ON THE SYMMETRY OF THE CONNECTION IN RELATIVITY
AND ECE THEORY:

by M.W. Evans
Civil List Scientist
www.aias.us

ABSTRACT

It is shown that the connection in relativity theory must always be anti-symmetric and that relativity theory must be based on a non-zero space-time torsion as in the Einstein Cartan Evans (ECE) field theory. . . .

. . .

1. INTRODUCTION

. . . In the ECE era it has also been recognized that there exists the Cartan Evans dual identity:

        D Ù T^~ = R^~ Ù q                 (3)

where T^~ is the Hodge dual of the torsion form, and R^~ is the Hodge dual of the curvature form. Neglect of torsion leads to a violation of the dual identity {2-10}, thus ending the Einsteinian era.

The fundamental reason for this error is the assumption of Einstein that the connection is symmetric. . . .

However, the concept of torsion was not fully developed until about 1922, when Cartan and Maurer gave the first structure equation:

        D Ù T = R Ù q                 (4)

In tensor notation this is equivalent to {1-10}:

        T^κ_μν = Γ^κ_μν − Γ^κ_νμ                 (5)

where Γ^κ_μν is the connection. If the latter is arbitrarily forced to be symmetric:

        Γ^κ_μν = ? Γ^κ_νμ                 (6)

the torsion vanishes. In the Einsteinian era therefore the torsion was seen es a complication which was arbitrarily removed. It was shown in paper 93 ff. on www.aias.us that this removal of torsion leads to a violation of geometry, i.e. a violation of the Cartan Evans dual identity. In Section 2 this conclusion is reinforced using a straightforward demonstration that the torsion must be antisymmetric, both in the torsion and the curvature tensors.

. . .

2. ANTISYMMETRY OF THE CONNECTION

Define the covariant derivative of a vector V of any dimension in any space-time as

. . .

where T^κ_μν is the torsion tensor:

        T^κ_μν = Γ^κ_μν − Γ^κ_νμ                 (10)

. . .

. . . These tensors are antisymmetric in their last two indices by construction

        T . . .                 (12)

        R . . .                 (13)

. . . The torsion is defined as:

        [D_μ,D_ν]V^ρ = − T^λ_μν D_λV^ρ + . . . = − (Γ^λ_μν−Γ^λ_νμ) D_λV^ρ + . . .                 (15)

and is antisymmetric, and it follows that

        Γ^κ_μν = − Γ^κ_νμ                 (16)