Time Dilation in Twin Paradox: Exploring the Puzzling Reality

[neopolitan]

When "checking the context" involves reading an entire 9-page thread, yes it is unreasonable. If you've spent any time on this forum you'll know it's extremely common for people to jump into respond to individual comments during the course of long discussions (as it is on every other discussion forum I've participated in).

That wouldn't be a bad thing to write, but it's unreasonable of you to expect that anyone jumping into a discussion would always have to preface their questions with such comments, or to interpret such questions/comments in totally thin-skinned and defensive ways like your paraphrase "you must be wrong. You must be so happy that I turned up to explain stuff for you." The point of this forum is explaining physics concepts, if someone occasionally explains something you already know (as has happened to me plenty of times), a simple "yes, I already knew about that" is fine, no need to scold the person or treat it as a personal attack (especially since the explanation could still be helpful to others reading the thread).

Maybe I have just been fortunate, but I have interacted with quite a number of people who are far more polite and self-aware (and by that I mean more polite and self-aware than both you and me). Clearly anyone who reads this exchange will realize that we have a history (one which included me being given your website address, by you).

How about you back out, let Fredrik answer the question I posed to him and I won't have the feathers on my thin skin ruffled unnecessarily? Is that possible?

As pro quid pro, I promise to not jump into any longer discussions that you are having in other threads.

cheers,

neopolitan

 

[Fredrik]

I am not totally convinced that you have to even consider simultaneity at all in order to prove that the twin paradox is a false paradox. The only reason I am bringing it up is because I disagree very strongly that "B" is forced to calculate that "A" ages 25.6 years during the turn around.

I'm embarrassed to say that I didn't even see post #79 until now. I guess I was too focused on the discussion with DaleSpam at that time. I should at least comment the statement that I just quoted.

You don't have to consider simultaneity to prove that it's a false paradox. We know that there can't be a paradox because the scenario that allegedly contains a paradox is just a triangle in Minkowski space. If three straight lines forming a triangle can cause a paradox, all of mathematics would fall with it.

We can also find out which one of the twins is younger simply by using the postulate that what a clock measures is the integral of $\sqrt{g_{\mu\nu}dx^\mu dx^\nu}$along the curve in Minkowski space that represents its motion. The result of the calculation is the same no matter what coordinate system we use, so we can choose one that's convenient. In any inertial frame, the expression above simplifies to $\sqrt{dt^2-dx^2}$. A's rest frame is an inertial frame, so let's use that. Now we can immediately see that the integral along A's world line is greater than the integral along B's world line, since dx=0 along A's world line, but not along B's world line.

So now we know that there's no paradox, and that B is younger when they meet again, and if we do the integrals (which is easy), we find the exact final ages of both twins. Simultaneity is clearly not needed for any of that.

What we need simultaneity for is to explain what's wrong with the calculation that says that A is younger.

 

[Fredrik]

How about you back out, let Fredrik answer the question I posed to him...

What question? Have I missed something again? This one?

Do you agree that there will be no discontinuity in the image of the Earth clock?

I assumed that was a rhetorical question or something. It's like asking if I agree that bears crap in the woods.

 

[neopolitan]

What question? Have I missed something again? This one?

I assumed that was a rhetorical question or something. It's like asking if I agree that bears crap in the woods.

Ok, we agree about bears, and about the paradox thing. I think it is just that you want to explain why bears don't use those japanese fish ponds full of koi, and I believe you really don't need to explain the wrong answer works once you can show how the right answer is generated. While I think that your approach can lead to some confusion, it is certainly not wrong.

So, nothing substantial to discuss here :)

cheers,

neopolitan

 

[phyti]

Jesse,


But when it comes to the spaceship itself, we would normally use something else against which to define "stationary".

cheers,

neopolitan

How about the distant star field, it works for inertial navigation.
It's the closest thing to fixed position we have.

 

[atyy]

How about the distant star field, it works for inertial navigation.
It's the closest thing to fixed position we have.

GR texts often acknowledge that such motion is absolute and "inertial" in some sense. In GR, an inertial frame is defined to be one in which the observer knows he is inertial using a purely local test, without looking out of the window, but by using an accelerometer.

I haven't seen this definition in SR texts, but even those define the test for non-inertiality using an accelerometer locally. So even if twin B starts off moving at a constant velocity relative to twin A, B will know at the turn-around that he has accelerated in an absolute sense, and will have enough knowledge not to apply the time dilation formula too naively.

