Power–Maxwell Electrodynamics

• Power–Maxwell electrodynamics is a one-parameter nonlinear generalization of Maxwell theory that modifies field equations and regularizes electromagnetic singularities.
• It employs a Lagrangian density expressed as (-F)^s, enabling new black hole solutions, holographic superconductivity, and spontaneous scalarization in gravitating systems.
• The theory influences energy conditions, thermodynamics, and particle dynamics, offering practical insights into gravitational phenomena and nonlinear electrodynamics.

Power–Maxwell electrodynamics is a one-parameter non-linear generalization of classical Maxwell theory, defined by a Lagrangian density that replaces the quadratic dependence on the field-strength invariant with a real power $q$ (or %%%%1%%%% or $p$, depending on convention). The theory is structurally central in studies of black hole solutions, electromagnetic self-energy singularities, spontaneous scalarization in gravitating systems, and phenomenology of holographic superconductors.

1. Lagrangian Formulation and Field Equations

The fundamental Power–Maxwell (PM) Lagrangian in four-dimensional spacetime is\newline

$$\mathcal{L}_{\rm PM}(F) = (-F)^s \qquad F = F_{\mu\nu} F^{\mu\nu}$$

where $$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$$ is the field strength, and $s$ (alternatively $q$ or $p$ in the literature) is a real-valued power parameter. For $s = 1$, the canonical Maxwell Lagrangian $-F$ is recovered. The action underlying PM electrodynamics, when coupled minimally to gravity, is

$$S = \int d^4x\, \sqrt{-g} \left( R - 2\Lambda + \mathcal{L}_{\rm PM}(F) \right)$$

with $R$ the Ricci scalar and $\Lambda$ the cosmological constant (Dariescu et al., 2023, Dariescu et al., 2022).

Varying the action yields the generalized Einstein and Maxwell equations,

$$G_{\mu\nu}+\Lambda g_{\mu\nu} = T_{\mu\nu}^{\rm PM}$$

$\nabla_\mu (F^{\mu\nu} (-F)^{s-1})=0$

with the stress tensor given by

$$T_{\mu\nu}^{\rm PM} = 2s F_{\mu\rho}F_{\nu}^{\ \rho} (-F)^{s-1} - \frac{1}{4}g_{\mu\nu} (-F)^s$$

This nonlinearity removes the superposition principle and modifies both the energy–momentum content and the field equations compared to standard electrodynamics.

2. Removal of Field Singularities and Energy Conditions

In Maxwell theory, the electric field of a point charge diverges as $E(r) \sim 1/r^2$ near $r=0$, yielding infinite self-energy. PM electrodynamics with $s<1/2$ regularizes this divergence (Panah, 2021). The static spherically symmetric solution for the electric field is

$E(r) \sim r^{-2/(2s-1)}$

which decays to zero at the origin for $s<1/2$, eliminating both the field singularity and the ultraviolet divergence in the self-energy. The energy density

$\rho(r) = (2s-1)\, (2E^2)^s$

remains integrable at $r=0$ for $s<1/2$. Comparison with Born–Infeld electrodynamics reveals that while Born–Infeld interpolates between $E=0$ and a saturation value, PM with $s<1/2$ produces vanishing field at the origin—a more regular outcome (Panah, 2021).

For the full gravitating system, the weak energy condition (WEC) $\rho\geq0$, $\rho+p_i\geq0$ is satisfied for $p>0$, while the dominant energy condition (DEC) $\rho\geq|p_i|$ restricts $0<p\le1$ (Liang et al., 16 Jun 2025).

3. Black Hole Solutions, Kiselev Geometry, and Effective Metrics

Power–Maxwell theory admits exact black-hole solutions distinct from Reissner–Nordström, notably the Kiselev geometry (Dariescu et al., 2022, Dariescu et al., 2023). For a static, spherically symmetric ansatz,

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

$f(r) = 1 - \frac{2M}{r} - k r^p$

the PM exponent $q$ relates to the geometric parameter $p$ by

• electric: $q = (p-2)/2p$
• magnetic: $q = (2-p)/4$

with $k$ fixed by the nonlinear charge. Thermodynamic analysis shows the presence of multi-horizon structures, horizon merging in extremality, and thermal instability ($C<0$) in regions $0<p\le2$, while heat capacity curves display Schottky-type peaks and features akin to Schwarzschild–de Sitter black holes (Dariescu et al., 2022). Regular, non-singular black holes can be constructed by appropriate modifications to the metric function and auxiliary fields, with stability depending on the PM power (Liang et al., 16 Jun 2025).

In PM electrodynamics, photons and null rays do not propagate on the background metric $g_{\mu\nu}$ but on an "effective metric" $h^{\mu\nu}$,

$$h^{\mu\nu} = L_F\,g^{\mu\nu} - 4L_{FF} F^{\mu\alpha}F^{\nu}_{\ \alpha}$$

Singularities or signature changes of $h^{\mu\nu}$ (e.g., $\Phi = L_F + 2F L_{FF}=0$) lead to causality violations or the existence of spacelike photon trajectories, especially for electric solutions with certain $p$ (Liang et al., 16 Jun 2025). This modifies both shadow structure and strong-lensing signatures.

