Twin paradox, virtual clock on ship with Earth time, discontinuity

[PeterDonis]

I am not sure what is meant by "equivalent".

He means having the twins age differently when they come back together.

A twin can never return younger unless he has been accelerated.

This is only true under a very restricted set of circumstances: spacetime must be flat, and you must insist that there are only two twins (i.e., scenarios like an outgoing traveler synchronizing his clock with an incoming traveler at the "turnaround" point are rule out) and that the stay at home twin is inertial the whole time. The rule does not generalize to scenarios where those conditions are not met.

The spacetime geometry rule, OTOH, which @PeroK has mentioned and which is discussed in more detail in the article I linked to in post #4 and its references, always works, no matter whether spacetime is flat or curved, no matter how many travelers there are, and no matter who is inertial and who is not.

Given these facts, @PeroK is perfectly correct that focusing on the acceleration is misplaced. The correct focus is on spacetime geometry and arc lengths along timelike curves in that geometry.

 

[Ibix]

This is only true under a very restricted set of circumstances: spacetime must be flat

And topologically trivial (just to add a further restriction). You can have a twin paradox with two eternally inertial twins in a cylindrical universe.

 

[Ibix]

But even in curved space-time it seems to be true that uniform motion does not cause a "paradox"?

In curved spacetime two inertial observers may meet more than once. Their elapsed times between meetings may be different.

 

[PeroK]

PS note that the maximum differential aging (for a given velocity) is achieved in the non-acceleration scenario. That is to say, the proper time in the purely inertial scenario (at some speed $v$) where the sum of two clock readings is taken will be less than the scenarios where there are two or four acceleration phases (up to the cruising speed $v$). In the latter scenarios, the clock travels at a lower relative speed for some of the journey. Although, as above, the difference can be made arbitrarily small.

This illustrates that the need for proper acceleration in the case where a single clock is used actually reduces the total differential aging, below a theoretical maximum - which can be achieved using two clocks.

This also illustrates that acceleration is actually a physical constraint of working with a single clock on both legs of the journey. And not a fundamental aspect of proper time measurements along different spacetime paths.

 

[Dale]

It took me a while to respond to this because I wanted to go through the paper you cited.

In Ship frame, Graphs 1, 2, 3, in Earth Frame Graph 4.

Let's assume a simple case, in which a Lorentz boost can be applied.

You can never use a Lorentz boost either to or from the ship frame. The ship frame is non-inertial and a Lorentz boost is a transform between two inertial frames. This is exactly what I was concerned that you were doing from your initial post.

You can use a Lorentz boost between the earth frame and any of the momentarily co-moving inertial frames of the ship. But none of the momentarily co-moving inertial frames are the ship's frame since the ship is not inertial and does not remain at rest in any of them.

Almost based on these calculations:

https://arxiv.org/pdf/1807.02148.pdf
https://guyleckenby.weebly.com/uploads/5/6/2/9/56292217/final_draft.pdf (Especially the formula on page 4, to calculate time jumps)

One thing to be aware of is that neither of these papers appear to be in the professional scientific literature. The first one appears to have been stored in arxiv for 5 years but still not accepted in any journal. It has some problems that makes me suspect that it will not be accepted. The second one appears to be a student's term paper for a class, so I don't think there was even an attempt to publish it. You should always be skeptical of such references. They are only valid if they are consistent with the rest of the scientific literature, which is difficult to know at the beginning.

Both papers have the same problem. Specifically, neither one actually defines the ship's frame. They both use some Lorentz transforms and some Rindler transforms and they try to hodgepodge together something that they claim represents the home time in the ship's frame, without actually constructing the ship's frame. The problem is that this hodgepodge produces a mapping that fails to be smooth and invertible. As such, it is not a valid reference frame and thus has no claim to represent anyone's perspective including the ship's perspective.

The problem is that although the requirement that any reference frame must be smooth and invertible does eliminate the hodgepodge approach, that requirement does not lead to a unique reference frame. In particular, there are an infinite number of different reference frames that disagree on simultaneity but would all be valid reference frames for the ship. All of those equally can claim to represent the ship's perspective, but they would give different plots, even for the same acceleration profile.

