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Well as you get closer to the speed of light, time slows down for you, so if he kept a stedy speed at close to the speed of light, it would feel like the same amout of time, but more time actally passed in the real world.

But doesn't that idea only apply if one body is completely inert?

I thought that the amount of time change had to do with how close one actually came to the speed of light.

I understand the idea that bodies move by merely changing position in relation to each other, but when something can be observed as the same by more than one reference point, doesn't that set the definition? Such as, if every reference point in the universe minus one observed it as the same, and you get infinity minus one, and since in any equation where you have an infinite you can eliminate all non-infinites, then wouldn't it hold true that Ann would be correct?

There's an important difference between Ann and Bill in this scenario, however. From a completely physical perspective, for the universe to be closed as it in this scenario, it must have some matter distribution. It makes quite a bit of difference how Ann and Bill move in regards to this matter, and so we should expect them to experience different durations of proper time.

In terms of geometry, there is no true symmetry between Ann and Bill because in the fundamental domain, their trajectories have different homotopy classes. Bill goes all the way around the universe, whereas Ann does not--this is a geometrical fact. It doesn't make it completely obvious as to who experiences more proper time before they meet again (in fact, Ann does, and in general, the one of zero homotopy class experiences more proper time than other classes), but it makes it clear that one cannot treat them as if their roles were interchangeable.

Hence the first paragraph of my previous post--there is a privileged frame of the cosmic dust or fluid. Geometrically, it is the frame corresponding to extremal size of the universe. I've assumed that "stationary" means "at rest with respect to the cosmic dust", as usual in questions like this. If the hypothetical universe is otherwise like our own, it's also the frame in which the cosmic background radiation is maximally isotropic. For simplicity, assume that the universe has the very simple geometry of R×S, with circular hypersurfaces of constant time for Ann. Bill zooms past at velocity v, measuring time along that trajectory; the hypersurfaces of constant time for Bill are helical. Whoops--geometry breaks the apparent symmetry.

To physically see that something fishy is going on, suppose both Ann and Bill try to measure the size of the universe by sending light signals into opposite directions. For Ann, the universe is exactly circular, so she will receive both signals at the same time and conclude that the universe is some size 2πR for some constant R. For Bill, we have two null geodesics going along θ = ±t/R, while Bill himself goes along θ = vt/R, meaning that Bill receives one of the signals at t = 2πR/(1+v) and another at t = 2πR/(1-v). He can then conclude that he is not at rest with respect to the privileged frame.

depending on how fast bill was traveling in relation to ann, and iff they are moving in the same direction or in opposite directions. You also have to take into account the condition of the watches, wither or not they were started at the "exact same time" The effects of spacial anomolies on time, and other uncalculated variables. (One cannot travel across a universe at near light speed without coming across a planet/star with a strong magnetic pull or a black hole.)

It's almost impossible to say who exactly has the watch with the largest time interval elapsed between start and finish....


I haet time paradoxes... Just divide by zero and call it good