Geodesics in general relativity

Remarks

For a timelike geodesic the parameter $\lambda$ is an affine function of the proper time $\tau$, and one usually takes $\lambda = \tau$. Example: in Minkowski space with $g = -dt^2 + dx^2 + dy^2 + dz^2$, all Christoffel symbols vanish. So the geodesics are the straight lines $x^\mu(\lambda) = a^\mu + b^\mu \lambda$ with affine parameter. With $b = (1, v, 0, 0)$ and $\abs{v} < 1$, it is the world line of a particle in uniform motion. Non-example: for $0 < a < 1$, the curve $t \mapsto (t, a\cos t, a\sin t, 0)$ in units with $c = 1$ is timelike, since $g(\dot\gamma, \dot\gamma) = -1 + a^2 < 0$. It is not a geodesic, since $\nabla_{\dot\gamma}\dot\gamma = (0, -a\cos t, -a\sin t, 0) \neq 0$; it is the world line of a particle in circular motion. Timelike geodesics are the critical points of the proper time functional, and light rays in geometric optics follow null geodesics.