S= J d4x, (2-101)
where the integration is performed over the region between any two spacelike
hypersurfaces. With variations taken in the dynamical variables za'and Aa which
vanish on these hypersurfaces, the action suffers the variation
6S = (-moga'b' + CeFab ) 6za'dr
+ i [- (47)-1 g12Fab;b + ] j Aad4x, (2-102)
provided that r is taken to be the proper time of the particle (and will henceforth
be assumed) such that
ga'b' ab' = -1, (2-103)
ga'b hab' = 0, (2-104)
ga'b/ a'iZb' g= -a'b' 2. (2-105)
Application of this action principle yields the dynamical equations
mnoxa' eFib' b', (2-106)
gl/2 Fabb 47a. (2-107)
By the fact that
1
Fabba (Rbac Fb + RbabcF a) 0, (2-108)
one can show via Eq. (2-107) that the current density is conserved.
Conservation of the stress-energy tensor.
The stress-energy tensor of the system is given by
Tab tickle + TIabd,
Particle field'
(2-109)