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4 The Field Equations Having introduced the h- and - elds, we now look to construct an invariant action integral which will provide a set of gravitational eld equations. 3) The eld R(a ^ b) is a bivector-valued linear function of its bivector argument a ^ b. Its action on bivectors extends by linearity to the function R(B ), where B is an arbitrary bivector and therefore, in four dimensions, not necessarily a pure `blade' a ^ b. Where required, the position dependence is made explicit by writing R(B x) or Rx(B ).

In addition, it is often necessary to enforce some gaugexing conditions. For the rotation gauge these conditions are applied by studying R(B ), so it is important to analyse its general structure. 38). 3) This derivation illustrates a useful point. Many derivations can be performed most e ciently by working with the Da , since these contain commuting partial derivatives. 39 However, the nal expressions take their most transparent form when the h- eld is included so that only fully covariant quantities are employed.

6) which summarises all the symmetries of R(B ). This equation says that the trivector @a ^ R(a ^ b) vanishes for all values of the vector b, so gives a set of 4 4 = 16 equations. These reduce the number of independent degrees of freedom in R(B ) from 36 to the expected 20. It should be clear from the ease with which the degrees of freedom are calculated that the present geometric algebra formulation has many advantages over traditional tensor calculus. 1 The Weyl Tensor A good example of the power of the present approach is provided by an analysis of the Riemann and Weyl tensors.