'' DEFINITIVE PROOF 5: CARTAN EVANS IDENTITY.''

March 08, 2009

Evans starts with the 1st Bianchi identity:

D Ù T^{a} =
R^{a}_{b} Ù q^{b} .

His assertion is that this identity can be ''dualized'' to yield a new identiy of differential geometry not yet known before

(*)
D Ù T ^{~ a} =
R^{~ a}_{b} Ù q^{b}

and named ''CARTAN EVANS IDENTITY'' where E. Cartan would immediately have renounced his authorship. So to be correct we have to discuss the validity of the ''EVANS IDENTITY'' (*) here.

Evans transforms his identity by a somewhat dubious way to tensor calculus to obtain

D_{μ}T^{κμν} = R_{μ}^{κμν}
(6)

This is Evans claim: According to Evans, Eq.(6) is an identity of general differential geometry equivalent to Eq. (*).

Therefore it should hold for all (pseudo-)Riemanian manifolds irrespective of their torsion. A great subclass consists of those manifolds with Christoffel connection, i.e. symmetric connection in the lower indices. Then the torsion vanishes, and Eq.(6) reduces to the condition

0 = R_{μ}^{κμν}
(6')

However, and this is Evans' problem, as his coworker Eckardt has obtained by computer evaluation, all non-trivial solutions of the Einstein equation fail to fulfil Eq.(6'). This means, Eckardt has shown that a large class of manifolds with zero torsion are counter examples to the validity of Evans' (transformed) ''EVANS IDENTITY'' (6).

This is the reason why Evans is making desperate efforts to preserve his assertion of the antisymmetry of all connections. If so then all ''classical'' manifolds with vanishing torsion were erroneous and would not yield counter examples for his EVANS IDENTITY (6).