Twin paradox, virtual clock on ship with Earth time, discontinuity

[PeterDonis]

these are different levels: Proper acceleration is something physical, that you can measure

Proper acceleration can be directly measured, but its effect on differential aging cannot; to explain that you have to use the object's worldline and spacetime geometry.

The only real "direct measure" that can be used to predict differential aging is the Doppler shift of light signals.

Within SR (where the twin paradox usually first comes up) the acceleration profiles fully determine the proper time intervals between two events

Even this is not true. The length of time that the traveling twin goes out before turning around also plays a role. So does the traveling twin's velocity relative to the stay at home twin during the outbound and inbound inertial legs.

 

[A.T.]

Proper acceleration can be directly measured, but its effect on differential aging cannot; to explain that you have to use the object's worldline and spacetime geometry.

That's why I said you have to use both in the part that you snipped.

Within SR (where the twin paradox usually first comes up) the acceleration profiles fully determine the proper time intervals between two events.

Even this is not true. The length of time that the traveling twin goes out before turning around also plays a role. So does the traveling twin's velocity relative to the stay at home twin during the outbound and inbound inertial legs.

I consider the inertial phases to be part of the acceleration profile. But it's true that you also need the initial relative velocity, so "fully determine" is not correct.

 

[Herman Trivilino]

I wonder if the OP thinks the traveling twin is actually younger than he was when his travels began. It's confusing to simply say the traveling twin is younger without completing the statement by saying the traveling twin is younger than the staying twin.

 

[Dale]

Proper acceleration is something physical, that you can measure. World lines and space-time are parts of a model. One must talk about both, and how they relate, not chose one or the other

Well, I disagree about your characterization that worldlines and spacetime cannot be measured. The issue I see here, however, is that once you have introduced the spacetime explanation there is nothing remaining to be explained by acceleration.

In fact, I would go the other way, I would put time dilation and proper acceleration on the same level. Both are explained as aspects of the worldline. Time dilation comes from the length of the worldline and proper acceleration comes from its bending.

You wouldn’t say that a line’s bending causes its length. But you can say that a bent line is shorter than a straight line between the same points in a simple flat space.

 

[Nugatory]

The basic version the paradox is supposed to be solvable using SR only, so the inertial twin is not actually on Earth.

A subtlety that goes unmentioned in every elementary presentation of the problem....

 

[PeroK]

A subtlety that goes unmentioned in every elementary presentation of the problem....

This thread has highlighted the flaw in taking the "home" twin off the Earth- specifically to ensure they experience no proper acceleration.

A good student would ask what happens if the home twin remains on Earth? And, we can all see the argument based on proper acceleration start to crumble. Or, at least, things get complicated. Exactly how do you deal with acceleration, as opposed to relative velocity?

What is the formula for dealing with proper acceleration? Other than to convert the acceleration profile to a relative velocity profile?

 

[A.T.]

You wouldn’t say that a line’s bending causes its length. But you can say that a bent line is shorter than a straight line between the same points in a simple flat space.

Yes, that is what I try to convey with the bike analogy:

Even if the steering doesn't directly affect the speed of the bike (odometer tick rate), the total distance measured by the odometer between two fixed points is affected by the steering-profile.

 

[A.T.]

... the student should focus on relative velocity in a suitable IRF or focus on spacetime geometry instead?

Saying "use a suitable IRF" doesn't answer the question, why the traveling twin's rest frame isn't "suitable", so this isn't answering the question raised by the paradox.

 

[PeroK]

Saying "use a suitable IRF" doesn't answer the question, why the traveling twin's rest frame isn't "suitable", so this isn't answering the question raised by the paradox.

The travelling twin's reference frame is not a single inertial reference frame. It's composed of two separate inertial reference frames. The travelling twin's rest frame is non-inertial even if we ignore the turnaround phase, as the homeward journey is not the same rest frame as the outbound journey. You can analyse the worldlines of both twins in any of these three IRF's: the home twin's rest frame; the outbound frame; or, the inbound frame. Or, in an arbitrary IRF. Or, using more powerful mathematics, you can show that the length of each worldline is invariant in any system of coordinates (inertial or otherwise).

You already asked this question in post #51 and I answered it in post #52:

How does this answer the question: "Why can't the traveling twin just do the same type analysis of the whole journey in his rest frame, as the at home twin can to in his rest frame?"

