Proof of Eq. (16) of UFT142
To: Myron Evans <myronevans123>
Pleasure! There are many UFT papers dedicated to rigorous proofs of geometry, and several are already classics. In the early days of ECE I studied Carroll’s chapter three and found it to be very condensed. I doubt whether any chemist or engineer could follow it, and only a few theoretical physicists.Yet this is supposed to be a textbook. In my experience people take one look at Cartan geometry and give up immediately. This is a pity because Cartan geometry is not that difficult. The rigorous and now classic proofs reduced the charlatans to silence about twelve years ago. So a textbook should aim at clarity and give all detail of important proofs. Leaving a proof as an exercise for the student is useless. In my experience at UNCC the students would not be able to handle Carroll’s book at all. Even a master’s class found Marion and Thornton to be very difficult. The huge readership of our sites shows that we have found how to teach properly. The UFT papers already give all detail of the important proofs, so a really good textbook can build on this. You have the technical ability to write such a textbook. For example the Cartan identity is proven in all detail in Appendix C of UFT15. In minimal notation it is D ^ T = R ^ omega = omega ^ R. This is very elegant, but very condensed. The UFT papers expand it to tensor and vector notation, and the ECE2 phase of development reduces the complexity of internal indices, so we achieve an elegant result – ECE2 covariance, which has already produced nearly a hundred papers and books. The Evans identity was proven in the same way in UFT137. So I decided to expand the proofs in to tensor notation and vector notation as you know. This early work is summarized in the appendices of UFT15, appendices which give tensor proofs of the first and second Maurer Cartan structure equations and the first and second Cartan identities. Later I developed the second Cartan identity in what has become a famous paper, UFT88, in co authorship with yourself. There is no way of simplifying further than these tensor proofs, which show results that are rigorously identical. Only then does the true nature of the identities emerge. The readership often finds the switching of dummy indices to be very difficult to understand. UFT99 is another classic paper which gives full details of the commutator method. If one googles "Proof of the Bianchi identity", UFT104 comes up on the second page of Google, showing that it has become another classic. The Evans identity is proven in UFT137. The Evans torsion identity is proven in UFT109. I also recommend UFT255, and UFT313 puts all the concepts together to give the Jacobi Cartan Evans identity. UFT100 is useful for its charts, as is "Criticisms of the Einstein Field Equation" (CEFE). PECE is a useful summary. If one plays around with google keywords one sees the UFT papers often appearing on the first or second pages. An ideal textbook would give all details of important proofs, an example is Appendix C of UFT15. This paper comes up on the first page of Google with keywords "spinning curving Cartan".
Thanks for the proof. I think I can now formulate what I wanted to explain.
I am beginning now the section with the Bianchi identity, Evans identity and Jacobi Cartan Evans identity. According to an earlier email, these are described in papers 109, 313 and 354. Can you recommend papers which are best suited to describe these identities in textbook manner?
Horst
Am 13.04.2018 um 12:51 schrieb Myron Evans:
Horst is writing what I regard as an important textbook, in which all details such as this proof can given. He asked for the proof of Eq. (16) of UFT142, a paper which gives simplified proofs of the first and second Maurer Cartan structure equation, the foundational equations of differential geometry and ECE theory. This proof was left as the proverbial "exercise for the student" by Sean Carroll, in a Harvard graduate class. Carroll gives only the proof of the first structure equation in his chapter three of "Spacetime and Geometry". All the proofs left out by Caroll are given in the UFT series.