Comments on a Detailed Proof of Note 122(10)

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. Horst’s 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

T^{κ}_{μν} = q^{κ}_{a} T^{a}_{μν}
= Γ^{κ}_{μν} − Γ^{κ}_{νμ}
(3)

with

T^{κ}_{μν} = − T^{κ}_{νμ} .
(4)

Therefore:

Γ

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

Γ^{κ}_{μν} = Γ^{κ}_{νμ} = 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.

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 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:

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.