Inertial Frames distinguished by proper times

[robphy]

No yourself - again you, like many, do not understand the difference between real time dilation (that shows up and is measurable in both one way and round trip experiments), and the apparent time dilation that results from the blind application of the LT transforms to a situation where you have failed to take into account the intrinsic asymmetry of the frames

The relationship between two spacetime events in the "at rest" system and the same two spacetime events in the moving system is directly determined by the invariance of the interval.

I need a clear definition (or better, a spacetime diagram) to continue in this discussion. (In addition, a response to my last post would be nice.)

I understand "time dilation" to mean the following.
An observer A computes the following ratio:
"elapsed-proper-time-on-A's-watch from O to T (events on A's inertial worldline)"
divided by
"elapsed-proper-time-on-B's-watch from O to T' (events on B's inertial worldline)"
where observer A regards T and T' as simultaneous.
That ratio is symbolized by gamma, the "time-dilation factor".

Please clearly compare and contrast this definition with your terms
"real time dilation"
and
"apparent time dilation".
It would help me if you first state (either "SAME" or "DIFFERENT") then use a similar spacetime language (e.g. events, readings on watches, relationships between those events (on the same inertial worldine? simultaneous? connected by a light signal?, etc...) as in my defintion.) A spacetime diagram would be fabulous.

 

[yogi]

robphy Sorry - I did not mean to ignor your post 20 - let me see if I can get across the concept by sticking to the simple case of a satellite in circular orbit about the Earth (whenever I do this I get a response from someone that the satellite is not an inertial system, and therefore SR isn't applicable ...and the thread gets sidetracked on that issue)..So whatever your views are on that, there is abundant authority that a clock in orbit is a free falling inertial system i.e., it is as good as any other inertial frame for the purposes of SR.

Now we sync two clocks S2 and S3 with an Earth clock E at the center of the earth. The two clocks S2 and S3 are put in orbit and both are compensated for the altitude. If all clocks transmit a pulse every second according to their local time, then the pulses transmitted by S2 and S3 will arrive at E at more than one second intervals as measured by the E clock. The pulses sent by E will arrive at S2 and S3 in shorter intervals than one second. This is actual time dilation - The two clocks in orbit have been accelerated to orbital velocity after they were synchronzed with the E clock that measures time in the ECRF. The orbital radius remains constant - there is no question as to which clock is running fast.

Now we correct the S3 clock as we do in GPS systems so that it runs at a faster rate - it now transmits signals that are received by E once every second. The S2 clock still runs slow

If S3 is used as a base for measuring time dilation between E and S3, none will be detected - they run in sync. If the S2 clock is used, S2 concludes that it receives pulses at a greater rate than it is transmitting, and concludes that E is running faster. Likewsie, E concludes from the slower pulse rate received by the signals sent from S2, that S2 is running slower - the situation is not reciprocal - in real time dilation there cannot be reciprocity.

Now we correct both S2 and S3 to run at the same speed so that each transmits pulses that are received by E at one second intervals - but instead of putting them in the same satellite, we put one in a polar orbit and the other in an equitorial orbit at the same altitude...so they will have varying relative velocity between one another at different places in their orbits. As measured by the Earth clock, both S2 and S3 are running at the same speed - but as each views the other, they will determine that the other clock is running slow - here there is reciprocity, but the measurments are apparent - the clocks run at same corrected speed during their respective orbits - but they measure an apparent slowing of the clock in the other satellite.

I hope this clarifies what i mean by apparent and actual time measurements

 

[robphy]

robphy Sorry - I did not mean to ignor your post 20 - let me see if I can get across the concept by sticking to the simple case of a satellite in circular orbit about the Earth (whenever I do this I get a response from someone that the satellite is not an inertial system, and therefore SR isn't applicable ...and the thread gets sidetracked on that issue)..So whatever your views are on that, there is abundant authority that a clock in orbit is a free falling inertial system i.e., it is as good as any other inertial frame for the purposes of SR.

