DFT-16c: The Newtonian Gravitational Potential as a Weak-Curvature Projection

MWells · Post by **MWells** » Mon Dec 08, 2025 10:24 pm

The case for gravity in DFT does not begin with force laws or particle exchange.
It begins from the structural rule of Dual-Frame Theory:

A scalar progression in the Natural Reference System appears, when projected, as both curvature and energy.

When the curvature produced by a bound motion is extremely weak, the projection into the S-frame does not lead to relativistic corrections or nonlinear field behavior.
Instead, it appears as a classical scalar potential.

In this regime, Newton’s inverse-square law is not asserted.
It is derived from spherical projection geometry.

1. Geometry of Weak Curvature

Let the T-frame curvature around a localized source be represented by a scalar function of radius:

$\kappa = \kappa(r)$

In the weak-field limit the curvature is small everywhere, and higher-order nonlinearities are negligible.
The S-frame potential is the integral of curvature with respect to radius:

$\Phi(r) \propto \int_{r}^{\infty} \kappa(r') \, dr'$

This expresses the fact that curvature accumulates in projection.

2. Radial Flux Conservation

Spherical symmetry imposes a purely geometric condition.
The radial flux of curvature through a sphere of radius r is

$F(r) = 4 \pi r^{2} \kappa(r)$

In the weak-field regime the curvature does not distort itself, and therefore

$4 \pi r^{2} \kappa(r) = \text{constant}$

Let that constant be 4 π κ₀:

$4 \pi r^{2} \kappa(r) = 4 \pi \kappa_{0}$

Canceling 4 π gives:

$\kappa(r) = \frac{\kappa_{0}}{r^{2}}$

Thus the curvature profile forced by spherical geometry alone is 1 / r².

This required no “law of gravity” input — it is a projection identity.

3. Projecting Curvature into a Potential

The S-frame potential is the integral of curvature:

$\Phi(r) \propto \int_{r}^{\infty} \kappa(r') \, dr'$

Substitute the curvature form:

$\Phi(r) \propto \int_{r}^{\infty} \frac{\kappa_{0}}{r'^{2}} \, dr'$

This evaluates to:

$\Phi(r) \propto \frac{\kappa_{0}}{r}$

Including the conventional sign:

$\Phi(r) = - \frac{\kappa_{0}}{r}$

Thus the Newtonian potential arises directly from curvature projection.

4. The Force Law

In the S-frame, force is the spatial gradient of potential:

$F(r) = - \frac{d\Phi}{dr}$

Differentiate the expression above:

$F(r) = - \frac{\kappa_{0}}{r^{2}}$

Therefore the force is inverse-square:

$F(r) \propto \frac{1}{r^{2}}$

No dynamics were assumed;
no graviton model was invoked;
the inverse-square law is purely geometric.

5. Identifying Curvature with Mass

The curvature strength κ₀ must scale with the source parameter physics calls mass.
Thus we identify:

$\kappa_{0} = G M$

Returning to the potential:

$\Phi(r) = - \frac{G M}{r}$

And the force becomes:

$F(r) = - \frac{G M}{r^{2}}$

These are exactly the Newtonian expressions obtained without assuming Newton’s theory.

6. Relation to General Relativity

General Relativity recovers the same results in the regime where curvature is small, static, and spherically symmetric.
In DFT this form arises from spherical projection, not from a metric field equation.

Thus:

DFT and GR agree in the weak-field limit
each uses a different language for curvature
the Newtonian form is the shared intersection

7. Why the Inverse-Square Law Matters

In textbooks, the inverse-square law is usually presented as empirical.
In this reconstruction it is:

a geometric necessity under projection,
independent of microscopic interpretation,
independent of dynamical or field-theoretic assumptions.

It follows only from:

spherical symmetry
curvature projection
flux conservation

8. Where Newton Breaks Down

Newton’s form is exact only when curvature is weak.
In strong curvature regimes:

κ(r) is no longer κ₀ / r²
potential is no longer -GM / r
Newton becomes the first term in a broader expansion

This explains precisely why Newtonian gravity is correct where it is, and incomplete where it fails.

Closing Summary

Curvature in the T-frame projects into potential in the S-frame.
Spherical projection forces curvature to fall as 1 / r².
Integrating yields a 1 / r potential.
Identifying curvature amplitude with GM reproduces Newton exactly.
No force postulate is needed; the law is geometric.

Thus, the Newtonian gravitational law is not assumed; it is derived from the projection structure of scalar curvature in DFT.

Appendix — Weak-Curvature Derivation of the Newtonian Form

The purpose of this appendix is to record a minimal reproducible derivation of the Newtonian form of gravity from the projection of weak curvature in Dual-Frame Theory.

This procedure contains no assumptions of forces, no differential equations of motion, and no metric ansatz.
Everything follows from the geometry of radial projection.

A. Weak Curvature as a Scalar Field

We begin with a radially symmetric curvature field in the T-frame:

$\kappa = \kappa(r)$

The S-frame energy potential is proportional to the radial integral of curvature:

$\Phi(r) \propto \int_{r}^{\infty} \kappa(r') \, dr'$

This expresses projection: potential is accumulated curvature.

B. Radial Flux Conservation

Curvature through a spherical surface of radius r is

$F(r) = 4 \pi r^{2} \kappa(r)$

In the weak-field regime curvature does not self-distort, so the flux is constant. Conservation follows because no curvature flows into or out of the spherical surface:

$4 \pi r^{2} \kappa(r) = 4 \pi \kappa_{0}$

Therefore

$\kappa(r) = \frac{\kappa_{0}}{r^{2}}$

This is purely geometric, not dynamical.

C. Projection into an S-Frame Potential

Project curvature into the potential:

$\Phi(r) \propto \int_{r}^{\infty} \frac{\kappa_{0}}{r'^{2}} \, dr'$

This integrates to

$\Phi(r) \propto \frac{\kappa_{0}}{r}$

Introducing the conventional sign:

$\Phi(r) = - \frac{\kappa_{0}}{r}$

D. Gravitational Force as Gradient of Potential

The force is the spatial derivative of potential:

$F(r) = - \frac{d\Phi}{dr}$

Differentiate:

$F(r) = - \frac{\kappa_{0}}{r^{2}}$

Thus the inverse-square law is recovered directly.

No inverse-square assumption was made.

E. Identifying Curvature Strength with Mass

To connect to physical units we set

$\kappa_{0} = G M$

Then the potential becomes

$\Phi(r) = - \frac{G M}{r}$

and the force becomes

$F(r) = - \frac{G M}{r^{2}}$

These are exactly the Newtonian expressions.

F. Where This Is Valid

The derivation is valid when curvature is:

small everywhere,
static,
spherically symmetric.

These are precisely the conditions under which Newtonian gravity is accurate.

G. Interpretation

This appendix establishes that:

Newton’s law is a projection identity, not a fundamental postulate.
Inverse-square curvature is forced by spherical geometry.
The 1/r potential is forced by integrating curvature.
GM/r is the projection amplitude.

Nothing in this derivation depends on

dynamical field equations,
particle exchange,
or quantization assumptions.

Gravity is a geometric bookkeeping of curvature in the weak-projection limit.