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Inhomogeneous (poincare) lorentz transormation 
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#1
Dec2911, 10:26 AM

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: [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\b eta}[/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...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 [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. 


#2
Dec2911, 10:37 AM

Mentor
P: 6,230

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] 


#3
Dec2911, 12:07 PM

Sci Advisor
P: 2,671

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



#4
Dec2911, 01:36 PM

P: 37

Inhomogeneous (poincare) lorentz transormation
Right! I knew it would have to have been something stupidly simple >.<
thanks guys. 


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