[Halc]

what's important is what we calculate to be the state at the given time "t" = "tau" that we have selected for the end of a given experiment

Yes! That's what's important. I did that, and you did not.
You did not calculate where the front ship (Fred) would be at tau. You did not in addition calculate where the front of the beam (attached to Ralph at the rear) would be at tau. They're different calculations, and you did neither.
You just declare things that are 'clearly' the way they must be. No calculations. You're totally skipping the important part.

Take your own challenge and answer your own question in post 6. Until then, you've absolutely nothing to back your contradictory assertions except you calling them 'clearly'.

Previously, I've given two arguments

No you've not. You've made two nonsense assertions. They're not arguments until you do what you say is important above: Calculate the state at given time tau (and be sure to include frame references).

Your argument 3
1) The equation you quote concerns a different scenario than the one you describe, so it just isn't applicable.
2) The equation concerns a different coordinate system than the one you're using.
3) You've not done any calculations, so you've not provided an argument.

I've shown mathematically all your assertions to be wrong. You've not shown any of my mathematics to be off. You've not responded to criticism at all, but instead just charge ahead with new assertions. That's the behavior of a troll. Please don't add yourself to that group Mike. You seem to be so desperately trying to pass their admissions test.
Show me explicitly where my mathematics is wrong and do the same calculation the correct way. Show that you've not devolved into a troll.

[MikeFontenot (OP)]

what's important is what we calculate to be the state at the given time "t" = "tau" that we have selected for the end of a given experiment

Yes! That's what's important. I did that, and you did not.
You did not calculate where the front ship (Fred) would be at tau.

The "state" that we need to calculate is what the leading clock reads at "t" = "tau".  There's no need to determine how far the two clocks have traveled at "t" = "tau".  That's because all I ultimately need to determine is what the leading clock reads at "tau", in the limit as "tau" goes to zero, and "A" goes to infinity (with the speed change always being from zero to 0.866 ls/s).  And in that case, the two clocks don't move at all between "t" = zero and "t" = "tau" (because the speed change is instantaneous) ... "tau" is essentially still zero.

The reason that I only need to determine what happens in the instantaneous speed change case, is that THAT is what is important in the twin paradox scenario, and THAT is what is needed to demonstrate an inconsistency between the twin paradox and the exponential version of the gravitational time dilation (GTD) equation.  Basically, the exponential GTD says that the home twin gets INFINITELY older during the traveling twin's instantaneous velocity change, which is inconsistent with the twin paradox outcome.  My GTD equation says that the home twin's age remains FINITE during the traveling twin's instantaneous velocity change, and is in complete numerical agreement with the twin paradox outcome.  There is NO inconsistency between my version of the GTD equation and the twin paradox: they exactly agree.

[MikeFontenot (OP)]

Even though I don't need to compute how far the clocks have moved from "t" = 0 until "t" = "tau", I'll show how to do it.

An acceleration "A" ls/s/s, from "t" = 0 to "t" = "tau" seconds, increases the rapidity "theta" linearly from zero to ("A" "tau") ls/s.  But rapidity ISN'T velocity.  Velocity "v" = tanh( "theta") ls/s.  So the distance "D" traveled by the clocks is

  D  =  integral from 0 to tau { v(u) du }

  D  =  integral from 0 to tau { tanh( A u) du } .

The two clocks are subjected to the same acceleration, so they each move the same distance "D".  The distance "L" between them doesn't change during the acceleration.

[MikeFontenot (OP)]

But some physicists are vague about whether the rapid variations in the current age of the distant person (according to the accelerating person) should be considered to be MEANINGFUL or not.  It is possible to show that those rapid age variations (including negative ageing) MUST be considered to be fully real and meaningful.

It can be shown to be meaningful to a given observer by showing how to construct an array of clocks which are mutually stationary wrt that observer.

Halc then said:

It is
1) Impossible to keep an array of clocks mutually stationary wrt some accelerating reference worldline. Some of them can be kept stationary if the acceleration schedule is known, but not all of them.

I, (Mike Fontenot) then replied:

That is not true.

Einstein said that a clock (and co-located "helper friend (HF)") whose acceleration "A" is the same as the designated observer's (the "DO's") acceleration (with both of them beginning the acceleration at the same instant, with synchronized clocks (and ages) at that instant), and leading (in the direction of the acceleration) the DO by the constant distance "L", will tic faster than the DO's clock, by the ratio exp(L*A).  (The "designated observer" (DO) is the person whose conclusion we want to know, about the current reading on each of the clocks in the array.)

