The Twin Paradox: Triplets Edition

[greswd]

A variant of the classic twin paradox.

There are three triplets, who have the boring names of Adam, Bob and Charles.

In Charles' "rest" frame, Adam and Bob get into identical rocketships, jet off in opposite directions and return a year later.

To Charles, Adam and Bob's paths are symmetrical.


At their reunion, who will be older and why?

This problem may remove some of the asymmetries of the original paradox, and I would like to hear from you guys.

 

[Nugatory]

Adam and Bob are the same age, and less aged than Charles.

 

[ghwellsjr]

Nugatory answered your first question but the reason why is simply that time dilation (the slowing of a clock) is based only on the speed of that clock as defined in any particular reference frame. Therefore, since Charles remains at rest in your chosen reference frame, his clock will not be time dilated and since Adam and Bob both move identically, their clocks will be time dilated to the same extent and so will end up with less time on them at the grand reunion.

But I'm curious, this is nothing more than two classic Twin Paradoxes, why did you think it would remove any asymmetries?

 

[greswd]

Nugatory answered your first question but the reason why is simply that time dilation (the slowing of a clock) is based only on the speed of that clock as defined in any particular reference frame. Therefore, since Charles remains at rest in your chosen reference frame, his clock will not be time dilated and since Adam and Bob both move identically, their clocks will be time dilated to the same extent and so will end up with less time on them at the grand reunion.

But I'm curious, this is nothing more than two classic Twin Paradoxes, why did you think it would remove any asymmetries?

now that you say it, it is like two twin paradoxes, but more complicated.

The two twin paradoxes are Adam-Charles and Bob-Charles, but now we also have to deal with Adam-Bob.

 

[greswd]

I'm still uncertain about the Doppler explanation as a resolution to the paradox.

The Time-Gap objection seems to be the most rational explanation, despite its bizarre predictions.

 

[ghwellsjr]

now that you say it, it is like two twin paradoxes, but more complicated.

The two twin paradoxes are Adam-Charles and Bob-Charles, but now we also have to deal with Adam-Bob.

I did deal with them. I said both their clocks are time dilatated to the same extent and so they end up with less time on them. Do you have any doubt about this conclusion?

 

[ghwellsjr]

I'm still uncertain about the Doppler explanation as a resolution to the paradox.

Have you made any progress in going through the Doppler explanation?

The Time-Gap objection seems to be the most rational explanation, despite its bizarre predictions.

Have you made any progress in going through the Time-Gap explanation?

 

[greswd]

I said both their clocks are time dilatated to the same extent and so they end up with less time on them. Do you have any doubt about this conclusion?

But isn't that only from Charles point of view? From Adam or Bob's point of view their other two brothers would age more slowly.

Have you made any progress in going through both explanations?

I'm currently reading
http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html

which is pretty informative. Still can't quite get the hang of it though.

 

[ghwellsjr]

I said both their clocks are time dilatated to the same extent and so they end up with less time on them. Do you have any doubt about this conclusion?

But isn't that only from Charles point of view? From Adam or Bob's point of view their other two brothers would age more slowly.

No, a point of view is what someone sees. Each person sees something different because they are at different points of the scenario at different times except at the beginning and at the end. Doppler analysis is how you determine what each person sees and has nothing to do with a frame of reference which is how we specify and calculate things like time dilation. No one can see time dilation. If they could, then because each person has a different speed in each different frame of reference, they would see a different time dilation and that doesn't make sense, does it? Remember, all frames of reference are equally valid and you can use any frame of reference to calculate what each person.

So if you are asking about each person's point of view, you're asking about what they actually see, correct? And this can be done most easily using Doppler analysis. Have you attempted to do this? Do you know the formula for the Relativistic Doppler factor? Do you know the formula for Velocity Addition?

Do you want to put some numbers on your example, like how fast do Adam and Bob travel? You already said they return after a year so I assume they travel away for a half year (according to the rest frame of Charles) and then instantly turn around and travel back at the same speed for the other half of the year?

