I'm reading a physics book and in the section on relativity they are using the Einstein summation convention, with 4vectors and matrices. They say that the transformations take the form: [tex]x^{\prime\mu}=x^{\nu}\Lambda^{\mu}_{\nu}+C^{\mu}[/tex] where it is required that [itex]\Lambda^{\mu}_{\nu}[/itex] satisfy the following relation: [tex]\eta_{\mu\nu}\Lambda^{\mu}_{\alpha}\Lambda^{\nu}_{\beta}=\eta_{\alpha\beta}[/tex] (note: I found the same thing on wikipedia, so you can see it in context if you like. http://en.wikipedia.org/wiki/Lorentz_transformation#Spacetime_interval it appears a tiny bit down from the section that the link takes you to.) My problem is that this seems impossible to satisfy by my current understanding, but I know I must be wrong, I just cannot see how. So we are summing over [itex]\mu[/itex] and [itex]\nu[/itex] in the above relation right? and we do this for all [itex]\alpha[/itex] and [itex]\beta[/itex] in order to satisfy all the components of the matrices. My problem is what happens when we get to the following situation?: [tex]\mu=0, \nu=1, \alpha=0, \beta=0[/tex] But, [itex]\eta_{01}=0[/itex], and [itex]\eta_{00}=-1[/itex]. So there is no possible values of the [itex]\Lambda[/itex]'s that will satisfy this because we now have 0=-1, which is a contradiction. Where did I go wrong with my thinking? Thanks.
In an inertial coordinate system, [tex]\eta_{\alpha\beta} = \eta_{\mu\nu}\Lambda^{\mu}_{\alpha}\Lambda^{\nu}_{ \beta} = -\Lambda^{0}_{\alpha}\Lambda^{0}_{ \beta} + \Lambda^{1}_{\alpha}\Lambda^{1}_{ \beta} + \Lambda^{2}_{\alpha}\Lambda^{2}_{ \beta} + \Lambda^{3}_{\alpha}\Lambda^{3}_{ \beta}[/tex]
You're not summing, you've just assumed 4 values for the 4 variables. Remember you have to sum over mu and nu.