Time Dilation Calculator
Calculate relativistic time dilation using t' = t·γ = t/√(1−v²/c²).
What Is Time Dilation?
Special relativity predicts that time, length, and mass all change for objects moving at significant fractions of the speed of light c = 3×10⁸ m/s. This calculator addresses Time Dilation, one of the three relativistic effects. The Lorentz factor γ = 1/√(1−v²/c²) governs all relativistic effects — it equals 1 at rest and increases without limit as v → c.
At everyday speeds (v << c), γ ≈ 1 and relativistic effects are negligible. At v = 0.1c, γ = 1.005 (0.5% effect). At 0.9c, γ = 2.294. At 0.99c, γ = 7.089. At 0.9999c, γ = 70.7. These effects are real and measurable in particle accelerators, cosmic ray muons, and GPS satellites.
Experimental verification: muons created in the upper atmosphere by cosmic rays (v ≈ 0.998c, γ ≈ 15) reach Earth's surface. Their lifetime is 2.2 μs, giving a laboratory lifetime of 2.2 μs, but at that speed they should travel only 660 m before decaying — yet they travel 10+ km. Time dilation extends their apparent lifetime by γ ≈ 15, allowing them to reach ground level.
GPS satellites move at 14,000 km/h relative to Earth, causing time dilation of about 7 μs/day (clocks run slow). Gravitational time dilation (general relativity) adds 45 μs/day (clocks run fast). The net +38 μs/day is corrected in GPS software — without this, GPS accuracy would degrade by 10 km/day.
Formula Reference Table
| Solve For | Formula | Notes |
|---|---|---|
| Dilated time | t' = t/√(1−v²/c²) = γt | γ = Lorentz factor |
| Lorentz factor | γ = 1/√(1−β²) | β = v/c (0 to 1) |
| Proper time | t = t'/γ | Measured in moving frame |
| Velocity from times | v = c·√(1−(t/t')²) | From measured time dilation |
| At 0.9c | γ = 2.294 | Time runs 2.3× slower |
| At 0.99c | γ = 7.089 | Time runs 7× slower |
3 Worked Examples
Cosmic ray muon at v=0.998c (γ≈15). Proper lifetime τ=2.2 μs. Expected range?
- Dilated lifetime = γτ = 15×2.2 = 33 μs
- Range = v×t' = 0.998×3×10⁸×33×10⁻⁶ = 9,880 m ≈ 10 km
- Without dilation: 0.998×3×10⁸×2.2×10⁻⁶ = 659 m — muons wouldn't reach ground
LHC proton at 99.9999991% of c (β=0.99999991). Find γ.
- γ = 1/√(1−0.99999991²)
- γ = 1/√(1−0.9999998²) ≈ 1/√(1.8×10⁻⁸) ≈ 7,461
- Mass increase: m' = 7,461×m_p = 7,461×1.673×10⁻²⁷ = 1.249×10⁻²³ kg
- KE = (γ−1)m_p c² = 7,460 × 938.3 MeV = 7.0 TeV ✓
Spaceship travels to Proxima Centauri (4.24 ly) at v=0.8c. Trip time for traveler vs Earth?
- γ = 1/√(1−0.64) = 1/0.6 = 1.667
- Earth time: t = 4.24/0.8 = 5.30 years
- Traveler time: t₀ = t/γ = 5.30/1.667 = 3.18 years
Real-World Applications
Common Mistakes to Avoid
The formula uses β = v/c. Input velocity as a fraction of c. v = 0.9 means 0.9c = 2.7×10⁸ m/s. Using km/s directly without dividing by c gives wrong γ.
Relativistic 'mass' increase means more momentum p = γmv and more energy E = γmc² — not that the object physically gets heavier in the gravitational sense.
At v = 300 m/s (fast aircraft), γ − 1 ≈ 5×10⁻¹³. Effects are completely negligible. Only above ~10% of c do relativistic effects become practically significant.
Time dilation: moving clocks run slow (t' > t). Length contraction: moving objects appear short (L' < L). They go in opposite directions.
Only the length along the direction of motion contracts. Transverse dimensions are unchanged. A circular object moving at 0.9c appears as an ellipse (contracted along motion direction, unchanged transversely).
Frequently Asked Questions
Related Physics Calculators
Formula Explorer connections
Interpretation: This relationship connects measurements between inertial frames or converts mass and energy using special relativity. Assumption: Use velocities relative to the speed of light and distinguish proper from observed quantities. Most formulas assume inertial frames and neglect gravity.