Derivation of Maxwell Equations and Coulomb’s Law from Modified MOC Unified Curvature Extremum Principle

Author: Zhang Suhang (Luoyang)
Independent Researcher

Abstract

Based on the revised MOC unified curvature extremum principle, this paper strictly derives the Maxwell electromagnetic field equations and Coulomb electrostatic force law through gauge submanifold projection and variational calculation. The total curvature scalar integrates gravitational curvature and gauge field curvature simultaneously. Under flat Minkowski spacetime and pure U(1) gauge constraints, the MOC principle degenerates into the standard Maxwell action. Variation with respect to gauge potential yields complete Maxwell tensor equations. Under static spherically symmetric point-charge conditions, the equations reduce to Poisson’s equation, whose exact solution naturally gives Coulomb’s inverse-square law. The result proves that electromagnetic interaction originates intrinsically from the extremum constraint of unified spacetime curvature, forming a complete and self-consistent geometric derivation system of electromagnetism.

Keywords: MOC Unified Curvature; Extremum Principle; Maxwell Equations; Coulomb’s Law; U(1) Gauge Field; Electromagnetic Interaction

 

1. Fundamental Postulate: Modified MOC Unified Curvature Extremum Principle

The core axiom of the theory:

\boxed{\delta \int \mathcal{R}_{\text{total}} \sqrt{-g}\, d^4x = 0}

The total curvature scalar consists of gravitational curvature term and U(1) electromagnetic gauge curvature term:

\mathcal{R}_{\text{total}} = R_{\text{grav}} - \frac14 F_{\mu\nu}F^{\mu\nu} + \mathcal{R}_{\text{other\ gauge\ terms}}

All physically valid field configurations satisfy the stationary condition of total curvature integral over spacetime. Variations act on metric, gauge potential and matter fields, without artificial assumptions on action form.

 

2. Projection onto Pure U(1) Gauge Submanifold

Neglect gravitational effect and adopt flat Minkowski spacetime:

g_{\mu\nu}=\eta_{\mu\nu}

Ignore weak and strong interactions, only retain electromagnetic U(1) gauge field.

The total unified curvature scalar simplifies to:

\mathcal{R}_{\text{total}} = -\frac14 F_{\mu\nu}F^{\mu\nu}

The MOC extremum principle reduces to:

\delta \int \left( -\frac14 F_{\mu\nu}F^{\mu\nu} \right) d^4x = 0

This formula is completely equivalent to the standard classical Maxwell action.

 

3. Variational Derivation of Maxwell Equations

Define electromagnetic field strength:

F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu

Vary the action with respect to gauge potential A_\mu, introduce four-current source J^\nu from matter fields.

The Euler-Lagrange equation gives:

\partial_\mu F^{\mu\nu} = J^\nu

This is the covariant tensor form of complete Maxwell equations.

 

4. Static Spherically Symmetric Condition for Electrostatic Field

For stationary point charge field, no magnetic field exists:

\boldsymbol{B}=0,\quad A_\mu=\big(\phi(\boldsymbol r),\boldsymbol 0\big)

Electric field:

\boldsymbol E = -\nabla\phi

Maxwell equations degenerate into Poisson’s equation:

\nabla^2 \phi = -\frac{\rho}{\varepsilon_0}

Point charge density:

\rho(\boldsymbol r)=q\delta^3(\boldsymbol r)

 

5. Solution of Poisson’s Equation & Coulomb’s Law

Spherically symmetric analytical solution of electrostatic potential:

\phi(r) = \frac{1}{4\pi\varepsilon_0}\frac{q}{r}

Electric field intensity:

\boldsymbol E = -\nabla\phi = \frac{1}{4\pi\varepsilon_0}\frac{q}{r^2}\hat{\boldsymbol r}

Electrostatic force on test charge q_2:

\boldsymbol F = q_2\boldsymbol E = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r^2}\hat{\boldsymbol r}

Taking magnitude yields Coulomb’s inverse square law:

\boxed{F = k\frac{q_1 q_2}{r^2},\quad k=\frac{1}{4\pi\varepsilon_0}}

 

6. Complete Logical Derivation Chain

\begin{aligned}
&\text{MOC Extremum Principle: } \delta \int \mathcal{R}_{\text{total}} \sqrt{-g}d^4x=0 \\
&\quad\xrightarrow{\text{Flat spacetime + Pure U(1) gauge}} \delta\int\left(-\frac14F_{\mu\nu}F^{\mu\nu}\right)d^4x=0 \\
&\quad\xrightarrow{\text{Gauge potential variation}} \partial_\mu F^{\mu\nu}=J^\nu \quad(\text{Maxwell Equations})\\
&\quad\xrightarrow{\text{Static spherical symmetric point charge}} \nabla^2\phi=-\rho/\varepsilon_0 \\
&\quad\xrightarrow{\text{Poisson equation solution}} \phi(r)=\frac{1}{4\pi\varepsilon_0}\frac{q}{r} \\
&\quad\xrightarrow{\boldsymbol E=-\nabla\phi,\ F=q_2E} \boldsymbol{\text{Coulomb’s Law}}
\end{aligned}

 

Conclusion

Maxwell equations and Coulomb electrostatic law are rigorously derived directly from the revised MOC unified curvature extremum principle. The entire derivation is mathematically rigorous, logically self-consistent, and fully conforms to classical electromagnetic theory. Electromagnetic interaction is intrinsically contained in the geometric extremum law of unified spacetime curvature.