hole solutions in $F(R)$ gravity

A. Rinc\'on; B. Eslam Panah; N. Heidari

REVIEW 2 major objections 5 minor 159 references

Accelerating electrically charged ModMax black hole solutions in $F(R)$ gravity

T0 review · 2 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read F(R)-ModMax gravity admits an exact accelerating electrically charged black hole solution whose thermodynamics and shadow depend on charge, acceleration, and curvature.

desk verdict The exact C-metric solution is a charge-rescaled disguise of a known F(R)-Maxwell black hole, and the thermodynamic section is built on an off-horizon mass; the shadow analysis is the most salvageable part. read the letter →

arxiv 2607.15914 v1 pith:ZET7K6R5 submitted 2026-07-17 gr-qc hep-th

B. Eslam Panah , N. Heidari , A. Rincón This is my paper · ORCID

classification gr-qchep-th

keywords acceleratingblackholesC-metricF(R)gravityModMaxelectrodynamicsnonlinearholethermodynamicsHawkingtemperatureshadow

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

F(R)-ModMax gravity couples modified gravity to a nonlinear electrodynamics that preserves Maxwell's duality and conformal symmetries. By inserting a C-metric ansatz and imposing constant Ricci scalar R0, this paper derives an exact four-dimensional accelerating, electrically charged black hole solution with explicit metric functions g(r) and X(θ). The solution reduces to known GR-ModMax, F(R)-Maxwell, and charged (A)dS limits, and its curvature singularity sits at r=0. The authors then extract closed-form Hawking temperature, modified area-law entropy, heat capacity, and shadow radius, and map how acceleration, charge, the ModMax parameter, and the F(R) parameter shift horizons, stability windows, and shadow size. The value of the construction is that it makes a difficult modified-gravity/NLED system analytically tractable and yields parameter-dependent predictions that could be compared with black-hole imaging.

What carries the argument

The load-bearing machinery is the C-metric ansatz, with conformal factor K=1+Ar cosθ, combined with the constant-curvature reduction R=R0. That reduction converts the fourth-order F(R) equations into Einstein-like equations with coupling 1+f_R0, which is what makes a closed-form solution possible. In the electric sector (P=0), the ModMax Lagrangian reduces to Maxwell's Lagrangian multiplied by e^γ, so the nonlinear-electrodynamics parameter acts as a charge rescaling. The thermodynamics are carried by the modified area-law entropy S = Area(1+f_R0)/4, while the optical analysis uses null-geodesic separation with a regularizing affine parameter to get the photon sphere and circular shadow.

What would settle it

Take f(R)=αR², the standard quadratic modification: Eq. (10) reduces to R0=0, so f_R0=0 and the solution collapses to the GR-ModMax case. A direct check—substituting any proposed nonzero root for a concrete f(R) into Eqs. (17)-(18) and into the full fourth-order field equations—would settle whether the claimed F(R) solution exists beyond this degenerate limit.

Watch

Extended reading notes

Core claim

The paper's central claim is that the F(R)-ModMax field equations, under the constant-curvature assumption R=R0 with R0=2 f(R0)/(f_R0-1), admit the C-metric solution g(r)=(1-A^2 r^2)(1-2m0/r + q^2 e^{-γ}/((1+f_R0)r^2)) - R0 r^2/12 and X(θ)=1+2m0 A cosθ + q^2 e^{-γ} A^2 cos^2θ/(1+f_R0). Every field equation is satisfied by these functions, with f_R0≠-1 required for physical solutions. The spacetime has a curvature singularity at r=0 and is not asymptotically (A)dS; in the pure electric sector the ModMax parameter enters only through the effective charge q e^{-γ/2}, making the solution formally equivalent to a Maxwell-charged F(R) black hole with rescaled charge. From this metric the paper der

Load-bearing premise

The result presupposes that the Ricci scalar takes a constant value R0 obeying R0=2 f(R0)/(f_R0-1) with f_R0≠-1, and the paper never exhibits a concrete f(R) model satisfying this condition; if no such model exists, the metric is not an F(R) black hole.

Editorial extensions

If this is right

The solution reproduces, as limits, the accelerating ModMax black hole in GR (f_R0=0), the accelerating charged F(R) black hole (γ=0), and charged (A)dS black holes (A=0, f_R0=0, γ=0, R0=4Λ).
Horizon geometry depends oppositely on charge and ModMax parameter: increasing q or |R0| removes horizons and favors naked singularities, while increasing γ or f_R0 enlarges the event horizon.
Large, physical black holes with positive temperature and entropy require A < sqrt(-R0/12), which forces R0<0; this selects the AdS-like branch.
Thermal stability (T>0, S>0, C>0 simultaneously) holds only in two disjoint radius intervals, and increasing acceleration shrinks these intervals until the phase transition between them disappears.
The shadow is circular with radius set by a balance: acceleration shrinks it, while γ and f_R0 enlarge it, giving a testable signature for future high-resolution imaging.

Reading between the lines

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

Because the electric-sector ModMax effect is only a charge rescaling, observations fitting this solution are also fit by F(R)-Maxwell with rescaled charge; distinguishing ModMax from Maxwell likely requires the magnetic or dyonic sector, where the √(S²+P²) term becomes nontrivial.
The constant-curvature reduction is the limiting step: for the quadratic model f(R)=αR², Eq. (10) forces R0=0 and the solution reduces to GR, so the paper's F(R) generality is demonstrated on a slice unless an explicit f(R) with a nonzero root is supplied.
The circularity of the shadow, which the paper notes is a C-metric property, implies these accelerating black holes cannot be told apart from static spherical ones by shadow shape alone; combining shadow size with lensing or quasi-normal-mode signatures would be the natural discriminator.
One extension the paper leaves implicit: imposing the constant-curvature root as a dynamical consistency condition, rather than a free dial, may select specific f(R) models and tie R0 to the black-hole mass and charge.

