The geodesic equation for a free particle on curved spacetime with metric g μ μ ν ν can be derived from the action S [ x , x ˙ ˙ ] = − − m c ∫ ∫ d t − − g μ μ ν ν x ˙ ˙ μ μ x ˙ ˙ ν ν If the spacetime-metric is g = ( − − 1 − − 2 ϕ ϕ ( x ) c 2 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) then, ignoring all contributions of order 1 c 2 the action becomes S [ x , x ˙ ˙ ] = − − m c ∫ ∫ d t c 2 + 2 ϕ ϕ ( x ) − − x ˙ ˙ 2 = − − m c ∫ ∫ d t ( c 2 + 1 2 c 2 ( 2 ϕ ϕ ( x ) − − x ˙ ˙ 2 ) + . . . ) ≈ ≈ ∫ ∫ d t ( − − m c 2 + 1 2 m x ˙ ˙ 2 − − m ϕ ϕ ( x ) ) which is the action that reproduces the Newtonian equations of motion of a particle in a gravitational potential ϕ ϕ ( x )