Non-linear Electrodynamics Black Hole

• Non-linear electrodynamics black holes are gravitational solutions modified by non-linear electromagnetic Lagrangians that regularize singularities and yield unique causal structures.
• They employ models like Born–Infeld and ModMax to alter geodesic stability, with Lyapunov exponents quantifying chaotic divergence and phase transitions in the black hole regime.
• Quantitative analyses reveal that parameters such as the non-linearity factor and angular momentum directly influence both thermodynamic behavior and the violation of classical chaos bounds.

Non-linear electrodynamics (NLED) black holes are gravitational solutions sourced by non-linear extensions of Maxwell theory. These objects arise in attempts to regularize singularities, model quantum electrodynamics (QED) corrections, and realize strongly coupled gauge sectors in gravitational or holographic settings. Their phenomenology significantly diverges from standard Reissner–Nordström black holes, modifying both the causal structure and dynamics of perturbations due to the non-linear field contributions.

1. Non-linear Electrodynamics in Black Hole Spacetimes

NLED black holes are constructed by coupling general relativity to a Lagrangian for the electromagnetic field that is non-linear in the field strength. Instead of the standard Maxwell Lagrangian $\mathcal{L} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu}$, one considers $\mathcal{L}(F, G)$ where $F = F_{\mu\nu}F^{\mu\nu}$ and $G = F_{\mu\nu}\star F^{\mu\nu}$. Prominent examples include the Born–Infeld theory and the ModMax model, the latter being a one-parameter family interpolating between Maxwell and conformally invariant NLED, preserving duality symmetry.

The generic static, spherically symmetric solution in $d=4$ is of the form

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

where the function $f(r)$ depends on mass $M$, NLED charge $q$, and model-dependent non-linearity parameters (e.g., the ModMax parameter $\eta$). In the ModMax case,

$$f(r) = \frac{r^2}{l^2} - \frac{2M}{r} + \frac{q^2 e^{-\eta}}{r^2} + 1$$

where $\eta$ controls the non-linear correction magnitude and $l$ is the AdS length scale (Bezboruah et al., 11 Aug 2025).

2. Geodesic Dynamics and Classical Instability

The motion of test particles (massless or massive) in NLED black hole backgrounds is governed by the geodesic Lagrangian. The effective potential for the radial motion is

$$\dot{r}^2 + V_{\rm eff}(r) = 0,\quad V_{\rm eff}(r) = f(r) \left( \frac{L^2}{r^2} + \delta_1 \right) - E^2,$$

with $\delta_1=1$ (massive) or $0$ (massless), conserved energy $E$, and angular momentum $L$. Unstable circular orbits at $r_c$ are critical points of $V_{\rm eff}$ with $V_{\rm eff}(r_c)=0,\ V_{\rm eff}'(r_c)=0$, and $V_{\rm eff}''(r_c)<0$.

Linearizing perturbations around such an orbit gives a local instability/chaotic divergence rate,

$$\lambda = \sqrt{-\frac{V_{\rm eff}''(r_c)}{2 \dot{t}^2}},$$

which is the Lyapunov exponent for orbit instability. The geodesic Lyapunov exponent encodes the exponential separation of nearby trajectories and measures the degree of local dynamical chaos in the strong-field region (Bezboruah et al., 11 Aug 2025).

3. Thermodynamic Phase Structure and Lyapunov Exponents

For NLED black holes in (A)dS spacetime, the phase structure can be probed using Lyapunov exponents. The thermal profile $\lambda(T)$ displays multivaluedness across first-order transitions (e.g., small/large black hole transitions at fixed charge and nonlinearity). At the critical point, the discontinuity $\Delta \lambda = \lambda_s - \lambda_l$ (difference in Lyapunov exponents between small- and large-black-hole phases) vanishes as $\Delta \lambda \propto |T - T_c|^{1/2}$, providing a dynamical order parameter with mean-field critical exponent $\delta=\tfrac{1}{2}$ (Bezboruah et al., 11 Aug 2025).

This behavior closely tracks the underlying free energy swallowtail and signals that geodesic instabilities (as measured by $\lambda$) encode global thermodynamic information, even in the presence of strong electromagnetic non-linearities. For ModMax black holes, the onset and magnitude of the Lyapunov discontinuity are controlled by the non-linearity parameter $\eta$ and probe angular momentum, with higher $\eta$ delaying the onset of chaos-bound violation to smaller horizon radii.

4. The Chaos Bound and Its Violation

In holographic and QFT contexts, the Maldacena–Shenker–Stanford (MSS) chaos bound places a universal limit on quantum Lyapunov exponents: $\lambda_L \leq 2\pi T$, where $T$ is the black hole Hawking temperature. For NLED black holes, the classical geodesic Lyapunov exponent $\lambda$ can exceed the surface gravity $\kappa = 2\pi T$ in regions of parameter space—specifically in the "small" black hole phase below a threshold horizon radius $r_0$. The violation region is tunable via both the NLED non-linearity ($\eta$) and the test particle's angular momentum $L$, with larger $L$ and smaller $\eta$ expanding the violation domain (Bezboruah et al., 11 Aug 2025).

This violation is not generic for all black hole types; it is induced by non-linear electrodynamics corrections and is absent in the pure Reissner–Nordström/Maxwell case.

5. Quantitative Structure: Table of Key Relations

Property	Expression (for ModMax–AdS)	Notes
Metric function $f(r)$	$r^2/l^2 - 2M/r + q^2 e^{-\eta}/r^2 + 1$	$\eta$ controls NLED strength
Lyapunov $\lambda$ for null geodesic	$$\lambda^2 = -\tfrac{r_c^2 f(r_c)}{2L^2} V''_{\rm eff}(r_c)$$	$L$: particle angular momentum
Lyapunov $\lambda$ for timelike geodesic	$$\lambda= \tfrac12 \sqrt{[r_c f'(r_c) - 2f(r_c)][-V''_{\rm eff}(r_c)]}$$	Depends on $L, M, q, \eta$
Chaos bound	$\lambda \leq 2\pi T$	$T$: Hawking temperature, $2\pi T = \kappa$
Discontinuity at phase transition, $\Delta \lambda$	$\propto |T - T_c|^{1/2}$	Order parameter, critical exponent $\frac12$

6. Broader Implications and Connections

NLED black holes, via their Lyapunov-instability spectrum, directly link dynamical chaos, critical phenomena, and quantum information bounds. The observed violation of classical chaos bounds in strong-field regimes highlights the role of high-order electromagnetic corrections and frames geodesic Lyapunov exponents as sensitive probes of nontrivial gravitational and electromagnetic microphysics. Such connections have analogues in studies of operator scrambling and out-of-time-order correlators (OTOCs) in quantum chaos and black hole information, as well as in extended discussions of generalized Lyapunov exponents and chaos bounds in quantum systems (Pappalardi et al., 2022, Khemani et al., 2018).

7. Open Problems and Research Directions

Open directions include rigorous characterization of the bound violation in other NLED models, analysis of gravitational perturbation Lyapunov exponents beyond geodesic probes, and computation of the full operator chaos spectrum in backgrounds with non-linear sources. The interplay of chaos indicators, phase structure, and energy scales in NLED black holes offers deep insights for holography, quantum gravity, and the nonlinear dynamics of strong-field theories (Bezboruah et al., 11 Aug 2025, Pappalardi et al., 2022).