View Full Version : Twin Paradox in S^3.
KingOrdo
Mar5-07, 04:26 PM
X and Y reside in the 3-sphere.
Y is accelerated to near the speed of light; say, 0.9c. He does not ever change direction. In a little while, he meets X, who happens to be residing on the great circle upon which Y is traveling. When they meet, they give each other high-fives. At that moment, the two are identical twins (Y was younger prior to his acceleration).
Y continues alone his great circle, unaccelerated, and therefore in an inertial frame. X is similiarly in an inertial frame. In a little while, they meet again. When they high-five for a second time, which is younger?
From the moment they high-fived for the first time, neither has undergone any accelerations; yet my understanding is that the reason the Twin Paradox can be resolved in the canonical case is that one of the twins underwent acceleration (when his spaceship turned around), and that is why there is an asymmetry between the two twins. But in this case there seems to be no difference, and it really is as accurate to say that X's time dilates with respect to Y as it is to say that Y's time dilates with respect to X.
I'm pretty sure this has to do with the question of absolute angular inertial frames, ie, determining whether a particle is spinning or not. Think of it this way: if two buckets of water are at the center of this great circle, one points towards X and one points towards Y, which will have a curved surface of water from the centrifugal force? If the one pointing towards Y has a flat surface, then Y is the one truly at rest. To figure out which it will be, you need to specify a metric, and since this is an artificial model with no masses, it's completely arbitrary. You can make anybody you want be the one at rest. But it's been a little while since I've done general relativity, so maybe someone else should verify (or refute) this.
Meir Achuz
Mar5-07, 05:11 PM
"Y continues alone his great circle, unaccelerated"
That can't happen. In circular motion there is centripetal acceleration.
I remember we have a thread on this sometime ago. This is known as twin paradox in compact spaces. Do a search. It has something to do with topology of the spaces concerned, more specifically, on "winding number".
"Y continues alone his great circle, unaccelerated"
That can't happen. In circular motion there is centripetal acceleration.
No. Travelling in a great circle, which is a geodesic, has no acceleration tangential to the spaces. That is what counts. For a two-dimensional surface, for example S^2, a geodesic is a curve in the surface whose acceleration is perpendicular to the surface. But in dealing with abstract manifold, there is no "outside" of the surface for it to be embedded in. So any component outside of tangent spaces do not count. So travelling along geodesic entails unaccelerated frame.
George Jones
Mar5-07, 05:58 PM
Elapsed proper time is found by integrating the metric (which, in general relativity, is a function of spacetime "position") along a worldline.Consequently, if X and Y are coincident at events p and q, there is no reason to expect the same elapsed proper times if they followed different worldlines between p and q, even if both observers have zero 4-acceleration.
This is the Cosmological Twin Paradox, which has been discussed here several times:
http://www.google.com/search?q=cosmological+twin+paradox+site%3Awww.phys icsforums.com
All twin paradox(s) disappear (whether cosmological or otherwise) by a straight forward application of the "invariance of the interval." Acceleration has nothing to do with any differential aging problem in SR. In any situation where observers in one frame set up an experiment involving two synchronized clocks in their frame to measure the time logged by a single clock in another frame as it journeys from one clock synchronized clock to the other, the single clock will always log less time than the two synchronized clocks. How the frames got into relative motion and how the single clock returns to the first clock (if it does) is immaterial - neither start-up acceleration nor turn around acceleration plays any part in determining the time difference. Einstein had in right in 1905 - he screwed it up later in 1918 by trying to make his principle theory into a constructive theory
KingOrdo
Mar5-07, 10:01 PM
All twin paradox(s) disappear (whether cosmological or otherwise) by a straight forward application of the "invariance of the interval." Acceleration has nothing to do with any differential aging problem in SR. In any situation where observers in one frame set up an experiment involving two synchronized clocks in their frame to measure the time logged by a single clock in another frame as it journeys from one clock synchronized clock to the other, the single clock will always log less time than the two synchronized clocks. How the frames got into relative motion and how the single clock returns to the first clock (if it does) is immaterial - neither start-up acceleration nor turn around acceleration plays any part in determining the time difference. Einstein had in right in 1905 - he screwed it up later in 1918 by trying to make his principle theory into a constructive theory
Yes, roger that; but I don't think that answers the question.
When X and Y high-five the first time, they are identical entities (by the definition of the thought experiment)--equivalently, you could consider them one body with a certain symmetry. When they high-five the second time, time dilation tells us that that symmetry is broken; i.e. one of the bodies will be 'older' than the other. But how can this be without postulating a preferred frame of reference, which is disallowed by the equivalence principle?
The paradox arises because it is mistakenly presumed that SR requires the moving clock to be running slow - it logs less time because it is measured to travel a spatial distance between the two synchronized clocks - it still runs at one second per second in its own frame - there are fewer seconds accumulated because part of the trip involves spatial motion and part of it involves temporal distance ct - SR does not say that the moving clock runs slower than either of the two clocks doing the measuring in the frame that was arbitrarily selected to be at rest - turn the situation around - put the two clocks in the other frame and measure how much time it takes for the remaining clock to pass between the two clocks - it is always the single clock that travels the spatial distance between two fixed clocks that logs less time - its a misnomer to say the moving clock runs slow - it runs the same - but it accumlates less time during the particular experiment.
KingOrdo
Mar5-07, 11:21 PM
The paradox arises because it is mistakenly presumed that SR requires the moving clock to be running slow - it logs less time because it is measured to travel a spatial distance between the two synchronized clocks - it still runs at one second per second in its own frame - there are fewer seconds accumulated because part of the trip involves spatial motion and part of it involves temporal distance ct - SR does not say that the moving clock runs slower than either of the two clocks doing the measuring in the frame that was arbitrarily selected to be at rest - turn the situation around - put the two clocks in the other frame and measure how much time it takes for the remaining clock to pass between the two clocks - it is always the single clock that travels the spatial distance between two fixed clocks that logs less time - its a misnomer to say the moving clock runs slow - it runs the same - but it accumlates less time.
Yes, I understand that. So then would you claim that both X and Y are identical at the time of the second high-five?
And if not, if one has 'aged' with respect to the other, which one is aged, and why?
A local demonstration of the great circle scenero is played out with each GPS satellite - ignoring the height correction - a GPS satellite in orbit experiences no acceleration - each time it passes over a particular location on the earth, it will appear to have lost some time (we can erect a tower to be at the same height of the orbiting satellite to avoid any height difference.) and we will find that the single clock in the tower serves the same function as two clocks in a one way journey - part of the orbiting clock's motion is temporal and part is spatial - the circumference of the orbit and the time lapse of the on board GPS clock have to combine in a pythagorean way to equal a single temporal value ct on the clock at the top of the tower.
Yes, I understand that. So then would you claim that both X and Y are identical at the time of the second high-five?
And if not, if one has 'aged' with respect to the other, which one is aged, and why?
If you know that two clocks are initially at rest, and one is then put in motion on a great circle, the one that has been put in motion will have aged the least as per Einstein 1905 part IV (analogous to the GPS satellite)
KingOrdo
Mar5-07, 11:53 PM
If you know that two clocks are initially at rest, and one is then put in motion on a great circle, the one that has been put in motion will have aged the least as per Einstein 1905 part IV (analogous to the GPS satellite)
The difference is that the GPS satellite is not in an inertial frame; it's orbiting, after all.
And motion is relative; in the thought experiment I provided neither of the two twins were put in motion (Y was not X's twin prior to Y's acceleration); they both were moving at constant velocity.
So I must ask again: "So then would you claim that both X and Y are identical at the time of the second high-five?
And if not, if one has 'aged' with respect to the other, which one is aged, and why?"
As you probably know, the traveler that circumnavigates the universe doesn't live any longer - if his round trip takes one week then he will experience 7 nights of sleep, 21 meals etc - he just covers a lot of territory in a short amount of time on his clock.
KingOrdo
Mar5-07, 11:56 PM
As you probably know, the traveler that circumnavigates the universe doesn't live any longer - if his round trip takes one week then he will experience 7 nights of sleep, 21 meals etc - he just covers a lot of territory in a short amount of time on his clock.
Indeed. "So then would you claim that both X and Y are identical at the time of the second high-five?"
Your post 13 - an orbiting satellite is a perfect example of an inertial frame - it feels no acceleration.
In order to properly answer the other questions, one would need to know what you mean by who is younger - In the usual twin paradox, it is clear which twin moved and which did not - and how you are going to measure age - usually it is done with two separated clocks or one clock that remains at rest to which the traveler returns - so in your story it could be that x is also traveling on a geodesic since there is no absolute rest frame in SR.
In other words, in your first post there is an ambiguity as to the initial conditions and as to how one measures which is older. Einstein created confusion when he said moving clocks run slow - you have to look at the experiment and see which observer is measuring age and how it is done. If you rely on Einstein 1905 part IV you could say only that if one of two initially synchronized clocks were accelerated to a high velocity and returned because of space curvature, the non accelerated clock would age most.
Have to Go
Yogi
KingOrdo
Mar6-07, 12:27 AM
Your post 13 - an orbiting satellite is a perfect example of an inertial frame - it feels no acceleration.
In order to properly answer the other questions, one would need to know what you mean by who is younger - In the usual twin paradox, it is clear which twin moved and which did not - and how you are going to measure age - usually it is done with two separated clocks or one clock that remains at rest to which the traveler returns - so in your story it could be that x is also traveling on a geodesic since there is no absolute rest frame in SR.
My understanding is that an inertial frame is a non-accelerating coordinate system. The satellite certainly is accelerating; that people living aboard the satellite aren't aware of this is of no concern.
And, Yogi: you're bringing up exactly my point. First, it's not that in the "usual twin paradox, it is clear which twin moved and which did not"--it's that it's clear which twin *accelerated* and which did not. In my example, neither accelerated; they're both candidates for time dilation modulo the other. And yes, they're both on the same geodesic.
K.J.Healey
Mar6-07, 01:19 AM
But there is not even a "first time" they're twins.
If they're both holding a digital clock to their chest with big bold letters:
When X see's Y right next to him its possible that their clocks both read 0.
But when Y see's X right next to him, Ys clock isnt at 0 yet, and X's is perhaps already past 0.
There is no simultaneity. (my numbers are not right, maybe even in the wrong order) But the point remains. Who is viewing them as twins (that their clocks are the same) X or Y? Cause by the second time they see eachother, they will both see eachother as OLDER than the other see's themselves, right? It has completely to do with relativity and WHEN you consider them to be "passing" eachother.
KingOrdo
Mar6-07, 02:17 AM
But there is not even a "first time" they're twins.
If they're both holding a digital clock to their chest with big bold letters:
When X see's Y right next to him its possible that their clocks both read 0.
But when Y see's X right next to him, Ys clock isnt at 0 yet, and X's is perhaps already past 0.
There is no simultaneity. (my numbers are not right, maybe even in the wrong order) But the point remains. Who is viewing them as twins (that their clocks are the same) X or Y? Cause by the second time they see eachother, they will both see eachother as OLDER than the other see's themselves, right? It has completely to do with relativity and WHEN you consider them to be "passing" eachother.
No, they are twins: that's the whole point of the thought experiment. When X and Y meet the first time, they are physically identical. That's the stipulation; indeed, as I mentioned, one could consider their first high-five as a single body with a symmetry.
And I'm quite aware that there's no simultaneity--indeed, that's what I'm arguing. But in classic relativity, we distinguish the 'younger' twin because of the accelerations he's undergone (the speeding up of the spaceship, its turning around, changes of direction, etc.) In this example, since the space is S^3 and BOTH X and Y are ALWAYS in inertial frames, what is the basis for making that judgment?
