Implications of the statement Acceleration is not relative

[GregAshmore]

By "ditch special relativity", I did not mean to say that SR should not be taught at all. I meant that SR should not be used to prove that the Twin Paradox is not a paradox.

Specifically, I mean that SR should not be used to prove that what I earlier called the "second aspect" of the Twin Paradox is not a paradox. That is the claim that when the episode is observed from the "permanently at rest" frame of the rocket, the Earth twin will be younger than the rocket twin. The paradox then is that both twins are "younger than the other", which can't happen in reality.

The problem with treating this aspect of the twin paradox in SR is that the case of the resting rocket cannot be considered. In SR, only observers in inertial frames can consider themselves to be at rest.

It has been pointed out that only the rocket experiences proper acceleration (which is, I believe, essentially the same thing as saying that the rocket frame is not inertial). If it can be shown that proper acceleration affects the operation of clocks, then there would be no need to consider the case of the permanently resting rocket. Is it claimed that proper acceleration affects clocks?

If proper acceleration does not affect clocks, it is necessary to consider the case of the resting rocket in order to prove or disprove the claim made about the case of the resting rocket. The case cannot be considered in SR; therefore the claim cannot be addressed within the confines of SR.

Unless, of course, it can be shown that the observer in the rocket cannot legitimately consider himself to be permanently at rest.

With respect to that, in my previous post I said, "But always in his mind is that goal, to understand how it is that the rocket can be legitimately understood to be at rest, and the Earth moving."

To which there was this reply:

Right. And as you now realize, in reality that goal was never reached. I don't know anyone who holds that for example the Earth is constantly "truly in rest".

And a few remarks later:

A much more pragmatic way of dealing with the issue would be (indeed, it's the common approach of textbooks):
- we do not need to consider the rocket to be in rest all the time
- just use SR for the problem

Harrylin, did you mean to say that the observer in the rocket cannot legitimately consider himself to be permanently at rest?

How would others in the discussion answer this question: Can the observer in the rocket legitimately consider himself to be permanently at rest? [edited to remove the misplaced word 'cannot']

 

[stevendaryl]

By "ditch special relativity", I did not mean to say that SR should not be taught at all. I meant that SR should not be used to prove that the Twin Paradox is not a paradox.

Well, there is nothing besides SR to prove that. As I said in a different post, General Relativity is the SAME theory as SR, in the case where there are no significant gravitational masses. GR doesn't tell us anything about the twin paradox beyond what SR tells us.

It has been pointed out that only the rocket experiences proper acceleration (which is, I believe, essentially the same thing as saying that the rocket frame is not inertial). If it can be shown that proper acceleration affects the operation of clocks, then there would be no need to consider the case of the permanently resting rocket. Is it claimed that proper acceleration affects clocks?

What you can prove from SR alone is that if two clocks start at the same starting point, travel at different velocities, and end up at the same end point, then the one that accelerates will have the shortest proper time.

 

[stevendaryl]

If proper acceleration does not affect clocks, it is necessary to consider the case of the resting rocket in order to prove or disprove the claim made about the case of the resting rocket. The case cannot be considered in SR; therefore the claim cannot be addressed within the confines of SR.

What you are saying is just not true.

 

[PeterDonis]

The problem with treating this aspect of the twin paradox in SR is that the case of the resting rocket cannot be considered. In SR, only observers in inertial frames can consider themselves to be at rest.

This is not true; you can use curvilinear coordinates in SR. I believe someone already pointed that out earlier in this thread. What you can't do is use curvilinear coordinates to describe a non-inertial frame and expect everything to work exactly the same as it does in an inertial frame. It won't. But as long as you're in flat spacetime, you're within the domain of SR.

For an example of a non-inertial coordinate chart in flat spacetime, which is perfectly valid within SR, try Rindler coordinates:

http://en.wikipedia.org/wiki/Rindler_coordinates

Note that the Rindler coordinate chart does not cover all of spacetime; that's one key difference between it (and most non-inertial charts) and an inertial chart.

Is it claimed that proper acceleration affects clocks?

No. But proper acceleration does make the "rest frame" of the accelerating object non-inertial.

How would others in the discussion answer this question: Can the observer in the rocket cannot legitimately consider himself to be permanently at rest?

