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A is a stationary observer watching B who is moving relative to A. Both of them hold two identical light clocks and each shoots light rays to estimate the lengths of their clocks. A's light ray will only have to travel a distance of the light clock but B's light ray will have to travel a lot further(longer than the length of the clock)to reach the end of its clock. Both observers measure the light ray travelling at the speed of light(according to Einstein's second postulate), but the time it took to reach the ends was different. So B's measurement of the length of the clock is slightly longer than A's measurement of his length of the clock. So this proves that observers in different reference frames will disagree on length in the perpendicular direction as well

There should be something wrong with this explanation but I'd like to know why. I am also aware of the general explanation for why length contraction can't effect the distances in the perpendicular direction using two cylinders, but I want to know why this logic is not valid.

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This is an example of how the principle of simultaneity can mislead to wrong answers.

Let's look at this problem from the perspective of $A$.

He uses the coordinates $x$ and $t$. For him, the time taken for light to reach of the ends of the clock is $l/c$, where $l$ is the length of the clock. $B$ measures the same length. But for $A$, he does not see the light reaching at the end of light clock of $B$. He assumes that the events of the light hitting the other ends of the observers is not simultaneous.

The same reasoning goes for $B$ too.

So, let's say that time taken for the light to reach the other end for $A$ be $t$ and for $B$ be $t'$. Thus we have: -

$t = t' = l/c$

So, if the observers meet after experiment, they will agree upon the time taken for light to reach the other end. But during the experiment, when they measure the time taken for light to reach the other point of the other observer, they get different results. This occurs because of the principle of simultaneity which states that the notion of simultaneity is relative . In a particularly specific reference frame, we could actually see the light hitting the ends at the same time.

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  • $\begingroup$ Thank you for your answer. To sum up your reasoning, are your saying that the reason why the two light rays hit the end of the clocks in different times is because simultaneity is defined differently between the two observers so A cannot conclude that B's measurements are longer? $\endgroup$
    – I am Einstein
    Commented May 18 at 7:58
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    $\begingroup$ Yes, in fact B concludes that A's measurement is longer and his measurement was shorter than A's measurement. Yet again, this conclusion is only limited to his reference frame. In future, try drawing spacetime diagrams, those diagrams are very good for understanding relativity. These paradoxes will never come up in your mind if you use spacetime diagrams. $\endgroup$
    – Ronny
    Commented May 18 at 8:12
  • $\begingroup$ yes, I can clearly see the whole picture when I draw a 3-dimentional spacetime diagram. Thank you. $\endgroup$
    – I am Einstein
    Commented May 18 at 8:24
  • $\begingroup$ Great! I am glad my response was helpful! $\endgroup$
    – Ronny
    Commented May 18 at 8:25

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