By introducing a specific but arbitrary coordinate system, show that among all timelike world lines that a particle could take to get from event P₀ to event P₁, the one or ones whose proper time lapse is stationary under small variations of path are the freefall geodesics. In other words, an action principle for a timelike geodesic P(λ) [i.e., x^α(λ) in any coordinate system x^α] is

where λ is an arbitrary parameter, which by construction ranges from 0 at P₀ to 1 at P₁.

5 for dt = 8 (-8 dd") d = 0, (25.19)