Einstein–Nonlinear Electrodynamics Models

• Einstein–Nonlinear Electrodynamics is a class of theories coupling Einstein gravity with nonlinear electromagnetic actions, introducing self-interacting gauge fields in dynamic spacetimes.
• Exact solutions include black point geometries for Λ=0 and non-asymptotically flat wormholes for Λ<0, derived using specific power-law exponents in the Lagrangian.
• Quantum probes reveal spin-dependent singularity resolution, where Dirac fields experience regular evolution while Klein–Gordon fields do not.

Einstein–Nonlinear Electrodynamics (NED) models comprise a class of gravitational theories in which the Einstein–Hilbert action is coupled not to standard Maxwell electromagnetism but to a nonlinear generalization of the electromagnetic sector. Such models explore the physical consequences of nonlinear corrections to classical electrodynamics in a fully dynamical spacetime, emphasizing how self-interacting gauge fields back-react on the geometry and generate new types of solutions that do not exist in standard general relativity.

1. Action Structure and Nonlinear Field Content

The general action for Einstein–NED theories is formulated in $D=2+1$ or higher dimensions as: $$I = \frac{1}{2} \int d^3x \, \sqrt{-g} \left[ R - 2\Lambda + \alpha |\mathcal{F}|^k \right]$$ where %%%%1%%%% is the Ricci scalar, $\Lambda$ is the cosmological constant, $\alpha$ is a coupling, and $k$ is a rational exponent. The electromagnetic field enters via the invariant $\mathcal{F} = F_{\mu\nu} F^{\mu\nu}$. Specific values of $k$ yield physically significant cases: for example, $k=1$ implies Maxwell theory, $k=1/2$ leads to a square-root Lagrangian, and $k=3/4$ yields conformally invariant theory.

A key feature is the allowance for unorthodox field configurations: instead of a purely radial electric field, the ansatz $F_{\mu\nu} = E_0 \delta^t_\mu \delta^\theta_\nu$ introduces an angular electric component, representing an electric field circulating around the axis of symmetry. The associated vector potential can be parameterized as $A_\mu = E_0(a_0 t, 0, b_0 \theta)$ with $a_0 + b_0 = 1$.

2. Exact Solutions: Black Points and Wormholes in 2+1 Einstein–Power–Maxwell Theory

The case $k=1/2$, $L_{NED} \propto \sqrt{|\mathcal{F}|}$, admits explicit analytic solutions for static, circularly symmetric metrics: $$ds^2 = -A(r) dt^2 + \frac{dr^2}{B(r)} + r^2 d\theta^2$$ with $B(r) = D - \frac{\Lambda}{3} r^2$ and

$$A(r) = [D - (\Lambda/3) r^2] \left( C + \frac{r \alpha |E_0|}{D \sqrt{2}} \right)^2$$

where $C$ and $D$ are integration constants.

(a) $\Lambda=0$: "Black Point" Solution

Setting $\Lambda = 0$, the geometry reduces to

$ds^2 = -r^2 dt^2 + \frac{dr^2}{D} + r^2 d\theta^2$

with $D>0$ (rescalable to $D=1$). Here, no horizon exists, but $g_{tt}$ vanishes at $r=0$, indicating a central "black point" singularity, which acts as a confining region for test particle geodesics.

(b) $\Lambda \neq 0$: Non-Asymptotically Flat Wormhole

For $\Lambda < 0$ and $D < 0$, the metric with $C=0$ can be cast as

$$ds^2 = -\left( \frac{\alpha |E_0|}{D\sqrt{2}} \right)^2 r^2 dt^2 + \frac{dr^2}{|D|(1/(r^2/r_0^2 - 1))} + r^2 d\theta^2$$

where the wormhole throat is at $r_0 = \sqrt{ 3|D|/|\Lambda| }$. This solution possesses the canonical wormhole properties: a finite redshift function, a "shape" function $b(r)$ satisfying $b(r_0)=r_0$ and $b'(r_0)<1$ (flare-out condition). Asymptotic flatness is lost; the geometry at large $r$ is controlled by the cosmological constant.

