Derivation of Kepler’s Third Law, reframed strictly through the lens of Time and Space Geometry.

 Here is the derivation of Kepler’s Third Law, reframed strictly through the lens of Time and Space Geometry.

In this view, we discard the concepts of "force," "acceleration," and "centrifugal effects." Instead, we view an orbit as a static object—a "rail" or geodesic—that exists within the curved geometry of spacetime. The planet is not fighting a force; it is simply following the straightest possible path through a distorted medium.

1. The Geometry of Time (The Potential)

In General Relativity, mass does not generate a force field; it generates a curvature in spacetime. The most dominant aspect of this curvature for orbital mechanics is the Time Dilation Gradient.

Mass (

$M$

) warps the rate at which time flows. The deeper you are in the gravity well (the closer

$r$

is), the slower time ticks relative to a distant observer. This gradient in the flow of time acts as the "potential."

We define the Time Factor (Geometric Potential)

$\Phi$

at a distance

$r$

as:

$\Phi \propto \frac{M}{r}$

This

$1/r$

relationship is the geometric signature of a point mass in 3D space. It represents the "slope" of the time dimension at that location.

2. The Geodesic Condition (Velocity as a Projection)

In a force-based Newtonian universe, we balance forces to find velocity. In a geometric universe, velocity is a projection.

To travel in a circle at radius

$r$

, an object must traverse a specific amount of space per unit of its own proper time. Because time itself is curved by the mass

$M$

, the relationship between the spatial step and the temporal step is fixed by the geometry of the well.

For a circular geodesic (the "orbit rail"), the orbital velocity

$v$

is not a variable caused by a push; it is a geometric requirement. The "slope" of the time well (

$M/r$

) dictates precisely how fast one must move through space to "keep up" with the curvature of time.

The geometric relationship between the spatial velocity and the time gradient is:

$v^2=\frac{M}{r}$

Note: This is the invariant definition of a circular orbit in this geometry. It is not derived from

$F=ma$

; it is derived from the metric of spacetime.

3. The Orbital Period (Geometry of a Circle)

The Period (

$T$

) is a purely geometric ratio: the total length of the closed path (Circumference) divided by the rate of traversal (Velocity).

$T=\frac{\text{Circumference}}{\text{Velocity}}=\frac{2\pi r}{v}$

4. The Geometric Synthesis

Now we combine the geometric constraints to eliminate the intermediate variable

$v$

(the velocity), leaving only the quantities of the orbit (

$r$

) and the mass (

$M$

).

Start with the geometry of the period:

$T=\frac{2\pi r}{v}$

Square both sides to remove the root, preparing for the substitution of the velocity squared:

$T^2=\frac{4{\pi }^2{r}^2}{v^2}$

Now, apply the Geodesic Condition (Step 2). We substitute the geometric definition

$v^2=M/r$

:

$T^2=\frac{4{\pi }^2{r}^2}{\left(\frac{M}{r}\right)}$

Simplify the denominator:

$T^2=\frac{4{\pi }^2{r}^2\cdot r}{M}$

Resulting in the Pure Geometric Law of Periods:

$T^2=\frac{4{\pi }^2}{M}{r}^3$

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Summary

We have arrived at the exact same law as Newton, but the narrative has fundamentally shifted:

1. Newtonian View: Gravity is a force pulling inward; Centrifugal force pushes outward. The orbit is where they balance.
2. Geometric View: Mass creates a gradient in time (
   $M/r$
   ). The orbit is a geodesic "rail" where the spatial speed (
   $v$
   ) is determined by that time gradient. The period is just the time it takes to traverse that rail.

The

$r^3$

term arises mathematically because the geometry of the orbit links

$r$

linearly to the period (

$T\propto r/v$

), while the time-well geometry links the velocity to the square root of the radius (

$v\propto 1/\sqrt{r}$

). The interaction of these two geometric ratios creates the cubic relation.