Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 2 / 35E / Solution

An affinely parametrized geodesic satisfies $d^2{x}^a/d{\lambda }^2+{\Gamma }^a{}_{bc}{T}^b{T}^c=0$, which is exactly ${\nabla }_{T}T=0$. The Levi-Civita connection is metric-compatible, sowhen both vectors are parallel transported. Their inner product is constant, including their lengths; this preservation in fact holds along any path with parallel vectors, not only a geodesic.

For the two-parameter family, mixed partial derivatives commute and ${\Gamma }^a{}_{bc}$ is symmetric in its lower indices. ConsequentlyUsing this equality and the supplied curvature-commutator convention,In components the geodesic deviation equation isIt describes the relative acceleration of nearby freely falling worldlines. A freely falling observer can remove the connection at an event by choosing a local inertial frame, but cannot remove the curvature-dependent tidal acceleration of neighboring freely falling particles. Measuring their separation and relative acceleration therefore measures the local gravitational tidal field.