jerromyjon said:
Then they could never reunite in reality because those realities can never coexist. If she traveled that speed in orbit around Earth and Bob sat still where she could fly by him periodically very close to Bob and see him age and he see her they would both appear younger to each other? When she stops nothing could have changed timewise.
I really do think it's worth while to explore how the questions about mutual time dilation and the twin paradox have very close analogies in Euclidean geometry.
As for mutual time dilation: You have two roads that run between points A and B. One road, R, goes straight from A to B, and the other road, R', takes a round-about path. The length of R between A and B is L. How do you compute the length of R'?
Well, let x measure the distance along R, with A being at x=0. At every point along R, you can look perpendicular to the road to see a corresponding point on R'. Let x' be the distance along R' from A to this point. Then Euclidean geometry tells us that
\delta x' = \sqrt{1+m^2} \delta x
where m is the relative slope between the two roads (slope = the tangent of the angle between them). So to compute the length of a path along R', just integrate: L' = \int \sqrt{1+m^2} dx. As long as m^2 > 0, L' > L. So the nonstraight road is longer.
At this point, you could point out a "road paradox": Someone traveling along road R' could just as well look perpendicular to the road to see a corresponding point on R. The distance x along R is related to the distance x' along R' via the formula:
\delta x = \sqrt{1+m^2} \delta x'
So the traveler along road R' would see that \delta x > \delta x'. So he should conclude that the distance along R is longer than the distance along R'. He should conclude L > L'. Paradox!
The resolution of the paradox is to realize that the recipe of finding a point x' on road R' that corresponds to a point x on road R gives different answers for R and for R'. If the line between x' and x is perpendicular to road R, then it WON'T be perpendicular to road R'. The two roads disagree about which points correspond. This is the Euclidean equivalent of the "relativity of simultaneity" in SR. R uses one convention for setting up corresponding points, and R' uses a different convention. The two conventions have different notions of which points correspond. They both correctly compute the relative lengths of segments of the other road, but they compare DIFFERENT segments.
So why does the calculation of relative length by R turn out to be right, and the calculation of relative length by R' turn out to be wrong? It's because R' is not straight. The convention that R' uses to figure out a corresponding point on R has sudden jumps whenever R' makes a turn. Because of these jumps, either a segment of R is skipped (when R' turns away from R), or a segment of R is counted twice (when R' turns toward the road). Because of these jumps, the length calculation by R' does not give the right answer. You can't use a nonstraight road to compute the length of a straight road (at least not in a straight-forward way). This is exactly analogous to the twin paradox: you can't use a noninertial path to compute the proper time for an inertial path without a lot of extra work.
The above is all about roads on a flat Earth, which is analogous to paths through spacetime in flat spacetime. Your question about orbits brings up curved spacetime, so the analogy would be roads on a curved Earth. Consider two roads that both run straight from the North Pole to the South Pole. Near the North Pole, the length of one road can be approximately computed in the same way as for a flat Earth, because the curvature is only noticeable when you've traveled a long distance. So right near the North Pole, you can calculate the length of one road relative to the other, and use the formula \delta x' = \sqrt{1+m^2} \delta x, and each road will conclude that the other road must be longer. But the resolution here is that the formula \delta x' = \sqrt{1+m^2} \delta x is only good for small distances, where you can ignore the curvature of the Earth. It can't be used to compute the length of a road running all the way from the North Pole to the South Pole.