Commentary on Evans' web note #122:

Gerhard W. Bruhn, Darmstadt University of Technology

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:


by M.W. Evans
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 T~ = R~ 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 T = R q                 (4)

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

        Tκμν = Γκμν − Γκνμ                 (5)

where Γκμν is the connection. If the latter is arbitrarily forced to be symmetric:

        Γκμν = ? Γκνμ                 (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 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 Tκμν is the torsion tensor:

        Tκμν = Γκμν − Γκνμ                 (10)

. . .

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

        T . . .                 (12)

        R . . .                 (13)

. . . The torsion is defined as:

        [Dμ,Dν]Vρ = − Tλμν DλVρ + . . . = − (Γλμν−Γλνμ) DλVρ + . . .                 (15)

and is antisymmetric, and it follows that

        Γκμν = − Γκνμ                 (16)

Independent of Evans' above consideration we have: Nobody denies that - by definition - the torsion tensor Tλμν = Γλμν − Γλνμ is antisymmetric in its two lower indices μ,ν. However, this antisymmetry of Tλμν does NOT IMPLY the antisymmetry of the connection coefficients Γλμν:

The torsion Tλμν is defined to be double the antisymmetric part of the connection Γλμν:

                Tλμν := Γλμν − Γλνμ = 2 Γλ[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:

                Cλμν = Cλνμ = ½ gλρ (∂μgρν + ∂νgρμ − ∂ρgμν ) .

In case of a non-constant metric gμν the Christioffel coefficients Cλμν cannot vanish completely.

Further arguments


Fundamental theorem of Riemannian geometry

Riemannian manifold

Pseudo-Riemannian manifold

Levi-Civita connection

Covariant derivative