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Time Dilation. The faster you travel the longer I have to wait for you to return?

by Semifaded
Tags: dilation, faster, return, time, travel
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Darwin123
#19
Dec9-12, 11:39 AM
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Quote Quote by Ibix View Post
Ghwellsjr's spacetime diagrams already demonstrate that this isn't true. Let his blue line be the worldline of the stay-at-home, one of the black legs be the worldline of the outbound cash, and the other black leg be the worldline of the inbound burrito. His diagrams accurately describe my scenario, and the proper time experienced by the cash on its trip plus the proper time experienced by the burrito on its trip total less than the proper time of the stay at home person, with no acceleration involved.

I note that you haven't posted maths or references to back up your claim that greater acceleration leads to a longer wait time for the stay at home twin. It seems wrong to me for reasons I articulated in my response to K^2. Could you let me know (with maths) what it is that you think happens? What is the parameter that determines when the Newtonian result (higher acceleration gets there in less time) gives way to your regime?
Most of these mathematical proofs using GR rely on one path being a geodesic and the other not a geodesic. The twin on earth is traveling a geodesic, which is the shortest path in space time between two point. The traveling twin is not on a geodesic. His path is not on the shortest path through space time.

Ghwellsjr's spacetime diagrams would not be satisfying because the diagram couldn't show the symmetry breaking mechanism. This is a case where a word could be worth a thousand pictures. Even if the analysis was mathematical, what would be useful is if the equation with the symmetry breaking mechanism was presented. At least one would get an idea of how the symmetry breaking occurred.

The problem is that there is no physical hypothesis that distinguishes between trajectories that are geodesics and trajectories that are not geodesics. One can always find a set of variables in which an observer is traveling a geodesic. After all, the physical laws are independent of the path of the observer. "Everything" is relative, or so people have been told.

What many people are asking is not whether there is mathematics that distinguish one twin from the other. They are asking for a physical hypothesis that distinguishes between the two observers. They want to know the "symmetry breaking" feature in the calculation. "Physical intuition" is not sufficient for distinguishing between a geodesic and a nongeodesic.

Many books on science for laymen say straight out that it is the rockets that break the symmetry. I am just generalizing what these books in "mainstream science" say. Instead of "rockets", I say "external force". What you seem to be saying is that the rockets have nothing to do with the twin on earth aging faster than the traveling twin. What is asked for is a physical hypothesis, not a mathematical proof. The mathematical proof is certainly worth a discussion on its own, but it is irrelevant here.

There are many problems in physics where a symmetry is broken. The question of what interaction physically breaks a symmetry is often a valid scientific question, regardless of scientific field.
K^2
#20
Dec9-12, 01:27 PM
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Quote Quote by Darwin123 View Post
Most of these mathematical proofs using GR rely on one path being a geodesic and the other not a geodesic. The twin on earth is traveling a geodesic, which is the shortest path in space time between two point. The traveling twin is not on a geodesic. His path is not on the shortest path through space time.
Longer path would mean longer proper-time, which would mean that twin that traveled aged more than the twin that remained in one place.

The fact is, twin that traveled has actually taken a shorter path. How? Geodesics are local minima. In fact, they don't even have to be minima, merely extrema, but I don't know if that's ever relevant in GR. At any rate, there can exist paths that are shorter.

For a simple analogy, think of an object in a glass of water placed in such a way that you can see the object both through the wall of the glass and through the surface. It looks like there are two copies of the object, one distorted more than the other. I'm sure you've seen this. In optics, light takes the "shortest" path as well. Fact that you can see two images of the same object tells you that there are two "shortest" paths between the object and your eye. Again, the path only needs to be locally shortest. Meaning that any small perturbation of the path has to increase the length.



Anyways, back to the twins. I'm not entirely sure whether it's fully equivalent to engine thrust, but imagine that the twin that traveled used gravity of a massive object, like a black hole, to turn around. That way, his trajectory is also a geodesic. It is also locally shortest. However, the question of which path is shorter remains.

We know the answer, of course. Special Relativity tells us that the twin that stayed put aged more. That means, his path was longer.
PeterDonis
#21
Dec9-12, 01:46 PM
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Quote Quote by K^2 View Post
The fact is, twin that traveled has actually taken a shorter path. How? Geodesics are local minima.
No, they are local *maxima*. But in flat spacetime, which is what seems to be the assumption in this thread, a local maximum is also necessarily a global maximum. The possibility of multiple extremal paths, which you refer to later on in your post, only exists in curved spacetime.

