Frame and Metric Perturbations in 4D - A Schwarzshild Solution with Torsion

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

[g] = [Ftranspose][eta][F],

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:

[C] = [Gamma].

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]:

[ C ] = [ Gamma ] - [ T ].

The Cartan matrix of curvature 2-forms will be computed by the formula

[Theta] = [dC]+[C]^[C].

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 ]}.

Click here for maple program and its output that demonstrates perturbations that generate the Schwarzschild solution with torsion.

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Last update 07/09/2001
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