Einstein clearly regarded the quantity "L" as being constant during the acceleration ... otherwise, he would have needed to tell us how to compute its value as the acceleration progressed, and he did not do that.

From the above, it's clear that an entire array of clocks (with their attending HF's) can be set up, with the n-th helper clock a distance n*L from the DO.  Einstein's exponential equation tells the DO, at each instant of his life (after the constant acceleration "A" begins), what the reading on each of the clocks in the array is (and how old the HF there is then).  The clocks in the DO's array aren't synchronized as the acceleration proceeds, but that doesn't matter: the fact that the DO (at each of his ages) can calculate the current reading on any of the clocks in the array (and the current ages of each of the HF's) establishes a "NOW at a distance" for him (the DO).  And that "NOW at a distance" guarantees that the DO's conclusion about the current age of a distant person must be considered to be meaningful for the DO.

[Question for Halc:  All of the above is standard special relativity: it came from Einstein himself.  I have introduced some terminology that I believe clarifies some things, but I haven't changed any of Einstein's results.  Do you nevertheless want to banish it to the "other theories" forum?]

I have proven that Einstein's exponential equation is incorrect.

[Another question for Halc: Does the fact that I have found an error in Einstein's exponential equation justify banishing this posting to the "other theories" forum?  My proof that the exponential equation is incorrect is fairly simple.  Have you tried to confirm or refute my proof?]

I eventually was able to derive the correct gravitational time dilation equation, and it can be used to construct the accelerating array of clocks and HF's that correctly establishes the "NOW at a distance" for the DO, and that makes the DO's conclusions meaningful to him.

[Another question for Halc:  Does the fact that I have derived the correct gravitational time dilation equation to replace the incorrect exponential equation justify banishing me to the "other theories" forum?  I think "other theories" SHOULD mean "theories other than special relativity", and that's definitely NOT the case with my results.]

P.S.:  I think what might be confusing you (Halc) is that you are probably looking at the scenario through the eyes of a perpetually-inertial observer. (I.e., you are interested in the conclusions of perpetually-inertial people, about what is happening to the separation between the accelerating clocks.)  In my analysis, I don't consider the conclusions of any perpetually-inertial observers at all, and don't need or want to.  I am only interested in the conclusions of the accelerating DO, and he says the spacing between the clocks is constant.  The acceleration "A" that each clock and their attendants (the DO and the HF's) are undergoing is produced by a rocket attached to each of them, and controlled by an attached accelerometer that always insists on a measured acceleration of exactly "A".

[Halc]

Question for Halc:  All of the above is standard special relativity: it came from Einstein himself.  I have introduced some terminology that I believe clarifies some things, but I haven't changed any of Einstein's results.  Do you nevertheless want to banish it to the "other theories" forum?

You consistently misunderstand both the situation described by Einstein, and also the meaning of the equation in question, so yes, here your post goes once again to this topic where you refuse to respond appropriately (not just with assertions) to errors being pointed out in your posts. You seem to totally disregard peer review, something that a professional physicist would never do.

Another question for Halc: Does the fact that I have found an error in Einstein's exponential equation justify banishing this posting to the "other theories" forum?

You calling it a fact doesn't make it a fact.

Have you tried to confirm or refute my proof?

Yes, I've refuted it. You ignore the errors I point out.

Another question for Halc:  Does the fact that I have derived the correct gravitational time dilation equation to replace the incorrect exponential equation justify banishing me to the "other theories" forum?

No, by all means post one. But calling it the 'correct' one implies that Einstein's equation is the incorrect gravitational time dilation equation when in fact that equation has nothing to do with gravity at all. That equation was not described until a later paper.

I'll try to point out some issues in this post. These have all been identified before.

There seem to be three different scenarios and you seem to want to apply the same equations to any of the three at times. This in itself is serious mistake. The 3 are:
Case B: Ships (clocks, observers, whatever) all having the same proper acceleration, starting simultaneously from a stop relative to some inertial frame in which both are initially stationary. I call this case B because it is exactly Bell's spaceship scenario. https://en.wikipedia.org/wiki/Bell%27s_spaceship_paradox
Case R: Rigid motion where there is one long spaceship with a clock/observer/whatever at either end. The proper acceleration measured at either end does not match the other.
Case T: The twins scenario, with only one guy accelerating and the other just sitting waiting for the reunion.

