If you throw up a clock in a field of gravity and it falls back down, will it show the same time as a clock that stayed at the surface? In other words, will the gravitational time dilation of the stationary clock exactly equal the sum of the smaller gravitational time dilation of the other clock (higher up in the field) and the dilation caused by its speed? Just to make things perfectly clear: clock 1 stays at the surface, clock 2 is thrown up and falls back down, so its time passes more slowly because of its speed, but also more quickly because it is higher up in the field of gravity. The question is: do the two dilations balance each other out exactly? I know this may look like a homework question, but trust me, I'm a 35 year old airline pilot :) The reason I'm asking is that I want to understand gravitational time dilation in the expanding universe. If I understood correctly, you can use two different coordinate systems for the universe: the coordinate system that we would normally use based on special relativity (time for distant (and therefore moving) objects passes more slowly), or the god-like cosmological coordinate system in which the time coordinate at any location in the universe is chosen to be local time as experienced by a local observer who is moving together with the expansion of the universe. The first view just looks at the universe as a collection of things that happen to be flying apart, while the second view is closely tied to the particular structure of our expanding universe. Both are just a choice of coordinates, nothing more. The two systems give different, but consistent views of the universe. Just one example: In the first system, very distant objects will never reach our age because time grinds to a halt as they approach (but never exceed) the speed of light. In the second system, they are the same age as us but their light will never reach us because space between us is expanding faster than the speed of light (even the light that is trying to come towards us, is actually retreating away from us due to the expansion of the universe). Both views result in the same observation: we never see them reach our age. It all seems to make perfect sense, except if I consider a universe that shrinks back together (like people used to think before they found out the universe is actually accellerating, but never mind that). In the event of a Big Crunch, the first coordinate system results in a twin paradox, while the second obviously has no such problem. I know the Big Crunch will probably never happen, but I just want to figure out why the two models seem to be contradictory in such a theoretically possible case. If the two clocks from my first question (one thrown up and caught, the other stationary) show the same time, that would solve my paradox since any observer would consider himself to be at the center of the universe's great field of gravity. But I'm not very good at this kind of math (yet) and somebody else can probably show me a calculation in a minute or so, or even better, give an argument why the clocks (hopefully) must show the same time.