The Twin Paradox: Triplets Edition

[ghwellsjr]

Your first two graphs are correct and illustrate what I'm saying. I have no idea what your third graph is depicting or why you think it describes what I'm saying.

 

[greswd]

Your first two graphs are correct and illustrate what I'm saying. I have no idea what your third graph is depicting or why you think it describes what I'm saying.

actually, I think the 2nd graph does not illustrate what you're saying.

you said that Adam spends half his time receiving signals at blueshifted frequency, and the 2nd graph does not illustrate that.

the 3rd graph illustrates that, but as you can see, it is a highly implausible, and almost impossible scenario.

basically I was trying to use this argument, as described in #24, to refute your resolution to the twin paradox

 

[ghwellsjr]

I did not originally see all the text that you later edited.

In both of your first two graphs, Adam travels along the top bent line and Charles remains stationary along the bottom line. Both graphs are shown for a frame in which both twins start out and end up together at rest. Neither graph shows the perspective of just one twin. The only difference between the two graphs is the first one shows the pulses being sent by Charles and received by Adam while the second one shows the pulses being sent by Adam and received by Charles. Your text for the second graph is wrong.

In the first graph, the light pulses are traveling upward from Charles to Adam. While traveling away from Charles for the first half of his trip lasting one year, Adam receives the light pulses red shifted and for the last half of his trip lasting one more year, he receives the light pulses blue shifted. Half of his time they are red and the other half they are blue.

In the second graph, the light pulses are traveling downward from Adam to Charles. For more than half of his time, Charles receives the light pulses red shifted and for the remaining time he receives the light pulses blue shifted. More than half of his time they are red and for less than half they are blue.

Also, the graphs are not to the scale of Adam traveling at 0.999c but they still make the important point. Graphs drawn to scale would be impossible to understand.

One further point I want to emphasize: you introduced light pulses being sent by each twin, which is OK, but it is not necessary. I'm assuming that they can just watch each others clock. Doppler applies to all information traveling between the twins, not just light pulses. But you could also assume that each twin periodically sends a coded message containing their current time. Light pulses can also be used if the twins agree on the rate they are sending them and keep track of the other ones pulses.

 

[ghwellsjr]

There is another detail that is wrong with your first two graphs (I'm ignoring your third graph as it is nonsense). You show the rate that Adam is sending out pulses to be about the same as the rate that Charles is sending out pulses so the total number of orange lines is the same in each graph. This is not correct. You need to show Adam's pulses in the second graph going out at a slower rate because his clock is time dilated which will result in fewer orange lines for the second graph compared to the first graph.

The article for "Twin paradox" on wikipedia shows the same two graphs you drew but drawn better (but still not perfectly). Don't be disturbed by the interchange of the two axes.

The graph on the left is for Charles sending out light pulses at a constant rate to Adam who receives them during the bottom half at a lower red shifted rate and during the top half at a faster blue shifted rate.

The graph on the right is for Adam sending out light pulses at the same constant rate to Charles but notice how they are much farter apart along the diagonals due to his time dilation caused by his speed in his original rest frame (the same as Charles rest frame). Also note that Charles receives the lower red shifted rate for way more than half the time and the higher blue shifted rate for just the final portion of the trip.

Remember that these graphs make it hard to see that the Doppler shifts are identical for both twins because they are drawn in the same frame.

 

[greswd]

Yeah I did make a few mistakes. Thanks for pointing it out.

I didn't count the no. of pulses drawn because I meant it more to be a conceptual diagram.



I'm going to repost #30 (not allowed to edit ), which may also clear up some of your confusion about #32.


Funny thing is why are Minkowski diagrams drawn with time as the y-axis?

 

[Nugatory]

Funny thing is why are Minkowski diagrams drawn with time as the y-axis?

It's pretty much an arbitrary convention, like the convention that we draw maps with North at the top. Every once in a while you'll come across a diagram drawn the other way, with time as the horizontal axis and distance as the vertical axis; these are perfectly correct and useful, but can be confusing the same way a map drawn with south at the top might be confusing.

