A.5.9 (HL)—Proper time interval and proper length

Identify proper time

The proper time interval $\Delta t_0$ is measured between two events that occur at the same position in the observer’s frame. It is the shortest time interval measured for those events. For a clock at rest in the frame, successive ticks occur at one location, so the clock measures $\Delta t_0$.

Identify proper length

The proper length $L_0$ is the length measured in the rest frame of the object. Its endpoints are measured simultaneously in that frame. It is the maximum length assigned to the object by inertial observers.

Choose from the event conditions

For time, ask: do the two events happen at the same place in this frame? For length, ask: is the object at rest in this frame, and are both endpoints measured at the same time? These conditions, not the observer’s label, determine whether the measurement is proper.

Common trap

Proper time is not simply the time measured by the “main” observer, and proper length is not the shortest measured length. The proper length is the rest-frame, longest length; moving observers measure a contracted length.