A **geodesic** of spacetime is a curve that is **straight** and uniformly parametrized, as measured in each local Lorentz frame along its way.

If the geosidesic is timelike, then it is a possible wordline for a freely falling particle, and its uniformly ticking parameter λ (called *affine parameter*) is a multiple of the particule's proper time, λ = κτ + μ.

This definition of geodesic translates into the abstract and coordinate-free language: a geodesic is a curve P(λ) that **parallel-transports its tangent vector u = dP/dλ along itself**.

Now defining a coordinate system {x^{α}(P)), along with basis vectors** e _{α}** = ∂/ ∂x

^{α}, we can define the tangent vector u and its components

and thus finally the component version of the abstract geodesic equation definition becomes

CQFD

This geodesic equation can be solved (in principle) when both initial data x^{α} and dx^{α} / dλ have been specified.