Proof of the Antisymmetry of the Gamma Connection

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

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.

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} .

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.

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}_{μ}dx^{μ} and
ω^{a}_{b} = ω_{μ}^{a}_{b}dx^{μ} .

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

(B8)
ω^{a}_{b} =
q^{a}_{ν}q^{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^{μ}Ù q^{b}

= (0) +
Γ_{μ}^{a}_{λ} dx^{μ}Ù dx^{λ}
= ½
(Γ_{μ}^{a}_{λ}−Γ_{λ}^{a}_{μ})
dx^{μ}Ù 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^{μ}Ù dx^{λ} .

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.

Fundamental theorem of Riemannian geometry