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Suppose that a rod has is length measured on a rest frame (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 than an observer in frame moving relative to as speed 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:
  • the left end of the rod is an observation event at a given position at a given time: and for or and for
  • the right end of the rod is an observation event at a given position at a given time : and for or and for
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
Measuring a length means to measure the difference for a single point in time in your frame ().
So what we want to obtain is for any given time .
In summary, we have:
By plugging those values into the Lorentz transformation, we can eliminate , and conclude that for any , 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.