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I was reading the book, "Relativity: The special and general theory" by Einstein. At some point it discusses the awkwardness of "relativity of simultaneity" with using universal time axis for all inertial frames. And to fix it uses Lorentz's transformation(the attached image shows two inertial frames(page 22 of the book).

image from page 22 of the book equation from page 22 of the book

To explain length dilation after this, it uses below example,

text from page 25 of the book text and equation from page 25 of the book

I am not able to derive the last line.

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    $\begingroup$ This is a question about special relativity, not general relativity. $\endgroup$ – Ben Crowell Apr 15 at 3:33
  • $\begingroup$ thanks for the correction @BenCrowell $\endgroup$ – Rahul Shaw Apr 15 at 4:28
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From the first equation, at time t , we can write $$x= x' \sqrt{1-\frac{v^2}{c^2}}+vt.$$ Substitute for $x'=0$ and $x'=1$, and do $x_{(x'=1)}-x_{(x'=0)}$. You will get the answer.

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  • $\begingroup$ I has to be considered as 1(the unit value) ? $\endgroup$ – Rahul Shaw Apr 16 at 11:07
  • $\begingroup$ Yes, it's not $I$, it is $1$. Actually the denominator term in the equation should be $\sqrt{1-\frac{v^2}{c^2}}$. Also the length of rod is $1$. $\endgroup$ – walber97 Apr 16 at 12:30

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