Fwd: mixed metric for paper 128

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Subject: Fwd: mixed metric for paper 128
Date: Sun, 22 Mar 2009 11:20:47 EDT

This is a perfect example of the power of the Cartan geometry in generalizing the metric to the tetrad. The latter contains much more information than the metric. It also illustrates the correctness of the computer algebra code in all detail. In papers 93, 95 and 120 it was found that the metrics obeyed metric compatibility, this was another check on the code. In paper 122 however it was found that the wrong connection has been used in general relativity so all the curvature tensors of that era are also wrong because of neglect of torsion.

For all investigated metrics the mixed metric comes out to be a unit matrix. This seems to be a general result. By writing down the terms of a 2×2-metric (see example below), even if the metric is asymmetric, this seems to be a consequence of the fact that the contravariant metric is the inverse of the covariant metric. Therefore in the example the sum can be rewritten to

g^0 alpha * g_alpha 1 = g^00 * g_01 + g^01 * g_11 = g_11/Det * g_01 – g_01/Det * g11 = 0

where Det ist the determinant of the matrix. The contravariant elements can be replaced by expressions with the determinant. Anyhow this must be generalizable for 4 (and any) dimensions, probably by inserting the definition of the inverse matrix expressed by sub-determinants.

Horst


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