Finite-Energy Electric Point Charge

• Finite-Energy Electric Point Charge is a concept where the self-energy of an isolated charge is regularized by modifying the classical Coulomb divergence.
• Nonlinear theories like Born–Infeld and BLTP introduce self-interaction and higher-derivative terms that soften the 1/r^2 singularity and yield finite electromagnetic energy.
• Coupling these electrodynamic models with gravity produces unique spacetime structures and potential charge quantization mechanisms, reshaping classical field theory outcomes.

A finite-energy electric point charge is defined as a classical or semiclassical field configuration in which the electrostatic (or in general, electromagnetic) self-energy of an isolated point-like charge is regularized to be finite, as opposed to the conventional Maxwell theory in which the self-energy diverges due to the $1/r^2$ singularity of the Coulomb field at the spatial origin. The construction of such finite-energy models is a central issue in both classical and quantum electrodynamics, touching upon foundational questions in field theory, general relativity, and particle physics.

1. Nonlinear Electrodynamic Theories Enabling Finite-Energy Point Charges

Nonlinear modifications to Maxwell's electrodynamics introduce self-interaction terms or higher-derivative (Podolsky-type) structures that soften or entirely remove the severe $1/r^2$ divergence of the Coulomb field. Theories in this class include:

• Born–Infeld Theory: The paradigmatic nonlinear model used since the 1930s to introduce an upper bound for $|E|$ and render the field energy of a point charge finite by replacing the Maxwell Lagrangian $\mathcal{L}_M = -\mathcal{F}$ with $$\mathcal{L}_{\rm BI} = \beta^2\left(1-\sqrt{1 - 2\mathcal{F}/\beta^2}\right)$$, where $\mathcal{F} = (1/4) F_{\mu\nu}F^{\mu\nu}$ (Gaete et al., 2014).
• Quartic and Polynomial Theories: Seeing as quantum electrodynamics (QED) corrections, e.g., the Euler–Heisenberg Lagrangian, provide quartic and higher-order terms in the field invariant $\mathcal{F}$, the inclusion of these terms (e.g., $\mathcal{L} = -\mathcal{F} + (y/2)\mathcal{F}^2$) produces a cubic field equation for $E(r)$ whose solution regularizes the field energy (Costa et al., 2013, Gitman et al., 2015, Breev et al., 2016).
• Logarithmic and Exponential Nonlinearities: The Lagrangian $$\mathcal{L} \sim -\beta^2\ln[1+\mathcal{F}/\beta^2]$$ yields bounded electric fields and finite energy for point charges, while exponential models may or may not share this property depending on the precise form of the nonlinearity (Gaete et al., 2013, Gaete et al., 2014).
• Bopp–Podolsky (BLTP) and Lee–Wick Theories: By extending the Maxwell Lagrangian with higher-derivative terms (e.g., $(1/2\kappa^2)(\partial_\alpha F^{\mu\nu})^2$), these theories lift the differential order of the field equations and replace the Green functions with regularized versions that eliminate the $1/r$ Coulomb divergence and result in finite field energy (Amorim, 8 Aug 2025, Barone et al., 2015).

The specific forms of the Lagrangian and constitutive relations dictate the asymptotic behavior of the electric field near the origin and thereby control the convergence of the self-energy integral.

2. Metric Structure and Singularities in General Relativistic Solutions

In general relativity, the divergence of the field energy of a point charge in pure Maxwell theory leads to strong spacetime singularities (as in the Reissner–Weyl–Nordström solution). Nonlinear or higher-derivative electromagnetic theories coupled to gravity—such as those with Born–Infeld or BLTP electrodynamics—replace the central singularity with a milder structure:

• Conical Singularities: For nonlinear “aether-law” Lagrangians derivable from a reduced Hamiltonian $\zeta(\mu)$, the resulting static, spherically symmetric, asymptotically flat spacetime exhibits a conical singularity at $r=0$ (i.e., a deficit angle around the time axis) (Tahvildar-Zadeh, 2010). The mass function $m(r)$ behaves as $O(r)$ near the origin, and the metric coefficient $e^\xi$ acquires a nonzero limiting value.
• Naked Point Defects: Both in BLTP electrodynamics and in certain degenerate-metric gravitational models, the charge appears as a naked singularity—i.e., one without an event horizon—but the field energy is finite and the potential is regular at $r=0$ (Amorim, 8 Aug 2025, Gera et al., 2021).
• Regular Geometry via Metric Degeneracy: In first order gravity formulations, the standard singularity is replaced with a metric phase where the lapse vanishes and the core region is noninvertible; the charge acquires a topological characterization (e.g., Euler characteristic $\chi=2$) (Gera et al., 2021).

In all these cases, by choosing appropriate boundary conditions (e.g., mass function $m(0)=0$), the transport of mass and charge is attributed entirely to the field configuration, avoiding the notion of an external “mechanical” mass.

3. Energy-Defining Conditions and Key Mathematical Structures

The models achieving finite energy for point charges enforce specific conditions on the underlying Lagrangian or the reduced Hamiltonian function. For general nonlinear “aether-laws” (Tahvildar-Zadeh, 2010):

Condition Graph	Description
(R1)	Weak field Maxwell limit (as $\mu\to0$)
(R2)	Monotonicity: $\zeta'(\mu)>0$, $\zeta-\mu\zeta'\ge0$
(R4)	Growth: $\int_0^\infty\mu^{-7/4}\zeta(\mu)d\mu<\infty$

These requirements ensure that: (i) the theory agrees with Maxwell in the weak field limit; (ii) the dominant energy condition holds; (iii) the self-energy integral $\int_0^\infty \zeta(\mu)r^2dr$ converges, yielding a finite ADM mass equal to the field energy.

