Relativity: Twin Paradox - Is Age Determinable?

[PeroK]

Thanks. Just to clarify: Are you assuming that the "Extended turnaround" will result in the same age difference on return as the "Simple turnaround"? Or why do you think that the effects of 1) and 2) must cancel?

From analysis in an IRF, all these scenarios depend only on B's spacetime interval - not on short-term acceleration profiles - isn't that the whole issue?

To within a day or two either way, the spacetime interval has a fixed length for all these scenarios. As long as B is traveling at 0.8c for almost all of the journey (in the Earth frame), then the differential ageing is determined (to within a day or two) solely by that.

This is the fundamental problem with the acceleration-based analysis. You can include all sorts of additional accelerations, but it doesn't make any significant difference - as long as the accelerations themselves are short lived.

 

[A.T.]

To within a day or two either way, the spacetime interval has a fixed length for all these scenarios.

You can set it up to have the same spacetime intervals for both scenarios. But then 1) and 3) of the extended turnaround are not identical to the acceleration in the the simple scenario, because they happen closer to A.

 

[jbriggs444]

You can set it up to have the same spacetime intervals for both scenarios. But then 1) and 3) of the extended turnaround are not identical to the acceleration in the the simple scenario, because they happen closer to A.

Presumably one would arrange for the all three proper accelerations at the extended turnaround to be equal in magnitude and proper duration. A might then disagree on their equality, but we are not particularly concerned with A's view of the turn-around. We are trying to wrap our heads around the difficulties with B's perspective.

Nonetheless, A should see that the velocity changes achieved by all three accelerations are equal in magnitude.

 

[A.T.]

Presumably one would arrange for the all three proper accelerations to be equal in magnitude and proper duration.

I was talking about the difference of acceleration 1 (or 3) to the single acceleration the simple scenario. Not a difference between 1 and 3.

I see no reason for the claim that the effects on differential aging of acceleration 1 and 2 have to cancel. It seems to be based on the wrong assumption that acceleration 3 is identical to the single acceleration in the simple scenario. In accelerated frames the clock rates are position dependent, so the distance plays a role.

 

[jbriggs444]

effects on differential aging of acceleration

Differential aging has no role to play in proper acceleration. It seems clear by symmetry that the same proper acceleration profile over proper time will achieve the same reversal of coordinate velocities in either direction.

Accelerations 1, 2 and 3 (and on up to 2n+1) at the extended turnaround are identical except in direction. Though they need not be for the scenario to achieve its pedagogical goal.

 

[A.T.]

Differential aging has no role to play in proper acceleration.

I'm referring to this :

Whatever "ageing" happens as a result of 1) must also happen as a result of 3). Therefore, the first "ageing" must be reversed by 2).

I don't see why 2) must "reverse the ageing" during 1).

 

[PeroK]

I was talking about the difference of acceleration 1 (or 3) to the single acceleration the simple scenario. Not a difference between 1 and 3.

I see no reason for the claim that the effects on differential aging of acceleration 1 and 2 have to cancel. It seems to be based on the wrong assumption that acceleration 3 is identical to the single acceleration in the simple scenario. In accelerated frames the clock rates are position dependent, so the distance plays a role.

The accelerations all take place at the same position in B's frame, by definition!

You can easily arrange for 1) and 3) to take place at the same position relative to A.

Fundamentally, though, look at what you are saying:

All accelerations for B must be physically unique in some way?? There's no such thing as periodic motion??

What if we have B execute SHM in A's frame? Each cycle of the motion is physically different? Or, the first is unique in some way?

 

[PeroK]

I'm referring to this :

I don't see why 2) must "reverse the ageing" during 1).

Because if it doesn't you get cumulative ageing of A (assuming multiple changes of direction) which is not supported by analysis in an IRF.

 

[A.T.]

You can easily arrange for 1) and 3) to take place at the same position relative to A.

But 1) and 3) do not take place at the same position relative to A as the acceleration in the simple case, assuming the same total space time intervals for both scenarios.

To clarify: Are you assuming that A will age the same amount in B's frame during these two accelerations?:
S1: Single acceleration in the simple scenario
E3: 3rd acceleration in the extended turnaround scenario

 

[jbriggs444]

Here is a drawing depicting extended turnaround in Euclidean geometry. The traveling twin B is on the right. The stay at home twin A on the left. Lines of (B-relative) simultaneity are drawn in the middle.

In order to shift from Minkowsky to Euclid, it was necessary to change the direction of the accelerations.

If you imagine yourself as a bug named B crawling up the right hand line, the progression of "simultaneous" positions sweeps both forward and backward on bug A's world line.

Of course, the multi-mapping of the point of intersection makes the naive B-relative coordinate chart invalid -- when extended as far as A's world line. The naive B-relative coordinate chart remains valid when applied over a sufficiently small world-tube surrounding B.

