The Paradox of Aging in Relativity: Resolving the Twin Paradox with a Twist

[harrylin]

Thanks for the input. I think distinguishing between inertial frames and others would bother anybody who is really thinking about it. I'm only just starting to read about general relativity, but hopefully the answers are there as you say.

While Fastchecker's answer was historically correct, regretfully GR does not give the answers that Einstein hoped for. In particular, in the link in #27 I briefly explained that Einstein's solution does not really work (it only works superficially), contrary to what Einstein claimed.

Compare also http://mathpages.com/home/kmath588/kmath588.htm :
The author seems to argue that inertia was built into the equations, so that GR doesn't really explain inertia.

 

[doaaron]

regretfully GR does not give the answers that Einstein hoped for

That's interesting. I wasn't aware of that. Is that an agreed upon fact in mainstream physics? I haven't yet finished my reading up on GR, but I'm wondering, is the mathematical result of GR similar to the "naive" approach for calculating the traveling twin's "rest" frame, and is that why it results in the problems that others have alluded to?

btw, I'm happy to hear any opinion whether it turns out to be right or wrong. thanks,
Aaron

 

[harrylin]

That's interesting. I wasn't aware of that. Is that an agreed upon fact in mainstream physics? I haven't yet finished my reading up on GR, but I'm wondering, is the mathematical result of GR similar to the "naive" approach for calculating the traveling twin's "rest" frame, and is that why it results in the problems that others have alluded to?

The mainstream opinion is roughly as depicted by Baez in the Physics FAQ link that I provided in #27.
Regretfully -IMHO- Baez didn't do a good job of sketching Einstein's original GR, while FactChecker gave a better sketch of that here.

And I don't know what "naive" approach you mean; but for sure Einstein's approach was not naive. And as I explained in my linked post, it even seemed to give the right answers as long as one did not look at it critically.

 

[doaaron]

And I don't know what "naive" approach you mean

I was just being lazy. In an earlier reply (I guess you missed it), DaleSpam introduced the wording to describe one possible approach to calculating what the traveling twin measures in his rest frame during acceleration/deceleration. This "naive" approach involves applying Lorentz transforms based on the traveller's instantaneous velocity for each part of his acceleration/deceleration. DaleSpam further pointed out that this approach yields some problems, and that there are many other ways (an infinite number of ways?) to do the calculation. I assume the word naive was used because its the first method any reasonable person would attempt before realising that it has problems.

I'm actually still confused about what is meant here. It seems that what people are saying is that:

1) in the traveling twin's "rest" frame, there is only one possibility for what he sees. If he has a clock with him, then for each tick of his clock, he can note down what he sees as the position of the stay at home twin.

2) in order for him to calculate the actual position of the stay at home twin, he needs to take into account the time it took for the light of the stay at home twin to reach him. The result of his calculation is what he measures.

3) Unfortunately, there is no consensus in the scientific community for the best way to go from what he sees to what he measures while he is accelerating/decelerating. Apparently, there are an infinite number of ways in which the calculation could be performed, and none are perfect. Essentially, this seems to imply that there is no consensus on how to define space-time coordinates in a non-inertial frame of reference.

have I got that right?best regards,
Aaron

 

[ghwellsjr]

I was just being lazy. In an earlier reply (I guess you missed it), DaleSpam introduced the wording to describe one possible approach to calculating what the traveling twin measures in his rest frame during acceleration/deceleration. This "naive" approach involves applying Lorentz transforms based on the traveller's instantaneous velocity for each part of his acceleration/deceleration. DaleSpam further pointed out that this approach yields some problems, and that there are many other ways (an infinite number of ways?) to do the calculation. I assume the word naive was used because its the first method any reasonable person would attempt before realising that it has problems.

I'm actually still confused about what is meant here. It seems that what people are saying is that:

1) in the traveling twin's "rest" frame, there is only one possibility for what he sees. If he has a clock with him, then for each tick of his clock, he can note down what he sees as the position of the stay at home twin.

2) in order for him to calculate the actual position of the stay at home twin, he needs to take into account the time it took for the light of the stay at home twin to reach him. The result of his calculation is what he measures.

