Derivation of the Law of Universal Gravitation from the MOC Unified Curvature Extremum Principle

Author: Zhang Suhang (Luoyang)

1. MOC Unified Curvature Extremum Principle (Postulate)

Let the spacetime manifold be equipped with a unified curvature scalar \mathcal{R}_{\text{total}}, composed of the gravitational curvature scalar R_{\text{grav}} and the gauge field curvature contribution R_{\text{gauge}}:

\mathcal{R}_{\text{total}} = R_{\text{grav}} + R_{\text{gauge}}

The MOC extremum principle states: physically permissible field configurations render the spacetime integral of the total curvature scalar stationary:

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

The variation is performed over all degrees of freedom (metric, gauge fields, matter fields), with no additional assumptions imposed on the form of the action.

 

2. Projection onto the Pure Gravitational Submanifold (Gauge Fields Vanish)

In regions governed solely by gravitational interaction (e.g., the vacuum exterior of a celestial body, where electromagnetic, weak, and strong forces are negligible), the gauge curvature contribution vanishes: R_{\text{gauge}} = 0. This gives:

\mathcal{R}_{\text{total}} = R_{\text{grav}} \equiv R

where R denotes the Riemannian curvature scalar of spacetime. The MOC extremum principle reduces to:

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

 

3. Variational Derivation of the Einstein Field Equations

Varying the above functional with respect to the metric g_{\mu\nu} yields via standard calculation:

\int \left( R_{\mu\nu} - \frac12 g_{\mu\nu}R \right) \delta g^{\mu\nu} \sqrt{-g}\, d^4x = 0

Including the contribution of the matter field action, the term 8\pi G\,\delta S_m/\delta g^{\mu\nu} is introduced on the right-hand side, leading to the final form:

\boxed{R_{\mu\nu} - \frac12 g_{\mu\nu}R = 8\pi G\, T_{\mu\nu}}

This is the Einstein field equation. Here G is Newton’s gravitational constant, emerging naturally from the coupling scale with the matter field.

 

4. Vacuum Spherically Symmetric Static Solution (Schwarzschild Metric)

In the exterior vacuum of a celestial body, T_{\mu\nu}=0. Imposing spherical symmetry and staticity, the Schwarzschild metric is the exact solution to the field equations:

ds^2 = -\left(1-\frac{2GM}{c^2 r}\right)c^2 dt^2
+ \left(1-\frac{2GM}{c^2 r}\right)^{-1} dr^2
+ r^2 d\Omega^2

 

5. Weak-Field Low-Velocity Newtonian Limit

Under the weak-field condition \dfrac{2GM}{c^2 r} \ll 1 and for particle velocities much smaller than the speed of light, the temporal component of the metric approximates to:

g_{00} \approx -\left(1+\frac{2\phi}{c^2}\right),\quad \phi(r) = -\frac{GM}{r}

where \phi(r) is the Newtonian gravitational potential.

 

6. Derivation of the Law of Universal Gravitation

In the Newtonian limit, the acceleration of a test particle in the gravitational field reads:

\boldsymbol{a} = -\nabla\phi = -\frac{d\phi}{dr}\hat{\boldsymbol{r}} = -\frac{GM}{r^2}\hat{\boldsymbol{r}}

The gravitational force acting on a test mass m is:

\boldsymbol{F} = m\boldsymbol{a} = -\frac{GMm}{r^2}\hat{\boldsymbol{r}}

Taking the magnitude yields the Law of Universal Gravitation:

\boxed{F = G\frac{Mm}{r^2}}

 

Summary: Complete Logical Chain

\begin{aligned}
&\text{MOC Extremum Principle: } \delta \int \mathcal{R}_{\text{total}} \sqrt{-g}\,d^4x = 0 \\
&\quad \xrightarrow{\text{Pure Gravitational Submanifold Projection}} \delta \int R \sqrt{-g}\,d^4x = 0 \\
&\quad \xrightarrow{\text{Variation}} \text{Einstein Field Equations} \\
&\quad \xrightarrow{\text{Vacuum Spherically Symmetric Static Solution}} \text{Schwarzschild Metric} \\
&\quad \xrightarrow{\text{Weak-Field Low-Velocity Limit}} \text{Newtonian Potential } \phi = -GM/r \\
&\quad \xrightarrow{\text{Acceleration } a = -\nabla\phi} \text{Law of Universal Gravitation } F = G\frac{Mm}{r^2}
\end{aligned}