Mathematical Grounding III
Curved Paths and Temporal Depth
1 · Circular motion and cadence compression
A clock moving at constant speed v around a circular path of radius r is always accelerating, even though its speed never changes.
Each instant, its direction shifts. That continual curvature keeps its cadence slightly compressed relative to a clock resting at the center.
In special relativity, the moving clock’s rate is:
Latex: t’ / t = \sqrt{1 - (v^2 / c^2)}
The proper inward acceleration is:
Latex: a = v^2 / r
Circular motion is motion pretending to be gravity: curvature of motion produces the same cadence compression as curvature from mass.
2 · Gravitational equivalence
General relativity predicts the same effect from real gravity.
In weak fields, time slows in proportion to the gravitational potential.
Latex: \Delta t / t \approx \Phi / c^2, \qquad \Phi = -GM / r
If we match the orbital potential v^2/2 to the magnitude of the gravitational potential |Φ|, both formulas give the same cadence slowdown.
A clock held in orbit and a clock at rest near a surface differ by the same cadence shift.
The geometry of circular motion and the geometry of gravity match.
This equivalence is the mathematical face of Temporal Depth.
3 · Light Frame relations (updated S and C entries)
In modern cadence geometry, we express this using the canonical TD law:
S31 — Temporal Depth (TD) Law
Latex: C_{TD}(r) = C_0 \left( 1 - \Phi / c^2 \right)
This replaces the earlier GR-only form and matches cadence invariance.
C05 — Temporal Depth (Concept)
Continuous curvature (gravity or circular motion) compresses cadence — the same effect we call gravitational time dilation.
TD is inward cadence curvature: rhythm squeezes inward as curvature grows.
4 · The Light-Frame Ratio (LFR)
The Light-Frame Ratio compares the rates of outward uncurvature (TS) to inward curvature (TD):
Latex: LFR(r) = (dR_{TS}/d\theta) / (dR_{TD}/d\theta), \qquad d\theta = dt / C_0
Here C0 = 1/c is light’s cadence constant — the fixed rhythm all Light Frames preserve.
At LFR = 1, TS and TD balance: local geometry is cadence-flat.
LFR > 1 means outward stretch dominates (TS).
LFR < 1 means inward depth dominates (TD).
This is the Light Frame’s geometric signature of curvature.
5 · Example magnitudes
Let’s check two ordinary cases.
Earth’s surface:
GM/r² ≈ 9.8 m/s².
TD compression is roughly one part in a billion — clocks on the ground run about 10^-9 seconds per second slower than clocks in orbit.
Low-Earth orbit:
v ≈ 7.7 km/s.
Substituting into the SR slowdown above gives nearly the same value.
Circular motion and gravitational TD match to parts per billion.
Satellites confirm this symmetry every day.
6 · Interpretation
Circular acceleration and gravitational potential are two aspects of Temporal Depth.
Straight-line motion produces Temporal Stretch (TS).
Curvature converts outward stretch (TS) into inward depth (TD).
Together they maintain the constant rhythm of light:
Latex: C_0 = 1 / c
Near mass, cadence compresses (TD grows) while TS relaxes.
Farther out, TD relaxes and TS expands.
This exchange keeps the cadence beat invariant across every Light Frame.
7 · Outlook
The local symmetry — where TS and TD balance — defines rest.
In the next step we widen the frame and let cadence drift.
When TS grows across cosmic distances, the balance tilts, and the universe itself joins the rhythm.