He can, but as I said above, he can't expect his "rest frame" to work like an inertial frame, because it isn't one.

 

[GregAshmore]

Greg, I'm telling you, T&W's objectors are buffoons. T&W are glad you see them as buffoons. They don't want you to understand the answer to the objectors' questions. They want you to simply understand the Twin Paradox by their preferred method which is to use Proper Clocks, which is their unique term for the spacetime interval.

Look at their summary on page 131 where they say that "each of the three [inertial reference] frames...has a perfectly consistent and nonparadoxical interpretation of the sequence of events". But then instead of showing how that works, they quote an objector, "all these different [inertial] reference frames sure do complicate the story" and they respond with "Exactly! These complications arise because observations from anyone frame are limited and parochial. All disagreements can be bypassed by talking only in the invariant language of spacetime interval, proper time, wristwatch time."

I will look again at the text.

But at least on the next page they say in very bold letters:

DO WE NEED GENERAL RELATIVITY? NO!

As you see in the above post, I disagree on this point. If a claim is made about the case of the resting rocket, that case must be considered in order to prove or disprove the claim, except as noted in the above post.

The math of Special Relativity is very simple and so are the explanations that T&W denigrate. In fact I presented it all in the thread that you referenced in your first post:

Go to the second page and post #23. Please study it. It has very simple math. I believe that you can understand it. If you have any questions, please ask. Please don't dismiss it just because T&W dismiss it.

The math of the Lorentz transform is very simple, and it is sufficient for simple problems such as the pole-in-barn paradox. (The interpretation of the results is not so simple, though.) But four-vectors and other, even more advanced math constructions do come up with regularity in these discussions.

I'll reread post #23.

 

[Mentz114]

The problem with treating this aspect of the twin paradox in SR is that the case of the resting rocket cannot be considered. In SR, only observers in inertial frames can consider themselves to be at rest.

As has been pointed out to you in earlier posts, both these sentences are false. It is possible to find coordinates in which the accelerating twin remains stationary but non-inertial. In these coordinates the accelerating twin has less proper time than the 'earth' twin. No problem. Changing coordinates will not change the invariant proper times.

 

[GregAshmore]

What you can prove from SR alone is that if two clocks start at the same starting point, travel at different velocities, and end up at the same end point, then the one that accelerates will have the shortest proper time.

Not quite. What you are proving is that when the inertial frame is considered to be at rest, and two clocks start at the same starting point...

You have not addressed the claim made about what happens when the rocket is considered to be at rest.

 

[stevendaryl]

As you see in the above post, I disagree on this point [about whether GR is needed]

But it isn't a matter of opinion. You're just wrong. GR adds nothing to the calculation that isn't already in SR. GR is the SAME theory as SR in the limit in which there are no significant masses present.

 

[stevendaryl]

Not quite. What you are proving is that when the inertial frame is considered to be at rest, and two clocks start at the same starting point...

That's completely wrong. It doesn't matter what you "consider" to be at rest.

 

[Mentz114]

GregAshmore, you are starting to talk nonsense, and ignoring any post that you can't refute by handwaving. It is possible to find coordinates in which the accelerating twin remains stationary but non-inertial. In these coordinates the accelerating twin has less proper time than the 'earth' twin. No problem. Changing coordinates will not change the invariant proper times.

Are you denying this ?

 

[stevendaryl]

You have not addressed the claim made about what happens when the rocket is considered to be at rest.

The issue is not whether anyone considers the rocket at rest, the issue is whether the rocket is inertial, or not. It doesn't change from inertial to noninertial or vice-verse based on how you think about it.

Think about this analogy: On a piece of paper, draw three points, not all in the straight line, and label them A, B and C. Take a blue pen, and draw a straight line from A to C. Take a green pen, and draw a line from A to B, and then to C.

Euclidean geometry says that the green curve will be longer than the blue curve.

Your asking what happens if the accelerating rocket considers itself to be at rest, is exactly like asking what happens if the green path considers itself to be straight.

 

[GregAshmore]

The issue is not whether anyone considers the rocket at rest, the issue is whether the rocket is inertial, or not. It doesn't change from inertial to noninertial or vice-verse based on how you think about it.