For now, I'll have to reserve comment on satellite/GPS issues... since this seems to complicate the situation beyond the originally posed question, which I tried to reformulate as a simple SR problem in my posts #4, #7, #9, #14, #18, and #20. (In SR [that is, flat spacetime on R⁴], I don't recall any need for a "compensation for altitude"... if one is needed, then it seems to me that the symmetry is already broken.) It seems to me that if your question can't be formulated as an SR problem, then it really isn't an SR problem. It may be a problem of inertial frames (i.e. geodesics) in a non-SR spacetime.

Now we sync two clocks S2 and S3 with an Earth clock E at the center of the earth. The two clocks S2 and S3 are put in orbit and both are compensated for the altitude. If all clocks transmit a pulse every second according to their local time, then the pulses transmitted by S2 and S3 will arrive at E at more than one second intervals as measured by the E clock. The pulses sent by E will arrive at S2 and S3 in shorter intervals than one second. This is actual time dilation - The two clocks in orbit have been accelerated to orbital velocity after they were synchronzed with the E clock that measures time in the ECRF. The orbital radius remains constant - there is no question as to which clock is running fast.

Now we correct the S3 clock as we do in GPS systems so that it runs at a faster rate - it now transmits signals that are received by E once every second. The S2 clock still runs slow

If S3 is used as a base for measuring time dilation between E and S3, none will be detected - they run in sync. If the S2 clock is used, S2 concludes that it receives pulses at a greater rate than it is transmitting, and concludes that E is running faster. Likewsie, E concludes from the slower pulse rate received by the signals sent from S2, that S2 is running slower - the situation is not reciprocal - in real time dilation there cannot be reciprocity.

Now we correct both S2 and S3 to run at the same speed so that each transmits pulses that are received by E at one second intervals - but instead of putting them in the same satellite, we put one in a polar orbit and the other in an equitorial orbit at the same altitude...so they will have varying relative velocity between one another at different places in their orbits. As measured by the Earth clock, both S2 and S3 are running at the same speed - but as each views the other, they will determine that the other clock is running slow - here there is reciprocity, but the measurments are apparent - the clocks run at same corrected speed during their respective orbits - but they measure an apparent slowing of the clock in the other satellite.

I hope this clarifies what i mean by apparent and actual time measurements

I'll have to ponder it... but I can't promise anything soon...
unless you'd like to continue the SR problem left off at post #20.
Are you trying to claim that "time dilation" (as I defined in my post #31) is not reciprocal between two inertial observers in SR [flat spacetime on R⁴]?

 

[yogi]

robphy: To my way of thinking, the GPS satellite analogy is the easy way to illustrate the notion of two relatively moving clocks - the corrections for altitude are put in because the G field is different at the height of the orbit. We can change this if you like to a thought experiment where the satellite orbits at sea level by constructing evacuated tunnels to circumscribe the Earth (an easy thought experiment but a difficult engineering task). With this model, we can forget about altitude compensation - the orbiting clock S2 does not run at the same rate as the E clock (the one that measures time in the non rotating Earth centered reference frame). The satellite clock S2 runs slower relative to the E clock and the velocity adjusted S3 clock runs faster than S2. If all three clocks transmit pulses at one second intervals as measured by their own clocks, the S2 pulses will be received by E at a slower frequency than S3 pulses, and the E transmitted pulses will be received in Sync with S3 but they will be blue shifted as to the uncompensated clock S2. As between S2 and E there is no ambiguity as to which clock is running faster (E) and which clock is running slower (S2)

I use this example because the distance between E and S2 remains the same - so there is no need to account for changes in length that arises in situations where clocks approach and recede from one another during the experiment. Nor is there any observed relativistic contraction in the radius of the Earth since the motion is transverse to the radial vector. There is a transverse Doppler effect - but it cannot account for the fact that when the satellite S2 clock is brought to rest in the Earth frame after many orbits, it will have logged less time than the E clock.

