= Solution
Let $\lambda$ be an <affine parameter> and $X=dx^\mu/d\lambda\,\partial_\mu$ the tangent to the <geodesic>. Since $\nabla_{\partial_\mu}\partial_\nu=\Gamma^\rho{}_{\mu\nu}\partial_\rho$, the product rule for the <covariant derivative> gives
$$
\nabla_XX=\left(\frac{d^2x^\rho}{d\lambda^2}
+\Gamma^\rho{}_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}\right)\partial_\rho.
$$
Independence of the coordinate basis vectors yields the \b[coordinate geodesic equation]
$$
\boxed{\frac{d^2x^\rho}{d\lambda^2}+\Gamma^\rho{}_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}=0.}
$$
The <affine parameter> condition matters: an arbitrary reparametrization instead allows a multiple of the tangent on the right, as in the <unparametrized geodesic equation>.
Back to article page
