213(4): Further Proof of the Tensorial Nature of the Christoffel Connection

Feed: Dr. Myron Evans
Posted on: Tuesday, March 20, 2012 8:47 AM
Author: metric345
Subject: 213(4): Further Proof of the Tensorial Nature of the Christoffel Connection

This is further proof with checks for self consistency. Cartan geometry is not at all difficult if a few rules of index placement are followed. The essential idea of the geometry is to assume that there exists a tangent space at point P to the base manifold. The tangent space is a flat or Minkowski space if it is assumed to be a four dimensional spacetime. The base manifold is the general space. Cartan’s original intent was to introduce his spinors (which he himself inferred in 1913) into Riemann Cartan geometry. Cartan inferred his identity having inferred the tangent spacetime and tetrad postulate. The latter is in fact not a postulate, it is very fundamental. What Cartan showed is that Riemann Christoffel geometry is not complete. It is very well known to mathematicians that the methods of Cartan can be greatly developed, but for physics his geometry seems to be sufficient. In any event an ECE type theory can be developed with a more abstract geometry, but that is best left to mathematical specialists. Cartan also introduced the wedge product, the exterior derivative, and the differential form, all are fundamental advances in mathematics. With Maurer he introduced the two Cartan Maurer structure equations, two more fundamental advances. His geometry can be reduced to just three equations:

T = D ^ q; R = D ^ omega and D ^ T := q ^ R

so it is supremely elegant and well worth studying. In UFT211 for example the Cartan identity is used to prove the antisymmetry of the Christoffel connection. The tragedy of twentieth century general relativity is that it was all based on an early form of geometry that was not only incomplete, but incorrect. It was then blown into Shavian superstition (science made superstition) by the media, the desire for fame and fortune replacing science. George Bernard Shaw himself clearly doubted Einstein’s claims and interviewed him on film. In order to derive the complete set of ECE field equations I inferred in four dimensions the Evans identity:

D ^ T tilde := q ^ R tilde

In four dimensions this is simply an example of the Cartan identity. The reason is that in four dimensions the Hodge dual T tilde of T is another antisymmetric tensor, and similarly the Hodge dual R tilde is another antisymmetric tensor. The commutator method of UFT137 for example isolates the connection and defines its anisymmetry very clearly, and the latest papers prove that the Christoffel connection transforms as a tensor so never has a symmetric component and never has an inhomogeneous component in its transformation law. These are all fundamental advances in mathematics. There was a severe mental block in the twentieth century in that the curvature and torsion were accepted to be antisymmetric, but the connection was incorrectly asserted to be symmetric in a display of “pathological science” (Langmuir’s term for dogmatism). Many scientists have rebutted EGR for almost a century, all have been ignored by an egotistic few. The damage to science has been immense.
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a213thpaprnotes4.pdf

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