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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 central object is an exact solution of the F(R)-ModMax field equations on the constant-curvature slice, but the paper's own conclusion confirms it is just the accelerating F(R)-Maxwell black hole of Ref. [136] with the replacement q → q e^{-γ/2}. The ModMax parameter rescales the charge in the pure-electric sector; it introduces no new dynamics. So the headline novelty is substantially overstated. That said, the derivation is honest and the reduction to the known GR and F(R) limits is clearly exhibited.

What is good: the action and field equations are written out, the C-metric ansatz is solved, and the resulting g(r), X(θ) do satisfy the field equations. The Kretschmann scalar discussion is standard but correct, and the photon sphere/shadow section is a reasonable extension of known methods. The paper is readable and the parameter dependence plots are instructive. This is genuine work, just not as new as claimed.

Soft spots: the thermodynamics are compromised. Eq. (33) is supposed to follow from g(r_+)=0, but direct algebra from Eq. (17) gives a different expression: the charge term in Eq. (33) carries an extra factor [1 - A^2(1+f_R0)r_+^2]/(1 - A^2 r_+^2), which is not 1 generically. So the mass m0 used in Eq. (34) does not put r_+ on the horizon; the Hawking temperature is evaluated off-horizon, and the heat capacity built on it inherits the error. On top of that, the horizon-area formula Eq. (39) does not match the actual area integral of the C-metric (8): for this metric, sqrt(gθθ gφφ) = r^2 sinθ/K^3, whose integral gives 4π r^2/(1 - A^2 r^2)^2, not the expression in Eq. (39). Entropy and heat capacity therefore need a full re-derivation. The constant-curvature reduction R=R0 with R0=2f(R0)/(f_R0-1) is legitimate but narrow; no concrete f(R) model is given, so the 'F(R) gravity' claim is only demonstrated on a slice.

Verdict: the algebraic solution may well be correct and the shadow part is probably salvageable, but the thermodynamic results should not be used. This is a conditional situation: fix Eq. (33), recompute temperature/entropy/heat capacity, and reframe the electric-sector solution as a specialization of Ref. [136]. It deserves a serious referee because the solution derivation is checkable and there is enough correct work to warrant revision rather than rejection.

Referee Report

2 major / 5 minor

Summary. The paper constructs an exact accelerating black hole solution in F(R) gravity coupled to ModMax nonlinear electrodynamics, using the C-metric ansatz (Eq. (8)). The metric functions are given in Eqs. (17)-(18). The authors then study the Kretschmann scalar, horizon structure, Hawking temperature, entropy, heat capacity (local stability), and shadow observables. The central claim is that this is the first exact accelerating ModMax black hole solution in F(R) gravity, valid for constant Ricci scalar R0 with f_R0 satisfying Eq. (10). For the pure electric sector, ModMax reduces to Maxwell with a rescaled charge, a point acknowledged in the paper.

Significance. If correct, the exact solution (17)-(18) is a useful extension of known accelerating black holes, and the paper has the virtue of being deductive rather than phenomenological: the metric functions can be checked by direct substitution into the field equations. The paper also correctly identifies the limitation that pure-electric ModMax is equivalent to Maxwell with an e^{-γ} rescaling, which tempers the claimed novelty. However, the thermodynamic analysis is undermined by a serious algebraic error in the horizon condition, and the Kretschmann scalar expression appears to omit the conformal factor of the C-metric. The shadow section is more phenomenogical but contains internal inconsistencies in the text.

major comments (2)

[IV, Eq. (33)] The mass parameter m0 is obtained by solving g(r_+)=0 with g(r) from Eq. (17). Direct algebra gives m0 = r_+ [1 - (A^2 + R0/12) r_+^2] / [2(1 - A^2 r_+^2)] + q^2 e^{-γ} / [2(1+f_R0) r_+]. Equation (33), however, contains the second (charge) term multiplied by [1 - A^2(1+f_R0) r_+^2] / (1 - A^2 r_+^2), which is not equal to 1/(1+f_R0) r_+ unless f_R0=0 or A=0. Consequently Eq. (33) does not place r_+ on the horizon. For example, with A=0.5, R0=-1, f_R0=0.1, q=1, γ=0, r_+=1, Eq. (33) gives m0≈0.995 but g(r_+)≈0.023, while the correct horizon mass is m0≈1.010. The Hawking temperature (34), the entropy (39)-(40), and the heat capacity (44) are all derived from this incorrect m0 and therefore do not refer to a horizon quantity. The entire thermodynamic and stability analysis in Sections IV and V must be recomputed with the correct horizon condition.
[III, Eqs. (24)-(25)] Equation (24) is presented as the Kretschmann scalar of the C-metric (8), but it contains no dependence on the conformal factor K(r,θ)=1+Ar cosθ. A conformal rescaling of the form used in (8) contributes K^{-6}-type terms and derivatives of K to the curvature invariants. In the limit A→0, Eq. (24) reduces to the spherically symmetric expression, which is consistent, but for A≠0 the displayed formula cannot be the Kretschmann scalar of (8). The explicit result (25) and the singularity analysis based on it should therefore be rederived or supported by a reference. This is load-bearing for the paper's singularity claims, since the behavior near r=0 and at the conformal boundary K=0 depends on the omitted K-dependent terms.

