Playground
Two analog clocks: moving frame (proper time) and stationary frame (dilated time). Adjust v/c to see the lab clock slow.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Dilated Timeoutput Time interval measured in the observer frame | s | T | 0 – 100 | |
| Proper Time Time interval in the clock's rest frame | s | T | 0 – 10 | |
| Velocity Relative speed of the clock | m/s | LT^-1 | 0 – 299792458 |
Deep dive
Derivation
Consider a light clock: a photon bouncing between mirrors separated by L. In the rest frame period is Δt0 = 2L/c. In a frame where the clock moves with velocity v, the photon traverses a diagonal of length sqrt(L^2 + (vΔt/2)^2), giving c·Δt/2 = sqrt(L^2 + (vΔt/2)^2). Solving yields Δt = 2L/(c·sqrt(1 - v^2/c^2)) = gamma·Δt0.
Experimental verification
Hafele–Keating (1971) atomic-clock flights, muon lifetime in storage rings (CERN, 1977) confirmed dilation to 0.2%, and GPS satellites correct for ~38 μs/day.
Common misconceptions
- Time dilation is reciprocal — each observer sees the other's clock as slow
- The effect applies to every physical process, not just clocks
- The 'twin paradox' is resolved by the traveling twin's acceleration
Real-world applications
- GPS satellite timing corrections
- Muon lifetime extension in cosmic rays
- Atomic clock comparisons on aircraft
- Particle collider beam lifetimes
Worked examples
Cosmic-ray muon
Given:
- Delta_t0:
- 0.0000022
- v:
- 296794533
Find: Delta_t
Solution
Δt = 7.09 × 2.2e-6 s ≈ 1.56e-5 s; travels v·Δt ≈ 4.63 km.
ISS astronaut
Given:
- Delta_t0:
- 31557600
- v:
- 7660
Find: Delta_t - Delta_t0
Solution
Over one year, ISS clock lags Earth by ~0.01 s.
Scenarios
What if…
- scenario:
- What if the clock is a biological heart?
- answer:
- Heartbeats slow in the observer frame by exactly gamma; the moving person ages slower as measured externally.
- scenario:
- What if v = 0.5c?
- answer:
- gamma ≈ 1.155; a 1-second proper interval becomes 1.155 s in the lab.
- scenario:
- What if both twins are inertial?
- answer:
- Neither twin ages differently permanently — the asymmetry requires one to accelerate and return.
Limiting cases
- condition:
- v → 0
- result:
- Δt → Δt0
- explanation:
- Classical limit: no dilation at rest.
- condition:
- v → c
- result:
- Δt → ∞
- explanation:
- A photon's clock would not tick at all.
- condition:
- v = 0.866c
- result:
- Δt = 2Δt0
- explanation:
- Moving clocks run at half speed.
Context
Albert Einstein · 1905
Einstein derived time dilation in his 1905 'On the Electrodynamics of Moving Bodies' paper from the two postulates of special relativity.
Hook
A muon lives 2.2 μs at rest — how far can it travel at 0.99c before decaying?
In the lab frame the muon's lifetime is gamma·Δt0. With gamma ≈ 7.09 and v = 0.99c, compute the lab-frame distance L = v·gamma·Δt0.
Dimensions:
- lhs:
- Δt → [T]
- rhs:
- gamma·Δt0 → [1]·[T] = [T]
- check:
- Both sides [T]. ✓
Validity: Valid for inertial frames and for any proper time interval. For accelerated clocks, integrate dtau along the worldline.