 

[neopolitan]

How about the distant star field, it works for inertial navigation.
It's the closest thing to fixed position we have.

While I agree with using the distant star field to a certain extent (since it is sort of fixed as a whole, as it probably consists of an aggregate of all possible trajectories), I'll point out that astronavigation (or celestial navigation) carried out on Earth using the stars also makes use of the fact that the Earth is moving. I suspect that you probably knew that, but it is worth pointing out that there are three informal frames used in star based navigation - the vessel, usually a ship; the Earth, mostly the rotation but with some daily corrections to take into account the orbit), and the stars which are taken to be fixed (but, as physics boffs, we know to not be fixed at all).

I am pretty sure that is the sort of navigation you are talking about, since there is another sort using gyroscopes to track the motion of a vessel which doesn't rely on the distant star field

There was someone who came out recently reporting that the twin paradox was solved due to the fact that we should use the universe as a whole as what would amount to a preferred frame, and in that sort of preferred frame, the traveling twin can be shown to age less rapidly. This is, of course, probably a horrible misrepresentation of what he said.

cheers,

neopolitan

 

[Al68]

If you agree that there is no real need to advance such a clock from 7.2 to 32.8 years at the turnaround, and that the 7.2 and 32.8 year figures are representative of incorrectly using two inertial frames as if they were one then we are not in disagreement.

Of course there is no need to advance such a clock. The "SR simultaneity rule" is not a law of physics, it's just true by convention. Those figures are obtained simply by using the SR simultaneity convention (t'=t-vx/c^2).

But if there is a clock at rest with earth, synched with Earth's clock in Earth's frame, and local to the turnaround, all observers would agree that it reads ~ 20 yrs just before, during, and after the turnaround, in every frame. 20 yrs is the turnaround time in Earth's frame. If this is all you're saying, I think everyone would agree (maybe).

I would even agree that the 7.2 and 32.8 yr figures don't represent anything in reality in an absolute sense. After all, they represent when signals would be received (adjusted for lightspeed delay) by observers in frames at rest with the ship's frames, but not local to the ship itself.

Also, we are not forced to even consider what the Earth's clock reads in the ship's frame. We could just consider a series of clocks at rest with Earth all along the trip, and call those readings "local Earth frame time". Each one would read ship time/0.6, assuming they are synched in Earth's frame. Of course these clocks are out of synch with each other in the ship's frame by vx/c^2, but we could just as rightfully claim that they represent reality.

Al

 

[atyy]

Of course there is no need to advance such a clock. The "SR simultaneity rule" is not a law of physics, it's just true by convention.

Although the sudden jump simultaneity convention works in this particular version of the twin paradox, it's kinda weird. George Jones posted a paper with a nice simultaneity convention that's a lot smoother for B (surprisingly simple too): https://www.physicsforums.com/showthread.php?p=1893032&highlight=accelerated#post1893032

 

[granpa]

why not consider 3 grids of synchronized clocks. one for Earth frame. one for outbound rocket frame. and one for inbound rocket frame.

 

[granpa]

The "SR simultaneity rule" is not a law of physics, it's just true by convention.

it may not be a law of physics but if you use something different then you will have to rewrite the laws of physics for that frame.

after all the whole point of relativity is that the laws of physics are the same in every frame. that is where the Einstein simultaneity rule comes from.

 

[Al68]

it may not be a law of physics but if you use something different then you will have to rewrite the laws of physics for that frame.

after all the whole point of relativity is that the laws of physics are the same in every frame. that is where the Einstein simultaneity rule comes from.

Of course, but we're never required to even consider what Earth's clock reads simultaneous with any event in the ship's frame. I didn't say the rule was wrong, just that we don't need to use it.

Al

 

[Al68]

why not consider 3 grids of synchronized clocks. one for Earth frame. one for outbound rocket frame. and one for inbound rocket frame.

I like this one. No need to worry about what a distant clock in a different frame reads, nothing "weird" happens at the turnaround, and the ship can just look out the window and see what time it is in Earth's frame (local to the ship) at any time. And that clock is the time that the Earth clock would read "now" if the ship decides to return to Earth's inertial frame (stop) near that clock.

