Proof of a Lorentz Transformation

[soothsayer]

Homework Statement

Show that the following is a Lorentz Transform:
$\Lambda _{j}^{i}=\delta _{j}^{i}+v^iv_j\frac{\gamma -1}{v^2}$
$\Lambda _{j}^{0}=\gamma v_j , \Lambda _{0}^{0}=\gamma , \Lambda _{0}^{i}=\gamma v^i$

where $v^2 =\vec{v}\cdot \vec{v}$, and $\delta _{j}^{i}$ is the Kronecker Delta.

Homework Equations

$\eta_{\mu\nu}=\eta_{\mu'\nu'}\Lambda_{\mu}^{\mu'} \Lambda_{\nu}^{\nu'}$
$\eta = \Lambda^T \eta \Lambda$

The Attempt at a Solution

I know how to go about proving a transform is a Lorentz transform, based on my "relevant equations", but I'm having a hard time setting the $\Lambda$ matrix up correctly. When I set up the matrix, I have terms in every cell, such as
$\Lambda_{1}^{1}=1+v^1 v_1 \frac{\gamma -1}{v^2}$
and
$\Lambda_{1}^{2}=v^2 v_1 \frac{\gamma -1}{v^2}$

and so on and so forth, but this feels wrong. I end up having to multiply two exceedingly complicated matrices along the way, which I know to be wrong (the professor hinted that excessive matrix multiplication was a sign you were doing the problem wrong.) How do I set things us? What I really want to know is, what is $\Lambda_{j}^{i}$? How do I handle the vector indices (vⁱ, v_j)?

 

[facenian]

I think you should do this, let $g_{\mu\nu}=\eta_{\mu'\nu'}\Lambda_{\mu}^{\mu'} \Lambda_{\nu}^{\nu'}$ then you sholud show that $g_{\mu\nu}=\eta_{\mu\nu}$. This you can do calculating for each case $g_{00},g_{0k}\, and\, g_{kl}$ using
$$g_{\mu\nu}=\eta_{\mu'\nu'}\Lambda_{\mu}^{\mu'} \Lambda_{\nu}^{\nu'}=\eta_{00}\Lambda_{\nu}^{0}\Lambda_{\mu}^{0}+\eta_{ij}\Lambda_{\nu}^{i}\Lambda_{\mu}^{j}$$
$$=-\Lambda_{\nu}^{0}\Lambda_{\mu}^{0}+\delta_{ij}\Lambda_{\nu}^{i} \Lambda_{\mu}^{j}=-\Lambda_{\nu}^{0}\Lambda_{\mu}^{0}+\Lambda_{\nu}^i\Lambda_{\mu}^i$$

 

[soothsayer]

Ok, that makes some sense to me. I'll give it a try, thank you!

The only part I couldn't follow is where you came up with the $\delta_{ij} \Lambda_{\nu}^{i} \Lambda_{\mu}^{j}$. Where did the delta come from? Sorry, I'm quite new at this sort of math.

 

[facenian]

The only part I couldn't follow is where you came up with the $\delta_{ij} \Lambda_{\nu}^{i} \Lambda_{\mu}^{j}$. Where did the delta come from? Sorry, I'm quite new at this sort of math.

assuming that latin indices take on values 1,2,3 while greek indices 0,1,2,3 then $\eta_{ik}=\delta_{ik}$ while $\eta_{00}=-1$

 

[soothsayer]

Ah, right, thank you!