s.t. 
in a lorentz metric.
(it is 
in a riemannian manifold).
The dirac algebra uses dirac matrices s.t. in a lorentz metric.
(it is in a riemannian manifold).
Treat 
as a spinor under local lorentz transformations and as a scalar
under global coordinate transformations.
if 
, 
, then 
and 
. with 
.