ModMax Black Holes

• ModMax black holes are charged solutions derived from a nonlinear deformation of Maxwell electrodynamics that preserves duality and conformal invariance.
• Their thermodynamic behavior features modified heat capacities and phase transitions, with the deformation parameter critically altering extremality and horizon formation.
• Dynamical analyses reveal shifts in photon sphere and shadow radii, impacting gravitational lensing and potential astrophysical observations.

ModMax black holes are charged black hole solutions in general relativity sourced by ModMax nonlinear electrodynamics, a distinguished extension of Maxwell theory engineered to preserve both conformal and electromagnetic duality invariance. The gravitational and phenomenological properties of these objects are deeply sensitive to the ModMax deformation parameter (commonly denoted as $\gamma$ or $\eta$), which controls the nonlinear screening of the electromagnetic charge. The resultant family of solutions has been extensively analyzed across a range of contexts, including asymptotically flat, (anti-)de Sitter, and modified gravity settings.

1. ModMax Nonlinear Electrodynamics and Black Hole Solutions

The ModMax model introduces a one-parameter deformation $\gamma$ of the Maxwell Lagrangian, yielding

$$\mathcal{L}_{\mathrm{ModMax}} = \tfrac{1}{2} \left[ S \cosh\gamma - \sqrt{S^2+P^2}\, \sinh\gamma \right]$$

with $S = F_{\mu\nu}F^{\mu\nu}$, $P = F_{\mu\nu} *F^{\mu\nu}$, and $\gamma\geq 0$ the nonlinearity parameter that uniquely preserves electromagnetic duality and conformal invariance during nonlinearization. In the purely electric, static, spherically symmetric case, this reduces the covariant field equations to an Einstein–ModMax system whose physically meaningful black hole solutions generalize the Reissner–Nordström metric: $$ds^2 = -f(r)\,dt^2 + f(r)^{-1} dr^2 + r^2(d\theta^2 + \sin^2\theta\,d\phi^2),$$

$$f(r) = 1 - \frac{2M}{r} + \frac{Q^2\, e^{-\gamma}}{r^2}.$$

The exponential suppression factor $e^{-\gamma}$ effectively "screens" the electromagnetic charge, sharply modifying the spacetime geometry and electromagnetic profiles relative to standard electrovacua (Baptista et al., 20 Jun 2025, Sucu et al., 8 Aug 2025).

2. Thermodynamic Structure and Phase Transitions

Thermodynamics of ModMax black holes is governed by standard relations, but is refined by the introduction of $\eta$0:

• Hawking temperature: $\eta$1
• Entropy: $\eta$2
• Mass: $\eta$3

Admitting a cosmological constant alters the lapse to $\eta$4, making the thermodynamic structure sensitive to extended phase space parameters and topology (Panah et al., 27 Dec 2025, Panah et al., 2024). The presence of $\eta$5 generically raises the critical value at which extremality and horizon formation transitions occur and suppresses the effect of the electric (or dyonic) field on the spacetime geometry.

ModMax black holes manifest both first- and second-order phase transition phenomena. Notably, the heat capacity at fixed $\eta$6,

$\eta$7

exhibits divergences signaling second-order (Davies-type) transitions, with the location and structure of stable/unstable branches strongly modulated by $\eta$8. In AdS backgrounds, the Hawking–Page and van der Waals–type transitions persist, with critical exponents and coexistence curves shifted accordingly (Bezboruah et al., 11 Aug 2025, Panah et al., 27 Dec 2025).

3. Dynamical Instability, Lyapunov Exponents, and Chaos Bounds

The Lyapunov exponent $\eta$9 characterizing the divergence rate of nearby geodesics (unstable circular orbits) can be extracted for massless and massive test particles. For photon (null) orbits at $\gamma$0: $\gamma$1 where $\gamma$2 is the effective potential. $\gamma$3 as a function of temperature $\gamma$4 (for fixed charge) encodes the black hole phase structure: the order parameter $\gamma$5 (difference of Lyapunov exponents across the transition) possesses a universal mean-field critical exponent $\gamma$6: $\gamma$7. This establishes $\gamma$8 as a probe of phase transitions (Bezboruah et al., 11 Aug 2025).

