By Richard Dawkins
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Extra resources for [Article] Good and Bad Reasons for Believing
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23) Γαµβ is called the Christoffel symbol of the second kind, and as may be expected it is intimately related to the metric tensor: Γµνσ = 1 µλ 2g (gλν,σ + gλσ,ν − gνσ,λ ) ≡ g µλ Γλνσ . 5. The ≡ sign defines the Christoffel symbol of the first kind, simply by raising one index with g µλ . 23) the Christoffel symbols define the connection between the base vectors of the tangent spaces at different positions. 24), but these play no role in GR. 22) and rename the dummy-indices: dA = (dAµ + Γµνσ Aν dxσ ) eµ ≡ (DAµ ) eµ .
31), and is left to the reader. Next follow a few definitions. 44) (XY ):σ = X:σ Y + X Y:σ . For example: 34 2 Geometry of Riemann Spaces (Aµ Bν ):σ = (Aµ,σ − Γαµσ Aα ) Bν + Aµ (Bν,σ − Γανσ Bα ) = (Aµ Bν ),σ − Γαµσ Aα Bν − Γανσ Aµ Bα . 46) Tµν:σ = Tµν,σ − Γαµσ Tαν − Γανσ Tµα . The recipe for tensors of higher rank should be clear by now. 45). The general pattern is T ∗:σ = T ∗,σ ± Γ-term for every index. 47) Q:σ = Q,σ . Covariant derivatives do not commute, unlike normal derivatives (X,αβ = X,βα for every X).
But which? The crucial step is to recognise that one should take T µν = ρuµ uν where ρ is now the rest mass density and uµ the 4-velocity. Since uµ = c−1 dxµ /dτ (γ, v i /c) we have T 00 = γ 2 ρ. 38) is to be replaced by Rµν = −(4πG/c2 )Tµν or equivalently Rµν = −(4πG/c2 )T µν . However, that leads to inconsistencies. The trouble is that Rµν:ν is in general nonzero, so that T µν:ν would also be nonzero. And as explained in a moment, conservation of mass or geodesic motion would no longer be guaranteed.