Because the rest frame is not a single inertial frame throughout. You either have an acceleration or a relay system.

 

[A.T.]

The travelling twin's rest frame is non-inertial..

Yes, that's the resolution of the paradox: His analysis in his rest frame based on velocities only, contradicts the analogous analysis done by the inertial twin, because he failed to take the acceleration of his rest frame into account.

 

[PeroK]

Yes, that's the resolution of the paradox: His analysis in his rest frame based on velocities only, contradicts the analogous analysis done by the inertial twin, because he failed to take the acceleration of his rest frame into account.

We'll have to disagree that acceleration is the only possible resolution to the twin paradox. The scientific literature says otherwise.

 

[Sagittarius A-Star]

Yes, that's the resolution of the paradox: His analysis in his rest frame based on velocities only, contradicts the analogous analysis done by the inertial twin, because he failed to take the acceleration of his rest frame into account.

The "Critic" in Einstein's "Dialog about Objections against the Theory of Relativity" (1918) made this error. Then the "Relativist" resolved the "paradox" by taking the pseudo-gravitation in the accelerated frame into account, see in the middle of the page:
https://en.wikisource.org/wiki/Translation:Dialog_about_Objections_against_the_Theory_of_Relativity

 

[A.T.]

We'll have to disagree that acceleration is the only possible resolution to the twin paradox.

If by "resolution" we mean "point out the error that lead to the contradiction (paradox)", then "failure to account for the frame acceleration" is the answer.

If we want to go beyond pointing out the error, into correcting the flawed analysis, then you indeed have multiple options. But IMO pointing out the error in the original analysis is the obligatory first step, that you cannot omit, because you don't like that it involves acceleration.

 

[Histspec]

Within SR (where the twin paradox usually first comes up) the acceleration profiles fully determine the proper time intervals between two events. So it's not just "one particular scenario", but I would always be clear, that this not the most general case.

This reminds me of Sommerfeld's take (first published in 1913) on the clock paradox in a comment on Minkowski's famous lecture "Space and Time", English translation in 1923 by W. Perrett and G.B. Jeffery, in: The Principle of Relativity, London: Methuen and Company, pp. 37-91:

As Minkowski once remarked to me, the element of proper time $d\tau$ is not a complete differential. Thus if we connect two world-points O and P by two different world-lines 1 and 2, then $$\int_{1}d\tau\ne\int_{2}d\tau$$
If 1 runs parallel to the t-axis, so that the first transition in the chosen system of reference signifies rest, it is evident that $$\int_{1}d\tau=t,\ \int_{2}d\tau<t$$
On this depends the retardation of the moving clock compared with the clock at rest. The assertion is based, as Einstein has pointed out, on the unprovable assumption that the clock in motion actually indicates its own proper time, i.e, that it always gives the time corresponding to the state of velocity, regarded as constant, at any instant. The moving clock must naturally have been moved with acceleration (with changes of speed or direction) in order to be compared with the stationary clock at the world-point P. The retardation of the moving clock does not therefore actually indicate “motion,” but “accelerated motion." Hence this does not contradict the principle of relativity.

This is one of the first instances when the clock hypothesis was clearly formulated, that is, time indicated by clocks only depends on the constant (momentary) velocity.
But nevertheless he viewed the final retardation of one clock as an indication of "accelerated motion", evidently because the length of the worldline is defined by the (proper) acceleration profile.

 

[PeroK]

This reminds me of Sommerfeld's take (first published in 1913) on the clock paradox in a comment on Minkowski's famous lecture "Space and Time", English translation in 1923 by W. Perrett and G.B. Jeffery, in: The Principle of Relativity, London: Methuen and Company, pp. 37-91:

The moving clock must naturally have been moved with acceleration (with changes of speed or direction) in order to be compared with the stationary clock at the world-point P. The retardation of the moving clock does not therefore actually indicate “motion,” but “accelerated motion." Hence this does not contradict the principle of relativity.

From the perspective of a modern student, that appears not to be true. We could have started with two clocks in uniform motion and accelerated one to a state of rest - in the "stationary" frame. Then the "accelerated" clock would be running faster than the "unaccelerated" clock.

And, this is precisely the situation that Hafele-Keating faced in their famous experiment. The atomic clocks on the Earth's surface were already in motion relative to an inertial reference frame in which the Earth is rotating. And, an aircraft that takes off in a westerly direction is slowing down in this reference frame.

