Conservation of Energy vs. 'Space-Time Action' ??

Imagine two masses, initially contacting, and suddenly jettisoned apart. As they travel apart, and then fall back together, they sweep out a certain 'area', in the spacetime x,t plane, forming a shape something like an ellipse. Is there a simple relation, between the area of that 'ellipse' swept out, in spacetime, and the initial total system energy? I started working through the equations, from Newton's laws in the reduced mass frame, and, whilst possible, nothing seems simple. Ultimately, at finite total energy (zero), the system starts at escape speed, so sweeping out an infinte spacetime area. So, plotting 'area swept out', vs. starting system energy, one begins at some negative bound-state value, and plots up to zero, watching as the function rises from zero area, before 'blowing up' at zero energy.