minor comments (5)

[III, Eq. (10)] The condition for the constant-curvature root also requires f_R0≠1; the paper only notes f_R0≠-1. Please state this, and ideally provide a concrete F(R) model for which a real root R0 of Eq. (10) exists, since the title claims F(R) gravity but no explicit f(R) is given.
[IV, Eq. (35)] The notation in Eq. (35) is ambiguous: the limit should be r_+→0, not r→0, since T is a function of r_+. Please correct.
[VI, Eqs. (75)-(80)] As written, Eqs. (77)-(78) give single numerical values for αph and βph (via the roots of the cubics (61) and (64)), but the shadow plots in Figs. 9 and 10 show circular boundaries. Please clarify what quantity is varied along the boundary of the shadow; if the shadow is circular, the azimuthal coordinate β should be a free parameter, not a single value fixed by Eq. (76).
[VI, Conclusions] There is a contradiction between the statement that the shadow is circular and independent of observer inclination (a 'limitation of C-metrics') and the later statement that the acceleration parameter induces 'a pronounced inward deformation, breaking circular symmetry.' Please reconcile these statements.
[Throughout] The paper contains many typographical errors and inconsistencies: 'accelearting', 'garvity', 'ModMad', 'simoulteneously', 'without the loss of generosity', incomplete references (e.g. [4] and [15]), and inconsistent use of R0 values in figure captions (e.g. R0=0.5 vs -0.5). A careful proofreading pass is needed.

Circularity Check

1 steps flagged · score 6.0 of 10

Electric-sector ModMax is Maxwell with a rescaled charge, so the claimed new F(R)-ModMax solution is a re-labeling of the known F(R)-Maxwell accelerating solution.

renaming known result [Section VII, Conclusions; see also Eq. (2), Eq. (7), Eq. (17), and Eq. (21)]
"A notable finding of this work is that the obtained solutions in the electric sector exhibit a formal equivalence to the f (R)−Maxwell framework through the identification of an effective charge Q = qe−γ/2. Rather than rendering the results trivial, this equivalence elucidates the specific contribution of the ModMax parameter to the spacetime structure."

With P = 0 in Eq. (2), the ModMax Lagrangian reduces to e^γ times the Maxwell Lagrangian for the electric sector, and Eq. (7) becomes the standard Maxwell equation with a rescaled coupling. Consequently, the metric function in Eq. (17), which contains q^2 e^{-γ}/(1+f_R0), is exactly Eq. (21) of the known F(R)-Maxwell accelerating black hole [136] under the charge reparametrization q → q e^{-γ/2}. The paper's own conclusion states this equivalence. Thus the claimed first F(R)-ModMax solution is not a structurally new solution; the new parameter γ is absorbed into the charge, so the output of the derivation is already contained in the input Lagrangian and in the prior Maxwell solution.

full rationale

The derivation is deductive: it solves the F(R)-ModMax field equations with the C-metric ansatz and a radial electric gauge field, with no parameter fitting to target observables and no imported uniqueness theorem. The constant-curvature restriction R0 = 2f(R0)/(f_R0 - 1) is a stated assumption, not a hidden use of the answer. However, the central claimed object is not independent of prior results: because the paper restricts to P = 0, ModMax is Maxwell up to an e^γ rescaling, and the solution (17)-(18) is the known F(R)-Maxwell accelerating solution [136] with the charge rescaled by e^{-γ/2}. This is a disclosed re-labeling of a known result, which I count as partial constructional circularity. Separately, Eq. (33) does not actually follow from g(r_+) = 0 unless f_R0 = 0 or A = 0, so the Hawking temperature and heat capacity are computed with an off-horizon mass; that is an internal consistency error, not a circularity, and I do not include it in the circularity score.

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

The central solution rests on the constant-curvature reduction of F(R) gravity, the pure-electric reduction of ModMax to Maxwell with e^{-γ} scaling, and the C-metric ansatz. The parameters R0 and f_R0 are introduced ad hoc (no concrete F(R) model is given), while γ, A, m0, and q are model/integration constants varied by hand. No new entities are postulated.

free parameters (6)

R0 (constant Ricci scalar) = none (varied: ±1, -0.4, -0.5)
R=R0 is assumed constant in Eqs. (9)-(10); R0 is a free dial that must satisfy the trace condition R0 = 2 f(R0)/(f_R0 - 1). No concrete f(R) model is specified.
f_R0 (derivative of f at R0) = none (varied: 0-0.8)
f_R0 = f'(R0) is treated as an independent control; the solution requires f_R0 ≠ -1. All thermodynamic and optical results are functions of this hand-chosen constant.
γ (ModMax parameter) = none (varied: 0-10)
Theory parameter of ModMax; enters the solution only through the effective charge q e^{-γ/2}; varied by hand in every scan.
A (acceleration parameter) = none (varied: 0-0.9)
C-metric acceleration/conical-deficit parameter; varied by hand; controls horizon and entropy divergences.
m0 (geometric mass) = 1 (plots)
Integration constant in (17); not fitted, set to 1 for numerical scans.
q (electric charge) = 0.3 (plots)
Integration constant in (14)/(17); varied near 0.3 in tables and figures.

