B Relativity: Twin Paradox - Is Age Determinable?

[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.

 

[PeroK]

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.

Yes, in a frame where the Earth-Mars system was moving (in the direction towards the Earth) then the traveller would be seen to decelerate to a slower speed, before accelerating back to the same speed as the Earth-Mars system. In this frame, more time would have elapsed on the traveling clock than a clock on Earth or Mars.

In that reference frame, the twin that traveled to Mars would be older.

 

[FactChecker]

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.

I see your point and stand corrected.

 

[FactChecker]

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.

My two cents:
The situation where both frames are inertial and no acceleration occurs is well known. There is no preferred inertial reference frame and symmetry holds. Both observers see the other as aging slower. That is true for the entire time that there is no acceleration. So it can not account for a mutually recognized differential aging without considering a change in velocity. The GR answer when there is acceleration is that the acceleration is equivalent to a gravitational field. Another observer far away is farther in the gravitational field and is effected more. Therefore, the acceleration causes a person far away to age (in the perspective of the accelerating observer) more than a nearby person. This same effect should be shown in SR when the path of an observer changes velocity.

 

[PeroK]

My two cents:
The situation where both frames are inertial and no acceleration occurs is well known. There is no preferred inertial reference frame and symmetry holds. Both observers see the other as aging slower. That is true for the entire time that there is no acceleration. So it can not account for a mutually recognized differential aging without considering a change in velocity. The GR answer when there is acceleration is that the acceleration is equivalent to a gravitational field. Another observer far away is farther in the gravitational field and is effected more. Therefore, the acceleration causes a person far away to age (in the perspective of the accelerating observer) more than a nearby person. This same effect should be shown in SR when the path of an observer changes velocity.

We work quite hard on PF to dispel the myths that:

a) Acceleration is the key to the twin paradox
b) To study acceleration we need GR, not just SR.

Quite explicitly, the twin paradox takes place in flat spacetime, which is the realm of SR. There is no gravity for which you would need GR.

Moreover, the twin paradox is about the lengths of different paths through the flat spacetime of SR. It's a geometric property that can be demonstrated with no acceleration. The "change in velocity" can equally well be achieved through an instantaneous change in IRF.

Therefore, the acceleration causes a person far away to age (in the perspective of the accelerating observer) more than a nearby person. This same effect should be shown in SR when the path of an observer changes velocity.

Here you are confusing an "accelerating reference frame", with the acceleration of an object in an IRF.

 

[FactChecker]

We work quite hard on PF to dispel the myths that:

a) Acceleration is the key to the twin paradox
b) To study acceleration we need GR, not just SR.

Quite explicitly, the twin paradox takes place in flat spacetime, which is the realm of SR. There is no gravity for which you would need GR.

To say that you have another way to solve it is not the same as saying that GR is false. If you want to argue against the GR equivalence principle, or how it is used then I will need to leave that argument for people who know more than I do.

Moreover, the twin paradox is about the lengths of different paths through the flat spacetime of SR. It's a geometric property that can be demonstrated with no acceleration. The "change in velocity" can equally well be achieved through an instantaneous change in IRF.

This reliance on an instantaneous change of IRF to say that acceleration is irrelevant seems ok to me. I can accept that the results are the same, but I do not agree that either necessarily invalidates the other. It seems like a leap of faith to tie an instantaneous IRF change to a physical end result like twins having different ages when the traveling one gets back to earth. That is switching from one IRF to another when the two do not agree with each other. But I can accept it. I have as easy a time accepting the equivalence principle in this simple application.

 

[PeroK]

To say that you have another way to solve it is not the same as saying that GR is false. If you want to argue against the GR equivalence principle, or how it is used then I will need to leave that argument for people who know more than I do.This reliance on an instantaneous change of IRF to say that acceleration is irrelevant seems ok to me. I can accept that the results are the same, but I do not agree that either necessarily invalidates the other. It seems like a leap of faith to tie an instantaneous IRF change to a physical end result like twins having different ages when the traveling one gets back to earth. That is switching from one IRF to another when the two do not agree with each other. But I can accept it. I have as easy a time accepting the equivalence principle in this simple application.

The equivalence principle has no relevance to the twin paradox. That is a fundamental misunderstanding.

The equivalence principle does not say that acceleration is equivalent to gravity. In particular, it definitely does not say that an accelerating object can be considered subject to a gravitational potential and subject to gravitational time dilation.

 

[FactChecker]

How does one even define an inertial reference frame without mentioning acceleration directly or in a disguised form (as a change in velocity wrt other reference objects)? How does one decide when to switch "instantly" from one IRF to another in determining the solution to the twins paradox? Without reference to some external knowledge or influence (eg. acceleration, reference to a third body, etc.), one can not distinguish the "stationary" twin from the "traveling" twin. IMHO, the attempts to completely ignore acceleration is flawed in the most fundamental ways.

 

[DaveC426913]

IMHO, the attempts to completely ignore acceleration is flawed in the most fundamental ways.

I didn't get the sense anyone was "completely ignoring" acceleration, but PeroK made a valid point, showing how it is not the acceleration itself that results in time dilation (I know you read this and responded already; just posting for clarity):

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 (*).

 

[FactChecker]

I didn't get the sense anyone was "completely ignoring" acceleration, but PeroK made a valid point, showing how it is not the acceleration itself that results in time dilation (I know you read this and responded already; just posting for clarity):

The change in the IRF is acceleration, either gradual or instantaneous. And that is the exact time when the aging of far away objects happens. At all other times, both observers see the other as aging slower.

I feel that this is going in circles and will leave the discussion to others. I just accept both ways of looking at it and am not convinced that there is a real conflict in the two views.

 

[DaveC426913]

The change in the IRF is acceleration, either gradual or instantaneous.

Yes. But (at the risk of being repetitive), the point is the acceleration itself does not result in the time dilation.

As witnessed in PeroK's example where an identical acceleration/deceleration curve can result in virtually no discrepancy in aging. It is the time spent moving at relativistic velocity that causes the discrepancy.

(A ship that accelerates at 5gs to .9c and then immediately decelerates back to rest may have a very small discrepancy, whereas a ship that accelerates at 5gs to .9c and stays there for a month will have a much larger discrepancy.)

And that is the exact time when the aging of far away objects happens.

No. The aging occurs during any time spent at relativistic velocities - whether for 1 second or for a month.
And the time spent getting to that speed can be arbitrarily short.