B Relativity: Twin Paradox - Is Age Determinable?

[Buckethead]

TL;DR Summary

time dilation with a one way twin

(I swore to myself I would never ask a relativity question again...oh well)

I don't know why I can't find anything about this in a search so I guess I'll just have to ask. Is a twin that takes off to Mars to stay, younger than a stay at home twin or is it ambiguous? I would think that this could be determined simply by sending the current time to each other and subtracting the data travel time using distance and c, determining in this way if the Martian's clock had slowed. In reading some twin paradox threads it seems this can be ambiguous and is not determinable and that times cannot be compared unless the clocks are in the same location. I'm assuming for the sake of this question the relative velocity between Earth and Mars is 0 and am ignoring any gravitational effects.

 

[Orodruin]

It is ambiguous since the clocks are not colocated.

 

[Buckethead]

That was going to be my guess, but why is my method of determining the clock differences not valid if there is no relative motion? I thought simultaneity issues didn't matter if there was no relative motion.

 

[Dale]

Is a twin that takes off to Mars to stay, younger than a stay at home twin or is it ambiguous?

It is ambiguous, or more precisely the answer is frame variant.

 

[Ibix]

To see if two people have experienced the same elapsed time you need to compare their watches at the beginning and end of the period. If, at either of those comparisons, the watches are not colocated then the relativity of simultaneity means that different frames have different views about what "the beginning and end of the period" actually means.

I thought simultaneity issues didn't matter if there was no relative motion.

But there is relative motion between all the people at rest in all the different frames you could be using.

If you specify a frame then there is a unique answer. But there's no One True Choice Of Frame whose answer is "more right" than any other.

 

[FactChecker]

Suppose that clocks on Earth and Mars are synchronized (we are assuming there is no relative motion) and remain synchronized through the entire experiment. Then a clock that traveled with the traveling twin can be compared with the Mars clock and the Earth clock will agree with the result as though they were collocated. The acceleration (and deceleration) of the traveling twin will slow down his clock. He will be younger.

 

[phinds]

The acceleration (and deceleration) of the traveling twin will slow down his clock.

No, it will decrease his aging, not slow down his clock. I think this is an important distinction for someone not yet solid on SR because most such people thing those are identical things and it causes them great confusion.

 

[Buckethead]

To see if two people have experienced the same elapsed time you need to compare their watches at the beginning and end of the period. If, at either of those comparisons, the watches are not colocated then the relativity of simultaneity means that different frames have different views about what "the beginning and end of the period" actually means.

I think I see. So it seems that the fact that there was relative motion between the time when the clocks were synchronized (colocated) and the time when the measurements took place that make the determination ambiguous. Is this correct?

So if the Martian determined that the Earth clock had slowed, the Earthling, making the same measurement could conflictingly determine that in fact it was the Martians clock that had slowed? They would not agree on their measurements?

 

[Ibix]

Suppose that clocks on Earth and Mars are synchronized (we are assuming there is no relative motion) and remain synchronized through the entire experiment.

But synchronised using what synchronisation convention?

The acceleration (and deceleration) of the traveling twin will slow down his clock.

It's not the acceleration that's important here, just the velocity. You can have someone pass Earth at constant speed, zero their clock as they pass, and find that when they pass Mars their clock is not in sync with the Mars clock - no acceleration needed.

He will be younger.

According to Einstein synchronised clocks, yes.

 

[Ibix]

So it seems that the fact that there was relative motion between the time when the clocks were synchronized (colocated) and the time when the measurements took place that make the determination ambiguous. Is this correct?

No. The point is that anyone in motion with respect to Eargh and Mars will regard their clocks as desynchronised. And they aren't wrong in any sense.

Under the simultaneity convention of Earth and Mars, the traveller is younger than his twin at home. But under other simultaneity conventions he may not be because "on Earth at the same time as the traveling twin gets to Mars" means different things to different frames.

 

[Buckethead]

According to Einstein synchronised clocks, yes.

Wait. I thought we just determined it was ambiguous?

 

[Buckethead]

Under the simultaneity convention of Earth and Mars, the traveller is younger than his twin at home. But under other simultaneity conventions he may not be because "on Earth at the same time as the traveling twin gets to Mars" means different things to different frames.