assumptions (5)

domain assumption Constant Ricci scalar R = R0 with trace condition R0 = 2 f(R0)/(f_R0 - 1)
Needed to reduce the F(R) field equations to the Einstein-like form (11); restricts the result to a special class of F(R) models.
domain assumption Pure-electric sector P=0: ModMax Lagrangian reduces to a Maxwell-like Lagrangian with e^{-γ} scaling
Used implicitly in Eq. (7) and the field equations (15)-(16); this is the property that makes the solution a charge rescaling of the Maxwell case. With magnetic components (P≠0) the reduction fails.
domain assumption Tracelessness of the ModMax energy-momentum tensor (5) in the electric sector
Assumed before Eq. (9) so that the trace of Eq. (3) yields the constant-curvature condition; standard for conformally invariant NLED but a model restriction.
domain assumption C-metric ansatz (8) with X(θ) of the form (18) solves the field equations (11)
The paper states the solution after 'careful consideration' without displaying the intermediate steps; correctness relies on this ansatz satisfying (15)-(16).
domain assumption f_R0 ≠ -1
Imposed after Eq. (18) to avoid a sign flip of the effective Newton coupling; also |f_R0| < 1 is used in the entropy discussion.

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Pith. "Pith review of Accelerating electrically charged ModMax black hole solutions in $F(R)$ gravity." pith.science (2026). https://pith.science/paper/ZET7K6R5

@misc{pith2026260715914,
  author       = {Pith},
  title        = {Pith review of: Accelerating electrically charged ModMax black hole solutions in $F(R)$ gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZET7K6R5}},
  note         = {Machine review of arXiv:2607.15914}
}

abstract

Using the $C-$metric in the context of $F(R)$ gravity coupled with the ModMax nonlinear electromagnetic field (the $F(R)-$ModMax theory), we derive an exact black hole solution in a four-dimensional spacetime. We then examine how various parameters influence the behavior of accelerating ModMax black holes. Treating this black hole as a thermodynamic system, we calculate the Hawking temperature and entropy for the accelerating ModMax black holes within the framework of $F(R)$ gravity. Subsequently, we explore the impact of the parameters in $F(R)-$ModMax theory on the Hawking temperature and entropy. We also assess local stability by analyzing the heat capacity. Additionally, we investigate both the angular shadow and the shadow radius of an accelerating black hole in the context of $F(R)-$ModMax gravity.

Figures

Figures reproduced from arXiv: 2607.15914 by the authors.

**Figure 2.** FIG. 2: The function [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FIG. 3: The Hawking temperature [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4: Entropy [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: FIG. 5: Entropy [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: FIG. 6: Entropy [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: FIG. 7: Entropy [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

**Figure 8.** Figure 8: FIG. 8: Entropy [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

**Figure 9.** Figure 9: FIG. 9: Angular shadow radius [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]

**Figure 10.** Figure 10: FIG. 10: Dependence of the shadow radius on the model parameters. The shadow radius is calculated with fixed [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

159 extracted references · 5 linked inside Pith

[136]

Wang, and J

Y. Wang, and J. Ren, Phys. Rev. D 106, 104046 (2022)

2022
[1]

This imposes that R0 must be negative

The large accelerating black holes in F (R)−ModMax theory include the positive entropy when A < q −R0 12 . This imposes that R0 must be negative
[2]

To have the positive entropy (or avoid of the divergence point in the entropy) of the accelerating ModMax black holes with arbitrary radius, we must consider the small value for A. 11 FIG. 4: Entropy S (Eq. ( 40)) versus r+ is plotted for various parameter values. V. HEA T CAP ACITY The heat capacity plays a significant role in determining thermal stabili...
[3]

These black holes in this area are non-physical objects

The first region devotes to the smallest accelearting ModMax black holes, i.e, r+ < r T1=0. These black holes in this area are non-physical objects
[4]

The second region is located in rT1=0 < r + < r C1=∞. Indeed, the accelearting ModMax black holes are physical objects and satisfy the local condition (because the temperature, entropy and the heat capacity are positive, simoultaneously) when their radius are in this area
[5]

These black holes are unstable because the heat capacity is negative, however their entropy and temperature ate positive

The third region belongs to rC1=∞ < r + < r C2=∞. These black holes are unstable because the heat capacity is negative, however their entropy and temperature ate positive
[6]

The fourth region is located in rC2=∞ < r + < r C2=0, and the black holes are physical objects and satisfy the local stablity condition because T , S, and C are positive, simoulteneously
[7]

In this area, the black holes are physical object but unstable

The fifth region is in the range rC2=0 < r + < r T =∞. In this area, the black holes are physical object but unstable

Show all 159 references

[8]

The temperature of these black holes is negative and so we encounter with non-physical objects

The sixth region is determined between rT =∞ and the forth root of the heat capacity (i.e., rT =∞ < r + < r C4=0). The temperature of these black holes is negative and so we encounter with non-physical objects
[9]

The black holes are physical objects but cannot satisfy the local stability condition

The seventh region is r+ > r C4=0. The black holes are physical objects but cannot satisfy the local stability condition. So, the accelerating ModMax black holes in F (R) garvity can satisfy the physical and local stability conditions, simoultenously, when their radius are in ...
[10]

A. G. Riess et al. [Supernova Search Team], Astron. J. 116, 1009 (1998)

1998
[11]

D. N. Spergel et al. [WMAP], Astrophys. J. Suppl. 148, 175 (2003)

2003
[12]

