The Twin Paradox Revisited: A Local Case Study

[yogi]

A similar but not identical situation to the cosmological twin problem occurs if there are two satellite clocks A and B in identical orbits but moving in opposite directions - If they cross at point X and sync their clocks, then when they have each completed one orbit and pass again at X, each will believe the other clock is running slow - but with only this information, you cannot tell which if either is running slow. Suppose clock A and B are originally on the same satellite and synchronized togther while traveling at orbital velocity v East to West and when point X is reached, clock B is quickly accelerated to -v so that it now travels at orbital velocity v West to East. Which if either of the two clocks will have logged a greater time when they meet again at X?
As with the cosmological twin paradox, the conditions under which the initial synchronization takes place must be unambiguously defined.

 

[Ich]

yogi - both will read the same time, regardless of synchronization conditions.

 

[JesseM]

If they cross at point X and sync their clocks, then when they have each completed one orbit and pass again at X, each will believe the other clock is running slow

Do you mean each will believe the other clock is running slow in their instantaneous inertial rest frames at the moment they pass? Or do you mean each will think the other is running slow throughout the entire orbit in some sort of non-inertial coordinate system where it sees itself as being at rest throughout the journey? If the second, you're wrong--as long as you're using the laws of physics correctly, they can't get incorrect predictions about what the other clock will read when they pass each other, and we already know (by considering things from the inertial frame of the Earth's center) that they will read the same time when they pass each other. So depending on how you define your non-inertial coordinate system, each would either have to see the other running slow at some moments and fast at others, or each would see the other running at the same rate as itself throughout the trip.

 

[yogi]

What procedure is available for synchronising "on the fly" - none that I am aware of - and that is the problem with the cosmological twin and the orbiting twin - when they simply pass each other initially, they can read each others time at one instant - but that does not sync the clocks ...to sync as per Einstein, both clocks must be at rest in the same frame initially. When so synced, there is no paradox because only one will have accelerated, whether it be to circumnavigate the universe via a geodesic or to orbit a massive body...and once put in motion by an initial acceleration, the path integral for the worldline of the accelerated twin will always include a spatial component which must be subtracted from the temporal component, whereas the non accelerated twin's worldline only includes a temporal element.

 

[JesseM]

What procedure is available for synchronising "on the fly" - none that I am aware of - and that is the problem with the cosmological twin and the orbiting twin - when they simply pass each other initially, they can read each others time at one instant - but that does not sync the clocks ...to sync as per Einstein, both clocks must be at rest in the same frame initially.

There's an ambiguity in the word "synchronization", you're right that it is often only taken to apply to clocks at rest in a single inertial reference frame, but in the cosmological twin paradox it just means that the twins set their clocks to read the same time at the moment they pass each other, it doesn't mean their clocks stay synchronized in any sense.

When so synced, there is no paradox because only one will have accelerated, whether it be to circumnavigate the universe via a geodesic or to orbit a massive body

In the cosmological twin paradox, neither twin accelerates, unless you imagine they both start off on Earth and then one accelerates away...but the paradox works equally well (and is conceptually simpler) if you imagine both have been moving inertially since the beginning of time. Plus, even if one twin accelerates away from the earth, there's no reason that the Earth would necessarily be at rest in the preferred coordinate system determined by the topology of the universe, so it's possible that after accelerating the traveling twin will be moving slower in this coordinate system, meaning that he will have aged more when he meets the earth-twin again, not less. If you think the question of whether the traveling twin accelerates away from the Earth or just passes it while moving inertially is at all relevant here, then I think you're still suffering from the same confusion about the importance of acceleration that you were on the twins paradox thread...if you do think that, then please address my question on that thread about what you meant by the word "moved":

yogi:
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.

JesseM:
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?

 

[yogi]

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 - which one accelerated identifies which one is moved. Einstein used the word "moved" in his 1905 description, and I am using it in the same way - why is it so difficult to understand when I use it, I quote again: "If one of two synchronous clocks at A is moved in a closed curve with constant velocity until it returns to A ...the traveled clock upon its arrival at A will be (1/2)t(v/c)^2 second slow"

 

[JesseM]

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?

 

[yogi]

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?

