Asymptotically flat gravitating spinor field solutions. Step 2 - the   compatibility of Dirac equations in a curve and a flat spaces
  Using the fact that a spin connection is defined to an accuracy of a vector it is shown that the spin connection should be modified in such a manner that Dirac equation in a curve space would be compatible with Dirac equation in a flat space. 
V. Dzhunushaliev
gr-qc