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

Gerhard W. Bruhn, Darmstadt University of Technology

Nov 16, 2008

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

        Γ^κ_μν = Γ^κ_νμ = 0 .                 (6)

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 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 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 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.

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