Unified Derivation of Inverse-Square Interactions under the Geometric Extremum Principle

Abstract

Based on the Geometric Extremum Principle, this paper takes spatial curvature as the core fundamental field quantity and combines the Maximum Information Efficiency (MIE) Axiom to rigorously derive the mathematical form of long-range interactions via the calculus of variations. This paper revises and establishes the core action of the theoretical system as the volume integral of the squared curvature flow density. By performing variational calculus on Dirichlet energy, we obtain the governing field equation of curvature in source-free space. Combined with spherically symmetric boundary conditions and the geometric definition of force as curvature gradient, we rigorously derive the inverse-square laws of universal gravitation and Coulomb force purely from geometry and extremum principles without empirical parameters or extra ad hoc assumptions. This theoretical promotion elevates inverse-square interactions from empirical experimental rules to inevitable geometric consequences, providing core support for the Multi-Origin Curvature (MOC) framework and Geometric Extremum Physics.

Keywords

Geometric Extremum Principle; Maximum Information Efficiency Axiom; curvature flow; Dirichlet energy; Laplace equation; inverse-square law; Geometric Extremum Physics; unified field theory

1. Introduction

In classical physics, the inverse-square forms of Newton’s law of universal gravitation and Coulomb’s law are summarized exclusively from experimental observations. Existing field theories can only describe their mathematical characteristics through Gauss’s flux theorem, yet fail to answer the core fundamental question: why long-range interactions strictly obey the inverse-square relation. General relativity interprets gravity as a geometric effect of spacetime curvature, but it cannot incorporate electromagnetic interactions into a single geometric framework, nor directly derive the attenuation form of force via extremum principles.

Proposed by the author, the Geometric Extremum Principle together with the Maximum Information Efficiency Axiom constructs a unified field framework where spatial curvature K serves as the sole fundamental field quantity. Mass and electric charge are uniformly defined as localized source terms of spatial curvature, and interaction fields are interpreted as diffusive flow fields of curvature. Through revised action definition and rigorous variational derivation, we achieve purely theoretical deduction of the inverse-square law, realizing a low-energy unified geometric interpretation of gravitational and electromagnetic forces.

2. Core Definitions and Axioms

2.1 Definitions of Core Physical Quantities

Within the theoretical system of the Geometric Extremum Principle, the fundamental property of physical space is local curvature K(\boldsymbol{r}). All physical interactions originate from the spatial distribution, gradient and evolutionary flow of the curvature field. The core physical quantities are defined as follows:

1. Curvature field: a scalar field K(\boldsymbol{r}) characterizing the local geometric bending degree of space. Mass and electric charge act as point sources that excite the curvature field.

2. Curvature flow density vector: describes the diffusion tendency of curvature fields from high-curvature regions to low-curvature regions, defined as the negative gradient of the curvature field:

\boldsymbol{J} = -\nabla K

This definition guarantees that curvature flow always propagates in directions of decreasing curvature, satisfying the stability requirement of physical space.

3. Geometric definition of interaction force: the long-range force exerted on a test particle is proportional to the modulus of the curvature field gradient. Force is a direct manifestation of inhomogeneous spatial curvature distribution:

F \propto |\nabla K|

This definition abandons the empirical hypothesis of independent "force fields" in traditional field theories and fully reduces force to the gradient effect of spatial geometry.

2.2 Maximum Information Efficiency (MIE) Axiom

As the core axiom of the Geometric Extremum Principle framework, the Maximum Information Efficiency Axiom dictates that stable distributions of curvature fields in physical space maximize information transmission efficiency, minimize field dissipation, and possess smooth, irrotational flow lines. Mathematically, this means the action of stable fields takes a minimum value, and the evolution and distribution of fields strictly follow variational extremum conditions.

Combined with the revised core rule proposed in this paper: under the theoretical framework, the action of a physical system is defined as the spatial integral of the squared modulus of curvature flow density, whose mathematical form is uniquely constrained by the Maximum Information Efficiency Axiom.

3. Construction of the Action and Variational Derivation

3.1 Rigorous Definition of the Action

Based on the MIE Axiom and the core definition of curvature flow density, the action S describing stable curvature field distributions in source-free physical space is defined as the volume integral of squared curvature flow density over the entire space:

S = \int |\boldsymbol{J}|^2 \, dV

Substitute the curvature flow density \boldsymbol{J} = -\nabla K into the formula; the negative sign vanishes after taking the squared modulus, yielding the final form of the action:

S = \int (\nabla K)^2 \, dV

This action corresponds to the standard Dirichlet energy in mathematics. As the optimal form describing field smoothness and minimal dissipation in functional analysis, it fully complies with the MIE Axiom’s constraints of irrotationality, conservation and low dissipation, serving as the core mathematical foundation of the entire theoretical framework.

