**Subject:** 131(1) : Some Fundamental Tetrad Properties

**Date:** Sun, 26 Apr 2009 07:46:00 EDT

These are some fundamental and important mathematical properties of the tetrad. It may be defined in any dimension n, and is an invertible n x n matrix. Invertible means that it has a well defined inverse, its adjoint divided by its determinant. This can only be done for a square matrix in a starightforward way, but mathematically may be possible for other types of matrix. I have never seen this procedure in a textbook, but it may exist in the mathematical literature. To simplify matters it is best to use a square matrix for the tetrad. This leads to the generally covariant equation of the fermion (eq. (20). To adhere to the rules on the tetrad, the generally covariant equation of the quark must be developed with n x n matrices relating spinors of dimension n. In the three quark model (SU(3)) the tetrad or wavefunction is a 3 x 3 matrix related to teh Gell-Mann matrices. In the four quark model it is a 4 x 4 matrix, in the five quark model it is a 5 x 5 matrix, and in the six quark model it is a 6 x 6 matrix. For the gravitational field it is a 4 x 4 matrix, and also for the electromagnetic field. So we use that ECE unifies the fields of physics quite straightforwardly provided the rules governing the tetrad are understood. Its a index is in an orthonormal space, its nu index in a general spacetime. The tetrad is a rank two tensor with mixed indices. Its covariant derivative is therefore of a rank two tensor. The dimensionality of mu in the covariant derivative may be 4, while that of a and nu may be 2, 3, 4, 5, 6 in physics. The dimensionality of mu may therefore be different form that of a and nu, which must be the same.

Attachment: a131stpapernotes1.pdf

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