[MikeFontenot (OP)]

According to the CMIF (Co-Moving-Inertial Frames) simultaneity method, an observer (he) who accelerates in the direction away from a distant person (she) will conclude that she rapidly gets YOUNGER during his acceleration. But I think I may have found a counterexample that shows that doesn't happen.

It is well-known that two stationary clocks at different positions in a gravitational field will run at different rates. The clock that is closer to the source of the gravitational field will run slower than the clock that is farther from the source of the field.

Because of the equivalence principle, it is also true that if two clocks that are separated by a fixed distance "d" ly are both accelerated with a constant equal acceleration of "A" ly/y/y, the trailing clock runs slower than the leading clock, by the factor exp(Ad).

So consider the following scenario:

At some instant, the perpetually-inertial "home twin" (she) is 20 years old, and is holding a display that always shows her current age. Facing her is the "helper friend" (the "HF") of an observer (he) who is "d" ly away to her right. Both the HF and he are also 20 years old, and are stationary wrt her at that instant. Like her, he and the HF are each holding a display that always shows their current ages.

Now, suppose that he and his helper then both start accelerating at a constant "A" ly/y/y toward the right. He knows that his helper friend (the HF) is then ageing at a constant rate that is slower than his own rate of ageing, by the factor exp(Ad).

An instant later, his display shows the time 20 + epsilon_1, where epsilon_1 is a very small positive number. He knows that HF's display shows the time 20 + epsilon_2, where epsilon_2 = epsilon_1 / exp(Ad).
She can still see HF's display (because HF has only moved an infinitesimal distance away from her, to her right). She will see that HF's display reads 20 + epsilon_1 / exp(Ad). And likewise, HF can still see her display. What does HF see on her display? Does HF see that she is now slightly younger than 20? No! It would clearly be absurd for someone essentially co-located with her to see her get younger. HF would see her display reporting that she was some very small amount epsilon_3 OLDER that she was at the instant before the acceleration. HF then sends a message to him, telling him that she was 20 + epsilon_3 right then. When he receives that message, he then knows that her current age, when he was 20 + epsilon_1, was 20 + epsilon_3. So he KNOWS that she didn't get younger when he accelerated away from her. That contradicts what CMIF simultaneity says.

[MikeFontenot (OP)]

[...]
What does HF see on her display? Does HF see that she is now slightly younger than 20?
[...]

Since my argument makes use of very small (unspecified) quantities,
it could be argued that time delays due to the speed of light might also need to be taken into account when describing what the HF sees on her display. But I think any such concerns can be addressed by pointing out that the separation "d" between him and her can be made arbitrarily large, and CMIF simultaneity says that the amount of negative ageing that occurs is proportional to their separation. Since the errors involved due to the finite speed of light between her and the HF are independent of the distance "d", those errors become negligible for sufficiently large "d".

 

[MikeFontenot (OP)]

A Possible Proof That Negative Ageing Doesn't Occur In Special Relativity

Michael Leon Fontenot


According to the CMIF (Co-Moving-Inertial Frames) simultaneity method, an observer (he) who accelerates in the direction away from a distant person (she) will conclude that she rapidly gets YOUNGER during his acceleration. But I think I may have found a counterexample that shows that such an accelerating observer does NOT conclude that.

It is well-known that two stationary clocks at different positions in a gravitational field will run at different rates. The clock that is closer to the source of the gravitational field will run slower than the clock that is farther from the source of the field.

Because of the equivalence principle, it is also true that if two clocks that are separated by a fixed distance "d" ly are both accelerated with a constant equal acceleration of "A" ly/y/y, the trailing clock runs slower than the leading clock, by the factor exp(Ad).

So consider the following scenario:

At some instant, the perpetually-inertial "home twin" (she) is 20 years old, and is holding a display that always shows her current age. Facing her is the "helper friend" (the "HF") of an observer (he) who is "d" ly away to her right. Both the HF and he are also 20 years old, and are stationary wrt her at that instant. Like her, he and the HF are each holding a display that always shows their current ages.

Now, suppose that he and his helper then both start accelerating at a constant "A" ly/y/y toward the right. He knows that his helper friend (the HF) is then ageing at a constant rate that is slower than his own rate of ageing, by the factor exp(Ad).

An instant later, his display shows the time 20 + epsilon_1, where epsilon_1 is a very small positive number. He knows that HF's display shows the time 20 + epsilon_2, where
epsilon_2 = epsilon_1 / exp(Ad). Epsilon_2 is less than epsilon_1, but is also positive. She can still see HF's display (because HF has only moved an infinitesimal distance away from her, to her right). She will see that HF's display reads 20 + epsilon_1 / exp(Ad). And likewise, HF can still see her display. What does HF see on her display?
Does HF see that she is now slightly younger than 20? No! It would clearly be absurd for someone essentially co-located with her to see her get younger. What HF would see her display reporting is that she was now some very small amount epsilon_3 OLDER that she was at the instant before the acceleration. HF then sends a message to him, telling him that she was 20 + epsilon_3 right then. When he receives that message, he then knows that her current age, when he was 20 + epsilon_1, was 20 + epsilon_3, which is greater than 20. So he KNOWS that she didn't get younger when he accelerated away from her. That contradicts what CMIF simultaneity says.

