Mod(A)Max Black Hole

• Mod(A)Max black hole is a solution of Einstein's equations with ModMax nonlinear electrodynamics that preserves conformal invariance and electromagnetic duality.
• It generalizes Reissner–Nordström spacetimes by incorporating a nonlinear parameter to modify causal structure, thermodynamics, and optical signatures across varied horizon topologies.
• Distinct parameter dependencies influence stability, emission rates, and phase transitions, offering observable markers that differentiate these black holes from classical cases.

A Mod(A)Max black hole is a solution of Einstein’s equations coupled to nonlinear electrodynamics of the ModMax type, possibly including anti-Maxwell (phantom) sectors, and optionally with nontrivial cosmological constant and horizon topology. These objects generalize the well-known Reissner–Nordström and Reissner–Nordström–(A)dS spacetimes by introducing a single nonlinear parameter that preserves conformal invariance and SO(2) electromagnetic duality, producing distinctive modifications in causal structure, thermodynamics, optics, quasinormal spectra, and phase transitions. This entry synthesizes the current technical understanding drawn from exact solutions, stability analysis, and phenomenological properties.

1. ModMax and Mod(A)Max Nonlinear Electrodynamics

The core of the ModMax (modified Maxwell) theory is the unique, one-parameter Lagrangian

$$\mathcal{L}_{\text{ModMax}} = S \cosh\gamma - \sqrt{S^2 + P^2} \sinh\gamma,$$

where $S = \frac{1}{4} F_{\mu\nu}F^{\mu\nu}$, $P = \frac{1}{4} F_{\mu\nu} \tilde F^{\mu\nu}$, and $\gamma$ is a real, dimensionless ModMax parameter. For $\gamma=0$, Maxwell’s theory is recovered; for $\gamma>0$, the electrodynamics is nonlinear but remains conformal and duality invariant. The so-called "ModAMax" branch ($\eta=-1$) flips the sign in front of the gauge Lagrangian, corresponding to a phantom or anti-Maxwell sector (Panah et al., 2024, Panah et al., 27 Dec 2025).

2. Metric Structure and Horizon Topology

The most general Mod(A)Max black hole studied to date employs a static, topological metric

$$ds^2 = -f(r)dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega_k^2,$$

with $k=+1,0,-1$ marking spherical, planar, or hyperbolic horizon topology. For the purely electric sector and in four dimensions, the metric function adopts the form

$$f(r) = k - \frac{m}{r} - \frac{\Lambda}{3} r^2 + \eta \frac{q^2 e^{-\gamma}}{r^2},$$

where $m$ is an ADM mass parameter, $q$ is the electric charge, $\Lambda$ is the cosmological constant, and $\eta=\pm1$ distinguishes ModMax ($+1$) from ModAMax ($-1$) (Panah et al., 2024, Panah et al., 27 Dec 2025).

The horizon structure depends on $\Lambda$, $\gamma$, and $q$. For AdS ($\Lambda<0$), two positive roots exist (inner and event horizons); for dS ($\Lambda>0$), up to three positive roots may occur (Cauchy, event, cosmological horizons).

3. Thermodynamics and Phase Structure

Mod(A)Max black holes obey the area law for entropy and possess distinctive first law and Smarr relations: $$S = \frac{A}{4} = \pi r_+^2,\ T = \frac{1}{4\pi}\left[\frac{k}{r_+} - \Lambda r_+ - \eta\frac{q^2 e^{-\gamma}}{r_+^3}\right],\ dM = T dS + \eta \Phi dQ + V dP,$$ with electric potential $\Phi = q e^{-\gamma}/r_+$, thermodynamic volume $V = \frac{4\pi}{3} r_+^3$, and pressure $P = -\Lambda/(8\pi)$ (Panah et al., 2024, Panah et al., 27 Dec 2025).

In the extended thermodynamics, van der Waals–like critical phenomena appear for $k=+1$, with the nonlinearity parameter $\gamma$ rescaling all charged terms. The heat capacity and free energy reveal phase transitions between small and large black hole branches. In the canonical ensemble, the heat capacity diverges at specific values of $r_+$, marking second-order phase transitions; sign changes delineate stable/unstable regimes (Panah et al., 27 Dec 2025).

For ModAMax ($\eta=-1$), all charge contributions to thermodynamic quantities are sign-flipped, resulting in altered stability and criticality loci (Panah et al., 27 Dec 2025).