4. Holographic Superconductors and Gauss–Bonnet Gravity

PM electrodynamics in anti–de Sitter backgrounds, coupled to charged scalars, is central to holographic superconductor models (Jing et al., 2011, Sheykhi et al., 2016). The Lagrangian in $d$ dimensions takes the form

$$S = \int d^d x \sqrt{-g}\Big[ R - 2\Lambda + \alpha \mathcal{L}_{GB} - b (F_{\mu\nu}F^{\mu\nu})^q - |\nabla\psi - i A \psi|^2 - m^2|\psi|^2 \Big]$$

where $\mathcal{L}_{GB}$ is the Gauss–Bonnet term and $q$ parametrizes the nonlinearity.

Key results include:

• The gauge field equation is

$$\nabla_\mu \left[ (F_{\rho\sigma}F^{\rho\sigma})^{q-1} F^{\mu\nu} \right] = 0$$

• Asymptotic analysis yields the restriction $1/2 < q < (d-1)/2$ for finite boundary values.
• With $q<1$ (sublinear regime), the critical temperature $T_c$ for scalar condensation rises, making superconductive phases easier to achieve. Larger $q$ suppresses $T_c$ and makes condensation more difficult.
• The critical exponent $\beta=1/2$ for the order parameter is universal (independent of $q,\,d,\,\alpha$), mirroring mean-field behavior (Jing et al., 2011, Sheykhi et al., 2016).

Gauss–Bonnet coupling ($\alpha>0$) generally reduces $T_c$, counteracting the enhancing effect of PM nonlinearity in the sublinear regime.

5. Spontaneous Scalarization and Modified Gravity

Einstein–Power–Maxwell models non-minimally coupled to real scalar fields exhibit spontaneous scalarization of charged black holes (Carrasco et al., 2024). The action is

$$S = \frac{1}{4}\int d^4 x\,\sqrt{-g}\left[ R - 2(\nabla\phi)^2 - f(\phi)(F^2)^n \right]$$

with $f(\phi) = \exp(-\alpha \phi^2)$ and typically $\alpha<0$.

Key points:

• The onset of scalarization is controlled by an effective tachyonic mass

$\mu_{\rm eff}^2 = -\frac{1}{2}\alpha X^n$

• Only for the special PM power $n=3/5$ (i.e., $\mathcal{L}\sim(F^2)^{3/5}$), the scalarized black holes are regular, entropically preferred, and the mass function and thermodynamic structure are well-defined.
• Scalarization maximizes the horizon area $S$ at fixed $(M,Q)$, making the new stable branches thermodynamically favored over the scalar-free solution.

6. Particle Motion, Orbital Structure, and Phenomenology

Charged-particle motion in PM black-hole backgrounds is governed by the generalized Lorentz equation,

$m \frac{Du^\mu}{d\tau} = q F^\mu_{\ \nu} u^\nu$

The effective potentials and geodesic equations depend explicitly on the PM parameter, shifting the innermost stable circular orbits (ISCO) and allowing new classes of stable orbits, both radial and inclined (Poincaré cones in the magnetic case) (Dariescu et al., 2023).

Nonlinear electrodynamics also modifies the radial electric potential, leading to non-$1/r$ tails, affecting charged-particle trapping, accretion dynamics, and electromagnetic shadow features. Observational signatures, including photon ring size and strong-lensing deviations, can trace the underlying nonlinearity (Dariescu et al., 2022, Dariescu et al., 2023).

7. Comparison, Dualities, and Limit Cases

Model	Lagrangian	Field Behavior ($r\to0$)	Energy Condition
Maxwell	$-F$	$E\sim 1/r^2\to\infty$	WEC, DEC
Born–Infeld	$(1-\sqrt{1+F/b^2})b^2$	$E\to E_{\rm max}$	WEC, DEC
Power–Maxwell ($s<1/2$)	$(-F)^s$	$E\to 0$	WEC, DEC ($p<1$)

At $p=1/2$, standard Ferrer–Plebanski duality between electric and magnetic branches is lost, but an auxiliary scalar formulation restores this symmetry (Liang et al., 16 Jun 2025).

Conformal invariance of the PM Lagrangian under $g_{\mu\nu} \to \Omega^2 g_{\mu\nu}$ occurs for $q=d/4$ in $d$-dimensions (Dariescu et al., 2022, Jing et al., 2011).

8. Thermodynamics and Stability

The thermodynamics of PM black holes departs from the Reissner–Nordström archetype:

• Temperature: $T = [2M/r_b^2 - p k r_b^{p-1}]/4\pi$,
• Entropy: $S = \pi r_b^2$,
• Heat capacity can be negative or feature Schottky-type peaks signifying transitions related to multi-horizon structure (Dariescu et al., 2022, Liang et al., 16 Jun 2025).

Regularized PM black holes admit regions of positive heat capacity, signaling local thermodynamic stability for certain $(p,\nu,\sigma)$. The first law and Smarr relations acquire nontrivial corrections as the Lagrangian depends explicitly on asymptotic charges and parameters (Liang et al., 16 Jun 2025).

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Power–Maxwell electrodynamics thus constitutes a rich and flexible framework for theoretical exploration across classical field theory, gravitation, holography, and nonlinear optics, providing mechanisms for curing classical divergences, constructing new phenomenological models, and realizing controlled deformations of both black hole and condensed-matter holographic systems. The underlying nonlinear parameter $q$ (or $s$ or $p$) critically determines both local field behavior and global spacetime or thermodynamic properties, with each regime ($q>1$, $q=1$, $q<1$, $q<1/2$) displaying qualitatively distinct physics (Panah, 2021, Dariescu et al., 2022, Liang et al., 16 Jun 2025, Dariescu et al., 2023, Carrasco et al., 2024, Jing et al., 2011, Sheykhi et al., 2016).