Length contraction

Proper length

The proper length $L_0$ of an object is its length measured in the object's rest frame. To measure the length of a moving object, an observer must record the positions of both ends at the same time in that observer's frame.

This simultaneity requirement is essential. Measuring a moving length is not just reading two endpoint positions whenever convenient. The endpoints must be located simultaneously according to the measuring frame.

Length contraction formula

If an object with proper length $L_0$ moves at speed $v$ relative to an observer, that observer measures its length along the direction of motion as

L=rac{L_0}{gamma}

where

gamma=rac{1}{sqrt{1-v^2/c^2}}

Since $gammageq1$, the measured moving length $L$ is shorter than $L_0$.

Direction matters

Length contraction occurs only along the direction of relative motion. Dimensions perpendicular to the motion are unchanged:

$L*perp=L*{0perp}$

If a spaceship moves in the x-direction, its x-length contracts, but its height and width do not contract in the same way.

Connection to time dilation

Consider unstable particles created high in the atmosphere. In Earth's frame, they live longer because of time dilation, allowing them to travel farther before decaying. In the particle's frame, the particle lifetime is its ordinary proper lifetime, but Earth and the atmosphere rush toward it. The atmosphere's thickness is length-contracted:

L=rac{L_0}{gamma}

Both descriptions agree on whether the particle reaches the ground. Time dilation and length contraction are complementary ways to describe the same spacetime facts.

Measuring a moving rod

Suppose a rod moves past a laboratory. To measure its length, lab observers use synchronized clocks at different positions and record where the front and back of the rod are at the same lab time. The distance between those simultaneous positions is the lab-measured length.

Observers riding with the rod use different simultaneity slices. They measure the rod at rest and obtain the proper length $L_0$. The disagreement comes from different definitions of simultaneous endpoint measurements.

Not physical crushing

Length contraction is not caused by mechanical compression in the object's own frame. A spaceship crew does not see their ship squashed. In their frame, the ship has proper length. They instead see outside objects contracted.

The contraction is a relationship between measurements in different frames, not a material stress effect.

Low-speed limit

At low speeds,

$gammaapprox1$

so

$Lapprox L_0$

This is why length contraction is not noticed in everyday life. At speeds near light speed, the effect becomes large. For $v=0.866c$, $gamma=2$, so the moving length is half the proper length.

Visual appearance versus measured length

Length contraction is not exactly what a camera would photograph. Photographic appearance also depends on light travel time from different parts of the object. A fast object may appear rotated or distorted rather than simply shortened. Length contraction refers to coordinate measurements after correcting for signal timing.

The big idea

Length contraction says moving objects are measured shorter along the direction of motion. Proper length is measured in the object's rest frame; moving length requires simultaneous endpoint measurements in another frame. The effect is inseparable from relativity of simultaneity and time dilation.