 

[PeroK]

But 1) and 3) do not take place at the same position relative to A as the acceleration in the simple case, assuming the same total space time intervals for both scenarios.

I really don't see why not. We're assuming that B executes some sort of periodic motion: half a cycle in the simple case and 1.5 cycles in the next case.

The difference in differential ageing between the scenarios must be less than the time for the periodic motion. As with all twin paradox scenarios, there are small variations based on how many acceleration phases B has.

To clarify: Are you assuming that A will age the same amount in B's frame during these two accelerations?:
S1: Single acceleration in the simple scenario
E3: 3rd acceleration in the extended turnaround scenario

Yes, of course, these are physically identical.

 

[A.T.]

But 1) and 3) do not take place at the same position relative to A as the acceleration in the simple case, assuming the same total space time intervals for both scenarios.

I really don't see why not. We're assuming that B executes some sort of periodic motion: half a cycle in the simple case and 1.5 cycles in the next case.

We also assumed the same total space time interval (path length) for all cases, so with more periods the amplitude (maximal separation has to be less).

To clarify: Are you assuming that A will age the same amount in B's frame during these two accelerations?:
S1: Single acceleration in the simple scenario
E3: 3rd acceleration in the extended turnaround scenario

Yes, of course, these are physically identical.

The aging of A in B's non-inertial frame depends on the spatial separation of A and B. Since the separation is less for E3 than for S1, A will age less in B's frame during E3 than during S1, even if B's proper acceleration and acceleration duration are the same for E3 and S1.

 

[PeroK]

The aging of A in B's non-inertial frame depends on the spatial separation of A and B. Since the separation is less for E3 than for S1, A will also age less, even if the proper acceleration and duration are the same.

You keep saying that but what prevents B from executing the same acceleration when he reaches the same distance from A? Why does B have to be closer? And why does be have to be significantly closer? If I specify the turnaround distance for B as $1m$ and I concede that E3 takes place $2m$ closer to Earth. I have no idea why, but let's accept that E3 must take place $2m$ closer to Earth than S1. These distances are negligible in the context of 4 light years. That is going to make a negligible variation to the $6.4$ years.

Please tell me why B cannot execute SHM as many times as he pleases? Back and forward the same mean distance from A? Why is SHM impossible in the twin paradox?

I only posted this idea to highlight an issue with the "acceleration causes ageing" interpretation. I didn't expect an argument on the physical feasibility of B changing direction more than once.

You must be fundamentally misunderstanding what I'm saying.

 

[PeroK]

The aging of A in B's non-inertial frame depends on the spatial separation of A and B. Since the separation is less for E3 than for S1, A will age less in B's frame during E3 than during S1, even if B's proper acceleration and acceleration duration are the same for E3 and S1.

Can you provide your analysis of the differential ageing assuming that A ages 6.4 years as a result of the first turnaround? What happens quantitatively if B changes direction linearly twice more? Why do subsequent changes of direction have mininal effect on the ageing of A?

Assume that any subsequent changes of direction of B take place in less than 1 day (in A's frame). Please show why no further significant ageing of A takes place, unless the overall journey itself is significantly extended (in A'a frame).

Please, let me see your analysis.

 

[A.T.]

You keep saying that but what prevents B from executing the same acceleration when he reaches the same distance from A? Why does B have to be closer?

I explained that here:

We also assumed the same total space time interval (path length) for all cases, so with more periods the amplitude (maximal separation has to be less).

 

[Dale]

That can't be right. The differential ageing relative to Terence can't depend on the number of changes of direction.

But it can depend on the distance between them at the turnaround.

 

[PeroK]

But it can depend on the distance between them at the turnaround.

What stops B making repeated changes of direction (over a relatively short time) in the vicincty of the initial turning point?

 

[PeroK]

I explained that here:

Obviously it's the same give or take a day or two for the various acceleration phases - as it always is for the twin paradox. It's a proper time of $6$ years (give or take an arbitrary variation for the turnaround(s)).

Obviously, if B does additional accelerations that will take a small amount of proper time. But that cannot explain additional differential ageing of 6.4 or 6.3 years or whatever.

 

[vanhees71]

But for the one way example here, there is no way to avoid a synchronization assumption, because that is the sole determinant of what the start event is for the Mars clock. There is only one incident of colocation. The interval beginnings are determined solely by a synchronization decision, which can be a physical procedure, thus invariant, but it is still a choice, and effectively defines a frame.

Then the problem is insufficiently defined. You have to clearly define everything in physical terms, i.e., in terms of physically defined events to begin with.

 

[A.T.]

Obviously it's the same give or take a day or two for the various acceleration phases - as it always is for the twin paradox. It's a proper time of $6$ years (give or take an arbitrary variation for the turnaround(s)).