 

[greswd]

Repost of #30

The orange lines represent individual light pulses. When we talk about frequency, we are referring to the frequency at which light pulses are received.
Here's the first graph, from Charles' perspective, and Charles is the one sending light pulses.

So it appears that Adam spends half the time receiving light pulses at a redshifted frequency, and half the time receiving blueshift.

http://imageshack.us/a/img145/5116/fesf.png Now for the 2nd graph, from Charles' perspective again.
As you can see, he's receiving light pulses from Adam.

http://imageshack.us/a/img829/7692/ccccx.png
Charles spends more time receiving pulses at a redshifted frequency.

Now what does Charles see? He is going to watch Adam's clock ticking slower than his own until it reaches one year because that is the time he sees on Adam's clock when he turns around, correct? So what time is on Charles's clock when that happens? Well, it would be the reciprocal of 0.02236627204, wouldn't it, which is 44.71017781 years. Now he sees Adam's clock ticking faster than his own for another year, correct? How much time progresses on his clock while that happens? It is the reciprocal of 44.71017781 which is 0.02236627204 years, correct? The sum of these two numbers, 44.73254408204 years, is how much time progresses on Charles's clock while he watches 2 years progress on Adam's clock.

Described with the 2nd graph. This is why I said that quote shown above is absolutely right.

After one year on Adam's clock, he sees that Charles's clock has advanced by 0.02236627204 years (8.1636892946 days), correct? Then he turns around and now he sees Charles's clock ticking 44.71017781 times his own so in one more year he sees Charles's clock advance by 44.71017781 more years for a total of 44.73254408204 years. So Adam sees Charles's clock advance by 44.73254408204 years while his own clock advances by just 2 years.

"So it appears that Adam spends half the time receiving light pulses at a redshifted frequency, and half the time receiving blueshift."

Now this is where the problem starts. The above two quotes are based on the 1st graph.
However, if we look at things from Adam's perspective, and Adam is receiving light pulses from Charles:
The spacetime graph should look like the 2nd graph. Which was previously from Charles' perspective instead of Adam'sBut this demonstrates symmetry. And as mentioned previously

The Doppler factors... are mutual but they don't apply symmetrically.

So we want to satisfy these two quotes:

After one year on Adam's clock, he sees that Charles's clock has advanced by 0.02236627204 years (8.1636892946 days), correct? Then he turns around and now he sees Charles's clock ticking 44.71017781 times his own so in one more year he sees Charles's clock advance by 44.71017781 more years for a total of 44.73254408204 years. So Adam sees Charles's clock advance by 44.73254408204 years while his own clock advances by just 2 years.

"So it appears that Adam spends half the time receiving light pulses at a redshifted frequency, and half the time receiving blueshift."

How do we draw a graph to describe that from Adam's perspective? It is this graph

http://imageshack.us/a/img4/1289/vvvvvi.png

It shows a highly implausible scenario.

BUT, it does show Adam receiving pulses at redshifted frequency for half of the time, and blueshifted frequency for the other half.
Look at the 2nd graph again. If this is from Adam's perspective, then Adam's spends more time receiving pulses at redshifted frequency.But based on what the last two quotes above say, we should draw the 3rd graph.Comparing the 3rd and 2nd graphs, you can see that the no. of pulses received at blueshifted frequency is much greater in the 3rd graph.

This overestimation of the no. of pulses received by Adam apparently solves the paradox.

 

[ghwellsjr]

I agree, your third graph is implausible.

I have a suggestion: let's change the scenario to one that you can accurately draw graphs for. So let's change the speed of Adam to 0.6c. This will make the redshift Doppler factor be 0.5 and the blueshift Doppler factor be 2.

Adam will go for one year away according to his clock and one year returning. He will emit a flash of light every month for a total of 24, 12 on the way out and 12 on the way back. He will see the flashes coming from Charles at one half that rate on the way out for a total of 6 and for double that rate on the way back for a total of 24 more and a grand total of 30.