This is simply the orbiting case of the round trip clock described in part 4 of Einstein's 1905 paper. The difference is that we have, in the GPS satellite metaphore, a scenerio where each clock can continuously interrogate the rate of the other clock in relation to its own passage of time. Moreover, we have the data that confirm the difference in the actual rates at which satellite clocks run in relation to the E clock.

 

[robphy]

robphy: To my way of thinking, the GPS satellite analogy is the easy way to illustrate the notion of two relatively moving clocks - the corrections for altitude are put in because the G field is different at the height of the orbit.

This obviously renders it a non-SR [that is, it is not a problem in a flat R⁴ spacetime] problem, in spite what you claim in your post #32. If it involves geodesics, it is still a problem concerning inertial frames... just not in SR. While some aspects may be analogous to SR, some are not... for example, this correction.

I entered this thread because of the initial question you posed and my rephrasing of it (in post #4) as an SR problem, which you agreed to.
The issue in post #20 (an implicit question to you, awaiting your response) was left unresolved. Your tone changed (in general) in #30. The question changed in #32 to a non-SR problem, as expained above, while abandoning the original question. Finally, my question in #33 was left unanswered.

For these points, I will retire temporarily from this thread. Thanks for the conversation.

 

[yogi]

Likewise for the conversation Robphy- but post # 30 was not directed to you.

My intent was to try to nail down the fundamental difference between apparent time dilation and actual time dilation in one way travel experiments, and I argued that a clock in an orbiting satellite is a perfectly good inertial frame - just as is a clock that is put in uniform linear motion. But in the latter case one doesn't have a convenient experimental platform because pulses transmitted between linearly moving frames must be Doppler compensated, and the changing distance between sources and receivers must also be accounted for. So my shift was intended to get some resolution or agreement as to the relative rate of time passage in the case of GPS and then carry this over to flat spacetime as per my original post.

I apologize if anything I said that may have offended you - I have always found your posts to be cordial and well thought out

Regards

Yogi

 

[Ich]

Yogi,

even if my posts lack the cordiality you seem to expect they could help you if you tried to understand SR. I became more aggressive when I found that you exhibit more and more a crackpot attitude that doesn´t fit in with this forum.
You claimed in former threads that an orbiting frame is a perfect (SR) inertial frame. You have been told that you´re wrong. Still you come back here and claim the same.
Likewise, you have been told repeatedly that initial acceleration woud not decide who is ageing more slowly. Still you ignore that.
In this thread, I tried (not for the first time) to give you clues to understand how SR works. You (obviously) took it as a personal attack and repeated even more aggressive your false claims.
That´s not how it works.
Just TRY to listen to the people you are discussing with. Then they´ll stay in tune.

 

[robphy]

Yogi,
Yes, I know that #30 wasn't directed to me.
I wasn't offended in any way from anything you said.
It's just that I felt that this thread was veering off the initial question without resolving it... and it wasn't clear [to me] where it was going. So, I was going to take a break from it.

robphy

 

[yogi]

Yogi,

.
You claimed in former threads that an orbiting frame is a perfect (SR) inertial frame. You have been told that you´re wrong. Still you come back here and claim the same...

There are many references that support what I have said - take a look at Road to Reality p 394, or Spacetime physics pp 26-31 if you want some authority as to the validity of orbiting satellites as free float inertial frames. In fact, they are preferable to Earth in many respects - and we normally consider the Earth as a good inertial frame for making relativistic experiments. I know there are limitations - they are not good inertial frames on a large scale where tidal forces are significant - but for the small volume inside a GPS satellite - they are as good as need be. Until you can sight me some authority to the contrary, I think I will go along with Penrose and Wheeler


As for your comment re the initial acceleration, I have said this is a direct consequence of Einstein's explanation in part 4 of his 1905 paper - I urge you to read it ... when two spaced apart clocks are synchronized and one is quickly accelerated to a constant velocity and then travels to the location of the other clock, the one which has been put in motion will be out of sync (read less time) with the other when they are compared. But Einstein doesn't stop there - he gives several more examples - concluding that a clock at the equator will run faster than one at the pole.

