The Dispersion Interpretation of Space-time Curvature
Now consider the derivatives of the metric tensor. These are known as the Christoffel field. Using comma "," notation for the partial derivatives with respect to the coordinates, the Christoffel symbols are the functions,
(39)
These derivatives are typically used to show the potentials of fictitious forces such as the Coriolis force. Using a Minkowski metric in Cartesian coordinates these derivatives vanish. However using the Refractive metric, the derivatives will only vanish if the permeability and permittivity four-vectors are constant! Otherwise these derivatives determine the geodesic motion of test particles that are solutions to the Dirac equation, on the polarizable space-time manifold. They can also be used to compute the curvature tensor, in terms of which the vacuum field equations of General Relativity exist.
Using the definition of the four-velocity,
(40)
where the invariant space-time interval is,
(41)
If the effect of the polarizability is small, then a solution to the Dirac equation is a plane wave solution of the form,
(42)
where 
are positive and negative solutions, and 
represents the
four Spinor solutions of the angular momentum. For this wave
function, free fall on the manifold is constrained by the
geodesic equation of motion,
(43)
One can always specify coordinates such that at any given point, for an instant of time the Christoffel symbols are zero. In other words, space-time is "flat" locally. These are referred to as geodesic coordinates [3].
This geodesic motion is analogous to a simple
harmonic oscillator consisting of a parallel connected inductor
and capacitor, ideally with zero resistance and a constant
quantity of electric charge oscillating between them. The
frequency of oscillation is determined by the values of
inductance 
and capacitance 
in the circuit. The energy stored by the
oscillator is also determined by these values. As they increase
the energy stored decreases in inverse proportion. The dispersion
of a wave packet is then analogous to the discharge of energy by
increasing the values of the inductor and capacitor in the
electronic oscillator circuit, while keeping the charge contained
therein constant