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The paragraph below is taken from the book An Introduction to Special Relativity by Robert Resnick.

Let A and B be the endpoints of a rod, for example, which is at rest in the S-frame. Then the primed (S') observer, for whom the rod is moving with velocity $-\vec v$, will measure the endpoint locations as $x_B'$ and $x_A'$, whereas the unprimed observer locates them at $x_B$ and $x_A$. Using Galilean transformations, however, we find that $x'_B=x_B-vt_B$ and $x'_A=x_A-vt_A$, so that $x'_B-x'_A=x_B-x_A-v(t_B-t_A)$. Since the two endpoints, A and B, are measured at the same instant, $t_A=t_B$ and we obtain $x'_B-x'_A=x_B-x_A$.

My question is why should we compare the left-hand side to the right-hand side at $t_A=t_B$?

Since the rod is eternally at rest in the unprimed frame, the markers $x_A$ and $x_B$ will have unchanging values no matter if $x_A$ and $x_B$ measured at the same instant or at different instants in the unprimed frame. Isn't it? If so, why does it matter when $x_A$ and $x_B$ are measured in the unprimed frame (to define its rest length)?

I can define the rest length in the unprimed frame by measuring both ends at 1.00 pm noon or one end at 1.00 pm and the other end at 1.30 pm. What's wrong?

N.B. Please note that my question js about Newtonian/Galilean relativity. Not special relativity.

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My question is why should we compare the left-hand side to the right-hand side at $t_A=t_B$?

Because that is how you measure the length of a rod in a frame in which the rod is moving. By measuring the points where the 2 ends of the rod are located at the SAME time.

Since the rod is eternally at rest in the unprimed frame, the markers $x_A$ and $x_B$ will have unchanging values no matter if $x_A$ and $x_B$ measured at the same instant or at different instants in the unprimed frame. Isn't it? If so, why does it matter when $x_A$ and $x_B$ are measured in the the unprimed frame (to define its rest length)?

It does not matter when $x_A$ and $x_B$ are measured. It matters when $x_A'$ and $x_B'$ are measured. That is the time referred to by $t_A$ and $t_B$. It is the time at which the primed coordinates are measured. These 2 need to be measured at the exact SAME time in the primed frame.

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  • $\begingroup$ "It does not matter when xA and Xb are measured. It matters when xA' and xB' are measured". But that means $t_A'=t_B'$ (not $t_A=t_B$) which, however, do not enter Resnick's equation. $\endgroup$ Commented Mar 23, 2021 at 14:00
  • $\begingroup$ There is no tA' or tB' that is different from tA or tB. This is newtonian physics you are talking about. Everyone has the same absolute time. So , it means the same thing. I used tA,tB . You used tA',tB' . They mean the same thing. The time of an event measured by primed observer is same as that measured by unprimed observer $\endgroup$ Commented Mar 23, 2021 at 14:02
  • $\begingroup$ So the point is, we require $t_B'=t'_A$ to define length in the prime frame in which the rod is moving. However, by invoking $t'=t$ (part of the Galilean transformation which says time is absolute), we also must have $t_B=t_A$. Thus, we set $t_B=t_A$. Is it correct? $\endgroup$ Commented Mar 23, 2021 at 14:06
  • $\begingroup$ Let us continue this discussion in chat. $\endgroup$ Commented Mar 23, 2021 at 14:36

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