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

[FactChecker]

This is not correct and not consistent with mainstream physics. Regardless of the rest of the universe, the different accelerometer readings make them not symmetric. GR is not necessary to explain this.

That surprises me. Without using objects external to a reference frame, how does one define acceleration of the entire reference frame? Isn't acceleration, velocity, and position all relative to something else? Then, without referring to the universe, how can inertia and the forces due to acceleration be explained? 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. I do know that there have been calculations done where the universe (a simplified model) was assumed to be spinning around a stationary bucket and the GR calculations said that the water would be pulled outward with the (approximately) correct force. Should I start a different thread for this? I don't want to hijack the thread.

 

[Fredrik]

general relativity is necessary for understanding the difference between inertial and non-inertial reference frames,

I disagree. The difference between SR and GR is that the metric in GR is a solution to Einstein's equation, and the metric in SR is the Minkowski metric, regardless of the matter content of spacetime.

Without using objects external to a reference frame, how does one define acceleration of the entire reference frame? Isn't acceleration, velocity, and position all relative to something else?

In the purely mathematical part of the theory, proper acceleration is defined as a measure of how much the world line deviates from being a geodesic. The physics is however in the correspondence rules, which tell us how to interpret the mathematics as predictions about results of experiments. They tell us in particular that the motion of an accelerometer that doesn't detect any acceleration is represented by a timelike geodesic.

Then, without referring to the universe, how can inertia and the forces due to acceleration be explained? 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.

It's quite possible that if the universe had been empty except for one accelerometer, that accelerometer would have to detect zero acceleration. But this is philosophical speculation of little value. It seems especially wrong to use it to argue against using SR in a scenario in which there are rockets, which by definition utilize the law of inertia to accelerate.

 

[FactChecker]

Regardless of what Einstein thought, SR distinguishes between inertial frames and accelerating frames to the same extent that GR distinguishes between locally inertial frames and accelerated frames. There are no preferred frames in SR. One can use accelerated frames throughout their study of SR if they wished.

It seems that it's not so much a question of distinguishing between inertial and accelerated frames as it is a question of objectively identifying which is which. How can that be done without referring to the universe and the associated space? SR takes that for granted and assumes that it is obvious which twin is accelerating. GR explains why the twin which is stationary wrt the universe is not the same as the twin that accelerates wrt the universe. Without that, the Twins are perfectly symmetric.

Reference?

(regarding the calculations of centrifugal force in a stationary bucket with a spinning universe): I read it several years ago and do not remember where.

Also this is highly off-topic as I see nothing in your post that even remotely addresses the OP's concerns.

He liked my reply. That's all I know.

This is just the same old song of "GR is needed to explain the twin paradox" which is wrong as others have pointed out. Many SR textbooks address this problem.

Do they answer the question of what causes one twin to feel acceleration and to be considered the accelerating twin? That seems like a GR subject. Without that, the Twins are symmetric.

 

[Fredrik]

It seems that it's not so much a question of distinguishing between inertial and accelerated frames as it is a question of objectively identifying which is which. How can that be done without referring to the universe and the associated space? SR takes that for granted and assumes that it is obvious which twin is accelerating.

SR relies on accelerometers, just like GR.

Do they answer the question of what causes one twin to feel acceleration and to be considered the accelerating twin? That seems like a GR subject. Without that, the Twins are symmetric.

A theory doesn't have to explain everything to be applicable. It just has to make some relevant predictions.

(I have to get some sleep, so I won't be able to discuss any of this further right now).

 

[Nugatory]

Although general relativity is necessary for understanding the difference between inertial and non-inertial reference frames...

That is just plain wrong. Special relativity works fine for both inertial and non-inertial reference frames (as it must, because the one is merely a coordinate transform away from the other) in the absence of gravity. General relativity works for this case and also for the more general case in which gravity is present.

 

[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.