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I am having difficulty in understanding problem number 14 in Goldstein's Classical Mechanics, 3rd edition, chapter 7 on special relativity. Here is the problem ---

A rocket of length $l_0$ in its rest system is moving with constant speed along the $z$ axis of an inertial system. An observer at the origin of this system observes the apparent length of the rocket at any time by noting the $z$ coordinates that can be seen for the head and tail of the rocket. How does this apparent length vary as the rocket moves from the extreme left of the observer w the extreme right? How do these results compare with measurements in the rest frame of the observer? (Note: observe, not measure).

How does this differ from the usual length contraction? What is the meaning of the hint given by asking the reader to "observe" not "measure", what is the difference here?

Manas Dogra
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I'd like to add to what 'PM 2Ring' wrote. The observer will measure the rocket to have a constant length no matter where it is in the observer's frame of reference (assuming it is moving at a constant velocity -- in which case it will be length contracted).

However, the observer will observe the rocket to be longer when it moves towards him and shorter when it moves away from him. This has nothing to do with relativity, just with the fact that there is a path difference between the light coming from each end of the rocket, which can make the rocket appear longer/shorter when it is moving at very high speeds. It may be a bit hard to visualise at first, make two diagrams of the rocket, separated by a small unit of time (in which case the rocket will have moved of course), and compare light pulses from the nose and tail.

It's a nuance of terminology, just have in mind that some people take measure to mean something different to observe. The difference should be explained whenever it makes a difference, which it clearly wasn't in the question.

Dedados
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The difference between measurement and observation is crucial in relativity.

When we observe the rocket, the finite speed of light affects our observation. In general, light from the head and the tail of the rocket will take a different amount of time to travel to the observer.

When we measure the rocket, we compensate for time delays caused by the finite speed of light. So if we measure two events A & B to be simultaneous we will only observe A & B to be simultaneous if the distances to A & B are identical in our frame.

As Alfred Centauri notes in the comments, it's not unusual for writers to use the term "observed" to refer to measured values, not the raw observed data. They assume that the reader knows that light travel time has to be compensated for. This unfortunate ambiguity confuses many people learning relativity.

PM 2Ring
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    FWIW, I've understood observe (in the SR context) to mean what is measure here, and see (or photograph) what is observe here. – Alfred Centauri Jun 03 '20 at 17:40
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    @Alfred Indeed, observe can be ambiguous, and some writers do use it in the way you describe. But I think that when observe is used in contrast to measure, then it's pretty clear that what is measured involves calculation, whereas observation involves the raw data. – PM 2Ring Jun 03 '20 at 17:47
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    PM 2Ring, yes, I agree. – Alfred Centauri Jun 03 '20 at 17:52
  • @PM2Ring So,it means the usual length contraction has to do with measurement,and this problem is to do with observation? – Manas Dogra Jun 03 '20 at 18:27
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    @Manas Yes, the usual length contraction has to do with measurement. The exercise asks you to determine both the observed size and the measured size, and to compare the two values. On a related note, see Penrose-Terrel rotation. The moving sphere is measured to be flattened by length contraction, but by observation it still looks like an undistorted sphere. – PM 2Ring Jun 03 '20 at 19:15
  • "See" is even more ambiguous! – m4r35n357 Jun 09 '20 at 11:28