The Twins Paradox: A Controversial Truth or a Perplexing Paradox?

[JesseM]

Jesse: From your post 117: "The only situation where you can talk objectively about which of two observers has aged less is where they first compare clocks at a single location, then move apart, then come back together to compare clocks again."

Simply not true - look again at what I have quoted from the 1905 paper - A and B are not together - they are at rest separated by a distance d, brought to sync - and then A moves to meet B. There is now an age difference. A's clock will have recorded less time than B's. Seems we have already established this about 10 times.

Can you provide that quote again? I've forgotten which post of yours included it. I am confident that Einstein would not say that in that situation A has objectively "aged less" than B in a frame-independent sense, if you think I'm wrong please show a specific quotation where he says something to that effect.

Let us assume Yogi and Jesse are on orbiting spacecraft - I am flying around the equator east to west and you are flying west to east. Each time we pass each other we check the other guys clock using two clocks in our own spacecraft - You will say poor Yogi - not only is he slow to understand SR - his clock is running slow also. And I will look out my window and measure the rate of your clock as it flies by and say - Jesse is to busy posting on the forums to set his clock rate up to speed.

Only in my instantaneous rest frame at any given moment can I say your clock is running at \sqrt{1 - v^2/c^2} times the rate of mine, but I can't integrate \int_{t_0}^{t_1} \sqrt{1 - v(t)^2/c^2} \, dv to find the elapsed time on your clock as my clock ticks from t_0 to t_1, since I am not in an inertial reference frame. Also, if we actually watched each other's clocks using light instead of calculating the rate of the clocks in our instantaneous rest frame, we would sometimes see each other's clocks ticking faster, sometimes slower.

Gonzo is at the North pole - at the same height above the Earth as you and I - he will see both our clocks running slow. Now each time we pass I will see your clock running slow and you will see mine running slow - but a funny thing happens -neither of us accumulates any age difference relative to the other - our situation is entirely reciprocal. So after many orbits we decide to land and we do so bu detouring to the North Pole - we compare clocks - Jesse and yogi's logged times will be the same - but Gonso's clock will read more than Jesse's and Yogi's Or do you have a different conclusion?

Yes, you're correct. But gonzo is the only one in an inertial reference frame here, so all of us must use his frame to calculate the actual time elapsed on all three of our clocks when we meet (assuming we're doing the calculations in SR, not GR).

 

[yogi]

Jesse - take a look at post #55 re your Q.

As far as you and I and Gonzo - none of us is in a truly inertial frame - we are accelerating due to the Earth's rotation and Gonzo is accelerating because of the Earth orbit around the Sun. Its just a matter of degree. Even if the measurments are perfect, you can still do the experiment - just assume you kink your orbit slightly each pass so that for an instant your path is straight. The apparent slowing of my clock will be detected by two clocks spaced apart in your spaceship.

 

[JesseM]

yogi, the quote you provided in post #55 was:

"If at points A and B there are stationary clocks which viewed in the stationary system, are synchronous, and if clock A is moved with the velocity v along the line AB to B then on its arrival the two clocks no longer synchronize, but the clock moved from A to B lags behind the other which has remained at B by (1/2)t(v/c)^2 ..."

There is absolutely nothing there about A "aging less" in an objective, frame-independent sense, in fact there is nothing about the relative rates of the clocks at all, all he says is that the clock A will be behind the clock B "on its arrival", which of course I agree with. The claim that A has objectively aged less, despite the fact that in A's frame it is B who has aged less, is totally ridiculous and conflicts with the basic assumptions of relativity, so I am sure Einstein never suggested such a thing.

Let me ask you this: do you agree that what's important in determining who has aged less when A and B meet is only the time intervals recorded by each frame, not the actual times on each clock when they meet? Suppose at B and E are synchronized in their own rest frame, and A is moving towards them at constant velocity, first passing E and then meeting B, and we don't assume that A synchronizes his clock with E at the moment they pass. Instead, at the moment A passes E, E's clock reads t_{EA} and A's clock reads some different time T_{AE}, and at the moment A reaches B's location, B's clock reads t_{BA} and A's clock reads T_{AB}. Do you agree that even if T_{AB} > t_{BA}, we can still say that A has aged less in the BE frame since (t_{BA} - t_{EA}) > (T_{AB} - T_{AE})? If so, are you claiming that not only would this mean that A aged less in the BE frame, it would mean that A aged less in a frame-independent sense? Please answer these questions yes or no.

