Dirac Equation - Dirac Equation in Curved Spacetime

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 e_aμ is the vierbein and D_μ is the covariant derivative for fermion fields, defined as follows

where σ_ab is the commutator of Dirac matrices:

and ω_μab are the spin connection components.

Note that here Latin indices denote the "Lorentzian" vierbein labels while Greek indices denote manifold coordinate indices.