George Jones
Mar6-07, 05:18 AM
And yes, they're both on the same geodesic.
They most certainly do not move along the same geodesic. The question is: Do they move along equivalent geodesics? Without doing a calculation, I can't say for sure, but I suspect the answer is no. In any case, there is no paradox.
If they move along different geodesics, then there can be an accumulated time difference because the two clocks experience the gravitational field in different ways.
The equivalence principal notwithstanding, the situation you described has to be analysed using general relativity; the concepts and intuition fron special relativity do not suffice.
It is an interesting calculation (not paradox), though.
KingOrdo
Mar6-07, 11:41 AM
They most certainly do not move along the same geodesic. The question is: Do they move along equivalent geodesics? Without doing a calculation, I can't say for sure, but I suspect the answer is no. In any case, there is no paradox.
Why can't they be on the same geodesic? Just imagine them in S^2: one is Cairo, one is Mexico City (assume both Cairo and Mexico City are on the Equator).
If they move along different geodesics, then there can be an accumulated time difference because the two clocks experience the gravitational field in different ways.
Again, that's the whole point: there is no gravitational field. From the moment of the first high-five (when the twins are identical) to the moment of the second high-five, both are in inertial frames of reference.
The equivalence principal notwithstanding, the situation you described has to be analysed using general relativity; the concepts and intuition fron special relativity do not suffice.
It is an interesting calculation (not paradox), though.
Well, no one seems to be able to say *why* it's not a paradox. Indeed, the calculation has to be done in GR because the spacetime's not flat. And you're begging the question: yes, elapsed proper time is different for X and Y. But *why* is that the case? It can be resolved in the actual Universe (apparently) because of the acceleration asymmetries undergone by the twin in the spaceship. But that would not be the case in either a matter-free compact space or, say, S^3 with a totally homogeneous distribution of matter. There would be NO grounds for distinguishing X from Y unless you posit--in direct opposition to the equivalence principle--that there exists a preferred frame of reference.
The paradox is in the apparent symmetry between the two twins. I pointed out in the second post how this symmetry is broken.
KingOrdo
Mar6-07, 12:33 PM
The paradox is in the apparent symmetry between the two twins. I pointed out in the second post how this symmetry is broken.
There is no "center of [a] great circle" in S^3. We're not talking about a higher-order embedding. Think of it this way: is there a center to the surface of the Earth (NOT the ball of the Earth)? There is not.
No unique center, but there are centers. The centers of the equator are the north and south poles. It is just a point that is the same distance to each point on the great circle. There is a little ambiguity about measuring distance, but you can always take the shortest path between two points.
KingOrdo
Mar6-07, 02:05 PM
No unique center, but there are centers. The centers of the equator are the north and south poles. It is just a point that is the same distance to each point on the great circle. There is a little ambiguity about measuring distance, but you can always take the shortest path between two points.
Indeed: and that's exactly the problem. It's just a pure matter of stipulation which person, X or Y, is at rest. So the Twin paradox is not resolved, unlike the R^3 case in which the symmetry is broken when one person's reference frame becomes non-inertial.
No, that's not what I meant. Think of it on a 2D sphere. X is on the equator, and Y is going around the equator at a constant velocity. But to Y, it is X that is moving. How do we determine which is really moving? (there must be an asymmetry here, unlike the case in flat space, because they will meet again and we'll be able to compare clocks, the whole point of this thread) Place an object at the north pole. Then this object is at rest with respect to X and Y, but they will disagree as to how it is spinning. We can set it to spin so that Y observes it to be at rest, or that X does. However, only in (at most) one of these cases will it have no centrifugal effects, and so be truly not spinning. This determines which of X and Y is really at rest, and so which ages more.
There may be some subtelty with observing this object across curved space, which is why I'm still not completely convinced this is the correct explanation.
KingOrdo
Mar6-07, 03:48 PM
No, that's not what I meant. Think of it on a 2D sphere. X is on the equator, and Y is going around the equator at a constant velocity. But to Y, it is X that is moving. How do we determine which is really moving? (there must be an asymmetry here, unlike the case in flat space, because they will meet again and we'll be able to compare clocks, the whole point of this thread) Place an object at the north pole. Then this object is at rest with respect to X and Y, but they will disagree as to how it is spinning. We can set it to spin so that Y observes it to be at rest, or that X does. However, only in (at most) one of these cases will it have no centrifugal effects, and so be truly not spinning. This determines which of X and Y is really at rest, and so which ages more.
I just don't see how this could possibly make a difference . . . use primitive test particles instead of extended entities (viz. people). Or, use a geometry in which the distance to circumnavigate the great circle is sufficiently less than the distance from the circle to the 'center' such that no signal can go from the center to the circle, even at c, in the time our sub-c test particle can circumnavigate the circle (imagine a squished globe, for the S^2 case).
And anyway, it doesn't resolve the paradox: the whole point is that it can't be just a stipulation of coordinates that makes one entity younger than the other. In the canonical example, one twin returns to Earth and occupies the same frame of reference as his 'twin'; yet he *still* is younger. It's not that he *appears* younger to some observers: he *is* younger in that frame of reference; it's coordinate-invariant. And your approach is tantamount to just stipulating a coordinate system, which, by the principles of relativity, can't work.
This has been discussed here before, as several people have mentioned. Unfortunately, I don't recall all the details of the previous discussions.
One of several threads on this was http://www.physicsforums.com/archive/index.php/t-51197.html
The literature seems to have some conflicting claims.
http://arxiv.org/abs/physics/0006039 Eur.J.Phys. 23 (2002) 277
for instance, claims that the winding number (homotopy class) can distinguish between the twins.
http://arxiv.org/abs/astro-ph/0606559 published in
http://adsabs.harvard.edu/abs/2006astro.ph..6559R claims otherwise
But there is broad agreement on the general results (one twin will be older) though there is some debate on whether this can be explained by homotopy alone or whether one needs the metric.
Note that one does not need to have a compact universe or a multiply connected topology to have two twins, both following geodesics, have different ages.
A simple case would be if one twin left earth at a high starting velocity, took a loop around a black hole, then came back and passed the Earth again at a high velocity.
That twin would have aged less than the twin who stayed on Earth.
Geodesics are only a local maximum of proper time. To be a global maximum of proper time, one must not only follow a geodesic, but pick the "right one".
To use an analogy I've used before, it's similar to the way that the shortest distance between two points on a curved surface is always a straight line, but a given straight line on a curved surface is not always the shortest distance between two points.
Consider
x ^^^^^ y
where x and y are separated by a tall mountain.
The path directly over the top of the mountain is a straight line. However, it is not the shortest distance between points x and y. The shortest distance is a different straight line - one that goes around the mountain, rather than over it.
And anyway, it doesn't resolve the paradox: the whole point is that it can't be just a stipulation of coordinates that makes one entity younger than the other. In the canonical example, one twin returns to Earth and occupies the same frame of reference as his 'twin'; yet he *still* is younger. It's not that he *appears* younger to some observers: he *is* younger in that frame of reference; it's coordinate-invariant. And your approach is tantamount to just stipulating a coordinate system, which, by the principles of relativity, can't work.
My point is that while there are no preferred velocities, there are preferred states of rotational motion. That is, no person can claim to be absolutely at rest, but they can claim to not be spinning. Determining this in an abstract situatuation like you've described does entirely come down to a (arbitrary) specification of coordinates. It's difficult to make precise, but I believe this is how you distinguish two people who undergo trips around closed loops.
KingOrdo
Mar6-07, 05:34 PM
Ahh . . . very interesting: so this is by no means an open-and-shut case; there's no consensus even in the literature.
There seems to be no dispute that one twin *will* be older in this experiment. However, no one is quite sure *why* that's the case. We know why--or, at least, we thought we knew why--in the canonical experiment (viz. non-inertial reference frames), but that can't be the explanation here. 'pervect''s black hole example could even be explained that way, since an orbit is, by definition, non-inertial (though it is freely falling).
But that can't be the case in a matter-free S^3 universe, since there's no gravitational force at all (make the test particles vanishingly small). Or, in a universe with perfectly homogeneous distribution of matter (which affects both test particles identically).
So it appears we're left with one of two options: either
(1) we posit that physical laws are not invariant modulo the geometry in which they are instantiated, or
(2) our currented accounting of the twins paradox is wrong.
Neither is attractive, but I see no other options. (2) also seems to imply that foundational relativistic principles (e.g. the equivalence principle) are wrong.
Thoughts?
JustinLevy
Mar6-07, 06:36 PM
'pervect''s black hole example could even be explained that way, since an orbit is, by definition, non-inertial (though it is freely falling).
I think you are still missing something here. Free fall, or a gravitational orbit, etc IS inertial motion. I think you may be confusing "coordinate" acceleration with physical acceleration.
KingOrdo
Mar6-07, 06:41 PM
I think you are still missing something here. Free fall, or a gravitational orbit, etc IS inertial motion. I think you may be confusing "coordinate" acceleration with physical acceleration.
No. Imagine you're on the Shuttle, orbiting the Earth. It may *appear* inertial to you, and indeed if you don't need hyper-precise measurements you can *assume* it's inertial, but it's not: it's an accelerated frame, which you can tell from the tidal forces due to the presence of the Earth. It's especially obvious if you substitute "neutron star" for "Earth".
Now, we're talking about curved spacetime, in which GR--not SR--holds, and global inertial frames are nonexistent. However, locally both X and Y are in inertial frames of reference. And hence, the paradox.
Ahh . . . very interesting: so this is by no means an open-and-shut case; there's no consensus even in the literature.
While there appears to be some disagreement in the literature, it's about rather fine details.
There is a general agreement about the broad details, which is that the two twins won't be the same age.
The disagreement as I read it is how to explain the age difference, i.e. to point to a particular mechanism. The first papers say that it can be addressed in terms of winding number (which is a topological property that doesn't even need a metric). The second papers say that this is not correct, that one needs more than the winding number to properly explain the age difference, that one needs the metric information.
Everyone agrees that there should be an age difference AFAIK.
As I said, I haven't really stuidied this particular matter very closely.
KingOrdo
Mar6-07, 08:05 PM
While there appears to be some disagreement in the literature, it's about rather fine details.
There is a general agreement about the broad details, which is that the two twins won't be the same age.
Right; and this was never in question in the first place--like I said, "There seems to be no dispute that one twin *will* be older in this experiment. However, no one is quite sure *why* that's the case."
I am curious to hear what people think about a potential resolution; the options I cited above seem to me to be the only way to patch a pretty glaring lacuna in our physics. I've always thought intuitively that the obstacles to quantum gravity were on the QM side-of-the-house . . . but perhaps relativity theory is the problem?
No. Imagine you're on the Shuttle, orbiting the Earth. It may *appear* inertial to you, and indeed if you don't need hyper-precise measurements you can *assume* it's inertial, but it's not: it's an accelerated frame, which you can tell from the tidal forces due to the presence of the Earth. It's especially obvious if you substitute "neutron star" for "Earth".
.
The literature does not agree with your definition. See Road to Reality by Penrose at p 394 ..."our insects falling trajectory and our astronaunts motion about the earth must both count as inertial motions"
Also see Spacetime Physics for a similar definition of a free float frame.
The fact that there are minor tidal effects should not obviate the thought experiment - The problem with the cosmological twins and the orbiting clocks in GPS is the same - the determination as to which twin ages the most will depend upon each orbiting twin setting up stations that allow each to determine how much distance the other twin has traveled between his two stations in a given amount of time as read by his own clocks. Previously this issue arose in a thread involving one clock in orbit and the other oscillating back and forth through the center of the earth so the two clocks meet periodically and compare times. Both are in inertial frames and one will have accumulated more time than the other on each successive hi 5.