 

[yogi]

Jesse --We have an entirely different interpretation of what Einstein has said - The only significant thing is Einstein's lack of explanation of why one of the two clocks objectively aged less - A or B. This is the whole point I have been trying (obviously w/o success), to get across. Einstein didn't explain it -but as I keep saying there is asymmetry in the one way twin problem because once A and B are in sync in the same rest frame, one must move so that they eventually meet - and the one that moves will have logged less time on his clock. If it is A that moves (by accelerating) it is of no consequence that in A's frame, B's clock appears to be running slow while they are in relative motion - that is the "apparenency" that always occurs when making measurments in a relatively moving frame -

if both A and B accelerated equally, A and B while in relative motion, would each believe the other clock is slow, but because there is symmetry in this example, there would be no real age difference when they compared clocks upon meeting. Einstein's whole point of part 4 of his 1905 paper was to express the physical meaning of the transforms - real measurable age difference - this is the transition from illusory observations to reality. He stated the result, but he didn't explain it.

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With regard to the non-inertiality of the frames in my orbiting spaceship description - I think it is insignificant. These are all thought experiments - we don't need to get bogged down with the fact that there is some slight curvature to the orbits during the sampling period - in one sense you can consider our spaceships to be non accelerating since they are in orbit, there are no G forces - and in another sense you can rely upon the fact that it is not necessary to take into account the curvature in GPS - we get the correct offset straight away from the relative velocity between the clock in the Earth centered reference frame and the satellite clock(s) via the Lorentz transforms - ergo it is not a general relativity problem.

 

[yogi]

Jesse - Your question - No - I don't think there is adequate information to make any conclusions about actual age difference - at least as I understand your question, I would have to know whether the relative velocity between A and the EB frame came about by acceleration of the EB frame or A.

 

[JesseM]

Jesse --We have an entirely different interpretation of what Einstein has said

No, your "interpretation" cannot possibly be valid, unless you imagine that Einstein himself disbelieved the principle that each reference frame's perspective was equally valid, and one would have to be grossly ignorant to assert such a thing, given everything else he wrote.

The only significant thing is Einstein's lack of explanation of why one of the two clocks objectively aged less - A or B.

No clock has objectively aged less, that's utter nonsense. I notice you didn't answer my question earlier, do you agree with my statement about the time intervals being all that matter? In other words, if E reads 2:00 and A reads 4:00 when they pass, and then A reads 5:00 and B reads 4:00 when they meet, do you agree that it doesn't matter that A has a greater time, all that really matters is that A saw an interval of 1 hour while the interval was 2 hours in the EB frame? And if you do agree on this, would you also say that this proves that A was objectively aging slower as he approached B? Please address this question.

if both A and B accelerated equally, A and B while in relative motion, would each believe the other clock is slow, but because there is symmetry in this example, there would be no real age difference when they compared clocks upon meeting. Einstein's whole point of part 4 of his 1905 paper was to express the physical meaning of the transforms - real measurable age difference - this is the transition from illusory observations to reality. He stated the result, but he didn't explain it.

Einstein made very clear that different frames define simultaneity differently--this was the whole point of section 1--so obviously that's how he would explain why this situation is symmetrical, because in A's frame B did not read the same time as E, B was in fact ahead of E, so the fact that B was ahead of A when they met is perfectly consistent with the idea that B was ticking more slowly than A in A's frame. If you don't like this answer, fine, but it would be foolish to deny that Einstein and other relativists would see it this way.

