in a Euclidean Manifold
The twin 'paradox' is somewhat clouded by what I call a "poof factor".
As already shown, this is actually impossible but we may, at least, imagine it to be possible.
Let A and B be the above brothers, C will be the distant cousin (1 light year away) and D an observer in an identical spaceship stationary at 1/2 light year down the path to C.
What does each observer actually see if he is always looking at everybody else?
When B stops back on A, everyone appears to be aging at a normal rate but B is two years younger than everyone else.
The "Rub"Here's the problem:
When B is passing D the situation appears to be logically symmetric but isn't physically symmetric. D must see B differently than B sees D for in the end D observes B to be younger while B observes D to be older. When does this occur from the viewpoint of either party (B,D) who are always observing the other?
Most science writers employ the poof factor and run over the problem like it doesn't exist. Well, if you sweep dirt under the rug it's clean ... isn't it?If B leaves A and D leaves from C and they pass in the middle, then you might have B seeing D and vice versa in the same way. But what about the direction of travel of A and C through the cosmos. Every bit of matter has an "acceleration history" so that to have a truly symmetric situation, the two ships must have equal but opposite accelerative histories when they pass, i.e. we have to subtract out their proper motions relative to the backdrop of galaxies (that is, all motion not generated by the Hubble expansion).
The absolute reference frame is just where everyone thinks it is after all.
The problem is one of logic/semantics. Obviously, we can always logically assert that "I'm not moving, you are!" or, "I'm not moving, the rest of the universe is!" and be right in the strictest logical sense. But since this argument is true for everyone always, it has no utility as a logical concept (except to generate the big "... so what?"). Some confuse the logical/physical aspect of theory which is all the same in the long run but which may be divergent in the short run.
Addendum 04-04-00:While it is quite true that one cannot detect the absolute reference frame by doing an experiment confined to the one inertial frame you are doing it in, we can detect it (in principle, down to uncertainty limitations) by observing the backdrop of stars. I'm here refering to the absolute frame which has physical utility.
This whole problem of logical/physical is 'gummed' up by the necessity of there being embodied in existence a backdrop, stage or reference frame in which to do business. I had the idea once that perhaps if one could "grab onto" the whole universe and throw it rearward, one might then be propelled forward thereby conserving linear momentum. But this is not possible because quantitive information cannot be transmitted at greater than light speed.
Imagine the universe as a ladder of infinite length. You are on the ladder. Now climb the ladder. You move up but does the universe move down? Of course not. A "compression" of the ladder moves down the ladder and an "expansion" moves up ahead of you just sufficient to balance the linear momentum books.Physical means logical with a condition attached (in this case the finite velocity of quantitive data transmission).