# Dirac equation in curved spacetime

As we know, the laws of physics in curved spacetime are obtained to lowest order by upgrading the flat space laws by substituting partial derivatives with the appropriate covariant derivatives. In the case of the Dirac equation however, we are dealing with a spinor, not a tensor, so it's not clear to me how to interpret the covariant derivative. I've seen $$\gamma^M( \partial_M + \Gamma_M )\psi$$ where $$\gamma^M$$ is some upgrade of the flat Dirac matrices and $$\Gamma$$ is the 'spin connection'.

I'm looking for what you think is a good pedagogical reference that carefully explains/motivates all this. I am comfortable with GR (but haven't done much tetrad stuff).

Ultimately, my goal is to understand the absorption problem for neutrino waves by a Schwarzschild BH (which I think means I won't be needing the extra technology of the Teukolsky equation, relevant to Kerr), so if you know of a good reference thereof, that would also be very appreciated.

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