Alternate twin paradox.

[RussDill]

The twin paradox is easily solveable with general relativity. However, there is another form of the paradox which I have not been able to solve. The paradox has one assumption, the universe is closed. Ok, so here's how it goes.

There are two probes, both with clocks strapped to them. Probe 1 and probe 2. They both start at point A, and head in opposite directions at very high speed (we'll say .9c, but you can use any near light speed to ease calculations).

Point A observes this, know that the universe is closed determines that they will meet at a point, B, at the "opposite end" of the universe. Knowing how large the universe is, point A makes a simple calculation on how long it will take them to get there, and what their clocks will say.

Probe 1 of course, has its own frame of reference. It sees point A disapearing into the distance at .9c, and point B approaching at .9c. It also sees probe 2 leaving at a speed greater than .9c.

Probe 1 sees that AB is a line segment traveling past it. Because this line segment is traveling at .9c, it calculates the time it will take to get to B, which is less than what A calculated (no problem yet, this will match up A's idea of what 1's clock should be when it reaches B). Next Probe 1 thinks about probe 2 and wonders when and where it will meet probe 2. Probe 2 caclulates where they will meet, it is not point B, but point B1. It also calculates that since time for probe 2 will be happening at a much slower rate, that when they will meet, probe 2's clock will be far behind probe 1's clock.

Probe 2 thinks the same, but opposite things, but thinks that they will meet at a point B2.

This problem is similar to objects traveling around at near light speed in a large circle (like an accelerator). That problem is easily solved because angular acceleration is involved. The above example, however, does not apear to have any forces, or any acceleration.

The only solution I see is a yet unknown (or just unknown to me) acceleration generated by traveling along the surface of a 4D hypersphere. Knowing the size of the universe would give the answer as to the strengh of such a force, but no clues as to the source of that force, the effect of that force, and why it would seem to provide a frame of reference as to what objects are moving, and what are not.

Also, I'm thinking I'm missing something really important, because imaging a point, and another point, making a line segment moving towards you at .9c does make sense when talking about relativity, it sure doesn't seem to make sense when talking about one point being here, and another point being at the opposite end of the universe, because it would make the universe smaller by just thinking about it.

 

[Ziggurat]

The twin paradox is easily solveable with general relativity. However, there is another form of the paradox which I have not been able to solve. The paradox has one assumption, the universe is closed. Ok, so here's how it goes.

Back up a second: the twin paradox is solveable in special relativity. You don't need general relativity at all, in either the posing of the question OR the solution to it, unless you want to make it needlessly complicated by introducing gravitational bodies.

Also, I'm thinking I'm missing something really important, because imaging a point, and another point, making a line segment moving towards you at .9c does make sense when talking about relativity, it sure doesn't seem to make sense when talking about one point being here, and another point being at the opposite end of the universe, because it would make the universe smaller by just thinking about it.

General relativity is strange, especially when you want to deal with the topology of space-time. But I think I have an idea of where the "paradox" may be. Simultaneity is a tricky thing is special relativity, and it only gets messier in GR. First off, we know that probe 1 MUST calculate that probe 2 will meet him at point B - the theory is self-consistent, if we think you've found a flaw it's almost certainly going to be in our own understanding of the situation. So let's take that as a given, and go from there.

OK, so what's the problem with probe 1 calculating that probe 2 meets probe 1 at B? Well, on the surface it looks like from Probe 1's reference frame, probe 2 would have to go faster than c to reach B at the same time. But we've actually made an assumption here: we've assumed that the observation of time at different locations (simultaneity) is independent of which way we wrap around our universe. But it is not.

First thing to keep in mind is that for our closed universe, there IS, in a sense, a rest frame for the universe. The mass density which causes the universe to be curved (and hence closed) defines a reference frame. Say the universe is a 4D sphere. Change reference frames with respect to that, and the laws of physics are the same, but now most of the mass in our universe is moving, and the universe isn't a perfect sphere anymore (It's lorentz contracted in one direction). That's what probe 1 is seeing: a warped universe. And that warping warps time as well, and not the same in every direction.

Now consider the two points, A and B, with clocks on them. At point A, probe 1 and 2 leave at 12:00. Suppose in A's reference frame, point B says 12:00 at the same moment (they're simultaneous). We already know from special relativity that if you ask the now moving probe 1 what the time is at point B, he won't agree with A (again, we're dealing with what they "oberve", not what they would see). That's nothing new. But here's the twist from GR: probe 1 will actually give you a different answer depending on the direction. In this closed universe, simultaneity depends not only on relative motion, but even the direction in which you're asking the question.

So basically, when probe 1 looks backwards, he observed that probe 2 left at the same time as he did. But when he looks forward, all the way around the universe and back at himself, he will observe himself and probe 2 having left at a different time. And if he uses that other time to track probe 2 approaching him, he'll calculate (correctly) that the two should meet at point B.

 

[Walter Wayne]

Probe 1 sees that AB is a line segment traveling past it. Because this line segment is traveling at .9c, it calculates the time it will take to get to B, which is less than what A calculated (no problem yet, this will match up A's idea of what 1's clock should be when it reaches B). Next Probe 1 thinks about probe 2 and wonders when and where it will meet probe 2. Probe 2 caclulates where they will meet, it is not point B, but point B1. It also calculates that since time for probe 2 will be happening at a much slower rate, that when they will meet, probe 2's clock will be far behind probe 1's clock.

Probe 2 thinks the same, but opposite things, but thinks that they will meet at a point B2.

