What Is Length Contraction and How Do You Calculate It?

Length contraction is the decrease in the measured length of an object moving relative to an observer, specifically along the direction of motion. In IB Physics HL, calculate it using $L=\frac{L_0}{\gamma}$, where $L_0$ is the object's proper length and $\gamma$ is the Lorentz factor.

The Mechanism

This is an Additional Higher Level (AHL) concept from A.5 Galilean and special relativity. The proper length $L_0$ is measured in the object's rest frame: the frame in which the object is stationary.

To measure a moving object's length, the observer must record the positions of its front and rear at the same time in that observer's frame. Since simultaneity depends on the reference frame, different frames can obtain different lengths.

An observer who measures the object moving at speed $v$ measures the contracted length $L$. The equations are

$$\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$

and

$$L=\frac{L_0}{\gamma}=L_0\sqrt{1-\frac{v^2}{c^2}}.$$

Here, $c=3.00\times10^8\ \text{m s}^{-1}$. Because $\gamma\geq1$, the measured moving length satisfies .

At everyday speeds, $v\ll c$, so $\gamma$ is extremely close to 1 and contraction is negligible. The effect becomes significant only at relativistic speeds.

For example, a spacecraft has proper length $L_0=100\ \text{m}$ and travels at $v=0.80c$. First calculate

$$\gamma=\frac{1}{\sqrt{1-0.80^2}}=1.67.$$

Therefore,

$$L=\frac{100}{1.67}=60\ \text{m}.$$

An Earth observer measures the spacecraft as $60\ \text{m}$ long, while an observer travelling with it measures its proper length of $100\ \text{m}$.

A common misconception is that the object physically compresses in its own frame. It does not: length contraction is a consequence of different inertial frames measuring space and simultaneity differently.

Exam Technique

For a Calculate question, identify the frame measuring the proper length, write the Lorentz factor, substitute without prematurely rounding, and state which observer measures each length. Never apply contraction to a dimension perpendicular to the motion.