General Relativity: An Introduction for Physicists · Problem 3.15

Vector calculus on manifolds — Hobson, Efstathiou & Lasenby

Physics · geodesic-equation-affine-parameter

Problem

An affinely parameterised geodesic

x^a(u)

satisfies

\frac{d^2 x^a}{du^2} + \Gamma^a{}_{bc}\frac{dx^b}{du}\frac{dx^c}{du} = 0.

Show that the form of this equation remains unchanged by an arbitrary coordinate transformation

x^a \to x'^a

. Find the form of the geodesic equation for a geodesic described in terms of some general (non-affine) parameter

\lambda

. Hence show that all affine parameters are related by a linear transformation with constant coefficients.