Twin Paradox Problem: Do Twins Age Differently?

[harrylin]

Nope, you got that wrong about Newton mechanics and I didn't mention optics or Huygens.

Newton thought that light consisted of particles that travel at limited speed in vacuum, and faster in glass. Perhaps you did not mean equations of mechanics but system transformation equations? Those did not include light speed at all and had nothing to do with it.

 

[TrickyDicky]

I said nothing about what Newton thought, and explicitly said that of course people knew light had finite speed long before.
Are transformation equations not equations of mechanics?
Do you mean you have never heard about the difference between Galilei relativity in classical mechanics and the special relativity of Einstein?

 

[harrylin]

I said nothing about what Newton thought, and explicitly said that of course people knew light had finite speed long before.
Are transformation equations not equations of mechanics?
Do you mean you have never heard about the difference between Galilei relativity in classical mechanics and the special relativity of Einstein?

Sure I did, and I also know very well some of the nonsense that is said about it. Newton did not include the idea of infinite light speed in any of his equations. However, that has nothing to do with the topic and I may have now traced your disagreement with George to another wrong phrasing that has nothing to do with what I think you meant to say. So, to limit the accumulation of waste of time, please tell me if this is correct:

When you wrote:
"As soon as you are talking about percentages of c you are using inertial frames",
did you mean that "0.9c" unwittingly refers to one or two inertial frames? If so, you may be right about that. I don't see how "0.9c" can be determined without referral to a reference system.

 

[TrickyDicky]

Newton did not include the idea of infinite light speed in any of his equations.

And I never said that he included it, but that classical mechanics assumed galileo relativity in which light speed is treated as infinite.

However, that has nothing to do with the topic and I may have now traced your disagreement with George to another wrong phrasing that has nothing to do with what I think you meant to say. So, to limit the accumulation of waste of time, please tell me if this is correct:

When you wrote:
"As soon as you are talking about percentages of c you are using inertial frames",
did you mean that "0.9c" unwittingly refers to one or two inertial frames? If so, you may be right about that. I don't see how "0.9c" can be determined without referral to a reference system.

Yes, something like that.

 

[zonde]

Certainly I was not clear enough in posts #8 and #27, but what was not clear??

I reread you post #8 and I can say that when I read it first time I didn't get that you are talking only about Earth frame.

At the moment that you make a turn-around, you:
1. can not influence what happens on earth
2. have only one inertial reference system at your disposal, which is the one of the Earth (ECI frame).
Next, after the turn-around you can decide to still indirectly use the ECI frame (just as astronauts always have done until now in real life), or set up a new inertial reference system by re-synchronizing your clocks. That system maps a different distant time as the other ones.

When that is understood, it is immediately clear that it's just a matter of switching reference frames, so that alternative scenario's with fly-by at the same velocities cannot have a different effect. There is no problem with that illustration, but it should not be presented as spooky action at a distance.

I said: "This "jump" ahead is a coordinate effect." I suppose this clearly says it's not physical effect.

Anyways, my objection is that you analyze "twin paradox" only from one inertial frame. I don't know what was historical role of that paradox but I suppose that all current discussion around "twin paradox" are concerned about consistency of SR.
Obviously to convince anybody about consistency of SR you have to present two alternative ways (involving different reference frames) how you can get to the same result i.e. you have to analyze "twin paradox" from perspective of both twins in parallel.

It's clear how it looks from perspective of first (stay at home) twin - second twin is time dilated by the same factor in forward and backward trip (the same speed). Acceleration does not come into the picture.

Now how it looks from perspective of second twin - first twin is time dilated by the same factor in forward and backward trip (the same speed). But contrary to the first case acceleration (because we have to switch coordinates) has effect. And result of this clearly unphysical effect is that we just add some accumulated time to Earth clock.

 

[zonde]

This is just another of the many ways to analyze the Twin Paradox and they all agree, as you pointed out, concerning their picture of Earth when they meet. And they all agree with the final outcome. And they all agree with everything else in between that is observable. They don't agree on what you are calling remote "now" which is another way of saying "coordinate time" but that is consistent with the calculation of the Proper Time on both clocks. The coordinate times can vary all over the place between these different frames but when you apply the time dilation you get the same Proper Time at each event no matter what frame you use.

You have proposed three inertial observers. You could have proposed analyzing what happens according to each of their rest frames and there would be no frame jumping and no acceleration. I hope you're not suggesting that these three inertial frames are not all equally valid and I hope you're not suggesting that an analysis based on jumping between two of those frames is somehow more valid or better suited to explaining what is "really" happening in the Twin Paradox, are you?

If you want to get some intuition about LT you would want to do some "jumping" between frames, right?

If the traveling twin actually knows physics, he would be aware that there is no such thing as the "real pace of the Earth twin's clock while traveling towards him". He would know that he can analyze the pace of both of their clocks from any inertial frame of reference and each one can assign different paces to their two clocks, none of which can be considered "real". What's real is the visual data that you call an illusion. Furthermore, each one of these inertial reference frames will agree on exactly what each twin sees throughout the entire trip. You can also analyze the scenario from non-inertial frames or jumping inertial frames and they can assign completely different paces to the two clocks but they will all agree on what each twin really sees.

