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.


1 Answer 1


The Wikipedia article "Dirac equation in curved spacetime" is rather short, but contains the basics and has links to longer articles.

In particular, the Arminjon paper cited in the Wiki article is quite readable and also has many useful references. (The original paper is by V Fock “Geometrisierung der Diracschen Theorie des Elektrons”, Zeit. Phys. 57, 261−277 (1929), but it is in German and not very accessible.)

It probably is necessary to learn the basics of tetrads, but this is not hard and is worthwhile even if you are not using Dirac.

On that note this arXiv paper by Yepez derives the spinor covariant derivative using tetrads, and has the bonus of providing a good introduction to vierbien spacetime.


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