Born–Infeld Electrogravity

• Born–Infeld electrogravity is a unified framework that combines gravity and nonlinear electromagnetism using determinantal and algebraic action structures.
• It regularizes electromagnetic self-energy and modifies black hole solutions by softening singularities and altering horizon properties.
• The framework supports both metric-affine and Kaluza–Klein formulations, yielding ghost-free corrections and rich phenomenology in strong-field regimes.

Born-Infeld electrogravity refers to a class of models unifying gravity and nonlinear electromagnetism through determinantal or algebraic structures in the action, typically inspired by Born–Infeld (BI) nonlinear electrodynamics. In these approaches, gravitational and electromagnetic interactions are treated on comparable footing—either via coupled modifications to both field equations, or through a single determinant that encodes both sectors. The resulting theories exhibit rich phenomenology including regularized field strengths, nonlinear couplings, modified black hole spacetimes, and characteristic deviations from general relativity (GR) coupled to Maxwell theory. Notably, these frameworks accommodate both metric and metric-affine (Palatini) formulations, with implications for curvature singularities, horizon structure, and strong-field behavior.

1. Fundamental Born–Infeld Structures

The core ingredient of Born–Infeld electrogravity is the replacement of the Maxwell action

$$\mathcal{L}_\text{Maxwell} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu}$$

with its nonlinear Born–Infeld analog. For pure electrodynamics,

$$\mathcal{L}_\text{BI}(F) = b^2\left[1 - \sqrt{1 + \frac{F}{2b^2} - \frac{G^2}{16 b^4}}\right],\quad F \equiv F_{\mu\nu} F^{\mu\nu},~ G \equiv F_{\mu\nu} \star F^{\mu\nu}$$

where $b$ is the field-strength scale.

These models exhibit a maximal field strength, yielding bounded electromagnetic self-energy. The energy–momentum tensor and generalized Maxwell equations acquire non-polynomial corrections in high-field regimes: $$T^{(\mathrm{BI})}_{\mu\nu} = \frac{F_{\mu\lambda} F_\nu{}^\lambda}{\sqrt{1+F/(2b^2)}} + b^2 g_{\mu\nu}\left(1 - \sqrt{1 + \frac{F}{2b^2} }\right)$$

$$\nabla_\nu\left(\frac{F^{\mu\nu}}{\sqrt{1+F/(2b^2)}}\right) = 0$$

The generalization to "electrogravity" occurs through an action that jointly modifies the gravitational and electromagnetic sectors, typically by combining the Ricci tensor and $F_{\mu\nu}$ in a determinantal form, such as

$$S = \frac{1}{\kappa} \int d^4x \left[ \sqrt{-\det\left(g_{\mu\nu} + \epsilon R_{(\mu\nu)} + \beta F_{\mu\nu} \right)} - \lambda \sqrt{-g}\right]$$

where $\epsilon$ and $\beta$ control the respective Born–Infeld scales and $\lambda$ the cosmological constant sector (Acuña et al., 7 Nov 2025, Afonso et al., 2021).

After eliminating auxiliary fields (when present), field equations relate the deformation of the metric sector to a composite of Ricci curvature and electromagnetic field strengths. These coupled deterministic forms underpin the theory's phenomenology and connect with various high- and low-energy limits.

2. Electromagnetic and Gravitational Sectors: Coupling and Unified Descriptions

Metric-Affine/Palatini Formulation: In the Palatini approach, the metric, affine connection, and electromagnetic potential are treated as independent variables. The equation of motion for the connection (after variation) enforces metric compatibility, fixing the connection to be the Levi–Civita connection of $g_{\mu\nu}$ on shell. The field equation for the metric is a constraint at the non-perturbative level but becomes dynamical once the connection is solved (Acuña et al., 7 Nov 2025, Afonso et al., 2021).

Two "Pictures" for Electrodynamics: The field equations admit two equivalent formulations:

• In the effective metric picture, electromagnetic field equations become standard Born–Infeld equations in a background corrected by curvature terms: $$\mathcal{G}_{\mu\nu} = g_{\mu\nu} + \epsilon R_{(\mu\nu)}$$ (Acuña et al., 7 Nov 2025).
• In the physical metric $g$ ("anomalous" BI), the Born–Infeld nonlinearities receive opposite sign under the square root, modifying the causal structure and the boundedness of field strengths.

Kaluza–Klein Theory and Higher-Dimensional Reduction: Reducing a five-dimensional Eddington-inspired Born–Infeld action with the Kaluza ansatz yields in four dimensions an action with both Einstein–Maxwell and additional nonlinear curvature–electromagnetic coupling terms. The determinant structure induces ghost-free, non-minimal couplings, and at low energies, recovers GR plus Maxwell, with leading corrections at order $\kappa$ (Fernandes et al., 2014).

The following table summarizes the main action formulations used in Born–Infeld electrogravity:

Action Structure	Curvature Dependence	Electromagnetic Nonlinearity
$$\sqrt{-\det[g_{\mu\nu} + \epsilon R_{(\mu\nu)} + \beta F_{\mu\nu}]}$$	Ricci (metric-affine)	Determinantal Born–Infeld in $F_{\mu\nu}$
$$S_{4}[g,A] = \sqrt{-\det[g_{\mu\nu} + \kappa(R_{\mu\nu} + 2 F_{\mu\lambda} F_\nu{}^\lambda)]}$$	Ricci (after KK)	Quadratic and higher-order curvature–$F^4$ terms
$\int d^4x \sqrt{-g} [R + L_{BI}(F)]$	Ricci scalar only	Standard (Hoffman) Born–Infeld

3. Black Hole Solutions and Singularity Structure

Born–Infeld electrogravity admits a variety of static, spherically symmetric black hole solutions, with key properties depending on the interplay of the gravitational and electromagnetic BI parameters ($\epsilon$, $\kappa$, $\beta$, $b$). These solutions interpolate between standard Reissner–Nordström metrics (in the Maxwell-Einstein limit) and regularized, but still singular, strong-field configurations in the nonlinear regime (Chen et al., 2023, Jana et al., 2015, Afonso et al., 2021, Acuña et al., 7 Nov 2025).