Fascinating!

 

[Histspec]

From the perspective of a modern student, that appears not to be true. We could have started with two clocks in uniform motion and accelerated one to a state of rest - in the "stationary" frame. Then the "accelerated" clock would be running faster than the "unaccelerated" clock.

In the context of Sommerfeld's example (two different worldlines connecting events O and P), his expression “retardation of the moving clock” wasn't simply about time dilation, it rather focuses on the difference in proper time intervals in the standard clock paradox. That is, Sommerfeld's description (like all other descriptions of the clock paradox at that time) focuses on a certain scenario in flat spacetime introduced by Einstein (1905) and Langevin (1911), which was criticized by contemporary crackpots like Gehrcke (who wrote a bunch of papers on that topic starting in 1912). That is:

a) We have two initially synchronous clocks 1 and 2 at position A, then clock 1 “moves” from A to B and comes back to A, with its time being retarded with respect to clock 2 that remained stationary at A.

b) The crackpot argument was: If the relativity principle is true, then clock 1 can also be considered “stationary” all the time, in which case the “moving” clock is 2 and its time must be retarded at reunion.

So relativistic physicists (like Sommerfeld) refuted point b) by showing that the asymmetry between the clocks (in the specified scenario) is caused by acceleration, thus the fact that only one clock is finally retarded doesn't violate the principle of relativity.

Now, it's certainly not surprising that an explanation that works perfectly in solving a scenario formulated in terms of specific boundary conditions, is incomplete or insufficient when it comes to more general scenarios with different boundary conditions. As long as one is aware of these limitations, this doesn't seem to be a problem.

 

[Dale]

If by "resolution" we mean "point out the error that lead to the contradiction (paradox)", then "failure to account for the frame acceleration" is the answer.

I see your point but I would always take “resolution” to include both pointing out the error and fixing it.

By far the vast majority of “fixing it” examples I have seen actually fix the triplets version and use the mapping between the triplets and twins scenarios to justify that as the fix. My favorite example of actually fixing the twins paradox is Dolby and Gull’s paper.

 

[A.T.]

I see your point but I would always take “resolution” to include both pointing out the error and fixing it.

I agree. My point was that one shouldn't omit the identifying the error first, just to avoid mentioning acceleration.

By far the vast majority of “fixing it” examples I have seen actually fix the triplets version and use the mapping between the triplets and twins scenarios to justify that as the fix. My favorite example of actually fixing the twins paradox is Dolby and Gull’s paper.

I don't have a favorite here. But I do see a conceptual difference between:

a) Just fixing the analysis the traveling twin attempted to do (single rest frame throughout), which leads to the pseudo gravity approach.

b) Proposing an alternative analysis based on multiple inertial frames, and justifying why it is equivalent (triplets).

I can understand why some see a) as a more direct fix, and consider b) as avoiding fixing the originally attempted approach, because it requires dealing with accelerated reference frame.

 

[Dale]

My point was that one shouldn't omit the identifying the error first, just to avoid mentioning acceleration

Agreed. I would never avoid mentioning acceleration. The concept of proper acceleration and accelerometers is important to how I understand general relativity.

I just dispute calling acceleration “fundamental” or any similar superlative.

 

[LikenTs]

Pondering about this interesting debate of accelerations in SR, a thought experiment has come to my mind.

Let's suppose two stars A and B. Along the line between them there are placed buoys with clocks at regular distances. All clocks, including the line and both stars, are synchronized. Stars A and B share clock rate, lengths and simultaneity criteria as they are frames at rest and inertial.

A ship, the twin, leaves from A to B, but this time it does so always accelerating, with continuous throttle and braking, coinciding with stops (v=0) at each buoy, for thousands of cycles, at an average speed v=0.8c.

Every time the ship stops, its comoving frame is at rest with respect to stars A and B, an it shares simultaneity and distances. So, it measures its time lag in each cycle and how far it is from both stars. For symmetry all cycles must have the same time offset. How does it matter the proximity of both stars? The ship is cyclically making the same accelerated movement between buoys. Therefore, in the ship they must see a contraction of terrestrial time that grows homogeneously throughout the round trip.