3) Unfortunately, there is no consensus in the scientific community for the best way to go from what he sees to what he measures while he is accelerating/decelerating. Apparently, there are an infinite number of ways in which the calculation could be performed, and none are perfect. Essentially, this seems to imply that there is no consensus on how to define space-time coordinates in a non-inertial frame of reference.

have I got that right?best regards,
Aaron

You're getting close. The thing you have to realize is that the position of the remote twin is not something that can be measured as if there is only one answer. It all depends on your definition of distance of which there are an infinite number and variety.

DaleSpam pointed you to a paper that describes using radar techniques. It's really very simple. This is basically how a laser range finder works. You send out a signal to the remote object at a measured time according to your clock and wait for the reflection or echo to get back to you for a second measurement of time. Then you assume that the signal travels at c in both directions and from this you can establish the distance away the object was at the average of those two times. Please note that this is based on the assumption (Einstein's second postulate) that the radar signal took the same amount of time to get to the object as it took for the echo to get back. The observer does this over and over again and from this he can build a non-inertial frame showing the distance away from him of the remote object as a function of his own time. It's really very simple.

If the observer also views the time on the remote object's clock, he can establish how the remote clock varies in time with respect to his own clock according to his assumptions.

Does that make sense to you?

 

[doaaron]

Hi ghwellsjr,thanks for the information. I got the gist of the idea in the paper DaleSpam sent, but what I really want to confirm is which of these two are correct,

a) There are a number (infinite?) of valid ways for defining how the traveling twin measures the remote twin's coordinates during acceleration

OR

b) There is no consensus on which is the correct way for the traveling twin to measure the remote twin's coordinates during acceleration because each one has its own problems.

I guess the options are not really exclusive of each other...thanks,
Aaron

 

[ghwellsjr]

Hi ghwellsjr,thanks for the information. I got the gist of the idea in the paper DaleSpam sent, but what I really want to confirm is which of these two are correct,

a) There are a number (infinite?) of valid ways for defining how the traveling twin measures the remote twin's coordinates during acceleration

What makes a method invalid is if it gives more than one answer for the remote twin's time. In other words, in a valid method, you can create the coordinate chart for the first twin looking at the second twin and then from that chart you can go back and create the second twin's chart for looking at the first twin. If the first chart has multiple times for the second twin's clock (repeated times) then it will be impossible to use that information for the second twin to create any chart because you can't tell which of his times include data on his chart.

OR

b) There is no consensus on which is the correct way for the traveling twin to measure the remote twin's coordinates during acceleration because each one has its own problems.

I guess the options are not really exclusive of each other...thanks,
Aaron

No, it's not because each one has its own problems, it's because there's no "correct" way. There are self-consistent ways and valid ways but two different ways can both be equally "correct".

Another point I want to clarify. In both your options you use the phrase "during acceleration" as if there is consensus and no problems up to when the acceleration starts and after the acceleration ends. If you work through an example using the radar method, you will see that there is consensus up to a point long before the acceleration starts and long after the acceleration ends. It's a fairly long period of time surrounding the acceleration where the methods deviate. That period of time is related to how far away the remote object is.

The radar method also works for an inertial observer and creates the same chart that we get through the normal inertial methods of Special Relativity. For an observer that starts out inertial but then has a relatively short period of acceleration, the normal inertial methods of Special Relativity produce the same kind of chart until he approaches his time of acceleration, again depending on how far away the remote object is.

 

[Ibix]

What makes a method invalid is if it gives more than one answer for the remote twin's time.

Pedantically, am I right to think that should be "if it gives more than one answer for the time of any event in space time"?

 

[Fredrik]

Pedantically, am I right to think that should be "if it gives more than one answer for the time of any event in space time"?

Events are assigned different time coordinates by different coordinate systems. It should be "...more than one result for the proper time of any timelike curve in spacetime".

 

[ghwellsjr]

What makes a method invalid is if it gives more than one answer for the remote twin's time.

Pedantically, am I right to think that should be "if it gives more than one answer for the time of any event in space time"?

I didn't say that very well, thanks for pointing that out. What I meant to say is that as the remote twin's Coordinate Time is advancing, the Proper Time on his clock must also advance. If the Proper Time goes backwards and then forwards again, the same Proper Time will occur along the remote twin's worldline more than once. Then it would be impossible to use the chart that was constructed by that method to create other charts.

