Twin Paradox Question

[garys_2k]

Help, please, I really can't quite get past the twin paradox thing in special relativity. I think I understand it, with time dialation and all, but here's what's puzzling:

Twin one leaves the earth in his 1-g accelerating rocket and lets it burn for three months. After only a few weeks he's travelling at relativistic speed relative to the earth and can quite clearly see through is telescope that the earth's clocks are wrong, they appear to run too slowly. He can confirm this by watching how fast the earth revolves around the sun, and since he used his own clock/calendar to decide when to shut off his engine he sees that the earth has not yet gone a quarter turn around the sun, but he knows that three months have elapsed according to his clock.

He turns his rocket around, facing the earth, and restarts his 1-g engine. After three months of this he is now stationary with respect to the earth again, having cancelled his outbound speed (let's assume he has a adjusted the thrust to maintain his 1-g acceleration even as he uses mass in the rocket). He notices that the clocks on earth now match his in terms of running rate, but again he can see that the earth has not gone another full quarter turn around the sun since he last checked. In fact, it has gone well less than a half turn since his journey began.

He again starts his rocket and runs it for three months, this time heading back home. After three more months he shuts it down and, yes, less than another quarter turn of the annual cycle has gone by since his last check. In fact, the earth is just a bit past half way around the sun since he origninally started. He again turns the ship around, facing away from earth, and restarts the engines. He reaches earth just as his speed relative to it drops to zero. He lands and climbs out.

Now, at this point, the "classic" twin paradox would have him meet his much older brother, as the traveller was supposed to have been exposed to the time slowing effects of his greater speed relative to earth. But if I got the above right, the astronaut should meet his YOUNGER twin on earth, as the earth (as seen from the astronaut's reference frame) was what travelled away and back.

I picked a 1-g constant accleration for the astronaut to remove general relativity effects.

So where did I mess up? Would an older twin (relative to earth) emerge from the rocket, or a younger one, or would they still be the same age relative to each other?

 

[Upchurch]

I picked a 1-g constant accleration for the astronaut to remove general relativity effects.

It doesn't matter how long you stretch out the acceleration. In fact, when I was in college I presented a research paper that showed that the longer you stretch out the acceleration (i.e. less acceleration over a longer period) the more difference there will be in the twins' ages.

So where did I mess up? Would an older twin (relative to earth) emerge from the rocket, or a younger one, or would they still be the same age relative to each other?

You messed up by trying to rationalize the entire trip only from the twin one's reference frame, which was, at some points, non-inertial. From the earthbound twin's frame, the rocket twin is also slowing down and, even though you can't explain it using only special relativity, it's slowing down even more during the acceleration.

 

[Ziggurat]

So where did I mess up? Would an older twin (relative to earth) emerge from the rocket, or a younger one, or would they still be the same age relative to each other?

Where you messed up is in your assumptions of simultaneity. Two events that happen simultaneously in one reference frame do NOT happen simultaneously in another reference frame. This is the non-intuitive aspect of special relativity. And when you change reference frames by accelerating, you're changing what is considered simultaneously. Here's a really nice explanation of the issue of simultaneity:
http://casa.colorado.edu/~ajsh/sr/simultaneous.html

First off, let me say that special relativity has no problem handling huge accelerations (huge gravity is different because it's a curvature of space-time), so we really don't need to limit ourselves to small accelerations. We'll simplify this by saying that the astronaut accelerates almost instantly (the same basics will apply regardless, but it makes the discussion easier). Now the astronaut heads out from earth, and after six months of his time, according to him six months has passed on earth. Now he comes to a stop with respect to earth. He changed reference frames. In doing so, simultaneity changes for him. When he was moving, six months on his clock happens at the same time (for him, not for earth!) as 3 months on earth, but now that he's stopped, six months on his clock happens at the same time as one year on an earth clock.

That seems a little strange, of course, but the other thing to keep in mind is the distinction between what one would actually "see" and what one "observes". The astronaut would not "see" any sudden change as he comes to a stop. But he wouldn't be seeing the clock at it's current time anyways because the light from the clock takes time to reach him. If you ask him how far away from him the earth was when the light from the clock reaching him now left earth, his answere WILL change suddenly when he comes to a stop, and that's how you can reconcile what he observes changing suddenly even though what he sees obviously should not.