I would like to engage in a discussion where the responses do not turn into a shouting match - if anything I have asserted is in violation of a confirmed experiment, please advise. If you have an explanation of why the two clocks in Einstein's illustration are reading different times at the end of the experiment, I would like to hear it. But remember, they are being compared in the same rest frame at the end of the experiment and the one which has been accelerated into uniform motion reads less (this is not a reciprocal situation just as the two readings on the twin's clocks (in a round trip journey) is not reciprocal.

 

[robphy]

There are many references that support what I have said - take a look at Road to Reality p 394, or Spacetime physics pp 26-31 if you want some authority as to the validity of orbiting satellites as free float inertial frames. In fact, they are preferable to Earth in many respects - and we normally consider the Earth as a good inertial frame for making relativistic experiments. I know there are limitations - they are not good inertial frames on a large scale where tidal forces are significant - but for the small volume inside a GPS satellite - they are as good as need be. Until you can sight me some authority to the contrary, I think I will go along with Penrose and Wheeler

I happen to have these on my shelf right now.
Certainly, as you say, it is valid to regard "orbiting satellites as free float inertial frames"... and yes, "they are preferable to Earth in many respects". In addition, I agree that "they are not good inertial frames on a large scale where tidal forces are significant - but for the small volume inside a GPS satellite - they are as good as need be."

So, for a single satellite, there's no problem... you can locally apply SR at any event.

For two satellites meeting at an event, there's no problem... you can locally apply SR at that meeting event.

The problem is that when you try to consider two satellites at different altitudes and you have to correct one because of its altitude, then you are now certainly outside "the small volume inside a GPS satellite". So, with the two satellites taken together in one frame, special relativity doesn't apply.

 

[yogi]

robphy, yes - what you say is quite correct- - I noticed after posting #39 that Rindler uses a strict definition of an ideal inertial frame as being one totally removed from all G fields (this may have been what hurkyl was referring to in a previous thread). Rindler, however, then goes on to say, as a practical matter we don't have access to this utopia so we do our experiments in a less than perfect environment. So with that understanding, perhaps we can ask the question of whether there is any difference between the principles involved in 1) the measured loss of time between a velocity uncompensated orbiting clock S2 and the ground E clock...and 2) the predicted time loss in the linear experiment where one clock is put in motion after being initially synchronized with a distant clock which remains at rest (as per Einstein part 4 of 1905).

 

[Ich]

Hi yogi, I´ve been away for two weeks.

I have one problem with our discussion: I already told you almost everything I have to tell, and you did ignore it. It´s ok for me to start again and discuss things until we come to a solution. But I will not tell everything a third time. So I expect that you address the points I make explicitly and that you tell me either that you agree or where you don´t (and why).

There are many references that support what I have said - take a look at Road to Reality p 394, or Spacetime physics pp 26-31 if you want some authority as to the validity of orbiting satellites as free float inertial frames. In fact, they are preferable to Earth in many respects - and we normally consider the Earth as a good inertial frame for making relativistic experiments. I know there are limitations - they are not good inertial frames on a large scale where tidal forces are significant - but for the small volume inside a GPS satellite - they are as good as need be. Until you can sight me some authority to the contrary, I think I will go along with Penrose and Wheeler

1. Nobody doubts that there is a local IF valid for the satellite. But it is never (not even for an infinitesimal time) valid at the center of earth. Therefore you can´t use SR to compare satellite and center time. GR will give you the correct result.
2. It is still possible and instructive to examine the problem in SR if you treat the satellite´s orbit as a polygon.

As for your comment re the initial acceleration, I have said this is a direct consequence of Einstein's explanation in part 4 of his 1905 paper - I urge you to read it ... when two spaced apart clocks are synchronized and one is quickly accelerated to a constant velocity and then travels to the location of the other clock, the one which has been put in motion will be out of sync (read less time) with the other when they are compared. But Einstein doesn't stop there - he gives several more examples - concluding that a clock at the equator will run faster than one at the pole.