Al

 

[Fredrik]

Of course, but we're never required to even consider what Earth's clock reads simultaneous with any event in the ship's frame.

We don't have to think about simultaneity to prove that there's no paradox, or to find the correct final ages of the twins, but we have to do it if we want to explain what's wrong with the calculation that says A is younger.

why not consider 3 grids of synchronized clocks. one for Earth frame. one for outbound rocket frame. and one for inbound rocket frame.

That's what my spacetime diagram does. The jump from 7.2 years to 32.8 years is the correction that's needed for the error that's introduced by simply switching from the first of B's inertial frames to the second.

No need to worry about what a distant clock in a different frame reads, nothing "weird" happens at the turnaround, and the ship can just look out the window and see what time it is in Earth's frame (local to the ship) at any time. And that clock is the time that the Earth clock would read "now" if the ship decides to return to Earth's inertial frame (stop) near that clock.

You only need one of those "grids" (inertial frames) for that, so I assumed that he had something else in mind when he started talking about three of them.

 

[Al68]

Although the sudden jump simultaneity convention works in this particular version of the twin paradox, it's kinda weird. George Jones posted a paper with a nice simultaneity convention that's a lot smoother for B (surprisingly simple too): https://www.physicsforums.com/showthread.php?p=1893032&highlight=accelerated#post1893032

It's only weird because looking at the coordinate distance of the ship from Earth (or equivalently, clock time on earth) in the ship's frame before, during, and after the turnaround is, IMO, just a weird way to look at it. It's like the only reason to look at it this way is to have an exercise in the simultaneity rule for its own sake.

I'll have to look at that other thread.

Al

 

[granpa]

It's only weird because looking at the coordinate distance of the ship from Earth (or equivalently, clock time on earth) in the ship's frame before, during, and after the turnaround is, IMO, just a weird way to look at it. It's like the only reason to look at it this way is to have an exercise in the simultaneity rule for its own sake.

I'll have to look at that other thread.

Al

as has already been pointed out several times, we don't need simultaneity to figure out how much less the twin will age. we need it to explain why the twin that ages less observes the other twin aging more slowly on the outbound and inbound parts of the trip. that is the whole point of the paradox.

 

[Al68]

We don't have to think about simultaneity to prove that there's no paradox, or to find the correct final ages of the twins, but we have to do it if we want to explain what's wrong with the calculation that says A is younger.

Sure, if someone makes the erroneous calculation.

The jump from 7.2 years to 32.8 years is the correction that's needed for the error that's introduced by simply switching from the first of B's inertial frames to the second.

Sure, if we care about Earth's clock reading simultaneous with the turnaround in the ship's frame.

You only need one of those "grids" (inertial frames) for that, so I assumed that he had something else in mind when he started talking about three of them.

Well, the other two grids could be used to look at the ship's frame(s) from Earth's pov.

Fredrik, do you believe that the "jump" in the coordinate position of the Earth (and Earth's clock, since it's directly dependent) in the ship's frame during the acceleration is more than just a result of changing the pov/reference frame?

Al

 

[Al68]

as has already been pointed out several times, we don't need simultaneity to figure out how much less the twin will age. we need it to explain why the twin that ages less observes the other twin aging more slowly on the outbound and inbound parts of the trip. that is the whole point of the paradox.

Only if the ship's twin looks at it in this "weird" way. If he ignores what time on Earth is simultaneous with local time, he could just look out his window at a clock in the grid. That won't tell him what time it is on Earth "now", but that's OK if he's not asking. It will tell him how much time will have elapsed on Earth if he chooses to return to Earth's frame (stop) near any of the clocks. That should make it clear to him that, since each clock shows shows more elapsed time than his own (by the gamma factor), that his twin on Earth will be older whenever and wherever he returns to Earth's frame.

Al

 

[granpa]

Only if the ship's twin looks at it in this "weird" way. If he ignores what time on Earth is simultaneous with local time, he could just look out his window at a clock in the grid. That won't tell him what time it is on Earth "now", but that's OK if he's not asking. It will tell him how much time will have elapsed on Earth if he chooses to return to Earth's frame (stop) near any of the clocks. That should make it clear to him that, since each clock shows shows more elapsed time than his own (by the gamma factor), that his twin on Earth will be older whenever and wherever he returns to Earth's frame.

