Past exam of the general relativity and cosmology course of the University of Oxford / 2017 / 1 / a / Solution

The covariant derivative differentiates a vector field while accounting for how its coordinate basis changes between neighboring points. For the Levi-Civita connection,The Christoffel symbol term makes this a tensor with one contravariant and one covariant index; ordinary partial derivatives of vector components do not generally have that transformation law. Equivalently, one compares the neighboring vector with the original after parallel transport.

For a curve $x^{\gamma }(\lambda )$ with tangent $U^{\gamma }=d{x}^{\gamma }/d\lambda$, contraction gives the covariant derivative along a curveIt is the rate of change measured relative to a parallel transport frame along the curve. It vanishes precisely when $v$ undergoes parallel transport along that curve. If $v=U$, it is the covariant acceleration, and its vanishing is the geodesic equation with an affine parameter.