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

[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.

 

[PeroK]

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.

If we insist on the Einstein light signal synchronization convention, then with this definition of simultaneity the events on Earth can be mapped to the events on Alpha Centauri and ages on Earth can be mapped to ages on Alpha Centauri.

The problem is that the Einstein convention is not the only one, although it's fairly universally used in SR. Taking this convention as the only choice will lead to problems if you continue your studies.

My worry about all your posts in this thread is that you are trying to nail down your own half-hearted acceptance of SR. You accept some things, but are trying to reestablish an absolute view of spacetime and simultaneity. For example, when you say "distorted view of simultaneity", you fail to fully accept the principle of relativity and give the galactic rest frame a special place. The laws of physics are equally valid in all inertial reference frames: no IRF has a "distorted view" of anything.

 

[Nugatory]

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

It is not clear, and as you've phrased it is unclear what that statement even means.

Let's try something more precise: It is clear that if we use Einstein clock synchronization to map between events on the earth twin's worldline and events on the segment of the traveling twin's worldline after their arrival at alpha centauri, then the event "signal received on traveler worldline" maps to the event "clock on earth worldline reads 2023".

That's a statement about one particular mapping between two non-overlapping sets of points in spacetime. It is really tempting to interpret that statement more strongly, to extract some useful information about the relative ages of the twins from it, but to do so we must make some additional assumptions (most likely, that Einstein synchronization is somehow more real/significant/relevant than other mappings, as opposed to being easy to use with clocks at rest relative to one another). We can get through elementary special relativity with that assumption, but it is an obstacle to understanding and has to be unlearned at some point.

 

[Sagittarius A-Star]

The habitat clock marks year 2023, since it is synchronized with Earth.

As others mentioned, also a non-standard clock synchronization is possible.

Einstein convention
See also: Einstein synchronization
This method synchronizes distant clocks in such a way that the one-way speed of light becomes equal to the two-way speed of light. If a signal sent from A at time ##t_{1}## is arriving at B at time ##t_{2}## and coming back to A at time ##t_{3}##, then the following convention applies:

##t_{2}=t_{1}+\tfrac{1}{2}\left(t_{3}-t_{1}\right)##.
...
Non-standard synchronizations
As demonstrated by Hans Reichenbach and Adolf Grünbaum, Einstein synchronization is only a special case of a broader synchronization scheme, which leaves the two-way speed of light invariant, but allows for different one-way speeds. The formula for Einstein synchronization is modified by replacing ½ with ε:
##{\displaystyle t_{2}=t_{1}+\varepsilon \left(t_{3}-t_{1}\right).}##
ε can have values between 0 and 1. It was shown that this scheme can be used for observationally equivalent reformulations of the Lorentz transformation, see Generalizations of Lorentz transformations with anisotropic one-way speeds.

Source:
https://en.wikipedia.org/wiki/One-way_speed_of_light#Non-standard_synchronizations

 

[LikenTs]

As others mentioned, also a non-standard clock synchronization is possible.

Source:
https://en.wikipedia.org/wiki/One-way_speed_of_light#Non-standard_synchronizations

But what physical sense does this anisotropic synchronization criterion have in this case?

Which means that light can take 2 years from Earth to Alpha Centauri but 6 years in the other direction.
It would force to change Lorentz transformations or dilations depending on the direction. I doubt until it is compatible with Maxwell's equations. For me it seems more like a theoretical mathematical exercise, which is compatible with twin paradox when they meet on closed loops. Maybe it can have physical sense in other scenarios but in this one I don't see any.

I think this is a problem of focusing on Mikowsky's mathematical model and the invariants as the only elements of reality, losing the physical perspective.

My worry about all your posts in this thread is that you are trying to nail down your own half-hearted acceptance of SR. You accept some things, but are trying to reestablish an absolute view of spacetime and simultaneity. For example, when you say "distorted view of simultaneity", you fail to fully accept the principle of relativity and give the galactic rest frame a special place. The laws of physics are equally valid in all inertial reference frames: no IRF has a "distorted view" of anything.

I mean that if the twins were in a flotilla of relativistic ships with low differences in speed, the frame of the flotilla would be the most appropriate to describe reality in it, and in the frame of the Galaxy they would have a distorted vision of the relationships between the 2 twins. This also happens with Galilean relativity. The frame of a train is not the most appropriate to describe what happens in a laboratory on land. Although the physical laws are covariant, the movements or phenomena are altered by the movement of the train . This is even worse in SR since there is not only spatial but also time distortion.