Al

it wouldn't seem weird to the people on the rocket. thear time would seem perfectly natural to them.

if the Earth twin accelerates and catches up with the moving twin and they spent the rest of their lives traveling at relativistic speed then Earth time would weird. its all relative.

 

[neopolitan]

It's only weird because looking at the coordinate distance of the ship from Earth (or equivalently, clock time on earth) in the ship's frame before, during, and after the turnaround is, IMO, just a weird way to look at it. It's like the only reason to look at it this way is to have an exercise in the simultaneity rule for its own sake.

I'll have to look at that other thread.

Al

Woot, that is a far more complex version of what I was thinking. Although I never thought about in terms of that prism-like effect in Figures 5 and 6, I do think I understand how it works.

Thanks for that link, Al.

cheers,

neopolitan

 

[Al68]

Woot, that is a far more complex version of what I was thinking. Although I never thought about in terms of that prism-like effect in Figures 5 and 6, I do think I understand how it works.

Thanks for that link, Al.

cheers,

neopolitan

Well, atyy provided the link, I only quoted his post.

Al

 

[Fredrik]

Fredrik, do you believe that the "jump" in the coordinate position of the Earth (and Earth's clock, since it's directly dependent) in the ship's frame during the acceleration is more than just a result of changing the pov/reference frame?

No.

(What else would it be? I can't even think of a wrong answer to that question.)

 

[Al68]

No.

(What else would it be? I can't even think of a wrong answer to that question.)

OK, something in another post made me wonder if I was missing something. My bad.

Al

 

[Dale]

We don't have to think about simultaneity to prove that there's no paradox, or to find the correct final ages of the twins, but we have to do it if we want to explain what's wrong with the calculation that says A is younger.

Q: What's wrong with the calculation that says A is younger?

A: It treats a non-inertial frame as an inertial frame.

No simultaneity needed.

 

[Fredrik]

Q: What's wrong with the calculation that says A is younger?

A: It treats a non-inertial frame as an inertial frame.

No, it doesnt. It doesn't even try to associate a coordinate system with B's world line. All it does is to combine the result of two calculations performed in two different coordinate systems. It must seem quite plausible to someone less experienced with relativity calculations that that should work, since the two frames agree that A's aging rate is 60% of B's. You clearly don't have to believe that B's path is a geodesic to think that the incorrect calculation looks correct.

Usually when someone makes the mistake of switching coordinate systems in the middle of a calculation, the reason why that doesn't work is that the two coordinate systems disagree about the specific thing you're calculating. How long did your plane trip take? Calculate it as local arrival time minus local departure time and you get the wrong result if the destination is in another time zone. The answer is wrong because the coordinate systems (time zones) disagree about the time. But in the twin paradox, the coordinate systems don't disagree about the aging rate. The only relevant thing they disagree about is simultaneity.

 

[granpa]

Q: What's wrong with the calculation that says A is younger?

A: It treats a non-inertial frame as an inertial frame.

No simultaneity needed.

but it is a fact that the moving twin perceives the stationary twin to be aging more slowly both on the outbound and the inbound parts of the trip. how do you explain this to a student?

 

[JesseM]

but it is a fact that the moving twin perceives the stationary twin to be aging more slowly both on the outbound and the inbound parts of the trip. how do you explain this to a student?

But he doesn't really "perceive" this in any direct sense (that certainly isn't what he sees), it's just that during each part of the trip, if he uses an inertial coordinate system where he is at rest during that phase, then in that coordinate system the other twin will be aging more slowly than himself. I would just explain to the student that you can't combine the elapsed ages for the inertial twin in the two coordinate systems for each leg of the trip, because the definition of simultaneity used in the first coordinate system is different from the definition in the second, so the inertial twin's age at the moment of the turnaround in the first one is very different from the inertial twin's age at the moment of the turnaround in the second one.

 

[granpa]

But he doesn't really "perceive" this in any direct sense (that certainly isn't what he sees),

it is exactly what he 'sees' as long as he takes light travel time into account. where are you getting the idea that it isnt. this is simple relativity.

 

[Fredrik]

I would just explain to the student that you can't combine the elapsed ages for the inertial twin in the two coordinate systems for each leg of the trip, because the definition of simultaneity used in the first coordinate system is different from the definition in the second,

What we're really discussing here is if the second half of the quoted text above needs to be included at all. DaleSpam is of the opinion that all we need to say is that there's no inertial frame in which B is stationary during the whole trip. "The End". My opinion is that this doesn't really explain why it's wrong to just use the time dilation formula on the two straight parts of B's world line separately. I think the only thing that can explain that is what you just said.

