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Suppose in the rest frame $S$, event $A$ occurs at $x = 0, t =0$ and event $B$ occurs at $x = 100, t = 80$.

Now suppose that frame $S’$ is moving with velocity $V$ with respect to $V$.

Now the question is what is the distance between the events in frame $S’$. Since the events occur at different times in the rest frame I was wondering if it is applicable to use the formula $L = L_0/\gamma$.

On the other hand if i transform the $x$ cords for both events using Lorentz transforms and subtract them i get $x_B’ - x_A’ = \gamma ( x_B - \beta ct_B - x_1 + \beta ct_A)$. In this instance since $t_B \neq t_A$ you will get a different answer than if you use Lorentz contradiction formula.

So since both approaches yield different answers, I was wondering which one is correct and what the fallacy with the other one is.

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  • $\begingroup$ note that $(c\Delta t)^2 - (\Delta x)^2$ is the same for all frames of reference $\endgroup$
    – JEB
    Commented Jan 23, 2020 at 4:44

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If the observer in S' wants to measure the distance between the two events, he would do so at the same time in his frame and the L0/𝛾 formula would hold.

In the second equation you wrote, he is calculating the distance between the two events but not at the same time in his frame, rather at the two times corresponding to tA and tB, so he gets a different result.

Don't forget - the S frame is moving with respect to S' so the S' locations corresponding to xA and xB are constantly changing.

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