The reason the traveling twin's path is shorter is the Lorentzian analog of the triangle inequality: two sides of a triangle are *shorter* than the third, if all three sides are geodesic segments. Physically, this corresponds to the fact that the motion of the traveling twin, who travels along two sides of the triangle, can't be geodesic everywhere; there must be at least one event where he undergoes non-geodesic motion. Only the stay-at-home twin, who moves along only one side of the triangle, can have a path that is geodesic everywhere, so his path is necessarily the longest.

Quote Quote by K^2 View Post
imagine that the twin that traveled used gravity of a massive object, like a black hole, to turn around. That way, his trajectory is also a geodesic. It is also locally shortest. However, the question of which path is shorter remains.

We know the answer, of course. Special Relativity tells us that the twin that stayed put aged more. That means, his path was longer.
Only if you assume that the gravity of the massive object is negligible throughout most of the spacetime; in other words, that the spacetime is flat everywhere except for a very small region around the massive object. Then in that flat spacetime, the path of the traveling twin is *not* a geodesic, which is why SR says the stay at home twin ages more. The massive object in this scenario just takes the place of the rocket or whatever it is that turns the traveling twin around; you're not actually making use of the curvature of spacetime around it except as a "rocket substitute".

If, OTOH, you want to have a scenario where the curvature of spacetime around the massive object is the primary effect, it is easy to construct one where the "stay at home" twin ages less. Just have the stay at home twin be in orbit about the massive object, and have the traveling twin be launched directly upward in such a way that he is pulled back down by the massive object's gravity to meet the stay at home twin after some integral number of orbits. Here both twins are traveling along geodesics, so it is clearer what the difference is between a local maximum and a global one.
PeterDonis
#22
Dec9-12, 01:59 PM
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Quote Quote by Darwin123 View Post
The problem is that there is no physical hypothesis that distinguishes between trajectories that are geodesics and trajectories that are not geodesics. One can always find a set of variables in which an observer is traveling a geodesic.
No, you can't. The path of the traveling twin cannot be geodesic everywhere. (At least, not in the standard version; as K^2 pointed out, and I amplified in my response to him, in curved spacetimes one can set up scenarios where there are multiple geodesic paths between the same pair of events, with different lengths. But in flat spacetime, meaning wherever gravity is negligible, there is only one geodesic path between any pair of events, and it is the path of maximal proper time.)

Physically, what "the path of the traveling twin cannot be geodesic everywhere" means is that the traveling twin has to turn around somehow. If spacetime is flat (i.e., gravity is negligible), the only way he can do that is by firing rockets, getting pushed by a laser, etc.--i.e., he has to experience a force, even if only for an instant (in the idealized version of the scenario that you've been discussing, where the turnaround is instantaneous). This means that an accelerometer carried with the traveling twin will read nonzero, even if only for an instant. The stay at home twin's accelerometer will read zero the entire time.

Quote Quote by Darwin123 View Post
"Physical intuition" is not sufficient for distinguishing between a geodesic and a nongeodesic.
No, but a direct physical measurement made with an accelerometer is. See above.
PeterDonis
#23
Dec9-12, 02:05 PM
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Quote Quote by Ibix View Post
with no acceleration involved.
This is not strictly true. Something has to change direction, and that requires acceleration. In your version, the "something" is information; nothing physical actual changes direction in your scenario. But you are describing the traveling worldline (burrito + cash) in two different inertial frames; the inertial frame in which the cash is at rest outbound is different from the inertial frame in which the burrito is at rest inbound. That is equivalent to having an acceleration somewhere; "acceleration" may not be the best word, but we don't really have another term for "switching inertial frames midstream".
Ibix
#24
Dec9-12, 04:09 PM
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Quote Quote by Darwin123 View Post
Many books on science for laymen say straight out that it is the rockets that break the symmetry.
You are reading too much into this statement, I think. The setup usually assigns specific tasks to each twin (sit on Earth, fly to Alpha C), for easy identification. But if you strip that away by putting both twins in rocket ships, performing symmetric accelerations so that they are separating at speed v, and then leaving them to coast for a while then there is nothing to identify one twin as the stay-at-home and the other as the traveller. You can only tell when one of the twins (chosen by a private agreement between the twins before the experiment begins) fires his engines to turn round and catch up with his sibling.