Yeah I have been reading

http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html

which is probably the best resource on the twin paradox. Still can't quite get the hang of it though.

 

[greswd]

Yup I know the formulas.

I always thought that observing time dilation is like watching high-speed footage. Something like this. Putting in some values would be good.But put yourself in Adam's shoes. Before you left, both you and Charles were handsome strapping young men.

After you've returned, you can still easily pick up babes, but Charles has become a lecherous old fool with Einstein-hair.

The question is, what happened in-between? It seems like the reverse effect of time dilation.

 

[ghwellsjr]

You said that Adam and Bob reunite with Charles after one year. Even if they traveled at an extremely high speed, they're only going to end up one year younger than Charles. If you want Charles to age by say 50 years, you better make the trip last 50 years. And then if you want Adam and Bob to age by just a couple years, they're going to have to travel faster than 99.9%c.

As I said before, time dilation applies to the one who is traveling at a high speed in a given frame. Adam and Bob are the ones who are traveling in [STRIKE]Bob[/STRIKE]
Charles's rest frame so they are the ones that age more slowly. Why does that seem like the reverse effect of time dilation?

 

[greswd]

If you want Charles to age by say 50 years, you better make the trip last 50 years. And then if you want Adam and Bob to age by just a couple years, they're going to have to travel faster than 99.9%c.

Adam and Bob are the ones who are traveling in Bob's rest frame so they are the ones that age more slowly. Why does that seem like the reverse effect of time dilation?

I think you meant to write "Charles"?


okay so we make them travel for 50 years before the reunion.
and both Adam and Bob are traveling at 0.99999999999999999c, as seen by Charles.




Adam can't feel time passing more slowly for himself. Time seems to be flowing normally.

However, to Adam, Charles was the same age before and perhaps a good 40 years older when they reunite. So it does seem like the reverse effect of time dilation to Adam.

 

[ghwellsjr]

I think you meant to write "Charles"?

Yes, thanks.

okay so we make them travel for 50 years before the reunion.
and both Adam and Bob are traveling at 0.99999999999999999c, as seen by Charles.

I don't have the computational power to deal with a number that close to 1. And I doubt that you do, too. Let's go with 99.9%c, OK?

Adam can't feel time passing more slowly for himself. Time seems to be flowing normally.

True, but he will feel an extreme acceleration. It will be much worse than getting punched in the face.

However, to Adam, Charles was the same age before and perhaps a good 40 years older when they reunite. So it does seem like the reverse effect of time dilation to Adam.

I told you, nobody can see time dilation--it's a calculation based on the speed in a given frame of reference. Adam and Bob are the only ones moving in your chosen frame of reference so they are the ones whose clocks are running slow in that frame.

Now at 0.999c, the speed that Adam and Bob are moving away from Charles, you need to use the Velocity Addition formula to calculate the relative speed between Adam and Bob. Can you do that? Tell me what you get.

 

[greswd]

I told you, nobody can see time dilation--it's a calculation based on the speed in a given frame of reference. Adam and Bob are the only ones moving in your chosen frame of reference so they are the ones whose clocks are running slow in that frame.

Now at 0.999c, the speed that Adam and Bob are moving away from Charles, you need to use the Velocity Addition formula to calculate the relative speed between Adam and Bob. Can you do that? Tell me what you get.

so unfortunately we can't see cool things like the relative velocity between Adam and Bob is 0.9999994994997501c.but the fact of the matter is, to Adam, Charles did age faster.

 

[harrylin]

now that you say it, it is like two twin paradoxes, but more complicated.

The two twin paradoxes are Adam-Charles and Bob-Charles, but now we also have to deal with Adam-Bob.

Good! Thus the usual twin paradox discussions are relevant such as the one that is still going on (and with links to earlier ones):
https://www.physicsforums.com/showthread.php?t=642784

Did you go through it? What is still unclear, when applying it to a triplet?

 

[ghwellsjr]

the relative velocity between Adam and Bob is 0.9999994994997501c.