Do you completely agree with everything I said in the above quote?

I agree with everything except statement in bold.

If you do, then please read this quote from post #42:

You just said that real meant within a given inertial frame of reference and now you want to talk about a frame that the traveling twin is a rest in. But it cannot be an inertial frame for the entire trip so how does that work?

I don't know why you want to make this so complicated. Let's do what you said and pick as our given inertial frame of reference the one in which the Earth twin is at rest and in which the traveling twin starts out and ends up at rest. In this frame the Earth twin's clock runs normally.

Now the traveling twin accelerates instantly to a speed of 90%c. Gamma at this speed is 2.294 (not 0.4359 as you claim in your linked diagram). That means that a clock traveling at 90%c will run slower by a factor of 2.294. The traveling twin's clock will run slower than the Earth twin's clock by this amount during his entire trip so when he gets back the Earth clock has advanced by 2.294 times whatever his clock advanced. This is what really happens according to your definition of real and it's exactly what I said would happen in post #5 and so I don't know why you say it's misleading.

Do you completely agree with everything I said in the above quote?

I will leave this without answer. There is some discussion involving definitions of real and I will try to stay out of this (within this thread).

If you do, then don't you think it is important to point out that whatever frame provides us with the simplest way to determine what will happen is just as valid as any other frame(s) and no other analysis based on any other frame(s) will provide us with any additional insight or information into what is happening or what any observer observes and so there is no point in discussing other frame(s) except to show that they all agree on what each observer observes throughout the entire scenario?

Yes, I agree. This is important. We can pick whatever frame we like for analysis. But if we want to demonstrate consistency of SR (for those who are not sure about this) we can compare any frames using LT.

 

[harrylin]

And I never said that he included it, but that classical mechanics assumed galileo relativity in which light speed is treated as infinite.

Yes, that's what I thought that you said; and I said that the Galilean relativity does not treat light speed at all.

Yes, something like that.

Good - then the discussion is back on track.

 

[TrickyDicky]

Galilean relativity does not treat light speed at all.

This is basic classical mechanics.
In which time is absolute, (a parameter, not a dimension as in Minkowskian SR), and space is Euclidean, this implies that light speed is infinite, there is no relativity of simultaneity, it takes no time for light signals to travel, now is the same now for anybody. Is this so hard to understand? Ask anyone versed on classical mechanics if you don't believe me.

 

[harrylin]

I reread you post #8 and I can say that when I read it first time I didn't get that you are talking only about Earth frame. [..]

The Earth is not an inertial frame, while in SR everything is analyzed with inertial frames; I next tried to clarify in post #27 that any inertial frame and any amount of switching between such frames can be used for the analysis, because that is how such transformations work. And we are free to map ourselves in any of those.

Anyways, my objection is that you analyze "twin paradox" only from one inertial frame. [...]

No, you took that message out of context: it just explains that the Earth doesn't need to be treated like an inertial frame, in response to an issue that TrickyDicky raised (but he next said that he meant a different issue, and that issue relates to the original purpose of the "twin" example). I already stressed in post #8 that one can choose to use any combination of inertial frames that one likes.

The past threads explain how the group property of the Lorentz transformations works by means of more than enough examples (linked to in post #3). Surely nobody wants to go through the motions again, as if it is not sufficiently explained?

[edit: I now removed my last remark which probably just related to how different people phrase the same thing differently]

 

[harrylin]

This is basic classical mechanics.
In which time is absolute, (a parameter, not a dimension as in SR), and space is Euclidean, this implies that light speed is infinite, there is no relativity of simultaneity, it takes no time for light signals to travel, now is the same now for anybody. Is this so hard to understand? Ask anyone versed on classical mechanics if you don't believe me.

I am rather versed in classical mechanics and that's what I referred to in my post [edit: post #108]; if you like we can discuss that in a topic on that issue (we must not hijack this thread; and perhaps this was already discussed in the past, in which case we can just continue an old thread on that topic).

 

[ghwellsjr]

If you want to get some intuition about LT you would want to do some "jumping" between frames, right?

The term "frame jumping" means explaining a scenario using two or more inertial frames where each is used for a different part of the scenario. For example, using one inertial frame for the traveling twin during his outbound portion of the trip and a second inertial frame during the inbound portion of the trip. I'm not aware of how the Lorentz Transformation process would be involved. Can you explain?

...
We can pick whatever frame we like for analysis. But if we want to demonstrate consistency of SR (for those who are not sure about this) we can compare any frames using LT.

Since the Lorentz Transformation process is pure algebra, how does it demonstrate consistency of SR?

EDIT: After posting this response to your response to me in post #111, I see that you provided the answers to my questions in post #110, so you don't need to answer here, I will compose another post in response to #110 (even though it was not directed at me).

