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I am reading in a book that when coupling a spinor to gravity, one replaces [itex] \partial_\mu \psi [/tex] by a covariant derivative [itex] D_\mu \psi [/tex] which must transform like a spinor under local lorentz transformations but as a vector under general coordinate transformations. (and does that mean that [itex] \gamma^\mu [/itex] must transform as a vector under GCTs ?)

Can someone explain to me the logic involved here? I know that the D contains an index "mu" which indicates that this should be a vector under GCT's, but I don't really understand the rationale. I don't really understand the physical distinction between the LLT's and the GCT's. I thought that there was only the GCT which included the LLT as special case but this doesn't seem to be the case.

And (and I know this is a different issue), how is it possible to do a GCT in the first place? Imean, the curvature had a physical impact on the geodesics of particles so if I do a GCT that turns a flat region into a curved one, the physics is changed, obviously. So how can the theory be invariant under GCTs?? I know that this is something that bothered Einstein but I never understood the resolution of this issue. Invariance under GCT does not make sense to me since physics seems to be changed.

What does it mean to say that [itex] D_\mu \psi [/itex] must transform as a vector under a GCT? It's not possible to define spinors under GCT?