I would reply to any criticism like this that eq. (14) is a new structure equation of differential geometry. The original two structure equations of Cartan, as you know, are:
T = D ^ q ; R = D ^ omega
and these remain the same. However, the Cartan identity
D ^ T := R ^ q
has a solution which is eq. (14), another definition of curvature, one that needs non zero T for non zero R. The original R is of course a solution of the Cartan identity also. The Evans identity is simply an example of the Cartan identity. In my opinion the commutator proof is very simple. It consists of
mu = nu
in which case the commutator becomes the null operator and both torsion and curvature vanish, reductio ad absurdum. The key point in that method is that there is a one to one correspondence between commutator and connection. In the old theory they simply omitted the torsion and this correspondence was incorrectly abandoned. So in order for T and R to exist the connection must be antisymmetric. If one takes the general case where the connection is hypothetically asymmetric then only its antisymmetric part contributes to T and R. Its symmetric part is zero. The symmetric part of a hypothetically asymmetric commutator is zero in precise analogy. The structure equations of Cartan are in the last analysis, definitions. The Cartan identity is an exact identity, and has two possible solutions. These are R = D ^ omega, and eq. (14). The original definition of curvature seems to have been given by Levi Civita and Ricci and co workers in about 1900 – 1905. At that time torsion was of course unknown. As eq. (12) shows, that definition consisted of grouping a particular combination of terms on the right hand side of the equation. Then as in UFT137 the three combinations are each made equal to a curvature tensor. So in the usual method a solution of the identity was CHOSEN in order to give R = D ^ omega. That is not the only solution, eq. (14) is the other possible one. The precisely correct groupings of terms are shown in eq. (12)’s right hand side. The correct grouping shows that if the connection is symmetric, the right hand side and left hand side vanish. Not only does the sum vanish, the curvatures vanish individually.
In a message dated 29/02/2012 08:21:38 GMT Standard Time,
The arguments about eq.(14) are convincing, but critcs will say that you have changed the definition of curvature. Then all equations containing curvature would have to be proven again. Is there an argument that this is not necessary? What are the consequences of changing this definition?
Betreff: 210(1): Proof that the Connection is Antisymmetric
This is the final version of note 210(1), proving conclusively in yet another way that the connection is antisymmetric. The note proves for the first time that curvature can be defined in terms of torsion as in eq. (14). Curvature is therefore constructed from torsion, which is the more fundamental quantity. The Einsteinian general relativity (EGR) is obviously incorrect because it violates the Cartan identity, and violates eq. (14). In EGR, torsion is incorrectly zero and curvature non-zero. This fact has now been proven in many ways, and funding of EGR should cease completely. In order to prevent this waste of public funds, the political will must be found to overrule advisors with vested interest. So the many government departments that follow this blog are advised accordingly. For mathematicians this note is not difficult to follow. For others however it is technically very difficult, which is why grant applications based on incorrect mathematics have been funded for such a long time. My aim is to give the geometrical truth, but I also advise governments all the time through this blog. The eight year feedback data bank shows this conclusively, and I advise as a British Civil List scientist, impartially and with no vested interest. I have discovered several new fundamentals of differential geometry, and eq. (14) is yet another new discovery. It is a solution of the Cartan identity. I often go over notes to improve them, and produce final versions. Thanks to Horst Eckardt and Douglas Lindstrom for critical comments on each note. They have the technical ability to understand all aspects of ECE and are also impartial with no vested interest in incorrect mathematics. No honest intellectual should ever have a vested interest in a theory that has been proven wrong so many times, in so many ways, by so many scientists – the EGR theory.