KingOrdo
Mar7-07, 12:06 AM
The literature does not agree with your definition. See Road to Reality by Penrose at p 394 ..."our insects falling trajectory and our astronaunts motion about the earth must both count as inertial motions"
Well, I've always used Tipler. But anyway, here is how Wikipedia defines things: "bodies are subject to so-called fictitious forces in non-inertial reference frames". And as we know, gravity is a fictitious force: as MTW says, 'space tells matter how to move'.
The fact that there are minor tidal effects should not obviate the thought experiment - The problem with the cosmological twins and the orbiting clocks in GPS is the same - the determination as to which twin ages the most will depend upon each orbiting twin setting up stations that allow each to determine how much distance the other twin has traveled between his two stations in a given amount of time as read by his own clocks. Previously this issue arose in a thread involving one clock in orbit and the other oscillating back and forth through the center of the earth so the two clocks meet periodically and compare times. Both are in inertial frames and one will have accumulated more time than the other on each successive hi 5.
Yes, we have all agreed that "Both are in inertial frames." That's the whole problem. The choice of which one gets older then is purely arbitrary, because there is no asymmetry: and relativity tells us that that cannot be right.
Does anyone have any ideas?
The difference is that the GPS satellite is not in an inertial frame; it's orbiting, after all.
This is a geodesic: it is an inertial frame. Only if there are rockets firing on the satellite does it go to a non-inertial frame
KingOrdo
Mar7-07, 01:34 AM
This is a geodesic: it is an inertial frame. Only if there are rockets firing on the satellite does it go to a non-inertial frame
No. A satellite is not an inertial frame. It is a very close approximation to one, but is technically only a freely falling frame. For a frame to qualify as inertial, a person in the frame cannot feel any forces; however, there is a small tidal force due to Earth's gravity which could be detected with a gradiometer. Cf., e.g. http://www.eftaylor.com/pub/chapter2.pdf. (There may also be a small pressure due to solar radiation.)
Again: any ideas for resolving the paradox?
To take the issue of satellites and free fall to its logical extreme, there is probably no such thing as a perfect inertial frame - this doesn't preclude making real experiments on earth or in space where there is no appreciable acceleration - a slight bit of acceleration does not destroy the experiment - it only introduces a small correction to the clocks - the use of free fall frames and orbiting satellites are well accepted by generally recogonized authorities - Tipler has some very strange ideas that are more metaphysical than scientific -
In the case of orbits or boomeranging through the earth (as Wheeler calls it), the answer to the question as to relative aging can probably best be approached by viewing the different trajectories from a thrid frame - such as the non rotating earth centered system - the time difference between two different orbits for example can be calculated relative to the NRECRF and the results compared.
In the cosmological case - there is no obvious 3rd frame where one could place synchronized clocks that are on the intersect of the paths of both x and y. The worldline of x needs to be better defined
From the moment they high-fived for the first time, neither has undergone any accelerations; yet my understanding is that the reason the Twin Paradox can be resolved in the canonical case is that one of the twins underwent acceleration (when his spaceship turned around), and that is why there is an asymmetry between the two twins. But in this case there seems to be no difference, and it really is as accurate to say that X's time dilates with respect to Y as it is to say that Y's time dilates with respect to X.
Which twin ages more always depends on the metric.
In the "canonical" case, it's specified that we're studying a Minkowski space-time, and we know a lot of shortcuts for analyzing things in a Minkowski space-time. (And the classical twin paradox appears when someone tries to use those shortcuts improperly)
You haven't specified any metric at all, so we cannot do the problem. I could assume that you intended a uniform metric on S^3 space, or even S^3 x R, but that still doesn't tell us everything we need to know.
Incidentally, you don't have to puzzle this out in your head. The 1+1-dimensional spacetime S^1 x R is readily accessible in the form of a cylinder. You can try drawing your example on, for example, a paper towel tube.
George Jones
Mar7-07, 07:25 AM
Hurkyl has made two points that I intended to post.
Which twin ages more always depends on the metric.
It is impossible to talk about stuff like elapsed proper time, timelike, spacelike, lightlike, etc. without a metric.
You haven't specified any metric at all, so we cannot do the problem. I could assume that you intended a uniform metric on S^3 space, or even S^3 x R, but that still doesn't tell us everything we need to know.
To to do this problem needed are:
a spacetime manifold (not just a space manifold like S^3), like, for example the S^3 x R of a closed FRW universe;
a metric on the spacetime manifold (not just on space) so that we can talk about time;
a pair of events where the observers are coincident;
a pair of distinct geodesic worldlines between these events.
Again, the geodesics are geodesic in spacetime, *not* geodesics in S^3.
Only then can someone point out where the asymmetry, if any, lies. In a highly symmetrical spacetime like you (KingOrdo) want to use, it may be possible to set up a situation in where which the two elpased proper times are the same.
Where there is difference in elapsed times, there is always an asymmetry in worldlines. In standard special relativity, it just happens that this asymmetry can be characterized by 4-acceleration. In the Barrow-Levin compact spacetime, the asymmetry is characterized by homotopy (winding number). Etc. Give a complete setup, and an asymmtry likely will be apparent.
If you can't give us a complete setup, stop saying that there is a problem, and that noone is answering your questions.
Also, in your spacetime, the curvature tensor is non-zero, so there are tidal forces on observers that move on geodesics, just as is the case for an observer freely falling in orbit about a planet or star. You can't use tidal forces to say that your example is different than an orbital example
Incidentally, you don't have to puzzle this out in your head. The 1+1-dimensional spacetime S^1 x R is readily accessible in the form of a cylinder. You can try drawing your example on, for example, a paper towel tube.
I not sure that this spacetime is the best example, as it is not simply connected. I think King Ordo wants to use a simply connected spacetime.
KingOrdo
Mar7-07, 10:22 AM
I'm afraid you guys are missing the point; someone earlier posted some very nice links to the literature that are interesting and perspicuous, and I recommend checking them out.
I'm not interested in *which* twin is older. Indeed, the reason that there's a problem in the first place is because of the stipulation that time dilation holds in situations like the one I described. Obviously we have no way to test that this is the case; it very well may be that time dilation is a contingent phenomenon. However, that's not the way we normally think in physics, and we have strong a priori grounds for believing that physical law is invariant with regard to the topology it's instantiated in.
The argument is this:
(1) Time dilation occurs in simply connected spaces (i.e. the Universe). We know this from experiment. The *explanation* is that one twin (the one that leaves Earth) undergoes accelerations. Fine: the twin asymmetry is due to the acceleration.
(2) There is no reason to think that the laws of physics would be different in a different, multiply connected universe: say, a compactified Kaluza-Klein spacetime R^3xS^1.
(3) In a compact spacetime the experiment *could* be set up such that X and Y undergo no accelerations. And so *that's* the stipulation of the experiment.
(4) But then the explanation that was given for the M^4 case doesn't work in the compact spacetime case.
(5) So either (1) or (2) is wrong. We have strong a priori grounds for (2), so the problem seems to lie in (1): acceleration is *not* the important criterion. But then what is it? (Plus, why do physical explanations change modulo the topology? Can that be right?)
The "winding number" argument has already been debunked in the literature (cf. earlier links), as it should intuitively: from the point-of-view of either observer, the other appears younger.
So what's the approach? Instead of looking for *real*, *testable*, *falsifiable* theory improvements, most people just want to *stipulate* that they'll find some asymmetry in the spacetime under consideration. That is, they just define away the problem. Well, I'm for one not willing to take this all on *faith*. And the literature makes apparent that this is a real problem upon which there is no professional consensus. We should be excited about this, because when lacunae are discovered in our theories, they can often lead to real discoveries. We should not try to sweep it under the rug.
So, again: any ideas?
For a frame to qualify as inertial, a person in the frame cannot feel any forces; however, there is a small tidal force due to Earth's gravity
Depends how big the satellite is...
George Jones
Mar7-07, 11:47 AM
(3) In a compact spacetime the experiment *could* be set up such that X and Y undergo no accelerations. And so *that's* the stipulation of the experiment.
You don't really mean this. For example, R^3 x S^1 is not compact. Any compact spacetime contains closed timelike curves. Here's a proof (http://www.physicsforums.com/showpost.php?p=1254758&postcount=82).
This does, however, provide a nice segue to ...
This thread seems to be stuck in a closed timelike curve, out which, I'm going to try my best to break.
KingOrdo
Mar7-07, 12:06 PM
You don't really mean this. For example, R^3 x S^1 is not compact. Any compact spacetime contains closed timelike curves. Here's a proof (http://www.physicsforums.com/showpost.php?p=1254758&postcount=82).
I'm giving you a multiply connected spacetime, per your request. Like I said, it's a *compactified* Kaluza-Klein manifold (equivalent to M^4xS^d, where d is 6, or 7, or 22, or whatever).
But we're getting rather far afield . . . again, I must ask: does anyone have any ideas?
There seems to be no dispute that one twin *will* be older in this experiment.
Have we really agreed on this?
The proper time measured between two events in spacetime is dependent on two things:
(i) the metric; and
(ii) the path taken between these two events.
In the original description of your problem, you have only specified (ii), but not (i).
Have I understood the problem correctly?
KingOrdo
Mar7-07, 07:47 PM
Have we really agreed on this?
The proper time measured between two events in spacetime is dependent on two things:
(i) the metric; and
(ii) the path taken between these two events.
In the original description of your problem, you have only specified (ii), but not (i).
Have I understood the problem correctly?
I have provided several topologies that should generate the paradox.
Now, if you want to claim that X and Y's proper time are in fact identical in these cases, that's a very interesting claim that I'd like to hear more about. It seems to imply that relativity is false--or, at best, incomplete. But that there's nothing wrong with that: it might just have to be expanded the way Newton's theory had to be expanded in the high velocity limit.
But the one 'way out' proposed in the literature--the 'winding number'--was demonstrated to fail for the very reason the accounting of the twin paradox does: X sees Y's winding number as one thing, and vice versa (cf. the earlier links); and that cannot be right. So something very subtle and interesting seems to be going on. What do we do? Is it time to discard the equivalence principle? I'm very interested to hear any ideas people have.
Well, I'm for one not willing to take this all on *faith*.
Neither is anyone else. The whole reason to use differential geometry is that you are guaranteed to get exactly the same answer, no matter what coordinates you use to do a computation.
(4) But then the explanation that was given for the M^4 case doesn't work in the compact spacetime case.
The explanation wasn't given for the M^4 case. The explanation was given for the "someone forgot about acceleration" case.
In the "someone forgot about the metric" case, the appropriate explanation is to remind them about the metric. :tongue:
FYI:
(1) Time dilation occurs in simply connected spaces (i.e. the Universe).
Time dilation is a coordiante-dependent phenomenon: it has no physical reality.
(Of course, it can be used to compute physical quantities -- e.g. the fact that an inertial worldline is the longest time between two timelike-separated points in Minkowski space)
And just to see if you're aware of it... if the spacebound twin travels to Mars and hangs out there long enough before travelling back to Earth, he will find out that he is older than the Earthbound twin. (Despite the fact the Earthbound twin travelled inertially the entire time)
The "winding number" argument has already been debunked in the literature (cf. earlier links), as it should intuitively: from the point-of-view of either observer, the other appears younger.
But why should your perception of the other's age be the same if you are looking forward around the universe or backward around the universe?
But the one 'way out' proposed in the literature--the 'winding number'--was demonstrated to fail for the very reason the accounting of the twin paradox does: X sees Y's winding number as one thing
(Assuming you project down onto space -- it doesn't make sense to speak of the winding number of a path that is not a closed curve)
Of course -- but the effects of a nonzero winding number are different in X's "coordinates" than they are in Y's "coordinates".
