ModMax Black Hole Solutions

• ModMax black hole is a charged black-hole solution sourced by a unique nonlinear electrodynamics that screens the electric charge via an exponential factor e^(–γ).
• The solution retains a Reissner–Nordström-like form while exhibiting modified horizon structure, thermodynamic properties, and photon sphere characteristics influenced by the ModMax parameter.
• It extends across various theories—such as F(R) and Gauss–Bonnet gravity—and predicts observable effects like altered shadows, phase transitions, and birefringence in the presence of dark matter or quintessence.

A ModMax black hole is a charged black-hole solution sourced by Modified Maxwell (ModMax) nonlinear electrodynamics, either in Einstein gravity or in extensions such as $F(R)$ gravity, Einstein–Gauss–Bonnet gravity, dRGT-like massive gravity, Kalb–Ramond gravity, and backgrounds containing quintessence, perfect-fluid dark matter, or a cloud of strings. In the purely electric, static, spherically symmetric sector, the characteristic deformation is that the Coulomb term is screened by a factor such as $e^{-\gamma}$, so many exact metrics take a Reissner–Nordström-like form with $Q^2\to e^{-\gamma}Q^2$; the same sector supports detailed analyses of horizons, thermodynamics, phase structure, photon spheres, shadows, scattering, and quasinormal modes (Guzman-Herrera et al., 2023, Panah, 2024, Hamil, 6 Jan 2026).

1. Field-theoretic basis and the canonical Einstein–ModMax solution

ModMax is presented as the unique one-parameter nonlinear electrodynamics that preserves conformal invariance and electromagnetic duality. In one standard Einstein–ModMax formulation, the action is

$$S=\int d^4x \sqrt{-g}\,\Bigl[\frac{1}{16\pi}R+\mathcal{L}_{\rm ModMax}(F,G)\Bigr],$$

with

$$\mathcal{L}_{\rm ModMax} =-\frac{\cosh\gamma}{4}F+\frac{\sinh\gamma}{4}\sqrt{F^2+G^2},$$

where $F\equiv F_{\mu\nu}F^{\mu\nu}$ and $G\equiv F_{\mu\nu}{}^\star F^{\mu\nu}$ (Guzman-Herrera et al., 2023). A closely related convention writes the Lagrangian in terms of the invariants $\mathcal{S}$ and $\mathcal{P}$, again with a single real, dimensionless ModMax parameter $\gamma$ (Barrientos et al., 2024).

For a static, spherically symmetric black hole with line element

$e^{-\gamma}$0

the Einstein–ModMax field equations admit the exact solution

$e^{-\gamma}$1

with ADM mass $e^{-\gamma}$2 and electric charge $e^{-\gamma}$3 (Guzman-Herrera et al., 2023). Several later treatments use equivalent parameterizations, such as $e^{-\gamma}$4 or $e^{-\gamma}$5, but the structural role of the ModMax parameter is the same: the electric sector is screened by an exponential factor (Awal et al., 10 Nov 2025, Belchior et al., 20 May 2026).

A recurrent point in the literature is that, within the purely electric static sector, ModMax often reduces Maxwell electrodynamics up to a rescaling of the electric charge. This is stated explicitly for higher-order curvature gravity with quintessence, where “the ModMax theory effectively reduces Maxwell electrodynamics up to a rescaling of the electric charge,” so that the resulting solution is a consistent subset of the broader nonlinear theory (Al-Badawi et al., 13 May 2026). This does not eliminate genuinely nonlinear effects in propagation or thermodynamic diagnostics, but it explains why many exact metrics retain Reissner–Nordström-type algebraic form.

2. Horizon structure, extremality, and representative geometries

For the canonical Einstein–ModMax black hole, the horizons are the roots of $e^{-\gamma}$6: $e^{-\gamma}$7 so the extremal limit is

$e^{-\gamma}$8

As $e^{-\gamma}$9 increases, the charge term is suppressed and the geometry approaches Schwarzschild (Guzman-Herrera et al., 2023, Baptista et al., 20 Jun 2025).

Representative ModMax black-hole geometries in the literature are summarized below.