4. Null Geodesics, Shadow, and Optical Signatures

The photon sphere radius $r_c$ and corresponding shadow radius

$R_s = \frac{r_c}{\sqrt{f(r_c)}}$

are determined by the condition $2f(r_c) = r_c f'(r_c)$. The explicit solution for the critical orbit reads

$$r_c = \frac{3m + \sqrt{9m^2 - 32q^2 e^{-\gamma}}}{4}$$

for the topological case (Panah et al., 2024).

The shadow and deflection properties are directly modified by the charge-screening $Q^2 \to Q^2 e^{-\gamma}$. Increasing $\gamma$ reduces the effective charge, leading to larger values for $r_c$, $R_s$, and hence the shadow, trending toward the Schwarzschild values as $\gamma \to \infty$ (Guzman-Herrera et al., 2023, Panah et al., 2024). Lensing angles, redshifts, and additional optical observables are similarly rescaled; explicit birefringence does not occur in ModMax for the spherically symmetric backgrounds, but distinct effective metrics persist for different polarization modes (Guzman-Herrera et al., 2023).

5. Quasinormal Modes and Dynamical Stability

Massless scalar, electromagnetic (vector), and Dirac perturbations reduce to wave equations

$$\frac{d^2\psi}{dx^2} + [\omega^2 - V_{\text{eff}}(r)]\psi = 0,$$

where the effective potential depends on the field spin and the modified metric function:

• Scalar: $V_s(r) = f(r)[\ell(\ell+1)/r^2 + f'(r)/r]$
• Electromagnetic: $V_{EM}(r) = f(r)[\ell(\ell+1)/r^2]$
• Dirac: Nontrivial supersymmetric form with squared and derivative terms (Panah et al., 2024).

Quasinormal spectra $\omega_{n\ell}$ show that increasing $\gamma$ lowers the real part (QN frequencies redshift) and increases the imaginary part (faster damping): the ringdown oscillates slower and damps more rapidly as nonlinearity is increased. For all parameters ($\gamma \ge 0, \Lambda, q$), $\Im \omega < 0$; thus, the Mod(A)Max (A)dS black hole is dynamically stable under these test-field perturbations (Panah et al., 2024).

In the eikonal limit ($\ell \gg 1$), the QNM frequencies are controlled by the photon sphere via

$$\omega \simeq \ell \Omega_c - i(n + \tfrac{1}{2})|\lambda|,$$

where $\Omega_c=\sqrt{f(r_c)}/r_c$ is the angular velocity, and $\lambda$ is the Lyapunov exponent, encoding instability timescales of null geodesics. The analytic dependencies of these quantities on ($\gamma, q, \Lambda$) are explicitly available (Panah et al., 2024).

6. Emission Rate, Hawking Radiation, and Parameter Dependence

The energy emission rate (integrated over the shadow radius) is

$$\frac{d^2E}{d\omega\,dt} = \frac{2\pi^2 R_s^2 \omega^3}{e^{\omega/T} - 1}.$$

The position and peak of the emission spectrum reflect both the nonlinear ($\gamma$) and cosmological ($\Lambda$) parameters:

• Increasing $\gamma$ narrows and enhances the emission peak, accelerating evaporation.
• Increasing $\Lambda$ in dS lowers both the temperature and shadow radius, reducing the emission rate (slower decay).
• In AdS, larger $|\Lambda|$ increases and blue-shifts the emission peak (Panah et al., 2024).

Such dependencies provide potentially observable signatures distinguishing ModMax and phantom sectors from classical Maxwell or GR black holes.

7. Extensions: Joule-Thomson, Heat Engine Cycles, and Phantom Sectors

Mod(A)Max (A)dS black holes in extended phase space support full thermodynamic machinery, including Joule-Thomson expansion and heat-engine efficiency calculations. The Joule-Thomson coefficient and inversion curves, as well as heat-engine efficiency, depend sensitively on both $\gamma$ and horizon topology ($k$), and differ sharply between ModMax ($\eta=+1$) and ModAMax ($\eta=-1$) branches (Panah et al., 27 Dec 2025).

The cooling–heating (inversion) lines, critical points, and maximum/minimum temperatures are all shifted by the nonlinear parameter, with $\gamma$ systematically suppressing charge contributions and driving all results toward the neutral AdS–Schwarzschild regime. Heat-engine efficiency generally rises with horizon size for $k=+1$, but falls for $k=0,-1$ (Panah et al., 27 Dec 2025).

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References: All equations and claims trace to (Panah et al., 2024, Panah et al., 27 Dec 2025), and associated references therein. Comprehensive treatments, stability analysis, and explicit formulae for all thermodynamic, optical, and dynamical characteristics can be found in those sources.