Consider flat RxS^1 again. Once around space in X-"coordinates" will be equivalent to going forward in time by some amount d_X. Similarly for Y. There's no reason to think d_X = d_Y: so when X and Y make their adjustments for the other's winding number, they are different.
(Incidentally, X and Y will also disagree on how long it is around the universe according to their respective coordinates)
I have provided several topologies that should generate the paradox.
But what about the geometry? You cannot even talk about the proper time an observer experiences until you have a geometry.
KingOrdo
Mar7-07, 10:09 PM
The explanation wasn't given for the M^4 case. The explanation was given for the "someone forgot about acceleration" case.
Yes, it was given for the M^4 case. The problem is generated in the first place because the resolution in M^4 does not work for other topologies. And that seems, a priori, wrong to most people.
But why should your perception of the other's age be the same if you are looking forward around the universe or backward around the universe?
That's exactly the point: it shouldn't. But in the cases at hand, you're postulating a physical change because of an arbitrary choice of coordinates, which is precisely what is disallowed by GR.
But what about the geometry? You cannot even talk about the proper time an observer experiences until you have a geometry.
I have mentioned on several occasions to assume for simplicity that this is a matter-free universe. If you're having trouble grasping exactly why the paradox reemerges in more complex topologies, I recommend checking out the several papers on the arXiv that were cited earlier.
Again, for what must be the tenth time: does anyone have any ideas? A good response would say, e.g.: 'X is what I think is going on in the complex topologies, and Y is why there is an asymmetry between the simple and complex cases.'
K.J.Healey
Mar7-07, 11:36 PM
How about commenting on the parts of Hurkyl's post that seem to answer the question such as:
Consider flat RxS^1 again. Once around space in X-"coordinates" will be equivalent to going forward in time by some amount d_X. Similarly for Y. There's no reason to think d_X = d_Y: so when X and Y make their adjustments for the other's winding number, they are different.
(Incidentally, X and Y will also disagree on how long it is around the universe according to their respective coordinates)
K.J.Healey
Mar7-07, 11:37 PM
When they both get back to a position where they see the other as next to themselves, they will both see the other as older, correct?
MeJennifer
Mar7-07, 11:46 PM
Time dilation is a coordiante-dependent phenomenon: it has no physical reality.
Perhaps you mean something else but the proper time interval differential between two objects is not coordinate dependent in the theory of relativity.
KingOrdo
Mar8-07, 12:10 AM
When they both get back to a position where they see the other as next to themselves, they will both see the other as older, correct?
No. That's the point. You're citing the paradox; one says, 'Hey, X moved relative to Y, yeah; but Y moved relative to X! So shouldn't they both see each other as older when Y gets back to Earth?' And the answer is, 'Of course not: that's a logical contradiction.' But then how do we resolve it? In the simple case, we resolve it by citing the fact that Y left an inertial frame of reference (when his rocket thrusted, for example, and when he turned around, etc.), while X never did (we ignore the Earth's gravity). Hence, there is a real asymmetry and the paradox disappears.
But in these complex cases, the example can be given (I gave it) in which Y undergoes NO accelerations; and then it really is just as accurate to say that Y's time dilates with respect to X as X's time dilates with respect to Y. And when they get back together and high-five, both should see each as older. But that's an obvious absurdity, and precisely what Einstein was trying to avoid in the first place. We can dispose of the paradox in simple cases, but not, it seems, in the complex ones. And the one proposed resolution in the literature, the 'winding number', was debunked (both papers available in the arXiv, cited earlier). And interesting, it would debunked on isomorphic grounds! X sees Y's winding number as something (say, z), and Y sees X's as z . . . like Einstein told us, it's all relative!
So there appears to be a real problem. One solution is to assert, a priori, that the physics of our Universe would be different if the geometry of our Universe were different. But that is a very high price to pay, and in my view we're better off taking a look at how our contingent theories might be false. . . .
Any ideas?
MeJennifer
Mar8-07, 12:40 AM
No. That's the point. You're citing the paradox; one says, 'Hey, X moved relative to Y, yeah; but Y moved relative to X! So shouldn't they both see each other as older when Y gets back to Earth?' And the answer is, 'Of course not: that's a logical contradiction.' But then how do we resolve it? In the simple case, we resolve it by citing the fact that Y left an inertial frame of reference (when his rocket thrusted, for example, and when he turned around, etc.), while X never did (we ignore the Earth's gravity). Hence, there is a real asymmetry and the paradox disappears.
But in these complex cases, the example can be given (I gave it) in which Y undergoes NO accelerations; and then it really is just as accurate to say that Y's time dilates with respect to X as X's time dilates with respect to Y. And when they get back together and high-five, both should see each as older. But that's an obvious absurdity, and precisely what Einstein was trying to avoid in the first place. We can dispose of the paradox in simple cases, but not, it seems, in the complex ones. And the one proposed resolution in the literature, the 'winding number', was debunked (both papers available in the arXiv, cited earlier). And interesting, it would debunked on isomorphic grounds! X sees Y's winding number as something (say, z), and Y sees X's as z . . . like Einstein told us, it's all relative!
So there appears to be a real problem. One solution is to assert, a priori, that the physics of our Universe would be different if the geometry of our Universe were different. But that is a very high price to pay, and in my view we're better off taking a look at how our contingent theories might be false. . . .
Any ideas?
Sorry but am I the only one who thinks there is no problem here at all? :confused:
The difference between the proper time interval of two observers between two space-time events is caused by the difference in path length. So what's the problem?
KingOrdo
Mar8-07, 12:57 AM
Sorry but am I the only one who thinks there is no problem here at all? :confused:
The difference between the proper time interval of two observers between two space-time events is caused by the difference in path length. So what's the problem?
Jennifer, the easiest way to envisage it is by just imaging that the spacetime is closed. Either it's because of the geometry, or--this way is easier--because of the topology of the manifold.
Here is a good precis on the arXiv (http://arxiv.org/PS_cache/physics/pdf/0006/0006039.pdf), although their proposed solution has been shown to be false.
Any ideas?
MeJennifer
Mar8-07, 01:13 AM
Jennifer, the easiest way to envisage it is by just imaging that the spacetime is closed. Either it's because of the geometry, or--this way is easier--because of the topology of the manifold.
Ok, so I imagine a closed universe. Which, by the way, implies that both the temporal and spatial components are topologically closed.
Now I still do not see what you see as a problem.
Care to explain what you think is a problem here?
Again, you can calculate the difference between the proper time interval of two observers between two space-time events by comparing the traversed path lengths.
KingOrdo
Mar8-07, 01:23 AM
Ok, so I imagine a closed universe. Which, by the way, implies that both the temporal and spatial components are closed.
Now I still do not see what you see as a problem.
Care to explain what you think is a problem here?
Jennifer, I'm just not sure I can make it any clearer. I'd refer you to my earlier posts; perhaps the first one is the most perspicuous.
Roukema and Bajtlik put it this way: "The paradox is the apparent symmetry of the twins' situations despite the time dilation effect expected due to their non-zero relative speed. It is difficult to understand how one twin can be younger than the other -- why should moving to the left or to the right be somehow favoured? Does the time dilation fail to occur?"
Perhaps time dilation *does* fail to occur. But that would mean physical law is not invariant of the universe in which it is instantiated. That price is, a priori, too high to pay, it seems. So however you slice it, an *explanation* is missing. Even if you believe--and this is really implausible--that "moving to the left" *does* matter, you still have to explain *why*. Most of us would, I think, claim that there's something analogous to the acceleration asymmetry in simple cases, but I've no idea what it is.
Any ideas?
MeJennifer
Mar8-07, 01:33 AM
For starters, if the path length between the two space-time events is identical for both observers the proper time interval differential is obviously zero.
Sorry, but I fail to see what the problem really is here.
KingOrdo
Mar8-07, 01:51 AM
For starters, if the path length between the two space-time events is identical for both observers the proper time interval differential is obviously zero.
Sorry, but I fail to see what the problem really is here.
The problem is with your first sentence. Read it and ask, 'Is that a consequence I'm really willing to accept?'
Any ideas?
MeJennifer
Mar8-07, 02:00 AM
The problem is with your first sentence. Read it and ask, 'Is that a consequence I'm really willing to accept?'
Huh? :confused:
This is basic relativity theory. Are you perhaps questioning the validity of SR or GR?
KingOrdo
Mar8-07, 02:10 AM
Huh? :confused:
This is basic relativity theory. Are you perhaps questioning the validity of SR or GR?
Yes, I know you're confused. But I don't know how else I can put it; like you said, "This is basic relativity theory". If my explanations aren't making sense, consult the arXiv; the relevant papers have been listed.
And yes we (physicsts) *are* questioning the "validity" of SR . . . well, at least we're not just going to take on *faith* that there's some explanation for what appears to be an anomalous result in an unusual topology (the example is easier to understand in the matter-free universe). *Evidence* and a *logical explanation* to the very interesting problem is both necessary and perhaps productive for the future of relativity theory. *One* explanation (the winding number theory) has been given, and that has shown to be false. So I will ask again:
any ideas?
MeJennifer
Mar8-07, 02:14 AM
Yes, I know you're confused.
Why do you think I am confused? About what?
Again, I know how to calculate, in principle, the proper time interval differential between two space-time events for two observers. I don't know what more there is to say. :smile:
Perhaps it is that you simply are not willing to accept the reality of the properties of space-time.
the example is easier to understand in the matter-free universe
A matter-free universe is flat, expanding and obeys a hyperbolic geometry.
KingOrdo
Mar8-07, 02:22 AM
Why do you think I am confused? About what?
Because you're not understanding the several really very basic examples I and other PF posters have given. And again, the papers I cited on the arXiv are especially good and clear in this regard. I recommend you consult them; and again, to keep it simple, imagine a matter-free universe of compact topology.
Again, I know how to calculate, in principle, the proper time interval differential between two space-time events for two observers. I don't know what more there is to say. :smile:
Well, I'm very pleased that you are able to calculate a proper time interval. That's a good first step, and really almost all you need to understand why the paradox persists in complex spaces.
A matter-free universe is flat, expanding and obeys a hyperbolic geometry.
Ah; now it's clear why you're not understanding the paradox. This statement of yours is false. There are lots of matter-free universes that are *not* flat; viz. the ones in question!
Perhaps it is that you simply are not willing to accept the reality of the properties of space-time.
Oh, yes: let's not worry about *evidence*; let's just take Jennifer's word for it that it's not a problem . . . we'll just *ignore* this lacuna because it's convenient. *One* good explanation has been provided; the 'winding number', and that was debunked in the literature. Again:
any ideas?
MeJennifer
Mar8-07, 02:24 AM
imagine a matter-free universe of compact topology.
A matter-free universe cannot be compact.
You are presenting a case that is completely impossible.
:smile:
KingOrdo
Mar8-07, 02:28 AM
A matter-free universe cannot be compact.
You are presenting a case that is completely impossible.
:smile:
Again, this is where you are making your mistake (although the paradox does persist in geometrically compact spaces (i.e. universes with matter)).
But one can have a compact matter-free universe (e.g. a compactified Kaluza-Klein manifold (equivalent to M^4xS^7, say)).
Again, any ideas?
MeJennifer
Mar8-07, 02:34 AM
But one can have a compact matter-free universe (e.g. a compactified Kaluza-Klein manifold (equivalent to M^4xS^7, say)).
Like a fish caught in a net and trying to wiggle out of it.
I see there is no point in arguing with you.
KingOrdo
Mar8-07, 02:41 AM
Like a fish caught in a net and trying to wiggle out of it.
I see there is no point in arguing with you.
Jennifer, what the heck? I'm asking a serious question about physics. I'm not looking for polemic, or faith-based arguments, or ad hominem attacks, or appeals to authority, or any other obfuscatory or magical mumbo-jumbo.