 

[Dale]

No, it doesnt. It doesn't even try to associate a coordinate system with B's world line.

Yes it does, specifically it tries to associate a coordinate system where B is at rest the whole time (B's world line is straight and vertical at all points). That is a non-inertial coordinate system.

All it does is to combine the result of two calculations performed in two different coordinate systems.

If you want to combine results performed in two different coordinate systems you must always properly transform your results from one into the other. Since you don't explicitly perform a coordinate transform you are implicitly working in a single coordinate system and that coordinate system is non-inertial.

 

[granpa]

so you accept that the moving twin sees the stationary twin aging more slowly during the outbound part of his trip and also during the inbound part of his trip. and you accept that it is the fact that he accelerates during the turn around that causes him to actually age less. so what exactly are you arguing?

 

[Dale]

but it is a fact that the moving twin perceives the stationary twin to be aging more slowly both on the outbound and the inbound parts of the trip. how do you explain this to a student?

Again, this is the same point I have been making all along. While the twins are not together they cannot make local comparisons of their clocks, so any "relative aging" claims are actually statements about the time coordinate in a given reference frame. Any reference frame where the stationary twin is aging more slowly on both legs is a non-inertial reference frame.

so you accept that the moving twin sees the stationary twin aging more slowly during the outbound part of his trip and also during the inbound part of his trip.

Only in a non-inertial reference frame.

and you accept that it is the fact that he accelerates during the turn around that causes him to actually age less.

The fact that he accelerates during the turn around is what indicates that his rest frame is non-inertial.

so what exactly are you arguing?

I am arguing that the spacetime geometric approach is completely sufficient for resolving the paradox because the metric is different in non-inertial reference frames and you cannot get a twin paradox in flat spacetime without using a non-inertial reference frame.

IMO, it is more important to teach a student to identify and avoid non-inertial reference frames than to teach them about confusing and arbitrary simultaneity conventions that can arise in non-inertial reference frames. Most students struggle with the relativity of simultaneity more than any other concept even in inertial reference frames. The extra confusion of simultaneity conventions in non-inertial frames is not necessary to the resolution and therefore should be avoided.

 

[granpa]

Again, this is the same point I have been making all along. While the twins are not together they cannot make local comparisons of their clocks, so any "relative aging" claims are actually statements about the time coordinate in a given reference frame.

they are statements about what each will actually see. while they are not accelerating

Any reference frame where the stationary twin is aging more slowly on both legs is a non-inertial reference frame.

that is exactly what we are saying. why do you say that as though it disagrees with what we are saying.
non-inertial=accelerating. acceleration=change in simultaneity.

 

[granpa]

if we assume that the laws of physics are the same in every frame then we must conclude that simultaneity is lost and that it follows the Einstein convention. so do you not believe that the laws of physics are the same in every frame?

 

[JesseM]

What we're really discussing here is if the second half of the quoted text above needs to be included at all. DaleSpam is of the opinion that all we need to say is that there's no inertial frame in which B is stationary during the whole trip. "The End". My opinion is that this doesn't really explain why it's wrong to just use the time dilation formula on the two straight parts of B's world line separately. I think the only thing that can explain that is what you just said.

If this is an accurate summary of DaleSpam's view I think I'd agree with you and disagree with DaleSpam--after all, if we want to integrate along a curve to find its length from point A to C in Euclidean space, we're free to pick some point B along the curve between A and B, and use one cartesian coordinate system to do an integral which gives us the length from A to B, and a different cartesian coordinate system to do an integral which gives us the length from B to C, and then add these two lengths. Here we are not using a single non-cartesian coordinate system, but rather adding results from two different cartesian coordinate systems, which is perfectly valid in this situation. If different inertial frames in SR agreed on simultaneity you could do something similar in the twin paradox, using one inertial frame to find the time elapsed on the inertial twin's clock between the moment the other twin departed and the moment the other twin turned around, and using a different inertial frame to find the time elapsed on the inertial twin's clock between the moment the other twin turned around and the moment they reunited. So knowing about the relativity of simultaneity is key to understanding why you can't just add partial elapsed times from two different frames in this way to get the total elapsed time.