A typical Evans flaw:
Comments on a Detailed Proof of Note 122(10)

Gerhard W. Bruhn, Darmstadt University of Technology

Nov 16, 2008

We read on Evans' blog under Detailed Proof of Note 122(10):
I will write this out in full so anyone with a training in linear algebra can understand that the connection must be anti-symmetric. I think that the ECE scientists should concentrate wholly on ECE, the older Einstein based cosmologies being so obviously wrong. Horstís animation sent to the blog today is the first one to be made from an ECE equation.

There is no Note 122(10). But at the beginning of Note 122(11) we read:

. . .

The torsion tensor is defined by

        Tκμν = qκa Taμν = Γκμν − Γκνμ                 (3)

with

        Tκμν = − Tκνμ .                 (4)

Therefore:


        Γκμν = − Γκνμ .                 (5)

The gamma connection cannot be symmetric. The only possibility is:

        Γκμν = Γκνμ = 0 .                 (6)

and therefore the physical science must be developed on the basis of torsion.


The conclusion (5) Þ (6) is WRONG. This is one of Evans' typical fallacies.

The antisymmetry of the torsion (4) does NOT at all imply the antisymmetry of the connection (5).

Let Cκμν denote (not completely vanishing) Christoffel symbols which are symmetric w.r.t. the lower indices:

(B1)                 Cκμν = Cκνμ .

The advantage of this (metric based) Levi-Civita or Christoffel connection (C.C.) is that it is always metric compatible.

Any other linear connection differs from C.C. by some additional tensor ½ Aκμν which is antisymmetric w.r.t. μ,ν.

(B2)                 Γκμν := Cκμν + ½ Aκμν .

Due to the antisymmetry of ½ Aκμν w.r.t. μ,ν this connection fulfils

(B3)                 0 = ½ Aκμν + ½ Aκνμ = (Γκμν + Γκνμ) − (Cκμν + Cκνμ) ,

hence

(B4)                 Γκμν + Γκνμ = Cκμν + Cκνμ not zero for at least one pair (μoo) .

Thus, the connection Γκμν is not antisymmetric w.r.t. μ,ν; since antisymmetry would require

(B5)                 Γκμν + Γκνμ = 0 for all μ,ν.

However, the torsion is antisymmetric w.r.t. μ,ν:

(B6)                 Tκμν = Γκμν − Γκνμ = Cκμν − Cκνμ + ½ Aκμν − ½ Aκνμ = 0 + ½ (Aκμν − Aκνμ) = Aκμν .

N.B. This asymmetric connection Γ will NOT be metric compatible in general.



How to change the torsion of a given manifold?

Let ½ Aκμν be some tensor antisymmetric w.r.t. its lower indices μ,ν. Then consider the connection Γμνλ as given by eq. (B2). We have to calculate the torsion Ta from

(B7)                 Ta = D Ù qa = d Ù qa + ωab Ù qb                 (1st Cartan structure equation)

using the 1-forms qa = qaμdxμ and ωab = ωμabdxμ .

Where to take the forms ωab from?

Remember the compatibility relation of frames (Carroll L.N. (3.132)) to obtain:

(B8)                 ωab = qaνqλb Γμνλ dxμ − qλb dqaλ = qλb Γμaλ dxμ − qλb dqaλ

By using these ωab 's we obtain

(B9)                 Ta = D Ù qa = d Ù qa + ωab Ù qb = d Ù qa + (qλb Γμaλ dxμ − qλb dqaλ)Ù qb
                                = (d Ù qa − qλb dqaλ) + qλb Γμaλ dxμÙ qb
                                = (0) + Γμaλ dxμÙ dxλ = ½ (Γμaλ−Γλaμ) dxμÙ dxλ

According to (B2) we have ½ (Γμaλ−Γλaμ) = Aμaλ, and so finally

(B10)                 Ta = D Ù qa = Aμaλ dxμÙ dxλ .

Thus, the change (B2) of the connection from C.C. to general C. has converted the torsion from Ta=0 to Ta=Aa without changing the manifold itself.


To illustrate this result for a manifold with metric:

A given field of (orthonormal) 'vielbeins' (tetrads in case of spacetime manifolds) can be imagined as system of local frames q attached to the different points of the manifold. The relative positions of such neighboring local frames are given by a connection and/or described by the torsion.

The metric defines a 'natural' choice of (orthonormal) local 'vielbein'-frames (tetrads in case of spacetime manifolds): The interrelation is the Levi-Civita or Christoffel connection: Neighboring 'natural' frames q are not 'distorted' relative to each other.

                          D Ù qa = 0.

The torsion is zero.

However, one can 'distort' these 'natural' local frames against each other and thereby define an alternative connection on the manifold: Then the 'torsion' T describes the reciprocal 'distorsion' of neighboring frames q.

(B7)                 Ta = D Ù qa .

The distorsion can be given by assigning a torsion T to the originally undistorted manifold, or by modifying the connection ω in eq. (B7). This change will neither affect the manifold nor the frame field qa. It is an interrelation between torsion Ta and connection forms ωab merely.



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