 

[garys_2k]

Re: Re: Twin Paradox Question

It doesn't matter how long you stretch out the acceleration. In fact, when I was in college I presented a research paper that showed that the longer you stretch out the acceleration (i.e. less acceleration over a longer period) the more difference there will be in the twins' ages.
You messed up by trying to rationalize the entire trip only from the twin one's reference frame, which was, at some points, non-inertial. From the earthbound twin's frame, the rocket twin is also slowing down and, even though you can't explain it using only special relativity, it's slowing down even more during the acceleration.

OK, I *tried* to eliminate GR effects by choosing 1-g acceleration for the astronaut, as that was the same as experienced by the earth-bound twin due to gravity.

Another assumption, likely wrong, on my part was that one of the relativities (GR, I think) states that there was no difference between force from gravity (as the earth-bound twin experiences) and from acceleration (as the astronaut experiences). By making them equal I could have them be a non-issue, but I guess that was an incorrect thought.

 

[sorgoth]

But whereas Cerulean concludes that his mirrors are all equidistant from him and that the light bounces off them all at the same instant, Vermilion thinks otherwise. From Vermilion's point of view, the light bounces off Cerulean's mirrors at different times and moreover at different distances from Cerulean. Only so can the speed of light be constant, as Vermilion sees it, and yet the light return to Cerulean all at the same instant.

...My head hurts. If he gets all the signals at the same time, HOW could the mirrors be at different distances? Or even bounce off at different times?

 

[Ziggurat]

Re: Re: Re: Twin Paradox Question

Another assumption, likely wrong, on my part was that one of the relativities (GR, I think) states that there was no difference between force from gravity (as the earth-bound twin experiences) and from acceleration (as the astronaut experiences). By making them equal I could have them be a non-issue, but I guess that was an incorrect thought.

You're remembering correctly, though I would phrase it differently: a common starting point for GR is that an observer cannot distinguish based only on local effects (ie, he can't look out at the stars, etc) between a *uniform* gravitational field and uniform acceleration. Some people incorrectly assume that this means that acceleration is something only GR can handle. But the main point of GR isn't that acceleration is like gravity, but rather how do you go from that local picture of gravity (which seems simple enough) to a situation where gravity is NOT uniform (since that's always going to be the case). If a person in deep space and a person on earth are motionless with respect to each other, one person feels a strong gravitational pull and the other person doesn't, and a simple "gravity = uniform acceleration" picture is clearly not going to be sufficient for a global description of both of them.

 

[garys_2k]

Re: Re: Twin Paradox Question

Where you messed up is in your assumptions of simultaneity. Two events that happen simultaneously in one reference frame do NOT happen simultaneously in another reference frame. This is the non-intuitive aspect of special relativity. And when you change reference frames by accelerating, you're changing what is considered simultaneously. Here's a really nice explanation of the issue of simultaneity:
http://casa.colorado.edu/~ajsh/sr/simultaneous.html

Thanks! Excellent link -- That does an excellent job and I think the dawn is beginning to break. When the astronaut PERCEIVES the earth as moving more slowly around the sun he's seeing it distorted due to the relative motion... Correct?

First off, let me say that special relativity has no problem handling huge accelerations (huge gravity is different because it's a curvature of space-time), so we really don't need to limit ourselves to small accelerations.

I was trying to keep the acceleration experienced by each twin the same, to eliminate confounding due to GR, but here's where I think the fundamental breakdown in my understanding lay: I assumed that the astronaut's observations DURING his acceleration phase were entireley controlled by SR, but since an accelerating reference frame is non-inertial (I think, boy do I feel thick!), those rules don't necessarily apply. The acceleration (necessary for a round trip, of course, is the hidden key of why one twin absolutely ages more than the other. Is that right?

We'll simplify this by saying that the astronaut accelerates almost instantly (the same basics will apply regardless, but it makes the discussion easier). Now the astronaut heads out from earth, and after six months of his time, according to him six months has passed on earth. Now he comes to a stop with respect to earth. He changed reference frames. In doing so, simultaneity changes for him. When he was moving, six months on his clock happens at the same time (for him, not for earth!) as 3 months on earth, but now that he's stopped, six months on his clock happens at the same time as one year on an earth clock.

