Unraveling the Mystery of Spin Connection in an Expanding Universe

[JorisL]

Hello,

I've worked through most of Carroll's appendix on the non-coordinate basis.
I see and agree how the spin connection and tetrad one-forms are useful while calculating.

However as an example he sets out to apply the formalism to a spatially flat, expanding universe.
$$ds^2 = -dt^2 +a^2(t)\delta_{ij}dx^idx^j = -e^0\otimes e^0 + \sum_i e^i\otimes e^i$$

The choice of the vielbein one-forms is clear in this case as is the calculation.

He states however that by using raising and lowering indices on the spin connection $\omega_{ab}$ we can show the following identities by using the antisymmetry. (I copied the expressions verbatim, i and j are different from 0)

$$\omega^0_{\,\, j} = \omega^j_{\,\, 0}$$

Here I wrote $\omega^0_{\,\, j} = \eta^{0a}\omega_{a j} = - \eta^{0a}\omega_{j a} = \omega_{j}^{\, \, 0}$
This is similar but not the same.

$$\omega^i_{\,\, j} = -\omega^j_{\,\, i}$$
For the other relation I can do exactly the same and get a similar result.
With the same difference in index position.

So I don't get the same relation. Is he abusing notation here?
Or am I overlooking something? It bothers me without measure whenever I encounter such a problem.

Thanks,

Joris

 

[WannabeNewton]

Here I wrote $\omega^0_{\,\, j} = \eta^{0a}\omega_{a j} = - \eta^{0a}\omega_{j a} = \omega_{j}^{\, \, 0}$

$-\eta^{0a}\omega_{ja} = -\eta^{00}\omega_{j0} = \omega_{j0} = \omega^j{}{}_0$

 

[JorisL]

Thanks, I don't know why I couldn't get it.
But now I can continue to do the mixmaster universe in the tetrad formalism.