Throughout this discussion I have recognized that the rocket frame is non-inertial.

Think about this analogy: On a piece of paper, draw three points, not all in the straight line, and label them A, B and C. Take a blue pen, and draw a straight line from A to C. Take a green pen, and draw a line from A to B, and then to C.

Euclidean geometry says that the green curve will be longer than the blue curve.

I see that. I have not questioned that; neither does the objector in Taylor & Wheeler.

Your asking what happens if the accelerating rocket considers itself to be at rest, is exactly like asking what happens if the green path considers itself to be straight.

Prove it. You drew your lines with my rocket in motion. I, in my rocket, have the right to consider myself to be permanently at rest. Prove to me that being non-inertial, yet always at rest, will result in a younger age.

That's all I am asking.

I already know the answer that Einstein gave. What I have been objecting to is that no proof is given in the typical treatment of the problem. If it is true that an answer can be given in SR, then it should be given. The objection should not be dismissed.

 

[Mentz114]

Prove to me that being non-inertial, yet always at rest, will result in a younger age.

It is a well known fact that an inertial worldline between two events has a larger proper interval than any non-inertial worldline connecting the events. I'll see if I can find a proof.

See section 6.6 of this lecture

http://physics.ucsd.edu/students/courses/winter2011/physics161/p161.14jan11.pdf

In summary, what we did here is extremize (in fact maximize) the proper time between two events to find the geodesics. Thus the geodesic is that path for which the maximum time passes on the wrist watch of the observer traveling that path.

 

[Dale]

If it is true that an answer can be given in SR, then it should be given. The objection should not be dismissed.

I did. I suspect that you didn't bother to read it, but stop acting as though the objection were summarily dismissed and no answer were given when you simply haven't bothered to read the answer given.

 

[Dale]

The case cannot be considered in SR; therefore the claim cannot be addressed within the confines of SR.

Sure it can. See here:
http://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html

How would others in the discussion answer this question: Can the observer in the rocket legitimately consider himself to be permanently at rest?

You can always make a coordinate system where any given object is permanently at rest. You just have to be very detailed about your specification of the coordinate system since there is no "standard" meaning.

You also cannot apply formulas derived for inertial frames in non-inertial frames. They are both legitimate, but not equivalent.

 

[stevendaryl]

Prove it. You drew your lines with my rocket in motion. I, in my rocket, have the right to consider myself to be permanently at rest. Prove to me that being non-inertial, yet always at rest, will result in a younger age.

The proof that an inertial path is the one with the longest proper time is essentially the same as the proof that a a straight line is the curve with the shortest length connecting two points. I'll go through both of them in parallel.

Euclidean case
In 2D Euclidean geometry, the formula for the length of a curve connecting two points is given by:

L = \int \sqrt{1+m^2} dx

where m = \dfrac{dy}{dx} is the slope of the curve y(x)

This formula is good using any Cartesian coordinate system, provided that the curve is never vertical (it breaks down in that case, because the slope becomes infinite).

Finding the path y(x) that makes L an extremum (either a maximum or a minimum), one uses the calculus of variations. The result is that the minimizing or maximizing curve satisfies:

\dfrac{d}{dx} ( \dfrac{m}{\sqrt{1+m^2}}) = 0
which has the solution that m is a constant.

So the curve with constant slope is the extremizing curve (the one making the distance either minimal or maximal--we can prove in this case that it is minimal).

Special Relativity case
In Special Relativity, the formula for the proper time of a spacetime path is given by:

\tau = \int \sqrt{1-\dfrac{v^2}{c^2}} dt

where v = \dfrac{dx}{dt} is the velocity of the path x(t)

This formula is good using any inertial coordinate system.

Finding the path x(t) that makes \tau an extremum (either a maximum or a minimum), one again uses the calculus of variations. In this case, the equation for the extremizing path is:

\dfrac{d}{dt} ( \dfrac{-\frac{v}{c^2}}{\sqrt{1-\frac{v^2}{c^2}}}) = 0
which has the solution that v is a constant.

So the path with constant velocity v is the path that makes the proper time maximal or minimal--we can prove in this case that it is maximal.