 

[robphy]

Just an FYI…

Almost based on these calculations:

https://arxiv.org/pdf/1807.02148.pdf
https://guyleckenby.weebly.com/uploads/5/6/2/9/56292217/final_draft.pdf (Especially the formula on page 4, to calculate time jumps)

One thing to be aware of is that neither of these papers appear to be in the professional scientific literature. The first one appears to have been stored in arxiv for 5 years but still not accepted in any journal.

I haven’t read the first paper…
but, out of curiosity, I just looked for it.

https://arxiv.org/abs/1807.02148
lists
https://doi.org/10.1139/cjp-2018-0788

J. Gamboa, F. Mendez, M.B. Paranjape, and Benoit Sirois. 2019. The “twin paradox”: the role of acceleration. Canadian Journal of Physics. 97(10): 1049-1063. https://doi.org/10.1139/cjp-2018-0788

 

[Dale]

Just an FYI…
I haven’t read the first paper…
but, out of curiosity, I just looked for it.

https://arxiv.org/abs/1807.02148
lists
https://doi.org/10.1139/cjp-2018-0788

J. Gamboa, F. Mendez, M.B. Paranjape, and Benoit Sirois. 2019. The “twin paradox”: the role of acceleration. Canadian Journal of Physics. 97(10): 1049-1063. https://doi.org/10.1139/cjp-2018-0788

Oops, you are right. The arxiv entry did not list the journal reference and I didn’t notice the DOI link which was listed as “related”. I wonder if there are substantial differences between the arxiv version and the published version, or why it was listed that way rather than the usual way.

In any case, the Canadian Journal of Physics is a very low ranked journal. Its Eigenfactor article impact score is the 19th percentile. And regardless, this doesn’t affect my substantive criticisms.

 

[Peter Strohmayer]

You measure the time along an inertial path from Earth to a turnaround point using one clock. Then you measure the time along an inertial path from the turnaround point using a second clock.

To compare two proper times, the two world lines must intersect twice. The stationary clock and the two moving clocks are three world lines (with three proper times) that do not intersect twice (A intersects B, B intersects C and C intersects A).

The fact that the sum of these times is equivalent to the times for a single clock, give or take the variation for the acceleration phase (which can be made arbitrarity small), should make a promising student sit up and take notice. Acceleration itself disappears from the equation and we are left with the simple sum of the proper time along two purely inertial paths.

The acceleration terms disappear from the equations just when it is no longer a physical process (infinitely short and therefore infinitely high accelerations).

you are directly working against the aims of PF.

By favouring equations that describe physical processes?

 

[A.T.]

You are saying that there is a generalized class of scenarios a that in them acceleration is necessary for the twin paradox.

As usual in these discussions, it seems that people are talking past each other, because they have a different idea what the "paradox" is, and what "acceleration" means (accelerated frame vs. accelerated clock).

The answer to the question, why the traveling twin cannot just do the same type analysis of the whole journey in his rest frame, as the at home twin can to in his rest frame, is the acceleration of the traveling twin's rest frame.

One should make very clear, that this doesn't imply, that the acceleration of a clock itself directly affects its tick rate. Instead, the tick rates of clocks can depend on their position, when analyzed in an accelerated frame.

I don't think one should try to avoid mentioning "acceleration", by refusing the directly answer the question above, and hiding behind alternative scenarios (like the inertial triplets). One should rather be more specific on what kind of acceleration is meant and what role it plays for the question above.

 

[PeroK]

As usual in these discussions, it seems that people are talking past each other, because they have a different idea what the "paradox" is, and what "acceleration" means (accelerated frame vs. accelerated clock).

That misses the point entirely. We run two experiments: one with a single accelerated clock; and one with two clocks used in relay. These two experiments result in approximately the same measurement for proper time of the round trip, with the relay experiment measuring slightly less time, depending on how long the acceleration phases take.

If acceleration is fundamental to the experiment, then the relay clocks should not record slightly less total time, as neither experiences any proper acceleration.

A good student should take note of this and use this to broaden their understanding of SR.