Charles will emit 30 flashes during the time that Adam is gone. Since he will be seeing Adam's flashes coming in at one half his rate and since he sees Adam turning around when he sees the 12th flash from Adam, he will be emitting his 24th flash at that moment. Then he sees the flashes coming in at double his rate for 12 more flashes from Adam while he is emitting 6 more. His total is again 30 while Adam's total is 24.

Please redraw your first two graphs using these numbers, make sure the slope of Adam's path is accurate for 0.6c and the rate of Adam's flashes are spaced farther apart by a factor of 1.25 compared to Charles's flashes to take into account his time dilation.

Then after you make those two graphs, make a third graph that is simply an overlay of the first two graphs. This is how you can show the Doppler effect and this one graph will show accurately the perspective of both Adam and Charles in terms of what they see compared to their own clocks.

 

[DrGreg]

greswd

The basic problem is that Adam isn't at rest in a single inertial frame for the entire journey, so it's difficult to cover the whole journey in a single diagram from his view.

Here's what the journey would look like in each of Adam's two frames.

An attempt to somehow cut-&-paste the two diagrams is shown on the right but as you can see it looks pretty weird as we are trying to reconcile two incompatible views.

(Note I've used different numbers to yours but you should be able to grasp the concept.)

 

[greswd]

greswd

The basic problem is that Adam isn't at rest in a single inertial frame for the entire journey, so it's difficult to cover the whole journey in a single diagram from his view.

True. This has been the source of much debate

It seems weird, as though Adam has to reference both his current frame, and his "past frame".

Plus, the 2nd and 3rd graphs are transformations, or "warps", of the 1st graph, so we could expect to them give the same result.

 

[greswd]

I would draw both graphs but they will look very similar to the ones I already have.

I'll clarify using the solution from Spacetime Physics.

This is from Adam's perspective, and Adam is receiving pulses from Charles.
Charles sends pulses to celebrate their mensiversaries.

Outbound leg:

Time between Charles' mensiversaries in Adam's frame: 1.25 mths

Additional distance pulse needs to traverse between mensiversaries: 1.25 × 0.6 c = 0.75 c mths

Time taken for pulse to travel additional distance: 0.75 mths

∴Time interval between pulses received: 1.25 + 0.75 = 2 mths

No. of pulses received by Adam after 12 mths: 12/2 = 6 pulsesOn the return leg:

Time interval between pulses received: 1.25 - 0.75 = 0.5 mths

No. of pulses received by Adam after 12 mths: 12/0.5 = 24 pulsesTotal no. of pulses received by Adam after 24 mths: 24 + 6 = 30 pulsesCharles has aged by 30 mths while Adam has only aged by 24 mths.

Paradox Solved...but not yet.Time between Charles' mensiversaries in Adam's frame: 1.25 mths


No. of Charles' mensiversaries in Adam's frame: 24/1.25 = 19.2 mensiversaries

Now how the heck does Adam manage to receive 30 pulses?The solution is given by this messed up diagram.

http://imageshack.us/a/img4/1289/vvvvvi.png

 

[pervect]

The diagrams in #37 all look correct to me at a glance. Where is the supposed "paradox", though?

As far as combining the diagrams goes - there is indeed a fundamental problem in finding coordinates for an observer who accelerates, or who changes velocity suddenly that's well known. It's not , as far as I know, referred to as a "paradox", but as a limit on the size of accelerated coordinate systems.You can represent the concept of "now" on a space-time graph. The blue line on #41 does this for Charlie. I'd suggest drawing lines of "now" for adam. They won't all be parallel, they'll change after the break where Adam accelerates. Observe that they must cross. Where they first cross defines the limits on the size of the coordinate system Adam can use.

 

[ghwellsjr]

I would draw both graphs but they will look very similar to the ones I already have.

Please do draw the graphs accurately for the new scenario where Adam is traveling at 0.6c.

Your problem is that you keep thinking that your graphs are from the perspective of just one of the twins and they're not. Both graphs are showing half of the perspective of both twins. Your first graph is showing Charles's perspective of emitting his light pulses and Adam's perspective of receiving them. Your second graph is showing Adam's perspective of emitting his light pulses and Charles's perspective of receiving them. But neither graph is showing the Doppler relationship between the rates that they are emitting pulses and the rates that they are receiving them from the other person. That's why I want you to redraw both graphs with the correct scaling and then superimpose them into a third graph.

