A better way to think about this is to think about the Dirac equation being obtained from the action of a fermionic field in curved spacetime. Then a lot of the concepts generalize from the Minkowski metric in a more straightforward manner. In flat spacetime we have

\[\int d^4x ~\bar{\Psi} (i \gamma^{\mu} \partial_{\mu} - m) \Psi\]

But in a curved spacetime we move to

\[\int d^4x ~\sqrt{-g} ~\bar{\Psi} (i \gamma^a e^{\mu}_a\nabla_{\mu}-m) \Psi\]

where the \(e_a^{\mu}\) are the vielbein, which allow us to establish a locally Minkowski frame where the standard Dirac matrices \(\gamma^{a}\) satisfy the Clifford algebra, and \(\nabla_{\mu}\)is the covariant derivative.