I have asked a question. The question has been asked extensively in the literature as well. I am soliciting opinions. If you do not understand it, or simply do not care, that is fine. But I would like to hear from people that have professional, considered opinions on the matter.
And again, there is such a thing as a matter-free compact manifold. Take the subject-mentioned S^3. You are confusing geometry with topology.
Again: any ideas?
MeJennifer
Mar8-07, 02:50 AM
And again, there is such a thing as a matter-free compact manifold.
Yes there is, but such a manifold is not a possible in both SR and GR.
If there is no matter the Riemann curvature tensor is zero and this implies that the manifold cannot possibly be compact. Furthermore, as I wrote before, a matter-free flat space-time must be expanding. Think for instance about the Milne model of matter free space-time.
I have asked a question.
You imply there is a problem without arguing why you think there is a problem.
My best guess as why you think there is a problem is that you perhaps fail to distinguish between an observer's indeterminism of the space-time path taken by an object moving relative to it and the factual space-time path taken.
KingOrdo
Mar8-07, 02:55 AM
Yes there is, but such a manifold is not a possible in both SR and GR.
If there is no matter the Riemann curvature tensor is zero and this implies that the manifold cannot possibly be compact. Furthermore, as I wrote before, a matter-free flat space-time must be expanding. Think for instance about the Milne model of matter free space-time.
I cannot keep going over things with you, though I certainly can recommend some excellent references. Though the paradox exists in universes with matter, it is easiest to envisage with test particles in a matter-free compact manifold. That is the original example given; and, in my mind, still the best.
You imply there is a problem without arguing why you think there is a problem.
You must be joking. The problem has been cited by me and others here at least a dozen times. And, again, on the arXiv. If you do not understand it I recommend the professional literature.
MeJennifer
Mar8-07, 03:03 AM
In general these types of "paradoxes" are generally caused by the misunderstanding of the relativity and equivalence principle.
All objects have a definite path in space-time but due to the relativity principle and the equivalence principle an observer cannot always determine (locally) which definite path is taken by objects.
KingOrdo
Mar8-07, 03:08 AM
In general these types of "paradoxes" are generally caused by the misunderstanding of the relativity and equivalence principle.
Indeed. Well, go tell all the professional physicists who are stumped by this one that the confusion is due to their "misunderstanding of the relativity and equivalence principle".
All objects have a definite path in space-time but due to the relativity principle and the equivalence principle an observer cannot always determine (locally) which definite path is taken by objects.
Well done.
Again: anyone have any ideas?
MeJennifer
Mar8-07, 03:15 AM
....all the professional physicists who are stumped by this one that the confusion is due to their "misunderstanding of the relativity and equivalence principle".
Would you care to provide some references in the literature by people like Einstein, Hawking, Penrose, Wald, Schutz, Thorne, Misner, Wheeler or Weinberg writing they are "stumped" by this "problem"?
KingOrdo
Mar8-07, 03:17 AM
Would you care to provide some references in the literature by people like Einstein, Hawking, Penrose, Wald, Shutz, Wheeler, Weinberg writing they are "stumped" by this "problem"?
No. Check the arXiv yourself.
Do any serious scholars--or any serious amateurs (i.e. people with non-crackpot theories)--have any ideas?
That's exactly the point: it shouldn't. But in the cases at hand, you're postulating a physical change because of an arbitrary choice of coordinates, which is precisely what is disallowed by GR.
No, I'm not.
Suppose you naïvely try to put inertial (t, x) coordinates on flat RxS^1. Any such coordinate chart will be periodic: the coordinates
(t_0, x_0)
and
(t_0 + d, x_0 + L)
refer to the exact same point of RxS^1, for some d and L.
In observer X's coordinates, let's choose L positive, and assume for simplicity that d is positive and large.
Suppose X meets Y at (0, 0), in X's coordinates. Let's call that event E.
Event E also has coordinates (-d, -L). So, if X looks to his left, he finds that X and Y met a long time ago. (so that preimage of Y is much older than the one he just met)
Event E also has coordinates (d, L). So, if X looks to his right, he finds that X and Y will not meet for a long time. (so that preimage of Y is much younger than the one he just met)
I strongly urge you to work it out yourself. Draw a space-time diagram in X's coordinates. Start with the polar coordinates on the cylinder RxS^1, (which will be inertial for an observer whose worldline is parallel to the axis of the cylinder), and do Lorentz transformation.
(Yes -- happily the formulae of SR will work in these coordinates)
If Y is travelling inertially rightward around the universe (in X's coordinates), then they will meet again, say, at (s, 0) -- X travelled the straight line (0, 0) --> (s, 0), so he ages s between meetings.
Y travelled the straight line (0, 0) --> (s + d, L). Equivalently, we can consider the straight line (-d, L) --> (s, 0). So, he ages:
\sqrt{(s + d)^2 - L^2}
which could be larger or smaller than s, depending on the actual values of everything.
If you're having trouble grasping exactly why the paradox reemerges in more complex topologies
I repeat, the (pseudo)paradox cannot emerge in a topology -- there is no such thing as the "age of an observer" or "inertial travel" or whatnot in a topology. You need a geometry before you can start talking about those things.
By specifying that Y follows a great circle, I presume this means circumnavigation of the Hubble sphere - which means a closed universe , and therefore defines a preferred frame (or at least a convenient frame) from which measurements of both x and y can be made to determine their respective cosmological world lines (their spacetime paths) with respect to a selected proper temporal interval as measured by a clock at rest wrt the CBR. Then use the principle of interval invariance to calculate which of the two clocks (X or Y) has accumulated more time during the two events that define the interval. Since the only events which are contained in both spacetime paths occur at the meeting points (the hi fives) the experiment will take a little while to collect the data - therefore:
Because of the practical importance of this subject, I suggest the contributors to this thread form a group to solicit money from the present administration to calculate from the information obtained over the course of the experimental period, which clock is older. We could call us the Cal-burton associates.
KingOrdo
Mar8-07, 03:55 PM
Hurkyl: I'm just not sure how much clearer I can make things for you. If my precis is confusing (and this is very possible), I really recommend getting into the literature (good summary of the problem here: http://arxiv.org/abs/physics/0006039).
By specifying that Y follows a great circle, I presume this means circumnavigation of the Hubble sphere - which means a closed universe , and therefore defines a preferred frame (or at least a convenient frame) from which measurements of both x and y can be made to determine their respective cosmological world lines (their spacetime paths) with respect to a selected proper temporal interval as measured by a clock at rest wrt the CBR.
Doesn't have to. Again, the problem is clearest in a matter-free universe, though that's not a necessary condition as long as X and Y are in inertial frames of reference. And it certainly doesn't have to be a spherical geometry (the 1+3 torus works just as well, e.g.) And I used 'great circle' rather than 'geodesic' because the original example was in S^3. Again, this is a thought experiment because the actual Universe isn't a compact space (well, probably not, anyway; and even if it is the experiment is practically impossible).
The only proposed resolution I can find in the literature (and none have been offered here) is the 'winding number' one, but that appears to have been debunked (cf. earlier link to the arXiv).
Any ideas? (Reminder, and pace some posters here, this isn't not a problem because Einstein/Feynman/Witten/et al. did/does not consider it so. We do not appeal to authority to resolve problems in physics; indeed, that's bad science. Please do not PM me saying 'Einstein wasn't worried about this, so there must be some obvious resolution.' I do not--and nor should you--take things on faith. Rigorous argumentation is what is welcome. Thanks!)
K.J.Healey
Mar9-07, 01:51 AM
From Wikipedia : (its a long article and this is merely part of it concerning this topic. Maybe this will clarify something. Maybe not.)
http://en.wikipedia.org/wiki/Time_dilation
Time dilation is symmetric between two inertial observers
One assumes, naturally enough, that if time-passage has slowed for a moving object, the moving object would find the external world to be correspondingly "sped up." But counterintuitively, Einsteinian relativity predicts the opposite, a situation difficult to visualize. This is based on an essential principle of the overall theory: if one object is moving with respect to another (at an unchanging velocity), the other is equally moving with respect to it.
...
But if motion is thus understood as purely relative, it can be divided-up between "mover" and "benchmark" in any way one pleases, even allowing them to completely switch roles. All that matters is the rate at which they are approaching, or departing from, one another, a grand total which re-distributing the speed-contribution of each one doesn't change. And if that is true, the consequences of relative motion predicted by the theory must also "add up" to an unchanging total effect. If A finds that B has undergone a slowdown-in-time during the period of relative motion, it must work out that B will also find that A has a relatively slower "clock." It seems an inconceivable situation: yet the math works out, and actual tests confirm it.
With respect to constant relative motion between two "clocks", a measurement of relative time must choose one clock as being "stationary" in spacetime, and that clock is the basis of a temporal coordinate system where time throughout is treated as synchronized with the stationary clock. The other "moving" clock is in motion with respect to this treated-as-stationary coordinated system, and its relative motion is the velocity value used in the applicable equations.
In the Special Theory of Relativity, the moving clock is found to be ticking slow with respect to the temporal coordinate system of the stationary clock. And as indicated, this effect is symmetrical: In a coordinate system synchronized, by contrast, with the "moving" clock, it is the "stationary" clocks that is found (by all methods of measurement) to be running slow. (Neglecting this principle of symmetry leads to the so-called twin paradox being regarded as paradoxical.)
Note that in all such attempts to establish "synchronization" within the reference system, the question of whether something happening at one location is in fact happening simultaneously with something happening elsewhere, is of key importance. Calculations are ultimately based on determining what is simultaneous with what.
It is a natural and legitimate question to ask how, in detail, Special Relativity can be self-consistent if clock A is time-dilated with respect to clock B and clock B is also time-dilated with respect to clock A. It is by challenging the assumptions we build into the common notion of simultaneity that logical consistency can be restored. Within the framework of the theory and its terminology, the short answer is that there is a relativity of simultaneity that affects how the specified "benchmark" moments of "simultaneous" events are aligned with respect to each other by observers who are in motion with respect to one other. Because the pairs of putatively simultaneous moments are differently identified by the different observers (as illustrated in the twin paradox article), each can treat the other clock as being the slow one without Relativity being self-contradictory. For those seeking a more explicit account, this can be explained in many ways, some of which follow.
-----------------------------------------
So like I said, at the time when each one relatively see's the high 5, the other will appear younger(as if they had been moving). You can't talk about age relative to a specific even that can be measured from either coordinate system without taking into account the differences in relative time OF that even in each coordinate system.
I don't see any problems with this scenario. Theres no difference than ANY other Twin Paradox (which by the way aren't always resolved with the "acceleration" argument. You can do this without acceleration.)
Remember the twins will NOT agree upon the time at which they high-fived.
Hurkyl: I'm just not sure how much clearer I can make things for you. If my precis is confusing (and this is very possible), I really recommend getting into the literature (good summary of the problem here: http://arxiv.org/abs/physics/0006039).
This certainly doesn't help -- in fact, it looks as if its conclusion is diametrically opposed to what you're trying to argue.
I'm going to assume that you agree with everything I said in my last post (since you haven't said otherwise). Since your thesis appeared to be that there is no asymmetry, but I've clearly demonstrated how asymmetry can occur, I'm confused as to why this discussion is still going on.
KingOrdo
Mar9-07, 10:51 AM
Healey01: You're misunderstanding the Twin paradox. The Twin paradox is *not* that "the twins will NOT agree upon the time at which they high-fived". That is trivally true. If you're finding the nature of relativity counterintuitive, this page might help: http://www.sc.doe.gov/Sub/Newsroom/News_Releases/DOE-SC/2005/THE_TWIN_PARADOX.htm. But the TP is resolved in SR by appeal to accelerations, and in GR by appeal to the fact that clocks run fast at large gravitational potentials, and vice versa.
