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A coordinate transformation from an initial domain of variables R =
{r,theta,phi,tau} will be given to a final range of {x,y,z,t}. An unperturbed
metric will be prescribed on the final state as [eta] = {-1,-1,-1,+1}. The
Jacobian matrix of the map will be used as the unperturbed (invertible) Frame
matrix of functions with arguments on the initial state. The Frame matrix will
be used to compute the right Cartan matrix of 1-forms [C] on the initial state.
The pullback metric [g], induced on the initial state by the compatibility
condition
will be used to compute the Christoffel connection [Gamma] in terms of the
metric on the initial state.
The two connections will be compared and will be identical for the unperturbed
case:
The curvatures induced by the two connections are also identical and equal to
zero.
Next, two types of perturbations will be applied. The first perturbation will
be to the metric on the final state. Again the Cartan Frame based connection
[C] and the Christoffel metrical based connection [Gamma] will be computed on
the initial state. The connections are no longer the same. The Cartan
connection can be decomposed into two parts, a metrical part [Gamma] and a Ricci
rotation part [T]:
The Cartan matrix of curvature 2-forms will be computed by the formula
As the Frame manifold has an inverse [G], the Cartan matrix of curvature 2-forms vanishes [Theta] = zero. Such spaces are called A4 spaces, and are said to be "flat". However, the total curvature formula can be written as [Theta] = [dC]+[C]^[C] = {[dGamma]+[C]^[Gamma]} - {[dT]+[C]^[T]}.
The first term will be defined as metric curvature relative to the Cartan
connection [C], and the second term will be defined as inertial curvature
relative to the Cartan connection [C].
As the total curvature [Theta] is zero, it is apparent that the metric curvature
is balanced by the inertial curvature. It is conjectured that this result is in
effect the proper definition of what intuitively is known as the principle of
equivalence.
The second type of perturbation will be on the Frame matrix, and will introduce
torsion to the Cartan connection. There are two species of Torsion (besides
left and right handedness). One species of Torsion will be due to "vorticity"
around a space-space axis, and the other will be due to a "vorticity" around a
space time axis. The Cartan connection again will be compared to the
Christoffel connection and the Ricci rotation terms [T] are not zero. The
conclusion holds even though the perturbation of the final state metric is zero.
Again, the metrical curvature is balanced by the inertial curvature to produce a
total zero Cartan curvature.
When both types of perturbations are present simultaneously interactions between
the two pertubations can occur, but the fundamental result of metric - inertial
equivalence still holds.
{[dGamma]+[C]^[Gamma]} = {[dT]+[C]^[T]}.
{[ metric curvature ]} = {[ inertial curvature ]}.
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