Why does length contraction seem to conflict with invariance of intervals?

Question

Suppose we have two simultaneous events, $A$ and $B$, separated by a distance $L$ (the simultaneity is in frame of reference $S$). Now suppose we have a second frame of reference $S'$ moving with speed $V$ relative to $S$.

According to the invariance of intervals, the interval between events $A$ and $B$ in frame $S'$ is also $L$, but since there is also a temporal separation between events (simultaneity isn't conserved), one finds that the spatial distance between events in $S'$ is larger than $L$. In contrast, according to the length contraction prediction of special relativity, the maximum length is in the rest frame, $S$. Why the inconsistency?

Answer 1

You need to review the derivation of Lorentz contraction. The result and your reasoning are correct, but wrongly interpreted.

The observer in the primed frame indeed measures $$ dx' > L $$ I've used a different notation for reasons that will become apparent. In fact, one can show the primed observer sees $$ dx' = \gamma(v) L $$ where $\gamma(v) \ge 1$ is a Lorentz factor.

However, to find the length of an object, he would need the simulatneous position of its endpoints (the events are not simultaneous in his frame). He needs to account for the fact that he knows the second endpoint at a later time, and during that time, it moved! He accounts for this by subtracting the amount it moved, $vdt'$, $$ L' = dx' - v dt' = \cdots= L / \gamma < L. $$ With that correction, he indeed sees a contracted length that is less than the proper length.