In that very limited sense, the symmetry is broken by the rockets firing, because that is when everyone can work out who will be the younger twin. But, physically, there is nothing special about the rockets firing. They don't "cause the other twin to age". They just enable the twins to meet up again, which they can't in inertial frames.

The analogy that I made earlier, of driving in a straight line from A-to-B compared to driving A-to-C-to-B is perfectly valid. It isn't the act of turning at C that makes the path lengths different, it is the decision to take a different route through space to the same destination. And if you (a third-party) are ignorant of the destination, you can't tell until one car turns which one is going to, so you cannot predict which will have the higher mileage.

That's exactly what ghwellsjr's spacetime diagrams are showing you. If you hide the top half of them, the first two graphs just look like mirror images of each other. It's only at the point that the rocket fires and one twin turns around that they start to look like different graphs.
Darwin123
#25
Dec9-12, 04:10 PM
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Quote Quote by PeterDonis View Post
Physically, what "the path of the traveling twin cannot be geodesic everywhere" means is that the traveling twin has to turn around somehow. If spacetime is flat (i.e., gravity is negligible), the only way he can do that is by firing rockets, getting pushed by a laser, etc.--i.e., he has to experience a force, even if only for an instant (in the idealized version of the scenario that you've been discussing, where the turnaround is instantaneous). This means that an accelerometer carried with the traveling twin will read nonzero, even if only for an instant. The stay at home twin's accelerometer will read zero the entire time.



No, but a direct physical measurement made with an accelerometer is. See above.
I agree with the sentences after the word, "Physically". The path of the traveling twin can not be a geodesic everywhere. However, geodesic is a mathematical term which isn't very useful to the experimenter or engineer. In terms of kinematics (i.e., without physical interaction), there is no reason that the twin on earth has to one a geodesic everywhere. What I meant by "physical" is a local measurement that can determine where the path is not geodesic.

Your comment about the accelerometer is very pertinent. I believe that you have proven my point. What breaks the symmetry is a type of acceleration, just as I said. However, it can't be any type of acceleration. A kinematic acceleration, meaning an acceleration not associated with an interaction, can't break the symmetry.

The physical quantity that determines where the path is not geodesic is whatever the accelerometer is measuring. I submit that the accelerometer is measuring the total force on the accelerometer applied by the surroundings.

The accelerometer is measuring the external force on it by other particles. The observer who is close to the accelerometer and stationary relative to the accelerometer is also under the influence of a force. There may be an accelerometer that is stationary and close to the observer on earth. That accelerometer does not detect an external force.

The quantity that the accelerometer is measuring is what I call the dynamic acceleration. I haven't found a reference that uses that phrase. However, I do see scientists refer to the external force. I think the external force is what breaks the symmetry in the twin paradox. The external force usually stated in the twin paradox is caused by the rockets.
Ibix
#26
Dec9-12, 04:24 PM
P: 378
Quote Quote by PeterDonis View Post
This is not strictly true. Something has to change direction, and that requires acceleration. In your version, the "something" is information; nothing physical actual changes direction in your scenario. But you are describing the traveling worldline (burrito + cash) in two different inertial frames; the inertial frame in which the cash is at rest outbound is different from the inertial frame in which the burrito is at rest inbound. That is equivalent to having an acceleration somewhere; "acceleration" may not be the best word, but we don't really have another term for "switching inertial frames midstream".
Agreed - especially about acceleration not being quite the right word. But if there are no rockets firing, it's not the rockets that are causing the difference, which Darwin123 was claiming and I was refuting. And it's not the transmission of the information that causes the difference either - it's the choosing to follow a different route to get to the same point that results in the journeys having different lengths (or different intervals, in SR language).
Darwin123
#27
Dec9-12, 06:05 PM
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Quote Quote by Ibix View Post
Agreed - especially about acceleration not being quite the right word. But if there are no rockets firing, it's not the rockets that are causing the difference, which Darwin123 was claiming and I was refuting. And it's not the transmission of the information that causes the difference either - it's the choosing to follow a different route to get to the same point that results in the journeys having different lengths (or different intervals, in SR language).
The word "choosing" implies that there is someone doing the choosing. Who is doing the choosing?

Maybe I was saying it wrong. I started out using the word acceleration. Maybe this isn't the right word either.

