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.