With regard to the non-inertiality of the frames in my orbiting spaceship description - I think it is insignificant. These are all thought experiments - we don't need to get bogged down with the fact that there is some slight curvature to the orbits during the sampling period

Slight curvature? They make a complete circle during the sampling period! If you were just looking at a very short time in which the curved orbit was close to a straight line, that'd be one thing, but a complete circle is about as far as a straight line as you can get.

in one sense you can consider our spaceships to be non accelerating since they are in orbit, there are no G forces

Sure there are, there will be centrifugal force on board the orbiting ship. edit: I just remembered that in orbit, the centrifugal force will be equal and opposite to the gravitational force...and of course the equivalence principle says that in an arbitrarily small region of spacetime, the laws of physics will look the same for an observer in free-fall as they do for one moving inertially. But if you want each observer to keep track of the movement of the other observer at moments other than the one where they are passing each other in orbit, then their coordinate systems cannot be arbitrarily small, and there will be effects which distinguish their coordinate system from an inertial one, like tidal forces and the fact that light seems to go faster in one direction than the other.

and in another sense you can rely upon the fact that it is not necessary to take into account the curvature in GPS - we get the correct offset straight away from the relative velocity between the clock in the Earth centered reference frame and the satellite clock(s) via the Lorentz transforms - ergo it is not a general relativity problem.

Nonsense, the GPS satellites certainly take into account the fact that clocks on the Earth's surface are not moving inertially--where are you getting your information? I suggest you take a look at http://relativity.livingreviews.org/Articles/lrr-2003-1/ on relativity and the GPS satellites...for example, look at section 2 (Reference Frames and the Sagnac Effect), where they show the time required for light to travel a certain path in a rotating reference frame (equation 7), and then say:

Observers fixed on the earth, who were unaware of Earth rotation, would use just \int d{\sigma}^{'} /c for synchronizing their clock network. Observers at rest in the underlying inertial frame would say that this leads to significant path-dependent inconsistencies, which are proportional to the projected area encompassed by the path.

 

[JesseM]

Jesse - Your question - No - I don't think there is adequate information to make any conclusions about actual age difference - at least as I understand your question, I would have to know whether the relative velocity between A and the EB frame came about by acceleration of the EB frame or A.

OK, thanks for addressing this question--but are you asserting that the history of which accelerated most recently is somehow relevant? In other words, if A is at first at rest relative to B and E and then accelerates towards them, this would lead you to a different conclusion about whether A had "objectively" aged less than B as it passed from E to B than if A had been moving at constant velocity towards the position of E and B since before E and B were moved (accelerated) from Earth to their current position in space? What if A last accelerated a million years ago while E and B last accelerated a million and one years ago, would this lead you to a different conclusion about who objectively aged less than if E and B last accelerated a million years ago while A last accelerated a million and one years ago?

 

[yogi]

Jesse:

The issue is not which frame accelerated most recently - it is simply this - two clocks in one frame are brought in sync - one is moved. It will be found to be out of sync with the one which did not move - and the amount of the discrepency is given in Part 4 of the 1905 paper.

With regard to the counter orbiting spacecraft - you again want to impose conditions that are not required - I do not have to wait for a complete orbit to determine that the other clock is running slow - I simply place two clocks at opposite ends of my spaceship, measure the distance between them L , and then use my two clocks to measure how long it takes your clock to travel the distance L between my two clocks. Based upon the time difference recorded by my clocks and the time recorded your clock - I will always believe that your clock is running slow.

Clocks in relative motion do record different absolute times whenever they have been synced at rest and one clock is put into motion - all evidenced by H and F airplane experiments, GPS, and the extended lifetime of high speed muons and pions that are created in the Earth reference frame and subsequently move relative thereto.

I don't really think we can learn anything further from each other - this thread has gone on way too long as it is.

 

[JesseM]

Jesse:

The issue is not which frame accelerated most recently - it is simply this - two clocks in one frame are brought in sync - one is moved. It will be found to be out of sync with the one which did not move - and the amount of the discrepency is given in Part 4 of the 1905 paper.