Einstein said that a clock (and co-located "helper friend (HF)")

Again, what possible purpose is served by this HF? If he reads the clock, what good is that? Is his job to assert things? I digress. The HF is not what's wrong with all this, it just seems entirely superfluous.

Einstein said that a clock whose acceleration "A" is the same as the designated observer's (the "DO's") acceleration

Just pointing out that they both have identical proper acceleration, making this case B. I've pointed this out before, but you often try to make it into one of the other cases.

with both of them beginning the acceleration at the same instant with synchronized clocks (and ages) at that instant), and leading (in the direction of the acceleration) the DO by the constant distance "L", will tic faster than the DO's clock, by the ratio exp(L*A).

Here is a great example of your biggest consistent mistake: Lack of frame references. You (and Einsten) say that they begin the acceleration simultaneously, but due to RoS, that is meaningless without a frame reference. So to be clear, they start simultaneously in frame F, the one inertial frame in which they both were initially at rest.
Since both clocks have identical acceleration profiles in frame F, they remain in sync in F, that is to say that at any moment, both clocks simultaneously read the same value. You say that the lead one ticks faster, but it doesn't in frame F. So you perhaps mean a different frame, but you don't specify it. Other suggestions might be the accelerated frame of DO or perhaps he accelerated frame of HF. You don't say, but you do say "leading the DO by the constant distance L" which is not true in any frame where HF ticks faster than DO.
See what I mean? You're mixing different frames in a single assertion. That whole description is an unworkable mess. Give the frame specifications so I can show which assertions are appropriate and which are wrong. There's no frame in which all of that can be correct.

Your reaction to such feedback is usually to dismiss the critique because you know it's just not possible that you've made a mistake. If so, yes, your posts very much belong in the lighter-side where we put stuff that's wrong.

The "designated observer" (DO) is the person whose conclusion we want to know, about the current reading on each of the clocks in the array.

Fine, but one has to specify which coordinate system is being used by DO. Is it F? Is it his own Rindler coordinates? Lass coordianates? Do you know which of these Einstein is using? He at least specifies his frames.

Einstein clearly regarded the quantity "L" as being constant during the acceleration

Does he say that? If so, does he say in which coordinates this is true, because it's true only in one of them. Making strawman claims of what he's asserting is another mistake.

Einstein's exponential equation tells the DO, at each instant of his life (after the constant acceleration "A" begins), what the reading on each of the clocks in the array is

This again utilizes simultaneity without specification of coordinate system. Your statement is not even wrong until you do this. An instant at DO is not that same instant at some other location without the frame specification.

I eventually was able to derive the correct gravitational time dilation equation, and it can be used to construct the accelerating array of clocks and HF's that correctly establishes the "NOW at a distance" for the DO, and that makes the DO's conclusions meaningful to him.

Here you're equating acceleration to gravity. That works only locally, but you're trying to apply it non-locally. There is no gravity in the accelerating case.

I think what might be confusing you (Halc) is that you are probably looking at the scenario through the eyes of a perpetually-inertial observer. (I.e., you are interested in the conclusions of perpetually-inertial people, about what is happening to the separation between the accelerating clocks.)  In my analysis, I don't consider the conclusions of any perpetually-inertial observers at all, and don't need or want to.

I'm not interested in any observers at all since they only can observe what's in their immediate presence. But the frame specifications are crucial, and they're missing in your 'proof', rendering it meaningless. Try to actually entertain the idea that I have a point here.

The acceleration "A" that each clock and their attendants (the DO and the HF's) are undergoing is produced by a rocket attached to each of them, and controlled by an attached accelerometer that always insists on a measured acceleration of exactly "A".

Ah good, that's the first in this post where you clearly specified constant proper acceleration. You're definitely case B then.

The post to which you said I was wrong was this comment of mine in another thread:
"It is impossible to keep an array of clocks mutually stationary wrt some accelerating reference worldline. Some of them can be kept stationary if the acceleration schedule is known, but not all of them."
That comment was about a case R (rigid motion) situation where the proper distance between the accelerating clocks is constant, and thus the proper acceleration of those clocks is not constant.

[MikeFontenot (OP)]

(I, Mike Fontenot, said:)
Another question for Halc: Does the fact that I have found an error in Einstein's exponential equation justify banishing this posting to the "other theories" forum?