Good, now can you calculate the Relativistic Doppler factor at that speed and also at the relative speed between Adam and Charles, 0.999c?

 

[Dale]

A variant of the classic twin paradox.

There are three triplets, who have the boring names of Adam, Bob and Charles.

In Charles' "rest" frame, Adam and Bob get into identical rocketships, jet off in opposite directions and return a year later.

To Charles, Adam and Bob's paths are symmetrical.


At their reunion, who will be older and why?

It will be helpful for you to think in terms of geometry. This scenario is equivalent to the following:

Take a square and draw a diagonal from one corner to the opposite. There are now three paths connecting the two corners, which is shortest and why?

 

[greswd]

Good, now can you calculate the Relativistic Doppler factor at that speed and also at the relative speed between Adam and Charles, 0.999c?

at 0.999c, 44.71017781 and 0.02236627204

and between Adam and Bob 1999 and 5.00250125×10^-4

 

[ghwellsjr]

Excellent.

I said in post #11 that Adam and Bob are going to age (about) a couple years so let's say they travel away for exactly one year according to their own clocks and then turn around and get back to Charles in exactly one more year. We'll first deal with what happens between Adam and Bob and when we get done with that we'll figure out what goes on between each of them and Charles.

Now according to the Doppler Analysis, Adam and Bob will each see the other ones clock running slower than their own by the factor of 5.00250125×10^-4 (which is [STRIKE]1/1900[/STRIKE] 1/1999). So the first question we want to answer is what time will each of them see on the other ones clock when they reach the point of turnaround? The answer is simple--we multiply 1 year by 5.00250125×10^-4 (or divide it by [STRIKE]1900[/STRIKE] 1999), which is just a little over four and a half hours.

The next question is what Doppler Factor will apply at the moment of turn around? How fast will they each see the other ones clock ticking immediately after they each turn around? What do you think?

 

[greswd]

Excellent.

I said in post #11 that Adam and Bob are going to age (about) a couple years so let's say they travel away for exactly one year according to their own clocks and then turn around and get back to Charles in exactly one more year. We'll first deal with what happens between Adam and Bob and when we get done with that we'll figure out what goes on between each of them and Charles.

Now according to the Doppler Analysis, Adam and Bob will each see the other ones clock running slower than their own by the factor of 5.00250125×10^-4 (which is 1/1900). So the first question we want to answer is what time will each of them see on the other ones clock when they reach the point of turnaround? The answer is simple--we multiply 1 year by 5.00250125×10^-4 (or divide it by 1900), which is just a little over four and a half hours.

The next question is what Doppler Factor will apply at the moment of turn around? How fast will they each see the other ones clock ticking immediately after they each turn around? What do you think?

so, they should see each other's clocks ticking very quickly during the turnaround, and this will offset the previous effect?

 

[ghwellsjr]

so, they should see each other's clocks ticking very quickly during the turnaround, and this will offset the previous effect?

No, that is not correct.

This would be correct for how Adam and Bob see Charles (but not how Charles sees Adam and Bob) so let's work on that relationship. Use the correct Doppler factors for how Adam and Bob see Charles's clock at the end of one year on their clocks and calculate how much they see Charles's clock progress. Then use the reciprocal factor for the return trip and see how much they see Charles's clock progress during their one year return. Add the two numbers together and you will have determined how much Charles has aged during their trips. What do you get?

Now think about what Charles sees when he looks at Adam and Bob. Using the same Doppler factors, figure out what time is on Charles's clock when he sees Adam and Bob reach one year and when he sees them turn around. Then for the remaining time that you calculated in the previous paragraph, you can figure out how much time on Charles's clock goes by while he's watching them return.

If you do all that correctly and if you understand what you are doing and why, you should be able to figure out why it is different when Adam and Bob look at each other. Can you do that?

 

[greswd]

No, that is not correct.