 

[harrylin]

I don't agree with this. For instance, the equations show that an object at rest at a constant r in the Schwarzschild vacuum feels a force - and thus is not inertial, nor moving ( relative to the field).

What you here call "at rest" is also what Einstein calls "at rest"; your disagreement seems to be with the definition of acceleration that he used.

Yes, but it will still be non-inertial. Are you saying that in this scenario the rocket feels no acceleration ?

That's correct. In Langevin's "twin" scenario, the space capsule feels no force during the voyage.

 

[harrylin]

[..] Specifically, I mean that SR should not be used to prove that what I earlier called the "second aspect" of the Twin Paradox is not a paradox. That is the claim that when the episode is observed from the "permanently at rest" frame of the rocket, the Earth twin will be younger than the rocket twin.

Indeed, a "permanently at rest" rocket frame in the sense as was meant by objectors means zero acceleration; their objection targeted GR, not SR and Einstein understood this very well. Some people seem to confound that issue with the question if we need GR to describe observations from the accelerating rocket - which is of course not needed, SR is fine for that.

The paradox then is that both twins are "younger than the other", which can't happen in reality.

Ehm no, that basic issue was taken care of in Einstein's 1918 paper, by means of an induced gravitational field. According to 1916GR, one may claim that the firing of the rockets doesn't accelerate the rocket at all but that instead this induces a gravitational field through the universe. That field makes that the stay-at home ages the right amount in comparison with the traveler. However, that solution opened a can of worms that nobody wants - so much, that it has been mostly ducked in the literature.

The problem with treating this aspect of the twin paradox in SR is that the case of the resting rocket cannot be considered. In SR, only observers in inertial frames can consider themselves to be at rest.

Quite so. In SR observers who are accelerating wrt inertial frames may consider themselves to be at rest in an accelerating frame; consequently they cannot consider themselves physically "in rest" in the sense of SR. Its laws of nature for inertial frames do not apply to that frame.

Is it claimed that proper acceleration affects clocks?

Some clocks are affected by such applied forces, but in particular atomic clocks are rather robust. For the typical twin paradox scenario's in which the turn-around only takes a relatively small time this is irrelevant. The assumption that acceleration doesn't affect the clocks is called the clock hypothesis which was probably first brought up in Einstein's 1905 paper.

[..] Unless, of course, it can be shown that the observer in the rocket cannot legitimately consider himself to be permanently at rest.

With respect to that, in my previous post I said, "But always in his mind is that goal, to understand how it is that the rocket can be legitimately understood to be at rest, and the Earth moving."

To which there was this reply: [..]

Harrylin, did you mean to say that the observer in the rocket cannot legitimately consider himself to be permanently at rest? [..]

I understand you to mean "permanently at rest" in the sense of Einstein-1918. There are many objections to make against Einstein's induced gravitational fields. Consider such things as cause and effect, speed of gravitational wave, etc. I have never seen a paper that even attempts to address those issues, let alone solve them.

 

[Dale]

There are many objections to make against Einstein's induced gravitational fields. Consider such things as cause and effect, speed of gravitational wave, etc. I have never seen a paper that even attempts to address those issues, let alone solve them.

Probably because most people don't believe they are issues. The cause is obviously the choice of coordinates, and this meaning of "gravitational fields" does not mathematically produce "gravitational waves" in the usual sense of a wave equation with a characteristic propagation speed.

 

[harrylin]

Probably because most people don't believe they are issues. The cause is obviously the choice of coordinates, and this meaning of "gravitational fields" does not mathematically produce "gravitational waves" in the usual sense of a wave equation with a definite propagation speed.

A change of coordinates isn't a gravitational field. I don't follow Einstein's Machian explanation of invoking a physical, induced gravitational field (for enabling the interpretation that the rocket is constantly "truly in rest"): "all the stars that are in the universe, can be conceived as taking part in bringing forth the gravitational field [..] during the accelerated phases of the coordinate system K' ".

 

[stevendaryl]

A change of coordinates isn't a gravitational field.

It depends on how you define "gravitational field". If you drop an object, and it follows a trajectory h = -\frac{1}{2}g t^2 for some constant g, then many people would call g the "acceleration due to gravity" or the "gravitational field". But it certainly is affected by a coordinate change. It can be made to vanish by careful choice of coordinates. If g=0 (no gravitational field), then a change of coordinates to accelerated coordinates can make g nonzero.