(Halc said:)
You calling it a fact doesn't make it a fact.

(I, Mike Fontenot, say:)
I told you what I got in the sequence of calculations (for the HF's age at the end of the acceleration) for each of my choices of the acceleration "A" and the duration of the acceleration "tau" (such that the velocity change is the same in each iteration of the sequence).  Clearly, in my calculations, the sequence was NOT converging to a finite limit ... the HF's age was clearly going to infinity as tau goes to zero, and that is inconsistent with the outcome of the twin paradox.  If you think my calculations are wrong, what do YOU get when you do that sequence of calculations?  If you're going to say my calculations are wrong, you need to show what the correct calculations are.

The results of my calculations, in Section 3 of the above referenced paper, are given in a table near the end of Section 3.  The exponential GTD equation says that the leading clock tics faster by the factor exp(A*d).  And so the change in the reading on the leading clock, during a time interval "tau", is just tau * exp(A*d).

Just do the cases where tau (the duration of the acceleration) equals 1.0, 0.1, and 0.01.  And in each case, "A" is whatever is required to keep the product of A*tau (which is the "rapidity") equal to 1.317 (corresponding to a velocity of 0.866).  So when you DECREASE tau by a factor of 10, you need to INCREASE "A" by a factor of 10.  The separation "d" (which I call "L" in other documents) is 7.52.

[MikeFontenot (OP)]

In Einstein's 1907 paper, https://einsteinpapers.press.princeton.edu/vol2-trans/319 , he gives his exponential gravitational time dilation equation.  Actually, he was working a special relativity problem (with accelerations and no gravitation) because he knew how to do that, and he was hoping the result would give him (via the equivalence principle) some help in his search for a gravitational theory.  So Einstein's equation was actually a time dilation equation for accelerating clocks that are separated by a fixed distance.  According to Einstein, for a pair of accelerating clocks separated by the distance "L" in the direction of the acceleration, the leading clock tics faster than the trailing clock by the factor

  R  =  exp(A*L),

where "A" is the acceleration.  In the iterations described below, I will limit myself to the case where "A" is constant during the acceleration.  I will show that the exponential time dilation equation is incorrect.  (I suspect that Einstein, and also physicists who came along later, didn't ever notice that the exponential equation is incorrect, because they never used it in the nonlinear range where its argument is large ... they only used it for very small arguments, where it is very nearly linear.)

Suppose that the two clocks are initially inertial (unaccelerated) at time "t" = 0, and both read zero at that instant.  Then, for t > 0, both clocks undergo a constant acceleration "A" (as determined by accelerometers attached to each of them, which control a rocket attached to each of them, so as to achieve the specified acceleration).  Their separation remains constant at "L" during the acceleration.  In all of my calculations below, I chose L = 7.520.  Let "tau" be the duration of the acceleration.  Therefore the reading on the leading clock, when the trailing clock reads "tau", is given by

  AC  =  tau * R  =  tau * exp(A*L) .

I first take the case where the duration "tau" of the acceleration is equal to 1.0.  I choose the magnitude of the acceleration "A" to be such that the velocity of the two clocks, after accelerating for a duration "tau" = 1.0, is 0.8660.  The product of the constant "A" and the duration "tau" gives the rapidity "theta", which is related to the velocity by the equation

  v  =  tanh(theta),

where tanh() is the hyperbolic tangent function, and is equal to

  tanh(theta)  =  { [exp(theta) - exp(-theta)] / [exp(theta) + exp(-theta)] } .

So, for v = 0.8660, the rapidity theta = 1.3170.  Since

  theta = A * tau,

  A  =  theta / tau  =  1.3170 / 1.0  =  1.3170.

So for the first case with tau = 1.0, we get

  R  =  exp(A*L)  =  exp( 1.3170 *  7.520 )  =  exp(9.90384)  =  20007.

The reading on the leading clock, at the end of the acceleration at tau = 1.0, is

  AC  =  tau * R  =  tau * exp(A*L)  = 1.0 * 20007  =  20007  =  2.0 * 10^4,

where 10^4 is just 10 raised to the 4th power.

So for the first calculation (with tau = 1.0), we have that at the end of the acceleration, the leading clock reads

  AC = 2.0 * 10^4.

For the second case, we increase the acceleration "A" by a factor of 10, and decrease the duration "tau" by the factor 10 (which still, as required, results in the same speed change as in the first case).  So we now have

  A  =  13.170

and

  tau  = 0.1.

