The Dirac Equation in Curved spacetime makes a difference between Lorentzian indicies and Covariant indicies. In the equation we find a $\partial_\mu$. Is this actually $e^a_\mu\partial_a$ where $e$ is the tetrad (or vielbein)? I.e. does this derivative look different than the regular derivative operator in flat space?
(To be clear I am not asking about the spin connection and covariant derivative, just if the partial will have addition factors from the tetrad.)