Similarly, in the BLTP model (Amorim, 8 Aug 2025), the presence of the parameter $\kappa$ (with $1/\kappa$ a length scale) replaces the potential near $r=0$ with $\varphi(0)=Q\kappa$, ensuring that the energy density is integrable.

For the Maxwell–Born–Infeld interpolations (Liu et al., 5 Sep 2025), the presence of the Maxwell term ensures electromagnetic asymmetry: the electric field energy is regularized, but the magnetic monopole (B-field) solutions inevitably feature a Maxwell term $(\kappa_1/2)B^2$ leading to divergent energy, thereby excluding finite-energy monopoles or dyons.

4. Physical Implications and Geodesic Structure

A finite-energy point charge solution radically alters not only the field configuration but also the physical structure and geodesic trajectories:

• For suitable parameter regimes (e.g., small mass-to-charge ratio $\varepsilon=m/|q|$), these spacetimes display no event horizons or trapped null geodesics (Tahvildar-Zadeh, 2010). All gravitational mass arises from the finite field energy.
• The electrostatic potential remains finite at the origin, but the direction of the field becomes undefined—modeling the “point defect” of the charge.
• Test particle and null geodesic analyses show that the conical or regularized geometry leads to qualitative differences from the Reissner–Nordström behavior. For example, non-radial null geodesics extend to infinity without encountering horizons or repulsive “bounces.”
• The charge distribution (as in the Einstein–Maxwell theory accounting for gravity) can be characterized by a unique radius (e.g., $r_c = \sqrt{r_e r_o / 2}$, with $r_e$ the classical electron radius and $r_o$ the Schwarzschild radius), and may be interpreted as a combination of the Planck length and the fine structure constant ($r_c = \ell_P \sqrt{\alpha_e}$) (Dekker, 2014).

A notable result is the possibility of associating the entire observed ADM mass with electromagnetic field energy.

5. Electromagnetic Asymmetry, Uniqueness, and Charge Quantization

The finite-energy constructions introduce several structural and physical phenomena:

• Electromagnetic Asymmetry: In “weighted” Maxwell–Born–Infeld theories, only electric point charges are regularized, while magnetic monopole (and dyon) solutions are not; the lower bound on the energy density from the Maxwell piece leads to divergence for monopolar B-fields (Liu et al., 5 Sep 2025).
• Birkhoff-Type Uniqueness: For the class of nonlinearly modified, spherically symmetric, asymptotically flat spacetimes, any solution to the coupled Einstein–nonlinear electromagnetic field equations is uniquely isometric to the constructed charged-particle spacetime (with vanishing magnetic charge), generalizing the standard Birkhoff theorem (Tahvildar-Zadeh, 2010).
• Charge Discreteness: In certain nonlinear models, the force between two point charges vanishes only when their charges are equal; this leads to the proposal that elementary point charges exist only in $2^n$ multiples of a fundamental charge—introducing a new mechanism for charge quantization (Breev et al., 2016).

6. Observational and Theoretical Consequences

Finite-energy point charge models have implications for both experimental and theoretical physics:

• Electron Mass and Radius Estimates: The regularized field energy can be linked to electron mass, with parameters chosen to reproduce the observed mass when the radius is set at the order of $10^{-13}$ cm or smaller, depending on the theory (Costa et al., 2013, Kruglov, 25 Jan 2024, Liu et al., 5 Sep 2025).
• Corrections to Coulomb's Law: The nonlinear theories generically yield $1/r^5$ corrections to the static interparticle potential, and these corrections may, in principle, be experimentally detected in high-precision measurements at small distances (Gaete et al., 2013, Gaete et al., 2014).
• Consistency with High-Energy and Gravitational Theories: The absence of singularities enhances the consistency of these classical models with candidate quantum gravity scenarios and suggests new approaches to regularization without arbitrary counterterms (Amorim, 8 Aug 2025, Gera et al., 2021).
• Topological Interpretation: In certain first-order gravity models, the electric charge is identified with a boundary correction to the Euler characteristic, relating gauge charges to topological indices (Gera et al., 2021).

These features represent both a conceptual advance (by resolving the classical divergence and mass renormalization puzzles) and provide new avenues for connecting field theory, spacetime structure, and quantum models.

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In summary, finite-energy electric point charge models are realized in nonlinear and higher-derivative extensions of Maxwell's electrodynamics, often coupled to general relativity. They feature mild (usually conical) or even regular spacetime singularities, bounded or softened electric fields, unique geometric structures, and—unlike classical Maxwell theory—finite electromagnetic self-energy that can be linked to the observable mass of elementary charges. The mathematical formulation centers on imposing appropriate growth, convexity, and energy conditions on the Lagrangian; the resulting physical implications include topological charge, electromagnetic asymmetry excluding monopoles, and modified interaction potentials—all testable or illustratable in diverse physical and mathematical contexts (Tahvildar-Zadeh, 2010, Costa et al., 2013, Gaete et al., 2013, Gaete et al., 2014, Barone et al., 2015, Gitman et al., 2015, Breev et al., 2016, Kruglov, 25 Jan 2024, Amorim, 8 Aug 2025, Liu et al., 5 Sep 2025).