I believe the standard resolution of the Twin “Paradox” is that, while at first glance it seems that each twin should be younger than the other, in reality one twin must have turned his spaceship around at some point and accelerated back in the direction of home. It is this asymmetry that defines him as the one who is younger.
But what happens if the universe is finite but unbounded–say a square toroid to avoid any curvature issues. Then a twin could travel all the way around the universe and meet back up with his brother, and neither would have experienced a second acceleration–what would be the resolution of the paradox in this case? Is it that the universe is so large that one cannot circumnavigate it?–Is this necessarily so?
(That brings up a second question: It has been pretty well established (I think) that the universe is “open” in the sense that dark energy will continue accelerating it apart for eternity. Does this necessarily mean that it is “open” in the sense of “infinite in physical extent,” or is it still possible that it has a shape like a square toroid, but that toroid is growing ever larger? I.e., I realize that the mass present (and, apparently, the quintessence) determine whether the universe is gravitationally open, closed, or flat; but is it possible for a gravitationally open space to be embedded in a geometric space that causes it to be finite in the same way that an “infinite” flat plane can be made finite by embedding it in a space where it is possible to join two pairs of orthogonal parallel edges without curvature.)
If you have them both at rest at start and finish, then one twin has clearlt accelerated and then decelerated, which breaks the symmetry.
If they’re not both stationary with respect to each other, then you’re comparing a moving twin to a stationary one – but that’s comparing two moving twins, or a stationary with a moving twin, depending uypon which reference frame you’re in. But I can already do that in an infinite unbounded universe.
But if initial/final acceleration is the problem, have them both accelerate away from each other and meet at the other side of the universe. Then both will have travelled symmetrically, but each should, I believe, still view the other as younger…
You don’t even need a weird universe for this – you can have a pair of twins accelerate away from each other, then decelerate after the same subjective time, and end up in the same reference frame. Sometimne later a third iobserver gathers up the reports from each and notes their arrival times, perhaps determined from some equidistant timer, like a variable star. Which twin got to his destination first?
I think they botgh arrive at the same time. There’s no reason to think one is any younger than the other. Since they’ve both followed identical schedules of acceleration and deceleration, they both have the same subjective time, with the same time dilation compared to a third person who stayed at the base.
Actually, Cal, bryanmcc has a point. A finite-but-unbounded universe does in face have a globally preferred reference frame, and whichever twin stays in that reference frame (or closer to that reference frame) will be genuinely older. The acceleration/deceleration of the travelling twin doesn’t break the asymmetry, since he only needs to accelerate at the rendezvous points.
I guess the real problem with this “paradox” is trying to apply the rules of special relativity in a situation where general relativity should be applied (i.e., if the reference frames are non-inertial).
Is there any reason to speculate that our universe has, or could have, this global preferred reference frame? (I know that 400 years of science has been regularly axing privileged position, a fact which I think needs to be further applied to fields such as SETI, evolution, and theology; but that trend does not itself negate the possibility of a preferred frame, if evidence presents itself.) For instance, the CMB seems to imply a universal reference frame to which it, at least, is at rest (doesn’t it?). What implications (if any) would this preferred reference frame have on, well, anything?
Suppose the universe is connected like a torus. Twin A accelerates to 99.99…99% (fast enough to get back on a round trip before twin B dies) of the speed of light and travels around the universe taking say 10 years. Just before he gets back twin B accelerates to the same speed reaching it just as twin A catches up to him.
Now they are clearly in the same inertial frame so which is older?
It seems to me it must be that since Twin B saw A moving away from him and 9+ years later he sees twin A approaching him, the topology of the universe connected with General Relativity must lead him to conclude that Twin A has gone through some other acceleration. I.e., if the universe is closed, General Relativity must be involved in some fashion.
There’s a couple of things I’ve never been able to understand about space.
How come if our orbital velocity is 30,000 mph, how come anything leaving our atmosphere doesn’t get left behind, like a cigarette thrown out of a car window?
and probably somewhat related,
If our sun is traveling at x speed in it’s own orbit around whatever, how come Earth isn’t traveling that fast as well? How come we are not flung outward, like a child on a rapidly rotated round-about?
Is it something to do with space being a vaccuum, and therefore there is no “wake”?
General Relativity is not needed, only special relativity. To help in visualizing this, make yourself a Minkowski Diagram on a sheet of paper, (in particular the fourth image on that page), with time running generally up/down, and space generally left/right. For a finite, unbounded universe, roll the paper into a tube so that the thick black lines line up across the seam. Notice that the blue space coordinate lines don’t meet themselves, but rather spiral around the tube. All reference frames in motion relative to the black system will have space coordinate lines that spiral. The black coordinate system is the preferred reference frame.
Assume that one of the twins is in the black system for (nearly) the ten years, and the other is in blue. The one in the black reference frame will be older, whether he’s twin A or twin B.
I’ll have a guess (assuming the circle is the spatial dimension, and there’s an unbounded time dimension also): Restrict it to the zero curvature (low mass) limit, with constant metric (no GR)?
