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The usual covariant derivative for the Dirac equation in curved space is:

$$D_\mu \psi = (\partial_\mu - {i \over 4} {\omega_\mu}^{ab} \sigma_{ab}) \psi$$

However, I think I found another possibility:

$$D_\mu \psi = (\partial_\mu - {i \gamma^5\over 4} {\omega_\mu}^{ab} \sigma_{ab}) \psi$$

This is because $\gamma^5$ gives the term a minus sign for the right-handed component of the spinor in the Weyl representation. Thus we have the transforms:

$$\psi_L \rightarrow e^{+i \epsilon^{ab} \sigma_{ab} } \psi_L$$

$$\psi_R \rightarrow e^{-i \epsilon^{ab} \sigma_{ab} } \psi_R$$

So they both transform correctly just with one the $i$ is replaced with $-i$ which is acceptable.

I currently can't see any reason to rule out the second possibility. Unless I'm missing something? So is this another alternative Dirac equation in curved space-time that could be true?

Edit: Investigating furthur, with the $\gamma^5$ in place instead of affect left and right handed particles differently, it would affect positive and negative particles differently leading to an apparent electromagnetic field from a rotating black hole.

Edit: I guess this symmetry would break the mass term(?) (but then so do weak interactions.):

$$m\overline{\psi}_R\psi_L$$

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    $\begingroup$ Well, the $\gamma^5 ~\sigma$ amounts to an ~ $\epsilon \sigma$, so ingrained parity breaking at the tangent space Lorentz level. It is not "ruled out", except by physics. You might as well ask why you can't add an extra $\epsilon$ to the Lorentz group parameters! $\endgroup$ – Cosmas Zachos Mar 26 at 15:55
  • $\begingroup$ Yep it seems like the $\gamma^5$ makes it more like an electromagnetic field than a gravitational field. Well to be more precise it would make particles and anti-particles move differently near a rotating black hole. Well... don't know if that would be ruled out though! $\endgroup$ – zooby Mar 26 at 16:44
  • $\begingroup$ If you're creating models where particles and antiparticles behave differently due to gravity, I'd be much more interested in studying their different behaviour in the early universe than I would near a black hole. $\endgroup$ – Jerry Schirmer Mar 26 at 16:54
  • $\begingroup$ @Jerry well the affects seem to only be relevant near rotating gravitational sources. So I'm not sure how much rotation was going on near the big bang. $\endgroup$ – zooby Mar 26 at 16:56

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