A question about the definition of proper length As I understand it, the proper length of an object is defined as the length of the object in its rest frame. In terms of the metric it is defined as the length of the spacetime interval between two space-like separated events, i.e. $$dl^{2}=\sqrt{ds^{2}}$$ (with the "mostly plus" signature).
Now, suppose that an observer is at rest in an inertial frame that is itself at rest with respect to a given object that the observer wishes to measure. Why is it the case that one considers the "simultaneous" length of the object, i.e. when $dt=0$, such that $$dl^{2}=\sqrt{dx^{2}+dy^{2}+dz^{2}}$$ Is it simply so that it agrees with the definition of spatial distance in Euclidean geometry or are there other intuitive reasons for why it must be the case (analogous to the definition of proper time in which the proper time of an object is equal to the coordinate time of an observer who is at rest [i.e. $dx=dy=dz=0$] with respect to the object, such that $d\tau=\frac{1}{c}\sqrt{-ds^{2}}=dt$)?!
 A: When two guys are in the same reference frame then events for both of you occur at the same time coordinate - ie they are simultaneous. What I mean by this is that both guys will ascribe the same time coordinate to when an event A occurred. Of course depending on how far they are from A means they have to use different values for their measurements but they should both agree. 
Now imagine a meter stick. In the meter sticks frame of reference let's suppose there are two events. Event A is one end of the meter stick being at x = 1m at t = 0 and event B is the other at x = 2m  at t = 0. According to your formula of the space time interval we can find the distance between these two events. Since t = 0 for both A and B in the sticks frame, we then get your formula for the length. (Edit: dt = 0 because the time for both events is t = 0)
Now if I am in the meter sticks frame, according to my first paragraph, A and B should be simultaneous events for me, because these are simultaneous events for the meter stick - and we are in the same frame.
I guess it can be kind of confusing.