Whatever physical interaction defines the path of the hypothetical observer is what breaks the symmetry. It doesn't have to be a canonical force. In the case of quantum systems, there may not be a canonical force. Maybe it is a measurement, or a decoherence, or some other type of interaction.

I was not claiming that the rockets cause the aging. I was claiming that the rockets caused the difference in ages. The question was what caused the asymmetry. I claim that the force of the rockets cause the asymmetry.

What physical thing do you think is the cause of the asymmetry? A space time diagram by itself is not enough. If you must present a diagram, circle the part of the diagram which has the physical cause of the asymmetry.

The concrete example in the OP's question involved twins, rockets, and aging. Later it turned in burritos, sidewalks and deliveries. The question was originally about the cause of asymmetry, not about twins or burritos.
ghwellsjr
#28
Dec10-12, 03:39 AM
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Quote Quote by Darwin123 View Post
...
I was not claiming that the rockets cause the aging. I was claiming that the rockets caused the difference in ages. The question was what caused the asymmetry. I claim that the force of the rockets cause the asymmetry.
The rockets cause a difference in the velocity of a spaceship. Depending on the arbitrarily selected IRF, this velocity change can result in an increase, a decrease, or no change in the speed of the spaceship and therefore a decrease, an increase or no change in the aging rate of the traveler. The aging rate is another word for time dilation and is not observable or measurable by either of the persons or anyone else that might be involved in the scenario.

Your concern over asymmetry or a broken symmetry is a red herring. It only seems relevant in the classic Twin Paradox because one twin remains inertial. But in general, we can have any number of observers starting at one location and traveling at different speeds in different directions and ending up in the same location (but not necessarily where they started) where there is no symmetry in the scenario at all and yet we can still analyze the scenario using any arbitrary IRF and determine how each one ages with respect to each other.
Quote Quote by Darwin123 View Post
What physical thing do you think is the cause of the asymmetry? A space time diagram by itself is not enough. If you must present a diagram, circle the part of the diagram which has the physical cause of the asymmetry.
I could present an asymmetrical scenario to you in which the twins end up the same age. What question would you ask in this situation?

A spacetime diagram is simply a way to present a whole lot of data in an easily graspable way. It is part of the discipline of the Theory of Special Relativity.

But you don't need to analyze scenarios like this using Special Relativity. You can do it simply with a Relativistic Doppler Analysis which shows physically what each person actually observes and measures. But you have to discipline yourself and not ask about physical causes beyond what can actually be measured and observed so I doubt that that would be satisfying to you either.
Quote Quote by Darwin123 View Post
The concrete example in the OP's question involved twins, rockets, and aging. Later it turned in burritos, sidewalks and deliveries. The question was originally about the cause of asymmetry, not about twins or burritos.
The OP did not present his question in the context of the Twin Paradox but he did ask about fetching burritos. He did not ask about asymmetry or its cause. He asked an ill formed question which didn't make sense to him (because it didn't make sense to anyone) and was straightened out to his satisfaction by post #7.

Warp was the one that turned this thread into yet another Twin Paradox issue and I corrected him on some false notions and I'm still trying to correct false notions.
bobc2
#29
Dec10-12, 05:53 AM
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Quote Quote by ghwellsjr View Post
...The aging rate is another word for time dilation and is not observable or measurable by either of the persons or anyone else that might be involved in the scenario...
I think this is questionable. One might think that through a little more carefully. You seem to be bringing philosophy into the mix (which I know you typically resent). By your apparent reasoning it would seem that no kind of observation or measurement is possible.
Vandam
#30
Dec10-12, 08:12 AM
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Quote Quote by ghwellsjr View Post
The aging rate is another word for time dilation and is not observable or measurable by either of the persons or anyone else that might be involved in the scenario.