This raises the following problems:

1. For non-accelerated motion, except turning, the SR predicts that the spacecraft should see time dilation on Earth during both legs. But our stumbling ship sees Earth time contraction during both legs. Accelerations can ideally be almost instantaneous with very short cycles between buoys. Geometrically the world line is almost the same and with the same average speed, but with small steps in the time axis. This seems to make a fundamental difference between the two types of travel. At least from the ship's perspective.

2. The stumbling ride apparently contradicts the principle of equivalence in GR, since it shows no dependence on proximity to stars A and B.

What is wrong with this?

 

[Ibix]

What is wrong with this?

That you are trying to use naive reasoning based on patching together inertial frames without doing the careful book-keeping necessary to get the right answer by this method. You need to pay attention to both the size and sign of the "time skips" you induce when you switch frames, and you show no evidence of even trying that.

I have no idea where you think the equivalence principle comes into this mess.

 

[PeroK]

What is wrong with this?

Time dilation is not the same as differential aging. The former is a coordinate effect and has no physical significance. The latter is an invariant quantity and physically meaningful.

One of the major insights of the non-acceleration, relay experiment is to highlight this: On both the fully inertial outbound and fully inertial inbound legs, the Earth clock is always time-dilated. And yet, the Earth clock shows more elapsed proper time at the end than the elapsed proper time along the two legs of the journey.

 

[Dale]

For symmetry all cycles must have the same time offset.

This is not actually a requirement. You would need to choose to enforce that symmetry in your design of the ship’s frame, if that is a feature you desire. One of the problems that you continue to face is the fact that “the ship’s frame” has no commonly accepted meaning, and the references you have based your work on do not resolve that issue.

Therefore, in the ship they must see a contraction of terrestrial time that grows homogeneously throughout the round trip

Since your premise is wrong in general the conclusion also fails in general. You have to design “the ship’s frame” to achieve this if you desire this feature.

For non-accelerated motion, except turning, the SR predicts that the spacecraft should see time dilation on Earth during both legs.

This is not true in general. It may be true for some very specific definitions of “the ship’s frame”, but you would have to demonstrate that.

This seems to make a fundamental difference between the two types of travel. At least from the ship's perspective

It certainly would make a difference in the workers compensation and passenger lawsuits from whiplash injuries.

The stumbling ride apparently contradicts the principle of equivalence in GR, since it shows no dependence on proximity to stars A and B.

Nonsense. The equivalence principle is irrelevant there.

What is wrong with this?

Primarily the failure to define “the ship’s frame”

 

[Ibix]

What is wrong with this?

Just to illustrate my previous answer, here's a Minkowski diagram of your ship stuttering its way from one planet to the other. The planets are shown stationary as blue worldlines and the ship as a red one alternating between 0.8c and zero. I've added bands in grey which show the sections of spacetime that a naive approach calls "during" each of the phases of its motion. The horizontal bands are "during" the zero speed phases and the sloped ones are "during" the 0.8c phases (notice where they overlap the red worldline of the ship).

Looking at the left hand "origin" planet, what the naive approach to the ship's simultaneity calls "during" the second phase has a gap after the first phase. "During" the third phase overlaps the second phase and has a larger gap from the end of the third phase to the beginning of the fourth. "During" the fifth phase is actually before any of the fourth phase. And it just gets worse - "during" the ninth phase is also "during" the sixth.

Looking at the right hand "destination" planet you can see the same pattern in reverse - the size of the discontinuities falls as you approach the planet.

Notice that the only place that there is no overlap or gap between one "during" and the adjacent ones is along the red worldline.

You can use this method to define what you mean by "what time it is on Earth, now", but every time you change speed you have to keep track of the gap (positive or negative) between the end of one "during" and the beginning of the next, and your clock will be jumping backwards and forwards. The lesson of the twin paradox, though, is ultimately that you'd be a fool to try it for anything much more complex than the vanilla scenario. Indeed, the discontinuities are incompatible with most of the analytical tools you would normally use.

 

[Dale]

@LikenTs please see Dolby and Gull’s paper for a definition of the ship’s frame that has the symmetry properties you mentioned, avoids the issues raised by @Ibix, and respects the second postulate:

https://arxiv.org/abs/gr-qc/0104077

It does not simply behave like a pair of SR inertial frames for the standard scenario. It also does not match the hodgepodge approach of your papers, but it fixes the issues that those papers don’t even acknowledge, let alone address

 

[Ibix]

@LikenTs please see Dolby and Gull’s paper for a definition of the ship’s frame that has the symmetry properties you mentioned, avoids the issues raised by @Ibix, and respects the second postulate:

https://arxiv.org/abs/gr-qc/0104077

Indeed - I was just trying to determine if I could re-draw the diagram above with D&G's simultaneity planes, but it's more fiddly than I can be bothered to implement. The planes change slope every time they cross the past or future lightcone of each acceleration event, and I don't get paid enough to debug that...