 

[TrickyDicky]

that's what I referred to in my post;

What post? And if so, why did you keep saying I was wrong if that was what you referred to in your post?

 

[harrylin]

What post? And if so, why did you keep saying I was wrong if that was what you referred to in your post?

Fixed now. If you start that topic, we can properly discuss it; I won't further hijack this thread with that topic.

 

[ghwellsjr]

Anyways, my objection is that you analyze "twin paradox" only from one inertial frame. I don't know what was historical role of that paradox but I suppose that all current discussion around "twin paradox" are concerned about consistency of SR.

Obviously to convince anybody about consistency of SR you have to present two alternative ways (involving different reference frames) how you can get to the same result i.e. you have to analyze "twin paradox" from perspective of both twins in parallel.It's clear how it looks from perspective of first (stay at home) twin - second twin is time dilated by the same factor in forward and backward trip (the same speed). Acceleration does not come into the picture.

Just because you analyze the Twin Paradox from a frame in which the Earth twin is at rest and all the time dilation applies to the traveling twin, this does not mean that this frame provides the Earth twin with any insight into the traveling twin's time dilation. Time dilation cannot be observed, it can only be calculated based on a reference frame. What can be observed by the Earth twin is the Relativistic Doppler which this frame will allow you to calculate (although there is an easier way). This is the only perspective that the Earth twin has of the traveling twin's clock.

In the same way, this frame will also allow you to calculate what the traveling twin sees of the Earth twin's clock. They both will see the other one's clock going slower than their own by exactly the same amount, and it's not the time dilation factor since it is different for each one in this frame.

Now how it looks from perspective of second twin - first twin is time dilated by the same factor in forward and backward trip (the same speed). But contrary to the first case acceleration (because we have to switch coordinates) has effect. And result of this clearly unphysical effect is that we just add some accumulated time to Earth clock.

Here is where you have chosen to jump frames so that the traveling twin is always at rest. But before doing that, I want to make sure you agree that we can use a frame in which the traveling twin is at rest during the outbound portion of the trip but for the entire trip. Do you agree with that?

In this case, the traveling twin will have no time dilation for the first half of the trip while the Earth twin has the same time dilation that the traveling twin had in the first frame where the Earth twin was at rest, correct? But then when the traveling twin turns around, he will have more time dilation than the Earth twin continues to have, correct? Do you see this as a legitimate explanation? And do you also see that the this frame does not provide either twin with any more perspective or insight or observation than they had with the first frame? And do you understand that even in this single inertial frame, we can calculate exactly what each twin can observe of each others clock during the entire trip and it will be exactly the same as what we calculated in the first frame?

Now we can go on to a third frame in which the traveling twin is at rest during the inbound portion of the trip but we will apply it during the entire trip. And all the same sorts of questions and answers apply, correct?

Finally, since you want to have the traveling twin and the Earth twin jump frames at the moment of turn around, I beg you to provide us with the details of the calculations. Let's assume that the traveling twin turns around after one year on his clock and is traveling at 90%c. Can you do that? And can you also show the calculations for what each twin sees of the other twin's clock during the entire scenario, please?

And then, to address your comments to me, I'd like you to show us how you use the LT in this process, OK?

 

[TrickyDicky]

Fixed now. If you start that topic, we can properly discuss it; I won't further hijack this thread with that topic.

Admitting your mistake doesn't hijack any thread. When you decided to answer something I wrote in response to ghwellsjr, not you, you must have not thought you were hijacking anything, curiously it only concerns you when it is clear what I said was right.

 

[D H]

This is basic classical mechanics.
In which time is absolute, (a parameter, not a dimension as in Minkowskian SR), and space is Euclidean, this implies that light speed is infinite, there is no relativity of simultaneity, it takes no time for light signals to travel, now is the same now for anybody. Is this so hard to understand? Ask anyone versed on classical mechanics if you don't believe me.

The highlighted text is nonsense. Newton was a champion of Rømer's calculation of the finite speed of light. Galilean relativity merely implies that the speed of light is not the same to all observers. The very purpose of the Michelson Morley experiment was to find the ether frame in which light did move at Maxwell's c by observing those variations in the speed of light that were dictated by Galilean relativity. The experiment famously failed to find those predicted variations in the speed of light.

There's nothing wrong in Newtonian mechanics with something going faster than the speed of light. Light was just a bunch of fast moving little particles to Newton. The finite speed of light had nothing to do with the nature of space-time in Newtonian mechanics. The concept of non Euclidean geometries didn't even arise until 100 years after Newton died, and that was just as a mathematical curiosity. Another 70 years passed before those concepts were shown to apply directly to our physical universe.

This discussion of the speed of light in Newtonian mechanics is ill-informed and is off topic, so stop debating it.

For that matter, this entire thread is in deep trouble. This site is not the place for anti-relativity crackpots.

