For uniform motion in flat spacetime, Δτ=T√(1−β²) with β=v/c. Both bars use the same time-unit scale, making the conditional relativistic comparison explicit.
Try the demonstration
β = v/c = 0.6; γ = 1.25. Over T = 10 time units, the moving ideal clock accumulates Δτ = 8. Difference T − Δτ = 2; rate factor √(1 − β²) = 0.8.
Exact constant-velocity proper time for an ideal clock in supplied flat spacetime. No acceleration, gravity, received-signal frequency, or source-derived universal clock law is simulated.
Set a speed and duration
The speed slider ranges from rest to 0.99c. The duration slider ranges from zero to twenty arbitrary time units. The dark bar shows inertial coordinate duration T, and the gold bar shows the moving clock's proper duration Δτ. Both use the same fixed zero-to-twenty scale.
The live summary also reports the Lorentz factor γ, the clock-rate factor, and the accumulated difference T−Δτ. The duration can be zero without a numerical singularity: both bars then vanish, while the speed-dependent rate factor remains well defined.
Derive the proper-time formula
Use a flat spacetime interval with a calibrated invariant speed c. Along a massive clock moving at constant v, the spatial displacement is dx=vdt. Its proper duration obeys:
Δτ=T√(1−β²)=T/γ, γ=(1−β²)−1/2.
The integral simplifies because the speed is constant. For a changing velocity, integrate √(1−v(t)²/c²) along the full path instead. The present controls do not model an acceleration profile, gravity, or a return journey.
A three-fifths-light-speed comparison
At β=0.6 and T=10, the rate factor is √(1−0.36)=0.8 and γ=1.25. The moving clock accumulates eight time units, giving a difference of two. At rest the factor is one and the bars agree. Increasing β reduces the moving clock's proper duration for the same coordinate interval.
This compares readings assigned using an inertial timing procedure. It is not the frequency an observer directly sees through arriving light pulses: received frequencies also include propagation and Doppler effects. Those require their own signal-exchange calculation.
An ideal clock is a physical assumption
A material clock can have extra responses to temperature, electromagnetic fields, deformation, or acceleration. The geometric formula describes its ideal operating regime after such effects are accounted for. Local processes remain regular when described by the moving clock's own proper time.
A recurrent carrier could record this duration if its unwrapped relative phase satisfies dφ=ω₀dτ with a calibrated rest frequency. That coupling is the required physical bridge. An arbitrary oscillator might instead respond to another source parameter, so its existence alone does not establish a proper-time law.
The SMM/DTT common-clock target
SMM motivates a common origin for apparatus and matter, while DTT separates the inner work maintaining them from their completed observations. A source derivation should show why distinct stable clocks respond to the same proper-time integral and why their rulers and signals share the same calibrated metric.
The six inner roles remain nested dependencies, not additional durations added to Δτ. The seventh role makes a clock record available and must retain enough information to count complete phase cycles. This demo supplies a relativistic benchmark for that programme rather than deriving it from source ontology. Reset restores the ten-unit, 0.6c example, also shown as the static fallback.
Continue with the articles
- A protected material clock
An encoded two-qubit clock can ignore branch activity while responding to resolution. A controlled coupling mismatch produces a calculable failure, making both the protection and its assumptions testable.
- Nested operations and clock independence
Clock independence must survive every admitted interleaving of branch and resolution operations. The orbit-algebra criterion removes branch effects from all such finite sequences and supplies an accumulated-error bound.
- Calibrating a conformal geometry
Causal order fixes a conformal class of metrics; an independently calibrated record density fixes the local scale. A reproducible clock then sets any remaining overall unit, provided the source supports the required protocol agreement.