= Timelike geodesic
{title2=$\nabla_u u=0$}
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 https://www.damtp.cam.ac.uk/user/tong/gr/grhtml/S1.html[Tong's geodesics in spacetime].
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