I Twin Paradox: Who Is Right, A or B?

[Will Flannery]

i do not believe it is possible to write a clear and complete description while using phrases like “He was changing frames”, and similar. The traveller isn’t switching frames, you are in your description of the problem.

I agree.

 

[Ibix]

# "if you pick the origin event of all frames to be the turnaround event"
-That is exactly what I did !

In the second half of the discussion in post #90, yes. The first half of the discussion uses a different origin, though. The whole thing will be simpler if you use the same origin throughout. Then you can get rid of the distinction between A and A'.

The purpose of my explanation was to analyze the trip from the traveler's perspective, and in particular the turnaround at the distant planet where he has to reverse direction

You can't create a "traveler's perspective" by stitching together inertial frames, though. Inevitably you end up with a situation where, according to such a Frankenstein setup, two people can shake hands with each other but be doing it at different times (potentially years apart) because of the oddities of such a description. That doesn't sound like anything I'd describe as anyone's perspective.

That was the focus, but if you try to do the Lorentz transform for that jump, the velocity is - v + (-v) = -2v which is greater than c and I think the Lorentz transform fails or doesn't make sense for that case,

Velocities don't add linearly in relativity. The velocity of the return leg as measured from the inertial frame of the outbound leg is about 0.88c (or 15c/17 to be precise). Google for "relativistic velocity addition".

I see all this as necessary ... :)

It isn't. Just define the traveller's two inertial frames with reference to the stay-at-home's frame. That makes the velocities $\pm$0.6c and if they all have a common origin at the turnaround then it's job done. Introducing frames with different origins and defining the traveller's return velocity relative to his outbound rest state just obscures a fairly straightforward setup.

Each spacetime diagram shows two frames, in the first the Earth frame and the frame for the traveler's outward journey, the 2 for the translated Earth frame and the traveler's home journey.

Each of your diagrams shows the coordinate grid from two frames, but this doesn't help to visualise the traveller's frame's description of the scenario. Draw your last diagram again with the traveller's frame's axes perpendicular instead of the stay-at-home's. The worldlines of the twins will look quite different.

 

[Will Flannery]

Velocities don't add linearly in relativity. The velocity of the return leg as measured from the inertial frame of the outbound leg is about 0.88c (or 15c/17 to be precise). Google for "relativistic velocity addition".

Good point.

It isn't. Just define the traveller's two inertial frames with reference to the stay-at-home's frame. That makes the velocities $\pm$0.6c and if they all have a common origin at the turnaround then it's job done. Introducing frames with different origins and defining the traveller's return velocity relative to his outbound rest state just obscures a fairly straightforward setup.

Well ... that would be three sets of axes on one graph. Too many. Better I think would be to do a graph similar to the one I did for the return trip, i.e. with the origin at the turnaround, except for the outbound trip, and have two graphs, which could be superimposed.

Each of your diagrams shows the coordinate grid from two frames, but this doesn't help to visualise the traveller's frame's description of the scenario. Draw your last diagram again with the traveller's frame's axes perpendicular instead of the stay-at-home's. The worldlines of the twins will look quite different.

Great Scott, I was ready to go on to 4-vectors today, got all my references lined up. But no ! This is my project for the day... wait a minute, the traveler must have 2 (or in my version 3) inertial frames, so you can't draw a single graph showing the entire trip from the traveler's perspective ?

 

[Bruce Harvey]

Surely, the point of the twins paradox was to contrast the relativity theory of Lorentz and Poincare with Einstein's theory. In the former, the slowing of clocks is a real physical effect derived from Maxwell's equations. In Einstein's theory, it is an artefact of observation depending on the two reference frames overlapping.
The argument that A's frame of reference distorts as he accelerates is spurious because it has no physical existence beyond the confines of his ship.
The first experimental observation of the slowing of a moving clock settled the case. We live in a Lorentz- Poincare universe, even if we have to drop the term "Aether" and substitute "fabric of spacetime".

 

[PeroK]

Surely, the point of the twins paradox was to contrast the relativity theory of Lorentz and Poincare with Einstein's theory. In the former, the slowing of clocks is a real physical effect derived from Maxwell's equations. In Einstein's theory, it is an artefact of observation depending on the two reference frames overlapping.
The argument that A's frame of reference distorts as he accelerates is spurious because it has no physical existence beyond the confines of his ship.
The first experimental observation of the slowing of a moving clock settled the case. We live in a Lorentz- Poincare universe, even if we have to drop the term "Aether" and substitute "fabric of spacetime".

