Black hole solutions in theory of ModMax-dRGT-like massive gravity

B. Eslam Panah

REVIEW 3 major objections 4 minor 1 cited by

Black hole solutions in theory of ModMax-dRGT-like massive gravity

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper derives exact charged black hole solutions in dRGT-like massive gravity with ModMax electrodynamics and confirms that their thermodynamics satisfy the first law and the Smarr relation.

desk verdict P=0 turns ModMax into Maxwell with a rescaled charge, so the paper's claimed new black hole family is the known dRGT-like massive gravity solution in disguise. read the letter →

arxiv 2507.05864 v1 pith:OQR5Y6NK submitted 2025-07-08 gr-qc hep-th

B. Eslam Panah This is my paper

classification gr-qchep-th MSC 83C5783D0583C22 PACS 04.70.-s04.50.Kd

keywords blackholesmassivegravityModMaxelectrodynamicsnonlinearholethermodynamicsSmarrrelationheatcapacityisoperimetricratio

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 show that exact, spherically symmetric charged black hole solutions exist when dRGT-like massive gravity is coupled to ModMax nonlinear electrodynamics, and that these black holes have a consistent thermodynamics. The central result is the metric function $\psi(r)=1-\frac{m_0}{r}-\frac{\Lambda r^2}{3}+\frac{q^2 e^{-\gamma}}{r^2}+m_g^2 C\left(\frac{c_1 r}{2}+c_2 C\right)$, which combines mass, cosmological constant, charge, and massive-gravity terms. The paper verifies the first law of thermodynamics in the non-extended phase space and, after identifying $P=-\Lambda/(8\pi)$, also the first law and Smarr relation in the extended phase space. It then uses the heat capacity to map regions of local stability and computes the isoperimetric ratio $R=1$, which obeys the reverse isoperimetric inequality. A reader would care because the ModMax parameter $\gamma$ shifts the horizon structure, temperature, and stability of the solutions.

What carries the argument

The load-bearing object is the combination of the dRGT-like massive tensor $\chi_{\mu\nu}$ and the ModMax Lagrangian $L=S\cosh\gamma-\sqrt{S^2+P^2}\sinh\gamma$, evaluated with the singular reference metric and with $P=0$. Under $P=0$, the Lagrangian reduces to $S e^{-\gamma}$ up to a sign, so the electrodynamics sector is Maxwell with charge rescaled by $e^{-\gamma/2}$; the massive sector contributes through the polynomials $u_1=2C/r$ and $u_2=2C^2/r^2$, producing the linear and constant terms in Eq. (21). This machinery turns the field equations into a single ordinary differential equation for $\psi(r)$.

What would settle it

Compare Eq. (21) with the known dRGT-like massive gravity black hole of [33] after the replacement $q\to q e^{-\gamma/2}$; if the two metrics are term-by-term identical, the electric-sector result contains no new physics beyond charge rescaling. A dyonic calculation with $P\neq0$ would be the decisive test of whether $\gamma$ changes the geometry in a way Maxwell charge cannot mimic.

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Extended reading notes

Core claim

The paper's central claim is that the exact electrically charged black hole solution of the action (1) is given by Eq. (21), $\psi(r) = 1 - \frac{m_0}{r} - \frac{\Lambda r^2}{3} + \frac{q^2 e^{-\gamma}}{r^2} + m_g^2 C\left(\frac{c_1 r}{2}+c_2 C\right)$, with the massive tensor built from the reference metric $f_{\mu\nu}=\mathrm{diag}(0,0,C^2,C^2\sin^2\theta)$. The authors compute the Hawking temperature, electric potential, entropy, and total mass, and show that these quantities satisfy $dM=T\,dS+U\,dQ$ in the non-extended phase space and $dM=T\,dS+U\,dQ+V\,dP+C_1\,dc_1+C_2\,dc_2$ plus the Smarr relation $M=2TS+UQ-2PV-c_1C_1$ in the extended phase space. They also prove the isoperimetric ratio is $R=1$. On the paper's own terms, this establishes a consistent thermodynamic description of charged black holes in ModMax-dRGT-like massive gravity, with the ModMax parameter controlling how much electric charge contributes through $Q=q e^{-\gamma}$.

Load-bearing premise

The argument rests on the decision to restrict the ModMax field to purely electric configurations ($P=0$); in that configuration the ModMax Lagrangian reduces to Maxwell's multiplied by a constant, so the parameter $\gamma$ is not physically independent.