I read and understood the paper. I agree with everything that Einstein said but not with your understanding that initial acceleration somehow decides which clock will read less time. An example:
3. Two synchronized clocks are accelerated by the same amount to +v and -v. After some time, clock 2 accelerates to +v. You bring both clocks slowly together and compare times: clock 2 shows less. So in this case it´s final acceleration which breaks the symmetry.

If you have an explanation of why the two clocks in Einstein's illustration are reading different times at the end of the experiment, I would like to hear it. But remember, they are being compared in the same rest frame at the end of the experiment and the one which has been accelerated into uniform motion reads less (this is not a reciprocal situation just as the two readings on the twin's clocks (in a round trip journey) is not reciprocal.

4. It´s instructive to see that even in a symmetric setup the time of the "moving" reference frame goes faster than your own, if you observe it at your position. I´m convinced that you are unaware of this fact.

 

[yogi]

Ich - your point 1 - a clock on a tower at the North pole can be used as one IRF, and a GPS satellite at the same orbital height as a second.

point 2 - you don't need to construct a polygon - the free falling orbiting frame works fine - it can be an ellipse with any eccentricity - it is a perfectly good IRF because for any experiment conducted therein inertia is isotropic

point 3 - I disagree - the situation is changed anytime any clock undergoes acceleration. But it is not the acceleration per se that affects the difference in time when the two clocks are later compared; acceleration is simply a means employed to change the speed between two clocks.

point 4 - I am fully convinced that observers in equivalent inertial frames will measure the apparent rate of the other guys clock to be running slower. But ...the whole post is not about apparent time dilation, it is about what happens when two at rest separated clocks are synchronized in the same frame, and one is put in motion (accelerated) and when it reaches the other clock, it reads less. Read part 4 of the 1905 paper again. In this description of what happens (I think Einstein used the word peculiar) there is only one acceleration - at the beginning. This is why I have consistently maintained over the course of numerous threads, you cannot sync two clocks on the fly - you can read the time of a passing clock by its hands when it is near - but you cannot be guaranteed that the other clock will be running at the same rate. At the end of the one way journey the two clocks read differently - the clock put in motion does not have to slow down at the end to be read - it can be read by the clock which has not moved and a comparison made as it passes by - but that is not true at the beginning - the initial conditions (namely synchronization followed by an acceleration) is vital to the conclusion reached by Einstien - namely, that one clock reads more than the other at the end of the one way voyage

 

[Ich]

1. Yes, that´s two IRFs. But one of those is only locally (and only for a short time) valid and does not include the north pole most of the time. What you try to do is to compare clocks in the two frames, using only SR. That is only possible if both clocks are always covered by both IRFs.
Relativity is a theory of relations (as the name implies). To calculate something, it is not sufficient that both observers are in a local IRF. You have to specify how these IRFs move relative to each other. And SR only deals with linear motion and global IFRs. Everything else is curved spacetime, and therefore you also start with a broken symmtery (see 2.).
2. Same problem. Gravity is not included in SR, and a IRF that travels in circles certainly does need gravity. To calculate "clock rates", you need GR, even if you can use weak field approximations. If you trie to get around that, you only confuse yourself. Why don´t you calculate circular motion the way SR allows it?
3. You disagree with what? I only said that the situation is symmetric until clock 2 accelerates. At this point you decide in which frame to compare clocks and which clock "really" loses time. Note: the decision is made as the last step of the experiment. Initial acceleration has nothing to do with it. Do you agree with that?
4.

the initial conditions (namely synchronization followed by an acceleration) is vital to the conclusion reached by Einstien - namely, that one clock reads more than the other at the end of the one way voyage

No, it is not. You can: a) sync two separated clocks at rest. b) sync a moving clock on the fly with the first clock when they meet. c) compare its reading with the second clock when they meet. The outcome is the same, and no acceleration is involved.

you cannot sync two clocks on the fly - you can read the time of a passing clock by its hands when it is near - but you cannot be guaranteed that the other clock will be running at the same rate.

