Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 10; Box 10.3; Problems 10.10, 10.11.
For a circular orbit, the Schwarzschild angular speed is obtained from the angular momentum:
We can write this as
Comparing this with the angular speed measured at infinity we have
The relation between and for a circular orbit is
In the Newtonian case, we can use Kepler’s law at all distances, so we have
Note that we can get the same result by defining the Newtonian energy
and then solving
In the Newtonian case, the potential energy has only one minimum.
For large , we can expand 5 to get (using the minus sign)
Thus the Schwarzschild case reduces to the Newtonian case for large .