What this calculator does
Length contraction is the companion effect to time dilation. An object moving at a substantial fraction of light speed measures shorter along its direction of travel, as observed from a frame it is moving relative to.
Only the direction of motion is affected. Dimensions across the motion are unchanged, so a fast-moving sphere would measure as a flattened shape rather than a smaller sphere.
The formula
Divide the rest length by the Lorentz factor, which is one over the square root of one minus speed squared over light speed squared. Equivalently, multiply the rest length by that square root directly.
| Term | Meaning |
|---|---|
| Proper length | The length measured in the object's own rest frame, which is always the largest measurement of it. |
| Lorentz factor (γ) | The same factor that appears in time dilation, here dividing rather than multiplying. |
| Rest frame | The frame in which the object is not moving. |
The inputs explained
| Field | What to enter |
|---|---|
| Proper length (at rest) (m) | The length of the object measured at rest, in metres. |
| Relative speed (m/s) | The relative speed in metres per second. |
When to use it
Seeing the other side of time dilation
The muon that survives to reach the ground can be explained either by its clock running slow or by the atmosphere being contracted in its frame. Both descriptions agree.
Working a relativity problem
Many questions can be attacked from either the length or the time side, and picking the easier one saves a great deal of algebra.
Understanding particle accelerator design
At the speeds reached in accelerators, contraction is substantial and has to be accounted for in the physics of collisions.
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 much does a 100 m object contract at speed?
A 100 metre object at a range of speeds.
| Speed | Contracted length | Lorentz factor γ |
|---|---|---|
| 100 million m/s | 94.2727 m | 1.0608 |
| 150 million m/s | 86.5825 m | 1.1550 |
| 250 million m/s | 55.1900 m | 1.8119 |
| 290 million m/s | 25.3498 m | 3.9448 |
Questions
Does the object actually get shorter?
It is genuinely measured shorter in that frame, but nothing is compressed or stressed. Both measurements are valid; length simply depends on the frame it is measured in, which is what relativity asserts.
Does it contract in every direction?
No, only along the direction of motion. Dimensions perpendicular to the travel are unaffected, so the shape changes rather than shrinking uniformly.
How does this relate to time dilation?
They are two views of the same geometry and always accompany each other. Where one observer explains something by a slowed clock, another explains it by a contracted distance, and the two accounts always agree on what actually happens.
Has it been observed?
Indirectly but conclusively. Particle accelerators and cosmic ray muons both behave exactly as the effect predicts, and the physics of those experiments would not work without it.
For the time side of the same effect, see the time dilation calculator. For relativistic energy, see the mass-energy equivalence calculator.