Measurement of Time Elapsed in an Expanding Non-Euclidian Space.

A non-Euclidian circle has a diameter that expands as a function of the radius according to the formula: circumference = 2 * phi * r^2 where phi is a constant, r is the radius and the circumference is the set of all points equal distance from r in this space. The radius of this circle increases with time at a rate of r/t. At a reference time and location, t1, x1, an object starts traveling along the circumference. The object only exists on the circumference so all linear coordinates and measurements are along the circumference. Sometime later, at t2, x2, it’s found that this object has travelled a distance, d, along the circumference at a rate of x/t. The angle from the observer to the starting (x1 to x2) point doesn't change as the circle expands even though the arc length does (the points are comoving). Given the distance, d, of the object at the end of the trip, and the velocity (x/t) that the object has travelled, compute the time, t1, at which the object started. Assume the circle as a radius of 0 at t0.