OK, it may be getting through! So what he perceives as three months on earth, by watching the earth revolve around the sun, is different from what the earth-bound twin sees. The earth-bound twin sees more time go by (on earth) compared to what the astronaut sees while observing earth as he was moving.

That seems a little strange, of course, but the other thing to keep in mind is the distinction between what one would actually "see" and what one "observes". The astronaut would not "see" any sudden change as he comes to a stop. But he wouldn't be seeing the clock at it's current time anyways because the light from the clock takes time to reach him. If you ask him how far away from him the earth was when the light from the clock reaching him now left earth, his answere WILL change suddenly when he comes to a stop, and that's how you can reconcile what he observes changing suddenly even though what he sees obviously should not.

OK, so let's go with the really fast acceleration model. As he travels outbound at relativistic speeds, nearing his six month point (just before he stops) he sees the clocks on earth (and the earth's revolutionary speed) going "too slow." After he comes to a stop the speeds are now alright, the clock ticks are the same as his, but he doesn't see the earth catch up for the "lost time" he saw it lose when travelling because the light rays from that time haven't reached him yet. In other words, he may believe that six months have gone by but when he looks back, after stopping, the quarter turn of the earth around the sun is an artifact of his being so distant from the earth. He'd have to sit around and wait for the later light to come out to where he was.

Does it sound like I'm closing in on this? I'm still confused about the effect of acceleration and how the astronaut's experiencing it will cause him to be (upon returning) to be the one that aged more slowly. But I think I'm getting a glimmer...

Thanks!

 

[garys_2k]

This page:

http://casa.colorado.edu/~ajsh/sr/wheel.html

explains the special relativity effects of acceleration. In fact, near the bottom they have duplicated my scenario pretty well.

Now I have to dig through that page to figure out what it all means! Thanks, everyone.

 

[Ziggurat]

It is indeed the traveling twin's acceleration that accounts for the difference in time he experiences compared to the earth-bound twin. People call it a "paradox" because it sounds like there should be some sort of symmetry between the two twins since, after all, relative motion is all the same, so the outbound twin's reference frame should be equivalent to the twin staying on earth. And it is, but it's that turnaround that makes their cases different. Here's another way of looking at the situation:

reference frame 1: earth-bound
reference frame 2: outbound at velocity v

Twin 1 stays on earth the whole time. Twin 2 starts out in reference frame 2, but then heads back to earth.

Let's watch the process from reference frame 1: twin 1 stays in our reference frame, aging at full speed. Twin 2 zooms off, aging at a slower rate, then zooms back, still aging at a slower rate, and arrives back at earth. Tin 1 is older.

Now let's watch from reference frame 2: Twin 2 is now still in our reference frame, and twin 1 (sitting on earth) is zooming away and aging more slowly. But here's where it gets different: the earth-bound twin never changes his speed, he keeps aging at a reduced rate, but now twin 2 zooms of at an even GREATER speed than twin 1 (compared to reference frame 2), so now his aging in reference frame 2 is even slower than twin 1. When he finally catches up with twin 1, he will have aged less overall. In other words, as expected, there isn't anything different between the actual reference frames themselves involved.

So here's the basic rule for time passing in special relativity: the LONGEST distance between two points in the time direction is a straight line. Regardless of which reference frame you choose, the earth-bound twin will travel a straight line through space-time from when the two twins depart to when they meet, and the astronaut twin will travel a curved (or bent) line.

 

[Ziggurat]

...My head hurts. If he gets all the signals at the same time, HOW could the mirrors be at different distances? Or even bounce off at different times?

The point is that it depends on which observer you ask. They are both moving with respect to each other. One observer will say all the mirrors are equal distance apart from each other, and that the light hits all mirrors simultaneously. The other observer (who is moving with respect to the first observer) will say that the mirrors are NOT the same distance apart, and that light does not hit all mirrors simultaneously. But both observers will agree that the light converges at a single point - for one oberver than point is in the same location as where the light was emitted, and for the other oberver that point has moved. Length and simultaneity are related concepts - they go together, and they get tweaked together when you change reference frames. The idea of length contraction in special relativity seems easier for people to grasp than the idea that simultaneity is also relative, but the two are really inseperable.

 

[69dodge]

Originally posted by garys_2k
This page:

http://casa.colorado.edu/~ajsh/sr/wheel.html

explains the special relativity effects of acceleration. In fact, near the bottom they have duplicated my scenario pretty well.