Just to be sure we speak of the same thing: "sync" means that you set the clock to a certain value, it has nothing to do with clock rates.

 

[Hurkyl]

2. It is still possible and instructive to examine the problem in SR if you treat the satellite´s orbit as a polygon.

Minor nitpick -- "thou shalt not use calculus" is not one of the postulates of SR. There is no reason to restrict one's self to polygonal curves aside from computational simplicity.

 

[Ich]

No, but it is what Einstein proposed in the paper yogi is referring to, and you can learn a lot when you calculate the effect of small translations and rotations. The result is the same, but you can see how it is achieved.

 

[yogi]

Ich - We are using the word sync differently - both definitions are valid - I am using it mean: "to render synchronous in operation" That is why I am claiming, in general, two clocks cannot be synchronized "on the fly" They must both be in the same reference frame (at rest wrt each other). When one is accelerated after using Einstein synchronization, I claim they will not be running in sync thereafter.

 

[yogi]

Post 44 - item 1. Exactly - each clock is a valid local IRF. If the orbit bothers you - forget it - consider the case where two spaced apart clocks are synchronized (using my definition that they are running at the same rate) and one is later accelerated to a uniform velocity v wrt the other. I claim the two clocks are no longer in sync and that to this extent the fames are no longer equivalent. Inertial experiments are the same when carried out in each frame, but the clock in the frame that was put into motion runs at a slower rate than the clock which remained in the rest frame in which the clocks were originally synchronized. I am aware there is an alternative explanation of why the two clocks do not read the same when the moving clock reaches the stationary clock - but I think it is fallacious

 

[Ich]

#47: good to know; I already got the impression that we´re not talking the same things. I suggest that we use the definition I gave, because it´s highly unusual to fiddle with clock rates. From Wikipedia: The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom. There is nothing to tune.

 

[Ich]

#48: Wonderful, a precisely stated claim. Let´s discuss it.

To avoid misunderstandings. we´re talking about SR here and not some personal theory how things should be.

You claim that the accelerated clock ticks slower than the one staying "at rest". That means, the amount of time lag is proportional to the time you let the clocks fly until you compare them.
How do you deal then with the following prediction of SR:
If you accelerate the second clock to match the velocity of the first clock immediatly before you compare them, the second clock will read less exactly the amount that you think the first one would.
What SR says is that moving frames are equivalent (regardless which one accelerated) until you decide in which frame you want to compare the clocks. That contradicts your claim.

 

[yogi]

Ich - I think you are trying to imply things I have not said. If you are saying that two separated clocks (say A and B) at rest wrt each other, and A is given a brief acceleration toward B, and travels most of the distance at constant velocity v, but just before reaching B, B is accelerated in the same direction to a velocity v, so now both A and B are moving together at velocity v relative to some object in the original frame of reference. Then A and B now run at the same rate because they are in the same frame (I hope we agree on that) but they do not read the same time - When B pulls alongside A, the time on the A clock will be less than the time on the B clock (do we agree on this)?

In fact, it is not necessary to wait until A is near B. For example, if A is at the origin of the X axis, and B is at X = 100 miles and both blast off in the direction of the + X axis at the same time with identical accelerations for identical time periods (their integrating accelerometers are set to cut off at the same velocity) Both will have reached a velocity v in the direction of the + X axis, so do you think there will be any difference in the reading on A's clock relative to B's clock now that they are traveling together in the same frame but still equally separated?

 

[Ich]

§1: Maybe I managed to misunderstand your claim. I agree with you that A will read less time.
What I had in mind was the following setup: A and B start at the same point, A being accelerated. A should "tick at a slower rate" from then. If you then accelerate B to match velocity with A, A and B should tick at the same rate. So if you bring A and B together, you would expect A to read less. SR says B will read less.

§2: We better don´t use accelerating frames. One has to be extremely careful with the setup and the calculations. For example, in your setup the distance between A and B would increase, and A will indeed read less time as he was at the bottom of a "gravity well". I think your §1 is enough to decide whether your view is consistent with SR´s.