In AdS spacetimes, the chaos bound $\gamma$9 (with $$\mathcal{L}_{\mathrm{ModMax}} = \tfrac{1}{2} \left[ S \cosh\gamma - \sqrt{S^2+P^2}\, \sinh\gamma \right]$$0 the surface gravity) can be violated for sufficiently small horizon radii (subthreshold $$\mathcal{L}_{\mathrm{ModMax}} = \tfrac{1}{2} \left[ S \cosh\gamma - \sqrt{S^2+P^2}\, \sinh\gamma \right]$$1), but $$\mathcal{L}_{\mathrm{ModMax}} = \tfrac{1}{2} \left[ S \cosh\gamma - \sqrt{S^2+P^2}\, \sinh\gamma \right]$$2 suppresses the violation region: increasing $$\mathcal{L}_{\mathrm{ModMax}} = \tfrac{1}{2} \left[ S \cosh\gamma - \sqrt{S^2+P^2}\, \sinh\gamma \right]$$3 or angular momentum shrinks the permissible window for exceeding the chaos bound. This establishes a direct connection between strong coupling, phase stability, and maximal entropy production in the nonlinear gauge sector (Bezboruah et al., 11 Aug 2025).

4. Geodesics, Photon Sphere, and Observational Signatures

The photon sphere, shadow radius, and ISCO for ModMax black holes are controlled by solutions of

$$\mathcal{L}_{\mathrm{ModMax}} = \tfrac{1}{2} \left[ S \cosh\gamma - \sqrt{S^2+P^2}\, \sinh\gamma \right]$$4

The effect of increasing $$\mathcal{L}_{\mathrm{ModMax}} = \tfrac{1}{2} \left[ S \cosh\gamma - \sqrt{S^2+P^2}\, \sinh\gamma \right]$$5 is to drive all characteristic radii (photon sphere $$\mathcal{L}_{\mathrm{ModMax}} = \tfrac{1}{2} \left[ S \cosh\gamma - \sqrt{S^2+P^2}\, \sinh\gamma \right]$$6, shadow $$\mathcal{L}_{\mathrm{ModMax}} = \tfrac{1}{2} \left[ S \cosh\gamma - \sqrt{S^2+P^2}\, \sinh\gamma \right]$$7, ISCO $$\mathcal{L}_{\mathrm{ModMax}} = \tfrac{1}{2} \left[ S \cosh\gamma - \sqrt{S^2+P^2}\, \sinh\gamma \right]$$8) outward, asymptotically approaching their Schwarzschild values as $$\mathcal{L}_{\mathrm{ModMax}} = \tfrac{1}{2} \left[ S \cosh\gamma - \sqrt{S^2+P^2}\, \sinh\gamma \right]$$9 (Ahmed et al., 2 Feb 2026, Awal et al., 10 Nov 2025). The critical impact parameter for the shadow scales as

$S = F_{\mu\nu}F^{\mu\nu}$0

with $S = F_{\mu\nu}F^{\mu\nu}$1 increasing with $S = F_{\mu\nu}F^{\mu\nu}$2.

Gravitational lensing, QPO frequencies, and deviations in the signature of astrophysical observables are correspondingly modulated by the ModMax parameter. VLBI/EHT shadow and GW ringdown/echoes provide direct probes of spectral and geometric deviations from the Reissner–Nordström case, with present constraints favoring $S = F_{\mu\nu}F^{\mu\nu}$3 (Pantig et al., 2022).