Time dilation not observable? Where do you get this? This is simply wrong.
In another thread I recently tried to explain and show you that time dilation is due to relativity of simultaneity.
http://www.physicsforums.com/showpos...2&postcount=38
The reason you do not understand this is probably because you are too much stuck to your calculator. (And I'm afraid at the end we will have to discuss what time-coordinates are, the meaning of space-like simultaneous events, etc etc .... and this is a no go in this forum. No wonder you battle to grasp the essence of SR...)
ghwellsjr
#31
Dec10-12, 08:32 AM
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Quote Quote by bobc2 View Post
Quote Quote by ghwellsjr View Post
...The aging rate is another word for time dilation and is not observable or measurable by either of the persons or anyone else that might be involved in the scenario...
I think this is questionable. One might think that through a little more carefully. You seem to be bringing philosophy into the mix (which I know you typically resent). By your apparent reasoning it would seem that no kind of observation or measurement is possible.
How can you say that? I also just said:
Quote Quote by ghwellsjr View Post
But you don't need to analyze scenarios like this using Special Relativity. You can do it simply with a Relativistic Doppler Analysis which shows physically what each person actually observes and measures. But you have to discipline yourself and not ask about physical causes beyond what can actually be measured and observed so I doubt that that would be satisfying to you either.
ghwellsjr
#32
Dec10-12, 08:58 AM
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Quote Quote by Vandam View Post
Quote Quote by ghwellsjr View Post
The aging rate is another word for time dilation and is not observable or measurable by either of the persons or anyone else that might be involved in the scenario.
Time dilation not observable? Where do you get this? This is simply wrong.
In another thread I recently tried to explain and show you that time dilation is due to relativity of simultaneity.
http://www.physicsforums.com/showpos...2&postcount=38
The reason you do not understand this is probably because you are too much stuck to your calculator. (And I'm afraid at the end we will have to discuss what time-coordinates are, the meaning of space-like simultaneous events, etc etc .... and this is a no go in this forum. No wonder you battle to grasp the essence of SR...)
And as I pointed out in that other thread, each of the IRF's that I drew in post #9 of this thread shows a different set of time-coordinates (and space coordinates) for each event which is illustrating the relativity of simultaneity. It also shows the relativity of time dilation. Neither of these are observable or measurable by the observers in the scenario, just like the one-way speed of light is not measurable. All three of these things are assigned by the definitions used in Special Relativity and after you assign them, then you can use the definitions and assignments to "read back" the same values you arbitrarily assigned to the events.

The essence of SR is that time is relative to the coordinate system or reference frame that you arbitrarily select. So is space. So is time dilation. So is simultaneity. Select a different coordinate system and all these characteristics change to different values. But what doesn't change are all the measurements and observations that each observer in the scenario makes. Each coordinate system preserves those measurements and observations. Maybe another way to say this is that the measurements and observations made at each event remain the same, even though the coordinates of each event take on different values in each reference frame.
Vandam
#33
Dec10-12, 10:21 AM
P: 126
Quote Quote by ghwellsjr View Post
And as I pointed out in that other thread, each of the IRF's that I drew in post #9 of this thread shows a different set of time-coordinates (and space coordinates) for each event which is illustrating the relativity of simultaneity. It also shows the relativity of time dilation. Neither of these are observable or measurable by the observers in the scenario, just like the one-way speed of light is not measurable. All three of these things are assigned by the definitions used in Special Relativity and after you assign them, then you can use the definitions and assignments to "read back" the same values you arbitrarily assigned to the events.

The essence of SR is that time is relative to the coordinate system or reference frame that you arbitrarily select. So is space. So is time dilation. So is simultaneity. Select a different coordinate system and all these characteristics change to different values. But what doesn't change are all the measurements and observations that each observer in the scenario makes. Each coordinate system preserves those measurements and observations. Maybe another way to say this is that the measurements and observations made at each event remain the same, even though the coordinates of each event take on different values in each reference frame.
Maybe you draw what you observe, measure, data, and I draw in my sketches why you observe and measure what you observe... (but if I understood you elsewhere, you seem not that much interested in what lays at the origin of the observations...?).
Let me put it to you this way. Say you have a forest full of trees... You can give me thousands of different coordinate systems with enless data lists of observations, from all over the place, and all plotted out in a different diagram. But as long as you do not tell me about the forest itself, I do not get it. This is what happens in SR discussions: data list talks. And where's the forest? (I'm glad that at least Bobc2 knows what the forest is in SR... and names it: 4D block universe)
PeterDonis
#34
Dec10-12, 10:35 AM
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Quote Quote by Darwin123 View Post
Igeodesic is a mathematical term which isn't very useful to the experimenter or engineer. In terms of kinematics (i.e., without physical interaction), there is no reason that the twin on earth has to one a geodesic everywhere. What I meant by "physical" is a local measurement that can determine where the path is not geodesic.
I already gave such a local measurement: the reading on an accelerometer. Zero acceleration measured = moving on a geodesic.

