Commentary on a section of
M.W. Evans' Annalen submittal Paper#89

APPENDIX ONE: THE STANDARD CARTAN STRUCTURE EQUATIONS AND IDENTITIES USED IN ECE THEORY.

Gerhard W. Bruhn, Darmstadt University of Technology

July 21, 2008
(Quotations from Evans' original text in
black, comments in blue)



. . .

The second structure equation of Cartan is:

                Ra = D Ù ωa = d Ù ωa + ωacÙ ωc                                       (1.5)

better written as

                Rab = d Ù ωab + ωacÙ ωcb                                                 (1.5')

. . .

The first identity of Cartan geometry is:

                DÙTa := RabÙ qb                                                                 (1.7)

. . .

Finally the second identity of Cartan geometry is a restatement of the second structure equation:

                DÙRa = DÙ(DÙ ωa)                                                 (1.11)

better written as

                DÙRab = DÙ(DÙωab)                                                 (1.11')

Note that for a Christoffel connection, i.e. if and only if the torsion vanishes, Eq.(1.11) becomes

                DÙRab = 0                                                                 (1.12)

Eq.(1.12) is always true, for general torsion, not only for a Christoffel connection. See e.g. F.W. Hehl's and Y.N. Obukhov's book [2, p.208] or G.W. Bruhn's web-note [3, Sect.2].

Where is the problem?

The problem is that Evans' New Math does not know the Poincaré formula dÙd = 0 (simply forgotten), and hence Evans cannot understand the proofs for DÙRab = 0 that are folklore of literature.

References

[1] M.W. Evans, APPENDIX 10: REBUTTAL OF G. BRUHN'S COMMENTS
    ON THE LORENTZ COVARIANCE OF THE B CYCLIC THEOREM
,
    Part of web-paper #89,
        http://www.aias.us/documents/uft/a89thpaper.pdf

[2] F.W. Hehl, Y.N. Obukhov, Foundations of Classical Electrodynamics −
    Charge, Flux and Metric
,
    Birkhäuser 2003, ISBN 0-8176-4222-6, ISBN 3-7643-4222-6

[3] G.W. Bruhn, Remarks on Evans' paper #100 - Section 2.,
        http://www.mathematik.tu-darmstadt.de/~bruhn/onMwesPaper100-2.html



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