It's a poor student who dismisses the relay experiment as invalid or unphysical.

 

[A.T.]

That misses the point entirely.

With different ideas on what the "paradox" is, both sides insist that other side is "missing" the point.

We run two experiments: one with a single accelerated clock; and one with two clocks used in relay. These two experiments result in approximately the same measurement for proper time of the round trip, with the relay experiment measuring slightly less time, depending on how long the acceleration phases take.

How does this answer the question: "Why can't the traveling twin just do the same type analysis of the whole journey in his rest frame, as the at home twin can to in his rest frame?"

 

[PeroK]

With different ideas on what the "paradox" is, both sides insist that other side is "missing" the point.How does this answer the question: "Why can't the traveling twin just do the same type analysis of the whole journey in his rest frame, as the at home twin can to in his rest frame?"

Because the rest frame is not a single inertial frame throughout. You either have an acceleration or a relay system.

The claim above that the relay system is invalid physics and only acceleration is valid physics is misinformation. Both experiments, with or without acceleration, are valid. That is the mainstream physics.

 

[A.T.]

Because the rest frame is not a single inertial frame throughout.

Yes, so the acceleration of the reference frame answers the question as posed.

The claim above that the relay system is invalid physics and only acceleration is valid physics is misinformation. Both experiments, with or without acceleration, are valid. That is the mainstream physics.

Yes, both are valid, because the acceleration of the clocks doesn't affect their tick rate.

Everything is much clearer, when you specify what acceleration is meant.

 

[A.T.]

I am talking about logic.

This is clearly a claim that acceleration is necessary for a twin paradox to occur. But when establishing that something is necessary the key thing to show is that in the absence of the necessary cause, the result does not occur.

1) From my experience, discussions that involve the words "cause" and "logic" tend to be the longest and fruitless.

2) Removing just one thing, to apply your causation logic, without changing anything else, is not trivial. It often relies on additional assumptions what can be changed, without being the "cause". If you really just remove the proper acceleration of the twin, then you have two inertial twins, that have two meetings, in flat space time. This not possible so it doesn't help us much.

Also, although you are speaking of accelerated frames, @Peter Strohmayer is not. He is clearly speaking of proper acceleration. In his words “he has been accelerated” and “accelerated motion”.

In the context of the original question, the accelerations of the traveling twin and of frame to be used are the same. But this is of course nothing general, and that's why I'm arguing for clearly distinguishing the acceleration of a frame vs. acceleration of a clock.

 

[Dale]

In the context of the original question, the accelerations of the traveling twin and of his rest frame are the same. But this is of course nothing general, and that's why I'm arguing for clearly distinguishing the acceleration of the frame vs. acceleration of a clock.

I agree with that. My objection is specific to @Peter Strohmayer's comments.

2) Removing just one thing, to apply your causation logic, without changing anything else, is not trivial. It often relies on additional assumptions what can be changed, without being the "cause".

That is not required. You do not need to change only the acceleration to show that it is not necessary. Any example where acceleration is absent and the differential aging is present is sufficient to show that acceleration is not necessary for differential aging.

 

[PeroK]

1) From my experience, discussions that involve the words "cause" and "logic" tend to be the longest and fruitless.

Given the importance of Minkowski geometry to more advanced studies in SR and given the pedagogic value of the non-acceleration, relay version of the twin paradox, it's difficult to understand your objections to what @Dale and I have posted here. We are not quibbling about cause or logic, but presenting textbook physics.

 

[A.T.]

That is not required. You do not need to change only the acceleration to show that it is not necessary. Any example where acceleration is absent and the differential aging is present is sufficient to show that acceleration is not necessary for differential aging.

This disproves that acceleration is a generally required condition, for all possible scenarios. But it doesn't disprove that it is the key distinction in a specific scenario. In the specific scenario of the original twins it is the only thing that distinguishes the two twins and their rest frames, so it must be relevant. Whether you want to call this a "cause" or "reason" is rather philisophical.