OK, the Martian and the Earthling will agree, the Martian is younger. This is real? The Martian actually is younger and will live longer?

 

[FactChecker]

But synchronised using what synchronisation convention?

By Einstein convention

It's not the acceleration that's important here, just the velocity. You can have someone pass Earth at constant speed, zero their clock as they pass, and find that when they pass Mars their clock is not in sync with the Mars clock - no acceleration needed.

Without taking acceleration into account, both twins think that the other twin is aging slower. It is the acceleration that breaks the symmetry and allows both twins to agree that the traveling twin aged more slowly (and by how much). By the time the traveling twin stops, they both must agree.

 

[Ibix]

This is real?

It depends on your simultaneity convention. So I'd say it's not real. It's comparable to the statement that you were doing 30mph. You really were doing 30mph relative to the surface of the Earth, but you were also doing a couple of hundred with respect to the Earth's center of mass, and about 20km/s with respect to the Sun. None of those figures is any less real than any other.

In the mutual rest frame of the Earth and Mars (to the extent there is such a thing...) the traveller is younger. In some frames, he's older. Neither answer is more real than the other.

 

[Ibix]

It is the acceleration that breaks the symmetry

No - as I pointed out, the same effect happens with an always-inertial observer passing Earth and Mars. The difference between the two frames here is that one of them has one clock (on the ship) and the other has two (Earth and Mars) and we do local comparisons between the ship clock and the Earth clock and the ship clock and the Mars clock.

 

[hutchphd]

No. The point is that anyone in motion with respect to Eargh and Mars will regard their clocks as desynchronised. And they aren't wrong in any sense.

Under the simultaneity convention of Earth and Mars, the traveller is younger than his twin at home. But under other simultaneity conventions he may not be because "on Earth at the same time as the traveling twin gets to Mars" means different things to different frames.

My twin brother moved to Boise. Are you saying that arbitrary observers cannot agree that we are the same age?

 

[Buckethead]

In the mutual rest frame of the Earth and Mars (to the extent there is such a thing...) the traveller is younger. In some frames, he's older. Neither answer is more real than the other.

I'm only interested in the Earthling and Martian since they have no relative velocity when all it said and done. I'm surprised at this answer because of all the discussion that acceleration has nothing to do with it and since the Martian stays, there is also no "turn around" point causing a shift in frames which is usually what is considered to be the reason for the traveler being younger. It certainly does seem that acceleration and not relative velocity is important. With no turn around point, what is causing the definitive aging difference.

 

[Ibix]

My twin brother moved to Boise. Are you saying that arbitrary observers cannot agree that we are the same age?

For that distance and the speeds involved the relativistic effects are a lot less than the difference in your birth times. But if you both had atomic clocks, synchronised before the move, then different observers would certainly have different opinions about the difference in their readings after it.

If my visualisation is correct you can't actually "swap ages" by changing frames unless you had a near-lightspeed removal service (same goes for the Mars trip). I could be wrong about that - I've just sketched a Minkowski diagram in my head, not done the full maths.

 

[Ibix]

there is also no "turn around" point causing a shift in frames which is usually what is considered to be the reason for the traveler being younger.

That's not why the traveller is younger. The traveller is younger because your elapsed time is the "length" of your path through spacetime, and the traveller took a shortcut, essentially. And this is why the Earth-to-Mars traveller is younger in the Earth frame - the route he took between the Earth's "now" when he left and the Earth's "now" when he arrived was shorter.

The frame change explains why claiming that "the stay at home twin is always moving for the traveller" and naively applying the time dilation formula gives the wrong answer.

 

[FactChecker]

I'm surprised at this answer because of all the discussion that acceleration has nothing to do with it and since the Martian stays, there is also no "turn around" point causing a shift in frames which is usually what is considered to be the reason for the traveler being younger.

It is not necessary to turn around to change the inertial reference frame. The acceleration leaving Earth and deceleration at Mars causes a shift in frames.

 

[hutchphd]

That's not why the traveller is younger. The traveller is younger because your elapsed time is the "length" of your path through spacetime, and the traveller took a shortcut, essentially. And this is why the Earth-to-Mars traveller is younger in the Earth frame - the route he took between the Earth's "now" when he left and the Earth's "now" when he arrived was shorter.

