Parallel transport, Christoffel symbols and the geodesic equation

To compare vectors at different points, choose a path and parallel-transport one vector along it. Different paths can give different comparisons. We use the Levi-Civita connection: it preserves the metric and has zero torsion. These conditions let the metric determine the transport rule. Partial derivatives of metric components measure their local rate of change in one coordinate while the other coordinates stay fixed.

For the Levi-Civita connection, Γⁱⱼₖ=(1/2)gⁱˡ(∂ⱼgₗₖ+∂ₖgₗⱼ−∂ₗgⱼₖ). Repeated indices are summed. In plain-text indices, ^ marks upper and _ marks lower; braces group indices. ∂_r means differentiation in r. In coordinates (r,θ), g=diag(1,r²) gives Γ^r_{θθ}=−(1/2)∂_r g_{θθ}=−r, since ∂_r(r²)=2r. Also Γ^θ_{rθ}=Γ^θ_{θr}=1/r. All other polar components vanish.

The vector derivative is ∇ⱼVⁱ=∂ⱼVⁱ+ΓⁱⱼₖVᵏ. The added term accounts for basis changes. In polar coordinates, ∇_θ V^r=∂_θ V^r−r V^θ. Here V^θ is an upper-index component, not V_θ. For r=3, V^θ=1 and ∂_θ V^r=0, this derivative is −3 even though the ordinary component derivative is zero.

A path is xⁱ(λ), where λ labels points along it. dxⁱ/dλ gives tangent components; d²xⁱ/dλ² measures their rate of change. Dots in the polar equations mean λ derivatives. For an affine λ, the geodesic equation is d²xⁱ/dλ²+Γⁱⱼₖ(dxʲ/dλ)(dxᵏ/dλ)=0. The tangent's covariant derivative along the path is zero. In spacetime, an ideal test body with no nongravitational force follows a timelike geodesic; light follows a null geodesic.

Coordinate acceleration need not vanish on a geodesic. The Christoffel term can cancel it. Christoffel symbols can also be nonzero in flat space, as the polar example shows. A freely falling capsule has zero proper acceleration even when its coordinate position has a nonzero second derivative.

Test a proposed path by substituting it into the geodesic equation. For a circle with fixed r>0 and θ˙≠0, the polar radial equation becomes −r(θ˙)²=0, which fails. The circle's tangent changes covariantly. Its appearance on a coordinate drawing cannot replace this test.