Framework	Metric function	Distinctive deformation
Einstein–ModMax	$Q^2\to e^{-\gamma}Q^2$0	Screened RN term
$Q^2\to e^{-\gamma}Q^2$1–ModMax	$Q^2\to e^{-\gamma}Q^2$2	Constant-curvature $Q^2\to e^{-\gamma}Q^2$3 sector
Topological Mod(A)Max AdS	$Q^2\to e^{-\gamma}Q^2$4	Horizon topology $Q^2\to e^{-\gamma}Q^2$5
KR + PFDM	$Q^2\to e^{-\gamma}Q^2$6	Lorentz violation and logarithmic PFDM tail
4D EGB–ModMax	$Q^2\to e^{-\gamma}Q^2$7	Gauss–Bonnet square-root branch

These examples illustrate two generic patterns. First, the electromagnetic contribution almost always enters through $Q^2\to e^{-\gamma}Q^2$8 or an equivalent screened combination. Second, the horizon polynomial is altered by curvature, topology, or environmental terms. In $Q^2\to e^{-\gamma}Q^2$9–ModMax theory, the horizon equation is generically quartic; for $$S=\int d^4x \sqrt{-g}\,\Bigl[\frac{1}{16\pi}R+\mathcal{L}_{\rm ModMax}(F,G)\Bigr],$$0 three real roots can appear (inner, event, cosmological), while for $$S=\int d^4x \sqrt{-g}\,\Bigl[\frac{1}{16\pi}R+\mathcal{L}_{\rm ModMax}(F,G)\Bigr],$$1 one finds two, one extremal, or none (Panah, 2024). In topological AdS constructions the discrete index $$S=\int d^4x \sqrt{-g}\,\Bigl[\frac{1}{16\pi}R+\mathcal{L}_{\rm ModMax}(F,G)\Bigr],$$2 changes the horizon geometry directly, while in perfect-fluid dark matter or string-cloud backgrounds additional logarithmic or deficit-angle terms modify the root structure (Panah et al., 27 Dec 2025, Belchior et al., 20 May 2026, Belchior et al., 20 May 2026).

Several limiting relations are standard. The Einstein–ModMax metric reduces to Reissner–Nordström for $$S=\int d^4x \sqrt{-g}\,\Bigl[\frac{1}{16\pi}R+\mathcal{L}_{\rm ModMax}(F,G)\Bigr],$$3 (Guzman-Herrera et al., 2023). In $$S=\int d^4x \sqrt{-g}\,\Bigl[\frac{1}{16\pi}R+\mathcal{L}_{\rm ModMax}(F,G)\Bigr],$$4–ModMax theory, the joint limit $$S=\int d^4x \sqrt{-g}\,\Bigl[\frac{1}{16\pi}R+\mathcal{L}_{\rm ModMax}(F,G)\Bigr],$$5, $$S=\int d^4x \sqrt{-g}\,\Bigl[\frac{1}{16\pi}R+\mathcal{L}_{\rm ModMax}(F,G)\Bigr],$$6, and $$S=\int d^4x \sqrt{-g}\,\Bigl[\frac{1}{16\pi}R+\mathcal{L}_{\rm ModMax}(F,G)\Bigr],$$7 recovers Reissner–Nordström–(A)dS (Panah, 2024). In higher-order curvature gravity with quintessence, taking $$S=\int d^4x \sqrt{-g}\,\Bigl[\frac{1}{16\pi}R+\mathcal{L}_{\rm ModMax}(F,G)\Bigr],$$8 removes the quintessence term, while $$S=\int d^4x \sqrt{-g}\,\Bigl[\frac{1}{16\pi}R+\mathcal{L}_{\rm ModMax}(F,G)\Bigr],$$9 returns the general-relativistic sector and $$\mathcal{L}_{\rm ModMax} =-\frac{\cosh\gamma}{4}F+\frac{\sinh\gamma}{4}\sqrt{F^2+G^2},$$0 returns Maxwell electrodynamics (Al-Badawi et al., 13 May 2026).

3. Thermodynamics, stability, and thermodynamic geometry

For the asymptotically flat Einstein–ModMax black hole, the event-horizon radius $$\mathcal{L}_{\rm ModMax} =-\frac{\cosh\gamma}{4}F+\frac{\sinh\gamma}{4}\sqrt{F^2+G^2},$$1 determines the mass, temperature, and entropy through

$$\mathcal{L}_{\rm ModMax} =-\frac{\cosh\gamma}{4}F+\frac{\sinh\gamma}{4}\sqrt{F^2+G^2},$$2

The first law takes the standard form

$$\mathcal{L}_{\rm ModMax} =-\frac{\cosh\gamma}{4}F+\frac{\sinh\gamma}{4}\sqrt{F^2+G^2},$$3

and the fixed-$$\mathcal{L}_{\rm ModMax} =-\frac{\cosh\gamma}{4}F+\frac{\sinh\gamma}{4}\sqrt{F^2+G^2},$$4 heat capacity is

$$\mathcal{L}_{\rm ModMax} =-\frac{\cosh\gamma}{4}F+\frac{\sinh\gamma}{4}\sqrt{F^2+G^2},$$5

Its zeros occur at the physical-limitation point $$\mathcal{L}_{\rm ModMax} =-\frac{\cosh\gamma}{4}F+\frac{\sinh\gamma}{4}\sqrt{F^2+G^2},$$6, while its divergence at $$\mathcal{L}_{\rm ModMax} =-\frac{\cosh\gamma}{4}F+\frac{\sinh\gamma}{4}\sqrt{F^2+G^2},$$7 signals a second-order phase transition (Panah, 8 Jul 2025). In the extended phase space, the same family satisfies

$$\mathcal{L}_{\rm ModMax} =-\frac{\cosh\gamma}{4}F+\frac{\sinh\gamma}{4}\sqrt{F^2+G^2},$$8

and the isoperimetric ratio is exactly $$\mathcal{L}_{\rm ModMax} =-\frac{\cosh\gamma}{4}F+\frac{\sinh\gamma}{4}\sqrt{F^2+G^2},$$9 (Panah, 8 Jul 2025).

