Proper length

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).

Proper length or rest length

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:

${\displaystyle L_0=\mathrm{\Delta }x.}$

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):

${\displaystyle L=\frac{L_0}{\gamma }.}$

In comparison, the invariant proper distance between two arbitrary events happening at the endpoints of the same object is given by:

${\displaystyle \mathrm{\Delta }\sigma =\sqrt{\mathrm{\Delta }{x}^2-c^2\mathrm{\Delta }{t}^2}.}$

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:

p. 407: "Note that the proper distance between two events is generally not the same as the proper length of an object whose end points happen to be respectively coincident with these events. Consider a solid rod of constant proper length l₀. If you are in the rest frame K₀ of the rod, and you want to measure its length, you can do it by first marking its endpoints. And it is not necessary that you mark them simultaneously in K₀. You can mark one end now (at a moment t₁) and the other end later (at a moment t₂) in K₀, and then quietly measure the distance between the marks. We can even consider such measurement as a possible operational definition of proper length. From the viewpoint of the experimental physics, the requirement that the marks be made simultaneously is redundant for a stationary object with constant shape and size, and can in this case be dropped from such definition. Since the rod is stationary in K₀, the distance between the marks is the proper length of the rod regardless of the time lapse between the two markings. On the other hand, it is not the proper distance between the marking events if the marks are not made simultaneously in K₀."