Nov 24, 2008

The reader has to have some patience until - at eqs.(15)/(16) - Dr. Evans arrives at the crucial point which is why he condemns all the preceeding science of the last century, the ''Einsteinian era'', to death. We sketch Evans' note before that point shortly:

AND ECE THEORY:

Civil List Scientist

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. . . .

. . .

. . . 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.

. . .

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_{λ}V^{ρ} + . . .
=
− (Γ^{λ}_{μν}−Γ^{λ}_{νμ})
D_{λ}V^{ρ} + . . .
(15)

and is antisymmetric, and it follows that

Γ^{κ}_{μν} = −
Γ^{κ}_{νμ}
(16)

The torsion T^{λ}_{μν} is defined to be double the
antisymmetric part of the connection
Γ^{λ}_{μν}:

T^{λ}_{μν} := Γ^{λ}_{μν}
− Γ^{λ}_{νμ} =
2 Γ^{λ}_{[μν]} .

However, this allows a non-vanishing symmetric part as well.
In case of
torsion-free *and* metric compatible connections
the connection coincides with its symmetric part which is given by the well-known Christoffel coefficients:

C^{λ}_{μν} = C^{λ}_{νμ}
= ½ g^{λρ} (∂_{μ}g_{ρν} +
∂_{ν}g_{ρμ}
− ∂_{ρ}g_{μν} ) .

In case of a *non-constant* metric g_{μν}
the Christioffel coefficients C^{λ}_{μν} cannot vanish completely.

Fundamental theorem of Riemannian geometry