Why do we need tetrads/vierbeins/frame-fields to describe fermions in curved space?

I'm learning about the frame formalism and read that to couple fermions to gravity you need to go to the frame-formalism.

As a motivation to learn more about frame-fields would someone sketch me why this is necessary? If possible give nice references.

• – Qmechanic Aug 11 '16 at 9:13

The Lorentz group $O(3,1)$ has spinor representations (actually $SL(2,\mathbb C)$, that is the universal cover of $O(3,1)$), as well known. The problem is that now, in general relativity, we want to deal with generic transformations. So we are working with G$L(4)$.
Roughly speaking, the associated Lie Algebra $\mathfrak{gl}(4)$ doesn't admit spinor representations. See for instance: No Spinor The way out is to go to a local inertial frame, in which you recover the flat space and you can define the spinors.