Then it will be obvious how during the first part of the trip, they each are seeing the other ones pulses coming in at exactly one half the rate they are each sending them, and during the last part of the trip, they are each seeing the other ones pulses coming in at exactly double the rate they are each sending them. Even though the Doppler factors are identical for the two twins (0.5 at the beginning and 2 at the end), they don't last the same percentages of the time for the two twins, 50-50 for Adam but 80-20 for Charles.

Keep in mind that there are only two things in this limited scenario that Adam and Charles can see--their own clock and the clock of the other twin (or light pulses coming at a fixed interval of time which amounts to the same thing). So when you talk about a point of view for one of the observers or the perspective of one of the observers, it must include those two things. That's why I want you to make a third graph which is a composite of the first two graphs. Please do it.

 

[Dale]

Here's the first graph, from Charles' perspective, and Charles is the one sending light pulses.

So it appears that Adam spends half the time receiving light pulses at a redshifted frequency, and half the time receiving blueshift.

This drawing is correct. It accurately reflects the physics and correctly predicts the measured observation that Adam spends half the time receiving redshift and half receiving blueshift. Any diagram which does not agree on the measured observation is WRONG and does not accurately reflect the physics.

Now for the 2nd graph, from Charles' perspective again.
As you can see, he's receiving light pulses from Adam.

Charles spends more time receiving pulses at a redshifted frequency.

Also correct, and similarly any diagram which does not agree on the measured observation is WRONG and does not accurately reflect the physics

However, if we look at things from Adam's perspective, and Adam is receiving light pulses from Charles:
The spacetime graph should look like the 2nd graph. Which was previously from Charles' perspective instead of Adam's

This is not correct. There is no transformation between Adam's and Charles' frames which will make the graph in Adam's frame look like the graph in Charles' frame. The actual math is more complicated than that and you need to be careful. See below.

How do we draw a graph to describe that from Adam's perspective? It is this graph.

Since this drawing does not correctly reproduce the measured observation we know immediately that it is wrong. However, it is helpful to look more carefully into why it is wrong. The problem is primarily that Adam's frame is non-inertial. This causes all sorts of problems, the primary problem being that it is not well defined, i.e. there is not one single coordinate system which is unambiguously implied when we say "Adam's frame", so you need to be more specific and describe in detail the process by which Adam assigns coordinates in "his frame".

Here is a reference for one reasonable method that Adam might use to assign coordinates in his frame ( http://arxiv.org/pdf/gr-qc/0104077v2.pdf ). Using this method you find that Charles' path has two bends. Furthermore, although this method is defined such that light travels at c, the time dilation formula from inertial frames does not apply. The combination of the non-inertial time dilation and the movement work out so that Adam and Charles each receive the correct amount of red and blue shifted signals.

 

[Dale]

Plus, the 2nd and 3rd graphs are transformations, or "warps", of the 1st graph, so we could expect to them give the same result.

The 2nd graph is not a transformation of the 1st graph, it is in the same coordinate system as the 1st graph and just shows a different set of light pulses.

The 3rd graph is not a legitimate transformation of the 1st. If you disagree then I encourage you to try to write down the coordinate transformation that takes you from the 1st to the 3rd graph. There is no such transformation. In any coordinate transform you will write you will either find that Charles' path has more than one bend in it or that the light paths have bends in them.

 

[ghwellsjr]

Now for the 2nd graph, from Charles' perspective again.
As you can see, he's receiving light pulses from Adam.

Charles spends more time receiving pulses at a redshifted frequency.