This certainly doesn't help -- in fact, it looks as if its conclusion is diametrically opposed to what you're trying to argue.
Again, its conclusion has already been debunked in the literature (I previously cited the link for you). However, that paper does provide a good summary of the problem at hand, despite the falsity of its ultimate conclusion.
And again: the fundamental reason why this variant on the Twin paradox is stumping so many people--including professionals in global GR, etc.--is that in order to resolve the Twin paradox, one twin has to be 'preferred' in some sense. And that is definitionally *impossible* in the compact space cases unless you want to discard a central tenant of relativity theory. And again:
any ideas?
But the TP is resolved in SR by appeal to accelerations
The most intuitive way to resolve the TP in SR is to calculate the length of the two worldlines in Minkowski space - and it is trivial that the one who went to Pluto and back has aged more as he has travelled two sides of an isosceles triangle whereas the one on Earth has travelled along the longer side of the triangle, which is shorter than the sum of the other two lengths (trivially).
Note: this resolution has nothing to do with accelerations!
KingOrdo
Mar9-07, 11:13 AM
The most intuitive way to resolve the TP in SR is to calculate the length of the two worldlines in Minkowski space - and it is trivial that the one who went to Pluto and back has aged more as he has travelled two sides of an isosceles triangle whereas the one on Earth has travelled along the longer side of the triangle, which is shorter than the sum of the other two lengths (trivially).
Note: this resolution has nothing to do with accelerations!
Again, that is *not* the Twin paradox. I really can't offer a primer on relativity theory--both because I have neither the time nor the talent--but I can certainly recommend some references; as mentioned previously, the papers in the arXiv are a good place to start. But, quoting from Wikipedia: "The perception of paradox, referred to as the twin paradox (sometimes called the 'clock paradox') is caused by the error of assuming that relativity implies that only relative motion between objects should be considered in determining clock rates. The result of this error is the prediction that upon return to Earth, each twin sees the other as younger -- which is clearly impossible." *That* is the Twin paradox, and it is resolved by citing a salient asymmetry between X and Y: the fact that X was in non-inertial frames of reference (e.g. when he turned his spaceship around).
Any ideas?
If you put coordinates and a metric on the space, then you'll see that, in contrast to the case of minkowski spacetime, there is a priveleged frame which is at rest. The twin in this frame will be older. Exactly why are you so averse to using a metric?
KingOrdo
Mar9-07, 11:38 AM
If you put coordinates and a metric on the space, then you'll see that, in contrast to the case of minkowski spacetime, there is a priveleged frame which is at rest. The twin in this frame will be older. Exactly why are you so averse to using a metric?
Again, I just can't explain it any clearer than I already have, nor than has been explained in the literature. I really do recommend checking out the papers on the arXiv, as they are especially perspicuous. I don't know who is "averse to using a metric"--I don't even know if you're talking to me--but the point to remember is that time dilation is a *real* phenomenon. When Y gets back to Earth, Y really is younger than X. It's a real, coordinate invariant phenomenon.
Any ideas?
"Coordinate invariant" does not mean you can just ignore coordinates, you need them to define the inertial reference frames. The topology is not enough.
KingOrdo
Mar9-07, 01:45 PM
"Coordinate invariant" does not mean you can just ignore coordinates, you need them to define the inertial reference frames. The topology is not enough.
Yes. That is precisely the point.
Any ideas?
Right, so you can't solve this problem until you specify a coordinate system and metric on your space. Yes, without any other information about the system, this will be arbitrary, but that can't be avoided. And once you do this, the twins will no longer be equivalent. Moreover, you'll be able to see explicitly that the familiar rule from minkowski space time that moving observers appear to age slower does not hold exactly in more complicated spacetimes.
KingOrdo
Mar9-07, 02:05 PM
Right, so you can't solve this problem until you specify a coordinate system and metric on your space. Yes, without any other information about the system, this will be arbitrary, but that can't be avoided. And once you do this, the twins will no longer be equivalent. Moreover, you'll be able to see explicitly that the familiar rule from minkowski space time that moving observers appear to age slower does not hold exactly in more complicated spacetimes.
StatusX, again: I can't make it any clearer. Consult the literature if you're not understanding why a problem arises. Links have been provided. All best,
Tom.
P.S. Anyone: any ideas?
Exercise for KingOrdo: resolve the following paradox.
We have two numbers, x and y. Which is bigger? This problem is perfectly symmetric, so we cannot say that x is bigger than y. So, x and y have to be equal, which is paradoxical!
P.S. Anyone: any ideas?
I already explained how to work out the problem in RxS^1. Either trying to understand it, or pick out an actual error is a good idea.
KingOrdo
Mar9-07, 06:24 PM
Hurkyl: unlike you, I am not going to engage in polemic or insult (mods, please?).
If you do not understand the phenomenon at hand--which admittedly may be due to my unperspicuous treatment--I recommend you read the several excellent posts made by others, and especially the professional literature (i.e. the arXiv). Regards.
Anyone: any ideas?
KingOrdo
Mar9-07, 06:49 PM
A-ha! I found a a proposed resolution to the paradox (and unlike the 'winding number' one it has not been, as far as I can find, refuted. Here's the link: http://arxiv.org/PS_cache/gr-qc/pdf/0101/0101014.pdf. But it seems to raise two problems:
(1) It only works in non-matter-free universes. Is that a price we're willing to pay? Should we just stipulate that there's something paradoxical about matter-free universes themselves?
(2) The mechanism to break the symmetry and resolve the paradox is fundamentally different than the one used in the simple cases (and indeed, in our own Universe). Does that seem intuitively right?
Any thoughts?
(1) It only works in non-matter-free universes. Is that a price we're willing to pay? Should we just stipulate that there's something paradoxical about matter-free universes themselves?
Nowhere in that paper did they suggest their universe had matter. To wit, there were using a perfectly flat metric, and their space-time is locally isometric to Minkowski space.
But why does it matter? The cosmological twin "paradox" is merely a pseudoparadox because the conclusion does not follow from the premises: it is a logically flawed argument. The conclusion does not follow from the hypotheses. The merit of this paper is that it vividly demonstrates the logical flaw, so as to help those still stuck on the paradox.
(2) The mechanism to break the symmetry and resolve the paradox is fundamentally different than the one used in the simple cases (and indeed, in our own Universe). Does that seem intuitively right?
I think you're asking if it's intuitive that global topology should affect things. Well, it depends on how you've developed your intuition -- if you've studied topology, for example, it would be obvious that it should have some relevance. OTOH, if you've studied other things and never had reason to leave the world of affine space, it would be more surprising.
Hurkyl: unlike you, I am not going to engage in polemic or insult
What insult? I thought it sufficiently likely that you were making exactly that mistake. (But buried underneat a bunch of other stuff so you don't see it)
A-ha! I found a a proposed resolution to the paradox
...
http://arxiv.org/PS_cache/gr-qc/pdf/0101/0101014.pdf.
...
Any thoughts?
But in any case, I'm glad you've finally understand this demonstration of the flaw in the cosmological twin paradox.
KingOrdo
Mar9-07, 11:44 PM
Nowhere in that paper did they suggest their universe had matter. To wit, there were using a perfectly flat metric, and their space-time is locally isometric to Minkowski space.
"A compact topology selects a preferred place and a preferred time so that some galaxy, if not our own, is at the center of the universe." "galaxy"=matter. This is the case for curved spaces, which is again I think the ones that bear especially important examination.
But why does it matter? The cosmological twin "paradox" is merely a pseudoparadox because the conclusion does not follow from the premises: it is a logically flawed argument. The conclusion does not follow from the hypotheses. The merit of this paper is that it vividly demonstrates the logical flaw, so as to help those still stuck on the paradox.
Well, if the paper is right, then yes: it is a pseudoparadox, just like the Twin "paradox" in the simple case is a pseudoparadox. Of course, if it fails--like the 'winding number' paper did--then the paradox would persist.
I think you're asking if it's intuitive that global topology should affect things. Well, it depends on how you've developed your intuition -- if you've studied topology, for example, it would be obvious that it should have some relevance. OTOH, if you've studied other things and never had reason to leave the world of affine space, it would be more surprising.
First, I don't know what "OTOH" is. Second, it is my no means obvious that global topology should change the physical laws of the universe. As a matter of fact, it's counter to standard experience: the laws of physics are the same on the surface of a sphere as they are in a flat space. If you want to argue for it, that's fine; you might be right. But that burden of proof is on you.
But in any case, I'm glad you've finally understand this demonstration of the flaw in the cosmological twin paradox.
Well, that's the question: is this the right way out? Are Barrow and Levin right where the other authors on the arXiv were wrong? The other papers postulated *different* mechanisms for resolving the case in complex spaces. At most one can be right. Do you think it's Barrow and Levin? Or the 'winding number' one (though that looks pretty conclusively refuted). But, again, please: *no faith-based arguments*. It's been nearly 100 posts and still no one is willing to say: 'here's the *right* way out: X, Y, and Z.' Or, 'Barrow and Levin are right; here's why I think that . . .'
Any ideas? Remember; operative word: *ideas*. . . .
Well, if the paper is right, then yes: it is a pseudoparadox, just like the Twin "paradox" in the simple case is a pseudoparadox. Of course, if it fails--like the 'winding number' paper did--then the paradox would persist.
If you mean "persists" in the sense that it's a logical worry, then you're wrong. The cosmological twin "paradox" is a pseudoparadox because it is a logically flawed argument -- the conclusion does not follow from the premise. Whether or not anyone has presented a counter-example you like is irrelevant.
"A compact topology selects a preferred place and a preferred time so that some galaxy, if not our own, is at the center of the universe." "galaxy"=matter. This is the case for curved spaces, which is again I think the ones that bear especially important examination.
If you're going to get technical, the twins are matter too, so you can't have
the twin paradox without matter. :tongue:
First, I don't know what "OTOH" is. Second, it is my no means obvious that global topology should change the physical laws of the universe. As a matter of fact, it's counter to standard experience: the laws of physics are the same on the surface of a sphere as they are in a flat space. If you want to argue for it, that's fine; you might be right. But that burden of proof is on you.
OTOH means "on the other hand".
And nobody said global topology changes physical laws. Global topology is relevant to global phenomena -- for example, what problems we might encounter when we take a local process like building coordinate charts isometric to Minkowski space, and try to extend globally across the entire universe.
I assume you've taken complex analysis? A lot of what you see in complex analysis is a simplified version of the issues we are seeing here. In the language of the paper you most recently linked, the fact that it is "impossible for H to synchronize her clocks" is the same sort of phenomenon as needing a branch cut for certain functions. And by golly, we see the same sort of pseudoparadoxes if we ignore that: as we trace counterclockwise around the unit circle, the imaginary component of log z is strictly increasing, so the imaginary component of log z must be larger at our ending point than at our starting point. This is seemingly paradoxical because our ending point can be our starting point if we go all the way around the circle.
But, again, please: *no faith-based arguments*.
Nobody's making a faith-based argument. People know there is no paradox because tensor analysis was defined precisely so that nothing depends on your choice of coordinates. (And that's why it was adopted for GR) So, when you analyze a situation in two different coordinates and get two different answers, we know one of the following is true:
(1) You made a mistake.
(2) The very foundation of mathematics is inconsistent.
And I don't mean (2) in the "oh, we got GR wrong" sense... I mean (2) in the "we now have a correct proof of 0 = 1" sense.
The "resolutions" people make of twin paradoxes are simply pedagogical devices: the author has a guess as to why people are confused by the twin pseudoparadox, and they try to make a vivid demonstration to help them out of their confusion.
