An important e-mail from MWE to the AIAS members and supporters

February 11, 2005 12:22 CET
PROOF OF THE TETRAD POSTULATE FROM FUNDAMENTAL MATRIX INVERTIBILITY OF THE TETRAD.

Note that this proof is valid for the vielbein of any dimension, in four dimensions it becomes the vierbein or tetrad. The Leibnitz Theorem holds for covariant differentiation as discussed by Carroll. It is nice to see that the Leibnitz Institute of the Bavarian Academy of Sciences in Munich has visited www.aias.us many times. The tetrad postulate is more properly the Cartan tetrad postulate. It is the geometrical basis of the Evans wave equation and Evans Lemma of geneally covariant unified field theory. Fuller details of the different proof of the tetrad postulate is given in the appendices of my new book.

MWE

Myron W. Evans' handwritten paper

http://www.atomicprecision.com/new/atetradpostulatefromfirstprinciples.pdf

PROOF OF THE TETRAD POSTULATE FROM FIRST PRINCIPLES

by Myron W. Evans

Consider the following basic properties of the tetrad:

q_ν^b q_b^ν = 1                                                 (1)

q_μ^a q_a^μ = 1                                                 (2)

q_a^μ q_ν^a = δ_ν^μ                                                 (3)

q_μ^a q_b^μ = δ_b^a                                                 (4)

where δ_ν^μ and δ_b^a are the Kronecker deltas. We now differentiate eqs. (1) to (4) covariantly using the Leibnitz Theorem:

q_b^ν D_ρq_ν^b + q_ν^b D_ρq_b^ν = 0                                                 (5)

q_μ^a D_ρq_a^μ + q_a^μ D_ρq_μ^a = 0                                                 (6)

q_a^μ D_ρq_ν^a + q_ν^a D_ρq_a^μ = 0                                                 (7)

q_μ^a D_ρq_b^μ + q_b^μ D_ρq_μ^a = 0                                                 (8)

Rearranging dummy indices in eqn. (5) (a → b, μ → ν)
Really executed: (b → a, ν → μ) in the left summand of (5). OK.

q_a^μ D_ρq_μ^a + q_ν^b D_ρq_b^ν = 0                                                 (9)

Rearranging dummy indices in eqn. (8) (μ → ν)
Really executed: (μ → ν) in the right summand of (8), then both summands swapped. OK.

q_b^μ D_ρq_μ^a + q_ν^a D_ρq_b^ν = 0                                                 (10)

Multiply equ. (9) by q_μ^a

D_ρq_μ^a + q_μ^a q_ν^b D_ρq_b^ν = 0                                                 (11)

Multiply equ. (10) by q_μ^b

D_ρq_μ^a + q_μ^b q_ν^a D_ρq_b^ν = 0                                                 (12)

It is seen that equ. (11) is of the form:

x + a y = 0                                                                 (13)

and equ. (12) is of the form

x + b y = 0                                                                 (14)

where

a ≠ b .                                                                 (14)

The only possible solution is:

x = 0 .                                                                 (15)

y = 0 .                                                                 (16)

This means

D_ρq_μ^a = 0                                                                 (17)

D_ρq_ν^b = 0 .                                                                 (18)

Equ. (18) is the tetrad postulate of Cartan.
It is true for all connections because no restriction on the connection is used in deriving it.