# Inhomogeneous (poincare) lorentz transormation

 P: 37 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: $$x^{\prime\mu}=x^{\nu}\Lambda^{\mu}_{\nu}+C^{\mu}$$ where it is required that $\Lambda^{\mu}_{\nu}$ satisfy the following relation: $$\eta_{\mu\nu}\Lambda^{\mu}_{\alpha}\Lambda^{\nu}_{\beta}=\eta_{\alpha\b eta}$$ (note: I found the same thing on wikipedia, so you can see it in context if you like. http://en.wikipedia.org/wiki/Lorentz...etime_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 $\mu$ and $\nu$ in the above relation right? and we do this for all $\alpha$ and $\beta$ in order to satisfy all the components of the matrices. My problem is what happens when we get to the following situation?: $$\mu=0, \nu=1, \alpha=0, \beta=0$$ But, $\eta_{01}=0$, and $\eta_{00}=-1$. So there is no possible values of the $\Lambda$'s that will satisfy this because we now have 0=-1, which is a contradiction. Where did I go wrong with my thinking? Thanks.
 Mentor P: 6,246 In an inertial coordinate system, $$\eta_{\alpha\beta} = \eta_{\mu\nu}\Lambda^{\mu}_{\alpha}\Lambda^{\nu}_{ \beta} = -\Lambda^{0}_{\alpha}\Lambda^{0}_{ \beta} + \Lambda^{1}_{\alpha}\Lambda^{1}_{ \beta} + \Lambda^{2}_{\alpha}\Lambda^{2}_{ \beta} + \Lambda^{3}_{\alpha}\Lambda^{3}_{ \beta}$$