KingOrdo
Mar10-07, 01:15 PM
If you mean "persists" in the sense that it's a logical worry, then you're wrong. The cosmological twin "paradox" is a pseudoparadox because it is a logically flawed argument -- the conclusion does not follow from the premise. Whether or not anyone has presented a counter-example you like is irrelevant.
No. It's a pseudoparadox in the simple case because there is a way to resolve it: one twin accelerates, and the symmetry is broken. If Barrow and Levin are right, then it's a pseudoparadox in the complex case. But if they're wrong, like the 'winding number' paper was, then the paradox--or appearance of paradox--persists.
If you're going to get technical, the twins are matter too, so you can't have the twin paradox without matter. :tongue:
Yes, that's true. Upon further reflection, I don't think it's an intelligible problem when using test particles.
And nobody said global topology changes physical laws. Global topology is relevant to global phenomena -- for example, what problems we might encounter when we take a local process like building coordinate charts isometric to Minkowski space, and try to extend globally across the entire universe.
Yes, you *are* claiming that physical law is variant on topology. If the mechanism for symmetry breaking in a complex space is essentially different than the mechanism in, say, the actual Universe (viz. acceleration), then--unlike in the actual Universe--law is not invariant with regard to topology. Now, I'm not saying you're *wrong*; indeed, a lot of (very smart) people believe exactly that. But we'll need some argument to overcome the prima facie implausibility.
I assume you've taken complex analysis? A lot of what you see in complex analysis is a simplified version of the issues we are seeing here. In the language of the paper you most recently linked, the fact that it is "impossible for H to synchronize her clocks" is the same sort of phenomenon as needing a branch cut for certain functions. And by golly, we see the same sort of pseudoparadoxes if we ignore that: as we trace counterclockwise around the unit circle, the imaginary component of log z is strictly increasing, so the imaginary component of log z must be larger at our ending point than at our starting point. This is seemingly paradoxical because our ending point can be our starting point if we go all the way around the circle.
The difference is, of course, that it's totally unintelligible to talk about the ages of points on the unit circle. Not so in the physical case. If you were right, then there would be no time dilation at all, even in the simple cases.
The "resolutions" people make of twin paradoxes are simply pedagogical devices: the author has a guess as to why people are confused by the twin pseudoparadox, and they try to make a vivid demonstration to help them out of their confusion.
No; you're confusing the counterintutiveness of time dilation with the Twin paradox itself. You are perfectly right in the former case; however, that's *not* what the Twin paradox is. The Twin paradox is *not*: 'Hey, this twin left Earth and came back to shake his brother's hand--when he did, he was younger than his brother!' It's clear that that's what you think it is, but you're wrong; again, I've been over this ad nauseum. I recommend checking out those links and especially the professional literature.
pervect
Mar10-07, 02:37 PM
IMO Hurkyl isn't saying anything that conflicts with the literature. He is trying (rather patiently) to correct some of King Ordo's misunderstandings of what the literature is saying as far as what the cosmological twin "paradox" is about and what it is not about.
KingOrdo
Mar10-07, 02:57 PM
IMO Hurkyl isn't saying anything that conflicts with the literature. He is trying (rather patiently) to correct some of King Ordo's misunderstandings of what the literature is saying as far as what the cosmological twin "paradox" is about and what it is not about.
But that's the problem: he's misconstruing the Twin paradox. He thinks (as do many people here, apparently) that the Twin paradox is just that it seems weird that when one twin leaves Earth and returns he's younger than his twin that stayed behind. But that's not paradoxical at all; rather, it's a straightforward implication of relativity theory (although it is a little weird, to be sure).
The Twin paradox is this: it's just as correct to say that the twin on Earth was the one that did the traveling, and the twin on the rocket stayed at rest. Therefore, when they get back together *both will see the other as older*, which is a logical contradicition. Now, it's a pseudoparadox in simple cases, because there is an asymmetry between the twins (viz. the one on the rocket accelerates). But that's not true in the complex cases. So an asymmetry needs to be found there, too. One candidate was the 'winding number'; however, that was debunked in the literature. Barrow and Levin have proposed another asymmetry. What do people think about this? Again, I know the topic is confusing, but the arXiv really does have several papers that make things pretty clear (links have been provided).
Any thoughts?
No. It's a pseudoparadox in the simple case because there is a way to resolve it
I've been using pseudoparadox in the technical sense (http://mathworld.wolfram.com/Pseudoparadox.html), rather than as a synonym for "aha, I'm no longer confused". (Of course, in this informal sense, this is a pseudoparadox for me, whether or not it's a paradox for you)
Yes, you *are* claiming that physical law is variant on topology. If the mechanism for symmetry breaking in a complex space is essentially different than the mechanism in, say, the actual Universe (viz. acceleration), then--unlike in the actual Universe--law is not invariant with regard to topology. Now, I'm not saying you're *wrong*; indeed, a lot of (very smart) people believe exactly that. But we'll need some argument to overcome the prima facie implausibility.
The symmetry demanded by Einstein was local, and it's still present even in these "complex spaces". It's not "broken".
But that symmetry was only demanded of the laws of physics -- it would be rather silly to demand everything be symmetric. The matter distribution, the metric, other fields we put on space-time... they aren't required, nor even expected to be symmetric.
But the diffeomorphism invariance of GR is not what's relevant here. We have the rather exceptional case that flat RxS^1 is locally isometric to 1+1 Minkowski space, and that flat Rx(S^1)^3 is locally isometric to 3+1 Minkowski space. We are interested in the question of whether a problem on RxS^1 can be treated as if it was a problem in 1+1 Minkowski space. This is a problem of piecing local information together, hoping to obtain global information, and whether we can do this is one of the big questions studied in topology.
That's how the winding number fits in, in the RxS^1 case -- if we follow observer X's path forward between meeting points, and then Y's path backwards between meeting points, we have a closed curve and can ask about its winding number. If that number is zero, we can treat the entire problem as if it were happening in Minkowski space. If that number is nonzero, then the global periodic nature of spacetime is relevant in an essential way -- in particular, the winding number is zero if and only if X and Y really do meet again, according to the Minkowski analysis.
To wit, if we take the cosmological twin paradox on RxS^1, then if the twins are both travelling inertially, they cannot possibly meet twice. In fact, that really should be the big clue that there are flaws in treating the situation with special relativistic methods.
The difference is, of course, that it's totally unintelligible to talk about the ages of points on the unit circle. Not so in the physical case. If you were right, then there would be no time dilation at all, even in the simple cases.
Age is a number. We're talking about numbers. :tongue:
In Barrow & Levin, they observe that "it becomes impossible for H to synchronize her clocks" -- that is because she needs to have a branch cut in her coordinates. (Or use multi-valued coordinate functions, or use the universal cover...)
No; you're confusing the counterintutiveness of time dilation
It's not counterintuitive. :tongue: (At least, it's not counterintuitive for me...)
KingOrdo
Mar10-07, 04:03 PM
I've been using pseudoparadox in the technical sense (http://mathworld.wolfram.com/Pseudoparadox.html), rather than as a synonym for "aha, I'm no longer confused". (Of course, in this informal sense, this is a pseudoparadox for me, whether or not it's a paradox for you)
No, you're quite wrong here; I've been using paradox in the precise, logical sense . . . you've been using it in the 'Hmm . . . that's weird.' sense. N.B. There's nothing wrong with your usage.
The symmetry demanded by Einstein was local, and it's still present even in these "complex spaces". It's not "broken".
No, it is broken. And it's broken in different ways; cf. the arXiv.
But that symmetry was only demanded of the laws of physics -- it would be rather silly to demand everything be symmetric. The matter distribution, the metric, other fields we put on space-time... they aren't required, nor even expected to be symmetric.
I'm not sure what you mean here; are you talking about symmetry as a component of a equivalence relation? Obviously some physical laws are symmetric, and some aren't.
But the diffeomorphism invariance of GR is not what's relevant here. We have the rather exceptional case that flat RxS^1 is locally isometric to 1+1 Minkowski space, and that flat Rx(S^1)^3 is locally isometric to 3+1 Minkowski space. We are interested in the question of whether a problem on RxS^1 can be treated as if it was a problem in 1+1 Minkowski space. This is a problem of piecing local information together, hoping to obtain global information, and whether we can do this is one of the big questions studied in topology.
That's how the winding number fits in, in the RxS^1 case -- if we follow observer X's path forward between meeting points, and then Y's path backwards between meeting points, we have a closed curve and can ask about its winding number. If that number is zero, we can treat the entire problem as if it were happening in Minkowski space. If that number is nonzero, then the global periodic nature of spacetime is relevant in an essential way -- in particular, the winding number is zero if and only if X and Y really do meet again, according to the Minkowski analysis.
This is why I have consistently directed you to the professional literature; this approach fails for the precise reasons it was implemented in the first place. I have been over this ad infinitum; again, if you are confused--and I don't blame you if you are (much of this is counterintuitive!)--consult the arXiv. The professionals say it much better than I do.
To wit, if we take the cosmological twin paradox on RxS^1, then if the twins are both travelling inertially, they cannot possibly meet twice. In fact, that really should be the big clue that there are flaws in treating the situation with special relativistic methods.
Why would we treat it using SR methods?
Age is a number. We're talking about numbers. :tongue:
No, it's not. Age is a property that has to do with the physical composition of the entity in question. In the simple case, forget about age: when the twin gets back to Earth he *will be physically different* than the one who stayed behind. And so too in the complex case. Again, it is best to think of the two twins at the first point of intersection as one body with a symmetry.
In Barrow & Levin, they observe that "it becomes impossible for H to synchronize her clocks" -- that is because she needs to have a branch cut in her coordinates. (Or use multi-valued coordinate functions, or use the universal cover...)
Just so.
So again, I must ask: any ideas? To quote pervect:
"There is a general agreement about the broad details, which is that the two twins won't be the same age.
The disagreement as I read it is how to explain the age difference, i.e. to point to a particular mechanism. The first papers say that it can be addressed in terms of winding number (which is a topological property that doesn't even need a metric). The second papers say that this is not correct, that one needs more than the winding number to properly explain the age difference, that one needs the metric information.
Everyone agrees that there should be an age difference AFAIK."
So: what is the proper resolution? Is the 'winding number' approach right, despite its apparent refutation in the literature? Or do Barrow and Levin have the right idea? Is there perhaps another idea we're missing? And please: I can't explain it any better than I already have; consult the literature if you're unsure about why the paradox may persist in the complex cases.
Any ideas?
The disagreement as I read it is how to explain the age difference, i.e. to point to a particular mechanism.
There is exactly one mechanism. If \gamma is the worldline of a twin between the two events where the twins meet, parametrized by u \in [0, 1], then the twin ages
\Delta \tau = \int_{\gamma} || \frac{d \gamma}{du} || \, du.
The amount each twin ages is a different integral, and thus can have different numerical values. We can compare those numbers to tell which twin ages more.
We can some prove some general facts about it \Delta \tau (such as that geodesics yield a locally maximal value of \Delta \tau) And if we make some assumptions, we can derive shortcuts and specialized theorems for \Delta \tau -- for example, the time dilation formula in an Minkowski inertial coordinate chart, or invoke the fact that in Minkowski space, there is only one geodesic between a pair of points to prove that an inertially travelling observer ages more than any other observer that he meets twice. But we should not mistake of assuming that the general case must also have such simple theorems.
MeJennifer
Mar10-07, 05:02 PM
There is exactly one mechanism. If \gamma is the worldline of a twin between the two events where the twins meet, parametrized by u \in [0, 1], then the twin ages
\Delta \tau = \int_{\gamma} || \frac{d \gamma}{du} || \, du.