I am not sure that we can say this for a closed universe. A closed universe has a curvature which will mean that geodesics are no longer describe by s²=t²-d². I don't believe one will get different results from looking forward, than from looking back.

By this I mean that t' != t/sqrt(1-v²), instead t' = k*t/sqrt(1-v²), where k is some factor due to the local curvature of space time. When integrated from point A to B, one will find that everything, k will make everything equal again .


It's late, but I will try to figure this out tomorrow.

Walt

 

[fishbob]

Probe 1 of course, has its own frame of reference. It sees point A disapearing into the distance at .9c, and point B approaching at .9c. It also sees probe 2 leaving at a speed greater than .9c.

Nope - Probe does NOT see Probe 2 leaving at a speed greater than C. I can't recall the formulas, but this is an easy calculation. Probe 1 sees Probe 2 at some speed between 0.9c and 1.0c.

 

[RussDill]

Nope - Probe does NOT see Probe 2 leaving at a speed greater than C. I can't recall the formulas, but this is an easy calculation. Probe 1 sees Probe 2 at some speed between 0.9c and 1.0c.

thats what I said, I said greater than .9c, I was hoping that I would not come across stupid enough to actually need to state that it was going less than c.

 

[epepke]

It's already been pointed out that you can solve the regular twin paradox with SR only, although the GR solution is nice, too. The linear acceleration aspects of GR are hardly more than SR.

I've thought about this paradox for a long time, and it seems to me that the problem is that the universe is expanding just fast enough that you can never get there from here. Which would mean that the speed of light is in some way related to the rate of expansion of the universe.

 

[Ziggurat]

I've thought about this paradox for a long time, and it seems to me that the problem is that the universe is expanding just fast enough that you can never get there from here. Which would mean that the speed of light is in some way related to the rate of expansion of the universe.

I don't think that's necessarily the case at all. First off, it's a closed universe, so it's not going to expand forever. With an artificial universe (ie, we can pick mass density, etc, and maybe even tweak things like G) it's not a problem to have it close to static compared to the time it takes to circumnavigate it. The problem, as with pretty much ALL relativity "paradoxes", is our intuitive notions of simultaneity break down. And in this case, they break down even more badly than they do in special relativity. In a closed universe, time is not necessarily a single-valued coordinate. That's easy to understand for space: imagine the path around the universe is 1 light year around. We start out at some point, label it x=0, and start moving around, labeling the points as we go along (x=0.1 light years, etc). When we loop back to the start, we find that x=0 is also x=1 light years. Our spacial coordinates are multi-valued.

OK, so how does this play out with the time coordinate? Let's take our universe, and put a circle of light bulbs around it, with the light bulbs at rest compared to A and B. We arrange so that in A's reference frame, all the lightbulbs flash at the same time as probe 1 and 2 leave. From special relativity, we know that the flashes do NOT happen at the same time in the reference frames of the two probes. The light bulbs ahead of probe 1 flashed earlier in probe 1's moving reference frame, and the probes behind it flashed later. That's all just special relativity. But here's the twist in this scenario: if we now ask, what about the light bulb at point B? In probe 1's moving reference frame, did it happen before or after probe 1 left point A? The answer to that depends on whether you consider B to be ahead of or behind the probe. The time coordinate in probe 1's reference frame is multi-valued, just like the space coordinate.

Here's another way to think about it: Take the space axis of the coordinate system of A and B, which marks all points in space at the same time. Now if we shift to a reference frame moving with respect to that reference frame, we have to tilt that axis. But if we tilt that axis, it won't close on itself when it wraps around the universe. There should be one reference frame in which time can be single-valued, but in any other reference frame, time is going to be multi-valued. This may look like it violates the idea of relativity, but it doesn't, because what should determine this special reference frame is the mass of the universe, which determines the geometry of the universe. The laws of physics are always the same in any reference frame, but it does make a difference whether the mass of the universe is stationary or not.

OK, so how does this resolve the apparent paradox? Probe 1 takes a certain amount of time to reach point B, where he meets with probe 2. Looking ahead of him, he calculates that it must have probe 2 longer to get to B than it took him. But (here's the twist) if he's looking forward (and not back), he also calculates that probe 2 left before he did (that's where the multi-valued part of all of this comes in). Probe 1 and 2 WILL meat at point B simultaneously, and they will both see each other approaching at speeds less than c. No paradox, other than the fact that we have to throw out the notion of time as being a single-valued coordinate. But that's OK, we already do that easily for space.

 

[Walter Wayne]

I have though about this, and I think the problem we are running into is with terminology. They intersect at events A and B, not points A and B. By the terminology used in the first section. Point A and Point B are lines in space time. Therefore observer at A, by symmetry sees both probes getting to point B by crossing identical distances and times.

From probe 1's point of view. He calculates that distance between him and Point B is contracted by the same amount as the distance between probe 2 and Point B. He also observes that probe 2's clock is running slower. However, what does he see the velocities as.

Point B is travelling towards him at .9 c. Probe 2 is travelling away from him at v > 0.9c. Point B is also travelling away from probe 2 at .9 c. Thus although the initial distances between the two probes and point B are equal, the point is running away from probe 2 in probe 1's point of view. Thus, even though probe 1 sees probe 2's clock as running slower, he sees probe 2 as having to travel a greater distance.

Walt

Edit: I may have to recant, I am doing the math on the above post and I am not getting agreement. For the math I am using a "cylindrical" universe, as I believe that a cylinder is mathmatically flat.

Edited to add: OK, not fun. While time can be uniquely defined for one observer in a "cylindrical" universe, this is not true for all observers. So this is exactly what Ziggurat is talking about. Thanks Ziggurat.