Consider this analogy:

Two people ride bikes on a perfectly flat plane, from point A to B. One is going straight (no steering), while the other one is doing some steering around. After meeting at B they compare their odometers, and find that the steering one has traveled more distance. But why?

I think it's okay to say, that the key difference is the steering. That's the only difference between them after all.

This doesn't imply that, steering is a general requirement to accumulate different distances in all possible scenarios. And pointing out, that on a curved surface they could accumulate different distances without either one of them steering, doesn't disprove that steering was the key distinction on the flat plane.

 

[PeterDonis]

In the specific scenario of the original twins it is the only thing that distinguishes the two twins and their rest frames, so it must be relevant.

But that specific scenario is not the only scenario under discussion in this thread. Perhaps @Peter Strohmayer would like to restrict discussion to just that one scenario (although he has made general claims--false ones--not just claims about that specific scenario), but this is not his thread. The issue that the OP of the thread is struggling with arises in other scenarios besides just that one.

 

[PeroK]

This disproves that acceleration is a generally required condition, for all possible scenarios. But it doesn't disprove that it is the key distinction in a specific scenario. In the specific scenario of the original twins it is the only thing that distinguishes the two twins and their rest frames, so it must be relevant. Whether you want to call this a "cause" or "reason" is rather philisophical.

Consider this analogy:

Two people ride bikes on a perfectly flat plane, from point A to B. One is going straight (no steering), while the other one is doing some steering around. After meeting at B they compare their odometers, and find that the steering one has traveled more distance. But why?

I think it's okay to say, that the key difference is the steering. That's the only difference between them after all.

This doesn't imply that, steering is a general requirement to accumulate different distances in all possible scenarios. And pointing out, that on a curved surface they could accumulate different distances without either one of them steering, doesn't disprove that steering was the key distinction on the flat plane.

Let's take a different scenario. One sets off in a rocket and accelerates at $1g$ for a year, cruises for 5 years (proper time), decelerates at $1g$ for a year, then reverses the journey.

The other stays on Earth and accelerates back and forwards at $1g$ for the whole duration of the traveller's journey.

When the traveller returns, he/she is younger because he/she had a different acceleration profile?

Eventually, with enough work, you can figure out from the acceleration profile who is younger. But, it's a helluva lot simpler to use the velocity profile relative to a suitable IRF.

In SR and GR, it's the four-velocity and not the four-acceleration that is the fundamental concept.

 

[PeroK]

PS and, in fact, in the usual twin paradox, the Earth twin is subject to a constant $1g$ proper acceleration. Why is that acceleration largely irrelevant, whereas the acceleration of the spaceship is the primary cause of differential aging? Could it be that proper acceleration is a red herring and instead the student should focus on relative velocity in a suitable IRF or focus on spacetime geometry instead?

What does the student gain by focusing on proper acceleration?

 

[Sagittarius A-Star]

Thanks for the responses and sorry for the delay. I have been studying accelerated frames in SR, to understand the key factor that indicated @Sagittarius A-Star, the importance of the sense of accelerations and dependence on distance.

So, this would be roughly the earth display clock on the ship, with non-instantaneous accelerations. The accelerations that matter are the far ones.

Graph 1

View attachment 331072

I think your graph 1 in posting #16 is OK. I assume, that $t'$ means the coordinate-time of the restframe of the ship, which is at the location of the ship equal to the proper time of the ship.

This non-inertial restframe can be approximated by a sequence of many different momentarily co-moving inertial reference-frames.

 

[A.T.]

Let's take a different scenario....

One can/should consider many scenarios. But it doesn't disprove the facts stated above about a specific scenario.

in fact, in the usual twin paradox, the Earth twin is subject to a constant $1g$ proper acceleration.

I'm referring to the twin paradox where one twin is considered inertial.

 

[PeterDonis]

I'm referring to the twin paradox where one twin is considered inertial.

Yes, we know that. But as I've already pointed out, that is not the only scenario under discussion in this thread.

 

[PeroK]

I'm referring to the twin paradox where one twin is considered inertial.

If you insist on the importance of proper acceleration in this scenario, you cannot consider an Earthbound twin to be inertial when they are subject to a proper acceleration of $1g$ throughout the experiment.

