Observations on a Mathematical Thread Linking Kinematic and Dynamic Ratios

J. Rogers, SE Ohio

Overview

When examining the fundamental equations of physics through the lens of unit-free ratios, an interesting mathematical consistency appears. This thread seems to connect the disparate worlds of Newtonian gravity, Special Relativity, and General Relativity. By stripping away the Jacobian scaling factors required for SI units, we can observe how these "different" laws may actually be expressions of a single geometric relationship.

1. The Proportionality of Force (

        $I_1 \cdot I_2$
      

)

If we consider a "natural" force as the interaction between two dimensionless densities, we find a curious simplification of Newton’s Law of Gravitation. In the substrate, where we deal with proportions of the total physical scale rather than kilograms and meters, "Force" can be expressed as:

        $$F_{natural} = I_1 \cdot I_2$$

Where

        $I$
      

is the unit-free ratio of mass to distance. In this form, the gravitational constant

        $G$
      

is no longer seen as a standalone parameter, but as the scaling factor required to translate this simple product into the misaligned units of the SI system.

2. The Identity of Velocity and Potential (

        $\beta^2 = 2I$
      

)

A particularly interesting result arises when we re-examine the classical relation for orbital or escape energy:

        $$\frac{1}{2}{v}^2=\frac{GM}{r}$$

If we express the velocity

        $v$
      

as a ratio of the maximum substrate flow (

        $\beta = v/c$
      

), the equation becomes:

        $$\frac{1}{2}\beta^2 c^2 = \frac{GM}{r}$$

Rearranging for

        $\beta^2$
      

:

        $${\beta }^2=\frac{2GM}{c^{2}r}$$

If we define

        $I$
      

as the "natural" mass-to-radius ratio (

        $m/r$
      

), we see that the term

        $\frac{2G}{c^2}$
      

acts as the unit-scaling bridge. In a unit-free environment, the relationship reduces to a striking identity:

        $${\beta }^2=2I$$

This suggests that, at the substrate level, the "Speed" of an object (

        $\beta^2$
      

) and the "Gravitational Potential" (

        $2I$

) are mathematically indistinguishable ratios.

3. The Unified Lorentz Factor (

        $\gamma$

)

The most notable part of this thread is found when we plug this identity into the Lorentz factor (

        $\gamma$

), which is the core of Special Relativity:

        $$\gamma =\frac{1}{\sqrt{1-{\beta }^2}}$$

By substituting

        $\beta^2 = 2I$
      

, we obtain:

        $$\gamma =\frac{1}{\sqrt{1-2I}}$$

This result is curious because it is identical to the formula for gravitational time dilation in a weak field (the Schwarzschild temporal component).

4. The Boundary Condition (

        $r = 2M$
      

)

A final interesting observation occurs at the mathematical limit of these formulas. In the substrate expression for time dilation, a singularity occurs when the denominator reaches zero. This happens when:

        $$1=2I$$

Substituting our definition of the natural ratio

        $I = m/r$
      

:

        $$1=2(m\mathrm{/}r)\hspace{0.25em} ⟹\hspace{0.25em} \mathbf{r}=\mathbf{2}\mathbf{m}$$

In the language of natural ratios, the point where "Speed" hits its limit (

        $\beta = 1$
      

) is the exact same point where the geometry hits the Schwarzschild radius (

        $r = 2m$
      

). This suggests that the "Speed of Light" limit and the "Black Hole" limit are the same geometric boundary, expressed through different coordinate lenses.

Conclusion

These derivations are interesting because they suggest that many of our "fundamental laws" are not independent discoveries, but different views of the same underlying tautology.

The thread reveals that:

• Force is a product of densities.

• Velocity squared is a measure of potential.

• Time dilation (

          $\gamma$
        

  ) is a geometric consequence of your position in that potential.

• The event horizon is the point where the unit-scaling ratio reaches unity.

The presence of constants like

        $c^2$
      

and

        $G$
      

in our standard formulas appears to be the "tax" paid for using instruments calibrated to different historical standards. When those calibrations are removed, we are left with a very simple, unit-free geometry where the orbit, the speed, and the time dilation are all functions of the same substrate proportion.