The equivalence principle is about this notion of gravitational field: there is no difference (other than the variation of g with location) between the effects of a g due to gravity and a g due to the use of accelerated cooridnates.

 

[Dale]

A change of coordinates isn't a gravitational field.

Remember that there is more than one possible meaning of the term "gravitational field" in GR. In the sense that Einsetin meant it (also my preferred sense) the "gravitational field" is the Christoffel symbols, and the Christoffel symbols do in fact change under a change of coordinates. So a change of coordinates does cause a gravitational field in the sense Einstein used the term.

[EDIT: stevendaryl made essentially the same point, but faster!]

 

[harrylin]

It depends on how you define "gravitational field". If you drop an object, and it follows a trajectory h = -\frac{1}{2}g t^2 for some constant g, then many people would call g the "acceleration due to gravity" or the "gravitational field". But it certainly is affected by a coordinate change. It can be made to vanish by careful choice of coordinates. If g=0 (no gravitational field), then a change of coordinates to accelerated coordinates can make g nonzero.

The equivalence principle is about this notion of gravitational field: there is no difference (other than the variation of g with location) between the effects of a g due to gravity and a g due to the use of accelerated cooridnates.

A change in coordinates can of course produce a fictitious gravitational field; but if you don't consider this a real, physical field, then you side with the objector ("Critic") of early GR. Einstein countered that "the accelerated coordinate systems cannot be called upon as real causes for the field". As I elaborated, the issue here is with Einstein's Machian explanation of a gravitational field that is induced by the distant stars.

[..] a change of coordinates does cause a gravitational field in the sense Einstein used the term. [..]

As you see here above, Einstein stated just the contrary.

 

[Dale]

A change in coordinates can of course produce a fictitious gravitational field; but if you don't consider this a real, physical field, then you side with the objector ("Critic") of early GR.

I don't worry too much about "real" or "fictitious" except where it is part of standard terminology (i.e. "real numbers" or "fictitious forces"), and I won't take either side of such a debate.

Einstein countered that "the accelerated coordinate systems cannot be called upon as real causes for the field". As I elaborated, the issue here is with Einstein's Machian explanation of a gravitational field that is induced by the distant stars.

As you see here above, Einstein stated just the contrary.

In the http://en.wikisource.org/wiki/Dialog_about_objections_against_the_theory_of_relativityhe also said "the gravitational field in a space-time point is still not a quantity that is independent of coordinate choice; thus the gravitational field at a certain place does not correspond to something 'physically real', but in connection with other data it does. Therefore one can neither say, that the gravitational field in a certain place is something 'real', nor that it is 'merely fictitious'." He also said in the same discussion "Rather than distinguishing between 'real' and 'unreal' we want to more clearly distinguish between quantities that are inherent in the physical system as such (independent from the choice of coordinate system), and quantities that depend on the coordinate system." and "the distinction real - unreal is hardly helpful".

My point remains, that the "gravitational field" he refers to by "A gravitational field appears, that is directed towards the negative x-axis. Clock U1 is accelerated in free fall, until it has reached velocity v. An external force acts upon clock U2, preventing it from being set in motion by the gravitational field. When the clock U1 has reached velocity v the gravitational field disappears" is the field from the Christoffel symbols and it is entirely determined by the choice of coordinates.

I have not read all of Einsteins writings, but I believe that this is the meaning he usually attributes to the term. If you believe that he refers to something besides the Christoffel symbols then please be explicit about what mathematical term you think he intends and why.

 

[Mentz114]

What you here call "at rest" is also what Einstein calls "at rest"; your disagreement seems to be with the definition of acceleration that he used.

I don't know what you are talking about. The case I gave refers to proper acceleration which is unambiguous.

That's correct. In Langevin's "twin" scenario, the space capsule feels no force during the voyage.

I still don't know what you mean. You are moving the gaoalposts and and being slippery, because you're wrong.

 

[GregAshmore]

I did. I suspect that you didn't bother to read it, but stop acting as though the objection were summarily dismissed and no answer were given when you simply haven't bothered to read the answer given.