This would be correct for how Adam and Bob see Charles (but not how Charles sees Adam and Bob) so let's work on that relationship. Use the correct Doppler factors for how Adam and Bob see Charles's clock at the end of one year on their clocks and calculate how much they see Charles's clock progress. Then use the reciprocal factor for the return trip and see how much they see Charles's clock progress during their one year return. Add the two numbers together and you will have determined how much Charles has aged during their trips. What do you get?

Now think about what Charles sees when he looks at Adam and Bob. Using the same Doppler factors, figure out what time is on Charles's clock when he sees Adam and Bob reach one year and when he sees them turn around. Then for the remaining time that you calculated in the previous paragraph, you can figure out how much time on Charles's clock goes by while he's watching them return.

If you do all that correctly and if you understand what you are doing and why, you should be able to figure out why it is different when Adam and Bob look at each other. Can you do that?

Analysing Adam-Charles, wouldn't the effects of Doppler shifting be mutual?

 

[ghwellsjr]

Analysing Adam-Charles, wouldn't the effects of Doppler shifting be mutual?

The Doppler factors that you calculated in post #18, 44.71017781 and 0.02236627204, are mutual but they don't apply symmetrically.

In our example, when they first depart, Adam and Charles both see each others clock ticking at 0.02236627204 times their own. 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.

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.

So can you see how even though the same two Doppler factors apply for both Adam and Charles in watching the other ones clock, they don't apply for the same length of time according to each observer and that's why they end up with different times on their own clocks when they reunite?

Do you have any questions on what happens between Adam and Charles? If not, can you see why we can't do a similar analysis between Adam and Bob?

 

[greswd]

Aha brilliant! You nearly had me there. :)

So first, with regards to how Charles sees Adam. At first I thought there was a flaw because to Charles, Adam took less time to return. Then i realized, this was what Charles saw, and not what actually happened in Charles' frame.

After drawing an ordinary displacement-time graph, i find that you're absolutely right.

Now with regards to how Adam sees Charles. I have encountered your explanation three times already, from three different textbooks. Heh.

Let's say that in Adam's frame Charles emits light pulses at a regular frequency. According to your explanation, Adam spends half his time receiving signals at a redshifted frequency, and the other half receiving signals at a blueshifted frequency.

after drawing a graph, it shows that this is not the case. Like Charles, Adam should spend more than half the time receiving signals at a redshifted frequency.

By overestimating the no. of signals received at a blueshifted frequency, the paradox is apparently solved.

You were quite clever in connecting both frames together in order to demonstrate the asymmetry. However using that explanation i could also say that Adam only makes a turnaround after seeing 44 years elapse on Charles clock, by which 2000 years have elapsed for Adam.
Do note that a displacement-time graph does show that 44 years elapses for Charles before he sees Adam make a turnaround. Just that the reason for that is different.

Now some have tried to solve the problem by saying that distances for Adam are length contracted. I turned to a Minkowski diagram for the solution. However, it produces a time gap.

Naturally, this raises three questions. is some part of Charles' life "event cloaked" to Adam?
Why does the time gap only apply one way? Why is the time gap just the right amount, but not more or less?



Other than the time-gap explanation, we can use the GR explanation. As Adam experiences acceleration, he perceives himself to be in a stronger gravitational field than Charles. Therefore time passes more slowly for him.

Given that most members know nuts about GR, I think you should stick to it.

 

[harrylin]

[..] Why does the time gap only apply one way? Why is the time gap just the right amount, but not more or less?

Other than the time-gap explanation, we can use the GR explanation. As Adam experiences acceleration, he perceives himself to be in a stronger gravitational field than Charles. Therefore time passes more slowly for him.

Given that most members know nuts about GR, I think you should stick to it.

Except for the calculations (nice!), this thread starts to look a little like the other spin-off from the last twin paradox thread:

https://www.physicsforums.com/showthread.php?t=646622

And already that one looked too much like all forgoing twin paradox threads.

Time gap: explained many times, such as here:

http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gap.html

In a nutshell: you are wondering why a Lorentz transformation works by "just the right amount".