Editorial extensions

If this is right

If Eq. (21) is correct, the theory contains exact spherically symmetric charged black holes whose horizon number ranges from zero to three depending on $\gamma$, $c_1$, $c_2$, and $C$.
The thermodynamic quantities satisfy the first law $dM=T\,dS+U\,dQ$ in the non-extended phase space, so the area-law entropy $S=\pi r_+^2$ is consistent with the first law.
In the extended phase space, with $P=-\Lambda/(8\pi)$, the first law and the Smarr relation $M=2TS+UQ-2PV-c_1C_1$ both hold.
The heat-capacity analysis implies that large black holes are locally stable in the situations examined, while medium black holes can switch between physical and non-physical regimes as $\gamma$ varies.
The isoperimetric ratio is $R=1$, so the solutions satisfy the reverse isoperimetric inequality.

Reading between the lines

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

Because the electric-only ($P=0$) sector reduces ModMax to Maxwell with the charge rescaled, the solution (21) is identical to the known dRGT-like massive gravity black hole with $q$ replaced by $q e^{-\gamma/2}$; the genuinely new regime would require magnetic or dyonic ModMax fields.
A dyonic calculation with $P\neq0$ is the natural next test: it would show whether $\gamma$ changes the geometry beyond a charge rescaling or introduces genuinely new terms.
The limit $\gamma\to\infty$ drives the total charge $Q=q e^{-\gamma}$ to zero while leaving the massive-gravity terms finite, which offers a route to model neutral remnants of charged black holes without fine-tuning.
Repeating the derivation with a different reference metric, for example one with $f_{tt}\neq0$, would change the massive polynomials and could modify the Smarr relation, delimiting how generic the extended-phase-space result is.

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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

X, you can skip this one unless you need a reminder about how P=0 kills ModMax. The paper derives charged black holes in dRGT-like massive gravity coupled to ModMax electrodynamics, then runs the standard thermodynamics: temperature, mass, heat capacity, first law, Smarr. The algebra is consistent and the identities check out. But the central claim—that γ is a new physical parameter—does not survive the P=0 restriction the paper makes just before Eq. (19). With P=0, the ModMax Lagrangian is a constant multiple of the Maxwell Lagrangian, so a field redefinition A_μ → e^{-γ/2} A_μ absorbs γ completely. The metric (21) depends on γ only through q^2 e^{-γ}, and every thermodynamic quantity inherits that combination. So the solution is the known dRGT-like massive gravity charged black hole with charge q e^{-γ/2}, and Q = q e^{-γ} is a relabeling, not a new charge. The plots showing 'γ effects' are just the Maxwell-dRGT solution at different charge values.

What is genuinely useful: the derivation is clean, the first law and Smarr checks are correctly done, and the catalog of horizon/temperature/heat-capacity behavior is complete. If the author had recognized the equivalence, this could have been a short note. As it stands, the framing overstates the novelty and misses the real opportunity: the dyonic or magnetic sector, where ModMax genuinely differs from Maxwell and where the combination with dRGT-like massive gravity would be new.

I recommend a desk reject. The paper is not incoherent and the math is fine, but the main result is a reparametrization of an existing family. A referee would spot the same thing quickly. The constructive fix is to extend the analysis to P ≠ 0 (magnetic or dyonic fields) before resubmission.

Referee Report

3 major / 4 minor

Summary. This paper considers dRGT-like massive gravity coupled to ModMax nonlinear electrodynamics. It sets P=0 in the ModMax Lagrangian to study electrically charged black holes, derives a static spherically symmetric solution, and computes curvature invariants, horizon structure, Hawking temperature, mass, charge, entropy, heat capacity, first-law checks, extended-phase thermodynamics, a Smarr relation, and the isoperimetric ratio. The paper reports that the ModMax parameter gamma and the massive-gravity parameters modify horizon structure, temperature, stability, and thermodynamic behavior, and it claims these are new black hole solutions with ModMax-driven effects.

