Special Relativity Calculator | Time Dilation, Length Contraction & Velocity Addition
Compute relativistic effects at any velocity from 0 to c: Lorentz factor γ, time dilation Δt, length contraction L, relativistic momentum p, kinetic energy Ek, total energy E = mc², and relativistic velocity addition. Enter velocity as a fraction of c.
| Speed | β | γ | Time dilation (%) | Length contraction (%) |
|---|---|---|---|---|
| 0.1c | 0.1 | 1.005 | 0.5038% | 0.5013% |
| 0.5c | 0.5 | 1.1547 | 15.47% | 13.4% |
| 0.9c | 0.9 | 2.2942 | 129.4% | 56.41% |
| 0.99c | 0.99 | 7.0888 | 608.9% | 85.89% |
| 0.999c | 0.999 | 22.366 | 2137% | 95.53% |
| 0.9999c | 0.9999 | 70.712 | 6971% | 98.59% |
What Is the Special Relativity Calculator | Time Dilation, Length Contraction & Velocity Addition?
Special relativity, formulated by Einstein in 1905, describes how space and time transform between inertial frames. The key parameter is β = v/c. The Lorentz factor γ ≥ 1 quantifies all relativistic effects: time dilation (moving clocks run slow), length contraction (moving objects are shorter), and the increase in momentum and energy. At everyday speeds β is tiny and γ ≈ 1, recovering Newtonian mechanics. As β → 1, γ → ∞ and it would take infinite energy to reach c.
Formula
γ = 1/√(1−β²) · Δt = γ·Δt₀ · L = L₀/γ · p = γmv · KE = (γ−1)mc² · u' = (u+v)/(1+uv/c²)
How to Use
- 1
Set the velocity β = v/c using the slider (0.001–0.9999) or type the value directly.
- 2
Enter the proper time Δt₀ in seconds — the time measured by the moving clock.
- 3
Enter the rest length L₀ in metres — the length in the object's rest frame.
- 4
Enter the object mass in kilograms to compute relativistic momentum and energy.
- 5
Optionally enter a second velocity u (fraction of c) for relativistic velocity addition.
- 6
Click "Calculate Relativistic Effects" to see all seven outputs and step-by-step working.
- 7
Use presets — Muon decay, GPS, LHC proton — to explore real-world examples.
Adjust the β slider or type a velocity fraction directly. Fill in proper time Δt₀, rest length L₀, mass, and optionally a second velocity u for relativistic addition. Press 'Calculate'.
Example Calculation
Example 1 — Muon decay: β = 0.9998, Δt₀ = 2.2 μs. γ = 1/√(1−0.9998²) ≈ 50. Dilated time Δt = 50×2.2 μs = 110 μs, long enough to reach sea level from 15 km. Example 2 — LHC proton: β = 0.999999991, γ ≈ 7454. A 1 km rest-length detector appears 1000/7454 ≈ 0.13 m long to the proton. Total energy E = γ×m_p×c² ≈ 6.5 TeV.
Understanding Special Relativity | Time Dilation, Length Contraction & Velocity Addition
Lorentz factor at key velocities
| Speed | β = v/c | γ (Lorentz factor) | Time dilation | Length contraction |
|---|---|---|---|---|
| 0.1c (Voyager max) | 0.1 | 1.005 | +0.5% | −0.5% |
| 0.5c | 0.5 | 1.155 | +15.5% | −13.4% |
| 0.9c | 0.9 | 2.294 | +129% | −56.4% |
| 0.99c | 0.99 | 7.089 | +609% | −85.9% |
| 0.999c (muon) | 0.999 | 22.37 | +2137% | −95.5% |
| 0.9999c (LHC p) | 0.9999 | 70.71 | +6971% | −98.6% |
Relativistic formulae reference
| Quantity | Formula | Newtonian limit (β≪1) |
|---|---|---|
| Lorentz factor γ | γ = 1/√(1−β²) | ≈ 1 + β²/2 |
| Time dilation | Δt = γ·Δt₀ | Δt ≈ Δt₀ |
| Length contraction | L = L₀/γ | L ≈ L₀ |
| Relativistic momentum | p = γmv | p ≈ mv |
| Kinetic energy | KE = (γ−1)mc² | KE ≈ ½mv² |
| Total energy | E = γmc² | E ≈ mc² + ½mv² |
| Velocity addition | u' = (u+v)/(1+uv/c²) | u' ≈ u + v |
Physical significance
- ›Muon decay confirmation: Cosmic-ray muons created at 15 km altitude survive to sea level because their proper lifetime (2.2 μs) is dilated by γ ≈ 22, demonstrating time dilation experimentally.
- ›GPS clock corrections: GPS satellites run fast by ~38 μs/day due to gravitational and special-relativistic effects; onboard clocks are pre-adjusted to match ground receivers.
- ›LHC proton energetics: LHC protons reach γ ≈ 7000, so their total energy E = γmc² is 7000 times the rest mass energy of 938 MeV — about 6.5 TeV per beam.
- ›Mass-energy equivalence: E = mc² emerges from setting v = 0 in E = γmc²; the rest energy of 1 kg is 9×10¹⁶ J — equivalent to ~21 megatons of TNT.
Frequently Asked Questions
Why can nothing travel faster than light?
As β → 1, the Lorentz factor γ → ∞. The relativistic kinetic energy KE = (γ−1)mc² also → ∞, meaning infinite energy would be required to reach c. For a massive object, c is an asymptotic limit, not a reachable speed.
What is proper time and why does it matter?
Proper time Δt₀ is the time interval measured by a clock that travels with the moving object — the shortest possible elapsed time between two events connected by that worldline. An observer in a different frame measures a longer dilated time Δt = γ·Δt₀.
Does length contraction mean objects physically shrink?
No — length contraction is a measurement effect between inertial frames. The object's proper length L₀ (measured in its rest frame) is unchanged. An observer in another frame measures L = L₀/γ due to the relativity of simultaneity when measuring the two ends at the 'same time'.
How does relativistic velocity addition prevent exceeding c?
Classical addition u+v can exceed c. The relativistic formula u' = (u+v)/(1+uv/c²) always gives |u'| < c when both u and v are < c. Even if u = v = 0.9c, u' = 1.8c/1.81 ≈ 0.994c, not 1.8c.
What is the relationship between kinetic energy and E = mc²?
Total energy E = γmc² includes both rest energy mc² and kinetic energy KE = (γ−1)mc². At low speeds, KE ≈ ½mv² (the Newtonian result). The rest energy mc² is the energy content of the mass even at rest, as demonstrated in nuclear reactions where small mass differences release enormous energy.
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