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 $\gamma_{\mu}(x)$?

  • 1
    $\begingroup$ Why should we do that? $\endgroup$ – Horus Sep 23 '15 at 15:23
  • $\begingroup$ Uh...you have to couple a contravariant something to the covariant $\partial_\mu$ in there toget something diffeomorphism invariant, no? $\endgroup$ – ACuriousMind Sep 23 '15 at 15:27
  • $\begingroup$ 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 $\endgroup$ – amateurRebel Sep 23 '15 at 15:30
  • $\begingroup$ Related post by OP: physics.stackexchange.com/q/208322/2451 $\endgroup$ – Qmechanic Sep 23 '15 at 15:34
  • $\begingroup$ @amateurRebel To whom is that comment meant for? $\endgroup$ – Horus Sep 23 '15 at 15:38

I believe I have answered the question myself in my work uploaded at


Any comments positive or negative are welcome and it would be really nice if someone can endorse me on arxiv.


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