Obviously, if B does additional accelerations that will take a small amount of proper time. But that cannot explain additional differential ageing of 6.4 or 6.3 years or whatever.

Where does this "additional differential ageing of 6.4 or 6.3 years" come from?

 

[PeroK]

Where does this "additional differential ageing of 6.4 or 6.3 years" come from?

This is getting just silly now. I've explained a simple scenario in excruciating detail and all you're doing in nitpicking the details.

I don't know what this is all about now.

 

[jbriggs444]

Where does this "additional differential ageing of 6.4 or 6.3 years" come from?

I've lost track. I think it was part of a reductio ad absurdum argument to the effect that one should not attribute differential aging to acceleration.

The following narrative is how I reconstruct it:

There were two claims. One was that the progress of the stay-at-home twin from the point of view of the traveling twin would always be in the forward direction. The other is that the "point of view" of the traveling twin is always accurately reflected by an instantaneously co-moving inertial frame.

If we accept the former claim then, during periods of forward acceleration (by B away from A), A's clock advances. In effect the former claim acts as a ratchet. [This claim is arguably correct -- in any valid coordinate chart, it will hold].

If we accept the latter claim then, during periods of reverse acceleration (by B toward A), A's clock advances by 6.3 or 6.4 years each time. [This claim is also arguably correct. If we look at the "time now" on A's clock in the after-acceleration frame, it will be 6.3 or 6.4 years advanced from the "time now" on B's clock in the before-acceleration frame]. I think that @PeroK proposed trip details to arrive at those numbers.

If one accepts both claims together, then one might conclude that the stay-at-home twin's elapsed time will have advanced by a total proper time equal to the number of turnarounds multiplied by 6.3 or 6.4. That conclusion is obviously false -- so something has gone wrong.

One way of looking at what went wrong is that the sequence of instantaneous tangent inertial frames do not fit together to create a valid coordinate chart covering A's world line. The first claim only holds for valid coordinate charts. The error in the analysis is pretending that the "traveler's frame" both covers A's world line and uses a synchronization convention that matches B's sequence of tangent inertial frames.

One can build an accelerated frame around B's world line and extend it to encompass A's world line. But the attribution of differential aging based on using that frame will come as much from the details of the frame as from B's acceleration profile.

 

[PeterDonis]

Ok, so it looks like I'm going to have to point out what @jbriggs444 predicted I would point out.

It seems logical that if the first turnaround caused A to age by 6.4 years, then so must the third change of direction.

It might seem logical, but it's not valid, because the implicit reference frame you are using is not valid. Once you have multiple turnarounds, or orbits, or whatever, the reference frame you are implicitly using to make statements like "A ages 6.4 years during the first turnaround) is not valid for such statements because it no longer validly covers A's worldline: the mapping from the frame's time coordinate to events on A's worldline is no longer one-to-one. (It is for the case of a single turnaround with no orbits, but only for that case.)

The deeper root cause of this problem is being unwilling to give up the intuition that there should be some fact of the matter about A's "rate of aging" relative to B. There isn't. That's what relativity tells us. The only invariant in the problem is the comparison of elapsed times when the twins meet again. There is no invariant that corresponds to A's "rate of aging" relative to B (or B's relative to A, for that matter). So statements like "A ages 6.4 years during the turnaround" aren't statements about physics; they're statements about some human's choice of coordinates. (And if the choice of coordinates isn't a valid coordinate chart, they're not even well-defined statements.) You can do all the physics without ever having to make such statements, so why make them at all?

 

[PAllen]

Then the problem is insufficiently defined. You have to clearly define everything in physical terms, i.e., in terms of physically defined events to begin with.

Well, you can use a physical procedure to define a frame. Einstein clock synchronization is a physical procedure, and if you specify two bodies with attached clocks performing it, the result of the procedure is frame independent, but at the same time, it effectively defines a frame based on those two bodies. The beginning events in a one way scenario are defined by a choice of bodies to perform this operation.

 

[PeterDonis]

It's about answering the question: "How does the whole process look like in the rest frame of the traveling twin?",

And if you insist on asking that question, even though, as I pointed out in my previous post just now, you can do all the physics without doing so, then you first have to construct a consistent "rest frame of the traveling twin" that covers all of A's worldline during the trip. And the frame @PeroK is implicitly using when he talks about A "getting younger" in a scenario with multiple turnarounds or orbits does not. There are multiple ways of doing so that do cover A's worldline, but none of them will have the property that "A gets younger" during any part of the trip.

 

[PeterDonis]

Einstein clock synchronization is a physical procedure

But it only works for a pair of bodies that are (a) in free-fall inertial motion, and (b) at rest relative to each other. That's a severe limitation.