Also correct, and similarly any diagram which does not agree on the measured observation is WRONG and does not accurately reflect the physics

If we are just illustrating the fact that "Charles spends more time receiving pulses at a redshifted frequency", then the graph is correct, but it describes a different scenario than the first graph because the two graphs show the same number of pulses (19) being transferred between the two twins which is not correct for a single scenario. This is why I'm urging greswd to redraw both graphs with the correct scaling and then to overlay them onto a third graph. If he does it correctly for the parameters I suggested in post #38, he will show 30 pulses in the first graph and 24 in the second graph. (Actually he will show 29 and 23 because the last ones happen when the twins reunite and won't be visible.) His problem with the second graph is that he is not showing the time dilation of Adam which will move the light pulses farther apart so that there will be fewer of them.

 

[DrGreg]

Plus, the 2nd and 3rd graphs are transformations, or "warps", of the 1st graph, so we could expect to them give the same result.

The 2nd graph is not a transformation of the 1st graph, it is in the same coordinate system as the 1st graph and just shows a different set of light pulses.

The 3rd graph is not a legitimate transformation of the 1st. If you disagree then I encourage you to try to write down the coordinate transformation that takes you from the 1st to the 3rd graph. There is no such transformation. In any coordinate transform you will write you will either find that Charles' path has more than one bend in it or that the light paths have bends in them.

To be fair to greswd, I think he/she was referring to my graphs in post #39, not his/her own graphs, so the comment is correct.

In case anyone is in any doubt, the 4th graph I drew is supposed to be a failed attempt to draw something from Adam's view. The graph makes little sense because of the gap in Charles's worldline (trajectory). What DaleSpam is saying is correct, if you "transform away bends" in Adam's worldline you'll end up with non-linear distortions to the rest of the diagram.

 

[ghwellsjr]

To be fair to greswd, I think he/she was referring to my graphs in post #39, not his/her own graphs, so the comment is correct.

Your graphs are good but they also don't show either twin's full perspective, they show Charles's perspective of sending the pulses and Adam's perspective of receiving them but they don't show the Doppler effect. It would be helpful if you would show another set of graphs in Charles's frame that shows Adam sending his pulses to Charles and then overlay them on to a third graph. Then it would be clear that the Doppler factors are the same for both at the beginning and at the end but switch at a different percentage of the time for both twins.

Then it would be really great if you could repeat these graphs for Adam's two frames.

But this is an exercise that greswd should perform since he is the one that claimed that he had a graph that contradicts the Doppler analysis that I presented earlier in this thread.

 

[Dale]

If we are just illustrating the fact that "Charles spends more time receiving pulses at a redshifted frequency", then the graph is correct, but it describes a different scenario than the first graph because the two graphs show the same number of pulses (19) being transferred between the two twins which is not correct for a single scenario. This is why I'm urging greswd to redraw both graphs with the correct scaling and then to overlay them onto a third graph. If he does it correctly for the parameters I suggested in post #38, he will show 30 pulses in the first graph and 24 in the second graph. (Actually he will show 29 and 23 because the last ones happen when the twins reunite and won't be visible.) His problem with the second graph is that he is not showing the time dilation of Adam which will move the light pulses farther apart so that there will be fewer of them.

Good catch, I completely missed that. As drawn, Adam and Charles are broadcasting at different frequencies.

 

[Dale]

To be fair to greswd, I think he/she was referring to my graphs in post #39, not his/her own graphs, so the comment is correct.

In case anyone is in any doubt, the 4th graph I drew is supposed to be a failed attempt to draw something from Adam's view. The graph makes little sense because of the gap in Charles's worldline (trajectory). What DaleSpam is saying is correct, if you "transform away bends" in Adam's worldline you'll end up with non-linear distortions to the rest of the diagram.

Sorry about any confusion that my comments may have caused. I was referring to greswd's drawings.

 

[greswd]

Sorry for bumping this ol' thread, I was quite busy.

First up, do you guys recommended that I ignore the 3rd diagram I presented?

Also, how would we apply the Doppler shifts between Adam and Charles?

 

[ghwellsjr]

Ignore your 3rd diagram and follow the suggestions I made in post #38:

I agree, your third graph is implausible.

I have a suggestion: let's change the scenario to one that you can accurately draw graphs for. So let's change the speed of Adam to 0.6c. This will make the redshift Doppler factor be 0.5 and the blueshift Doppler factor be 2.