 

[Ibix]

But what physical sense does this anisotropic synchronization criterion have in this case?

No synchronisation convention has any physical sense. It's always about personal choice, usually made for convenience.

It would force to change Lorentz transformations or dilations depending on the direction.

More precisely, you would not be using inertial frames as defined by Einstein. So the Lorentz transforms would be replaced by other coordinate transforms.

I doubt until it is compatible with Maxwell's equations.

Of course it is. It's just a coordinate change. Laws of physics don't care what coordinates you use, although the maths may be more or less messy.

think this is a problem of focusing on Mikowsky's mathematical model and the invariants as the only elements of reality, losing the physical perspective.

The invariants are the mathematical expression of things we can measure. They are the physical perspective.

they would have a distorted vision of the relationships between the 2 twins.

They have a different view. It isn't distorted though, since that would imply that there was an undistorted viewpoint one could have. And that would imply a priviledged viewpoint.

The rest of what you wrote in that paragraph seemed reasonable.

 

[Sagittarius A-Star]

But what physical sense does this anisotropic synchronization criterion have in this case?

The twins cannot detect any difference between the isotropic and anisotropic synchronization / stipulated one-way-speeds. Therefore they cannot find out their age difference independently of a synchronization convention, until they meet again.

 

[Nugatory]

But what physical sense does this anisotropic synchronization criterion have in this case?

It has the exact same physical significance that Einstein synchronization or any other choice of coordinates, does: none.
No matter which synchronization convention we choose we will calculate the same invariants - which includes the results of direct observations and experimental results.

I think this is a problem of focusing on Minkowsky's mathematical model and the invariants as the only elements of reality, losing the physical perspective.

The invariants ARE the elements of physical reality. They're the things that are experienced and observed.
You also are confusing two different things: Minkowski's mathematical model of flat spacetime which can be described with whatever coordinates are convenient for the problem at hand, and Minkowski x,y,z,t coordinates which are often used in special relativity problems. We're trying to talk you out of focusing on the second.

 

[LikenTs]

The invariants ARE the elements of physical reality.

Not all elements of reality are frame invariants. The real length of a bar, in its rest frame, is a physical element of reality, but it is not frame invariant. Although everyone can calculate the proper length in the rest frame and agree. In that sense it is an invariant.

SR plus Einstein's synchronization postulate (Isn't that the standard or official SR from Einstein?) allow us to affirm that halfway through the symmetrical trip the traveler twin has already gained half the age respect to his twin's inertial frame. It is not frame invariant, but everyone can calculate it in the rest frame of the twins ( Stationary brother's inertial frame)

 

[Sagittarius A-Star]

SR plus Einstein's synchronization postulate (Isn't that the standard or official SR from Einstein?)

Yes, it is standard. But Einstein himself also wrote:

The latter can now be determined by establishing by definition that the "time" needed for the light to travel from A to B is equal to the "time" it needs to travel from B to A.

Source (page 142):
https://einsteinpapers.press.princeton.edu/vol2-trans/156

 

[PeterDonis]

Not all elements of reality are frame invariants.

Wrong. If you actually believe this, then you will have serious problems trying to understand relativity.

The real length of a bar, in its rest frame, is a physical element of reality, but it is not frame invariant.

Wrong. The length of the bar in its rest frame is an invariant. It is measured by a definite physical process that gives a definite answer. All observers will agree on it.

The length of the bar in some other frame, in which it is not at rest, has a different value because it is a different invariant, measured using a different procedure. And all observers will agree on that too.

Notice that in both cases above, I linked the relevant invariant to a specific physical process that measures it. There is no corresponding physical process involved with simultaneity. For example:

SR plus Einstein's synchronization postulate (Isn't that the standard or official SR from Einstein?) allow us to affirm that halfway through the symmetrical trip the traveler twin has already gained half the age respect to his twin's inertial frame.

But this is not a measurement; there is no way for the twin to measure "what time it is on Earth" while he is away from Earth. He can only measure the time by his own clock and the Doppler shift of light he is receiving from Earth.

 

[Sagittarius A-Star]

I'll study it.

