Clifford Calculus
Since three dimensional Clifford algebra is identical to the Pauli quaternion algebra we would expect the calculus to be identical and indeed this is the case. As with quaternions the moving orthonormal frame is convenient. The big difference here is that the orthonormal frame is a triad of basis vectors in ordinary three dimensional space. Each basis vector is of unit length. Time intervals are defined by the propagation of light signals across that unit length.
As before coordinates are defined as linear combinations of the basis vectors with differential form coefficients:
(Here, we use 
for the coordinate differential one-forms
rather than 
for the latter now represents a bivector.)
As we move away from the origin the basis vectors will rotate such that
where 
is a vector-bivector object which effects a small
rotation and a small Lorentz tranform. The Hermitian conjugate
operation is mathematically identical to the Pauli quaterion
Hermitian conjugate. In terms of the Clifford algebra basis it
appears as:
where A,B...H are ordinary scalars or differential form coefficients.
Again, the order of the product of the differentials must be preserved when calculating these expressions.
Example: Friedmann metric
The coordinates of a space time event are:
where r and t are scaled by the 'radius' a(t).
Differentiate this to find the change in basis vectors at neighboring points:
The basis vectors rotate according to the vector-bivector one form object :
(a(t)t denotes the derivative of a(t))
The curvature vector-bivector two form is
where
The total volume V of this universe:
Let M be the total (constant) mass of the universe. The energy density is:
The Energy density three form is:
Equating this energy density to the Einstein three form determines the function a(t):
