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2.4 Spacetime curvature

The main message of the previous two Sections 2.2 and 2.3 is that one must have an a priori idea of how vectors and higher rank tensors are moved from point to point in spacetime. Another manifestation of the complexity associated with carrying tensors about in spacetime is that the covariant derivative does not commute. For a vector we find
∇ ρ∇σV μ − ∇ σ∇ ρV μ = R μνρσV ν, (55 )
where μ R νρσ is the Riemann tensor. It is obtained from
μ μ μ μ τ μ τ R νρσ = Γνσ,ρ − Γνρ,σ + Γ τρΓνσ − ΓτσΓ νρ. (56 )
Closely associated are the Ricci tensor R νσ = R σν and scalar R that are defined by the contractions
μ νσ R νσ = R νμσ, R = g Rνσ. (57 )
We will later also have need for the Einstein tensor, which is given by
1 G μν = Rμν − 2-Rg μν. (58 )
It is such that μ ∇ μG ν vanishes identically (this is known as the Bianchi identity).

A more intuitive understanding of the Riemann tensor is obtained by seeing how its presence leads to a path-dependence in the changes that a vector experiences as it moves from point to point in spacetime. Such a situation is known as a “non-integrability” condition, because the result depends on the whole path and not just the initial and final points. That is, it is unlike a total derivative which can be integrated and thus depends on only the lower and upper limits of the integration. Geometrically we say that the spacetime is curved, which is why the Riemann tensor is also known as the curvature tensor.

To illustrate the meaning of the curvature tensor, let us suppose that we are given a surface that can be parameterized by the two parameters λ and η. Points that live on this surface will have coordinate labels xμ(λ,η). We want to consider an infinitesimally small “parallelogram” whose four corners (moving counterclockwise with the first corner at the lower left) are given by x μ(λ,η), x μ(λ, η + δη), x μ(λ + δλ,η + δη), and xμ (λ + δ λ,η). Generally speaking, any “movement” towards the right of the parallelogram is effected by varying η, and that towards the top results by varying λ. The plan is to take a vector μ V (λ,η) at the lower-left corner μ x (λ,η), parallel transport it along a λ = const curve to the lower-right corner at xμ(λ,η + δη) where it will have the components Vμ (λ, η + δη), and end up by parallel transporting V μ at x μ(λ, η + δη) along an η = const curve to the upper-right corner at xμ(λ + δλ, η + δη). We will call this path I and denote the final component values of the vector as μ VI. We repeat the same process except that the path will go from the lower-left to the upper-left and then on to the upper-right corner. We will call this path II and denote the final component values as V μ II.

Recalling Equation (32View Equation) as the definition of parallel transport, we first of all have

μ μ μ μ μ ν ρ V (λ,η + δ η) ≈ V (λ, η) + δηV ∥ (λ,η ) = V (λ, η) − Γ νρV δηx (59 )
V μ(λ + δλ,η) ≈ V μ(λ,η) + δ V μ(λ,η) = V μ(λ,η) − Γ μ V νδ xρ, (60 ) λ ∥ νρ λ
δηxμ ≈ xμ(λ,η + δη ) − x μ(λ,η), δλx μ ≈ xμ(λ + δλ,η ) − x μ(λ, η). (61 )
Next, we need
Vμ ≈ Vμ(λ, η + δη) + δ V μ(λ, η + δ η), (62 ) I λ ∥ VμII ≈ Vμ(λ + δλ, η) + δηVμ∥ (λ + δλ, η). (63 )
Working things out, we find that the difference between the two paths is
μ μ μ μ ν σ ρ ΔV ≡ VI − VII = R νρσV δλx δηx , (64 )
which follows because δ δ x μ = δ δ xμ λ η η λ, i.e. we have closed the parallelogram.

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