Quote Quote by Darwin123 View Post
Your comment about the accelerometer is very pertinent. I believe that you have proven my point. What breaks the symmetry is a type of acceleration, just as I said. However, it can't be any type of acceleration. A kinematic acceleration, meaning an acceleration not associated with an interaction, can't break the symmetry.
A "kinematic acceleration" is not the same as acceleration measured by an accelerometer, so yes, I think I agree.

Quote Quote by Darwin123 View Post
The physical quantity that determines where the path is not geodesic is whatever the accelerometer is measuring. I submit that the accelerometer is measuring the total force on the accelerometer applied by the surroundings.
This is one way of looking at it, yes.

Quote Quote by Darwin123 View Post
The accelerometer is measuring the external force on it by other particles. The observer who is close to the accelerometer and stationary relative to the accelerometer is also under the influence of a force. There may be an accelerometer that is stationary and close to the observer on earth. That accelerometer does not detect an external force.
Huh? Any accelerometer at rest on the surface of the Earth will give a nonzero reading.

Quote Quote by Darwin123 View Post
The quantity that the accelerometer is measuring is what I call the dynamic acceleration. I haven't found a reference that uses that phrase.
The standard term in relativity is "proper acceleration", although many texts just use "acceleration", and that's what they usually mean by it if there is no further qualification.

Quote Quote by Darwin123 View Post
However, I do see scientists refer to the external force. I think the external force is what breaks the symmetry in the twin paradox. The external force usually stated in the twin paradox is caused by the rockets.
In the standard twin paradox, where one twin has to fire rockets to turn around, yes, the force generated by the rockets, which causes the traveling twin to feel acceleration, read a nonzero reading on his accelerometer, etc., is what breaks the symmetry. However, as I noted in other posts, in curved spacetime (unlike flat spacetime, which is what the standard twin paradox is set in), one can have scenarios where both twins are moving on geodesics the whole time, i.e., they never feel any acceleration, but still they age differently. So "external force breaks the symmetry" won't work in all cases.
PeterDonis
#35
Dec10-12, 10:39 AM
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Quote Quote by Ibix View Post
But if there are no rockets firing, it's not the rockets that are causing the difference, which Darwin123 was claiming and I was refuting. And it's not the transmission of the information that causes the difference either - it's the choosing to follow a different route to get to the same point that results in the journeys having different lengths (or different intervals, in SR language).
If there are no rockets firing, then whatever it is that is following the "different route", it can't be the traveling twin, or indeed any single object. The only word I can come up with for whatever it is that does follow the different route is "information". Perhaps that's not the best word, but we have to have some word for whatever it is that picks out the "route in spacetime" whose length is to be evaluated, in cases where no single object follows that route.
Ibix
#36
Dec10-12, 12:30 PM
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Quote Quote by PeterDonis View Post
If there are no rockets firing, then whatever it is that is following the "different route", it can't be the traveling twin, or indeed any single object. The only word I can come up with for whatever it is that does follow the different route is "information". Perhaps that's not the best word, but we have to have some word for whatever it is that picks out the "route in spacetime" whose length is to be evaluated, in cases where no single object follows that route.
I don't disagree with this. I am just trying to argue that it is the whole route that matters, not just the corners. Darwin123 seems to me to be arguing the converse.

Perhaps I need a different example. Consider twins at rest at a space station. They leave together in identical rockets at velocity +v. At time t1, one twin fires his motors and turns round, returning to the space station at velocity -v before braking to a relative stop. The other twin carries on until time t2, when he also turns around and returns at -v before stopping at the space station. Both twins do identical accelerations, but it's easy to show that the difference in ages when they meet up is [itex]\Delta t=2(t_2-t_1)(1-1/\gamma)[/itex], which is zero only if they turn around at the same time or they don't travel at all. So acceleration isn't the only thing that matters. Both the amount of time between accelerations and the accelerations matter.

In the limited context of the classic twin paradox, you only need to know which twin accelerated to determine everything. So in this narrow circumstance, I agree one could argue that acceleration is the key. However, this isn't a useful view in general. In general, you need the complete history of both twins - i.e., their routes through spacetime.

Acceleration (or at least a frame change) is necessary for the worldlines to cross again. But it doesn't cause the age difference, any more than corners cause the triangle inequality. Different paths through a spacetime with a Minkowski geometry causes that.


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