In the above, I asked

"What does HF see on her display?".

And I answered

"HF would see her display reporting that she was some very small amount epsilon_3 OLDER that she was at the instant before the acceleration."

Since the above argument makes use of very small (unspecified) quantities, it could be argued that time delays due to the speed of light might also need to be taken into account when describing what the HF sees on her display.

But I think any such concerns can be alleviated by pointing out that the separation "d" between him and her can be made arbitrarily large, and CMIF simultaneity says that the amount of negative ageing that occurs is proportional to their separation. Since the errors involved due to the finite speed of light between her and the HF are independent of the distance "d", those errors become negligible compared to the change in her displayed age seen by the HF, for sufficiently large "d".

If the CMIF simultaneity method is incorrect, as the above proof contends, what is the alternative? I am aware of only a single alternative that (like the CMIF method) obeys the principle of causality. I describe that alternative in my paper, “A New Simultaneity Method for Accelerated Observers in Special Relativity”, on the viXra repository:

http://viXra.org/abs/2106.0133

I've put the above possible proof that there is no negative ageing (and thus that the CMIF simultaneity method can't be correct) on the viXra repository:

http://viXra.org/abs/2106.0142

There is another argument that shows that the HF ("Helper Friend") can't conclude that the home twin (she) is less than 20 years old when the HF is 20 + epsilon_2. We can require that she transmits NO light messages to him when she is 20 years old or younger. Suppose the HF receives a light message from her when he is 20 + epsilon_2 years old. By the requirement, she must have been older than 20 years old when she sent that message. When the HF receives that message, he knows that she must be older than when she sent the message, so she must definitely be older than 20 years old when the HF is 20+epsilon_2. Therefore, she did NOT get younger, according to him, when he accelerated away from her.

A still simpler argument is that, if the HF ever concluded that she got younger when he accelerated away from her, he would be concluding that she was less than 20 years old at that instant of his acceleration. But the HF was co-located with her when she was less than 20, and he couldn't be two places a that same instant.

It seems to me that, once the distant accelerating observer has a way to set up an array of clocks (with attending observers) that he can use to define his concept of "NOW" (analogous to how Einstein did it for perpetually-inertial observers), it becomes impossible for the home twin to age negatively, according to the distant accelerating observer. It's true that those clocks aren't synchronized as they are in the perpetually-inertial case, but they don't have to be, since the distant accelerating observer knows exactly how the rates of those clocks compare to his own clock.

I suspect that the same type of argument can be used to show that the essentially instantaneous (positive) ageing of the home twin (according to the traveler who instantaneously changes his velocity) also cannot occur. If these arguments are correct, then the commonly-used CMIF (Co-Moving Inertial Frames) simultaneity method can't be correct.

[Halc]

Topic has been merged because
1) It violates posting guidelines for the main sections of the forum
2) It’s the same as this prior topic with which it has been merged

A Possible Proof That Negative Ageing Doesn't Occur In Special Relativity

SR does not conclude or assert any particular preferred coordinate system, and thus SR would not make any frame-independent statement such as “Negative aging doesn’t occur”.

SR does not bother discussing something like the CMIF coordinate system, which is a coordinate system based on a time-like worldline, because such a coordinate system does not seem useful for any useful purpose for making any predictions.

It is well-known that two stationary clocks at different positions in a gravitational field will run at different rates. The clock that is closer to the source of the gravitational field will run slower than the clock that is farther from the source of the field.

Because of the equivalence principle, it is also true that if two clocks that are separated by a fixed distance "d" ly are both accelerated with a constant equal acceleration of "A" ly/y/y, the trailing clock runs slower than the leading clock, by the factor exp(Ad).

Equivalence principle is applied incorrectly in multiple ways. In gravity, the thing further down experiences greater proper acceleration if it is to maintain the same proper distance from the thing higher up.
If you have say two observers (Ted at the head, higher up, Bob at the bottom) both with identical proper acceleration, moving at the same velocity at some time T, then in the CMIF of either observer’s worldline, the proper separation between the two after local time T will increase, which is why the string breaks in Bell’s spaceship scenario. You should know these basics.

If the acceleration is 1G and the separation 1 LY at T, then relative to Ted’s CMIF of Ted, Bob ages backwards.

