Dirac Equation in Curved Spacetime
The Dirac equation in curved spacetime can be written by using vierbein fields and the gravitational spin connection. The vierbein defines a local rest frame, allowing the constant Dirac matrices to act at each spacetime point. In this way, Dirac's equation takes the following form in curved spacetime:
Here eaμ is the vierbein and Dμ is the covariant derivative for fermion fields, defined as follows
where σab is the commutator of Dirac matrices:
and ωμab are the spin connection components.
Note that here Latin indices denote the "Lorentzian" vierbein labels while Greek indices denote manifold coordinate indices.
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