Nov 17, 2008
We read on Evans' blog:
Subject: 122(13) :
Proof of the Antisymmetry of the Gamma Connection
There is no Note 122(10). But at the
beginning of Note 122(13) we read:
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:
. . .
Γκμν = Γκνμ = 0 . (6)
The conclusion (5) Þ (6) is WRONG. This is one of Evans' typical fallacies.
Let Cκμν denote (not completely vanishing) Christoffel symbols which are symmetric w.r.t. the lower indices:
(B1) Cκμν = Cκνμ .
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) ,
(B4) Γκμν + Γκνμ = Cκμν + Cκνμ not zero for at least one pair (μo,νo) .
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.
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 co-frame 1-forms qa = qaμdxμ and ωab = ωμabdxμ .
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
Ta = D Ù qa =
d Ù qa +
ωab Ù qb =
d Ù qa +
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 .
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 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.
Fundamental theorem of Riemannian geometry