# Inhomogeneous (poincare) lorentz transormation

1. Dec 29, 2011

### spacelike

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:
$$\eta_{\mu\nu}\Lambda^{\mu}_{\alpha}\Lambda^{\nu}_{\beta}=\eta_{\alpha\beta}$$
(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 $\mu$ 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.

2. Dec 29, 2011

### George Jones

Staff Emeritus
In an inertial coordinate system,
$$\eta_{\alpha\beta} = \eta_{\mu\nu}\Lambda^{\mu}_{\alpha}\Lambda^{\nu}_{ \beta} = -\Lambda^{0}_{\alpha}\Lambda^{0}_{ \beta} + \Lambda^{1}_{\alpha}\Lambda^{1}_{ \beta} + \Lambda^{2}_{\alpha}\Lambda^{2}_{ \beta} + \Lambda^{3}_{\alpha}\Lambda^{3}_{ \beta}$$

3. Dec 29, 2011

### Matterwave

You're not summing, you've just assumed 4 values for the 4 variables. Remember you have to sum over mu and nu.

4. Dec 29, 2011

### spacelike

Right! I knew it would have to have been something stupidly simple >.<

thanks guys.