# Relativity: Twin Paradox - Is Age Determinable?

• B
In summary, two twins, one staying on Earth and the other traveling to Mars and back, will have different perceptions of time due to the relativity of simultaneity. This means that their clocks will not be in sync and they will not agree on their measurements of time. The traveling twin will experience less elapsed time due to their velocity, but it is the acceleration that breaks the symmetry and allows both twins to agree on the aging process.
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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.

It is ambiguous since the clocks are not colocated.

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.

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.

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.

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.

FactChecker said:
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.

Ibix said:
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?

FactChecker said:
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?
FactChecker said:
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.
FactChecker said:
He will be younger.
According to Einstein synchronised clocks, yes.

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.

Ibix said:
According to Einstein synchronised clocks, yes.
Wait. I thought we just determined it was ambiguous?

Ibix said:
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?

Ibix said:
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.

hutchphd
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.

FactChecker said:
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.

Ibix said:
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?

Ibix said:
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.

hutchphd said:
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.

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.

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
Ibix said:
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.

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.

hutchphd said:
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

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.

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hutchphd
FactChecker said:
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.

phinds said:
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.

FactChecker said:
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.

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.

phinds said:
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...

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?

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FactChecker
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.

DaveC426913 said:
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.
DaveC426913 said:
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.

Ibix said:
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.

FactChecker said:
The twin who traveled to Mars will be younger.
How much younger?
FactChecker said:
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.

Ibix said:
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.

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