Deriving the Metric
The contravariant four-vector for the relativistic energy and momentum of a free particle as a wave solution of the Dirac equation is,
(28)
The coordinate four-vector is,
(29)
If at each point we define a space-time
manifold by its events and its shape as determined by its
polarizability as hypothesized, then the gradient derivatives of
the polarizability with respect to the parameter 
are tangent to
the manifold at each point. The derivatives taken in Cartesian
coordinates describe the permeability and permittivity at the
point P as a pair of four-vectors tangent to the
manifold. The gradients of capacitance and inductance. The speed
of light, like permeability and permittivity is independent of
the inertial coordinate system in which it is measured. It is
determined by the inner product of these two vectors.
(30)
By decomposing these two four-vectors into two
unit four-vectors 
and 
, and normalizing the invariant, we have the inner
product,
(31)
which determines the components of the Refractive metric tensor [1].
(32)
To derive these components the field strength is broken into the source field and the polarization. Then solved for the components which are added to the identity tensor.
(33)
These are the equations that determine the refractive index at each set of coordinates.