Proper length
Proper length or rest length refers to the length of an object in the object's rest frame.
The measurement of lengths is more complicated in the theory of relativity than in classical mechanics. In classical mechanics, lengths are measured based on the assumption that the locations of all points involved are measured simultaneously. But in the theory of relativity, the notion of simultaneity is dependent on the observer.
A different term, proper distance, provides an invariant measure whose value is the same for all observers.
Proper distance is analogous to proper time. The difference is that the proper distance is defined between two spacelike-separated events (or along a spacelike path), while the proper time is defined between two timelike-separated events (or along a timelike path).
The proper length or rest length of an object is the length of the object measured by an observer which is at rest relative to it, by applying standard measuring rods on the object. The measurement of the object's endpoints doesn't have to be simultaneous, since the endpoints are constantly at rest at the same positions in the object's rest frame, so it is independent of Δt. This length is thus given by:
However, in relatively moving frames the object's endpoints have to be measured simultaneously, since they are constantly changing their position. The resulting length is shorter than the rest length, and is given by the formula for length contraction (with γ being the Lorentz factor):
In comparison, the invariant proper distance between two arbitrary events happening at the endpoints of the same object is given by:
So Δσ depends on Δt, whereas (as explained above) the object's rest length L0 can be measured independently of Δt. It follows that Δσ and L0, measured at the endpoints of the same object, only agree with each other when the measurement events were simultaneous in the object's rest frame so that Δt is zero. As explained by Fayngold:
In special relativity, the proper distance 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. In such a specific frame, the distance is given by
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