The Dolby & Gull paper refers under Figure 8 to the spacetime-metric of a uniformly accelerated frame in radar coordinates (Lass coordinates). Instead, the 2nd linked paper in your posting #19 uses in equation (8) on page 4 the spacetime-metric of a uniformly accelerated frame, expressed in Kottler–Møller coordinates.

 

[LikenTs]

I think I'm beginning to have a clearer vision of the problem and SR in general.

In particular, I understand the thought experiment that I proposed in post #80, about the ship that makes stops during its trip.

The key factor is the two ways of measuring other frame's time. One can follow one foreign clock and in that case both inertial frames measure the slowdown of the other. Or one can follow the synchronized clocks of another frame while passing through them and in this case both frames observe that the other frame's time goes faster. The Lorentz transformation is more focused on the first type of time measurement.

If we suppose a universal inertial frame of convenience, for example the galactic frame, and we imagine that it contains a mesh or net of synchronized clocks, all relativistic ships could see their time offset with respect to galactic time when passing through these clocks. If they also stop close to one of these, they share the simultaneity criterion with the galactic frame, so they can assign that time to that of a clock on Earth.

Fortunately, that mesh is not necessary. A ship, with a local clock, an accelerometer (more possibly gyroscopes) and a computer, can reconstruct dynamically or in real time its speed and position with respect to the galactic frame, and with this can build a display that shows the galactic time. In fact, the ship's frame is also the most privileged for this computation, since all other frames see things with a signal delay and require greater infrastructures.

What this inertial or galactic time display would show on all relativistic ships is always a higher ratio than their local clocks and an increasing time offset. The greater the faster they go or the longer they travel (without direct dependence on acceleration). This inertial clock display would only go at the same rate as the local clock when the ships stop (And they would know at all times how old their relatives are).

The way to understand this inertial or galactic time display is that the simultaneity criterion of the Galaxy is adopted. While in most of the graphs that I put above the criterion of simultaneity of the ship is adopted. The computer can also reconstruct a second display to follow Time on Earth (or any Galactic place) with the ship's simultaneity criteria. This one does show dependence on accelerations and distances and can even go backwards in time.

Here is the graph of both virtual clocks on the ship, that of universal or galactic time, with galactic's frame simultaneity , in blue, and the time on Earth according to the simultaneity of the ship, in black. At the return point, green, the stopping time is somewhat exaggerated, a slope of 45 degrees. At that point in the ship is observed the same ratio of the 3 clocks , the local one and the two virtual ones. You can also see how the virtual ones coincide at the stops.

Some reference articles:

About the reconstruction of the world line from the ship.

Differential aging from acceleration, an explicit formula
https://arxiv.org/abs/physics/0411233
Journal reference: Am.J.Phys. 73 (2005) 876-880

On the two ways of measuring time and the different relevance for both twins
http://kirkmcd.princeton.edu/examples/clock.pdf

 

[PeterDonis]

inertial frames

What inertial frames are you talking about? You are talking about ships that accelerate and decelerate, i.e., they are not inertial. That means there is no inertial frame in which they are always at rest. You don't appear to be properly taking that very important fact into account. For example, you talk about "the" Lorentz transformation, but a Lorentz transformation is only valid between two inertial frames, and at best there is only one that applies to your entire scenario.

If we suppose a universal inertial frame of convenience, for example the galactic frame

If you are talking about our actual galaxy, there is no such thing. You can construct a frame in which the center of the galaxy is at rest, but it will not be an inertial frame. Gravity is present, and in the presence of gravity there are no global inertial frames.

If you are talking about an imaginary case where there is no gravity, but you have labeled some region of space as "the galaxy" and are choosing an inertial frame in which that region is at rest, then you can use such a frame in your imaginary case, but many conclusions that you reach will not generalize.

Here is the graph of both virtual clocks on the ship, that of universal or galactic time, with galactic's frame simultaneity , in blue, and the time on Earth according to the simultaneity of the ship, in black.

How are you calculating these lines?

 

[LikenTs]

What inertial frames are you talking about? You are talking about ships that accelerate and decelerate, i.e., they are not inertial. That means there is no inertial frame in which they are always at rest. You don't appear to be properly taking that very important fact into account. For example, you talk about "the" Lorentz transformation, but a Lorentz transformation is only valid between two inertial frames, and at best there is only one that applies to your entire scenario.