The rest of the post seems to rest on these invalid assertions, so it ends here. The CMIF method isn’t ‘incorrect’ as you conclude, but merely useless.

From the OP:

Because of the equivalence principle, it is also true that if two clocks that are separated by a fixed distance "d" ly are both accelerated with a constant equal acceleration of "A" ly/y/y,

Under the equivalence principle, neither coordinate nor proper acceleration is identical between the two clocks. The upper.lead clock undergoes lower acceleration (of both kinds) than the lower/trailing clock.

Now, suppose that he and his helper then both start accelerating at a constant "A" ly/y/y toward the right.

But here you're giving them identical acceleration of one kind. You fail to specify if this is constant proper or coordinate acceleration, but the two points of measurement in the former example (with the equivalence principle) will maintain constant proper separtion, and will not do so with what you're doing here. Your conclusions seem to be based on the two situations being equivalent.

What you're essentially doing with the distant guy constant proper acceleration is to set up Rindler coordinates based on his acceleration. Under that, 'she' will not age negatively if she exists within his Rindler frame, and she will age negatively in those coordinates if not.

If the acceleration is constant coordinate acceleration, then the Rinder coordinates do not apply, and everybody's ages is a function of the chosen coordinate system and not a function of anything he is doing.

[puppypower]

[Long rambling off-topic post removed]

[MikeFontenot (OP)]

A Possible Proof That Negative Ageing Doesn't Occur In Special Relativity

Michael Leon Fontenot


According to the CMIF (Co-Moving-Inertial Frames) simultaneity method, an observer (he) who accelerates in the direction away from a distant person (she) will conclude that she rapidly gets YOUNGER during his acceleration. But I think I may have found a counterexample that shows that such an accelerating observer does NOT conclude that.
[...]

It's now looking like to me that Einstein's 1907 method of defining an array of clocks for an accelerating observer agrees completely with the CMIF method. If so, that means that CMIF is the law of the land, no further assumptions required. I can also now see what the error in my proof (that negative ageing doesn't occur) was. Details to follow.

[Bored chemist]

A Possible Proof That Negative Ageing Doesn't Occur In Special Relativity

Had anyone said that it did?

[Halc]

Had anyone said that it did?

Andromeda 'paradox' is one of the more classic examples.
I am walking one way, and simultaneously (relative to my immediate inertial frame), the Andromeda race has launched a fleet against Earth.  Now I turn around and walk the other way, and simultaneously (relative to my new immediate inertial frame), the Andromeda race is still in the process of making the decision to launch the fleet a month from now. By my act of turning around, I have 'caused' the Andromeda race to age (relative to me) negative one month.
These sorts of RoS things seem to keep Mike up at night, and all his threads are about ways to resolve this problem that isn't actually a problem.

[MikeFontenot (OP)]

Andromeda 'paradox' is one of the more classic examples.
I am walking one way, and simultaneously (relative to my immediate inertial frame), the Andromeda race has launched a fleet against Earth.  Now I turn around and walk the other way, and simultaneously (relative to my new immediate inertial frame), the Andromeda race is still in the process of making the decision to launch the fleet a month from now. By my act of turning around, I have 'caused' the Andromeda race to age (relative to me) negative one month.

That was a very good response.

These sorts of RoS things seem to keep Mike up at night, and all his threads are about ways to resolve this problem that isn't actually a problem.

I personally don't see it as a problem at all ... I prefer the CMIF simultaneity method to all other alternatives.  But the problem that I see is that many physicists DON'T accept the CMIF method, or at least the part of CMIF that says that the distant person can get younger, according to the accelerating observer.  Also, my recent proof seems to show (contradicting the CMIF method) that the distant person doesn't ever get younger (according to the accelerating person), and I'm still having trouble finding the error in that proof, if there is an error.

[Halc]

But the problem that I see is that many physicists DON'T accept the CMIF method

Citation needed. From what I see, most physicists simply find no utility in a coordinate system based on a worldline with haphazard motion. Such a coordinate system does not foliate spacetime and thus doesn't assign unique coordinates to most events.
But they very much find utility in coordinate systems based on say continuous linear acceleration, despite having most of the same anomalies mentioned above.

the part of CMIF that says that the distant person can get younger

Depends on what you mean by getting younger.

A(bstract): If it is simply an abstract assignment of arbitrary 'current times' to various points on a worldline, there's nothing preventing this abstract action. There's no contradiction to say that it's Monday for me at your-now, but to assign the time of <your-now plus five minutes> to what is last Friday to me. It's just your abstract assignment of numbers to my worldline which concerns me not one hoot.