Hey, you promised not to promote your non-mainstream views on here!

I promise not to use the forum to promote my own work or give answers contrary to the standard model

 

[vanhees71]

We have to drop the term Aether because no one could ever observe it (i.e., finding a preferred frame of reference in vacuo). "Fabric of spacetime" sounds like some PR text on the cover of (usually pretty misleading) popular-science books. What's called "relativity" is nothing else than a theory about how to describe space and time and how to define reference frames, but I don't want to get into this discussion again, because that's an unnecessarily hot topic in these forums ;-)).

 

[PeterDonis]

Surely, the point of the twins paradox was to contrast the relativity theory of Lorentz and Poincare with Einstein's theory.

No, the point of the twin paradox was to contrast relativity with Newtonian mechanics.

In the former, the slowing of clocks is a real physical effect derived from Maxwell's equations. In Einstein's theory, it is an artefact of observation depending on the two reference frames overlapping.

You are mistaken. The difference in aging of the twins when they meet up again is a real affect, not "an artefact of observation" in relativity, period.

The argument that A's frame of reference distorts as he accelerates is

Not an argument that relativity makes, so you are attacking a straw man.

The first experimental observation of the slowing of a moving clock settled the case.

Between relativity and Newtonian mechanics, yes.

@PeroK has already pointed out that you previously agreed not to promote your non-mainstream views here. If you post about them again, you will receive a warning.

 

[Bruce Harvey]

I was careful not to promote my own ideas. I just quoted from history. Lorentz's derivation of the contraction from Maxwell's equations is back in print thanks to Dover. He identifies Maxwell'e wave equation in electric potential and Possion's equation as special cases of the same equation and in a few lines derives the Lorentz contraction. The increase in mass and slowing of clocks follows.

 

[PeterDonis]

I was careful not to promote my own ideas.

Not careful enough:

Surely, the point of the twins paradox was to contrast the relativity theory of Lorentz and Poincare with Einstein's theory.

We live in a Lorentz- Poincare universe, even if we have to drop the term "Aether" and substitute "fabric of spacetime".

In any case, what you said about the twin paradox was incorrect.

 

[Elroch]

Both. With a one way trip the answer is frame dependent.

There is an objective answer to the question of whether A or B is younger (age in proper time) when they meet. Thus, with all due respect, this answer simply cannot be frame dependent.
With the planets having relative velocity zero and with the watches being synchronised (by say one twin sending the time and the other using that time to set their watch with a correction for the known lag due to transit - doesn't matter which), A (the traveling twin) is younger. The reason is clear in a space-time diagram drawn in the common initial stationary frame, where A's path is shorter according to the proper time pseudometric.
Thus, as has been explained in a slightly different way by others, B is right.

 

[Motore]

There is an objective answer to the question of whether A or B is younger (age in proper time) when they meet.

Not for a one way trip (which means they don't meet).

 

[Nugatory]

With a one way trip the answer is frame dependent.

There is an objective answer to the question of whether A or B is younger (age in proper time) when they meet. Thus, with all due respect, this answer simply cannot be frame dependent.

But in a one-way trip they don't meet so your comment does not apply to that case.

"A is younger than B" implies that at the same time that A's age is T, B's age is something less than T. If the two twins are not colocated then "at the same time" and any comaparisom of their ages is frame-dependent.

 

[vanhees71]

The aging is given by the proper time and thus frame-independent. Frame-dependent quantities are auxilliary tools to get physical observables which are all frame-independent.

 

[Dale]

There is an objective answer to the question of whether A or B is younger (age in proper time) when they meet. Thus, with all due respect, this answer simply cannot be frame dependent.

Yes, the age at reunion is frame independent. What is frame dependent is the initial age and the rate of aging.

 

[Ibix]

There is an objective answer to the question of whether A or B is younger (age in proper time) when they meet.

In the original formulation of the question on this thread the twins were born on different planets and meet up later. So yes, there's an invariant answer to the question of which one is older when they meet. There is no invariant answer to the question of whether they were born at the same time.

