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Old Oct 1st 2016, 02:10 PM   #1
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Join Date: Oct 2016
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Tensor calculation in Lorentztransformations

Hi,
I have difficulties understanding the following relations. Given the Minkowski metric $\displaystyle \eta_{\alpha\beta}=diag(1,-1,-1,-1) $ and the line segment $\displaystyle ds^2 = dx^2+dy^2+dz^2$, then how can i see that this line segment is equal to $\displaystyle ds^2 = \eta_{\alpha\beta}dx^\alpha dx^\beta $. Further, we want the line segment to be unchanged under this metric. And i don't understand why the following equivalences hold true: $\displaystyle ds^2 = ds'^2 $ if and only if $\displaystyle c^2d\tau^2 = c^2d\tau'^2$
and $\displaystyle \Lambda^{\alpha}{}_{\gamma} \Lambda^{\beta}{}_{\delta} \eta_{\alpha}{\beta} = \eta_{\gamma}{\delta} \iff \Lambda^T \eta \Lambda = \eta

$
I am thankful for any kind of hints and tips and tricks. Thank you very much.
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