Nov 16, 2008
We read on Evans' blog under Detailed Proof of Note 122(10):
There is no Note 122(10). But at the beginning of Note 122(11) we read:
. . .
The torsion tensor is defined by
Tκμν = qκa Taμν = Γκμν − Γκνμ (3)
Tκμν = − Tκνμ . (4)
The gamma connection cannot be symmetric. The only possibility is:
Γκμν = Γκνμ = 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 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 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:
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.