The Cartan Identity


The rigorous proof of this identity of standard differential geometry was first given in paper 15, and then in more detail in paper 102. In my shorthand notation it is:

D ^ T := R ^ q


D ^ := d ^ + omega ^

Here d ^ is the exterior derivative, T is the torsion form, R is the curvature form, omega the spin connection and ^ is the wedge product. The tensor formats of R and T are given in Eqs. (9.6) and (9.7) of paper 102, and the tensor format of the identity in Eq. (9.12). The latter is proven to be Eq. (9.20) of this paper, an exact identity, consisting of the cyclic sum of three curvature tensors. This sum is identically equal to the cyclic sum of the definitions of the same curvature tensors. Without the torsion, differential geometry collapses, and all of standard cosmology collapses, because it neglects torsion. Many papers of the ECE series show this using computer algebra, otherwise the hand calculations are far too laborious. It took until 2007 to uncover this catastrophe for standard cosmology. We have replaced the obsolete standard cosmology with one correctly based on spacetime torsion. Every pair of indices mu and nu of the Cartan identity obey the identity, and so

D ^ T tilde := R tilde ^ q

where tilde denotes Hodge dual transformation. The Cartan identity is Hodge dual invariant. The wave and field equations of physics in ECE are based directly on this well known textbook geometry. The latter can be developed into a more abstract format. However for engineering the opposite course of action is needed, reduction to vector format, and this has been done in the ECE engineering model powerpoint slides available on this site.


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