 

[PeroK]

In the original formulation of the question on this thread the twins were born on different planets ... There is no invariant answer to the question of whether they were born at the same time.

Hey, maybe them twins ain't twins!

 

[Ibix]

Hey, maybe them twins ain't twins!

Whether it's a twin paradox or not is frame dependent.

 

[Elroch]

In the original formulation of the question on this thread the twins were born on different planets and meet up later. So yes, there's an invariant answer to the question of which one is older when they meet. There is no invariant answer to the question of whether they were born at the same time.

Yes.
But if they are essentially stationary relative to each other there is an objective answer to the question of which was born first according to either of the clocks they carry with them (because these clocks run at the same speed and can be synchronised at any time in a unique, consistent way). The special nature of a proper time clock for each person makes it seem natural to use the frame that agrees with both.
Of course we do this all the time on our own planet. If you ask someone "how old are you?", they rarely answer "with respect to a clock moving at which speed?".

 

[PeroK]

Yes.
But if they are essentially stationary relative to each other there is an objective answer to the question of which was born first according to either of the clocks they carry with them (because these clocks run at the same speed and can be synchronised at any time in a unique, consistent way). The special nature of a proper time clock for each person makes it seem natural to use the frame that agrees with both.
Of course we do this all the time on our own planet. If you ask someone "how old are you?", they rarely answer "with respect to a clock moving at which speed?".

This is missing the point about the relativity of simultaneity.

Your age is essentially the length of your worldline (though spacetime) since you were born. That is an invariant. As long as you define the event local to you to where the worldline is measured and your age calculated.

(But, your age at the time when, say, Betelgeuse goes supernova is frame dependent. Because that event is not colocated with you.)

The amount the traveling twin has aged between leaving Earth and arriving at planet X is also an invariant.

But:

The amount that the Earth has aged between these two events is frame dependent. In the Earth's rest frame it is greater than the above. But, in a reference frame in which the Earth (and planet X) are moving, the traveling twin may have initially decelerated and aged more between these events than either the Earth or planet X.

It's only when the traveller returns to Earth that all frames agree on the differential ageing. Not all frames agree at the turnaround point.

 

[Herman Trivilino]

But if they are essentially stationary relative to each other there is an objective answer to the question of which was born first according to a clock either carries with them (because these clocks run at the same speed and can be synchronised at any time in a unique way).

They are synchronized in their common rest frame, but they will not have been synchronized in the rest frame of the traveling twin. You don't really have a twin paradox here. What you have is the paradox of time dilation, that is, how can each observe that the other's clock is running slow? The solution lies in the understanding that each of the twins have different notions of simultaneity. If the staying twin observes the twins being born at the same time, the traveling twin, once he's in motion, will observe that in the rest frame he now occupies, they were born at different times.

 

[Ibix]

But if they are essentially stationary relative to each other there is an objective answer to the question of which was born first according to either of the clocks they carry with them (because these clocks run at the same speed and can be synchronised at any time in a unique, consistent way).

Using Einstein synchronisation is a matter of choice. Using Einstein synchronisation in a particular frame is a matter of choice. And any other choice amounts merely to picking a different coordinate system which, apart from making the maths more complicated, has no consequences.

It is an objective fact that if you use Einstein's synchronisation procedure and if you take the twins to be at rest then you will say that they are twins. But the twins may have made different choices - so they may not agree with you. There are rather a lot of subjective caveats to your "objective answer".

Of course we do this all the time on our own planet. If you ask someone "how old are you?", they rarely answer "with respect to a clock moving at which speed?".

And "I was only doing 30mph relative to the car I crashed into" is not a defence against a charge of doing 60 in a 30 limit, although I have yet to see a law that specifies the reference frame in which speed limits are set. We adopt standard conventions all the time, but they should not be elevated to the status of "objective".

 

[Dale]

Hey, maybe them twins ain't twins!

It must have been a painful delivery if they are

 

[Dale]

But if they are essentially stationary relative to each other there is an objective answer to the question of which was born first according to either of the clocks they carry with them (because these clocks run at the same speed and can be synchronised at any time in a unique, consistent way).

That doesn’t change the fact that their starting ages are frame dependent.

Also, it isn’t terribly informative since (regardless of the motion of the twins or the clocks) there is always an objective answer of which was born first according to any given pair of synchronized clocks at the births. The problem is that different pairs of synchronized clocks at the births disagree on their respective objective answers.