Adam will go for one year away according to his clock and one year returning. He will emit a flash of light every month for a total of 24, 12 on the way out and 12 on the way back. He will see the flashes coming from Charles at one half that rate on the way out for a total of 6 and for double that rate on the way back for a total of 24 more and a grand total of 30.

Charles will emit 30 flashes during the time that Adam is gone. Since he will be seeing Adam's flashes coming in at one half his rate and since he sees Adam turning around when he sees the 12th flash from Adam, he will be emitting his 24th flash at that moment. Then he sees the flashes coming in at double his rate for 12 more flashes from Adam while he is emitting 6 more. His total is again 30 while Adam's total is 24.

Please redraw your first two graphs using these numbers, make sure the slope of Adam's path is accurate for 0.6c and the rate of Adam's flashes are spaced farther apart by a factor of 1.25 compared to Charles's flashes to take into account his time dilation.

Then after you make those two graphs, make a third graph that is simply an overlay of the first two graphs. This is how you can show the Doppler effect and this one graph will show accurately the perspective of both Adam and Charles in terms of what they see compared to their own clocks.

 

[greswd]

Sorry I meant Adam and Bob. oops. Why is it that they both return having aged the same?

 

[ghwellsjr]

Sorry I meant Adam and Bob. oops. Why is it that they both return having aged the same?

You haven't done a correct set of graphs for Adam and Charles. If you do that and then add in a similar graph for Bob and Charles but flipped upside down, then you will be almost there. All you will need to do at that point is extend the light paths for Adam and Bob so that they go past Charles and reach all the way to the other person.

Or did you want to forget about graphs and pick up where we left off in the middle of page 2?

 

[greswd]

You haven't done a correct set of graphs for Adam and Charles. If you do that and then add in a similar graph for Bob and Charles but flipped upside down, then you will be almost there. All you will need to do at that point is extend the light paths for Adam and Bob so that they go past Charles and reach all the way to the other person.

Or did you want to forget about graphs and pick up where we left off in the middle of page 2?

graphs are very important, special relativity becomes very clear with a minkowski diagram. Plus as Einstein himself said, a geometric understanding is necessary to make sense of his general theory.

 

[ghwellsjr]

graphs are very important, special relativity becomes very clear with a minkowski diagram. Plus as Einstein himself said, a geometric understanding is necessary to make sense of his general theory.

If you're going to make a graph, it has to be done correctly. Your set of graphs haven't been done correctly. If you would do the graphs correctly for Adam and Charles, as I have suggested, and then add in Bob, as a flip side of Adam or a mirror image of Adam, then you will see how Adam and Bob have aged the same amount when they rejoin Charles.

If you don't want to make the graphs correctly, I will. And I will also explain the scenario without the use of graphs.

By the way, your graphs are not Minkowski diagrams, they are simply conventional position versus time graphs. And I'm not saying that simply because you are interchanging the time versus distance axes that is more common for a Minkowski diagram.

 

[Dale]

By the way, your graphs are not Minkowski diagrams, they are simply conventional position versus time graphs. And I'm not saying that simply because you are interchanging the time versus distance axes that is more common for a Minkowski diagram.

What is the difference?

 

[ghwellsjr]

Minkowski diagrams have at least two sets of axes to show how each event has two sets of coordinates for two different reference frames. All of the graphs that greswd presented have only one set of axes corresponding to the frame in which Charles remains at rest and in which Adam and Bob start out at rest and end up at rest.

People were drawing position versus time graphs long before Minkowski or Einstein or Lorentz or even Maxwell. I don't think Minkowski gets backwards credit for all those graphs just because they only have one set of axes.

 

[DrGreg]

Minkowski diagrams have at least two sets of axes

That's news to me. Do you have a textbook that uses that specific definition?

 

[ghwellsjr]

That's news to me. Do you have a textbook that uses that specific definition?

It's very hard to find a specific definition anywhere. Is there an official definition that you can point me to or that you want to provide even without reference?

If someone draws a graph of position versus time, does that automatically make it a Minkowski diagram? Would you call greswd's graphs on this thread Minkowski diagram's?