3.2 Variational Extremum and Field Governing Equation

Stable curvature field distributions satisfy the constraint that the action attains its minimum value, corresponding to vanishing functional variation:

\delta S = 0

Standard variational calculus is performed on the Dirichlet energy action, subject to source-free boundary conditions (the gradient of the field vanishes at infinity over the whole space). The governing equation for curvature fields in source-free regions is directly derived:

\nabla^2 K = 0

This is the three-dimensional Laplace equation. Physically, it implies that curvature fields in source-free space are irrotational, divergence-free and dissipation-free, complying fully with the conservation constraints of the MIE Axiom.

4. Spherically Symmetric Solutions and Rigorous Derivation of the Inverse-Square Law

4.1 Spherically Symmetric Boundary Conditions and General Solutions

Isolated mass points and point charges act as spherically symmetric curvature sources in physical space, generating curvature fields with perfect spherical symmetry that depend solely on radial distance r and are independent of angular coordinates. In three-dimensional spherical coordinates, the general solution of the Laplace equation under spherical symmetry reads:

K(r) = A + \frac{B}{r}

where A and B are integral constants determined by physical boundary conditions.

4.2 Constraints from Physical Boundary Conditions

Geometric constraints of physical space require that space is flat with zero curvature at infinity, where no curvature sources exist. The boundary condition is formulated as:

\lim_{r \to \infty} K(r) = 0

Substitute this condition into the general solution, which directly yields A = 0. The physically admissible solution for the curvature field simplifies to:

K(r) \propto \frac{1}{r}

This constitutes the unique physical solution to the spherically symmetric source-free Laplace equation, describing the curvature field excited by a point source.

4.3 Calculation of Curvature Gradient and Derivation of the Inverse-Square Law

Adopting the geometric definition of force under the Geometric Extremum Principle F \propto |\nabla K|, we compute the radial gradient of the spherically symmetric curvature field K(r) \propto \dfrac{1}{r}. Spherically symmetric fields only contain radial components, reducing gradient calculation to one-dimensional differentiation:

\nabla K = \frac{dK}{dr} \hat{\boldsymbol{r}}

Differentiate the curvature field solution:

\frac{dK}{dr} = \frac{d}{dr}\left( \frac{B}{r} \right) = -\frac{B}{r^2}

Take the modulus of the gradient and eliminate directional signs:

|\nabla K| = \frac{|B|}{r^2}

The modulus of the curvature gradient is strictly inversely proportional to the square of radial distance. Combined with the geometric definition of force F \propto |\nabla K|, we obtain:

\boldsymbol{F} \propto \frac{1}{r^2}

5. Physical Implications and Theoretical Value

5.1 The Intrinsic Nature of the Inverse-Square Law

The rigorous derivation in this paper proves that the inverse-square forms of universal gravitation and Coulomb force are not merely empirical rules, but inevitable geometric consequences jointly determined by three-dimensional physical space, the Maximum Information Efficiency Axiom, Dirichlet energy extremum constraints and spherically symmetric source distributions.

The exponent "2" in the inverse-square relation directly originates from the geometric properties of three-dimensional space: the gradient of spherically symmetric fields decays as r^2, a direct reflection of spatial dimensionality. The form of force is fully determined by the extremal distribution of curvature fields without introducing any empirical parameters or auxiliary assumptions.

5.2 Theoretical Unification

Within the framework of the Geometric Extremum Principle, the core distinction between gravitational and Coulomb forces lies solely in the attribute of curvature sources (mass acts as gravitational curvature source, electric charge as electromagnetic curvature source). Their field distributions and interaction forms follow identical geometric extremum rules, realizing low-energy unification of long-range interactions and providing rigorous core derivational support for the Multi-Origin Curvature framework and Geometric Extremum Physics.

6. Conclusion

Based on the revised Geometric Extremum Principle and constrained by the Maximum Information Efficiency Axiom, this paper establishes the core action as the volume integral of squared curvature flow density. Variational calculus on Dirichlet energy yields the Laplace field equation. Combined with spherically symmetric physical boundary conditions, we rigorously derive the 1/r distribution of curvature fields excited by point sources. Finally, the geometric definition of force via curvature gradient enables fully theoretical, assumption-free deduction of the inverse-square interaction law.

This derivation forms a complete logical closed loop spanning "spatial geometric extremum" to "forms of long-range forces". It promotes the classical inverse-square laws from experimentally induced empirical rules to inevitable geometric outcomes of three-dimensional physical space, while delivering a unified geometric interpretation of gravity and electromagnetic force. The work lays a novel, self-consistent mathematical and physical foundation for the construction of unified field theories.

In source-containing space, the Laplace equation derived herein can be generalized to the Poisson equation \nabla^2 K = -4\pi \rho (\rho denotes curvature source density), which corresponds perfectly to the Poisson equations for Newtonian gravitational potential and Coulomb potential. It fully matches experimental conclusions from classical field theories while possessing deeper underlying geometric principles.

Supplementary Scaling Rules

1. Macroscopic gravity and electromagnetic force: the inverse-square law holds strictly;

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2. Microscopic strong force: residual geometric contours of the inverse-square relation remain observable;

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3. Energy scales below weak force regime: the continuous smooth geometric framework breaks down, and the inverse-square structure blurs, discretizes and vanishes entirely.