What this calculator does
Special relativity says a moving clock runs slow as measured by an observer it is moving relative to. The effect is real and measured routinely, but it only becomes noticeable at speeds that are a serious fraction of light speed.
At everyday speeds the factor is so close to one that it is undetectable without atomic clocks. At half light speed it is about 1.155, and it climbs steeply from there, heading for infinity as the speed approaches that of light.
The formula
Calculate the Lorentz factor, one divided by the square root of one minus the speed squared over light speed squared, then multiply the proper time by it. The result is the time elapsed in the stationary observer's frame.
| Term | Meaning |
|---|---|
| Proper time | Time measured by a clock travelling with the moving object, which is always the shortest measurement between two events. |
| Lorentz factor (γ) | The stretching factor between frames. It is 1 at rest and rises without limit as speed approaches c. |
| c | The speed of light in vacuum, exactly 299,792,458 metres per second. |
The inputs explained
| Field | What to enter |
|---|---|
| Proper time elapsed (moving clock) (s) | The time elapsed on the moving clock, in seconds. |
| Relative speed (m/s) | The relative speed in metres per second. Light speed is about 3 × 10⁸, so meaningful effects need values in that range. |
When to use it
Understanding GPS corrections
Satellite clocks run at a different rate from ground clocks, and the correction must be applied continuously or positions would drift badly.
Explaining muon detection at sea level
Muons created high in the atmosphere should decay before reaching the ground, and they arrive because time runs slow in their frame.
Working through a relativity problem
The twin paradox and similar questions all reduce to applying the Lorentz factor correctly.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the Lorentz factor grow with speed?
A range of speeds as a fraction of light speed.
| Speed | Lorentz factor γ | Speed as a fraction of c |
|---|---|---|
| 100 million m/s | 1.0608 | 0.3336×c |
| 150 million m/s | 1.1550 | 0.5003×c |
| 250 million m/s | 1.8119 | 0.8339×c |
| 290 million m/s | 3.9448 | 0.9673×c |
Questions
Is time dilation real or just an illusion?
Real and measured. Atomic clocks flown on aircraft return showing less elapsed time than identical clocks left on the ground, by exactly the predicted amount. GPS depends on accounting for it.
Which clock actually runs slow?
Each observer sees the other's clock running slow, which sounds contradictory but is not, because they also disagree about simultaneity. The asymmetry in the twin paradox comes from one twin accelerating to turn around, breaking the symmetry between the frames.
Why is the effect invisible in daily life?
Because the speed enters as a ratio to light speed and then gets squared. Even a fast aircraft gives a Lorentz factor differing from 1 in about the thirteenth decimal place.
What happens at the speed of light?
The factor becomes infinite, which is one way of seeing that no object with mass can reach light speed. It would require infinite energy, so the limit is not a practical obstacle but an absolute one.
For the accompanying effect on measured length, see the length contraction calculator. For the energy equivalent of mass, see the mass-energy equivalence calculator.