A. G. Riess et al. [Supernova Search Team], Astrophys. J. 607, 665 (2004)

2004
[13]

D. J. Eisenstein et al. [SDSS], Astrophys. J. 633, 560 (2005). 82, 451 (2010)

2005
[14]

De Felice, and S

A. De Felice, and S. Tsujikawa, Living Rev. Rel. 13, 3 (2010)

2010
[15]

Fujii, and K

Y. Fujii, and K. Maeda, Cambridge University Press, 2007, doi:10.1017/CBO9780511535093

2007 doi
[16]

Ashtekar, and E

A. Ashtekar, and E. Bianchi, Rept. Prog. Phys. 84, 042001 (2021)

2021
[17]

Rovelli, Living Rev

C. Rovelli, Living Rev. Rel. 1, 1 (1998)

1998
[18]

Krssak, etb al., Class

M. Krssak, etb al., Class. Quant. Grav. 36, 183001 (2019)

2019
[19]

Wang, Int

A. Wang, Int. J. Mod. Phys. D 26, 1730014 (2017)

2017
[20]

Bonanno, and M

A. Bonanno, and M. Reuter, Phys. Rev. D 62, 043008 (2000)

2000
[21]

B. Koch, I. A. Reyes, and Á. Rincón, Class. Quant. Grav. 33, 225010 (2016)

2016
[22]

Rincón, and G

Á. Rincón, and G. Panotopoulos, Phys. Rev. D 97, 024027 (2018)

2018
[23]

Eichhorn, [arXiv:2003.00044]

A. Eichhorn, [arXiv:2003.00044]

2003 arXiv
[24]

T. P. Sotiriou, and V. Faraoni, Rev. Mod. Phys
[25]

Nojiri, and S

S. Nojiri, and S. D. Odintsov, Phys. Rept. 505, 59 (2011)

2011
[26]

Capozziello, and M

S. Capozziello, and M. De Laurentis, Phys. Rept. 509, 167 (2011)

2011
[27]

Faraoni, and S

V. Faraoni, and S. Capozziello, ”Beyond Einstein Gravity: A Survey of Gravitational Theories for Cosmology and Astrophysics”, Springer, (2001), ISBN 978-94-007-0164-9

2001
[28]

Nojiri, S

S. Nojiri, S. D. Odintsov, and V. K. Oikonomou, Phys. Rept. 692, 1 (2017)

2017
[29]

M. B. Varela, and O. Bertolami, Phys. Rev. D 106, 124059 (2022)

2022
[30]

Starobinsky, phys

A. Starobinsky, phys. Lett. B 91, 99 (1980). 23

1980
[31]

De Felice, and S

A. De Felice, and S. Tsujikawa, Living. Rev. Rel. 13, 3 (2010)

2010
[32]

Capozziello, Int

S. Capozziello, Int. J. Mod. Phys. D 1, 4831 (2002)

2002
[33]

Nojiri, and S

S. Nojiri, and S. D. Odintsov, Phys. Rev. D 68, 123512 (2003)

2003
[34]

Capozziello, V

S. Capozziello, V. F. Cardone, S. Carloni, and A. Troisi, Int. J. Mod. Phys. D 12, 1969 (2003)

1969
[35]

S. M. Carroll, V. Duvvuri, M. Trodden, and M. S. Turner, Phys. Rev. D 70, 043528 (2004)

2004
[36]

Hu, and I

W. Hu, and I. Sawicki, Phys. Rev. D 76, 064004 (2007)

2007
[37]

E. J. Copeland, M. Sami, and S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006)

2006
[38]

Nojiri, and S

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

2007
[39]

Clifton, P

T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, Phys. Rept. 513, 1 (2012)

2012
[40]

Katsuragawa, and S

T. Katsuragawa, and S. Matsuzaki, Phys. Rev. D 97, 064037 (2018)

2018
[41]

Parbin, and U

N. Parbin, and U. D. Goswami, Mod. Phys. Lett. A 36, 2150265 (2021)

2021
[42]

Capozziello, and A

S. Capozziello, and A. Troisi, Phys. Rev. D 72, 044022 (2005)

2005
[43]

Capozziello, A

S. Capozziello, A. Stabile, and A. Troisi, Phys. Rev. D 76, 104019 (2007)

2007
[44]

Cooney, S

A. Cooney, S. DeDeo, and D. Psaltis, Phys. Rev. D 82, 064033 (2010)

2010
[45]

M. K. Cheoun, C. Deliduman, C. Gungor, V. Keles, C. Y. Ryu, T. Kajino, and G. J. Mathews, JCAP 10, 021 (2013)

2013
[46]

A. V. Astashenok, S. Capozziello, and S. D. Odintsov, JCAP 12, 040 (2013)

2013
[47]

Feola, X

P. Feola, X. J. Forteza, S. Capozziello, R. Cianci, and S. Vignolo, Phys. Rev. D 101, 044037 (2020)

2020
[48]

A. V. Astashenok, and S. D. Odintsov, MNRAS 493, 78 (2020)

2020
[49]

A. V. Astashenok, S. Capozziello, S. D. Odintsov, and V. K. Oikonomou, Phys. Lett. B 816, 136222 (2021)

2021
[50]

Sarmah, S

L. Sarmah, S. Kalita, and A. Wojnar, Phys. Rev. D 105, 024028 (2022)

2022
[51]

G. J. Olmo, and D. Rubiera-Garcia, Class. Quantum Gravit. 37, 215002 (2020)

2020
[52]

