Feed: Dr. Myron Evans
Posted on: Wednesday, February 29, 2012 2:17 AM
Subject: New Structure Equation
|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?