# Derivation of Dirac equation in curved spacetime

In all the Literature I have read, the covariant Dirac equation in curved spacetime is given as

\begin{equation} \left(i\hbar\gamma^{\mu}(x)\left[\frac{\partial}{{\partial}x^{\mu}}-{\Gamma}_{\mu}(x)\right]-mc\right)\psi(x)=0 \end{equation}

Where $\gamma^{\mu}(x)$ are the contravariant forms of curvature dependent Dirac matrices $\gamma_{\mu}(x)$ defined as

\begin{equation} \gamma_{\mu}(x)\gamma_{\nu}(x)+\gamma_{\nu}(x)\gamma_{\mu}=2g_{\mu\nu}\end{equation}

None of the references I have gives any justification for using contravariant form of the curvature dependent Dirac matrices. My question is why do we use the contravariant form $\gamma^{\mu}(x)$ and not the covariant form $\gamma_{\mu}(x)$?

• Why should we do that? – Horus Sep 23 '15 at 15:23
• Uh...you have to couple a contravariant something to the covariant $\partial_\mu$ in there toget something diffeomorphism invariant, no? – ACuriousMind Sep 23 '15 at 15:27
• Because then we can derive the Dirac equation in curved spacetime from the energy-momentum relation in curved spacetime, just like Dirace equation in flat spacetime is derived from energy-momentum relation in flat spacetime. See my other question link – amateurRebel Sep 23 '15 at 15:30
• Related post by OP: physics.stackexchange.com/q/208322/2451 – Qmechanic Sep 23 '15 at 15:34
• @amateurRebel To whom is that comment meant for? – Horus Sep 23 '15 at 15:38