 

[harrylin]

Admitting your mistake doesn't hijack any thread. When you decided to answer something I wrote in response to ghwellsjr, not you, you must have not thought you were hijacking anything, curiously it only concerns you when it is clear what I said was right.

No, I tried to be polite by not explaining that it was exactly what I earlier referred to as nonsense (which it is). Now the hijacking of this thread by your topic continued, as D_H felt it necessary to explain this to you in this thread on the twin paradox, which has nothing to do with it.

 

[harrylin]

[..] this entire thread is in deep trouble. This site is not the place for anti-relativity crackpots.

This thread is mostly a discussion among SR specialists of how to best explain beginner questions on SR such as the OP, with some noise due to discussions about words. Where is any reference to anti-relativity crackpots??

 

[robinpike]

No, that's not true. Not in any objective sense.

Here's an analogy. Suppose you have a system of roads, and every road has markers on it every 100 meters. You have two roads that meet at a point, diverge, and then come back together at a second point. You can compare the number of markers along the two different roads between the first meeting and the second meeting. You might find that one road has a greater number of markers between the two points than the other road. Does that mean that the road with the greater number of markers must have its markers closer together, or that the road with the small number of markers must have its markers farther apart? No, it's just that the distance between two points depends on the path taken. The number of markers is an accurate measure of these two distances.

In SR, the analogy of "number of markers along a road" is "number of ticks of a clock along a spacetime path". Two spacetime paths meet at some point, diverge, and then come back together later. The number of clock ticks between the two points is different for the two different paths. Does this mean that one clock's ticks come closer together, or that the other clock's ticks come farther apart? No, it's just that the proper time between two points depends on the spacetime path taken.

That is a good try at trying to solve the problem... but it fails - for it simply replaces the change in the rate of ticks of the clock, with a change in the length of the spacetime path.

The same problem still persists, but now becomes: if the initial acceleration reduces the length of the traveling twin's spacetime path as compared to the stay at home twin's spacetime path, how does the traveling twin use acceleration at the end of his journey to return to the stay at home twin's spacetime path?

(And on a point of understanding your description of a spacetime path, not sure how the shorter spacetime path can return to the stay at home twin's longer space time path? Is the stay at home twin's spacetime path curved or something?)

 

[Nugatory]

The same problem still persists, but now becomes: if the initial acceleration reduces the length of the traveling twin's spacetime path as compared to the stay at home twin's spacetime path, how does the traveling twin use acceleration at the end of his journey to return to the stay at home twin's spacetime path?

It's not that the acceleration/deceleration reduces the length of the path, it's that it sets the traveling twin on a different path. I've attached a picture that shows the different paths followed by the twins through space-time, how they start out on the same path , and how the traveling twns acceleration and decelerations change his path through spacetime.

Now, because these are paths through space-time, not just space, it's not that surprising that the two twins measure different times along the two different paths. Basically if you travel one of those paths, your wristwatch will tick off the 'distance' traveled along the path.

What is more surprising at first glance, and sometimes throws people, is that the traveling twin appears to be taking the longer path through space-time. You have to remember that in space-time, the distance between points is calculated as $s^2=x^2-t^2$, not the $s^2=x^2+y^2$ that the Pythagorean theorem tells you to expect in space.

 

[arindamsinha]

This is one of the most intriguing paradoxes in relativity. Majority opinion today appears to be that the paradox is resolved within SR, considering the asymmetry between the two observers (stationary vs. traveling twins). A reasonably vocal minority opinion considers this as not resolved, though, perhaps with some justification.

Ultimately, for a real clock difference to show up (e.g. as in GPS satellites), there has to be some asymmetry between the two observers/bodies in question. The GPS satellite represents the traveling twin, though it never reverses course, and keeps traveling. Yet the GPS satellite shows measurable velocity time dilation (ignoring the gravitational TD part).

In my humble opinion, SR being a kinematic theory without a preferred frame of reference, does not admit of such an asymmetric solution within its framework. A mutual acceleration between two bodies can be considered kinematics, but ascribing acceleration to one body and not the other, where there is no preferred frame, is a study in dynamics, i.e. we need to get into the physical reason behind such an asymmetric acceleration. This takes it outside the domain of a kinematic theory.

GR does take into consideration momentum conservation and creates a preferred frame (though many would not admit this!), at least in scenarios where we are considering two bodies, one of which is massive and the other very small.

Turns out GR gives a more satisfactory explanation to the observed experimental phenomena, by introducing the preferred frame (the CG of a large mass) when considering the time dilation of much smaller masses in the vicinity.

 

[harrylin]

The problem with Relativity's explanation for the Twin paradox, is that, once back on earth, for the traveling twin's clock to have a lesser time than the stay at home twin's clock, it can be deduced that the rate of time on the traveling twin's clock must have slowed down at some point during the journey. [..] any further acceleration can only cause the clock's rate of time to slow down even more...

As I noted in post #103, likely you would agree with a change rephrasing as follows:

"it can be deduced that according to any inertial coordinate system the clock rate of the traveling twin's clock must have slowed down at some point during the journey."