Significance. Within the chosen P=0 sector, the algebraic derivation is straightforward and the non-extended first-law checks are internally consistent (Eqs. (33)-(36)). However, the central claim of new ModMax physics is not supported: the P=0 sector is field-redefinition equivalent to Maxwell electrodynamics, so gamma merely rescales the charge. The reported gamma-effects are therefore plots of the known dRGT-like massive gravity Maxwell solutions at different effective charges. The extended-phase section also contains an incorrect mass formula (Eq. (43)) as printed. On the positive side, the paper gives an exact solution and explicit consistency checks that are reproducible and standard, but these do not rescue the central novelty.

major comments (3)

[Sec. II, Eqs. (3), (7), (9); Sec. III, Eq. (21); Secs. IV-V] Setting P=0 in the ModMax Lagrangian (3) makes the electric sector a field redefinition of Maxwell electrodynamics. For P=0, L=S cosh(gamma) - sqrt(S^2) sinh(gamma), which is a constant multiple of S on each branch; hence the field equation (9) and stress tensor (7) are Maxwell's with F_{\mu\nu} rescaled by an exponential factor. Consequently, Eq. (21) is exactly the dRGT-like massive gravity Maxwell solution with q_eff = q e^{-gamma/2}, and the temperature (25), charge (26), mass (29), and heat capacity (31) depend on gamma only through this rescaling. All reported gamma-effects, such as horizon multiplicities, temperature roots, mass extrema, and stability regions, are therefore the known Maxwell-dRGT family evaluated at a different charge. The central claim that ModMax introduces new physics in the electric sector is not supported. Genuine ModMax nonlinearity requires P different from 0, which the paper explicitly excludes before Eq. (19).
[Sec. II, after Eq. (3); Eq. (9)] The P=0 truncation is also inconsistent with the electric branch of the stated Lagrangian. For the electric solution of Sec. III, S=F_{\mu\nu}F^{\mu\nu}/4 = -q^2/(2r^4) < 0 with the metric signature of Eq. (10), so P=0 gives L=S e^{\gamma}, not S e^{-\gamma}. The field equation (9), the stress tensor (7), and the electric field E(r)=q e^{-\gamma}/r^2 therefore follow from the opposite branch, or from a different sign convention for gamma. Rescaling gamma does not restore physical novelty, because the same reparametrization argument applies, but the derivation as written does not match the stated action for electrically charged black holes.
[Sec. V, Eq. (43)] The extended-phase mass formula (43) does not reproduce the mass (39) when S=pi r_+^2 and Q=q e^{-\gamma} are substituted, and its derivative with respect to S does not yield the temperature (45). For example, the printed first term behaves as 1/(2r_+) rather than the r_+/2 term of Eq. (39), and the c_1-dependent piece in Eq. (43) is independent of S, whereas Eq. (39) is linear in r_+^2 proportional to S. Thus Eqs. (44)-(50), including the Smarr relation, are not consequences of Eq. (43) as stated, and the extended-phase first-law check needs to be redone with the correct M(S,Q,P,c_1,c_2).

minor comments (4)

[Sec. III, Eq. (17)] The gauge potential should be written as a one-form A=h(r)dt rather than A_mu = h(r) delta_t^mu, which mixes vector and covector notation.
[Sec. IV, Eq. (26)] In Eq. (26), F is used both as the field-strength two-form and as the Maxwell invariant; the flux integral should be written explicitly with the two-form or its Hodge dual, especially because the charge normalization q e^{-\gamma} differs from the effective charge q e^{-\gamma/2} appearing in the metric.
[Sec. II and Sec. VI] There are typographical errors, for example 'Aditionally' before Eq. (7) and 'In addition. Furthermore' in Sec. VI.
[References] References [75] and [79] are duplicate entries of the same Kastor-Ray-Traschen paper and should be merged or cross-referenced.

Circularity Check

3 steps flagged · score 6.0 of 10

For P=0, ModMax is a constant rescaling of Maxwell; Eq. (21) and all thermodynamic quantities depend on γ only through q²e^{-γ}, so the claimed ModMax effects reduce to a charge redefinition of the known Maxwell-dRGT-like black hole.

renaming known result [Section II, Eq. (3) and the sentence before Eq. (19)]
"L = S coshγ − √S2 + P 2 sinhγ, (3) ... Whereas we are interested to get the electrically charged black holes, so we can consider P = 0 in the ModMax’s Lagrangian (3). ... So, the ModMax field equation (Eq. (5)), turns to ∂µ(√−ge−γFµν) = 0. (9)"