Another way of saying this is that all the information that is contained in one chart should be contained in any other chart and there should be a method to go from one chart to any other chart.

(I probably still am not saying this very well.)

 

[Ibix]

Thanks, Fredrik and George. I feel like there's something I'm not quite getting, though. I can see the problem with the naive approach of stitching together two sets of standard SR inertial coordinate systems. The issue George is explaining is (I think) easiest to see by adding a second traveller who goes out and back further and faster than our existing twin and returns home at the same time (so the space-time diagram in the stay-at-home frame is a vertical line with two isosceles triangles on it, sharing a base and pointing the same way). According to the naive approach, just before turnaround the slower twin says that the faster twin has already turned around, but just afterwards says that the faster twin has yet to turn around. This is a problem because it means there is no invertible relationship between the fast twin's proper and co-ordinate times.

I think this is a restatement of the idea that coordinate charts can only be combined if any overlap is "smooth". Presumably this isn't a smooth overlap, but I'm not grasping what makes it non-smooth. It isn't that the coordinates aren't equal in the overlap region, as Fredrik pointed out. Is it that the coordinate basis vectors aren't parallel in the overlap region? Or am I conflating unrelated concepts?

 

[Dale]

Isn't acceleration, velocity, and position all relative to something else?

No. Velocity is relative, but acceleration is invariant. Note, there are two distinct concepts of acceleration in relativity. One is called proper acceleration, and it is the acceleration measured by an accelerometer. The other is coordinate acceleration, which is relative to a given coordinate system. However, regardless of anything else, proper acceleration is well defined and invariant.

In other words, I don't see how to explain accelerometer readings other than by saying that an object is accelerating relative to the universe.

The laws of physics do not distinguish between different reference frames moving with different velocities, but the laws of physics do distinguish between different reference frames moving with different accelerations. So when you say "moving with velocity v" you have to specify the reference frame, but when you say "moving with (proper) acceleration a" you do not need to specify a frame.

 

[pervect]

OK, in the diagram below, A is a spacetime diagram of the coordinate system of a stationary observer with the axis (t,x) drawn, and B is the space-time diagram of the coordinate system of an observer with the axis (t', x') drawn. Hopefully this is familiar.

[add]I should explain the diagram anyway. Time, t, runs up the page, as usual. There is one spatial dimension,x, that runs left and right. A curve of constant t is horizontal for a stationary observer as in diagram A, and almost horiziontal for a moving observer as shown in B.

If not try http://www.sparknotes.com/physics/specialrelativity/kinematics/section3.rhtml , it might not be the best reference but they had the diagram I want you to compare to B in the drawing below, labelled Figure %: Minkowski or spacetime diagram.

Now we combine A and B into C, the "patched together" coordinate system. We note that point P has two different time coordinates. This is the problem, points are supposed to have unique coordinates.

If we draw the set of points at some time T , which we can call the "time axis", and which are the horizontal or near-horizontal lines in the diagrams above, then a point must have only one time, it's not allowed for it to be assigned two different times. But we can see that point P does lie on two different time-axis, it has two different times associated with it.

 

[Dale]

1) in the traveling twin's "rest" frame, there is only one possibility for what he sees. If he has a clock with him, then for each tick of his clock, he can note down what he sees as the position of the stay at home twin.

More than that. There is only one possibility for what he sees in any frame, whether it is his or someone else's, whether it is inertial or not. In other words, for any frame to be physically valid it must reproduce what any covered observer physically sees. The explanations may differ, but the end result must be invariant.

The rest of what you say seems on target. You might consider the word "calculate" or "infer" instead of "measure", but your meaning is clear given how you defined "measure" earlier

 

[FactChecker]

No. Velocity is relative, but acceleration is invariant.

Mathematically, the derivative of a relative function is relative. We need to make a distinction between acceleration and the derivative of velocity. There is a physical reason why acceleration can be felt and measured as an invariant whereas the derivative of velocity is not invariant. For that, we are saying that acceleration is measured in the space and metric tensor defined by the universe. The Twins "Paradox" attempts to make the twins appear symmetric. And the derivative of relative velocity between the twins is symmetric. That makes the paradox deceptive. It is the physics of one twin accelerating in the universe and the other not that makes the twins asymmetric.