I don't understand what the problem is. In 1905, right from the start, it was already deduced that:

"we conclude that a balance-clock at the equator must go more slowly, by a very small amount, than a precisely similar clock situated at one of the poles under otherwise identical conditions."*
- section 4 of http://www.fourmilab.ch/etexts/einstein/specrel/www/

And in 1911 it was remarked that the space traveller is :

"without possibility of coming back to inform us of the result of his voyage, since any attempt of the same kind could only transport him increasingly forward [in time]".
- p.50 of http://en.wikisource.org/wiki/The_Evolution_of_Space_and_Time

Perhaps the "problem" that you mean is that SR only explains the consequences of the "special" relativity principle; SR gives no interpretation of what "really" is happening. Thus it could happen that, despite both using Minkowski spacetime, Langevin explained it with the ether model while stevendaryl here explains it with the block universe model.

* this SR analysis did of course not account for the gravitational potential

 

[zonde]

Just because you analyze the Twin Paradox from a frame in which the Earth twin is at rest and all the time dilation applies to the traveling twin, this does not mean that this frame provides the Earth twin with any insight into the traveling twin's time dilation. Time dilation cannot be observed, it can only be calculated based on a reference frame.

As for me that's fine.

What can be observed by the Earth twin is the Relativistic Doppler which this frame will allow you to calculate (although there is an easier way). This is the only perspective that the Earth twin has of the traveling twin's clock.

Not sure I understand. Are you saying that the only perspective that I can have about something is what I can observe about this something?

Here is where you have chosen to jump frames so that the traveling twin is always at rest. But before doing that, I want to make sure you agree that we can use a frame in which the traveling twin is at rest during the outbound portion of the trip but for the entire trip. Do you agree with that?

In this case, the traveling twin will have no time dilation for the first half of the trip while the Earth twin has the same time dilation that the traveling twin had in the first frame where the Earth twin was at rest, correct? But then when the traveling twin turns around, he will have more time dilation than the Earth twin continues to have, correct? Do you see this as a legitimate explanation? And do you also see that the this frame does not provide either twin with any more perspective or insight or observation than they had with the first frame? And do you understand that even in this single inertial frame, we can calculate exactly what each twin can observe of each others clock during the entire trip and it will be exactly the same as what we calculated in the first frame?

Now we can go on to a third frame in which the traveling twin is at rest during the inbound portion of the trip but we will apply it during the entire trip. And all the same sorts of questions and answers apply, correct?

Mostly yes. I just want to point out that perspective is not the same thing as observation. We can have different interpretations (perspectives) about the same observations.

Finally, since you want to have the traveling twin and the Earth twin jump frames at the moment of turn around, I beg you to provide us with the details of the calculations. Let's assume that the traveling twin turns around after one year on his clock and is traveling at 90%c. Can you do that? And can you also show the calculations for what each twin sees of the other twin's clock during the entire scenario, please?

And then, to address your comments to me, I'd like you to show us how you use the LT in this process, OK?

I consider spacetime diagrams much more convenient tool to demonstrate "frame jumping" and role of LT in this. Anyways for me spacetime diagrams come first and then math. And you will have to wait a bit.

 

[zn5252]

Please refer to the beautiful explanation of this paradox in this book :

https://www.amazon.com/dp/0716723271/?tag=pfamazon01-20

You will then be able to sleep at night...

 

[Dale]

In my humble opinion, SR being a kinematic theory without a preferred frame of reference, does not admit of such an asymmetric solution within its framework.

I disagree. SR does not have a preferred frame, but it does have a preferred class of frames, inertial frames. In that class of frames you can distinguish kinematically which twin is accelerating.

 

[stevendaryl]

That is a good try at trying to solve the problem... but it fails - for it simply replaces the change in the rate of ticks of the clock, with a change in the length of the spacetime path.

The same problem still persists, but now becomes: if the initial acceleration reduces the length of the traveling twin's spacetime path as compared to the stay at home twin's spacetime path, how does the traveling twin use acceleration at the end of his journey to return to the stay at home twin's spacetime path?

(And on a point of understanding your description of a spacetime path, not sure how the shorter spacetime path can return to the stay at home twin's longer space time path? Is the stay at home twin's spacetime path curved or something?)

There are two different aspects to time dilation and the twin paradox: The first is understanding how mutual time dilation (each twin views the other twin's clock to be running slow) can produce an asymmetric result (the traveling twin's clock advances less for the whole trip than the stay-at-home twin). The second is the non-Euclidean metric of Minkowsky space. The first aspect has a direct analogy with Euclidean geometry. The second does not.

