Gravitational orbits in one dimension

Abstract

The analytic solution of a body falling an arbitrary distance toward a gravitational source is presented. This problem has apparently received little attention in textbooks. The solution can be extended smoothly through the singularity at the origin to form a class of trajectories that we call singular orbits. Trajectories in phase space have singular branches, yet cycle in a finite time. We calculate the period of oscillation about the singularity confirming that the singular orbits obey Kepler’s third law. Viewed as geodesics in a Newtonian 1+1 spacetime, the geodesic deviation equation is derived and solved analytically, providing an excellent example of curvature physics in Newtonian spacetime. The results are used to estimate the duration during which a freely falling local frame can be considered inertial. A numerical investigation with damping included shows that a final state is reached in which the particle is confined to the origin, but acquires infinite speed during each pass. We show via some examples, the pedagogical applications of the solution.