 

[Ibix]

Now we combine A and B into C, the "patched together" coordinate system. We note that point P has two different time coordinates. This is the problem, points are supposed to have unique coordinates.

If we draw the set of points at some time T , which we can call the "time axis", and which are the horizontal or near-horizontal lines in the diagrams above, then a point must have only one time, it's not allowed for it to be assigned two different times. But we can see that point P does lie on two different time-axis, it has two different times associated with it.

What's confusing me is that different coordinate values for the same point aren't always a problem. For example, you can't cover S² with one coordinate chart. The textbook solution is something like stereographic projection, where you define two charts each excluding one point and overlapping everywhere else. Obviously the co-ordinates aren't equal in the overlap region - so why is the fact that the coordinates aren't equal in the overlap bad in this twin paradox case? I can see that it is bad (there exist timelike paths that cannot be parameterised by their proper time, as Fredrik noted in #44), but is there a way to tell from the definition of the charts that it's bad?

Perhaps I should start a new thread, as I'm not sure this is entirely on topic.

 

[Fredrik]

(there exist timelike paths that cannot be parameterised by their proper time, as Fredrik noted in #44),

That's not what I noted. Every timelike curve can be parametrized by proper time. I was just saying that different coordinate systems assign different coordinates to events (I didn't mean to suggest that this is a problem), and that we would have a real problem (the theory would be nonsense) if there had been two valid ways to calculate a coordinate-independent quantity like the proper time of a timelike curve, and those ways yield different results.

 

[PhoebeLasa]

If, for a given observer, the current age of a distant object is arbitrary, then it is a meaningless concept. If it is a meaningless concept, then we shouldn't even be talking about the basic time dilation result at all (that a distant clock moving wrt us is ticking slower than our own clocks).

 

[ghwellsjr]

If, for a given observer, the current age of a distant object is arbitrary, then it is a meaningless concept. If it is a meaningless concept, then we shouldn't even be talking about the basic time dilation result at all (that a distant clock moving wrt us is ticking slower than our own clocks).

No, meanings come from definitions. Different definitions produce different meanings. Each different reference frame is a different definition for Coordinate Time and that's why they produce different a "current age" for each different reference frame. What's the problem?

 

[PeterDonis]

There is a physical reason why acceleration can be felt and measured as an invariant whereas the derivative of velocity is not invariant. For that, we are saying that acceleration is measured in the space and metric tensor defined by the universe.

You may be saying something true here, but I'm not sure. The true statement is that the metric determines what states of motion are freely falling and what states of motion are not freely falling, i.e., accelerated. But we don't measure acceleration by measuring the metric; we measure it with an accelerometer. That is, we don't have to determine acceleration indirectly (by measuring the metric and then calculating something about our state of motion); we can determine it directly. The acceleration we directly measure must be consistent with other measurements we make that tell us about the metric, of course.

 

[Evanish]

I was thinking about this scenario a while back.

I never came up with a solution I really liked so I kept thinking about it.

Now I'm thinking that maybe the distance between the two observers changes in such a way that it makes up for the time difference.
https://lh5.googleusercontent.com/1TJ6R_4zbJIYyHSrPRL3SyEgfOGRDSHjGWkADjKAqIvYYyKgpWdZHzsA6-ZhwtDVDuV2Qg98mFNlOcilqEjld211clA1oTTzkiVydwNA4x9d898GmUw
For simplicity's sake let's assume there is another person at the start who is observing all this. We can call him Mike. When Fred is moving away from Mike at a constant velocity the distance Mike observes him at is increasing in such a way that events between the two reference frames match up. Basically space is lengthening.

Picture that the three people have a device that produces a radio pulse, senses the radio pulse of the others and measures the distance they are away from each other. Mike would observe that Fred's time is slower, and if he used the speed of light plus measured distance between himself and Fred to calculate when the two pulses happened in relationship to each other he would find that the two pulses happened simultaneously. This is because the distance between them would have lengthened by the amount necessary to make it happen.

I wish I was better at math so I could describe this more clearly. At any rate interesting stuff.