 

[JesseM]

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?

No, the cosmological twin paradox cannot be localized, because without the universe having an unusual topology there's no way that two clocks can move away from each other inertially and then meet again to compare their readings, without either one having accelerated since they last were at the same location. So can you answer my question about the cosmological twin paradox in my last post? Your other question is off-topic on this thread, so I'll address it on the twins paradox thread.

 

[yogi]

Your post 37: When I say nil wrt to acceleration - I am saying the acceleration per se does not have anything to do with the clock rate difference. It tells us only which clock moved - and Einstein tells us that the clock that moves accumulates less time wrt to the clock which has not been accelerated.

I do not agree that the twin that accelerated away from Earth could age more, when compared to the stay at home twin - the SAHT has remained in the same inertial system - and will always accumulate more time than the clock which flies away - irrespective of the state of the Earth's motion wrt anything else. We do not know if there is a preferred coordinate system defined by the mass of the universe.

And no, i do not agree with your last statement - knowing which clock moved is critical to determining which clock runs slower in SR

 

[yogi]

Jesse - your post 39 - as we already discussed at length - we do not have to have the twins meet to determine which has aged the most - we can set up milestones like Clock B takes off and when he reaches Altair he sends a message as to his (B's) clock reading. And when the tranmission arrives at earth, the SAHT compares this with his A clock - and knowing the light travel time to Altare, can calculate what A clock read when the message was sent.

Let us carry the experiment a bit further - we program B's flight path using an auto pilot that insures B flies at the same velocity wrt to his heading , but also provides a slight centripetal acceleration to the spacecraft so that B flies a large circle. A knows B's velocity and he also knows the curvature of the flight path - so he knows where B is at every point in the trip - B transmits his clock reading every hour - when B returns the B clock will read less than A - but it isn't a sudden change - the difference is accumulating during the entire trip. The continuous but constant gradual radial acceleration that returns B to Earth means that the aging differential accumulates proportionately in accordance with the lapsed time of the flight.

 

[yogi]

So - back to the cosmological twin paradox - and local analogy - consider a first clock J in circular Earth orbit and a second clock K in a highly elliptical orbit - they pass once neither having any idea of the others past history - later on after the J clock has made several orbits and k has made only one, their paths again cross - would you say that there will be a difference in the clock readings - and if so can you say which one has traveled the greater distance. Would you say that the mass about which the two clocks orbit acts as a sort of preferred frame?

 

[JesseM]

Your post 37: When I say nil wrt to acceleration - I am saying the acceleration per se does not have anything to do with the clock rate difference. It tells us only which clock moved - and Einstein tells us that the clock that moves accumulates less time wrt to the clock which has not been accelerated.

No, Einstein says nothing like that, that's your own weird misinterpretation. You quoted him as saying that if a clock is "moved in a closed curve with constant velocity" it will show less time than one that moves inertially, which of course is true, but this cannot be generalized to the statement that if two twins were initially at rest with respect to each other and then one accelerated, the one that accelerated will automatically be the one whose clock shows less time when they meet (in the cosmological twin paradox this is not true, for example).

I do not agree that the twin that accelerated away from Earth could age more, when compared to the stay at home twin - the SAHT has remained in the same inertial system - and will always accumulate more time than the clock which flies away - irrespective of the state of the Earth's motion wrt anything else. We do not know if there is a preferred coordinate system defined by the mass of the universe.

The "preferred coordinate system" has nothing to do with the mass of the universe, it has to do with the fact that in this paradox we are imagining a "compact" universe where if you travel far enough in one direction you will return to your point of origin. It has been established by physicists that this leads to a preferred coordinate system--look at this paper by John Barrow and Janna Levin which says:

Twins traveling at constant relative velocity will each see the other's time dilate leading to the apparent paradox that each twin believes the other ages more slowly. In a finite space, the twins can both be on inertial, periodic orbits so that they have the opportunity to compare their ages when their paths cross. As we show, they will agree on their respective ages and avoid the paradox. The resolution relies on the selection of a preferred frame singled out by the topology of the space.

