What is the proper time of a vertically moving inertial clock?

[KDP]

TL;DR Summary

What is the elapsed proper time of vertically moving inertial clock in Schwarzschild geometry?

Hi. I am looking for an equation for the round trip elapsed proper time of a clock that is initially moving vertically straight up with a given initial velocity, reaches apogee and then returns to the starting location under gravity. I would like to use the external Schwarzschild geometry of a non rotating black hole to keep things as simple as possible. At all times during the the experiment the clock is moving inertially, so no rockets or thrusters involved (and no horizontal motion allowed).

 

[PeroK]

Is there any reason you can't do the calculation yourself?

 

[KDP]

Is there any reason you can't do the calculation yourself?

Getting too old, I guess...

 

[PeroK]

Getting too old, I guess...

The calculation for freefall is here, for example:

https://heiup.uni-heidelberg.de/reader/download/534/534-69-86465-1-10-20190919.pdf

 

[Dale]

I also like this paper on the topic.

https://arxiv.org/abs/gr-qc/0311038v2