Yesterday I skimmed some posts, believing that I had "the gist" of the argument, without stopping to digest all the details. It's possible I missed your "gist", which would be sloppy of me. However, I don't think I missed what I was looking, for. I don't recall seeing it in posts prior to yesterday, either.

Before I say what I was looking for, I will accept a good deal of the criticism in recent posts. I have made some overly broad and not well thought out statements, resulting in errors. I realized that as I went through the mental exercise of constructing the spacetime diagram for the twin paradox, rather than accepting it as a finished product.

In my previous post, I asked for a SR solution of the problem in which the non-inertial rocket is always at rest. I don't recall anyone presenting such a solution. My sense is that my calls for such a solution have been rebuffed.

Every SR solution that I have seen has the rocket in motion. Certainly, in every SR spacetime diagram I have seen, the rocket is in motion; it is the reversing motion of the rocket which produces the knee in the rocket's worldline, and thus the difference in proper times. The article discussing acceleration in special relativity also speaks of the accelerating object as in motion. (http://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html)

I, in my rocket, claim that I am not in motion. I have been sitting in my rocket in the same position throughout the episode. You claim that I will be younger than my twin at the end of the episode. As proof, you present to me a diagram that shows me in motion. I reject it. I categorically deny that the diagram applies to me. The diagram shows me in motion; I have not moved.

I do not deny that my experience has been non-inertial. I do not deny that the experience of my twin has been inertial. I do not even deny that I have experienced "proper acceleration", because you have told me that I can experience proper acceleration while remaining motionless.

I deny that I have been in motion. Therefore, I insist on a solution that does not put me in motion.

What would a diagram of such a solution look like? There would have to be a point representing my position and time at the beginning of the episode, another point representing my position and time at the end of the episode, and a (possibly curvilinear) path that connects the two points. Whatever the shape of the path, the value of the position coordinate must not vary from the beginning to the end of the path, because I do not move.

It is maintained that SR can be used to solve this problem. I will attempt to draw the spacetime diagram. For convenience, I'll make my time and position axes orthogonal, in the usual manner with time positive toward the top of the page. And as usual I will set my starting position and time at the origin.

Because I do not move, my worldline must be parallel to the time axis; in this case it will be coincident with the time axis. I realize that this is the worldline of an inertial object. I am not inertial. It doesn't seem right that my worldline should be inertial, but that's how it must be, because I do not move.

Now to draw the earth. The Earth is moving in my frame, and inertial in its own frame. I cannot find any worldline that will satisfy both conditions, and also meet me at the end of my worldline. I cannot complete the spacetime diagram.

Being unable to represent in a spacetime diagram the perfectly legitimate scenario of myself at rest throughout the episode, and the Earth reversing, I conclude that SR is not suitable for solving the problem.

I may be wrong in my conclusion, as has been claimed. If so, I would like to see a SR solution which has me in one position throughout the episode.

 

[ghwellsjr]

I would like to see a SR solution which has me in one position throughout the episode.

How about this one:

from this thread.

 

[GregAshmore]

How about this one:

from this thread.

I haven't read it yet. I was just coming on to see if I could save myself from the same mistake I've made before: seeing the thing in my head without checking it on paper. Some people learn slowly.

 

[Dale]

However, I don't think I missed what I was looking, for. I don't recall seeing it in posts prior to yesterday, either.
...
In my previous post, I asked for a SR solution of the problem in which the non-inertial rocket is always at rest. I don't recall anyone presenting such a solution.
...
What would a diagram of such a solution look like?
...
I may be wrong in my conclusion, as has been claimed. If so, I would like to see a SR solution which has me in one position throughout the episode

See post 80.

Because I do not move, my worldline must be parallel to the time axis; in this case it will be coincident with the time axis. I realize that this is the worldline of an inertial object. I am not inertial. It doesn't seem right that my worldline should be inertial, but that's how it must be, because I do not move.

Don't forget, your frame is non-inertial so straight lines do not correspond to inertial worldlines. Inertial worldlines are geodesics, which is not the same thing as a straight line when you are using non-inertial coordinates.

Another, more accurate, way to say it is that your coordinate lines are bent. So lines of constant coordinates are not straight lines and straight lines don't have constant coordinates. A similar thing happens, e.g. in polar coordinates. Since the coordinates are curved the equation r=mθ+b does not represent a straight line.