The frame change explains why claiming that "the stay at home twin is always moving for the traveller" and naively applying the time dilation formula gives the wrong answer.

My point is that this would also be true if the Martian trip were a round trip. It is not the lack of co-location that causes an issue. In fact any movement apart can be done symmetrically.

 

[Dale]

I'm only interested in the Earthling and Martian since they have no relative velocity when all it said and done.

That is fine. The answer is still frame variant, but by specifying the frame you can at least get a unique answer.

In this frame the person who traveled from Earth to Mars is younger.

It certainly does seem that acceleration and not relative velocity is important.

Why do you think this? The same individual who accelerates is also the same individual who has a non-zero velocity relative to the specified frame. So there is no way your “and not ...” claim can be justified here.

 

[Dale]

It is not the lack of co-location that causes an issue

It is the lack of co-location that makes the age difference frame variant. If they were co-located then all frames would agree on their age difference. But since they are not collocated different frames disagree

 

[FactChecker]

When something is true, it can often be proven in multiple ways. Accelerations (including instantaneous changes in velocity) match the path which indicates that the traveling twin is younger. You can't have one without the other. But for the actual physical processes to be have been slower in the traveling twin when they reach a directly comparable state, there must be more profound going on than a simple graph on paper -- even if the graph on paper gives the correct result.
EDIT: I want to correct the prior statement. This graph on paper is a very direct representation of the trade-off between motion in the time axis and motion in the spatial dimensions. As such, it is profound and can account for the different ages.

The spatial separation of Earth and Mars is not enough to prevent comparing the twins. Suppose there is also a "pseudo-twin" on Mars who was born at the exact same time according to Einstein-synchronized clocks. Because there is never any relative motion between Earth and Mars (OP assumption), the Earth twin and the Mars pseudo-twin age identically (by Einstein synchronized clocks). Eventually, the traveling twin slows to a stop at Mars and compares his physical age (how his physical processes have progressed) with that of the "pseudo-twin". Their inertial frames are now identical and there is a moment of truth -- were the physical processes of one slower than the other or not? If so, then one is younger. If not, they are the same age. The traveling twin is younger than the Mars pseudo-twin because of the acceleration (or, if you like, the path which indicates acceleration). Likewise, the traveling twin is younger than the Earth twin.

 

[phinds]

Their inertial frames are now identical and there is a moment of truth -- were the physical processes of one slower than the other or not? If so, then one is younger. If not, they are the same age.

It sounds like you are continuing to promote the falacious point of view that things slow down for the traveler IN HIS FRAME. That is not true. Neither his clock nor his biological processes slow down in his frame, he's just taking a different path through space-time so the NUMBER of ticks of his clock is different but not the rate at which they occur.

I think we need to be careful not to promote this very misleading point of view.

 

[FactChecker]

It sounds like you are continuing to promote the falacious point of view that things slow down for the traveler IN HIS FRAME. That is not true.

I have not made that mistake. By the time he comes to a stop on Mars, his frame is identical to the other twin and to any "pseudo-twin" on Mars. In an extreme case, where one twin has aged decades more than another, their physical difference will be obvious and undeniable. The fact that the traveling twin could not detect any slowing of his age as he traveled does not change the fact that, when he comes to a stop on Mars, he is much younger. He can say that the twin on Earth and any "pseudo-twin" on Mars aged extremely rapidly during his accelerations.

 

[phinds]

I have not made that mistake.

Well, it seems to me that you have stated it in a way that makes it seem you have and that is certainly likely to be confusing to a newbie to SR.

The fact that the traveling twin could not detect any slowing of his age as he traveled does not change the fact that, when he comes to a stop on Mars, he is much younger.

I agree that he is younger, but he can't detect any slowing of his aging because there IS no slowing. His different age is not due to any slowing of his biological processes, it's because he took a different path through space-time and but the way you state it you make it sound like there IS a slowing of his aging process.