For P=0, the square root in (3) becomes |S|, so the ModMax Lagrangian is a constant multiple of S, i.e. a pure rescaling of the Maxwell Lagrangian (the factor is e^{-γ} or e^{γ} depending on the sign branch of S). Equation (9) is therefore Maxwell’s equation with a rescaled gauge potential, and γ can be absorbed into the charge or field redefinition. The paper chooses the electric sector explicitly, so the 'ModMax' input it uses is Maxwell by construction; γ is not an independent nonlinear-electrodynamics parameter in this sector.
renaming known result [Section III, Eqs. (19) and (21)]
"Eqrr = Eqtt = rψ′(r)+ψ(r)−1+Λr2−m2gC(c2C+c1r)+ q2e−γ r2 , (19) ... ψ(r) = 1 −m0 r − Λ 3 r2+q2e−γ r2 +m2gC(c1r 2 +c2C), (21)"

The field equations and the metric function contain γ only through the product q²e^{-γ}. Defining q_eff = q e^{-γ/2} makes Eq. (21) exactly the known charged black hole solution of dRGT-like massive gravity with a Maxwell field of charge q_eff. Consequently all horizon-structure statements in Fig. 1—naked singularities, extremal cases, number of roots—are properties of that one-parameter Maxwell family at different q_eff values, not of a genuinely new ModMax sector.

1 more flagged steps

renaming known result [Section IV, Eqs. (26)-(27); Section V, Eqs. (40)-(46)]
"The electric charge, electric potential, and entropy in the extended phase space of black hole solutions in the theory of ModMax dRGT-like massive gravity are given by Q = qe−γ, (40) U = q r+ , (41) S = πr2 +, (42)."

The thermodynamic mass (29) and temperature (25) contain q²e^{-γ}, and the first-law work term is U dQ with Q = q e^{-γ} and U = q/r+, so UQ = q²e^{-γ}/r+. In the extended mass (43), the combination πQ²e^{γ}/S equals q²e^{-γ}/r². Thus the pair (q,γ) collapses into a single effective charge q_eff = q e^{-γ/2} = Q e^{γ/2}. The first-law and Smarr checks verify the standard Maxwell-dRGT-like family at a rescaled charge; γ labels no new branch of solutions.

full rationale

The derivation from the stated action is algebraically self-contained: there are no fitted parameters, no load-bearing self-citations, and the first-law/Smarr computations are internally consistent. The circularity lies in the physical claim that γ is a new ModMax parameter. The paper explicitly sets P=0 to obtain electrically charged black holes; for P=0 the ModMax Lagrangian (3) is a constant multiple of the Maxwell invariant S, and the field equation (9) is Maxwell with a rescaled potential. Hence Eq. (21) depends on γ only through q²e^{-γ}, and the same combination appears in the temperature (25), charge (26), mass (29), heat capacity (31), and extended-phase quantities (40)-(46). With q_eff = q e^{-γ/2}, all reported γ effects—horizon roots, temperature roots, mass extrema, stability regions—are exactly the Maxwell-dRGT-like black hole at a different charge. This is a renaming of a known result rather than a new ModMax-dependent solution family; genuine ModMax behavior would require P≠0, which the paper explicitly excludes.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The model contains many adjustable couplings: gamma, m_g, c1, c2, C, Lambda, and q. In the electric-only sector, gamma is not an independent parameter; it can be absorbed into q, so the central thermodynamic analysis depends only on combinations like q^2 e^{-gamma}. No new particles, fields, or forces are introduced.

free parameters (7)

gamma (ModMax parameter) = unconstrained; appears as e^{-gamma} in q^2 e^{-gamma}
Dimensionless parameter in the ModMax Lagrangian, Eq. (3). All claimed effects of gamma are equivalent to rescaling the electric charge q, so it is not an independent physical parameter in the P=0 sector.
c1 (massive gravity coupling) = e.g., -10 or 1 in figures
Arbitrary coefficient in the massive gravity action, Eq. (1). Values are chosen by hand for the numerical plots and thermodynamic analysis.
c2 (massive gravity coupling) = e.g., 23 or 0.2 in figures
Arbitrary coefficient in the massive gravity action, Eq. (1). Values are chosen by hand for the numerical plots and thermodynamic analysis.
C (reference metric constant) = e.g., 0.4 or 0.1 in figures
Positive constant in the fiducial metric, Eq. (11). It sets the scale of the massive gravity terms and is chosen by hand in the figures.
m_g (graviton mass) = e.g., 0.8 or 0.4 in figures
Graviton mass in the massive gravity action, Eq. (1). Treated as an input parameter and varied by hand.
q (electric charge) = e.g., 1 in figures
Integration constant from the Maxwell/ModMax field equation. It is a free parameter of the solution family and is set to 1 in the plots.
Lambda (cosmological constant) = e.g., -0.5 in figures
Cosmological constant in the action, Eq. (1). Treated as an input parameter and chosen by hand in the figures.