5. Extensions: Topology, Modified Gravities, and Exotic Sectors

The ModMax black hole construction generalizes in several directions:

• Topological Black Holes and AdS/CFT: The metric acquires the generic form $S = F_{\mu\nu}F^{\mu\nu}$4 for spherical, planar, or hyperbolic topology (Panah et al., 27 Dec 2025, Panah et al., 2024). The phase behavior and heat engine characteristics, including Joule–Thomson effect, depend strongly on $S = F_{\mu\nu}F^{\mu\nu}$5 and topology.
• Einstein–Gauss–Bonnet Gravity: Adding higher-curvature corrections modifies the horizon structure, admits stable remnant solutions, and affects QNMs. Here, increasing $S = F_{\mu\nu}F^{\mu\nu}$6 pushes the ISCO outwards and decreases the real part of QNM frequencies, increasing their damping rate (Hamil, 6 Jan 2026).
• $S = F_{\mu\nu}F^{\mu\nu}$7 Gravity and dRGT Massive Gravity: In $S = F_{\mu\nu}F^{\mu\nu}$8–ModMax theories, the factor $S = F_{\mu\nu}F^{\mu\nu}$9 renormalizes both mass and charge, modifying stability and global structure of horizons. Massive gravity provides additional degrees of freedom, further enriching thermodynamics and permitting nonstandard horizon structures (Panah, 2024, Panah, 8 Jul 2025).
• Kalb–Ramond and Phantom Branches: Coupling to Lorentz-violating fields introduces new parameters ($P = F_{\mu\nu} *F^{\mu\nu}$0), enhancing the range of possible radii for photon spheres and ISCOs. The "phantom" branch ($P = F_{\mu\nu} *F^{\mu\nu}$1) allows for negative energy contributions, thermal instability (negative specific heat throughout), and distinctive lensing/deflection behaviors (Ahmed et al., 5 Aug 2025, Ahmed et al., 11 Mar 2026).

6. Linear Response: Quasinormal Modes, Scattering, and Absorption

ModMax black holes are linearly stable to scalar, electromagnetic, and gravitational perturbations. QNM frequencies in the eikonal limit are set by photon sphere data: $P = F_{\mu\nu} *F^{\mu\nu}$2 with $P = F_{\mu\nu} *F^{\mu\nu}$3 (angular frequency) and $P = F_{\mu\nu} *F^{\mu\nu}$4 (Lyapunov exponent) derived from the photon sphere. Increasing $P = F_{\mu\nu} *F^{\mu\nu}$5 leads to reduced oscillation frequencies and increased damping. Scattering cross sections for test fields interpolate between Reissner–Nordström and Schwarzschild as charge screening increases, with interference fringes in scattering and absorption curves narrowing with higher $P = F_{\mu\nu} *F^{\mu\nu}$6 (Baptista et al., 20 Jun 2025, Panah et al., 2024).

7. Swampland and Consistency Constraints

Black holes in ModMax electrodynamics satisfy the Weak Gravity Conjecture (WGC): the extremal bound $P = F_{\mu\nu} *F^{\mu\nu}$7 enhances the allowed charge-to-mass ratio above the standard RN value, while the existence of a well-defined photon sphere until extremality ensures the weak cosmic censorship conjecture (WCCC) is preserved up to the enlarged extremal bound. In the presence of a cosmological constant or $P = F_{\mu\nu} *F^{\mu\nu}$8 corrections, the critical value for horizon formation is further increased (Gashti et al., 16 Apr 2025).

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In summary, ModMax black holes exemplify the intersection of nonlinear gauge field dynamics and strong gravity, with the screening parameter $P = F_{\mu\nu} *F^{\mu\nu}$9 providing a tunable interpolant between Maxwell and Schwarzschild limits. Their rich thermodynamic and dynamic behavior, persistence of duality and conformal invariance, and generalized stability/consistency features render them a central object of study in high-energy, gravitational, and observational astrophysics contexts (Bezboruah et al., 11 Aug 2025, Panah et al., 27 Dec 2025, Ahmed et al., 2 Feb 2026, Awal et al., 10 Nov 2025, Hamil, 6 Jan 2026, Baptista et al., 20 Jun 2025, Gashti et al., 16 Apr 2025).