 

[metastable]

The deeper root cause of this problem is being unwilling to give up the intuition that there should be some fact of the matter about A's "rate of aging" relative to B. There isn't. That's what relativity tells us. The only invariant in the problem is the comparison of elapsed times when the twins meet again. There is no invariant that corresponds to A's "rate of aging" relative to B (or B's relative to A, for that matter). So statements like "A ages 6.4 years during the turnaround" aren't statements about physics; they're statements about some human's choice of coordinates. (And if the choice of coordinates isn't a valid coordinate chart, they're not even well-defined statements.) You can do all the physics without ever having to make such statements, so why make them at all?

I’m confused on this point. If A and B are both radioactive, won’t their relative compositions differ when they meet again? Won’t this problem now affect A & B’s invariant mass in addition to their elapsed time?

 

[PeterDonis]

If A and B are both radioactive, won’t their relative compositions differ when they meet again?

Yes, that's a consequence of the invariant I described: the comparison of elapsed proper times. But you're adding an element to the problem that nobody in this thread was including. See below.

Doesn’t this now affects A & B’s invariant mass?

This is just quibbling. Nobody has been talking about radioactive objects, or indeed objects undergoing any kind of change. We're just talking about the twin paradox. Throwing in a complication like what will happen to radioactive substances is irrelevant to the topic of the thread. If you want to know what happens to the invariant mass of a radioactive object over time, start a separate thread.

 

[PeroK]

Ok, so it looks like I'm going to have to point out what @jbriggs444 predicted I would point out.
It might seem logical, but it's not valid, because the implicit reference frame you are using is not valid. Once you have multiple turnarounds, or orbits, or whatever, the reference frame you are implicitly using to make statements like "A ages 6.4 years during the first turnaround) is not valid for such statements because it no longer validly covers A's worldline: the mapping from the frame's time coordinate to events on A's worldline is no longer one-to-one. (It is for the case of a single turnaround with no orbits, but only for that case.)

The deeper root cause of this problem is being unwilling to give up the intuition that there should be some fact of the matter about A's "rate of aging" relative to B. There isn't. That's what relativity tells us. The only invariant in the problem is the comparison of elapsed times when the twins meet again. There is no invariant that corresponds to A's "rate of aging" relative to B (or B's relative to A, for that matter). So statements like "A ages 6.4 years during the turnaround" aren't statements about physics; they're statements about some human's choice of coordinates. (And if the choice of coordinates isn't a valid coordinate chart, they're not even well-defined statements.) You can do all the physics without ever having to make such statements, so why make them at all?

I thought that was my whole point. That A's rate of "ageing" (it was always in quotes in my earlier posts) relative to B is meaningless.

I still think the whole idea that "acceleration of B causes A to age" is not a valid concept. Even if you can justify it with a caveat that "it only works once". It's not an explanation for differential ageing that has any physical significance, as far as I can see.

Perhaps my argument against it overlooked deeper problems with coordinate systems. But, if B makes an elaborate interstellar journey then the differential ageing can still be simply computed by the integral of the speed in A's frame. Attempts to attribute differential ageing to acceleration and time dilation in B's frame are fundamentally flawed.

 

[PAllen]

But it only works for a pair of bodies that are (a) in free-fall inertial motion, and (b) at rest relative to each other. That's a severe limitation.

So what? That is exactly what is needed to specify the OP scenario to make it fully defined.

 

[DaveC426913]

What I was attempting to analyse was the "acceleration causes ageing" interpretation of the twin paradox. I was trying to highlight an issue with this interpretation.

Ah. Then we are in agreement.

 

[Dale]

What stops B making repeated changes of direction (over a relatively short time) in the vicincty of the initial turning point?

Nothing

 

[Bruce Wallman]

It only depends on the velocity of the traveling twin. If that person gets anywhere near the velocity of c (compared to the universe), that person will suffer from time dilation and will lose some heartbeats, etc in aging. So that person will be younger. However, the twin on Earth is also traveling at a decent speed within the universe. So it needs to be that the one going to Mars has a much faster velocity relative to the universe and something significant against c. I do not think acceleration has anything to do with time dilation directly.

 

[Ibix]

If that person gets anywhere near the velocity of c (compared to the universe)

This is not correct. In a standard twin paradox, where one twin is inertial and one twin travels out-and-back then it's the speed of the traveller relative to the inertial observer that matters. In the "one-way" version under discussion here there is no unique answer.

"Speed compared to the universe" is not a well-defined concept.

 

[Mister T]

If that person gets anywhere near the velocity of c (compared to the universe), that person will suffer from time dilation and will lose some heartbeats, etc in aging.

All that matters is the relative speed of the twins. And the speed need not be anywhere near $c$. Modern clocks are precise enough to see the effect when the speed is a very tiny fraction of $c$.