 

[PeterDonis]

Now I'm thinking that maybe the distance between the two observers changes in such a way that it makes up for the time difference.

"Distance" is frame-dependent, so in a sense it does change depending on which observer's frame you use. But you seem to be leaving out relativity of simultaneity; you can't do a proper analysis without including that.

Mike would observe that Fred's time is slower, and if he used the speed of light plus measured distance between himself and Fred to calculate when the two pulses happened in relationship to each other he would find that the two pulses happened simultaneously.

If the two pulses were emitted simultaneously according to Mike, then they can't have been emitted simultaneously according to Fred, because simultaneity is relative, i.e., frame-dependent.

My advice: draw a spacetime diagram. First draw it in the frame of the observer who remains at rest at the starting point. Then transform to Mike's and Fred's frames (note that there are two for each of them, one for the constant velocity segment outbound and one for the constant velocity segment returning) to see how things look there.

 

[ghwellsjr]

My advice: draw a spacetime diagram. First draw it in the frame of the observer who remains at rest at the starting point. Then transform to Mike's and Fred's frames (note that there are two for each of them, one for the constant velocity segment outbound and one for the constant velocity segment returning) to see how things look there.

I already drew these diagrams in post #6 except that I made the accelerations instantaneous. There's no point in making things more complicated by spreading the accelerations out over time.

Note that there are actually just three diagrams but it is Mike that remains at rest (he gets one frame) and there are two more frames, since Bob's starting frame is the same as Fred's ending frame and vice versa.

You can copy my diagrams and draw in the paths of the radio signals in all three diagrams and see that they start and end at the same Proper Times for all observers no matter which frame you use.

If you want to see how a particular observer establishes the distance away of the other observers, just use a radio signal emitted by one observer that echoes off another observer and have the first observer use the time interval between sending and receiving as the two significant points of measurement. By assuming that the radio signals took the same amount of time to get to the other observer as it took for the return signal to get back and by assuming that the signals travel at c, the first observer can establish a time and distance to the other observer. Repeating these measurements over and over again will allow each observer to create a chart for each of their reference frames, even for the non-inertial observers. The concept is extraordinarily simple but it is tedious. Try it, you'll like it.

 

[PeterDonis]

I already drew these diagrams in post #6

Yes, I see you did. I should have expected that.

 

[Evanish]

Thanks ghwellsjr. I'll try to do what you suggested.

 

[Dale]

Mathematically, the derivative of a relative function is relative.

That is simply not correct. Whether or not something is relative is a question of the laws of physics, not merely whether it is a derivative of a relative function.

In fact, the derivative of a relative function can be invariant. This is easiest to see in Newtonian physics where the usual formulation of Newton's 2nd law is clearly invariant under translations and Galilean boosts, but not under accelerations. The velocity is relative but the derivative of velocity is not.

 

[Dale]

I think this is a restatement of the idea that coordinate charts can only be combined if any overlap is "smooth". Presumably this isn't a smooth overlap, but I'm not grasping what makes it non-smooth.

Think about the surface of the Earth (just the surface, so it is a 2D curved space). Now, even without any coordinates defined you can still do a lot of things. You can talk about the length of a road, or the angle of intersection between two roads. You can name points and give directions by reference to named points (on Main street, half a mile past Bob's Hardware).

Now, if you want, you can map physical points on the surface of the Earth to mathematical points in R2. This mapping between points on the Earth and points in R2 is called a chart, and the result can be put on a piece of paper (a cartographer's map).

There are many different ways to do that mathematical mapping (cartographers call them projections) and all are equally valid. Some preserve distances, some preserve angles, some are just easy to draw or read, and so forth.

The charts all have three important features
1. They map from open subsets of the surface of the Earth to open subsets of R2
2. They are smooth, meaning that nearby points on the Earth are also nearby in R2
3. They are invertible, meaning that for every point in the open subset of R2 there is one and only one point in the open subset of the surface

So the actual problem is not non smoothness (feature 2), it is non invertibility (feature 3).

