C = T + M: Directional vs Symmetric Timing
This demo visualizes the CBF budget law \( C = T + M \) in action. Each particle carries an internal oscillator and a motion budget. You can switch between two clock rules: CBF (directional), which tilts the tick rate with the observer’s motion, and SR (symmetric), which uses the Lorentz factor from the SR relative speed.
Note: this demo focuses on local pacing only. To see how frames actually align their ticks across space, try the companion Beat-Matching Demo, which visualizes phase alignments and measured vs. predicted beat periods.
The Budget Law \(C = T + M\)
In CBF, every entity splits a per-tick budget: \(T\) for translation (motion, propagation) and \(M\) for maintenance (internal ticking, constraints). The effective oscillator frequency an observer sees is proportional to the available maintenance share. When motion demands more \(T\), less \(M\) is available, so the observed clock slows.
Key idea: timing effects are budget effects. Change the move/maintain split and you change the clock.
Two Timing Rules in This Demo
| Mode | Rule | Meaning |
|---|---|---|
| SR (symmetric) | \( \displaystyle \frac{d\tau}{dt}=\sqrt{1-w^2}, \quad w=\frac{v_p - v_\text{obs}}{1 - v_p v_\text{obs}} \) | Classic time dilation from special relativity based only on relative speed \(w\). Direction does not matter. |
| CBF (directional) | \( \displaystyle \frac{d\tau}{dt}= \max\!\big(0.01,\; 1 + v_\text{obs} - v_p\big) \) | Directional pacing: moving with the observer increases the tick rate, against decreases it. Encodes C = T + M as a simple tilt. |
The demo lets you toggle between these rules and watch how perceived clock rates change.
What to Look For
- Pulse aging: Each particle has its own internal pulse or tick counter. When the observer’s frame speed increases, the clocks in the lower-speed frames appear to age faster—their pulses accumulate more quickly. This visualizes how different frames advance through their budgets at different rates, providing an intuitive resolution of the twin paradox within CBF.
- Directional vs. symmetric timing: In CBF mode, the observer’s motion changes the budget split \( C = T + M \). More translation effort (T) leaves less maintenance time (M), stretching the local tick period. In SR mode, the pacing depends only on the Lorentz factor \( \sqrt{1 - w^2} \), making both directions equivalent.
- Observer comparison: Switching between CBF and SR shows that the asymmetric aging comes directly from the directional budget rule, not from relative speed alone.
- Want to see synchronization? This demo focuses on local pacing only. To see how frames actually align their ticks across space, try the companion Beat-Matching Demo, which visualizes phase alignments and measured vs. predicted beat periods.
Controls
| Control | Default | Effect |
|---|---|---|
| Clock rule | CBF (directional) | Switch between CBF and SR timing. |
| Observer speed \(v_\text{obs}\) | 0.30 c | Tilts rates in CBF mode, changes \(w\) in SR mode. |
| Particle speed (new) | 0.60 c | Sets \(v_p\) for newly added particles. |
| Rest frequency \(f_0\) (new) | 2.0 Hz | Base oscillator, scaled by the timing rule. |
| Scene speed | — | (If present) visual only, makes motion more noticeable. |
Tip: click the canvas to spawn a particle at the cursor with current “new particle” settings.
FAQ
Is this showing “time symmetry”?
No. This particular demo intentionally contrasts a directional pacing rule (CBF) with a symmetric one (SR). It is about how the budget split changes the observed clock rate, not about reciprocity proofs.
Where does the Lorentz factor appear?
In SR mode through \( \gamma(w)=1/\sqrt{1-w^2} \). The demo reports \(w\) and uses \( \sqrt{1-w^2} \) for the rate \(d\tau/dt\).
How does this tie back to C = T + M?
CBF says motion and internal ticking draw from the same budget. When relative conditions demand more translation \(T\), internal maintenance \(M\) is reduced, so the observed clock slows. The directional rule is a minimal way to make that split visible.