Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 10; Problem 10.5.
And now for a general relativistic look at the standard physics problem of throwing an object up in a gravitational field. Suppose we start at radius and throw up the object so that it comes to rest momentarily at before turning around and falling back to . What is the total proper time (as measured by the object) in this trip?
We begin by working out the energy . For an object at rest at :
The radial equation of motion in the Schwarzschild metric is
For radial motion, so we get
To find the time that elapses on the upward leg of the journey, we must evaluate
This is, as Moore says, a bit of a nasty integral. Using software produces a bit of a jumble, so I resorted to the old-fashioned method of looking the integral up in a table, which produced a somewhat nicer result. We get
At the upper limit, the first term is zero and the second term is
so the result is
Here we’re assuming that the arctan lies in the range . The total time is twice this, so