 Length contraction

Suppose that a rod has is length $L$ measured on a rest frame $S$ (or maybe even better: two identical rulers were manufactured, and one is taken on a spaceship, a bit like the twin paradox).

Question: what is the length $L^{\prime }$ than an observer in frame $S^{\prime }$ moving relative to $S$ as speed $v$ observe the rod to be?

The key idea is that there are two events to consider in each frame, which we call 1 and 2:Note that what you visually observe on a photograph is a different measurement to the more precise/easy to calculate two event measurement. On a photograph, it seems you might not even see the contraction in some cases as mentioned at en.wikipedia.org/wiki/Terrell_rotation

Measuring a length means to measure the $x_2-x_1$ difference for a single point in time in your frame ($t2=t1$).

So what we want to obtain is $x_2^{\prime }-x_1^{\prime }$ for any given time $t^{\prime }2=t^{\prime }1$.

In summary, we have:

By plugging those values into the Lorentz transformation, we can eliminate $t_{2}and{t}_1$, and conclude that for any $t_2^{\prime }=t_1^{\prime }$, the length contraction relation holds:

The key question that needs intuitive clarification then is: but how can this be symmetric? How can both observers see each other's rulers shrink?

And the key answer is: because to the second observer, the measurements made by the first observer are not simultaneous. Notably, the two measurement events are obviously spacelike-separated events by looking at the light cone, and therefore can be measured even in different orders by different observers.