Table: Two Key Geometries

Regime	Metric Feature	Singularity Type
$\Lambda = 0$	"Black point"	Central, pointlike
$\Lambda < 0$	Wormhole throat	Nonsingular (at $r_0$)

3. Conformal Invariance and Naked Singularities

Specializing to $k=3/4$ ensures local conformal invariance of the matter sector in $2+1$ dimensions. The exact solution for $\Lambda=0$ is

$$A(r) = \left[ 1 + \alpha (E_0^2)^{3/4} 2^{1/4} \sqrt{r} \right]^4, \qquad B(r) = \left[ 1 + \alpha (E_0^2)^{3/4} 2^{1/4} \sqrt{r} \right]^{-2}$$

The Kretschmann scalar diverges at $r=0$, producing a timelike naked singularity, unshielded by any event horizon.

Quantum probing reveals a dichotomy:

• For spin-0 Klein–Gordon fields, normalizability is preserved but essential self-adjointness fails; the singularity remains quantum mechanically "bad".
• For spin-1/2 Dirac fields, the operator is essentially self-adjoint; quantum evolution is unique, and the singularity is regular from the spinorial perspective.

4. Geodesic Structure and Particle Confinement

The field ansatz and nonlinear NED form ensure an energy–momentum tensor with only radial pressure ($T_{rr}$ nonzero), imposing significant consequences for geodesic motion. For neutral particles in the $\Lambda=0$ background: $\dot{r}^2 = \alpha_0^2 - \beta_0^2 r^2 - 1$ implies a maximal $r$; i.e., test particles cannot escape beyond a finite radius. Charged and massive particles are similarly confined, a direct outcome of the chosen energy–momentum structure.

5. Singularities and Quantum Resolution

Classical singularities (notably for $k=3/4$) invite the question of their status under quantum field probes. The quantum mechanical status is determined by the essential self-adjointness of the wave operator acting on the relevant Hilbert space:

• Klein–Gordon fields encounter a quantum singularity: the operator is not self-adjoint, evolution is ambiguous.
• Dirac fields experience an essentially self-adjoint operator; the singularity is "healed" for fermionic probes.

This result demonstrates that the quantum fate of classical singularities depends on the probe spin, indicating a spin-dependent "severity".

6. Broader Implications and Phenomenology

Einstein–NED models in $2+1$ dimensions, as exemplified here, illuminate several phenomena:

• Distinct matter sources (power-law NED vs. scalar fields) can yield identical or very similar metric structures, highlighting matter–geometry degeneracy in lower dimensions.
• Confining properties of the spacetime, observable for both charged and neutral geodesic motion, directly follow from the localization of energy-momentum in the radial sector.
• The existence of wormhole solutions sustained by ordinary (non-exotic) NED matter + cosmological constant, without recourse to matter violating standard energy conditions, provides new pathways for traversable wormhole construction.
• The dependence of singularity resolution on field spin suggests subtleties in the link between classical and quantum gravity.

These models also serve as valuable analogs for higher-dimensional NED–gravity interactions, with potential applications ranging from quantum gravity toy models to condensed matter analogs.

7. Summary Table: Key Features

Setting	Solution Type	Horizon?	Geometry	Quantum Status
$\Lambda = 0,\, k=1/2$	"Black point"	No	Conformally flat	Singular
$\Lambda < 0,\, k=1/2$	Wormhole	No	Non-asymptotically flat WH	Regular
$\Lambda = 0,\, k=3/4$	Conformal, naked sing.	No	Timelike naked singularity	Spin 0: singular; Spin 1/2: regular

These explicit $2+1$ Einstein–NED solutions thus provide a controlled laboratory for the interplay of geometric, topological, and quantum effects in low-dimensional nonlinear field theories (Mazharimousavi et al., 2013).