The usual related issues also apply:

https://www.physicsforums.com/showthread.php?p=4111027
https://www.physicsforums.com/showthread.php?p=4114579

Cheers,
Harald

 

[ghwellsjr]

Let's say that in Adam's frame Charles emits light pulses at a regular frequency.

Why do you say "in Adam's frame"? If Charles emits light pulses at a regular frequency, it has to be in his frame.

According to your explanation, Adam spends half his time receiving signals at a redshifted frequency, and the other half receiving signals at a blueshifted frequency.

This is true if Adam is traveling at the same speed going and returning and if Charles remains at rest.

after drawing a graph, it shows that this is not the case. Like Charles, Adam should spend more than half the time receiving signals at a redshifted frequency.

If you drew a graph that shows this not to be the case, then you drew the graph wrong. Your statement is wrong.

By overestimating the no. of signals received at a blueshifted frequency, the paradox is apparently solved.

I did not overestimate anything. You are confused. As a result, the rest of your post is of no consequence. Please try to figure out where you are confused so we can continue the Doppler analysis.

 

[greswd]

if you say that drawing a graph doesn't contradict, could you link me to an image? We wouldn't want to be describing diagrams with words.

My guess is you had it from charles perspective, whereas from Adam's perspective, you would get different results.

 

[Dale]

if you say that drawing a graph doesn't contradict, could you link me to an image? We wouldn't want to be describing diagrams with words.

My guess is you had it from charles perspective, whereas from Adam's perspective, you would get different results.

You are the one that claimed to have a drawing.

However, one possible set of correct diagrams can be found at:
http://arxiv.org/abs/gr-qc/0104077

See figures 8 and 9, but I recommend reading the entire article.

 

[ghwellsjr]

if you say that drawing a graph doesn't contradict, could you link me to an image? We wouldn't want to be describing diagrams with words.

There is no graph, no analysis of any type that can contradict what the Doppler analysis shows because it describes what each observer actually sees and measures. All other analyses must agree with what I have described. It is such a simple description, I don't know why a graph is needed. But if you made a graph that you believe contradicts, maybe you should show it to us and maybe we can help you find your mistake.

Or you could state what you think is the mistake in the Doppler analysis. So far, the only thing you have indicated is that Adam will spend a different amount of time (according to his own clock) traveling a particular distance in both directions (going and coming) at the same speed but you haven't said why or given any support for such a notion.

Keep in mind that the Doppler analysis does not depend on identifying any reference frame where coordinate time needs to be defined according to synchronization of clocks. There is no coordinate time assigned in the Doppler analysis but if you choose to assign a coordinate system, it will provide exactly the same results as the Doppler analysis--how could it not?

My guess is you had it from charles perspective, whereas from Adam's perspective, you would get different results.

Of course each has a different perspective but not different results. But they agree that when they see Adam turning around, they see his clock reading one year. And they agree when Adam gets back to Charles, they both see Adam's clock at two years and they both see Charles's clock at 44.73254408204 years. That's the whole point of the analysis--to show how even though they both see the other ones clock Doppler shifted by the same factors, they don't both see those Doppler shifts applied in the same proportions. They still both agree that Adam's clock advanced to one year when he turned around and advanced to two years when they reunited while Charles's clock advanced to 44.73254408204 years.

What part of that do you reject and why?

 

[greswd]

Ok I haz graphs, crudely made with powerpoint.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, this time from Adam's perspective.
As you can see, he's receiving light pulses from Charles.

http://imageshack.us/a/img829/7692/ccccx.png

He spends more time receiving pulses at a redshifted frequency.Follwing our interpretation of the first graph, Adam spends half the time receiving pulses at redshifted frequency, and the other half receiving pulses at blueshifted frequency.
But this contradicts the 2nd graph. I can describe what ghwellsjr was saying with this 3rd graph, which looks implausible.

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

Comparing the 2nd and 3rd graphs, we can see where the overestimation comes into play.

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 1st graph. This is why I said that you were 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.

Described by 3rd graph.

 

[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?