 

[Elroch]

I don't have any problem agreeing with your unambiguous statements.

1. "their starting ages are frame dependent"
2. "there is always of which was born first according to any given pair of synchronized clocks at the events"
3. " different pairs of synchronized clocks at the births disagree on their respective objective answers"

I think the phrases "terribly informative", the implication implied by "since" and the nature of the "problem" referred to at the end need a little clarification. Let me illustrate with an analogous statement (in italics to indicate it is not my view).

The time the Earth has existed is frame dependent. It isn't terribly informative since there is always an objective answer of how long the Earth has existed given a choice of frame. The problem is that different choices of frame disagree on their respective objective answers.

I emphasise that I am not disagreeing with the physics that has been described. I would suggest it is correct to say that picking the stationary frame is no more informative than using any other (just simpler). i.e. nothing more can be inferred about observations in the real world (which is all that really matters, right?)

As no-one else has done so, perhaps a sketch of a couple of freehand (sorry) space time diagrams will throw light on the central point to someone. They always do to me.

The first is the story in the rest frame of the two planets. Those familiar with the special relativistic pseudometric will immediately see the blue path has shorter proper time associated with it. The second diagram is in a frame chosen to coincide with the velocity of the traveling "twin". Here the blue path has visibly larger proper time. This illustrates the traveling twin has only "aged less" in some frames.

 

[Ibix]

The first is the story in the rest frame of the two planets. Those familiar with the special relativistic pseudometric will immediately see the blue path has shorter proper time associated with it. The second diagram is in a frame chosen to coincide with the velocity of the traveling "twin". Here the blue path has visibly larger proper time. This illustrates the traveling twin has only "aged less" in some frames.

Worth noting that those two spacetime diagrams don't show the same thing. In both frames you have the red and blue lines starting simultaneously at $t=0$ and $t'=0$, so you've cut them off at different events. This is misleading, since the whole point is that the twins are only born at the same time in at most one frame. It is this difference in birth times, coupled with the paths of different "lengths", that resolves the apparent paradox.

I would also say that your last sentence is confusing at best. If you mean that the traveling twin has aged less than the other during the journey in some frames, then I agree. However, if you mean that the traveling twin has aged less when they meet up in some frames, then I disagree - that's true in all frames.

 

[Elroch]

Worth noting that those two spacetime diagrams don't show the same thing.

Yes, while that was key, it is worth emphasising. It is of course only possible for the proper times to be different in the two diagrams because they start at different points in the lives of the twins (or they are different pairs of twins).

In both frames you have the red and blue lines starting simultaneously at $t=0$ and $t'=0$, so you've cut them off at different events. This is misleading, since the whole point is that the twins are only born at the same time in at most one frame. It is this difference in birth times, coupled with the paths of different "lengths", that resolves the apparent paradox.

You rightly draw attention to something that I should have clarified. The two different horizontal axes cannot possibly both correspond to both births for the same pair of twins. They can represent two different sets of twins being born at the same time according to a stationary observer in one case and a moving one in the other. Alternatively, they can correspond to two pairs of points in the lives of the "twins" which are respectively considered simultaneous to the two different observers.

I would also say that your last sentence is confusing at best. If you mean that the traveling twin has aged less than the other during the journey in some frames, then I agree. However, if you mean that the traveling twin has aged less when they meet up in some frames, then I disagree - that's true in all frames.

Suppose the second diagram does show the birth of the twins at time zero in the moving observer's frame (or just a chosen time in his frame from which he wishes to see how A and B age). Then he infers that A (the traveling twin) ages more from there to when they meet up, and he is right.

 

[Dale]

The time the Earth has existed is frame dependent. It isn't terribly informative since there is always an objective answer of how long the Earth has existed given a choice of frame. The problem is that different choices of frame disagree on their respective objective answers.

I am not sure I get your point. It sounded at first like you were trying to make an objectionable analogous statement. But then your analogy was not at all objectionable to me.

 

[Elroch]

I am glad it was not inadvertently objectionable! You achieve a very good tone in your contributions.

 

[vanhees71]

Yes, the age at reunion is frame independent. What is frame dependent is the initial age and the rate of aging.

I don't understand the latter statement. The proper time of each twin is a scalar and thus frame independent, right?