 

[Dale]

But it doesn't disprove that it is the key distinction in a specific scenario

I agree completely.

However claims that “A twin can never return younger unless he has been accelerated” and “The acceleration is of fundamental importance” cannot be justified based on a single specific scenario. Restricting the scenarios under consideration correspondingly restricts the claims that you can make.

Claims about “unless he has been accelerated” imply that you are looking at other scenarios, specifically including ones without acceleration. And claims about “fundamental” imply that all scenarios are considered. And neither are correct here.

 

[A.T.]

If you insist on the importance of proper acceleration in this scenario, you cannot consider an Earthbound twin to be inertial when they are subject to a proper acceleration of $1g$ throughout the experiment.

The basic version the paradox is supposed to be solvable using SR only, so the inertial twin is not actually on Earth.

 

[A.T.]

However claims that “A twin can never return younger unless he has been accelerated” and “The acceleration is of fundamental importance” cannot be justified based on a single specific scenario. Restricting the scenarios under consideration correspondingly restricts the claims that you can make.

Yes, one should be careful with generalizations. But this applies to both sides of this debate.

In general, the acceleration profiles are one factor that plays a role for the accumulated proper times between two events. One should neither rely completely on them, nor should one dismiss them as irrelevant (especially in cases where they are the only distinct factor, and thus fully determine the proper time intervals).

 

[PeroK]

Yes, one should be careful with generalizations. But this applies to both sides of this debate.

In general, the acceleration profiles are one factor that plays a role for the accumulated proper times between two events. One should neither rely completely on them, nor should one dismiss them as irrelevant (especially in cases where they are the only distinct factor, and thus fully determine the proper time intervals).

This is no way to learn SR. And also, if I may say, an element of wilful contrariness has crept in.

We start with scenario 1. Twin A inertial (not on Earth), twin B accelerates. Therefore, the explanation for the twin paradox in these precise circumstances is acceleration. Because there is nothing else.

Then we have scenario 2 where twin A is put on Earth. Now we have a different explanation in this precise scenario. Different acceleration profiles. Whatever.

Then we produce the relay experiment. Which, not surprisingly, the student who has been fed a diet of "acceleration is fundamental" rejects as invalid. At this point the poor student digs in their heels and sticks to their guns: in the twin paradox, acceleration is fundamental. They cannot be shifted. The seed has been sown and it's practically impossible to get the student to revise their thinking. A good student may feel duped about what they have been taught - it was nothing to do with acceleration all along. It was all to do with spacetime geometry. Why didn't you say that in the first place?

You can see the evidence in this thread where the OP has focused their efforts on the physical nature of acceleration in order to understand SR. The whole debate arose in an effort to broaden their thinking away from acceleration. It can't help to encourage the OP to focus on acceleration, which is a dead end in terms of trying to understand SR and GR.

 

[PeterDonis]

the acceleration profiles are one factor that plays a role for the accumulated proper times between two events

But how does it play a role? Answer: it plays a role by determining the worldline of the object in spacetime, and that, combined with the spacetime geometry, is what determines how much the object ages between any two events.

Given that, for general understanding it seems much better to focus on the general thing that applies in any scenario, the object's worldline and the spacetime geometry, than to focus on one particular thing that happens to be a factor in determining the object's worldline in one particular scenario.

 

[A.T.]

But how does it play a role? Answer: it plays a role by determining the worldline of the object in spacetime, and that, combined with the spacetime geometry, is what determines how much the object ages between any two events.

Yes, but these are different levels: Proper acceleration is something physical, that you can measure. World lines and space-time are parts of a model. One must talk about both, and how they relate, not chose one or the other.

Given that, for general understanding it seems much better to focus on the general thing that applies in any scenario, the object's worldline and the spacetime geometry, than to focus on one particular thing that happens to be a factor in determining the object's worldline in one particular scenario.

I don't think you have to always use the most general and and complete explanation. Within SR (where the twin paradox usually first comes up) the acceleration profiles fully determine the proper time intervals between two events. So it's not just "one particular scenario", but I would always be clear, that this not the most general case.