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The Riemann Curvature Tensor and the Einstein Equation

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Riemannian Curvature February 26, 2013 We now generalize our computation of curvature to arbitrary spaces. 1 Parallel transport around a small closed loop We compute the change in a vector, wα, which we parallel transport around a closed loop. The loop starts at a point P, which we take to have coordinates (0; 0), progresses along a direction uα a distance λ to a α point at (λ, 0). Next, we transport along the direction v a distance σ to a point at (λ, σ), giving a curve C1 from (0; 0) to (λ, σ). Rather than returning in the direction −uα by λ then −vα by σ, we perform a second α α transport, interchanging the order: first v by σ, then u by λ, giving curve C2 from (0; 0) to (λ, σ). We then compare the expressions for wα (λ, σ) along the two curves. This gives the same result as if we had transported around the closed loop C1 − C2, but the computation is easier this way. Start with two directions, uα (0; 0) and vα (0; 0). Parallel transport each in the direction of the other to get uα (0; σ) and vα (λ, 0): dvα vα (λ, 0) = vα (0; 0) + (0; 0) λ dλ α β µ α = v (0; 0) − u (0; 0) v (0; 0) Γ µβ (0; 0) λ and duα uα (0; σ) = uα (0; 0) + (0; 0) σ dλ α β µ α = u (0; 0) − v (0; 0) u (0; 0) Γ µβ (0; 0) σ Now take a third vector, wα (0; 0) and transport it to wα (λ, σ) in two different ways, first along uα and then vα, then in the opposite order. First, α α β µ α w (λ, 0) = w (0; 0) − u (0; 0) w (0; 0) Γ µβ (0; 0) λ then α α β µ α w (λ, σ) = w (λ, 0) − v (λ, 0) w (λ, 0) Γ µβ (λ, 0) σ α β µ α = w (0; 0) − u (0; 0) w (0; 0) Γ µβ (0; 0) λ β λ τ β µ ρ σ µ α δ α − v (0; 0) − u (0; 0) v (0; 0) Γ τλ (0; 0) λ w (0; 0) − u (0; 0) w (0; 0) Γ σρ (0; 0) λ Γ µβ (0; 0) + u @δΓ µβ (0; 0) λ σ = wα (0; 0) β µ α −u (0; 0) w (0; 0) Γ µβ (0; 0) λ β µ α −v (0; 0) w (0; 0) Γ µβ (0; 0) σ λ τ µ β α +u (0; 0) v (0; 0) w (0; 0) Γ τλ (0; 0) Γ µβ (0; 0) σλ β ρ σ µ α +v (0; 0) u (0; 0) w (0; 0) Γ σρ (0; 0) Γ µβ (0; 0) σλ 1 β µ δ α −v (0; 0) w (0; 0) u @δΓ µβ (0; 0) λσ λ τ µ β δ α 2 +u (0; 0) v (0; 0) w (0; 0) Γ τλ (0; 0) u @δΓ µβ (0; 0) λ σ β ρ σ µ δ α 2 +v (0; 0) u (0; 0) w (0; 0) Γ σρ (0; 0) u @δΓ µβ (0; 0) λ σ λ τ β ρ σ µ α 2 −u (0; 0) v (0; 0) Γ τλ (0; 0) u (0; 0) w (0; 0) Γ σρ (0; 0) Γ µβ (0; 0) λ σ λ τ β ρ σ µ δ α 3 − u (0; 0) v (0; 0) Γ τλ (0; 0) u (0; 0) w (0; 0) Γ σρ (0; 0) u @δΓ µβ (0; 0) λ σ Now that all quantities are expressed at P = (0; 0), we may drop these (0; 0) arguments. Keeping only terms to second order: wα (λ, σ) = wα − uβwµΓα λ − vβwµΓα σ C1 µβ µβ β µ δ α λ τ µ β α −v w u @δΓ µβλσ + u v w Γ τλΓ µβσλ β ρ σ µ α +v u w Γ σρΓ µβσλ Now repeat in the opposite order (just interchange uα with vα and σ with λ), wα (λ, σ) = wα − vβwµΓα σ − uβwµΓα λ C2 µβ µβ β µ δ α λ τ µ β α −u w v @δΓ µβλσ + v u w Γ τλΓ µβσλ β ρ σ µ α +u v w Γ σρΓ µβσλ The difference between these gives the change in wα under parallel transport around the loop δwα = wα C1−C2 = wα (λ, σ) − wα (λ, σ) C1 C2 α β µ α β µ α λ τ µ β α = w − u w Γ µβλ − v w Γ µβσ + u v w Γ τλΓ µβσλ β ρ σ µ α β µ δ α +v u w Γ σρΓ µβσλ − v w u @δΓ µβλσ α β µ α β µ α λ τ µ β α −w + v w Γ µβσ + u w Γ µβλ − v u w Γ τλΓ µβσλ β ρ σ µ α β µ δ α −u v w Γ σρΓ µβσλ + u w v @δΓ µβλσ so α α α β µ α β µ α β µ α β µ α δw = w − w − u w Γ µβλ + u w Γ µβλ − v w Γ µβσ + v w Γ µβσ µ β δ α β δ α +w u v @δΓ µβ − v u @δΓ µβ σλ µ λ τ β α β ρ σ α λ τ β α β ρ σ α +w u v Γ τλΓ µβ + v u Γ µρΓ σβ − v u Γ τλΓ µβ − u v Γ µρΓ σβ λσ µ β ν α ν β α = w u v Γ µβ,ν − v u Γ µν,β σλ µ λ τ β β α ν β σ α β ν σ α +w u v Γ τλ − Γ λτ Γ µβ + v u Γ µβΓ σν − u v Γ µν Γ σβ λσ µ α α σ α σ α ν β +w Γ µβ,ν − Γ µν,β + Γ µβΓ σν − Γ µν Γ σβ v u λσ The change in wα per unit area is our measure of the curvature, δwα = wµ Γα − Γα + Γσ Γα − Γσ Γα vν uβ λσ µβ,ν µν,β µβ σν µν σβ µ α β ν = w R µβν u v We see that the change in wα depends linearly on wα, and also on the orientation of the infinitesimal loop spanned by uα and vα. The rest of the expression, α α α σ α σ α R µβν ≡ Γ µβ,ν − Γ µν,β + Γ µβΓ σν − Γ µν Γ σβ 2 is called the Riemann curvature tensor. It may be thought of as a trilinear operator which takes an oriented unit area element, 1 Sβν = uβvν − uν vβ 2 α α βν α to give a linear operator, T µ = R µβν S . This linear operator gives the infinitesimal change in w per unit area, in the limit of zero area, when wα is transported around the area element with orientation Sβν . α Since all elements of this construction are defined geometrically from within the manifold, R µβν is intrinsic to the manifold. To see that it is also a tensor, we could recompute the same construction in α different coordinates. Since the entire construction is perturbative, we would find the components of R µβν changing linearly and homogeneously in the transformation matrix. However, there is an easier way to prove α α that R µβν is a tensor. Consider two covariant derivatives of the vector w , α α β α DµDν w = Dµ @ν w + w Γ βν α β α ρ β ρ α α β α ρ = @µ @ν w + w Γ βν + @ν w + w Γ βν Γ ρµ − @ρw + w Γ ρν Γ νµ Because we use covariant derivatives, this object is necessarily a tensor. Now take the derivatives in the opposite order and subtract, giving the commutator. This is also a tensor, α α α [Dµ;Dν ] w = DµDν w − Dν Dµw α β α β α ρ β ρ α α β α ρ = @µ@ν w + @µw Γ βν + w Γ βν,µ + @ν w + w Γ βν Γ ρµ − @ρw + w Γ ρν Γ νµ α β α β α ρ β ρ α α β α ρ −@ν @µw − @ν w Γ βµ − w Γ βµ,ν − @µw + w Γ βµ Γ ρν + @ρw + w Γ ρν Γ µν α α β α ρ α β α ρ α = @µ@ν w − @ν @µw + @µw Γ βν + @ν w Γ ρµ − @ν w Γ βµ − @µw Γ ρν α β α ρ α β α ρ − @ρw + w Γ ρν Γ νµ + @ρw + w Γ ρν Γ µν β α β ρ α β α β ρ α +w Γ βν,µ + w Γ βν Γ ρµ − w Γ βµ,ν − w Γ βµΓ ρν β α = w R βνµ β α β α Since the result is w properly contracted with R βνµ, and w is a tensor, R βνµ must also be a tensor. 2 Symmetries of the curvature tensor α α α Recall that parallel transport of w preserves the length, w wα of w . This means that the transformation, α α µ α µ α βν δµ + T µσλ w = w + w R µβν S σλ = wα + δwα α α α must be an infinitesimal Lorentz transformation, Λ β = δβ + " β.
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