# Learning Cartan’s Geometry

I am very pleased to hear this, someone in a Department of Economics doing the learning work that standard physicists should be doing. There is a glossary of a kind in the attached book by Lar Felker. The first three chapters of Sean M Carroll are good too. Review papers UFT100 and UFT200 may also be useful as a guide, and also the ECE Engineering Model and some excellent work by the AIAS colleagues. A short glossary is given as follows.

1) The tetrad postulate is the requirement that the complete vector field be independent of the way in which it components and basis elements are described. This is very fundamental. In ECE theory all the wave equations of physics emerge from the tetrad postulate, including all of quantum mechanics minus the Heisenberg indeterminacy.
2) The first and second Cartan structure equations define the torsion and curvature respectively as vector and tensor valued two-forms of differential geometry:

T = D ^ q; R = D ^ omega

In ECE theory these define the relation between fields and potentials. They can be translated into the Riemannian torsion and curvature. The Riemann torsion and curvature are both given automatically and simultaneously by the action of the commutator of covariant derivatives on a vector or more generally a tensor of any kind and any rank in any dimension. If the Christoffel connection is symmetric both the curvature and torsion vanish because the commutator vanishes. So in order for the Cartan structure equations to produce a finite torsion and curvature, the Christoffel connection must be antisymmetric in its lower two indices. This inference refutes the entire twentieth century thought in general relativity. GENERAL RELATIVITY MUST INCLUDE FINITE TORSION.
3) The Cartan identity is

D ^ T := R ^ q = q ^ R

and in ECE theory defines half of the field equations of gravitation and electromagnetism. The Evans identity is:

D ^ T tilde := R tilde ^ q = q ^ R tilde

and defines the other half of the field equations in four dimensions. The Evans identity holds only in four dimensions, but the Cartan identity holds in any dimension.
4) Elie Cartan’s original geometry has been greatly developed in many directions by mathematicians, not least among whom was his son, Henri Cartan. Obviously they all accept the correctness of Cartan’s original geometry. This is so obvious that it hardly needs to be written down. Within its definitions, Cartan’s geometry is rigorously correct and self consistent. For example I have shown that in tensor notation the Cartan identity is exactly correct, one side is exactly the same as the other.

Many of the UFT papers show how the form notation is translated into tensor notation and then vector notation for engineers. Everything in ECE theory is based on Cartan geometry, whose correctness is very well known. I have broken out the proofs in all detail to help the readers and give much more detail than Sean Carroll, who himself gives an exceptional amount of detail. Any “attack” on Cartan geometry is total utter nonsense, as in the wikipedia article written by enemies of mine. I have no idea why this anomosity developed, that is their problem, not mine, or Cartan’s. Incidentally, in a letter to John B. Hart, Sean Carroll accepted ECE as a plausible theory. As for any theory, it must be tested against experimental data. Since it produces all the equations of physics (at least the correct ones), it has passed all its experimental tests so far. Curently I am making a whole series of new predictions of the fermion equation with Horst Eckardt.

In a message dated 29/11/2013 18:43:34 GMT Standard Time, writes:

Dear Myron. I agree you should encourage nascent students, and I especially agree with your comment on your current technical abilities. This is something I have noticed in myself, having been “forced” back through the differential and integral calculus and matrix algebra for my studies, and now enjoy and teach them at high levels. Regarding Cartan geometry, I have been reading your work for many years now, and convinced myself without great depth of understanding that your maths were right.

So, to deepen my understanding, I am learning Cartan-based differential geometry using a nice little (but rigorous) Dover book. And it is not difficult.

So, to the Colas Chabauds of the world, enforce the necessity of Cartan differential geometry. The only thing I feel I am missing at this point, so am committed to build, is a glossary of relevant operators and functions (unless I have missed that in all your work).

Very best,

Steve Bannister

Evans_Equations_%20Rev3.pdf