So there's something that has bothered me for quite some time.(adsbygoogle = window.adsbygoogle || []).push({});

I know the normal equations of time dilation and length contraction are the following:

[tex]\Delta t'=\gamma \Delta t[/tex] and [tex]L'=\frac{L}{\gamma}[/tex]

where the primed variables are in reference frame S' and the unprimed variables are in reference frame S. With frame S' moving with velocityvrelative to frame S.

Ok, so far so good. Now when we try to look at the same thing with Lorentz transformations (in other words: derive the above from Lorentz transformations)

we have:

[tex]t'=\gamma(t-\frac{v}{c^{2}}x)[/tex]

[tex]x'=\gamma(x-vt)[/tex]

When deriving time dilation we simply take it that the event which lasts durationtin reference frame S, is atx=0. Therefore we immediately get:

[tex]t'=\gamma t[/tex]

[tex]\Delta t'=t'_{2}-t'_{1}=\gamma t_{2}-\gamma t_{1}[/tex]

[tex]\Delta t'=\gamma \Delta t[/tex]

Great, that gives us time dilation as expected.. Now the problem I'm having is with length contraction. So starting again from the lorentz equation:

[tex]x'=\gamma(x-vt)[/tex]

This time we consider that we measure the length att=0, leaving us with:

[tex]x'=\gamma x[/tex]

Now to get length:

[tex]L'=x'_{2}-x'_{1}=\gamma x_{2}-\gamma x_{1}=\gamma L[/tex]

That means that according to Lorentz transformations:

[tex]L'=\gamma L[/tex]

However, according to the formula for length contraction which I wrote at the top:

[tex]L'=\frac{L}{\gamma}[/tex]

These are totally opposite!

what am I doing wrong?

Thanks

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# Quick question about lorentz transforms

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