 

[yogi]

§1: Maybe I managed to misunderstand your claim. I agree with you that A will read less time.
What I had in mind was the following setup: A and B start at the same point, A being accelerated. A should "tick at a slower rate" from then. If you then accelerate B to match velocity with A, A and B should tick at the same rate. So if you bring A and B together, you would expect A to read less. SR says B will read less.

After B accelerates to the same speed as A (same direction) then A and B will run at the same rate (I guess we agree on that). They are separated in space, but both are in the same frame at rest wrt each other (I assume we agree on that) My question is: "How do you propose to bring them together?" What if they are not brought together - each simply interrogates the other with radio signals? If they are not brought together, which clock will read more time?

 

[Ich]

We agree on both points.

My question is: "How do you propose to bring them together?" What if they are not brought together - each simply interrogates the other with radio signals? If they are not brought together, which clock will read more time?

To compare clocks, you either interrogate them with radio signals (same procedure as Einstein suggested in his paper) or you bring them together by slow transport (v<<c). It is a feature of SR that both procedures will give the same result: A will read more time.

 

[yogi]

Ich: That is interesting - let's see - we could actually do an orbiting version of what you suggest - let's launch a GPS satellite clock that has been corrected for height but not velocity - the two clocks are in sync in the Earth frame before launch - A is accelerated into orbit and flys for one year, during which time A will run slower than the B clock on earth. One year later we launch the B clock into an identical orbit (after correcting for altitude) and now A and B are side-by-side, so both are running at the same rate, but there will be a difference in the lapsed time accumulated on the A clock and the B clock during the one year that passed between the two launches. Are you saying the A clock will read more time, or are you saying the analogy is flawed?

I am assuming in the linear case that you proposed, the conclusion that A will read more time was arrived at using the methodology adopted by Einstein (1918) and Born to explain the twin thing. If so, I will comment upon that.

 

[Ich]

No, you could not do an orbiting version of this. I can explain later why not and how SR explains the effect in circular motion.
But for now I strongly suggest that we stop complicating things until we got the basics right.
I don´t know the methodology of Einstein and Born. It is simply the old simultaneity thing: Until A and B join frames, each one is equally right (or wrong) to say that the other´s clock is ticking slower. When B suddenly accelerates, A did not change his view of things. That means, B is still (nearly) at the same position, and his clock (nearly) shows the same time as before the acceleration. So the result that B shows less time still holds.
But B´s notion of simultaneity changed drastically. "Now" A´s clock is ahead of his, and this won´t change if the clocks are brought together.
WARNING: the following may be unintelligible. Ignore it, if you can´t make sense of it. It´s kind of a metaphor, but not too far from truth.
There is nothing important happening with A´s and B´s clock at this time. It is more like B suddenly recognized that his time was "flowing in the wrong direction" all the time. But which direction is the right one is decided only when you decide in which frame you want to compare clocks. If it would have been B´s frame, A´s time would have been flowing in the wrong direction.

 

[yogi]

Until A and B join frames, each one is equally right (or wrong) to say that the other´s clock is ticking slower. When B suddenly accelerates, A did not change his view of things. That means, B is still (nearly) at the same position, and his clock (nearly) shows the same time as before the acceleration. .

If I correctly picture what you painted - A and B are synchronized at the origin of an x-y coordinate system and A first accelerates to a velocity v along the positive x-axis - the A clock will run slower as long as this condition persists - for example, A could travel for a long time at v = 0.5c relative to B and would wind up with less accumulated time when arriving at Altair (we assume Altair is at rest relative to the origin of the coordinate system). Now just before A reaches Altair, B quickly accelerates to 0.5c wrt the coordinate axis 0,0 in the same direction (along the + axis toward Altair) - and you say that B's clock shows the same (nearly) time as it did before B commenced accelerating (OK agreed). So during A's long journey 1) the A clock either ran slower than the B clock, or 2) the spatial distance D between the origin and Altare is contracted from A's point of view so his clock only recorded a time L/v. (where L is the contracted length). Either way, before B accelerates, do you agree that A's clock will have recorded less time as A nears Altair (not yet decelerating) than clocks at rest with respect to the origin of the coordinate system where B has remained at rest? If so, then I do not understand how you arrive at a result that predicts B will show a Real (not apparent) lesser time than A after B completes his short duration of acceleration.

