As we have seen, a (charged) matter distribution on flat spacetime is described by the fields Ja, Fab, and Tab, where Ja determines the charge density, Fab describes the electromagnetic field, and Tab determines the four-momentum density. The background geometry of M determines ∇a and the constant tensor fields gab and εabcd. Nature does not, of course, allow Ja, Fab, and Tab to vary in an arbitrary manner, but imposes constraints in the form of conservation laws and equations of motion, which may be expressed in the form of field equations. In this chapter we shall discover the field equations satisfied by Ja, Fab, and Tab.

Conservation Laws

Consider a congruence whose curves are the particle world lines of a uniform dust distribution with particle-number current ja. Let us select one particular curve, l0, and three neighboring curves, l1, l2, and l3, which are joined to l0 by three space-like connecting vectors aa, ba, and ca (Fig. 8.1).

According to an observer with four-velocity va orthogonal to aa, ba, and ba, the volume V = εabcd vaabbccd will always contain the same number of particles. Note that va is determined uniquely by the condition that it is orthogonal to aa, ba, and ba, but this does not mean that it is Liepropagated along the congruence.