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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.

[tex]ds^2 = -dt^2 +a^2(t)\delta_{ij}dx^idx^j = -e^0\otimes e^0 + \sum_i e^i\otimes e^i [/tex]

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)

[tex]\omega^0_{\,\, j} = \omega^j_{\,\, 0}[/tex]

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.

[tex]\omega^i_{\,\, j} = -\omega^j_{\,\, i}[/tex]

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

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# Spin connection

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