## Key Properties of the Commutator

The ECE theory is basically of great simplicity, it makes use of well known mathematical properties in a completely original way. One of these well known properties is the commutator of covariant derivatives (see for example S. P. Carroll, “Spacetime and Geometry: an Introduction to General Relativity” (Addison Wesley, New York, 2004, and online notes circa 1997). The commutator is by definition antisymmetric in its indices, labelled mu and nu. It acts on any tensor (notably on a vector) to produce tensors and non-tensorial connections in which appear mu and nu. Whenever mu and nu appear together in a tensor or connection thus generated, that tensor or connection is individually antisymmetric in mu and nu by definition. If mu were equal to nu all terms disappear because there is no symmetric commutator by definition. The symmetric commutator operator is a null operator, generating null tensors and null connections. The symmetric connection used in the old physics is zero, it is simply wrong. This is the powerful new antisymmetry law of ECE theory now being developed in papers 122 to 136. One of these is a conference paper. The antisymmetry law means that the connection is itself antisymmetric, and so the torsion is identically non-zero. Neglect of torsion has been shown in papers 93 onwards to lead to complete disaster for the standard model, its gravitational physics and its cosmology. These have had to be rewritten from scratch in papers 93 to 136. The Hodge dual of the commutator operator is another commutator operator. This produces a variation on the Cartan identity,

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

becomes

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

(for notation see this site). These are examples of the same rigorously correct identity. In the same way the Hodge dual of a two-form in four dimensions is another two-form in four dimensions. The Hodge dual operator switches indices, from example from 01 to 23. These are indices of the same two-form.