By the "non-Euclidean metric of Minkowsky space", I mean this: In Euclidean geometry, if you have a line segment that runs from point A to point B, and the displacement in going from A to B is X in the x-direction and Y in the y-direction, then then length of the line segment L is given by L² = X² + Y². In contrast, if point A and B are points in Minkowsky space, and the separation is X in the x-direction and T in the time-direction, then the proper time of the inertial path connecting A and B is given by L² = (cT)² - X². That minus sign is the reason moving clocks run slower, instead of faster. I don't know of a really good way to understand why spacetime has that minus sign in its metric, other than working with it and seeing how it fits together. For example, if you assume that light has speed c in all directions in one frame, and you assume that getting a straight rod moving in a direction perpendicular to its length doesn't change that length, then you automatically get time dilation for a moving "light clock" formed by a pair of mirrors on either end of a moving rod, with a pulse of light bouncing back and forth between them.

But a lot of the trouble have with the twin paradox is not in accepting time dilation, it's in understanding how MUTUAL time dilation can lead to an asymmetric result in the elapsed times of the two twins. This is where the analogy with Euclidean geometry helps.

You have two roads that intersect at point A. The "slope" of the second road relative to the first road is m. (Slope is the tangent of the angle between the roads). If s₁ is the distance along the first road, and s₂ is the distance along the second road, then we can relate the two as follows:

Imagine being at a point on the first road at distance marker s₁ , and looking in the perpendicular direction toward the second road to see what the "corresponding" distance marker is there. Euclidean geometry predicts that

$ds_2 = ds_1 \sqrt{1+m^2}$

So $ds_2 > ds_1$

If you move $ds_1$ along the first road, then the corresponding distance marker on the second road changes by a greater amount, $ds_2$. But this effect is completely mutual! A traveler on the second road comparing the distance marker $s_2$ to the corresponding distance marker $s_1$ on the first road (which he sees by looking in a direction perpendicular to his road) will likewise find:

$ds_1 = ds_2 \sqrt{1+m^2}$

So $ds_2 > ds_1$

That seems like a contradiction. The first traveler thinks that the markers on the second road are increasing faster, and the second traveler thinks that the markers on the first road are increasing faster. How can they both be right? It's because while each traveler sets up a correspondence between markers on one road and markers on the other road, they use a DIFFERENT correspondence. The traveler on the first road associates two points on a line perpendicular to the first road, while the traveler on the second road associates two points on a line perpendicular to the second road. Those are two different correspondences.

Now, if the first road continues straight, but the second road makes a turn and comes back to meet the first road, then the first traveler can compute the relationship between distance markers on the two roads as follows:

$s_2$ at finish = $s_2$ at start + $\Delta s_2$

where $\Delta s_2$ is computed by

$\Delta s_2 = \int \sqrt{1+m^2} ds_1$

If the slope $m$ is greater than 0 anywhere along the route, then

$\Delta s_2 > \Delta s_1$

This is the same mystery as in the twin paradox. Slope is relative, each traveler thinks that the other road has a nonzero slope. But when the two roads get back together, the difference in distance markers along the two roads is objective. How is that possible? Why can't the traveler on the "bent" road use the analogous formula to conclude that $\Delta s_1 > \Delta s_2$?

The answer is that when the second road makes a turn, its notion of the "perpendicular" direction changes, and its notion of the "corresponding" point on the other road also changes. The formula

$ds_2 = ds_1 \sqrt{1+m^2}$

doesn't take into account these abrupt changes of the correspondence. From the point of view of the traveler on the "bent" road, the corresponding distance marker on the first road leaps forward or backward suddenly when he makes the turn. This is exactly what happens in the twin paradox to the traveling twin. During turn-around, the traveling twin's notion of how old the stay-at-home twin changes abruptly.

 

[ghwellsjr]

There are two different aspects to time dilation and the twin paradox: The first is understanding how mutual time dilation (each twin views the other twin's clock to be running slow) can produce an asymmetric result (the traveling twin's clock advances less for the whole trip than the stay-at-home twin). The second is the non-Euclidean metric of Minkowsky space. The first aspect has a direct analogy with Euclidean geometry. The second does not.
...
This is exactly what happens in the twin paradox to the traveling twin. During turn-around, the traveling twin's notion of how old the stay-at-home twin changes abruptly.

Exactly?? I'd feel a lot better about your analogy is if you had pointed out at the end what you pointed out at the beginning, that it gets it only half right. So if someone remembers what you pointed out earlier that there is that minus sign in the metric for the Minkowski space, then they might not be disturbed by your last two sentences. When the traveling twin turns around the time on the stay-at-home clock jumps forward over a positive gap but in your analogy the distance jumps backwards over a negative gap.

I would also feel a lot better if you had emphasized that your analogy applies to the explanation that uses two inertial frames and requires both twins to jump between them at the moment of turn-around and that this is just one of many different explanations that all account for the final clock comparison equally well.

For example, you showed by analogy how the traveler that continues in a straight line can calculate the other traveler's path without "jumping frames". You could have also shown how you could have continued to use the "frame" of the other traveler even after he made the bend to calculate the distance of both paths. In fact, I maintain that you only use one frame, that of the straight-line traveler and that you are making an incorrect claim that the other one can actually see the gap in distance (and by analogy time) which is not true.

