Why length contracts according to the Lorentz transformation

Question

To get the length contractions we use the rest length measured in the moving frame x’ and time in the rest frame t. Because the measurement happens instantly t=0. But why don’t we use t’=0, which will give the opposite result (length “dilation”) since the measurement in the moving frame also happens instantly? For example, I’m in a moving spaceship S’ with a meterstick in front of me. I measured the length of the meterstick x’=1m and t’=0, using the inverse LT the Earth observer should see x=γx', which means the length is actually bigger. In other words, how do we know whether we should use the primed t or the unprimed time?

Answer 1

You're right that the spatial-distance between the two events $(x',t')=(0,0)$ and $(x',t')=(l,0)$ (where the primed coordinates are the coordinates in the inertial frame attached to the spaceship) would be observed to be $\gamma l$ by an observer on the earth. However, this does not contradict length contraction. This is because the observer on the earth would not observe these two events to be simultaneous and thus, would not conclude that the spatial distance between these two events constitutes the length of the meter-stick. The events associated with the measurement of the length of the meter-stick by the observer on earth would be of the form $(x,t) =(0,0)$ and $(x,t)=(l/\gamma,0)$ -- which, in turn, would not be simultaneous in the spaceship frame of reference. See this related answer of mine.

Answer 2

You are looking at the derivation of length contraction as though it describes a relationship between two events, but it is rather a relationship between two worldlines. Instead of focusing on the transformation of a pair of events, one on the left and the other on the right of a rod, look at the transformation of the left and right worldlines.

So if the left worldline in the primed frame is $\vec r’_L=(t’,0,0,0)$ then the right worldline is $\vec r’_R=(t’,L,0,0)$.

Then if we Lorentz transform those two worldlines and express the transformed worldlines in terms of the unprimed coordinates we get $\vec r_L=(t,vt,0,0)$ and $\vec r_R=(t, vt + L/\gamma,0,0)$

Then the length of the rod is the separation between the worldlines which is $L/\gamma$ as expected. Again, the key is to treat it as a transformation of worldlines rather than a transformation between events.