Kalita, and B

S. Kalita, and B. Mukhopadhyay, Astrophys. J. 909, 65 (2021)

2021
[53]

Chiba, T

T. Chiba, T. L. Smith, and A. L. Erickcek, Phys. Rev. D 75, 124014 (2007)

2007
[54]

S. A. Appleby, and R. A. Battye, Phys. Lett. B 654, 7 (2007)

2007
[55]

Tsujikawa, Phys

S. Tsujikawa, Phys. Rev. D 77, 023507 (2008)

2008
[56]

Negrelli, et al., Phys

C. Negrelli, et al., Phys. Rev. D 101, 064005 (2020)

2020
[57]

LIGO Scientific and Virgo collaborations, Phys. Rev. Lett. 116, 061102 (2016)

2016
[58]

Event Horizon Telescope collaboration, Astrophys. J. Lett. 875, L1 (2019)

2019
[59]

Born, and L

M. Born, and L. Infeld, Proc. R. Soc. Lond. 144, 425 (1934)

1934
[60]

E. S. Fradkin, and A. A. Tseylin, Phys. Lett. B 163, 123 (1985)

1985
[61]

G. W. Gibbons, and C. A. R. Herdeiro, Class. Quantum Grav. 18, 1677 (2001)

2001
[62]

Heisenberg, and H

W. Heisenberg, and H. Euler, Z. Phys. 98, 714 (1936)

1936
[63]

Schwinger, phys

J. Schwinger, phys. Rev. 82, 664 (1951)

1951
[64]

Ayon-Beato, and A

E. Ayon-Beato, and A. Garcia, Gen. Relativ. Gravit. 31, 629 (1999)

1999
[65]

Yajima, and T

H. Yajima, and T. Tamaki, Phys. Rev. D 63, 064007 (2001)

2001
[66]

V. A. De Lorenci, et al., Phys. Rev. D 65, 063501 (2002)

2002
[67]

Dymnikova, Class

I. Dymnikova, Class. Quantum Grav. 21, 4417 (2004)

2004
[68]

Corda, and H

C. Corda, and H. J. Mosquera Cuesta, Mod. Phys. Lett. A 25, 2423 (2010)

2010
[69]

J. G. Russo, and P. K. Townsend, JHEP 06, 191 (2024)

2024
[70]

J. G. Russo, and P. K. Townsend,” Black holes and causal nonlinear electrodynamics ” . [arXiv:2601.07789]

arXiv
[71]

Ibrahim, et al., Astrophys

A. Ibrahim, et al., Astrophys. J. Lett., 574, L51 (2002)

2002
[72]

H. J. Mosquera Cuesta, and J. M. Salim, Mon. Not. R. Acad. Sci., 354, L55 (2004)

2004
[73]

G. J. Olmo, and D. Rubiera-Garcia, Phys. Rev. D 84, 124059 (2011)

2011
[74]

S. H. Mazharimousavi, and M. Halilsoy, Phys. Rev. D 84, 064032 (2011)

2011
[75]

S. H. Hendi, B. Eslam Panah, and R. Saffari, Int. J. Mod. Phys. D 23, 1450088 (2014)

2014
[76]

M. E. Rodrigues, et al., Phys. Rev. D 94, 024062 (2016)

2016
[77]

G. G. L. Nashed, and E. N. Saridakis, Phys. Rev. D 102, 124072 (2020)

2020
[78]

E. F. Eiroa, and G. Figueroa-Aguirre, Eur. Phys. J. Plus. 137, 478 (2022)

2022
[79]

Sekhmani, S

Y. Sekhmani, S. K. Maurya, M. K. Jasim, A. Al-Badawi, and J. Rayimbaev, Phys. Dark Univ. 46, 101701 (2024)

2024
[80]

Y. B. Shi, and R. J. Yang, Eur. Phys. J. Plus 140, 438 (2025)

2025
[81]

G. W. Gibbons, and D. A. Rasheed, Nucl. Phys. B 454, 185 (1995)

1995
[82]

Bandos, K

I. Bandos, K. Lechner, D. Sorokin, and P. K. Townsend, Phys. Rev. D 102, 121703 (2020)

2020
[83]

B. P. Kosyakov, Phys. Lett. B 810, 135840 (2020)

2020
[85]

Guzman-Herrera, and N

E. Guzman-Herrera, and N. Breton, JCAP 01, 041 (2024)

2024
[86]

Eslam Panah, A

B. Eslam Panah, A. Rincon, and N. Heidari, [arXiv:2411.02907]

arXiv
[87]

Jafarzade, Z

K. Jafarzade, Z. Bazyar, and M. Jamil, Phys. Lett. B 864, 139390 (2025)

2025
[88]

Eslam Panah, B

B. Eslam Panah, B. Hazarika, and P. Phukon, Prog. Theor. Exp. Phys. 2024, 083E02 (2024)

2024
[89]

S. I. Kruglov, Int. J. Mod. Phys. 31, 2250025 (2022)

2022
[90]

Barrientos, A

J. Barrientos, A. Cisterna, M. Hassaine, and K. Pallikaris, Phys. Lett. B 860, 139214 (2025)

2025
[91]

Flores-Alfonso, B

D. Flores-Alfonso, B. A. Gonzalez-Morales, R. Linares, and M. Maceda, Phys. Lett. B 812, 136011 (2021)

2021
[92]

Bandos, K

I. Bandos, K. Lechner, D. Sorokin, and P. K. Townsend, JHEP 03, 022 (2021)

2021
[93]