 

[FactChecker]

To an outside observer, stationary with respect to Earth, the traveling twin's aging and all physical processes have slowed. The traveling twin has no way of detecting a change, and in his physics, it has not changed. But the fact remains, that when he comes to a stop on Mars, he is younger than any "pseudo-twin" on Mars. The traveling twin can say that the twin on Earth and the pseudo-twin on Mars have aged rapidly. He would say that the Earth twin aged rapidly as the traveling twin slowed down at Mars and that the pseudo-twin on Mars aged rapidly when the traveling twin accelerated leaving Earth. As long as they all understand the effects of relativity, there are no disagreements.

 

[hutchphd]

No, it will decrease his aging, not slow down his clock. I think this is an important distinction for someone not yet solid on SR because most such people thing those are identical things and it causes them great confusion.

While I applaud your concern I think no one else is making this strange distinction in the name of clarity...

 

[DaveC426913]

So, to clarify:

1. The twin who travels to Mars will - in every way - experience the passage of time normally. He will age at one second per second, and his clock will tick at one second per second.
2. When he communicates with Earth, the Earth clock will be ahead of his by some amount, including his twin, his twin's clock and anything else he thought to calibrate before he left.
3. The Earth twin and the Mars twin will agree that the Mars twin has not aged as much as the Earth twin has (though they may disagree on their observation of how that came about).

Agree?

 

[Ibix]

Let's do some maths. The real numbers are complicated, time varying, and messy. So I'm going to do this with a constant distance of 1ly (in the Earth frame) and a transit velocity of 0.8c, implying $\gamma=5/3$. The traveller sets off at time zero. With those numbers he arrives at "Mars" 1.25 years later having experienced 0.75 years of time.

According to the Earth frame the stay-at-home has aged 1.25 years and the traveller 0.75. But this cannot be verified directly! It can only be verified by asserting that clocks on "Mars" are synchronised with clocks on Earth and comparing the traveller's clock to the local ones (or some equivalent process involving actually communicating).

Other frames do not agree that the clocks are synchronised. They agree that the time when the ship left was zero. They agree that the traveller's elapsed time was 0.75 years. But they do not agree with the procedure for comparing the traveller's age to the stay-at-home's - the clocks are out of sync.

A frame moving at $v$ (with gamma factor $\gamma_v$) will say the arrival event occurs at time $t'=\gamma_v(1.25-v/c)$, at which time the Earth is at $x'=-v$, implying an elapsed time on Earth of $t=\gamma_v(\gamma_v(1.25-v/c)-v^2/c^2)$, which implies an age difference of $\gamma_v(\gamma_v(1.25-v/c)-v^2/c^2)-0.75$, which is clearly frame dependent.

 

[Ibix]

When he communicates with Earth, the Earth clock will be ahead of his by some amount,

The problem is that the signal travel time is finite, so you have to break the time between the traveller leaving Earth and the Earth receiving a message saying "one month has passed" into the times before and after the traveller sent the signal. And different frames do that in different ways.

3] The Earth twin and the Mars twin will agree that the Mars twin has not aged as much as the Earth twin has (though they may disagree on their observation of how that came about).

That's coordinate dependent, and quite hairy because the Mars twin needs to use a non-inertial frame. It's plausible that they will eventually agree (typically some time after the arrival), but that does depend on their choice of coordinates and, in particular, their choice to use the Earth rest frame.

 

[FactChecker]

That's coordinate dependent, and quite hairy because the Mars twin needs to use a non-inertial frame. It's plausible that they will eventually agree (typically some time after the arrival), but that does depend on their choice of coordinates and, in particular, their choice to use the Earth rest frame.

Regardless of how the traveling twin got to Mars, he knows how he has aged, how his physical processes progressed, and how anything physical he had that could constitute a clock progressed. Once he is there, he can determine what he thinks the elapsed time was. A Mars observer, Einstein-synchronized with Earth, can determine how much time has elapsed in the Earth inertial reference system. They can be compared. The twin who traveled to Mars will be younger. The difference can be extreme and not in doubt.

 

[Ibix]

The twin who traveled to Mars will be younger.

How much younger?

The difference can be extreme and not in doubt.

The age difference can also be quite small, and then which is older is clearly frame dependant since the variation in what "simultaneous" means is quite large.

In the inertial frame where the traveller was at rest in the crossing it is obviously true that the traveller is always older than the stay at home.

