These entries are grossly misleading and mathematically incorrect. In the barebones notation of differential geometry the so called “first Bianchi identity” is:

R ^ q =? 0

where the question mark denotes that it is mathematically incorrect. Here R is shorthand for the Cartan curvature form, a differential form, and q is shorthand for the Cartan tetrad. The second Bianchi identity (upon which the Einstein field equation is based directly) is

D ^ R =? 0

and is also incorrect. Here:

D ^ := d ^ + omega ^

where d ^ denotes the exterior derivative of differential geometry, and omega the Cartan spin connection. Here ^ denotes the well known wedge product of differential geometry, known and taught for eighty years.

It is now known the standard model’s incorrectly named first and second Bianchi identities are mathematically incorrect, and are not valid approximations. The reason is that they both use an incorrect connection symmetry where the lower two indices of the connection are made equal. We denote this error by:

mu =? nu

It is now known that the connection must take the antisymmetry of the well known and taught commutator of covariant derivatives. There is no symmetric commutator, in other words the symmetric commutator is zero. The symmetric connection is therefore zero. This means that spacetime torsion, denoted T, must be non-zero in any spacetime, in any dimension.

The correct Cartan Bianchi identity has been well known and well taught for eighty years, and is:

D ^ T := R ^ q = q ^ R

and is written out in tensor notation and proven rigorously in many entries of this site, for which a search facility is provided. It is invariant under Hodge dual transformation, which gives;

D ^ T tilde := R tilde ^ q

It was found by computer algebra from 2007 onwards (e.g. papers 93, 95, 96, 117, 120 etc. on this site) that all metrics of the Einstein field equation violate the Cartan Bianchi identity and its Hodge dual invariant. In retrospect this is obvious because Einstein made the basic blunder of omitting torsion. The true second Bianchi identity is developed in paper 88, and is the straightforward result:

D ^ (D ^ T) := D ^ (R ^ q)

In form notation it is simple, in tensor notation it is very complicated. However both notations are equivalent and valid.

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