So

  R  =  exp(A*L)  =  exp( 13.170 *  7.520 )  =  exp(99.0384)  =  1.028 * 10^43 ,

and

  AC  =  tau * R  =  0.1 * R  =  1.028 * 10^42.

Note that when we increased the acceleration by a factor of 10, the reading of the leading clock didn't increase by a factor of 10, it increased by ten raised to a power that increased by a factor of about ten.

For the third case, we increase the acceleration "A"  again by a factor of 10, and again decrease the duration "tau" by the factor 10 (which still, as required, results in the same speed change as in the first and second case).  So we now have

  A  =  131.70

and

  tau  = 0.01.

So

  R  =  exp(A*L)  =  exp( 131.70 *  7.520 )  =  exp(990.38)  =  1.31 * 10^430 ,

and

  AC  =  tau * R  =  0.01 * R  =  1.31 * 10^428.

Note that, again, when we increased the acceleration by a factor of 10, the reading of the leading clock didn't increase by a factor of 10, it increased by ten raised to a power that increased by a factor of roughly ten.

So we get the following table:

  tau            AC

  1.0       2.0 * 10^4

   0.1      1.0 * 10^42

   0.01    1.0 * 10^428

Clearly, this iteration is NOT approaching a finite value for the leading clock's reading, as tau goes to zero.  The leading clock's reading is clearly diverging as tau goes to zero.  I.e., the leading clock's reading goes to infinity as tau goes to zero.

So when we use this method to determine by how much the home twin's (her) age increases when the traveling twin (he) instantaneously changes his velocity by 0.866 when he reverses course at the turnaround, it tells us that the home twin gets INFINITELY older, which is not true.  From the time dilation equation for an inertial observer (which the home twin IS), we KNOW that both she and he have a finite age at their reunion.  Therefore the exponential gravitational time dilation equation CAN'T be correct.

[Halc]

You quote a statement of mine denying the fact that you've proven something about Einstein's equation, but then you post stuff irrelevant to that demonstration. You seem focused on a different scenario than the one Einstein describes.

I told you what I got in the sequence of calculations (for the HF's age at the end of the acceleration) for each of my choices of the acceleration "A" and the duration of the acceleration "tau" (such that the velocity change is the same in each iteration of the sequence).  Clearly, in my calculations, the sequence was NOT converging to a finite limit ... the HF's age was clearly going to infinity as tau goes to zero

Your calculation is dependent on the arbitrary coordinate system you choose to make such a measurement. You consistently refuse to specify this, but I'm guessing the inertial frame of all objects after all accelerations cease. And yes, your accelerations cease in your scenario.

That results in all cases in the velocity of the two clocks, at the end of the acceleration, being 0.866 ls/s.

This is important because Einstein did not specify this. His scenario involves eternal acceleration of all objects once they start. He also isn't computing ages using any inertial frame since nothing is inertial.

and that is inconsistent with the outcome of the twin paradox.

If course it is. The twin paradox is Case T, not Case B. It involves only one entity accelerating, not both of them. This is what I mean by you attempting to use formulas of one scenario with the wrong scenario.

If you think my calculations are wrong, what do YOU get when you do that sequence of calculations?

I went straight for zero duration, an instant change of velocity of all objects to .866c, then did a simple Lorentz transform. Relative to the new inertial frame of the rear clock (which still reads 0), the front clock now reads just over 13 seconds, and is just over 15 ls distant, exactly twice its former distance (relative to the inertial frame in which they were stationary before the velocity change) of 7.52. Both clocks are stationary in this new frame after the rear clock does its acceleration.

If you're going to say my calculations are wrong, you need to show what the correct calculations are.

Well, I didn't actually see any calculations. But infinite seems wrong in most plausible coordinate systems I can think of.

The results of my calculations, in Section 3 of the above referenced paper, are given in a table near the end of Section 3.

If it isn't posted here, I don't see it. I don't do links to personal material.

The exponential GTD equation says that the leading clock tics faster by the factor exp(A*d).  And so the change in the reading on the leading clock, during a time interval "tau", is just tau * exp(A*d).

