Length contraction

An object moving relative to an observer is measured to be shorter along its direction of motion than it is in its own rest frame. This difference in measured length is called length contraction.

The object's length in its own rest frame is its proper length, $$l_0$$. If the object moves at speed $$v$$ relative to an observer, its length $$l$$ measured in the observer's frame is:

Here, $$c$$ is the speed of light in a vacuum. For $$0 < v < c$$, the square-root factor is less than one, so $$l < l_0$$. As $$v$$ approaches $$c$$, the factor becomes smaller and the contraction becomes greater.

For example, if an object moves at $$0.6c$$ relative to an observer, the formula gives $$l = 0.8l_0$$. An object with a proper length of $$2.0,\text{m}$$ is therefore measured to be $$1.6,\text{m}$$ long in that observer's frame.

Measuring the moving length

To measure the length of the moving object, the observer must record the positions of its two endpoints at the same time in the observer's frame. The distance between those positions is the measured length. Recording the endpoints at different times would also include some distance travelled by the object between the measurements.

Only the dimension parallel to the motion contracts. Lengths perpendicular to the direction of motion are unchanged.