assumptions (6)

domain assumption dRGT-like massive gravity action and massive tensor, Eqs. (1) and (6), are the correct starting point.
The paper assumes the ghost-free dRGT massive gravity framework with the reference metric (11) without deriving it from a more fundamental theory.
domain assumption ModMax Lagrangian, Eq. (3), is the nonlinear electrodynamics theory with the desired duality and conformal symmetries.
The paper adopts the ModMax Lagrangian from the literature and does not justify it from a more fundamental principle.
ad hoc to paper P=0 truncation of ModMax for electrically charged black holes.
The paper restricts to electric configurations, which makes the ModMax Lagrangian reduce to Maxwell times a constant. This is the step that trivializes the ModMax parameter and is not acknowledged as a reduction.
domain assumption Area law for entropy, S = A/4.
The paper assumes the Bekenstein-Hawking area law, Eq. (28), as the entropy of the black hole.
domain assumption Ashtekar-Magnon-Das (AMD) approach for total mass, M = m0/2.
The paper uses the AMD method, Eqs. (29) and references [72,73], to identify the total mass of the black hole.
domain assumption Extended phase space pressure P = -Lambda / 8 pi.
The standard black hole chemistry identification, Eq. (37), is assumed for the extended thermodynamics analysis.

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Cite this review

Pith. "Pith review of Black hole solutions in theory of ModMax-dRGT-like massive gravity." pith.science (2026). https://pith.science/paper/OQR5Y6NK

@misc{pith2026250705864,
  author       = {Pith},
  title        = {Pith review of: Black hole solutions in theory of ModMax-dRGT-like massive gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OQR5Y6NK}},
  note         = {Machine review of arXiv:2507.05864}
}

read the original abstract

This paper explores the properties of black holes using a new model of nonlinear electrodynamics called modified Maxwell (ModMax), in conjunction with nonlinear massive gravity known as dRGT-like massive gravity. We start by deriving the exact black hole solutions within the framework of ModMax-dRGT-like massive gravity and analyze how the parameters of both ModMax and dRGT-like massive gravity influence the characteristics of these black holes. Additionally, we calculate the thermodynamic quantities for these black holes in the non-extended phase space and investigate how the parameters of ModMax and dRGT-like massive gravity affect these quantities. We confirm that these quantities satisfy the first law of thermodynamics. We also examine local stability by analyzing heat capacity and assess how ModMax and the massive parameters influence phase transitions and physical limitation points. Next, we expand our analysis to the extended phase space, demonstrating that these thermodynamic quantities satisfy both the first law of thermodynamics and the Smarr relation in this context. Finally, we examine the isoperimetric ratio of black holes in ModMax-dRGT-like massive gravity.

Figures

Figures reproduced from arXiv: 2507.05864 by the authors.

**Figure 2.** ). In this scenario, the temperature remains positive, indicating that the black holes are always physical objects. iii) by considering special values for parameters of massive gravity, the temperature of medium black holes is positive for the ModMax field and negative for the Maxwell field. In other words, medium black holes in the Maxwell-dRGT-like massive gravity (see dashed line in [PITH_FULL_IMAGE:figures/ful… view at source ↗

**Figure 3.** ). Our findings reveal that the asymptotic limit of the mass (i.e., lim M r+→∞ = r+ 2 − Λr 3 + 6 + m2 gCr+ 2 c1r+ 2 + Cc2 ) depends on the cosmological constant, and the parameters of dRGT-like massive gravity. In addition, for large values of massive gravity’s parameters, there are three critical points (two minima and one maximum) for total mass (see the left panel in [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: FIG. 5: Heat capacity [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 4.** Figure 4: ). Notably, increasing the ModMax parameter reduces the number of phase transition points from two to one. Additionally, as γ increases, the physical limitation points decrease from three to one. 2- For small values of the massive gravity parameters, only large black …

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