 

[FactChecker]

The age difference can also be quite small, and then which is older is clearly frame dependant since the variation in what "simultaneous" means is quite large.

Once the traveling twin has stopped at Mars, there is only one reference frame (Earth and Mars) and they all agree on what Einstein-synchronized clocks would define as simultaneous.

 

[Ibix]

Once the traveling twin has stopped at Mars, there is only one reference frame

There are always infinitely many frames. You may choose to stop using any frame except the Earth's rest frame, but that doesn't mean they don't exist (to the extent frames exist at all).

 

[FactChecker]

There are always infinitely many frames. You may choose to stop using any frame except the Earth's rest frame, but that doesn't mean they don't exist (to the extent frames exist at all).

The observers of interest are the twins. They end up at fixed relative positions and will be in the same inertial reference frame. I think that we should say that the definition of "simultaneous" for that reference frame (Einstein-synchronized) is the only one to consider since both twins are in it. They will agree that the traveling twin is younger.

 

[Ibix]

I think

Exactly - that's your decision. Others are available. That's why this is not an invariant fact.

 

[PeterDonis]

I think that we should say that the definition of "simultaneous" for that reference frame (Einstein-synchronized) is the only one to consider since both twins are in it.

But they're not co-located. There is no invariant fact about simultaneity for observers who are not co-located. The fact that they are at rest relative to each other means that they can choose to Einstein synchronize their clocks and use that simultaneity convention. But no physical facts depend on that choice; they could choose to use some other frame and it wouldn't change any invariants.

 

[FactChecker]

Exactly - that's your decision. Others are available. That's why this is not an invariant fact.

If you are saying that they can choose to disagree by picking other methods of synchronization, then I will not argue. But I think that Einstein-synchronization has some serious logical advantages in this application. In a more extreme example, like a trip to Vega at a speed of 0.99c, the age difference could be 24 years (48 years for a round-trip, see How Fast Is It - 05 - General Relativity II - Effects at 18:08 ). Suppose someone picks a synchronization that says the twins are the same age. They would be arguing that a stationary twin with grey hair and no teeth is the same age as a middle-aged traveling twin. How would they explain the sudden aging of the Earth twin if the traveling twin then completed a round-trip and was standing beside an old man? They would have to say that he didn't age slower going one way but did age much slower going the other way. IMHO, that would require some justification. So I think that they are not logically free to pick any method of synchronization that they want.

 

[Dale]

The twin who traveled to Mars will be younger.

The correct way to say this is “The twin who traveled to Mars will be younger in the Earth-Mars frame”. Neglecting the specification of the frame makes it ambiguous.

If you are saying that they can choose to disagree by picking other methods of synchronization, then I will not argue. But I think that Einstein-synchronization has some serious logical advantages in this application.

The point isn’t whether or not to use Einstein synchronization. The point is in which frame to use it. In the earth-mars frame you get one answer, but in other frames using Einstein synchronization gives you a different answer.

Einstein synchronization is frame dependent.

 

[phinds]

While I applaud your concern I think no one else is making this strange distinction in the name of clarity...

Perhaps not in this thread but we see that confusion here a lot, and it is not a strange distinction. We've had a fair number of people come here thinking that biological processes actually DO slow down, so the distinction is important.

 

[FactChecker]

The correct way to say this is “The twin who traveled to Mars will be younger in the Earth-Mars frame”. Neglecting the specification of the frame makes it ambiguous.

The point isn’t whether or not to use Einstein synchronization. The point is in which frame to use it. In the earth-mars frame you get one answer, but in other frames using Einstein synchronization gives you a different answer.

Einstein synchronization is frame dependent.

I stand corrected. I agree that I should have specified Einstein-synchronization in the Earth/Mars frame. That is a reasonable request. But suppose that another choice of frame is made in which the Mars twin and Earth twin are the same age. Then one would have to consider the possibility that the traveling twin returns to Earth and is standing beside a twin that is physically much older. One would have to justify that the age remained unchanged when traveling in one direction but changed drastically when returning. I think that choice of frame would be hard to rationalize.

EDIT: Using the Earth/Mars frame gives equal age changes on both the departing trip and a possible return trip.