Well, by definition, during the proper interval tau, any clock advances by exactly tau. You're probably talking about 'as measured by something not in the presence of the clock in question', but for that to have meaning, a coordinate system needs to be specified. In any one inertial coordinate system, if zero time has gone by, then all clocks advance by zero seconds regardless of their motion or acceleration rates. So either you're changing frames without specifying such an action, or you're choosing a non-inertial frame. Your refusal to do this renders all your conclusions meaningless.

In Einstein's 1907 paper, https://einsteinpapers.press.princeton.edu/vol2-trans/319 , he gives his exponential gravitational time dilation equation.  Actually, he was working a special relativity problem (with accelerations and no gravitation) because he knew how to do that, and he was hoping the result would give him (via the equivalence principle) some help in his search for a gravitational theory.  So Einstein's equation was actually a time dilation equation for accelerating clocks that are separated by a fixed distance.  According to Einstein, for a pair of accelerating clocks separated by the distance "L" in the direction of the acceleration, the leading clock tics faster than the trailing clock by the factor

  R  =  exp(A*L),

where "A" is the acceleration.  In the iterations described below, I will limit myself to the case where "A" is constant during the acceleration.  I will show that the exponential time dilation equation is incorrect.  (I suspect that Einstein, and also physicists who came along later, didn't ever notice that the exponential equation is incorrect, because they never used it in the nonlinear range where its argument is large ... they only used it for very small arguments, where it is very nearly linear.)

Every bit of the above is rubbish, and just a repeat of the assertions made in earlier posts (mostly meaningless assertions given the still absent frame references), all of which have been refuted, and without response to the refutations. I'll not bother doing it again, but rather comment on the remaining bit.

Suppose that the two clocks are initially inertial (unaccelerated) at time "t" = 0, and both read zero at that instant.  Then, for t > 0, both clocks undergo a constant acceleration "A" (as determined by accelerometers attached to each of them, which control a rocket attached to each of them, so as to achieve the specified acceleration).  Their separation remains constant at "L" during the acceleration.  In all of my calculations below, I chose L = 7.520.  Let "tau" be the duration of the acceleration.
Therefore the reading on the leading clock, when the trailing clock reads "tau", is given by
AC  =  tau * R  =  tau * exp(A*L) .

That cannot be true. For one, you have a self-contradiction going on. If the lead clock 'tics faster' as you put it, it will finish its tau-seconds of acceleration first, meaning there will be a period of time where one ship is accelerating and not the other. This contradicts your assertion (that Einstein certainly doesn't make) that the separation distance remains constant. Anyway, the formula you give will not in any way give the reading on the leading clock, when the trailing clock reads "tau". I did that above (the number approached as acceleration increases without limit) and got about 13 seconds. You don't seem to suggest a different figure, only giving a demonstration that the formula above isn't going to give the right answer, which it indeed doesn't.

So we get the following table:
  tau            AC
  1.0       2.0 * 10^4
   0.1      1.0 * 10^42
   0.01    1.0 * 10^428
Clearly, this iteration is NOT approaching a finite value for the leading clock's reading, as tau goes to zero.

Indeed. They're also all wrong since the right column should decrease from about 14 down to 13, and not those huge numbers. What do you suggest are the correct values? I notice you don't work that out. Are my numbers wrong?

So when we use this method to determine by how much the home twin's (her) age increases when the traveling twin (he) instantaneously changes his velocity by 0.866 when he reverses course at the turnaround it tells us that the home twin gets INFINITELY older, which is not true.

Correct again.You seem amazed that using a perfectly good formula for an inapplicable situation gives wrong answers. What were you expecting?

From the time dilation equation for an inertial observer (which the home twin IS), we KNOW that both she and he have a finite age at their reunion.  Therefore the exponential gravitational time dilation equation CAN'T be correct.

Wrong conclusion, twice. 1, That formula has nothing to do with gravity. 2, it also has nothing to do with either your scenario with the decreasing tau  or the twins scenario with only one clock accelerating. It is for a different scenario, one that you've never considered. If you read Einstein's paper (or my posts above), maybe you'd notice that.

[MikeFontenot (OP)]

[...] you have a self-contradiction going on. If the lead clock 'tics faster' as you put it, it will finish its tau-seconds of acceleration first, meaning there will be a period of time where one ship is accelerating and not the other.

No.  The leading clock stops accelerating when that clock reads
  tau * exp(L*A), and the trailing clock reads "tau".

Also, you have never responded to my statement that, if Einstein had believed that the separation "L" isn't constant, then he wouldn't have said that the tic rate ratio was exp(L*A), without specifying how "L" varied with time.