In $F\equiv F_{\mu\nu}F^{\mu\nu}$0–ModMax theory, the thermodynamic quantities are modified by the constant-curvature factor $F\equiv F_{\mu\nu}F^{\mu\nu}$1. The entropy becomes

$F\equiv F_{\mu\nu}F^{\mu\nu}$2

the ADM mass is

$F\equiv F_{\mu\nu}F^{\mu\nu}$3

and the Hawking temperature is

$F\equiv F_{\mu\nu}F^{\mu\nu}$4

The first law

$F\equiv F_{\mu\nu}F^{\mu\nu}$5

is verified explicitly. Local stability is controlled by the heat capacity $F\equiv F_{\mu\nu}F^{\mu\nu}$6, while global stability is analyzed through the Helmholtz free energy $F\equiv F_{\mu\nu}F^{\mu\nu}$7. The HPEM thermodynamic Ricci scalar diverges exactly at the physical-bound point and at the phase-transition points given by the poles of $F\equiv F_{\mu\nu}F^{\mu\nu}$8 (Panah, 2024).

Topological Mod(A)Max AdS black holes extend the same structure to nontrivial horizon topology and pressure. For

$F\equiv F_{\mu\nu}F^{\mu\nu}$9

one has

$G\equiv F_{\mu\nu}{}^\star F^{\mu\nu}$0

together with the extended first law and Smarr relation. For $G\equiv F_{\mu\nu}{}^\star F^{\mu\nu}$1, the critical point satisfies

$G\equiv F_{\mu\nu}{}^\star F^{\mu\nu}$2

whereas for $G\equiv F_{\mu\nu}{}^\star F^{\mu\nu}$3 or $G\equiv F_{\mu\nu}{}^\star F^{\mu\nu}$4 no Van der Waals-type critical point exists in the physical region (Panah et al., 27 Dec 2025).

Thermodynamic geometry is used widely in the ModMax literature. In the quintessence-corrected higher-curvature case, the Ruppeiner scalar diverges precisely at the zeros of the heat capacity, while the Weinhold curvature does not generally coincide with those singularities (Al-Badawi et al., 13 May 2026). In $G\equiv F_{\mu\nu}{}^\star F^{\mu\nu}$5–ModMax theory, the HPEM curvature distinguishes bound-point and critical-point singularities through different sign changes (Panah, 2024).

4. Geodesics, birefringence, photon spheres, and shadows

Null propagation in ModMax is not exhausted by the background metric. For general nonlinear electrodynamics, wave fronts propagate according to an effective metric

$G\equiv F_{\mu\nu}{}^\star F^{\mu\nu}$6

and for ModMax one finds

$G\equiv F_{\mu\nu}{}^\star F^{\mu\nu}$7

Hence one polarization sees the background geometry, while the other sees a distinct optical geometry (Guzman-Herrera et al., 2023). In the static spherically symmetric ModMax black-hole background, the equatorial optical metrics are

$G\equiv F_{\mu\nu}{}^\star F^{\mu\nu}$8

$G\equiv F_{\mu\nu}{}^\star F^{\mu\nu}$9

This yields birefringence and branch-dependent lensing, redshift, and shadow observables (Guzman-Herrera et al., 2023).

For the Einstein–ModMax background, the unstable circular photon orbit is located at

$\mathcal{S}$0

The critical impact parameters are

$\mathcal{S}$1

so the branch-1 shadow is reduced by an additional factor $\mathcal{S}$2 relative to branch 2 (Guzman-Herrera et al., 2023). For scalar-wave absorption, the low-frequency limit is universal,

$\mathcal{S}$3

while the high-frequency limit approaches the capture cross section

$\mathcal{S}$4

Numerically, increasing $\mathcal{S}$5 screens the charge contribution and causes cross-section curves for different $\mathcal{S}$6 to coalesce (Baptista et al., 20 Jun 2025).

