ModAMax: Phantom Electrodynamics in AdS

• Modification Anti-Maxwell (ModAMax) is a nonlinear electromagnetic model where the kinetic term is reversed, leading to phantom behavior with altered causal and thermodynamic properties.
• It extends the ModMax framework and connects to theories like Born-Infeld and Euler-Heisenberg by employing a sign flip that generates novel AdS black hole solutions with diverse horizon topologies.
• Key features include rich phase structures, modified Joule–Thomson behavior, and topological classifications that offer fresh insights into gravitational stability and thermodynamic phase transitions.

Modification Anti-Maxwell (ModAMax) refers to a family of nonlinear electromagnetic models in which the kinetic term is reversed in sign relative to the ordinary Maxwell Lagrangian, generalizing the "phantom electrodynamics" paradigm to higher-order, covariant, and integrable frameworks. These models arise as solutions of the @@@@1@@@@ theory with the sign flip η = –1, or as phenomenological limits of more general nonlinear actions including the famous Born-Infeld and Euler-Heisenberg Lagrangians. ModAMax black holes in AdS spacetime admit rich phase structures, thermodynamic behaviors, and topological classifications not accessible in standard Maxwell or even conventional nonlinear models, particularly when the horizon topology parameter k = ±1, 0 is varied.

1. Formulation of ModAMax Nonlinear Electrodynamics

The prototypical ModAMax Lagrangian in the context of AdS black hole solutions is obtained from the generalized ModMax model by setting the deformation parameter γ ≥ 0 and the "phantom" branch parameter η = –1. The electromagnetic action density is

$$L(\mathcal{S}, \mathcal{P}) = \mathcal{S} \cosh \gamma - \sqrt{\mathcal{S}^2 + \mathcal{P}^2} \sinh \gamma,$$

where, for purely electric fields (as in the black hole sector), $$\mathcal{S} = \frac{1}{4} F_{\mu\nu} F^{\mu\nu},\quad \mathcal{P} = 0,$$ yielding the effective Lagrangian

$L_{\text{ModAMax}} = \mathcal{S} e^{-\gamma},$

but crucially, in the field equations and metric functions the Maxwell term appears with negative sign:

$$f(r) = k - \frac{m}{r} - \frac{\Lambda r^2}{3} - q^2 e^{-\gamma} / r^2,\qquad \eta = -1.$$

This "phantomization" reverses the sign of the energy density for electric fields, implying radically altered causal and thermodynamic properties (Panah et al., 27 Dec 2025).

2. Black Hole Solutions and Topological Structure

Static, asymptotically AdS black hole solutions in Einstein–ModAMax gravity possess topological horizon geometries parameterized by k = +1 (spherical), k = 0 (planar), or k = –1 (hyperbolic). The general metric ansatz is

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

with blackening function for ModAMax:

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

The electromagnetic potential is obtained by integrating the field equations:

$$A_\mu dx^\mu = -q/r \, dt,\qquad Q = q/(4\pi),\qquad \Phi_{\text{hor}} = q e^{-\gamma} / r_+.$$

A defining feature of the phantom branch η = –1 is the negative energy contribution of the field, with direct implications for thermodynamic quantities and stability.

3. Thermodynamics and Phase Structure

The mass (enthalpy), Hawking temperature, entropy, and electric chemical potential are given by:

• $$M = \frac{1}{8\pi}\left[k r_+ - \Lambda r_+^3/3 - q^2 e^{-\gamma} / r_+\right],$$
• $$T = \frac{1}{4\pi}\left[k/r_+ - \Lambda r_+ - q^2 e^{-\gamma} / r_+^3\right],$$
• $S = r_+^2/4,\qquad \Phi = q e^{-\gamma} / r_+.$ The first law, $dM = T dS + \Phi dQ + V dP,$ is verified by direct differentiation.

