Editing Proper length (section)

== Proper length or rest length ==
The ''proper length''<ref name=fayngold /> or ''rest length''<ref name=franklin /> 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 Δ''t''. This length is thus given by:

:<math>L_{0} = \Delta x.</math>

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]]):

:<math>L = \frac{L_0}{\gamma}.</math>
 
In comparison, the invariant proper distance between two arbitrary events happening at the endpoints of the same object is given by:

:<math>\Delta\sigma = \sqrt{\Delta x^2 - c^2 \Delta t^2}.</math>

So Δ''σ'' depends on Δ''t'', whereas (as explained above) the object's rest length ''L''<sub>0</sub> can be measured independently of Δ''t''. It follows that Δ''σ'' and ''L''<sub>0</sub>, 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 Δ''t'' is zero. As explained by Fayngold:<ref name=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 ''l''<sub>0</sub>. If you are in the rest frame ''K''<sub>0</sub> 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 ''K''<sub>0</sub>. You can mark one end now (at a moment ''t''<sub>1</sub>) and the other end later (at a moment ''t''<sub>2</sub>) in ''K''<sub>0</sub>, 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 ''K''<sub>0</sub>, 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 ''K''<sub>0</sub>."