 

[Ibix]

The charts all have three important features
1. They map from open subsets of the surface of the Earth to open subsets of R2
2. They are smooth, meaning that nearby points on the Earth are also nearby in R2
3. They are invertible, meaning that for every point in the open subset of R2 there is one and only one point in the open subset of the surface

So the actual problem is not non smoothness (feature 2), it is non invertibility (feature 3).

Got it. Somehow I'd missed the invertibility requirement, even though (on a re-read) that's what George was getting at back in #42. And it's kind of an obvious requirement for a chart, really. Now you've pointed it out, anyway.

Thank you.

 

[FactChecker]

FactChecker said: ↑
Mathematically, the derivative of a relative function is relative.

That is simply not correct. Whether or not something is relative is a question of the laws of physics, not merely whether it is a derivative of a relative function.

Please explain that. For instance: If the velocities from Twins 1 and 2's perspective always numerically satisfy v₁ = -v₂ , then why wouldn't we always have the numerical identity d/dt ( v₁ ) = d/dt ( -v₂ ) = -d/dt ( v₂ )?

 

[Dale]

Please explain that. For instance: If the velocities from Twins 1 and 2's perspective always numerically satisfy v₁ = -v₂ , then why wouldn't we always have the numerical identity d/dt ( v₁ ) = d/dt ( -v₂ ) = -d/dt ( v₂ )?

Yes, clearly. But if the velocities always satisfy d/dt ( v₁ ) = -d/dt ( v₂ ) then that does not imply that v₁ = -v₂. The question is, which is a valid law of physics. In relativistic physics neither would be a valid law of physics, but in Newtonian physics you could have a scenario where d/dt ( v₁ ) = -d/dt ( v₂ ) derived from a valid law of physics, e.g. Hooke's law or Newton's law of gravitation.

If you built a scenario using Hooke's law and Newton's laws such that d/dt ( v₁ ) = -d/dt ( v₂ ) and you transformed it to a different frame where Hooke's law and Newton's laws were still satisfied then you would find that d/dt ( v₁ )' = -d/dt ( v₂ )' . If you built a scenario using Hooke's law and Newton's laws such that v₁ = -v₂ and you transformed it to a different frame where Hooke's law and Newton's laws were still satisfied then you would find that v₁' ≠ -v₂'. Veocities are relative, accelerations are invariant.

 

[Ibix]

Got it. Somehow I'd missed the invertibility requirement

The naive solution to a reference frame for the traveling twin doesn't work because the map is not invertible - there is a patch of space-time that is covered by both coordinate systems and a patch covered by neither. It seems like a naive solution to that would be to split the space-time diagram along some line through the turnaround event, such as the stay-at-home twin's "now" plane through that event, and use one set of coordinates before the line and one set after.

I think that's invertible, but fails as a coordinate chart due to not being open - you cannot have an open ball anywhere along the boundary.

Is that right?

You need to go with something more sophisticated, such as the Dolby and Gull paper linked upthread.

 

[Dale]

The naive solution to a reference frame for the traveling twin doesn't work because the map is not invertible - there is a patch of space-time that is covered by both coordinate systems and a patch covered by neither.

Exactly.

It seems like a naive solution to that would be to split the space-time diagram along some line through the turnaround event, such as the stay-at-home twin's "now" plane through that event, and use one set of coordinates before the line and one set after.

Yes, you can make that kind of a split as long as you are careful and clear about it.

I think that's invertible, but fails as a coordinate chart due to not being open - you cannot have an open ball anywhere along the boundary.

Is that right?

Yes, that is true, but usually you can get around that kind of problem by careful specifications of the boundaries (i.e. use only strict inequalities)

 

[FactChecker]

Yes, clearly. But if the velocities always satisfy d/dt ( v₁ ) = -d/dt ( v₂ ) then that does not imply that v₁ = -v₂. The question is, which is a valid law of physics. In relativistic physics neither would be a valid law of physics, but in Newtonian physics you could have a scenario where d/dt ( v₁ ) = -d/dt ( v₂ ) derived from a valid law of physics, e.g. Hooke's law or Newton's law of gravitation.

