Consider two observers, Jane and Susan, both in the vacuum region surrounding a 15 compact spherical object of geometrized mass m and radius less than 4m, but greater than 2m.
Jane is in a circular orbit of radius r = 4m. Susan is fired radially from the surface of the compact object with a velocity that is less than the escape velocity. Through careful calculation, the timing of the launch of Susan is such that when she arrives at r = 4m, Jane is just arriving there as well in her circular orbit. At this spacetime event common to both Jane and Susan they synchronize their respective clocks. Susan continues on her journey radially away from the system, and eventually begins to fall back towards the system. Suppose that by the appropriate choice of the initial speed of Susan, when she falls back to r = 4m, Jane happens to be at the same location in her orbit, and Susan and Jane are again at the same spacetime event, and Jane has completed N orbits in the interim. At this second common point in their two world lines, they compare their clocks and determine the difference in their two clocks' times.
Calculate elapsed time on the clocks of each of Jane and Susan, the difference in the times on the two clocks, and the maximum value of Jane's distance from the centre of the spherical object.
Now consider a third observer, Lucy, equipped with an appropriate rocket so that she hovers at the position in Jane's orbit throughout the flight of Susan away from Jane until they meet again as described above. Calculate the elapsed time on Lucy's clock through this period of Susan's flight and compare this to those of Susan and Jane. Provide an explanation of the various elapsed times, and why they differ, if they do in fact differ.