Understanding Lorentz Group Generators: Derivation & Step in Eq 15

[VVS]

Hi,
I am trying to understand the derivation of the Lorentz generators but I am stuck.
I am reading this paper at the moment: http://arxiv.org/pdf/1103.0156.pdf
I don't understand the following step in equation 15 on page 3:
$\omega^{\alpha}_{\beta}=g^{\alpha\mu}\omega_{\mu\beta}$
I don't understand how this can be true. I mean g is not the identity matrix.

 

[stevendaryl]

Hi,
I am trying to understand the derivation of the Lorentz generators but I am stuck.
I am reading this paper at the moment: http://arxiv.org/pdf/1103.0156.pdf
I don't understand the following step in equation 15 on page 3:
$\omega^{\alpha}_{\beta}=g^{\alpha\mu}\omega_{\mu\beta}$
I don't understand how this can be true. I mean g is not the identity matrix.

That's sort of by definition. $g^{\alpha \mu}$ raises indices on a tensor and $g_{\alpha \mu}$ lowers them. Note that in general $\omega^{\alpha}_\beta \neq \omega_{\alpha \beta}$

In inertial coordinates, $g^{t t} = +1$, $g^{x x} = g^{y y} = g^{z z} = -1$, and $g^{i j} = 0$ if $i \neq j$. So in inertial coordinates, $\omega^{\alpha}_{\beta} = \pm \omega_{\alpha \beta}$.