General Relativity: An Introduction for Physicists · Problem 3.16

Vector calculus on manifolds — Hobson, Efstathiou & Lasenby

Physics · geodesic-intrinsic-derivative-alternative-form

Problem

x^\mu(\lambda)

is an affinely parameterised geodesic, show that

\frac{Du^\mu}{D\lambda} = 0,

where

u^\mu = dx^\mu/d\lambda

. Hence show that the geodesic equations can be written as

\frac{du_\mu}{d\lambda} = \frac{1}{2}(\partial_\mu g_{\nu\sigma})u^\nu u^\sigma.