Evans stubborn: Confirms his former flaw:
Proof of the Antisymmetry of the Gamma Connection

Gerhard W. Bruhn, Darmstadt University of Technology

Nov 17, 2008

We read on Evans' blog:

Subject: 122(13) : Proof of the Antisymmetry of the Gamma Connection
Date: Mon, 17 Nov 2008 07:07:41 EST
This note will expand some more on note 122(10), which is an imporatnt one worthy of a paper on its own. The straightforward proof is given of the fact that the commutator of covariant derivatives acting on any tensor in any spacetime in any dimensions always produces an anti-symmetric connection. The use of an ad hoc symmetric connection in the Einsteinian era was incorrect. The animations being produced for TV by the obsolete Einsteinian scientists are mathematically incorrect. we uncovered this in papers 93 ff., and this proof adds to the already very severe international criticism of standard model physics The computer can be made to produce an awful lot of total nonsense if the starting equations are not right. Animations are very ueful to see if the equations are working, but of course the equations must be based on the right geometry. The computer is only a calculating machine, and must always be controlled by the scientist. This is obvious but worth mentioning.

There is no Note 122(10). But at the beginning of Note 122(13) we read:

. . .

By construction:

Γ^κ_μν = − Γ^κ_νμ . (5)

so: ???

Γ^κ_μν = Γ^κ_νμ = 0 . (6)

Q.E.D.

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 C^κ_μν denote (not completely vanishing) Christoffel symbols which are symmetric w.r.t. the lower indices:

(B1) C^κ_μν = C^κ_νμ .

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 A^κ_μν which is antisymmetric w.r.t. μ,ν.

(B2) Γ^κ_μν := C^κ_μν + ½ A^κ_μν .

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

(B3) 0 = ½ A^κ_μν + ½ A^κ_νμ = (Γ^κ_μν + Γ^κ_νμ) − (C^κ_μν + C^κ_νμ) ,

hence

(B4) Γ^κ_μν + Γ^κ_νμ = C^κ_μν + C^κ_νμ not zero for at least one pair (μ_o,ν_o) .

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

(B5) Γ^κ_μν + Γ^κ_νμ = 0 for all μ,ν.

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

(B6) T^κ_μν = Γ^κ_μν − Γ^κ_νμ = C^κ_μν − C^κ_νμ + ½ A^κ_μν − ½ A^κ_νμ = 0 + ½ (A^κ_μν − A^κ_νμ) = A^κ_μν .

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

How to change the torsion of a given manifold?

Let ½ A^κ_μν be some tensor antisymmetric w.r.t. its lower indices μ,ν. Then consider the connection Γ_μ^ν_λ as given by eq. (B2). We have to calculate the torsion T^a from

(B7) T^a = D Ù q^a = d Ù q^a + ω^a_b Ù q^b (1st Cartan structure equation)

using the co-frame 1-forms q^a = q^a_μdx^μ and ω^a_b = ω_μ^a_bdx^μ .

Where to take the forms ω^a_b from?

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

(B8) ω^a_b = q^a_νq^λ_b Γ_μ^ν_λ dx^μ − q^λ_b dq^a_λ = q^λ_b Γ_μ^a_λ dx^μ − q^λ_b dq^a_λ

By using these ω^a_b's we obtain

(B9) T^a = D Ù q^a = d Ù q^a + ω^a_b Ù q^b = d Ù q^a + (q^λ_b Γ_μ^a_λ dx^μ − q^λ_b dq^a_λ)Ù q^b
= (d Ù q^a − q^λ_b dq^a_λ) + q^λ_b Γ_μ^a_λ dx^μÙ q^b
= (0) + Γ_μ^a_λ dx^μÙ dx^λ = ½ (Γ_μ^a_λ−Γ_λ^a_μ) dx^μÙ dx^λ

According to (B2) we have ½ (Γ_μ^a_λ−Γ_λ^a_μ) = A_μ^a_λ, and so finally

(B10) T^a = D Ù q^a = A_μ^a_λ dx^μÙ dx^λ .

Thus, the change (B2) of the connection from C.C. to general C. has converted the torsion from T^a=0 to T^a=A^a without changing neither the co-frame field q^a nor the manifold itself.

To illustrate this result for a manifold with metric (= pseudo-Riemannian manifold):

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 q^a = 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) T^a = D Ù q^a .

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 q^a. It is an interrelation between torsion T^a and connection forms ω^a_b merely.

Links

Fundamental theorem of Riemannian geometry

Riemannian manifold

Pseudo-Riemannian manifold

Levi-Civita connection

Covariant derivative

Evans stubborn: Confirms his former flaw: Proof of the Antisymmetry of the Gamma Connection