[MikeFontenot (OP)]

(I, Mike_Fontenot, said:)

 The leading clock stops accelerating when that clock reads
  tau * exp(L*A), and the trailing clock reads "tau".

(And now I'm adding:)

It is sometimes simpler to just assume that the constant acceleration goes on forever, and we just choose to assess the situation at any instant that we want.  But for now, we can stick to the idea that they stop accelerating when the trailing clock reads tau, and the leading clock reads tau * exp(L*A).  (And in that last sentence, I am specifying a SINGLE instant in their accelerating reference frame ... they both agree about that.)
 

(Then, Halc says:)

If they start at the same time and the rear clock tics slower (thousands of times slower according to your post), then when the lead clock reaches tau, the rear clock will will not yet be at tau. You cannot have it both ways. Do they stop at the same time or is there time dilation going on?

(end of Halc's statement).

You've misunderstood me.  The leading observer does NOT consider the stopping time to be tau on HIS watch  But he DOES expect the trailing clock to read tau when they stop, and his own clock to read tau * exp(L*A) when they stop.   A "NOW-at-a-distance" instant, according to these two accelerating observers, consists of any arbitrary instant "t" in the life of the trailing observer, and the corresponding instant

  t * exp(L*A)

in the life of the leading observer.  They each consider that to be the SAME instant in their accelerating reference frame.

So for the instant immediately after the acceleration has started, they both know that the reading on each of their watches will be zero.  For the trailing observer, that reading is just "tau", which is zero.  For the leading observer, that reading is tau * exp(L*A), which is also zero.

But for some later instant, tau will be some non-zero number, say "T".  So the trailing clock will read "T", and the leading clock will read

  T * exp(L*A),

and they both agree that that is the SAME instant in their accelerating reference frame, even though their clocks don't read the same thing then.  So they each know how to end the acceleration at an agreed-upon instant.

Does that help?

[Halc]

The leading observer does NOT consider the stopping time to be tau on HIS watch  But he DOES expect the trailing clock to read tau when they stop

Now you've changed the story. Post 5 (where you first describe this particular case) is pretty clear that the lead clock stops accelerating when its own clock reads tau:

and who then simultaneously start accelerating (according to their accelerometers) at a constant "A" ls/s/s
...
In each of the cases, the acceleration "A" and the duration "tau" of the acceleration are chosen such that the product "A tau" is constant for all cases at 1.317 ls/s.  That results in all cases in the velocity of the two clocks, at the end of the acceleration, being 0.866 ls/s.

The proper acceleration is specified to be the same for both objects, and the resulting celerity (1.317c) or speed relative to S (0.866c) is identical for both, which cannot happen if the lead object accelerates longer than tau as measured on its own clock.

Second problem is that the lead observer is not in the presence of the rear clock, so the moment that the remote clock reads tau is frame dependent, and you refuse to specify a frame.

Third problem is that for something accelerating (in the first case) at 31½ million g, in the inertial frame in which the front observer is stationary (what you call his CMIF), the rear clock is always unaccelerating since time moves backwards for it. So by your specification above, and assuming you're using the front observer's own CMIF (you never say), the condition of the read clock reading tau will never occur.

Get your story straight Mike. Use frame references. I tire of pointing out your mistakes only to have you move the goal posts into a new but still inconsistent story.

A "NOW-at-a-distance" instant, according to these two accelerating observers, consists of any arbitrary instant "t" in the life of the trailing observer, and the corresponding instant

  t * exp(L*A)

OK, can you identify the frame in which this is true? The event of the rear clock reading 1 second and the second event of the front clock reading ~5½ hours (from 1st line of your table) can be made simultaneous by selection of an inertial frame where neither observer is stationary. Do you want to use that one? The clocks will be an awful lot further apart in that frame than just L after the front one shoots off at 31 million g for several hours.

[MikeFontenot (OP)]

I realized this morning that I was wrong about the separation being constant ... the separation increases with time.  (I had a diagram showing that behavior, tacked up on the wall over my desk, that I had produced probably about 20 years ago.)  That's good, because it still results in the exponential equation being infinite for instantaneous velocity changes ... i.e., it doesn't change my conclusion that the exponential equation is wrong.

Whether or not I can still derive a replacement equation is unknown, but I'm cautiously optimistic.  I won't have much time to work on it, though, until tax-prep time is over.