The measurement of lengths is more complicated in the theory of relativity than in classical mechanics.
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.
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 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.
where The two formulae are equivalent because of the invariance of spacetime intervals, and since Δt = 0 exactly when the events are simultaneous in the given frame.
Two events are spacelike-separated if and only if the above formula gives a real, non-zero value for Δσ.
It is, however, possible to define the proper distance along a path in any spacetime, curved or flat.
Along an arbitrary spacelike path P, the proper distance is given in tensor syntax by the line integral
should be dropped with a metric tensor that is normalized to use a distance, or that uses geometrized units.