If you built a scenario using Hooke's law and Newton's laws such that d/dt ( v₁ ) = -d/dt ( v₂ ) and you transformed it to a different frame where Hooke's law and Newton's laws were still satisfied then you would find that d/dt ( v₁ )' = -d/dt ( v₂ )' . If you built a scenario using Hooke's law and Newton's laws such that v₁ = -v₂ and you transformed it to a different frame where Hooke's law and Newton's laws were still satisfied then you would find that v₁' ≠ -v₂'. Veocities are relative, accelerations are invariant.

I am just saying this:

Velocity of Twin 2 in Twin 1 reference frame == - Velocity of Twin 1 in Twin 2 reference frame
d/dt ( Velocity of Twin 2 in Twin 1 reference frame ) = d/dt ( - Velocity of Twin 1 in Twin 2 reference frame ) = - d/dt( Velocity of Twin 1 in Twin 2 reference frame )
The derivative of Twin 2 velocity in Twin 1 reference frame == - The derivative of Twin 1 velocity in Twin 2 reference frame.
So the derivative of velocity is just as relative as the velocity itself is.

I think this is the basic reason that people see the Twins' situations as symmetric and thus the Twins "Paradox". To address the difference between the derivative and acceleration, one should point out that the Universe provides the space that acceleration is defined wrt. The preference for inertial reference frames to define acceleration bothered Einstein greatly and motivated his work on GR. He stated that in no uncertain terms.

 

[Dale]

So the derivative of velocity is just as relative as the velocity itself is.

No, it isn't, as explained above.

Your justification is completely irrelevant because it does not address what makes something "relative". It is like saying that an apple is just as sweet as a cherry because they are the same color.

How do you know that velocity is relative? You cannot use the justification that velocity is the 0th derivative of velocity. You also cannot use the justification that if v=-v then v is just as relative as -v. So what is it about velocity that makes it relative?

 

[pervect]

What's confusing me is that different coordinate values for the same point aren't always a problem. For example, you can't cover S² with one coordinate chart. The textbook solution is something like stereographic projection, where you define two charts each excluding one point and overlapping everywhere else. Obviously the co-ordinates aren't equal in the overlap region - so why is the fact that the coordinates aren't equal in the overlap bad in this twin paradox case? I can see that it is bad (there exist timelike paths that cannot be parameterised by their proper time, as Fredrik noted in #44), but is there a way to tell from the definition of the charts that it's bad?

Perhaps I should start a new thread, as I'm not sure this is entirely on topic.

It really depends on how carefully you think as to what sort of trouble you get yourself into. I don't think you'll find much guidance from textbooks on how to deal with situations where you assign multiple coordinate labels to the same points. I'm not aware of any textbooks or papers that cover this issue, which means you are sort of on your own if you go this route. (Or find some papers to talk about it, perhaps0. The fact that you have multiple charts in a manifold (which I think is what you're referring to) isn't really the same thing as giving a point multiple coordinates.

Let's go back to high school geometry for a second. There are two general approaches to it. One is the coordinate-free approach, based on Euclid''s axioms, such as "a straight line can be drawn between any two points". You don't need to use coordinates in this approach. The contrasting approach is one of analytic geometry, where you assign coordinates to every points, and use algebra to solve your geometry problems.

In the later case, you often implicitly assume that there is a 1:1 correspondence between your numbers, and the geometry. If you remove this requirement, and aren't careful, your analytic approach won't give the same answer to the geometrical questions you ask that the coordinate-free approach gives.

But it should - mathematically, the coordinate free approach and the coordinate based approach are supposed to represent the same underlying concept.

So, if you are able to do the coordinate-free approach, and the coordinate-based approach, you have at least a chance of spotting any errors you might make by the non-standard approach of assuming you can assign points multiple coordinates.

If you are totally relying on the analytic approach to geometry (I've seen PF posters do this, they seem unable to grasp the idea that one can do geometry without coordinates), you can easily confuse yourself into incorrect conclusions when you relax the rule that every point must have unique coordinates. You can't rely on the uniqueness and existence theorems of your algebraic problems to say anything about the uniqueness and existence of the geometrical problems. You can also (and I've seen this happen on PF) confuse yourself when you have a situation where you don't assign coordinates to every point, because you have some points that you don't assign coordinates to.

I think the cleanest thing to say about accelerated observers is this: "The" coordinate system of an accelerated observer exists locally, but it doesn't cover all of space-time, only a partial region of it.