I am trying to uderstand the behavior of Dirac spinors in terms of the scalar perturbations of the FRW metric and I was wondering:
Can the perturbation of the spin connection δωabμ be determined by perturbing the first structure equation dea=−ωab∧eb ? More explicitly, can i use the following
d(δea)=−δωab∧ˉeb−ˉωab∧δeb
?
In the above, the background quantities are being denoted with a bar.