In matter-dressed or modified-gravity backgrounds, the photon-sphere condition usually retains the form

$\mathcal{S}$7

but the resulting shadow radius depends sensitively on the extra sector. In higher-order curvature gravity with quintessence, exact analytic expressions for $\mathcal{S}$8 are available for $\mathcal{S}$9; higher-order curvature corrections and quintessence significantly enhance the shadow size, whereas the electric charge has the opposite effect, and quintessence has a more pronounced impact on the shadow than the charge (Al-Badawi et al., 13 May 2026). In perfect-fluid dark-matter backgrounds, the shadow radius $\mathcal{P}$0 also controls the eikonal quasinormal frequency through $\mathcal{P}$1 (Ahmed et al., 8 Feb 2026).

5. Modified-gravity and matter-dressed ModMax black holes

The phrase “ModMax black hole” now denotes a broad class of exact solutions obtained by combining the same screened electrodynamic sector with modified gravitational dynamics or environmental sources.

In constant-curvature $\mathcal{P}$2 gravity, the exact electrically charged black hole is

$\mathcal{P}$3

with entropy and mass rescaled by $\mathcal{P}$4 (Panah, 2024). In the same $\mathcal{P}$5 setting with quintessence dark energy, the lapse function becomes

$\mathcal{P}$6

so the ModMax screening competes directly with the quintessence tail $\mathcal{P}$7 (Al-Badawi et al., 13 May 2026).

In 4-dimensional Einstein–Gauss–Bonnet gravity, the negative branch of the exact solution is

$\mathcal{P}$8

The horizon condition implies

$\mathcal{P}$9

and extremality gives

$\gamma$0

The entropy acquires the Gauss–Bonnet correction

$\gamma$1

and the minimum mass

$\gamma$2

is interpreted as a stable remnant (Hamil, 6 Jan 2026).

Lorentz-violating extensions are especially active. In Kalb–Ramond gravity with perfect-fluid dark matter, the lapse is

$\gamma$3

where $\gamma$4 and $\gamma$5. The perfect-fluid dark matter term introduces an additional logarithmic correction that is subleading at large $\gamma$6 but significant at intermediate scales (Belchior et al., 20 May 2026). In Einstein–bumblebee gravity with a cloud of strings, one has

$\gamma$7

with

$\gamma$8

so Lorentz violation, strings, and ModMax screening enter nontrivially and independently (Belchior et al., 20 May 2026). In the dyonic Kalb–Ramond plus string-cloud solution,

$\gamma$9

and all shadow-size observables inherit the conical-deficit factor $e^{-\gamma}$00 (Ahmed et al., 11 Mar 2026).

Massive-gravity versions preserve the same screened electric term while adding massive couplings. A representative AdS ModMax–dRGT black hole has

$e^{-\gamma}$01

so the screening factor directly enters the enthalpy, equation of state, thermodynamic geometry, and heat-engine efficiency (Hassanabadi et al., 26 Jun 2026).

6. Perturbations, quasinormal modes, and current interpretation

Linear perturbations of ModMax black holes are typically reduced to Schrödinger-type equations in the tortoise coordinate. For topological ModMax (A)dS black holes, the scalar, electromagnetic, and Dirac effective potentials are

$e^{-\gamma}$02

with $e^{-\gamma}$03. In the eikonal limit,

$e^{-\gamma}$04

where $e^{-\gamma}$05 and $e^{-\gamma}$06 is the Lyapunov exponent of the unstable null orbit (Panah et al., 2024). In that model, both $e^{-\gamma}$07 and $e^{-\gamma}$08 decrease monotonically as $e^{-\gamma}$09 increases from zero (Panah et al., 2024).

The dependence on $e^{-\gamma}$10 is not universal across all embeddings. In the Kalb–Ramond phantom-sector construction, Padé-averaged WKB and time-domain analyses show that increasing either $e^{-\gamma}$11 or $e^{-\gamma}$12 raises $e^{-\gamma}$13 and $e^{-\gamma}$14, with the strongest sensitivity in the phantom branch $e^{-\gamma}$15 (Sekhmani et al., 25 Jul 2025). This model dependence is important: the screened charge term is common, but the surrounding gravitational sector controls how the perturbation barrier is reshaped.

Emission-rate calculations likewise tie thermodynamic and optical sectors together. For topological ModMax (A)dS black holes,

$e^{-\gamma}$16

so the shadow radius and Hawking temperature jointly determine the high-frequency emission profile (Panah et al., 2024). In dyonic Kalb–Ramond black holes with a string cloud, the same geometric-optics limit gives $e^{-\gamma}$17, and the shadow radius carries the same conical-deficit prefactor that appears in asymptotic observables (Ahmed et al., 11 Mar 2026).

A plausible implication is that “ModMax black hole” names a robust RN-like core plus a large family of curvature- and matter-dressed exact solutions. The core sector is governed by the screened combination $e^{-\gamma}$18; the surrounding theory then determines whether that screening primarily shifts horizons, alters entropy and criticality, produces birefringence, changes shadow size, or reshapes the quasinormal spectrum.