The equation of state for extended phase space thermodynamics (identifying pressure with cosmological constant) is:

$$P = \frac{T}{2r_+} - \frac{k}{8\pi r_+^2} - \frac{q^2 e^{-\gamma}}{8\pi r_+^4}.$$

Critical points (phase transitions) are determined by solving $\partial_v P = \partial^2_v P = 0$ (v = 2 r_+), which for η = –1 yields

$$r_c = \left(6 q^2 e^{-\gamma} / k \right)^{1/4},\qquad P_c = k^2/(96\pi q^2 e^{-\gamma}),$$

with the sign structure leading to "exotic" branch behavior and critical parameters (Panah et al., 27 Dec 2025).

4. Topological Classification and Universality

Thermodynamic phase branches are encoded via the topological charge (winding number) assigned to zeroes of the vector field $\phi^a = (\partial_s f, -\cot\theta \csc\theta)$ in the (s, θ) plane, $s = S/S_c$, $\theta$ an auxiliary angle. Each zero is assigned a charge $w_i = \pm 1$: $+1$ for locally stable (minimum free energy), $–1$ for unstable (saddle).

For ModAMax AdS black holes, the canonical ensemble (fixed $Q$) generically supports multiple branches differing from the Maxwell and ModMax (η=+1) cases:

• For spherical topology $k=+1$, either no small-black-hole phase exists (if the sign structure forbids a physical $r_c$), or a first-order transition is possible only for specific $(\gamma, q)$ ranges.
• For hyperbolic $k=-1$, criticality is suppressed and only a single stable branch is present.
• For planar $k=0$, the negative phantom charge term prevents conventional van der Waals transitions.

These features are mapped onto topological classes labeled by total winding number $W$: $W = +1,\, 0,$ or $-1$, according to the number and nature of stable branches and their appearance/disappearance in temperature or chemical potential scans. For ModAMax, $W$ is sensitive to the underlying field parameters and topology (Panah et al., 27 Dec 2025); this universality is independent of spacetime dimension $d$ and higher curvature order $k$ (Wang et al., 2024).

5. Joule–Thomson Expansion and Thermodynamic Cycles

ModAMax black holes admit a nontrivial Joule–Thomson "inversion curve" structure, defined by the coefficient

$$\mu_J = \frac{1}{C_{P,Q}}\left[T (\partial V/\partial T)_P - V\right],$$

and associated inversion temperature $T_i(P_i)$,

$8\pi P_i r_+^4 + 6k r_+^2 + 9 q^2 e^{-\gamma} = 0.$

In the ModAMax branch (η = –1), the sign reversal induces regimes with either suppressed cooling (absence of a physical $T_i$) or exotic heating/cooling boundaries depending on $k$.

Within black-hole heat-engine cycles, efficiency is bounded by Carnot's limit but is strongly modulated by $\gamma$ and topology:

$$\eta = \frac{W}{Q_H} = \frac{1 - P_C/P_H}{1 + \text{corrections}(k, q, \gamma, S_1, S_2, P_H)},$$

where the negative phantom charge term acts to suppress overall efficiency and can even eliminate engine cycles for forbidden parameter regions (Panah et al., 27 Dec 2025).

6. Stability, Physical Interpretation, and Observational Consequences

The sign flip in ModAMax profoundly changes local and global thermodynamic stability:

• The negative energy density for electric fields leads to altered specific heat, with stabilization or destabilization contingent on $(k, \gamma, q)$.
• In many parameter regions, small-black-hole phases and swallowtail transitions (characteristic of van der Waals-like behavior) are absent.
• The interplay of $\gamma$ and horizon topology $k$ enables a broad landscape of thermodynamic behaviors, some of which may lack a conventional event horizon or physical branch entirely.

From a topological perspective, each ModAMax black hole solution is a defect in the off-shell free-energy landscape, classified by quantized winding number. The ModAMax variant extends the universality of topological thermodynamics to include negative-energy branches, exemplifying the power of topological invariants to encode macroscopic phase structure driven by microscopic modifications of field theory (Panah et al., 27 Dec 2025).

A plausible implication is that, in physical models incorporating phantom electrodynamics, gravitational instability and dynamical evaporation could be qualitatively distinct from standard AdS black holes, inviting further study in holographic contexts and gravitational wave phenomenology.