So, whichever twin is at rest in this preferred frame, he will have aged more than the twin moving in it when they meet again. Thus, if the Earth is moving in this preferred frame with some finite velocity v, and then the Earth twin accelerates briefly so that his velocity is now 0 in this frame, and then both continue to move inertially away from each other, when they meet again it will be the Earth twin who is older, even though he was the one who accelerated.

And no, i do not agree with your last statement - knowing which clock moved is critical to determining which clock runs slower in SR

In a way that's true, because in SR there is no analogue of the cosmological twin paradox, so if two twins start out at the same position then meet again at the same position, one must have turned around (accelerated), and he will be younger. The problem is that you mistakenly generalize this to cases like the one where the two twins started out at completely different positions, and say that the one who "moved" aged less as they come together and meet, even though this statement is perfectly meaningless.

Let me put it this way--do you agree that if we know the complete set of initial conditions at some time t in a particular frame (the position and velocity of both objects at t, the time on their own clocks, etc.) and we want to make some predictions about what will happen later, then since the laws of relativity are completely deterministic, these initial conditions at t are sufficient to make a unique prediction about the future? Do you agree that what happened before t is irrelevant, including the question of which of two clocks accelerated before t?

 

[yogi]

Yes - i will agree to that.


But I will take issue with your statement that it is meaningless to say that: "the one that moved aged less" - Doubt about which aged most would be valid if a third frame is introduced - which was somehow a better reference than clock A or B. I cannot say that it does not exist - but w/o it, I still stand by the notion that Two clocks can be separated by any distance and synced as long as they are not in relative motion at the time the sync operation is performed - then if one is moved to the location of the other, the one that moved will have aged less.

The reason I have for saying this is the same as what we debated on the other thread - it boils down to finding the physical reason ...why the observational time differences of the LT translate to real time differences - the leap Einstein made when going from reciprocal symmetry between inertial frames in relative motion, to the assertion that real age differences occur where one clock moves relative to the frame of another clock with which it had been correlated (synced). Experiments confirm its veracity, - but what is left out is why.

In the cosmological case - there is by definition, no acceleration - so it can be argued that neither clock is any better than the other - for the purpose of making a comparison - but to my way of thinking - this doesn't create a paradox - each can be considered as moving relative to the reference frame of the other - rather than a 3rd reference frame. It can be argued then that if the clock A moves further in the reference frame of B, than B moves in the reference frame of A, then A should log less time than B when they meet. In other words, neither is necessarily at rest in the absolute sense - both are moving - each in the frame of the other, and the one that moves the most relative to the frame of the other, ages less.

In the local case we can know which moves the most in the fame of the other by bringing two clocks together and accelerating one more than the other

Granted, the cosmological case is different than two local clocks in orbit in different directions that periodically pass by each other - but just as with the local twins, I do not see the paradox - the rationale for the asymmetry that leads to the age difference between two local twins should be applicable to the cosmological twins. In the comological case, of course, we don't how far each has moved in the fame of the other, so I suppose it could only be inferred from periodic readings taken each time the two clocks pass each other.

 

[yogi]

Let me ask how you would deal with the following for the local case:

We start with 3 clocks A,B and C at rest and in sync in an inertial frame - A and B push off in opposite directions with A's velocity relative to C being twice that of B's velocity relative to C (C remains at home in the original frame). A and B travel identical circles in opposite directions (passing each either once which we will ignor) and eventually wind up back at C with A having made two loops and B having made one (there is a small acceleration for each that makes them follow the curved path, but that does not effect the outcome as between (A relative to C) and (B relative to C) so long as the tangential velocity of A and B along the circle remains constant. Einstein gives the prescription for the time difference for A relative to C and for B relative to C upon A and B's simultaneous arrival at C, and from that we get the time difference between A and B. So far we have just used Einstein's desync formula twice - once for A relative to C and once for B relative to C.

But when we try to arrive at the same result (the time differential between A and B) w/o considering C, things get complex. For one thing, the relative velocity between A and B is not constant, and we cannot really claim that A or B is inertial. But the problem could be resolved by breaking the paths into short segments and adding up all the differences. So it would seem that the addition of the clock C serves the same function in the local case as the introduction of a rest frame in the cosmological twin scenereo - a convenience, but not absolutely necessary?