Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 10; Problem 10.12.
We can generalize an earlier problem by working out the period of a circular orbit as measured by three different observers. The first observer is attached to the orbiting object which circles the mass at a radius . The second observer is at rest at radius , while the third observer is at infinity.
From the formula for angular momentum as measured by the orbiting object:
we can calculate the angular speed from , so
The period is then
Defining the velocity as , we get
The period and velocity as measured at infinity are found from the angular speed at infinity
Finally, the period as measured by the observer at rest at is found from the relation between and , as before.
In this case, and so
Thus and . The condition could be explained by time dilation, since to the observer at rest, the orbiting clock would run slow, so less time would appear to elapse on it than on the stationary clock. The condition is a consequence of the difference between the proper time for an observer at rest at a finite distance from the mass and the time at infinity, which is given by the time coordinate .
Since the square roots must all be real, we must have . This ensures that , but for , . I’m not entirely sure what the resolution of this apparent paradox is, but in the frame of the orbiting object, its own velocity is, of course, zero, so at best the expression for above must be considered an artificial velocity which cannot be measured in any physical sense. If the speed of the object is measured by an external observer (such as the stationary observers at and infinity) the value always seems to be less than 1.