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How to raise and lower the indices of dirac matrices in curved spacetime? Do we just use $g_{\mu \nu} $ ?

your are probably talking of \(\mu\) alone. Could you be more precise ?

@igael From what I understand the dirac matrices are constructed in curved spacetime with multiplying the standard gamma matrix with vielbein, say I want to lower the index of curved gamma matrix then do I use $ g_{\mu \nu} $ or flat metric? Also in flat space we have $\epsilon_{12} = 1$ and $\epsilon_{21} = -1$ in flat space. How do I go from $\epsilon_{r \theta}$ to $ \epsilon^{r \theta}$ ?

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