Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 10; Problem 10.4.
The radial equation of motion in the Schwarzschild metric is
We can use this to derive an equation for , the rate of change of with respect to the Schwarzschild time coordinate. The coordinate isn’t the time as measured by any particular object (that time is the proper time in the reference frame of the object) so we wouldn’t expect it to be the same as .
To get the equation, we can use
where the last line uses the definition of . Plugging this into the top equation we get
As approaches , .
In the special case where we drop an object from rest at , we can work out both and . In this case, motion is radially inward so . To find , we use the fact that for an object at rest at :
We have therefore
From 1 we get
Comparing the two, we see that
For the case where the object is released from rest at , the speed at is
which agrees with the earlier calculation done by a different method.