 Timelike geodesic

A timelike geodesic has timelike tangent and vanishing covariant acceleration in an affine parameter. Choosing proper time gives $g(\dot{\gamma },\dot{\gamma })=-1$.

Physically, an ideal massive test particle in free fall follows a timelike geodesic when nongravitational forces and appreciable self-gravity can be neglected. The normalization above uses $c=1$; in SI units it is $g(u,u)=-c^2$. Zero covariant acceleration does not mean constant coordinate velocity: a dropped object accelerates relative to the ground. Conversely, a person standing on the ground is pushed away from free fall by the supporting surface. A photon follows a null geodesic, whose parameter cannot be proper time. See Tong's geodesics in spacetime.