S. I. Kruglov, Phys. Lett. B 822, 136633 (2021)

2021
[94]

S. M. Kuzenko, and E. S. N. Raptakis, Phys. Rev. D 104, 125003 (2021). 24

2021
[95]

A vetisyan, O

Z. A vetisyan, O. Evnin, and K. Mkrtchyan, Phys. Rev. Lett. 127, 271601 (2021)

2021
[96]

P. A. Cano, and A. Murcia, JHEP 08, 042 (2021)

2021
[97]

Ballon Bordo, D

A. Ballon Bordo, D. Kubiznak, and T. Perche, Phys. Lett. B 817, 136312 (2021)

2021
[98]

Bokulic, I

A. Bokulic, I. Smolic, and T. Juric, Phys. Rev. D 103, 124059 (2021)

2021
[99]

Babaei-Aghbolagh, et al., Phys

H. Babaei-Aghbolagh, et al., Phys. Lett. B 829, 137079 (2022)

2022
[100]

C. A. Escobar, and R. Linares, Phys. Rev. D 106, 036027 (2022)

2022
[102]

R. C. Pantig, L. Mastrototaro, G. Lambiase, and A. Ovgun, Eur. Phys. J. C 82, 1155 (2022)

2022
[103]

Bokulic, T

A. Bokulic, T. Juric, and I. Smolic, Phys. Rev. D 106, 064020 (2022)

2022
[104]

Lechner, P

K. Lechner, P. Marchetti, A. Sainaghi, and D. P. Sorokin, Phys. Rev. D 106, 016009 (2022)

2022
[105]

Ortaggio, Eur

M. Ortaggio, Eur. Phys. J. C 82, 1056 (2022)

2022
[106]

Nastase, Phys

H. Nastase, Phys. Rev. D 105, 105024 (2022)

2022
[107]

Ali, and K

A. Ali, and K. Saifullah, Annals Phys. 437, 168726 (2022)

2022
[108]

J. B. Jimenez, D. Bettoni, and P. Brax, JHEP 02, 009 (2023)

2023
[109]

Ferko, and A

C. Ferko, and A. Gupta, Phys. Rev. D 108, 046013 (2023)

2023
[110]

S. M. Kuzenko, and I. N. McArthur, JHEP 05, 127 (2023)

2023
[111]

Rathi, and D

H. Rathi, and D. Roychowdhury, JHEP 07, 026 (2023)

2023
[112]

H. M. Siahaan, Int. J. Mod. Phys. D 32, 2350099 (2023)

2023
[113]

H. M. Siahaan, Commun. Theor. Phys. 76, 065402 (2024)

2024
[114]

Barrientos, A

J. Barrientos, A. Cisterna, D. Kubiznak, and J. Oliva, Phys. Lett. B 834, 137447 (2022)

2022
[116]

T. Hale, D. Kubiznak, J. Mensikova, R. B. Mann, and J. Yang, Phys. Rev. D 111, 104004 (2025)

2025
[117]

Eslam Panah, Contrib

B. Eslam Panah, Contrib. Sci. Tech Eng. 1(3), 25 (2024)

2024
[118]

Kinnersley, and M

W. Kinnersley, and M. Walker, Phys. Rev. D 2, 1359 (1970)

1970
[119]

J. F. Plebanski, and M. Demianski, Annals Phys. 98, 98 (1976)

1976
[120]

O. J. C. Dias, and J. P. S. Lemos, Phys. Rev. D 67, 064001 (2003)

2003
[122]

Emparan, G

R. Emparan, G. T. Horowitz, and R. C. Myers, JHEP 01, 007 (2000)

2000
[123]

Emparan, and G

R. Emparan, and G. Milanesi, JHEP 08, 012 (2009)

2009
[124]

V. E. Hubeny, D. Marolf, and M. Rangamani, Class. Quantum Gravit. 27, 025001 (2010)

2010
[125]

Ferrero, J

P. Ferrero, J. P. Gauntlett, J. M. P. Ipina, D. Martelli, and J. Sparks, Phys. Rev. D 104, 046007 (2021)

2021
[126]

P. S. Letelier, and S. R. Oliveira, Phys. Rev. D 64, 064005 (2001)

2001
[129]

Anabalon, F

A. Anabalon, F. Gray, R. Gregory, D. Kubiznak, and R. B. Mann, JHEP 04, 096 (2019)

2019
[130]

Eslam Panah, and Kh

B. Eslam Panah, and Kh. Jafarzade, Gen. Relativ. Gravit. 54, 19 (2022)

2022
[131]

Gregory, and A

R. Gregory, and A. Scoins, Phys. Lett. B 796, 191 (2019)

2019
[132]

Astorino, Phys

M. Astorino, Phys. Rev. D 95, 064007 (2017)

2017
[133]

Appels, R

M. Appels, R. Gregory, and D. Kubiznak, JHEP 05, 116 (2017)

2017
[134]

Jafarzade, J

Kh. Jafarzade, J. Sadeghi, B. Eslam Panah, and S. H. Hendi, Ann. Phys. 432, 168577 (2021)

2021
[135]

Gregory, Z

R. Gregory, Z. L. Lim, and A. Scoins, Front. Phys. 9, 666041 (2021)

2021
[137]

Astorino, Phys

M. Astorino, Phys. Rev. D 88, 104027 (2013)

2013
[138]