In this paragraph I speak in general of two inertial frames. Both see each other's time slowed down or faster, depending on what kind of time measurement they do. Even so, if what they measure is the time of the other's clock mesh, any accelerated frame always sees that the inertial frame's mesh goes at a higher rate.

If you are talking about our actual galaxy, there is no such thing. You can construct a frame in which the center of the galaxy is at rest, but it will not be an inertial frame. Gravity is present, and in the presence of gravity there are no global inertial frames.

I am ignoring in this case the gravitational fields, since that would be the scope of GR. But surely a galactic frame with standard time for low gravitational fields could be defined, and the computer on the ship could make adjustments according to GR. So everyone on the ship will know, seeing the inertial virtual clock, the time lag with Earth, even when the ship orbits a black hole. I think this type of virtual clock will be mandatory on relativistic ships, if they are ever developed.

How are you calculating these lines?

The blue line, Galactic or Universal Time virtual clock on ship, is easy, T = T' . gamma (1.66 for v=0.8c). Being T' the local clock on ship. It is what you would calculate by integrating the proper time of world line from the inertial frame. The black line, the time of the clock on Earth according to the criterion of simultaneity in the ship, is calculated in the accelerated frame (There are several documents cited above), but the key is that when the ship is inertial during both legs the ship sees the Earth clock going slower, while at the return point it sees a rapid jump in time on the earth clock. The trip is better understood anyway by adopting the galactic simultaneity criterion, and thinking that the own simultaneity criterion is distorted with respect of the Galaxy. The traveling twin sees on the inertial clock display that he is progressively falling behind in galactic time during the trip.

 

[PeterDonis]

In this paragraph I speak in general of two inertial frames.

Ok. But I don't see the point since there is only one global inertial frame in your scenario.

Both see each other's time slowed down or faster, depending on what kind of time measurement they do.

This is a very sloppy way of talking, which I suspect is causing many of your problems.

It is impossible for observer A, spatially separated from observer B, to directly measure "what time it is now for observer B". The only direct observation observer A can have from observer B is receiving light signals from B, and those can only show what B's clock read at the event where they were emitted. Observer A can of course calculate all manner of things, but those are calculations, not direct observations. Your use of the word "see" and "time measurement" to refer to such calculations obfuscates this important point.

if what they measure is the time of the other's clock mesh, any accelerated frame always sees that the inertial frame's mesh goes at a higher rate.

I'm not sure what this means.

I am ignoring in this case the gravitational fields, since that would be the scope of GR.

Ok.

But surely a galactic frame with standard time for low gravitational fields could be defined

You can define a frame in which the center of the galaxy is at rest, and you can define coordinate time in such a frame. But you cannot assume that this frame will work like a global inertial frame in SR. It won't.

the computer on the ship could make adjustments according to GR.

If the computer knows the spacetime geometry, and knows where it is in that spacetime geometry, it can compute appropriate adjustments, yes.

everyone on the ship will know, seeing the inertial virtual clock, the time lag with Earth

If they also know Earth's location in the defined coordinates, and are using that simultaneity convention, then yes, they can compute this according to that convention. But, as has been said many times now, this is just a convention. It has no physical meaning. It is certainly not a direct observation. But you appear to really, really want to treat it as such. That won't work.

The blue line, Galactic or Universal Time virtual clock on ship, is easy, T = T' .

The blue line in your chart, the one titled "Universal Time Simultaneity", is not a 45 degree line, which is what ##t = t'## describes.

The black line, the time of the clock on Earth according to the criterion of simultaneity in the ship, is calculated in the accelerated frame (There are several documents cited above)

Which specific paper's definition are you using for "the accelerated frame".

when the ship is inertial during both legs the ship sees the Earth clock going slower, while at the return point it sees a rapid jump in time on the earth clock.

Your diagram doesn't show this. This is a reasonable description of one possible "rest frame" for the traveling twin in the standard twin paradox, but your diagram doesn't look like a diagram of the standard twin paradox.

The trip is better understood anyway by adopting the galactic simultaneity criterion, and thinking that the own simultaneity criterion is distorted with respect of the Galaxy.

Perhaps this is true for you. (And even that implies that your understanding is correct, which, given how long this thread has gone on, is not obvious.) That doesn't mean it will be true for everyone.