Time dilation and length contraction calculator
Everything relativistic follows from one number, the Lorentz factor, which is 1 at rest and diverges at c.
years
Lorentz factor
2.294 157 339
- Dilated time
- 2.294 157 3 years
- Contracted length
- 43.588 989
- Speed
- 269 813 212.2
- Relativistic kinetic energy
- 1.1631306e+17
- Classical kinetic energy for comparison
- 3.6399585e+16
- Total energy
- 2.0618858e+17
Show the working
- Speed as a fraction of c = 0.9
- Proper time = 1
- Proper length = 100
- Rest mass = 1
- Speed = 0.9 * c = 269 813 000
- Lorentz factor = 1 / sqrt(1 - 0.9^2) = 2.294 16
- Dilated time = 1 * 2.294 16 = 2.294 16
- Contracted length = 100 / 2.294 16 = 43.589
- rest_energy = 1 * c^2 = 8.98755e+16
- Total energy = 2.294 16 * 8.98755e+16 = 2.06189e+17
- Relativistic kinetic energy = (2.294 16 - 1) * 8.98755e+16 = 1.16313e+17
- Classical kinetic energy for comparison = 1 * 269 813 000^2 / 2 = 3.63996e+16
Values are shown to six significant figures here; the arithmetic itself runs on exact fractions and is only rounded where a number is printed.
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The formula
- gamma = 1 / sqrt(1 - v^2/c^2)
- dilated time = gamma x proper time
- contracted length = proper length / gamma
- E = gamma m c^2, KE = (gamma - 1) m c^2
At 0.9c the Lorentz factor is 2.294: clocks run at 44% of the rate and lengths shrink to 44% along the direction of travel.
The classical kinetic energy row shows how badly Newtonian mechanics undercounts at high speed.