Key Properties:

• The central curvature singularities present in Reissner–Nordström black holes are "softened": curvature invariants diverge more mildly—e.g., as $r^{-6}$ or $(\rho-1)^{-1}$—rather than as $r^{-8}$, but geodesic incompleteness persists (Acuña et al., 7 Nov 2025, Afonso et al., 2021).
• In anomalous BI cases ($\beta$ real), the singular locus shifts to a sphere of finite radius rather than the center ($r \to 0$).
• Black hole horizons, photon spheres, and thermodynamical characteristics (temperature, entropy) all receive finite corrections dependent on the BI scale. For example, the horizon radius for the modified Reissner–Nordström-like solution with $\alpha=4\kappa b^2 = 1$ shifts according to

$$r_h^2 = \frac{(M \pm \sqrt{M^2 - q^2})^4 - \kappa q^2}{(M \pm \sqrt{M^2 - q^2})^2}$$

(Jana et al., 2015).

Boson Stars and Compact Solitons: When Born–Infeld electrodynamics is minimally coupled to scalar fields, static charged boson star solutions are found to exist up to higher charge values than in the Maxwell case, with lower mass and compactness at a given charge. The electromagnetic self-energy remains bounded, leading to regularization of the core fields (Jaramillo et al., 2023).

4. Light Propagation and Gravitational Lensing

A novel feature emerges in the behavior of photon trajectories:

• In Born–Infeld-coupled backgrounds, the effective metric for photon propagation depends both on the metric and on the $F_{\mu\nu}$ field, leading to two types of photon spheres: one for test-photon (null geodesics) propagation and another for "BI-photons" propagating inside the nonlinear electromagnetic medium (Jana et al., 2015).
• For sufficiently strong nonlinearity (small $\beta$ or $b$), certain spacetimes develop two photon spheres (inner and outer), leading to more complex lensing phenomena and multiple relativistic images (Chen et al., 2023).
• The gravitational lensing angle in the strong-deflection limit for light with impact parameter $b$ is modified to

$$\alpha(b) = -\bar a \ln(b/b_c - 1) + \bar b + \mathcal{O}((b/b_c - 1)\ln(b/b_c - 1))$$

with $b_c$ determined by the effective metric at the photon sphere.

In backgrounds with Born–Infeld naked singularities, photons can traverse the $r=0$ region (effective potential vanishes), resulting in transparency through the singularity, appearance of additional inner relativistic images, and the absence of a central shadow when imaging a celestial sphere (Chen et al., 2023).

5. Physical Interpretation and Comparisons with General Relativity

Born–Infeld electrogravity, through nonlinear self-coupling and determinant-based unification, provides a natural mechanism for regularizing certain pathologies endemic to GR-Maxwell theory. In the perturbative expansion of pure BI electrodynamics, electromagnetic forces appear as first-order effects, while gravitational phenomena emerge as effective metric corrections at second order in weak fields—aligning the induced metric with the linearized Einstein field equations (Chernitskii, 2010).

However,

• For physical parameter ranges, curvature divergences, while softened, are not entirely removed; spacetime continues to be geodesically incomplete, and singularities persist either at the origin or a finite radius depending on the "picture" (Acuña et al., 7 Nov 2025).
• The determinant structure in the action frequently ensures ghost-freedom, but in the generic metric–affine determinantal model, the "wrong" sign in the square root of the nonlinear electrodynamics sector yields unbounded electric fields and may jeopardize stability and well-posedness via the absence of field strength bounds and potential ghosts (Afonso et al., 2021).

A tabular summary (parameter regimes for spherically symmetric solutions):

Sector BI Parameter Limit	Black Hole Structure	Singularity Behavior
$\kappa \to 0$, $b^2 \to \infty$	Reissner–Nordström	$r^{-8}$ divergence (RN)
$\kappa>0$, $b^2$ finite	LN-modified RN	$r^{-6}$ or finite $r_*$
$\alpha=1$ ($b^2=1/4\kappa$)	RN-like (+mod.)	shifted locus, milder

6. Nonlinear Unification, Limitations, and Future Directions

The determinantal approach yields a consistent, covariant unification of gravitational and electromagnetic sectors, automatically generating nonminimal couplings and higher-order interaction terms in the metric and field equations. In Kaluza–Klein or metric–affine reductions, nontrivial corrections to black hole interiors, horizon structure, and regularity arise without the ad-hoc addition of higher-derivative terms.

Nevertheless:

• The regularization of singularities is incomplete at the classical level.
• The anomalous (wrong-sign) BI sector in some metric–affine models implies the absence of an upper bound on field strength and does not ensure finite self-energy (Afonso et al., 2021), contrary to BI's original motivation.
• Quantum and higher-derivative corrections—not included in current classical frameworks—may be required for true singularity resolution and ultraviolet completion.

Born–Infeld electrogravity thus provides a rich laboratory for exploring unified nonlinear dynamics, strong-field phenomenology, and potential implications for the cosmic censorship hypothesis, compact object structure, and observational signatures (e.g., black hole lensing, shadow morphology). Future extensions likely involve quantization, inclusion of further matter couplings, and interplay with string-inspired corrections.