Mathematical Grounding II — The Cadence Lorentz
How γ Becomes the Rhythm of Motion
Last time we watched the rhythm of light itself — redshift stretching, blueshift compressing — and found that the two halves always balance.
That symmetry told us something subtle: motion doesn’t tear light’s timing apart; it only bends the geometry of its rhythm.
Now we need a way to measure how much that rhythm bends.
Einstein already gave us the tool; cadence geometry just turns it sideways.
The cadence form of the Lorentz factor
In ordinary relativity, the Lorentz factor
LaTeX: \gamma = \frac{1}{\sqrt{1 - (v/c)^2}}
tells how much a moving clock slows relative to one at rest.
In cadence geometry, it becomes the Cadence Lorentz Factor:
LaTeX: \gamma_{\text{cad}} = \frac{1}{\sqrt{1 - (v/c)^2}}
Here, γ is not “time slowing.”
It is rhythm tilting:
stretch behind (recede)
compress ahead (approach)
Light keeps its cadence slope constant:
LaTeX: C_0 = c^{-1} = 3.33564095\ \mathrm{ns,m^{-1}}
Every increase in speed is a deeper tilt of that rhythm.
The Light Frame flexes so the universe’s metronome never misses a beat.
The balance point (Optical Arc)
At γ = 2 (v ≈ 0.866c), the red and blue halves balance perfectly.
The ship’s internal time per leg is half the external mission time — the classical balance point.
LaTeX: t_{\text{Earth}} = \frac{D}{v}
LaTeX: \tau_{\text{Ship}} = \frac{t_{\text{Earth}}}{\gamma}
At γ = 2 → v ≈ 0.866c:
LaTeX: 2\tau_{\text{Ship}} = t_{\text{Earth}}
Two halves of one optical arc — the redshifted outbound and the blueshifted inbound — form a single cycle of cadence.
This is the heart of Optical Arc:
motion and light share one beat of the universe’s rhythm.
Cadence Lorentz
Idea:
Lorentz factor re-read as cadence tilt.
Rather than “time slows,” rhythm redistributes:
shrink ahead (approach), stretch behind (recede).
LaTeX: \gamma_{\text{cad}} = 1/\sqrt{1 - (v/c)^2}
Role: Bridges Einstein’s time-dilation term to cadence geometry.
What this means
At this midpoint speed, traveler and observer share one optical beat of the universe’s rhythm. Below it, the red half (the stretch side) dominates; above it, the blue half (the compression side) does.
But through it all, light’s cadence remains fixed — a single, unbroken measure.
In a closed 10-light-year frame, the geometry ends here — perfectly balanced.
In the next post, we widen the range and let the frame drift — where that balance begins to slide, and the universe itself joins the rhythm.