Denoting by γa the Minkowski space gamma matrices with respect to the Lorentz tetrad {ea}, and covariant derivative Da, then the gammas are covariantly constant.
Start with the massless Dirac equation γbDbΨ=0
Act again with the Dirac operator γaDaγbDbΨ=0 So, since D annihilates γ γaγbDaDbΨ=0 so 12{γa,γb}DaDbΨ+12γaγb[Da,Db]Ψ=0 (1) But {γa,γb}=2ηab and [Da,Db]Ψ=RabΨ Where Rab is the spin-curvature (antisymmetric in a and b). Rab satisfies the identity −γbRab=Rabγb=12γbRab where Rab is the Ricci tensor (in the Lorentz tetrad). so (1) becomes [DaDa+14γaγbRab]Ψ=0 i.e. [DaDa−14R]Ψ=0
This post imported from StackExchange Physics at 2014-03-22 17:27 (UCT), posted by SE-user twistor59