 

[Ibix]

One would have to justify that the age remained unchanged when traveling in one direction but changed drastically when returning. I think that choice of frame would be hard to rationalize.

Not really. The frame you are talking about is the one where the outbound twin and stay-at-home have equal and opposite velocities. On the return leg, the traveling twin has to travel very fast to catch up with the Earth, so a large age difference is completely predictable.

 

[FactChecker]

The explanations that I have seen for the twins paradox have the difference in age primarily as a function of the distance traveled and the change in velocity, not the absolute velocity.

 

[Ibix]

change in velocity, not the absolute velocity.

Acceleration really has nothing to do with it and there's no such thing as absolute velocity. If you choose an inertial frame, though, then for a complete twin paradox (not the one way version we were initially discussing) the velocity in that frame (or rather the gamma factor and the time spent at that gamma factor) are the only important things.

I'll try to draw Minkowski diagrams later.

 

[Buckethead]

Why do you think this? The same individual who accelerates is also the same individual who has a non-zero velocity relative to the specified frame. So there is no way your “and not ...” claim can be justified here.

OK, good. I was looking for the asymmetry and here it is. It was the fact that the traveller had a velocity at some point relative to the eventual frame that both he and the Earthling whould share, wheras the Earthling never had this. Got it! Thanks Dale.

 

[PeroK]

The fact that the traveling twin could not detect any slowing of his age as he traveled does not change the fact that, when he comes to a stop on Mars, he is much younger. He can say that the twin on Earth and any "pseudo-twin" on Mars aged extremely rapidly during his accelerations.

There are a number of problems with your analysis in this thread. You have a problem here:

Consider a third traveller, who accelerates very rapidly, then decelerates very rapidly without having traveled very far from Earth. The acceleration and deceleration phases could be identical to those of a space traveller, who continued at their cruising relativistic speed for some time (before decelerating).

This third traveller will have experienced minimal differential ageing despite having experienced the same acceleration and deceleration as the space traveller.

This shows that nothing special happens during an acceleration. There is no rapid ageing.

The differential ageing is entirely a function of the time spent traveling at relativistic speeds (*).

(*) PS More generally, it depends on the entire path through spacetime; and not on the periods of acceleration.

 

[Buckethead]

Thank you all very much for your in depth analyses of this and the discussions. I have a much clearer picture now. And the thing to remember (that I keep forgetting) is "velocity relative to the frame...velocity relative to the frame". The traveler had a velocity relative to the Earth Mars frame and this was the determining factor as to why he aged (when seen from the Earth Mars frame). Any other frame would give a different result as to the age difference because of relativity of simultaneity and synchronization issues, but again, I was really just interested in the age as measured in the Earth Mars frame. My head feels much clearer.

 

[Nugatory]

Is a twin that takes off to Mars to stay, younger than a stay at home twin or is it ambiguous?

The ambiguity (“incomplete specification” might be a better term) will be easier to see if we rephrase the question. Both twins zero their clocks while together on earth, and then Mars-twin starts their journey. On arrival, Mars-twin looks at their clock and sees that it reads $T$. You are asking whether at the same time that Mars-twin’s clock reads $T$, Earth-twin’s clock reads something less than $T$ (Earth-twin is younger), greater than $T$ (Earth-twin is older), or the same (both twins still the same age).

Clearly the answer depends on how we define “at the same time”.

I would think that this could be determined simply by sending the current time to each other and subtracting the data travel time using distance and c, determining in this way if the Martian's clock had slowed.

That is one sensible way of defining “at the same time”. In effect we are taking the reading on Earth-twin’s clock when the signal from Mars reaches Earth, subtracting the light travel time, and we have the time on Earth-twin’s clock when the signal left Mars. Using this definition and with your sensible simplifying assumptions (“the relative velocity between Earth and Mars is 0 and am ignoring any gravitational effects”) we will find that Mars-twin is younger.

However, suppose I am moving relative to Earth and Mars and I try using the same technique for comparing the clock readings (receive signals from both twins reporting their clock readings, subtract light travel time to determine when signal was sent) I will get different answers and may even find that Earth-twin is younger. There’s no paradox here, it’s just that my definition of “at the same time” will be different from that of someone who (as are the twins) is at rest relative to Earth and Mars.