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I am little bit confused about the idea of length contraction based on my textbook which did not elaborate this topic much.

lets say there are two guys named A and B. A is at rest and B is in motion relative to A with velocity v. A measures that a rod is $2$m long and tries to measure what its length would be observed with respect to B so he calculates the measurements and ends up having a result $2\gamma$m which is less than what he himself observed. By this example I tried to explain that the concept of length contraction is that the length calculated/measured by A which he thinks that B will be able to observe. I mean its like A asking himself "if I know its length is 2m, lets try and calculate how long will it be observed by B".

Is this understanding correct? Or am I thinking the opposite that is with respect to A, B will be able to measure the length of the rod to be $2/\gamma$m?

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  • $\begingroup$ Does this provides the answer you seek? : physics.stackexchange.com/questions/142370/length-contraction $\endgroup$
    – K.R.Park
    Feb 17, 2022 at 10:08
  • $\begingroup$ @K.R.Park it didn't exactly answer this question but gave a clue about another confusion I was having. Thanks for the link $\endgroup$
    – MSKB
    Feb 17, 2022 at 10:17
  • $\begingroup$ A "contracted" length is a "false" length, where the two ends of an object are not measured at the same value of time coordinate by the other observer. It is not a fundamental concept (although it is frequently misused as one); it is derived by applying the Lorenz Transform, and is a major source of confusion in SR teaching. I suspect you have an old or bad textbook. The best way to learn SR is through the spacetime interval: en.wikipedia.org/wiki/Spacetime#Spacetime_interval. $\endgroup$
    – m4r35n357
    Feb 17, 2022 at 11:09
  • $\begingroup$ The textbook I was talking about used Lorentz transformation in order to derive the equations regarding length contraction which seemed little vague as it didn't correspond explicitly to theoretical explanations as provided by the author $\endgroup$
    – MSKB
    Feb 17, 2022 at 11:11
  • $\begingroup$ @m4r35n357 Length contraction is an idea that emerged directly from experiment. That's as fundamental as ideas get in physics. Physics is not mathematics. The Lorenz transform is a more sophisticated mathematical idea that came later, so to say that length contraction is derived from it is nonsense. $\endgroup$
    – John Doty
    Jul 9, 2022 at 22:21

3 Answers 3

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For your example of A and B you seem to be right, the only thing is that $\gamma$ is greater than 1 when you're moving so for B to see a smaller length (which is what happens) you would need to divide by $\gamma$ giving you an expression of $\frac{2}{\gamma}$m for the length that B sees the 2m stick to be so your idea of the moving person seeing the length getting contracted and this being able to be calculated by A is correct. But A can also see length contraction, if B was also holding a 2m long ruler then A would see it contracted to $\frac{2}{\gamma}$m. The qualitative idea of length contraction is that a person in one frame that sees something moving will see that thing being contracted.

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What it means is that the distance between any two stationary points on a line will always be longer than the distance between them in any frame that is moving along that line.

So if the two points are the ends of a stationary 2metre rod in A's frame, the distance between them in B's frame (in which the rod is moving) will be less than 2m. Note that the length contraction formula only applies if the motion is along the line of the rod.

The length contraction effect occurs because of the relativity of simultaneity. In the frame in which the rod is moving, the position of leading edge of the rod is measured at the same time as the position of the trailing edge; however, in the rest frame of the rod that equates to measuring the position of the leading edge before measuring the position of the trailing edge, giving the trailing edge a chance to move forward in the meantime, thus making the distance between the two positions shorter.

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If you throw a rod the rod 1: appears to move and 2: appears to become shorter.

Usually we compare lengths of same rod when it appears to move and when it appears to not move. We say that a rod that appears to move at speed 0.87c appears to be contracted by 50%, which means contracted compared to the length at speed zero.

A way to check the length of a moving rod is to measure the time that the passing of the rod takes.

If a rod appears to be moving, then that rod appears to be shortened from its normal length. That is the length contraction phenomenon.

There is one length contraction phenomenon and and then there are all kinds of length contraction calculations.

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