What's the Difference Between Proper Time and Proper Length?

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The Lorentz factor is

For proper length, simultaneity is essential: the moving observer must record the positions of both endpoints at the same time in that observer's frame. Length contraction applies only to the dimension parallel to the relative velocity; dimensions perpendicular to the motion are unchanged. These conditions follow from the relativity of simultaneity.

Because $\gamma\geq 1$, proper time is the shortest time interval between the events, whereas proper length is the greatest measured length of the object.

For example, suppose $\gamma=1.25$. If a particle's lifetime in its own rest frame is $\Delta t_0=4.0\,\mu\text{s}$, a laboratory observer measures

If a spacecraft has proper length $L_0=4.0\,\text{m}$, the laboratory measures

A common misconception is that “proper” means the value measured by the observer. It does not: it means the value measured under the quantity's defining condition.

Exam Technique

Before selecting an equation, identify the frame. Ask whether the events occur at one position for proper time, or whether the object is at rest for proper length. In an IB Explain question, state the defining condition as well as the formula; naming the moving observer alone is insufficient.