Do you believe that the "frame jumping" explanation is the only valid one for understanding the Twin Paradox?

What you should be pointing out if you want to make this simple, like Einstein claimed his theory is, is that in any inertial frame that you want to pick, you integrate the Proper Time rate of each clock over the entire scenario and you use the Time Dilation formula to calculate that Proper Time rate based on the instantaneous speed of each clock. It's so simple when you do that and to suggest that you need to something more complicated or that doing it the more complicated way (frame jumping) somehow is closer to the traveling twin's reality is just wrong. It doesn't matter which frame(s) you use, none of them change anything that any observer can perceive or see or measure or observe or know. Any observer is free to use any frame they want, not just the one or ones that they happen to be at rest in.

And as I have asked others to do, I would like to see your frame jumping explanation done with the specific example of the the traveling twin traveling for one year at 90%c and then returning. Show the numbers, calculations for the entire scenario. Show us how you do it including how you determine what each twin actually sees of the other twins clock during the entire scenario, not just at the end. And I will be happy to show you the same scenario based on any frame you desire.

 

[stevendaryl]

Exactly?? I'd feel a lot better about your analogy is if you had pointed out at the end what you pointed out at the beginning, that it gets it only half right. So if someone remembers what you pointed out earlier that there is that minus sign in the metric for the Minkowski space, then they might not be disturbed by your last two sentences. When the traveling twin turns around the time on the stay-at-home clock jumps forward over a positive gap but in your analogy the distance jumps backwards over a negative gap.

In both cases, comparing the path parameter of one path to the path parameter of the other path involves a discontinuity when the path "bends". The numerical value of the discontinuity depends on the metric, of course.

 

[D H]

This is exactly what happens in the twin paradox to the traveling twin. During turn-around, the traveling twin's notion of how old the stay-at-home twin changes abruptly.

No. I like how the Usenet Physics FAQ at http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gap.html puts it: That's just an accounting error.

The problem is that to arrive at that sudden jump in the age of the stay at home twin, you are ignoring that the co-moving inertial frames of the outbound and inbound travelers are different frames. They are different tangent spaces. Suppose you and I are standing at different points on the surface of the Earth and we each point north with one hand, east with the other. We aren't pointing in the same directions. You can't travel along the surface using my unit vectors. In a sense, my directions and yours are incomparable. At least not directly. You need a transformation to make them comparable.

One way to make those differently calculated ages comparable is to ask what the outbound and inbound traveler see. For example, the traveling twin might have a very good telescope aimed at the Earth. Or she might just be in regular communication with the stay-at-home twin. Now there is no sudden jump in the age of the stay at home twin at the turnaround point. What there is instead is a sudden jump in the rate at which the stay at home twin ages.

 

[stevendaryl]

No. I like how the Usenet Physics FAQ at http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gap.html puts it: That's just an accounting error.

I wouldn't say that it's an "accounting error". It's not an error at all. At any given time, the traveling twin has an instantaneous inertial reference frame. In each of those frames, there is a different notion of "the current age of the stay-at-home twin". Changing frames means changing your notion of what events are simultaneous, which changing your notion of what the "current" age of the distant twin is. That's not an error, it's just a fact. I don't know why Baez would call it an "error".

The problem is that to arrive at that sudden jump in the age of the stay at home twin, you are ignoring that the co-moving inertial frames of the outbound and inbound travelers are different frames.

That doesn't make any sense. How does saying "When the traveler changes frames, his notion of the age of the stay-at-home twin jumps suddenly" ignore the fact that the traveler changes frames?


I don't quite understand the Physics FAQ on this point. They have a very nice spacetime diagram showing the "gap" here:
http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_vase.html#gap

So in what sense is the gap an "accounting error"?

 

[Austin0]

I wouldn't say that it's an "accounting error". It's not an error at all. At any given time, the traveling twin has an instantaneous inertial reference frame. In each of those frames, there is a different notion of "the current age of the stay-at-home twin". Changing frames means changing your notion of what events are simultaneous, which changing your notion of what the "current" age of the distant twin is. That's not an error, it's just a fact. I don't know why Baez would call it an "error".

That doesn't make any sense. How does saying "When the traveler changes frames, his notion of the age of the stay-at-home twin jumps suddenly" ignore the fact that the traveler changes frames?I don't quite understand the Physics FAQ on this point. They have a very nice spacetime diagram showing the "gap" here:
http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_vase.html#gap

So in what sense is the gap an "accounting error"?

Consider what happens at turn around. In practice the traveler frame would have to resynchronize its system clocks.
As a quick fix let's assume that there is a previously moving frame headed to earth, which by chance has the same proper time as the traveler at his location. So if we assume a line of virtual traveler observers stretching back to Earth they can simply adjust their proper time to the now co-moving clocks of the other frame which are proximate.
What would that adjustment be??