Being unable to represent in a spacetime diagram the perfectly legitimate scenario of myself at rest throughout the episode, and the Earth reversing, I conclude that SR is not suitable for solving the problem.

Your inability to solve this reflects your own personal limitation, not a limitation of SR. It has been made abundantly clear to you that SR is not limited in this way.

 

[GregAshmore]

See post 80.

I have downloaded the paper.

Don't forget, your frame is non-inertial so straight lines do not correspond to inertial worldlines. Inertial worldlines are geodesics, which is not the same thing as a straight line when you are using non-inertial coordinates.

Another, more accurate, way to say it is that your coordinate lines are bent. So lines of constant coordinates are not straight lines and straight lines don't have constant coordinates. A similar thing happens, e.g. in polar coordinates. Since the coordinates are curved the equation r=mθ+b does not represent a straight line.

I'll have to work on this to understand it.

Your inability to solve this reflects your own personal limitation, not a limitation of SR.

I fully expect to find that you are correct in this.

It has been made abundantly clear to you that SR is not limited in this way.

Well, no, it hasn't been made clear. It ought to be clear, I'm sure. The fact that it isn't clear is much more a factor of my response to what has been said than a factor of the content.

I have the impression that some of the contributors on this forum are teachers by trade, so what follows may be of interest. If not, no need to read further.

I've been trying to figure out why I have had so much trouble learning relativity. In particular, I have never had the experience of repeatedly thinking that I understand a subject, only to discover that I am profoundly wrong.

One reason, no doubt, is the bizarre premises that we are called on to accept. However, that was much more of a stumbling block at the beginning than it is now. At this point, I can "suspend disbelief" and treat the problem as an exercise in abstract logic. The "truth" or "reality" of the premises can be evaluated later.

So perhaps I'm just not good at abstract logic. Maybe. I'm sure I'm no Einstein, at any rate. But I'm not profoundly stupid, either. So how do I repeatedly find myself in the position of being profoundly wrong?

I had an "aha" moment on this a couple of weeks ago, which was reinforced and clarified last night. It has to do with my pattern of learning.

The entire subject of relativity is completely hands-off for me. I'll never see, much less operate, a particle accelerator. I learn new things all the time in my work, but in every case I can test my understanding of what ought to happen against what actually happens when I act on my understanding.

In addition, many aspects of relativity are hypothetical (hands-off) for everyone, at least in our lifetimes. We'll never travel at relativistic speeds. So none of us have the opportunity to directly test our understanding by experiment. (We have indirect experimental evidence to support what is predicted; that's not the same thing as making the prediction come to pass.)

I have made the mistake of thinking that "unable (in practice) to test by experiment" means "completely unable to verify". In my usual method of learning, I form a mental picture of what ought to happen, then I test it by experiment. Because I am unable to test, I have been in the habit of stopping after forming the mental picture.

In my work, I may do calculations after forming a mental image and before conducting an experiment. (I nearly always did when I designed machinery; I rarely do now that I work in a larger company and only write software.) The calculations are viewed as a means of avoiding failure in the experiment; they are never seen as a verification of anything. The calculations are never an end in themselves; they are a means of getting to the end, which is a functioning piece of equipment.

In relativity, for someone in my situation, the calculations are both the verification and the deliverable. That's what finally penetrated my thick skull last night. I can no more put something on this forum without verifying it by calculation than I can deliver an untested product to a customer.

Now maybe we'll see fewer dumb statements by me on the forum.

I did find this interesting, for perspective. Errors are never acceptable. But if even professionals have trouble, I should not be surprised if I have trouble, too.

From the paper referenced in post #80:

The path through this confusion existed already in Einstein’s original paper[9], and was popularised by Bondi in his work on ‘k-calculus’. It lies in the correct application of ‘radar time’ (referred to as ‘Marzke-Wheeler Coordinates’ in Pauri et al.[10]). This concept is not new. Indeed Bohm[1] and D’Inverno[2] both devote a whole chapter to k-calculus, and use ‘radar time’ (not under that name) to derive the hypersurfaces of simultaneity of inertial observers. However, both authors then apply this definition wrongly to the traveling twin.