If you want to compare clocks of two observers, this must be done at one event, i.e., when they meet. Then you can compare the clock readings of the observers.

Take as an example the most simple practical setup for a test of the twin paradox: Some unstable particle/nucleus in a storage ring. It's lifetime is defined (sic!) as the mean proper time of this particle it takes for the particle to decay measured from a time where the particle's existence has been established with certainty. This lifetime can be compared to a clock reading at rest wrt. the laboratory. The measured mean lifetime in the lab is longer by a Lorentz $\gamma$ factor.

 

[Ibix]

If you want to compare clocks of two observers, this must be done at one event, i.e., when they meet. Then you can compare the clock readings of the observers.

Unfortunately, the setup in this version of the thought experiment doesn't allow that. The twins don't start in the same place - that's the whole problem. So there's an invariant answer to how old the traveller was when he left his planet and an invariant answer to how old both were when they meet. But "how old was the inertial twin when the traveller started travelling" depends on your simultaneity criterion, as does "how fast does each one age".

 

[vanhees71]

Then you cannot compare the age of the twins to begin with and the question doesn't make sense at all, and it's not a twin paradox at the usual sense.

 

[PeroK]

Then you cannot compare the age of the twins to begin with and the question doesn't make sense at all, and it's not a twin paradox at the usual sense.

In other words:

Hey, maybe them twins ain't twins!

 

[Ibix]

Then you cannot compare the age of the twins to begin with and the question doesn't make sense at all, and it's not a twin paradox at the usual sense.

Indeed. It's actually nearer the cosmic ray muons experiment.

 

[vanhees71]

The cosmic-ray muons experiment is an interesting example for a "one-way twin paradox"-like setting, which is well defined, because what's fixed here is the travel distance of the muons. The experiment is simply measuring the rate of cosmic muons as function of their momentum, or their $\gamma=E/(mc^2)$ factor, relative to Earth at two different heights, i.e., at some fixed distance $L$ (as measured relative to Earth). The travel time (measured relative to Earth) is of course $t=L/v=L c p/E$. The lifetime of the muon is $\tau$ and by defnition measured in their rest frame, i.e., it's measured in terms of the muons' proper time. The SRT prediction then is that the lifetime as measured with respect to Earth is $\tau'=\gamma \tau$, i.e., one should get
$$N=N_0 \exp[-t/(\gamma \tau)].$$
muons, and that's confirmed by experiment.

https://en.wikipedia.org/wiki/Experimental_testing_of_time_dilation#Frisch–Smith_experiment

 

[Dale]

The proper time of each twin is a scalar and thus frame independent, right?

Yes, but the clocks are spatially separated so the initial ages depend on the frame’s definition of simultaneity, and the rate of aging is a frame-variant ratio of coordinate time and proper time

If you want to compare clocks of two observers, this must be done at one event, i.e., when they meet.

At the beginning of this scenario the twins are spatially separated. You certainly can compare such clocks (that is indeed the point of a simultaneity convention), but the result is frame-variant as I said.

 

[vanhees71]

Ok, but then we are not discussing a twin paradox anymore.

 

[Dale]

Ok, but then we are not discussing a twin paradox anymore.

Agreed. Hence @PeroK’s funny comment

Hey, maybe them twins ain't twins!

 

[mitochan]

Say twin pilot A and B are on board the same product of VERY LONG spaceship with synchronized clocks in each way outfitted in everywhere in ship. They are at cockpits situated at the top end of the ships. Along the same line course they approach with inertial motion, pass nearby. A and B observe they share cokpit clock time = 0 when the cockpits are nearby.

The cockpit of B ship pass the tail end of A ship, then and there the B top cockpit clock time < the A tail end clock time
A judges "If now B recoils back with same speed to see me again, then I will be older than B by 2* ( RHS - LHS of the above inequality ) when we meet."

The cockpit of A ship pass the tail end of B ship, then and there the A top cockpit clock time < the B tail end clock time
B judges "If now A recoils back with same speed to see me again, then I will be older than A by 2* ( RHS - LHS of the above inequality ) when we meet."

Such a scenario comes to my mind inspired by the title of OP. If which pilot will turn back is a kind of chicken race, the looser who recoils back is younger when they meet. If they both do not want to lose, they both keep going and will not meet again forever so the race is a draw or no match.