It seems clear to me that they would be turning back their clocks all along the line with the maximal amount being at earth.
SO relative to their new proper times the Earth time has jumped ahead but of course this does not imply any unrealistic shift or change in Earth time whatsoever.

Does this track??.

It also tells us nothing about the relative temporality of the traveler and his inertial twin (proper times) or anything meaningful about simultaneity . This is just a comparison of coordinate time and relative clock synchronicity.

 

[stevendaryl]

Consider what happens at turn around. In practice the traveler frame would have to resynchronize its system clocks.
As a quick fix let's assume that there is a previously moving frame headed to earth, which by chance has the same proper time as the traveler at his location. So if we assume a line of virtual traveler observers stretching back to Earth they can simply adjust their proper time to the now co-moving clocks of the other frame which are proximate.
What would that adjustment be??

It seems clear to me that they would be turning back their clocks all along the line with the maximal amount being at earth.
SO relative to their new proper times the Earth time has jumped ahead but of course this does not imply any unrealistic shift or change in Earth time whatsoever.

Of course not.

It also tells us nothing about the relative temporality of the traveler and his inertial twin (proper times) or anything meaningful about simultaneity.

I don't know what you think it should be telling you. My point is that for the traveling twin to keep track of the age of the stay-at-home twin, he must also account for jumps due to changes in inertial frames.

 

[Austin0]

It seems clear to me that they would be turning back their clocks all along the line with the maximal amount being at earth.
SO relative to their new proper times the Earth time has jumped ahead but of course this does not imply any unrealistic shift or change in Earth time whatsoever.



It also tells us nothing about the relative temporality of the traveler and his inertial twin (proper times) or anything meaningful about simultaneity . This is just a comparison of coordinate time and relative clock synchronicity.

Of course not.
I don't know what you think it should be telling you. My point is that for the traveling twin to keep track of the age of the stay-at-home twin, he must also account for jumps due to changes in inertial frames.

You are the one who apparently thinks it is telling you something significant.

I wouldn't say that it's an "accounting error". It's not an error at all. At any given time, the traveling twin has an instantaneous inertial reference frame. In each of those frames, there is a different notion of "the current age of the stay-at-home twin". Changing frames means changing your notion of what events are simultaneous, which changing your notion of what the "current" age of the distant twin is. That's not an error, it's just a fact. I don't know why Baez would call it an "error".

That doesn't make any sense. How does saying "When the traveler changes frames, his notion of the age of the stay-at-home twin jumps suddenly" ignore the fact that the traveler changes frames?I don't quite understand the Physics FAQ on this point. They have a very nice spacetime diagram showing the "gap" here:
http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_vase.html#gap

So in what sense is the gap an "accounting error"?

Here you are attributing factual reality to the simultaneity as indicated by a hypersurface of clock synchronicity.
you seem to think there is an unambiguous ,definite age (proper time) that can be determined for separated locations (twins). This seems outside the principles of SR as I understand them.
You are assuming actual synchronicity of the clocks throughout the traveler system if you assign any meaning to the sudden difference in readings that occurs when the traveler clocks are resynchronized. But that change only occurs in the traveler frame. It can have no effect on either the traveler or the Earth twin or their relative ages.

Which are fundamentally unknowable in any meaningful sense until they are reunited imo.

Those ages are frame dependent evaluations so it is just a matter of picking your assumptions.
You could, equally validly, just take the Minkowski diagram at face value, in which case it is clear that at turn around the Earth twin is exactly half the total dt older than at the start.

DO you think that would have meaning?

 

[stevendaryl]

You are the one who apparently thinks it is telling you something significant.

I don't know what gave you that impression, but you're completely off track here.

Here you are attributing factual reality to the simultaneity as indicated by a hypersurface of clock synchronicity.

I didn't do any such thing. What I said was that the noninertial twin cannot use the time dilation formula to compute the age of the stay-at-home twin, unless he takes into account the jumps due to changes of frames. The spacetime diagram explains why.

 

[arindamsinha]

I disagree. SR does not have a preferred frame, but it does have a preferred class of frames, inertial frames. In that class of frames you can distinguish kinematically which twin is accelerating.

Dalespam, not trying to challenge your deeper knowledge in relativity, but I do think there is something to discuss here.

Agreed there are inertial frames in SR, but the whole point in SR is that none of them are 'preferred'.

Any kinemtic acceleration will have to be completely mutual between two frames. This means that any and all time dilation will be mutually equal, and no experiment should be able to establish actual measurable velocity time dilation between two bodies moving with a relative velocity w.r.t each other, since the relative velocities are also mutually equal.

This is clearly not the case in experiments, as actual velocity time dilation does provenly exist.

The moment we try to establish that one body has the acceleration and the other doesn't, we are forced to consider 'why', and that leads to dynamics. Kinemtics cannot ponder the question 'why' of any preferred movement. This is why I feel the SR solutions of the twin paradox go outside the boundary of the SR theory itself.

Very interested to hear your opinion (or that of others) on the above logic.