 

[LikenTs]

That you are trying to use naive reasoning based on patching together inertial frames without doing the careful book-keeping necessary to get the right answer by this method. You need to pay attention to both the size and sign of the "time skips" you induce when you switch frames, and you show no evidence of even trying that.

I suppose you are right. But I think that understanding this failure would be enlightening. My reasoning is, If a ship travels to a nearby star that has its clock synchronized with Earth and stops there, it is not necessary for the twin to turn around and complete his trip to know that he is already 2 years younger than his twin on Earth (on completion he would be 4 years younger). It is true that it is not an invariant for every frame. Some passing relativistic traveler may disagree (basically because he sees the brother on earth on another axis of simultaneity), but it is an "invariant" for both stars and twins, "stationary" observers. They can establish communication on TV, and discounting the time of the signal, see that the traveller twin is 2 years younger. Or simply the twin can compare his wristwatch with that of the star at same place and see a difference of 2 years. Years later he can continue the journey to another nearby star aligned with the previous two, and if the conditions are the same (distance and speed), when he stops at the third star he will be 4 years younger than his brother on Earth, according to all "stationary" observers. And so on , 2, 4, 6,.. years until he decides to make the reverse trip, stopping at each star on the way back. And already on Earth, if he has visited N aligned stars he would be 4N years younger than his brother. Being this an invariant for everybody.

I have no idea where you think the equivalence principle comes into this mess.

I was referring to pseudo gravity. When accelerating in direction of star B ship is in a pseudo gravitational field, with star B above, and its time speeds up according to ship. An approach for the non-inertial frame.

@LikenTs please see Dolby and Gull’s paper for a definition of the ship’s frame that has the symmetry properties you mentioned, avoids the issues raised by @Ibix, and respects the second postulate:

https://arxiv.org/abs/gr-qc/0104077

It does not simply behave like a pair of SR inertial frames for the standard scenario. It also does not match the hodgepodge approach of your papers, but it fixes the issues that those papers don’t even acknowledge, let alone address

I'll study it.

 

[Nugatory]

it is not necessary for the twin to turn around and complete his trip to know that he is already 2 years younger than his twin on Earth.

That’s not what they know. What he knows is that there is an arbitrary and physically meaningless definition of “already” in which the earth twin is two years older. To get a sense of how arbitrary and meaningless that is, consider that the traveller could just as naturally and correctly use the time dilation formula and know that the earth twin is still younger.

Before you respond to this post, try stating what you mean by “already”. I expect that it seems so obvious to you as not to need stating, but it does.

 

[LikenTs]

That’s not what they know. What he knows is that there is an arbitrary and physically meaningless definition of “already” in which the earth twin is two years older. To get a sense of how arbitrary and meaningless that is, consider that the traveller could just as naturally and correctly use the time dilation formula and know that the earth twin is still younger.

Before you respond to this post, try stating what you mean by “already”. I expect that it seems so obvious to you as not to need stating, but it does.

The traveler arrived at Alpha Centauri and has been living there for a decade in a planetary habitat. The terrestrial TV transmission arrives with 4 years of delay, and he watches the news in the year 2019. The habitat clock marks year 2023, since it is synchronized with Earth. He also keeps the atomic clock that traveled with him in the relativistic ship and it marks year 2021. So it is clear that his brother is 2 years older than him. He also knows that if any relativistic ship is crossing Alpha Centauri it will have a distorted view of simultaneity and it will set his brother in the past or future with respect to the concept of now shared in Alpha Centauri, Earth, neighboring stars and in almost all the Galaxy.

 

[Ibix]

So it is clear that his brother is 2 years older than him.

What frame did he measure the distance in before he added the light travel time? Did he use an orthogonal coordinate system where one-way light speed is isotropic or not?

He will get different values of how out of date the TV broadcasts are depending on the answers to those questions. So he will still see the TV showing 2019, but may have different opinions about what that means about "now" on Earth.