Expansions of the affinity, metric and geodesic equations in Fermi normal coordinates about a geodesic

Fermi normal coordinates about a geodesic form a natural coordinate system for the nonrotating geodesic (freely falling) observer. Expansions of the affinity, metric, and geodesic equations in these coordinates in powers of proper distance normal to the geodesic are calculated here to third order, fourth order, and third order, respectively. An iteration scheme for calculation to higher orders is also given. For generality, we compute the affinity and the geodesic equations in an arbitrary affine manifold, and compute the metric in a Riemannian manifold with arbitrary signature.