 

[Ich]

If I correctly picture what you painted - A and B are synchronized at the origin of an x-y coordinate system and A first accelerates to a velocity v along the positive x-axis - the A clock will run slower as long as this condition persists - for example, A could travel for a long time at v = 0.5c relative to B and would wind up with less accumulated time when arriving at Altair (we assume Altair is at rest relative to the origin of the coordinate system).

No, you start again mixing "observer" and "observer´s rest frame". It may sound like nitpicking, but it is crucial to understand the difference:
A will read less time than a clock positioned at Altair which is synchronized with B in their common IF. It will not read unambiguously less than B´s clock, because B himself is not at Altair, and comparing times at different positions is a very special thing in relativity.

Now just before A reaches Altair, B quickly accelerates to 0.5c wrt the coordinate axis 0,0 in the same direction (along the + axis toward Altair) - and you say that B's clock shows the same (nearly) time as it did before B commenced accelerating (OK agreed).

Yes.

So during A's long journey 1) the A clock either ran slower than the B clock, or 2) the spatial distance D between the origin and Altare is contracted from A's point of view so his clock only recorded a time L/v. (where L is the contracted length). Either way, before B accelerates, do you agree that A's clock will have recorded less time as A nears Altair (not yet decelerating) than clocks at rest with respect to the origin of the coordinate system where B has remained at rest?

Again, don´t compare clocks at different positions. That´s where all the trouble comes from. I agree that A will read less time than a clock synchronized with B when A passes it, eg the Altair clock when A is at Altair.

If so, then I do not understand how you arrive at a result that predicts B will show a Real (not apparent) lesser time than A after B completes his short duration of acceleration.

And again, B is not at Altair. A will read more time than a clock at Altair which is synchronized with B after B´s acceleration. B´s "simultaneity plane" or however you call it shifted during acceleration.
If you then bring A and B slowly together, the result stays the same: B will Really read less than A.

 

[yogi]

All the clocks in the coordinate frame can be synchronized prior to A's initial acceleration - So if we add a clock at Altair (call it D) it can be synced with A and B and will read the same as A and B before any accelerations have taken place. When A is accelerated, B and D are still in sync (reading the same time and logging time at the same rate).

1) When A arrives at D, A will read less than D. Do we agree on this?
I think from your post above, the answer must be yes

2) If Yes, do you agree that B and D still read the same time (prior to B's acceleration). If not, how did they get out of sync?

3) If yes to no 2 above, do you agree that A reads less than B immediately prior to B's acceleration?

- probably not - but in any case you conclude that after B's acceleration B will read less than A...and that is where we part company

Invariably the analysis of these interesting problems skips from actual real times (local times or proper, whatever you want to label them) logged by a clock to an apparent observation that typically involves a rapid shift in the slope of the plane of simultaneity

...so in the distant inertial system of A, the time on B's clock has rapidly changed as viewed by A - actually B's clock would have to lose a lot of time (run backwards) during a short period of acceleration - because prior to the acceleration, B clock should read the same as D clock - but physically the B clock cannot run backwords just to accommodate the book-keeping. While some folks are comfortable with such abstractions, I am not. To me the interest in these problems is in finding an explanation that is consistent with a physical reality.

 

[Hurkyl]

so in the distant inertial system of A ... because prior to the acceleration, B clock should read the same as D clock

The B clock and the D clock have never read the same time, as measured by any inertial reference frame in which A is stationary for its journey.

To me the interest in these problems is in finding an explanation that is consistent with a physical reality.

Coordinate charts are not physical reality. It is not inconsistent with reality for things to run backwards according to a coordinate chart. (Though technically such a thing is a generalization of a coordinate chart)