The amount each twin ages is a different integral, and thus can have different numerical values. We can compare those numbers to tell which twin ages more.
Exactly right, and really, that is all there is to say about this "paradox".
KingOrdo
Mar10-07, 05:06 PM
I can make it no more perspicuous for you. Consult the arXiv if you need clarification.
So again, I must ask: any ideas? To quote pervect:
"There is a general agreement about the broad details, which is that the two twins won't be the same age.
The disagreement as I read it is how to explain the age difference, i.e. to point to a particular mechanism. The first papers say that it can be addressed in terms of winding number (which is a topological property that doesn't even need a metric). The second papers say that this is not correct, that one needs more than the winding number to properly explain the age difference, that one needs the metric information.
Everyone agrees that there should be an age difference AFAIK."
So: what is the proper resolution? Is the 'winding number' approach right, despite its apparent refutation in the literature? Or do Barrow and Levin have the right idea? Is there perhaps another idea we're missing? And please: I can't explain it any better than I already have; consult the literature if you're unsure about why the paradox may persist in the complex cases.
Any ideas?
I can make it no more perspicuous for you. Consult the arXiv if you need clarification.
I'm not aware of anything I need clarified, except maybe precisely what you think, why you think that way, and what criteria an answer must satisfy before you would find it acceptable.
KingOrdo
Mar10-07, 06:01 PM
I'm not aware of anything I need clarified, except maybe precisely what you think, why you think that way, and what criteria an answer must satisfy before you would find it acceptable.
I can make it no more perspicuous for you. Consult the arXiv (and my earlier posts) if you need clarification.
So again, I must ask: any ideas? To quote pervect:
"There is a general agreement about the broad details, which is that the two twins won't be the same age.
The disagreement as I read it is how to explain the age difference, i.e. to point to a particular mechanism. The first papers say that it can be addressed in terms of winding number (which is a topological property that doesn't even need a metric). The second papers say that this is not correct, that one needs more than the winding number to properly explain the age difference, that one needs the metric information.
Everyone agrees that there should be an age difference AFAIK."
So: what is the proper resolution? Is the 'winding number' approach right, despite its apparent refutation in the literature? Or do Barrow and Levin have the right idea? Is there perhaps another idea we're missing? And please: I can't explain it any better than I already have; consult the literature if you're unsure about why the paradox may persist in the complex cases.
Any ideas?
JustinLevy
Mar10-07, 06:58 PM
The symmetry demanded by Einstein was local, and it's still present even in these "complex spaces". It's not "broken".
No, it is broken. And it's broken in different ways; cf. the arXiv.
You are misunderstanding the papers. And you are misunderstanding the posters here trying to teach you.
As pervect said:
"IMO Hurkyl isn't saying anything that conflicts with the literature. He is trying (rather patiently) to correct some of King Ordo's misunderstandings of what the literature is saying as far as what the cosmological twin "paradox" is about and what it is not about."
In short, people are trying to be very patient with you and help answer your questions. But you continue to ignore or misunderstand all the help presented to you.
I can understand that you do not believe you are misunderstanding anything, but please entertain the possibility to allow this discussion to move forward.
So again, I must ask: any ideas?
Yes, I have an idea to help this discussion. To clear up some misunderstanding and help everyone see where the root problem is coming from ... and to prevent the discussion from continuing in circles indefinitely ... KingOrdo, please answer these questions:
1) As pervect mentioned, even in a non-closed universe, two distinct inertial paths can cross in two places.
a] So before moving onto closed spaces, do you understand that there is no paradox about how much proper time elapsed on these two world lines?
b] If so, please explain your understanding of the resolution of this "paradox" to give others a starting point to build explanations from.
2) Do you agree that the question of how much proper time elapsed requires a geometry, ie. that until a geometry is defined we cannot ask for the distance between spacetime points? If not, please explain why.
3) Do you agree that specifying a geometry does not specify a coordinate system (ie. the description is still coordinate invarient)? If not, please explain why.
4) Do you agree that once the geometry is specified, there is a unique answer to how much proper time elapsed along a path in spacetime? And therefore there is no "paradox"? If not, please explain why.
KingOrdo
Mar10-07, 07:59 PM
I simply can't make it any clearer for you. If the fundamentals of relativity theory are still hazy to you, I can recommend some excellent references. Also, consult the arXiv (and my earlier posts) if you need clarification.
So again, I must ask: any ideas? To quote pervect:
"There is a general agreement about the broad details, which is that the two twins won't be the same age.
The disagreement as I read it is how to explain the age difference, i.e. to point to a particular mechanism. The first papers say that it can be addressed in terms of winding number (which is a topological property that doesn't even need a metric). The second papers say that this is not correct, that one needs more than the winding number to properly explain the age difference, that one needs the metric information.
Everyone agrees that there should be an age difference AFAIK."
So: what is the proper resolution? Is the 'winding number' approach right, despite its apparent refutation in the literature? Or do Barrow and Levin have the right idea? Is there perhaps another idea we're missing? And please: I can't explain it any better than I already have; consult the literature if you're unsure about why the paradox may persist in the complex cases.
Any ideas?
JustinLevy
Mar11-07, 12:57 AM
If you are not willing to entertain the possibility that you are misunderstanding, then you can never learn. In that case, it is pointless to even try to have a discussion about this with you. Is this what you are telling us?
I truely hope not. So please go back and answer the questions asked here:
http://www.physicsforums.com/showpost.php?p=1269327&postcount=105
KingOrdo
Mar11-07, 01:56 AM
JustinLevy, I simply can't make it any clearer for you. If the fundamentals of relativity theory are still hazy to you, I can recommend some excellent references. Also, consult the arXiv (and my earlier posts) if you need clarification.
So again, I must ask: any ideas? To quote pervect:
"There is a general agreement about the broad details, which is that the two twins won't be the same age.
The disagreement as I read it is how to explain the age difference, i.e. to point to a particular mechanism. The first papers say that it can be addressed in terms of winding number (which is a topological property that doesn't even need a metric). The second papers say that this is not correct, that one needs more than the winding number to properly explain the age difference, that one needs the metric information.
Everyone agrees that there should be an age difference AFAIK."
So: what is the proper resolution? Is the 'winding number' approach right, despite its apparent refutation in the literature? Or do Barrow and Levin have the right idea? Is there perhaps another idea we're missing? And please: I can't explain it any better than I already have; consult the literature if you're unsure about why the paradox may persist in the complex cases.
Any ideas?
MeJennifer
Mar11-07, 03:12 AM
Any ideas?
Yes, to close this topic, it's going nowhere.
KingOrdo
Mar11-07, 03:15 AM
Yes, to close this topic, it's going nowhere.
Indeed; but not for want of trying on my part. I've been extremely patient, to the tune of five pages of posts. . . .
JustinLevy
Mar11-07, 03:26 AM
JustinLevy, I simply can't make it any clearer for you.
Why do you insist on evading the questions? That would make your position abundantly clear. Which currently it is not as you keep stating things that contradict GR as well as the papers you are referring to.
The previous questions and related points have been brought up by other posters and you continue to ignore them or resist acknowledging them. In doing so you ignore the very discussion you repeatedly seek. So if you do not answer those questions, it will be nearly impossible for any poster to help you.
This is a simple request, and the questions are not difficult or time consuming. So please answer the specific questions previously given to you.
KingOrdo
Mar11-07, 03:28 AM
JustinLevy, I simply can't make it any clearer for you. If the fundamentals of relativity theory are still hazy to you, I can recommend some excellent references. Also, consult the arXiv (and my earlier posts) if you need clarification.
So again, I must ask: any ideas? To quote pervect:
"There is a general agreement about the broad details, which is that the two twins won't be the same age.
The disagreement as I read it is how to explain the age difference, i.e. to point to a particular mechanism. The first papers say that it can be addressed in terms of winding number (which is a topological property that doesn't even need a metric). The second papers say that this is not correct, that one needs more than the winding number to properly explain the age difference, that one needs the metric information.
Everyone agrees that there should be an age difference AFAIK."
So: what is the proper resolution? Is the 'winding number' approach right, despite its apparent refutation in the literature? Or do Barrow and Levin have the right idea? Is there perhaps another idea we're missing? And please: I can't explain it any better than I already have; consult the literature if you're unsure about why the paradox may persist in the complex cases.
Any ideas?
JustinLevy
Mar11-07, 03:36 AM
Yes, to close this topic, it's going nowhere.Indeed; but not for want of trying on my part.
How can you possibly say that? It is going no where directly because of your lack of trying.
I have asked repeatedly for you to answer some simple questions. You refuse to acknowledge questions were even asked of you, let alone actually try to answer them.
So stop avoiding the questions, as these relate to the heart of the matter.
Please answer the following:
1) As pervect mentioned, even in a non-closed universe, two distinct inertial paths can cross in two places.
a] So before moving onto closed spaces, do you understand that there is no paradox about how much proper time elapsed on these two world lines?
b] If so, please explain your understanding of the resolution of this "paradox" to give others a starting point to build explanations from.
2) Do you agree that the question of how much proper time elapsed requires a geometry, ie. that until a geometry is defined we cannot ask for the distance between spacetime points? If not, please explain why.
3) Do you agree that specifying a geometry does not specify a coordinate system (ie. the description is still coordinate invarient)? If not, please explain why.
4) Do you agree that once the geometry is specified, there is a unique answer to how much proper time elapsed along a path in spacetime? And therefore there is no "paradox"? If not, please explain why.
KingOrdo
Mar11-07, 03:44 AM
JustinLevy, I simply can't make it any clearer for you. If the fundamentals of relativity theory are still hazy to you, I can recommend some excellent references. Also, consult the arXiv (and my earlier posts) if you need clarification.
So again, I must ask: any ideas? To quote pervect:
"There is a general agreement about the broad details, which is that the two twins won't be the same age.
The disagreement as I read it is how to explain the age difference, i.e. to point to a particular mechanism. The first papers say that it can be addressed in terms of winding number (which is a topological property that doesn't even need a metric). The second papers say that this is not correct, that one needs more than the winding number to properly explain the age difference, that one needs the metric information.
Everyone agrees that there should be an age difference AFAIK."
So: what is the proper resolution? Is the 'winding number' approach right, despite its apparent refutation in the literature? Or do Barrow and Levin have the right idea? Is there perhaps another idea we're missing? And please: I can't explain it any better than I already have; consult the literature if you're unsure about why the paradox may persist in the complex cases.
Any ideas?
MeJennifer
Mar11-07, 03:48 AM
You have now repeated that last posting 6 times.
To me that is trolling. Hopefully a moderator can take some action here.
KingOrdo
Mar11-07, 04:17 AM
I've asked for moderators to intervene pages ago. I ask a simple question and get only evasion. I can explain myself no better than I already have. Any insult/evasion/irrelevance/etc. will be met by boilerplate. You should expect that. I am asking a serious question about a serious topic.
JustinLevy
Mar11-07, 04:51 AM
I ask a simple question and get only evasion.
You did not get evasion. You got answers from several posters which you preceded to ignore or misunderstand. When people took the time to help resolve this misunderstanding you refused to help in anyway.
I can explain myself no better than I already have.
Yes you can. You can answer the simple questions directly addressed to you to help others understand where the common ground lays and where the disagreement occurs.
The questions are not an insult, nor an evasion, nor irrelevant.
The are simple, reasonable, and relevant. Your evasion of them makes me question your motives here. If you are not here to just troll, please stop evading the questions.
ZapperZ
Mar11-07, 05:59 AM
Obviously, this thread is going nowhere. Questions have either been answered, or not been addressed, or issues not clear, etc.. etc.
After 8 pages of responses, I believe it is time to stick a fork into this one. Please do not repost this question in another thread.
Zz.
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