Astorino, Phys

M. Astorino, Phys. Lett. B 760, 393 (2016)

2016
[139]

F. L. Carneiro, S. C. Ulhoa, and J. W. Maluf, Gravit. Cosmol. 28, 352 (2022)

2022
[140]

Eslam Panah, Phys

B. Eslam Panah, Phys. Lett. B 868, 139711 (2025)

2025
[141]

Sekhmani, et al., [arXiv:2507.19088]

Y. Sekhmani, et al., [arXiv:2507.19088]

arXiv
[142]

Ahmed, A

F. Ahmed, A. Al-Badawi, and I. Sakalli, [arXiv:2508.03226]

arXiv
[143]

Sekhmani, et al., [arXiv:2509.16782]

Y. Sekhmani, et al., [arXiv:2509.16782]
[144]

Eslam Panah, Prog

B. Eslam Panah, Prog. Theor. Exp. Phys. 2024, 023E01 (2024)

2024
[145]

Zhang, and R

M. Zhang, and R. B. Mann, Phys. Rev. D 100, 084061 (2019)

2019
[146]

Appels, R

M. Appels, R. Gregory, and D. Kubiznak, Phys. Rev. Lett. 117, 131303 (2016)

2016
[147]

Anabalon, M

A. Anabalon, M. Appels, R. Gregory, D. Kubiznak, R. B. Mann, and A. Ovgun, Phys. Rev. D 98, 104038 (2018)

2018
[148]

de la Cruz-Dombriz, A

A. de la Cruz-Dombriz, A. Dobado, and A. L. Maroto, Phys. Rev. D 80, 124011 (2009)

2009
[149]

T. Moon, Y. S. Myung, and E. J. Son, Gen. Relativ. Gravit. 43, 3079 (2011)

2011
[150]

Eslam Panah, Phys

B. Eslam Panah, Phys. Lett. B 787, 45 (2018)

2018
[151]

Cognola, E

G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov, and S. Zerbini, JCAP 02, 010 (2005)

2005
[152]

Grenzebach, V

A. Grenzebach, V. Perlick, and C. Lammerzahl, Int. J. Mod. Phys. D 24, 1542024 (2015)

2015
[153]

Zhang, and J

M. Zhang, and J. Jiang, Phys. Rev. D 103, 025005 (2021)

2021
[154]

Karshiboev, et al., Commun

K. Karshiboev, et al., Commun. Theor. Phys. 76, 025401 (2024)

2024
[155]

T. C. Frost, and V. Perlick, Class. Quant. Grav. 38, 085016 (2021)

2021
[156]

T. C. Frost, Phys. Rev. D 105, 064064 (2022)

2022
[157]

Batic, H

D. Batic, H. Kittaneh, and M. Nowakowski, Phys. Rev. D 104, 124029 (2021)

2021
[158]

Carter, Phys

B. Carter, Phys. Rev. 174, 1559 (1968). 25

1968
[159]

Y. K. Lim, Phys. Rev. D 103, 024007 (2021)

2021
[160]

Heidari, and B

N. Heidari, and B. Eslam Panah, Phys. Lett. B 866, 139530 (2025)

2025
[161]

Grenzebach, V

A. Grenzebach, V. Perlick, and C. Lammerzahl, Phys. Rev. D 89, 124004 (2014)

2014
[162]

P. C. Li, M. Guo, and B. Chen, Phys. Rev. D 101, 084041 (2020)

2020
[163]

T. T. Sui, Q. M. Fu, and W. D. Guo, Phys. Lett. B 845, 138135 (2023)

2023
[164]

J. B. Griﬀiths, and J. Podolsky, Int. J. Mod. Phys. D 15, 335 (2006)

2006
[165]

Perlick, and O

V. Perlick, and O. Y. Tsupko, Phys. Rep. 947, 1 (2022)

2022

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [136]

Wang, and J

Y. Wang, and J. Ren, Phys. Rev. D 106, 104046 (2022)

2022

[2] [1]

This imposes that R0 must be negative

The large accelerating black holes in F (R)−ModMax theory include the positive entropy when A < q −R0 12 . This imposes that R0 must be negative

[3] [2]

To have the positive entropy (or avoid of the divergence point in the entropy) of the accelerating ModMax black holes with arbitrary radius, we must consider the small value for A. 11 FIG. 4: Entropy S (Eq. ( 40)) versus r+ is plotted for various parameter values. V. HEA T CAP ACITY The heat capacity plays a significant role in determining thermal stabili...

[4] [3]

These black holes in this area are non-physical objects

The first region devotes to the smallest accelearting ModMax black holes, i.e, r+ < r T1=0. These black holes in this area are non-physical objects

[5] [4]

The second region is located in rT1=0 < r + < r C1=∞. Indeed, the accelearting ModMax black holes are physical objects and satisfy the local condition (because the temperature, entropy and the heat capacity are positive, simoultaneously) when their radius are in this area

[6] [5]

These black holes are unstable because the heat capacity is negative, however their entropy and temperature ate positive

The third region belongs to rC1=∞ < r + < r C2=∞. These black holes are unstable because the heat capacity is negative, however their entropy and temperature ate positive

[7] [6]

The fourth region is located in rC2=∞ < r + < r C2=0, and the black holes are physical objects and satisfy the local stablity condition because T , S, and C are positive, simoulteneously

[8] [7]

In this area, the black holes are physical object but unstable

The fifth region is in the range rC2=0 < r + < r T =∞. In this area, the black holes are physical object but unstable