What does "moved" mean? Does it mean "accelerated", or does it just mean that A is moving relative to the rest frame of B and E? If acceleration isn't relevant, why did you say "at least as I understand your question, I would have to know whether the relative velocity between A and the EB frame came about by acceleration of the EB frame or A"? If acceleration is relevant, yet it doesn't matter who was accelerated most recently, then in every possible experiment both A and EB will have been accelerated at some point in their past, it's not like there are any clocks in the universe which have been traveling at constant velocity since the Big Bang. So what does this lead you to conclude about which one was "moved", and which clock is "really" ticking slower?

With regard to the counter orbiting spacecraft - you again want to impose conditions that are not required - I do not have to wait for a complete orbit to determine that the other clock is running slow - I simply place two clocks at opposite ends of my spaceship, measure the distance between them L , and then use my two clocks to measure how long it takes your clock to travel the distance L between my two clocks. Based upon the time difference recorded by my clocks and the time recorded your clock - I will always believe that your clock is running slow.

In this case, you're right that the curvature of the orbit will be negligible, as long as your ship is small. But then what was the point of bringing an orbit into the picture at all? Why not just imagine that your ship is moving inertially, and mine is too?

Finally, what does this have to do with the claim that there is an absolute truth about which clock was ticking slower? Don't you agree that if my ship has two clocks on either end, and I look at the time on my clocks as one of your clocks passes both, I will say that your clock ran slow?

Clocks in relative motion do record different absolute times whenever they have been synced at rest and one clock is put into motion - all evidenced by H and F airplane experiments, GPS, and the extended lifetime of high speed muons and pions that are created in the Earth reference frame and subsequently move relative thereto.

None of these experiments show evidence for an "absolute" truth about which clock is running slow, all of them are perfectly consistent with the idea that each inertial frame sees clocks in other frames running slow, and that there is no way to settle which frame's point of view is the correct one. If you think Einstein would have disagreed, then you are completely misunderstanding his paper.

I don't really think we can learn anything further from each other - this thread has gone on way too long as it is.

I think that if you will clarify some of the ambiguities in your arguments, like what you mean by "moved" and why you think there is an absolute truth about which of two clocks is running slow, then this argument could still end up being a productive one.

 

[JesseM]

Following up some comments from the Cosmological Twin Paradox thread:

Jesse. Moved means - they are initially brought to rest in one frame - and one of the clocks is accelerated - we know which is accelerated because some agency is required to bring about a velocity change of one clock only. But the acceleration has nil to do with the reading of the clocks

If it has nil to do with the reading of the clocks, then why did you bring it up? Do you agree that in the cosmological twin paradox, if the twins are originally both on Earth (which we assume is moving inertially) and then one twin accelerates briefly and then flies away from the Earth at constant velocity, the twin who accelerated may have aged more rather than less when the two twins meet again, since the Earth may not be at rest in the preferred coordinate system defined by the topology of the universe? Do you also agree that in SR, the question of which of two clocks was "moved" is irrelevant to the question of which clock "really" aged less in a particular time-interval?

Let us see if the cosmological twin problem can be localized - say that we have two clocks A and B in orbit about the Earth - Clock A was built on Alpha and was in sync with all alpha clocks before it was launched a million years ago and eventually captured by the Earth's G field into an east-west circular orbit. Clock B was built on Earth and was in sync with Earth clocks before being launched into a west-east circular Earth orbit. As they pass each other every 2 hours - will one or the other of these clocks appear to be gaining time?

Why did you bother to say that B was built on Alpha Centauri and A was built on Earth? This is totally irrelevant to the problem. Again, past history doesn't matter in relativity, including the question of which one accelerated (or which one 'moved' in your terminology). So all that matters is that we have two clocks, one orbiting west-east and the other orbiting east-west, if we know their specific velocities and the size of their orbit this will tell us whether one clock has gained any time from one moment they pass to another, and if so how much. If they are both orbiting at the same speed in the rest frame of the Earth's center, then neither will gain time; if they are orbiting at different speeds relative to the center (which would be true if they were orbiting at the same speed relative to the surface), then the one that's orbiting faster in the rest frame of the Earth's center will get show less elapsed time between successive passings.