Proper Length Between Two Events
In special relativity, the proper length between two spacelike-separated events is the distance between the two events, as measured in an inertial frame of reference in which the events are simultaneous. So if the two events occur at opposite ends of an object, the proper length of the object is the length of the object as measured by an observer which is at rest relative to the object.
In any inertial frame of reference, the proper length L is
,
where
- Δt is the difference in the temporal coordinates of the two events,
- Δx, Δy, and Δz are differences in the linear, orthogonal, spatial coordinates of the two events, and
- c is the speed of light.
Two events are spacelike-separated if and only if the above formula gives a real, non-zero value for L.
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