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Length contraction

| nosplit | ↑ parent "Lorentz transformation" | words: 373 | descendant words: 409 | descendants: 1
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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).
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Question: what is the length than an observer in frame moving relative to as speed observe the rod to be?
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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 https://en.wikipedia.org/wiki/Terrell_rotation
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Measuring a length means to measure the difference for a single point in time in your frame ().
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So what we want to obtain is for any given time .
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In summary, we have:
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By plugging those values into the Lorentz transformation, we can eliminate , and conclude that for any , the length contraction relation holds:
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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?
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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.
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Ancestors

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