# Dirac equation in curved spacetime

In mathematical physics, the Dirac equation in curved spacetime generalizes the original Dirac equation to curved space.

It 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:[1]

$i\gamma^a e_a^\mu D_\mu \Psi - m \Psi = 0.$

Here eaμ is the vierbein and Dμ is the covariant derivative for fermionic fields, defined as follows

$D_\mu = \partial_\mu - \frac{i}{4} \omega_{\mu}^{ab} \sigma_{ab}$

where σab is the commutator of Dirac matrices:

$\sigma_{ab}=\frac{i}{2} \left[\gamma_{a},\gamma_{b}\right]$

and ωμab are the spin connection components.

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