From that page:

Does this mean you go faster than the speed of light? No. From the point of view of a person at rest on Earth, you never go faster than the speed of light. From your own point of view, distances along your direction of motion are Lorentz-contracted, so distances that are vast from Earth's point of view appear much shorter to you. Fast as the Universe rushes by, it never goes faster than the speed of light.

I don't quite get this. Sure, when you're moving, distances along your direction of motion are Lorentz-contracted, but initially, before you started moving, they weren't Lorentz-contracted. So, from your point of view, as you accelerate, far-away objects approach you very quickly, much quicker than the speed of light, right?

 

[Ziggurat]

From that page:I don't quite get this. Sure, when you're moving, distances along your direction of motion are Lorentz-contracted, but initially, before you started moving, they weren't Lorentz-contracted. So, from your point of view, as you accelerate, far-away objects approach you very quickly, much quicker than the speed of light, right?

In a sense, yes, but in another sense, no. Here's the tricky part: when you ask how far away an object is, you mean how far away it is NOW. But if what counts as now keeps changing (the space plane in those diagrams keeps tilting) because you're accelerating, and the object is moving with respect to you so that it isn't in the same place if you change what is considered "now", then far-away objects will be getting closer to you "faster" than the speed of light. But this is ONLY an effect of changing reference frames (in other words, you're not in an inertial reference frame). If halt your acceleration and ask how fast stuff is moving, it's never faster than c. And even as you are accelerating, nothing in your immediate vicinity is moving faster than c. So you're right in a sense, but you shouldn't take that to mean that the speed of light is violated in any way.

 

[garys_2k]

It is indeed the traveling twin's acceleration that accounts for the difference in time he experiences compared to the earth-bound twin. People call it a "paradox" because it sounds like there should be some sort of symmetry between the two twins since, after all, relative motion is all the same, so the outbound twin's reference frame should be equivalent to the twin staying on earth. And it is, but it's that turnaround that makes their cases different. Here's another way of looking at the situation:

reference frame 1: earth-bound
reference frame 2: outbound at velocity v

Twin 1 stays on earth the whole time. Twin 2 starts out in reference frame 2, but then heads back to earth.

Let's watch the process from reference frame 1: twin 1 stays in our reference frame, aging at full speed. Twin 2 zooms off, aging at a slower rate, then zooms back, still aging at a slower rate, and arrives back at earth. Tin 1 is older.

Now let's watch from reference frame 2: Twin 2 is now still in our reference frame, and twin 1 (sitting on earth) is zooming away and aging more slowly. But here's where it gets different: the earth-bound twin never changes his speed, he keeps aging at a reduced rate, but now twin 2 zooms of at an even GREATER speed than twin 1 (compared to reference frame 2), so now his aging in reference frame 2 is even slower than twin 1. When he finally catches up with twin 1, he will have aged less overall. In other words, as expected, there isn't anything different between the actual reference frames themselves involved.

So here's the basic rule for time passing in special relativity: the LONGEST distance between two points in the time direction is a straight line. Regardless of which reference frame you choose, the earth-bound twin will travel a straight line through space-time from when the two twins depart to when they meet, and the astronaut twin will travel a curved (or bent) line.

BING! Light has been lit, thanks, Ziggurat! I do see the difference, what with that direction reversal and how it impacts the change of reference point. I'm thinking "rocket is a reference frame," but that wasn't true. It described one frame on the outbound trip and a completely different one inbound. Therefore, the earth twin's and the astronaut's situations are NOT the same. I drew a spacetime curve and I do see what you mean about the distances being fundamentally different, what with separation being (x-t)^2 and all.

Thanks!

 

[Ziggurat]

Re: Re: Twin Paradox Question

From the earthbound twin's frame, the rocket twin is also slowing down and, even though you can't explain it using only special relativity, it's slowing down even more during the acceleration.

Just for the record, nothing about the twin paradox, even using finite accelerations, is beyond special relativity. Special relativity has absolutely no problem handling acceleration. The math can get ugly (you REALLY need calculus) if you want to handle the acceleration quantitatively, so it makes things much simpler to consider an instantaneous change in speed. But there's really nothing "hidden" in the accelerating part of the whole process, and going from instant velocity change to finite acceleration doesn't change the basics of the problem at all.