“Analysis of the Isomorphism between Coulomb Force and Universal Gravitation – Based on the MOC Unified Curvature Framework” by Zhang Suhang (Bosley Zhang).

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Analysis of the Isomorphism between Coulomb Force and Universal Gravitation – Based on the MOC Unified Curvature Framework

Author: Zhang Suhang (Bosley Zhang)

Abstract

Within the framework of the revised Multi-Origin Unified Curvature (MOC) extremal principle, this paper compares and analyzes the mathematical structure and physical essence of the Coulomb electrostatic interaction and the universal gravitational interaction. Both forces are endogenously derived from the MOC unified curvature extremal principle. Under static spherically symmetric conditions, they exhibit an identical inverse-square law form. The core difference between them arises solely from the distinct degrees of freedom of their field sources: the gravitational field source (mass) is always positive, manifesting as purely attractive; the electromagnetic field source (electric charge) can be positive or negative, naturally yielding both attraction and repulsion. This paper demonstrates that the two forces possess a profound geometric isomorphism within the MOC framework, providing crucial support for a unified description of the four fundamental interactions.

Keywords: MOC unified curvature; Coulomb force; universal gravitation; inverse-square law; gradient force of a field; isomorphism

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I. Introduction

In classical physics, both the Coulomb force and universal gravitation are described by the inverse-square law, and the striking similarity in their mathematical forms has long been noted. However, their physical essences are placed in different theoretical frameworks: gravitation is described by the spacetime geometry of general relativity, while electromagnetism is described by the Maxwell gauge field theory. Based on the MOC unified curvature extremal principle, this paper traces both forces back to a single geometric origin and reveals the fundamental source of their isomorphism and differences.

II. Unified Derivation Path of the Two Forces in the MOC Framework

2.1 Core Postulate: The Revised MOC Unified Curvature Extremal Principle

The sole core postulate adopted in this paper is:

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

where the total curvature scalar \mathcal{R}_{\text{total}} includes both the gravitational curvature term of spacetime and the U(1) electromagnetic gauge curvature term. By projection onto submanifolds, gravitational and electromagnetic interactions can be derived separately.

2.2 Derivation of Universal Gravitation

Projecting the MOC principle onto the pure gravitational submanifold, ignoring gauge field contributions, the Hilbert action is recovered:

\delta \int R_{\text{grav}} \sqrt{-g} \, d^4x = 0

Variation with respect to the metric yields the Einstein field equations. The static spherically symmetric vacuum solution (Schwarzschild metric) naturally leads, in the weak-field slow-motion limit, to the Newtonian gravitational potential:

\phi_{\text{grav}}(r) = -\frac{GM}{r}

The force on a test particle is given by the negative gradient of the potential:

\boldsymbol{F}_{\text{grav}} = m\boldsymbol{a} = -m\nabla\phi_{\text{grav}} = -G\frac{Mm}{r^2}\hat{\boldsymbol{r}}

The direction of the force always points toward the source mass, manifesting as purely attractive universal gravitation.

2.3 Derivation of the Coulomb Force

Projecting the MOC principle onto the pure U(1) gauge submanifold in flat spacetime, ignoring gravitational and other gauge field contributions, the standard Maxwell action is recovered:

\delta \int \left(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\right) d^4x = 0

Variation with respect to the gauge potential yields the Maxwell tensor equations. Under static spherically symmetric conditions for a point charge, these reduce to the Poisson equation, whose analytic solution gives the electrostatic potential:

\phi_{\text{elec}}(r) = \frac{1}{4\pi\varepsilon_0}\frac{q}{r}

The force on a test charge is:

\boldsymbol{F}_{\text{elec}} = q_2\boldsymbol{E} = -q_2\nabla\phi_{\text{elec}} = \frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r^2}\hat{\boldsymbol{r}}

The direction of the force is determined by the sign of the product q_1q_2: like charges repel, opposite charges attract.

III. Isomorphism Analysis: Unity of Form and Essence

3.1 Isomorphism of Mathematical Forms

The mathematical expressions of the two forces share an identical structure:

· Universal gravitation: F_{\text{grav}} = G\frac{Mm}{r^2}
· Coulomb force: F_{\text{elec}} = k\frac{q_1q_2}{r^2}

Both obey the inverse-square law, both potentials are of the 1/r form, and both forces are the product of the negative gradient of the potential and the field source (mass/charge). This isomorphism is not coincidental; it directly reflects the fact that both are gradient forces of static spherically symmetric fields.

3.2 Homology of Physical Essence

Within the MOC framework, both forces originate from the same geometric extremal principle:

· Universal gravitation is the low-energy limit manifestation of the Riemannian curvature of spacetime.
· The Coulomb force is the static limit manifestation of the U(1) gauge field curvature.

The inverse-square structure of both forces is a natural consequence of flux conservation (Gauss’s law) in three-dimensional Euclidean space, and its geometric root is embedded in the overall structure of the MOC unified curvature.

IV. Fundamental Source of the Difference: Degrees of Freedom of the Field Sources

The core difference between the two forces arises solely from the nature of their field sources:

1. Gravitational field source (mass): mass is always non-negative, so the sign of the gravitational potential is fixed, and the force direction is always attractive; there is no degree of freedom for repulsion.
2. Electromagnetic field source (electric charge): charge can be either positive or negative, so the direction of the Coulomb force is determined by the combination of source signs, naturally yielding both attraction and repulsion.

Coulomb repulsion is not an independent or additional physical effect; it is simply the inevitable result of the charge degree of freedom within the inverse-square law framework. The derivation proceeds in complete isomorphism with that of gravitation, differing only by the additional freedom to choose the sign of the field source.

V. Conclusion

Within the MOC unified curvature framework, this paper rigorously demonstrates the profound isomorphism between the Coulomb force and universal gravitation: both are gradient forces of static spherically symmetric fields, both obey the inverse-square law, and both are endogenously derived from the same geometric extremal principle. The core difference between them lies solely in the degrees of freedom of their field sources: mass is always positive, leading to purely attractive gravitation; charge can be positive or negative, leading to a Coulomb force that includes both attraction and repulsion.

This conclusion further supports the central position of MOC theory as a unified field theory: the formal differences among the four fundamental interactions can be interpreted as manifestations of the same unified curvature projected onto different submanifolds, and the isomorphism between the Coulomb force and universal gravitation provides key evidence for this unified picture.

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