M(etaphysical): It is a very different thing to make a metaphysical statement that M1) I have an actual age, and M2) that this metaphysical designation of my current age is a function of non-local (or even local) events. I don't think you're making this (very idealistic) assertion, but your wording is open to interpretation. You calling it some kind of 'law of the land' suggests the latter interpretation. Your statement of "Time has to move to the future" suggests that you assert at least M1, in which case your CMIF method is completely wrong. The actual age of the distant person is whatever it actually is and only the one preferred coordinate that corresponds to reality, and any alternate 'according to X' coordinate system is simply wrong by definition.

[MikeFontenot (OP)]

It's now looking like to me that Einstein's 1907 method of defining an array of clocks for an accelerating observer agrees completely with the CMIF method. If so, that means that CMIF is the law of the land, no further assumptions required. I can also now see what the error in my proof (that negative ageing doesn't occur) was. Details to follow.

That was apparently wishful thinking on my part. I still can't find an error in my proof that negative ageing doesn't occur ... the error that I THOUGHT I saw didn't "pan out".

Then, I did an analysis which approximates instantaneous (and large) velocity changes (in both directions). When the acceleration was directed TOWARD the distant person (she), the analysis said she got much older during the acceleration ... consistent with the CMIF simultaneity method. But when the acceleration was directed AWAY from the distant person (she), the analysis said her age changed very little, and she didn't get younger ... which is NOT consistent with the CMIF simultaneity method. In fact, the those two results, taken together, are clearly inconsistent with the standard twin "paradox" scenario: suppose he keeps instantaneously changing his velocity, first accelerating toward her, then immediately away from her, over and over. According to my results using the array of clocks (and "helper friends", HF's), each of those accelerations TOWARD her produces a large increase in her age (according to him), but each of those accelerations AWAY from her produces NO decrease in her age, and very little change in her age at all. So he can make her be as old as he wants to, by continuing to do those paired accelerations, over and over. So when he quits doing all those paired accelerations, and then accelerates toward her one last time, and then coasts back to their reunion (with the usual slow ageing by her, according to him), his computation of her age at his arrival won't be what is required by the twin "paradox" result. (They HAVE to agree with each other about their respective ages at the reunion). So this analysis produces an absurd and inconsistent result, and it can't be correct. But Einstein's gravitational time dilation equation had been well-tested, so how could it produce an erroneous result when its acceleration equivalent (using the equivalence principle) is used? Is it possibly because the gravitational time dilation equation is not intended to be used with gravity being switched on and off, and therefore we can't use it to describe what happens when the observer switches his acceleration on and off?

Or maybe I've just made a mistake in my analysis somewhere. I'll give a few facts about my analysis that will allow anyone who wishes, to check my work. I use an acceleration of +-1 ls/s/s (equivalent to about 30.5 million g's), and the acceleration lasts only 1.317 seconds. That results in a velocity of +-0.866, which gives the convenient gamma = 2.0. When computing distances traveled by various observers, I integrate the velocity v with respect to t, the accelerating person's time. The relation between velocity v and "rapidity", theta, is v = tanh(theta). Theta is simply related to acceleration A and time t: theta = A t. (Rapidity is a non-linear equivalent of velocity: rapidity can vary from -infinity to +infinity, whereas velocity can't exceed the speed of light in magnitude.) The integral of tanh is equal to ln(cosh). But you can get a rough idea of the magnitude of the results by just integrating theta rather than v to get approximate distances.

[MikeFontenot (OP)]

I've had a new thought that is seeming "air tight" to me:

If some person (the "HF") is momentarily co-located with the home twin (she), the HF cannot possibly witness her age to instantaneously change at that instant, either positively of negatively. That would be an absurdity.

The distant accelerating observer (the "AO") is able to say that the HF always shares his (the AO's) notion of "NOW". Therefore the AO must also conclude that the home twin's age didn't instantaneously change (either positively or negatively) at that instant.

If the above argument is correct, then the CMIF simultaneity method can't be correct.

[MikeFontenot (OP)]

I've had a new thought that is seeming "air tight" to me:

If some person (the "HF") is momentarily co-located with the home twin (she), the HF cannot possibly witness her age to instantaneously change at that instant, either positively of negatively. That would be an absurdity.

The distant accelerating observer (the "AO") is able to say that the HF always shares his (the AO's) notion of "NOW". Therefore the AO must also conclude that the home twin's age didn't instantaneously change (either positively or negatively) at that instant.

If the above argument is correct, then the CMIF simultaneity method can't be correct.

The way the accelerating observer (the "AO") defines his "NOW" instant at distant locations comes directly from the gravitational time dilation equation, via the equivalence principle. It says that a "helper friend" (HF) who always is accelerating exactly as the AO is accelerating, with acceleration A, will age at a rate that is a fixed known ratio of the AO's rate. The given HF and the AO are always a constant distance "d" apart. If the chosen HF is BEHIND the AO (compared to the direction of the acceleration), that HF will age SLOWER than the AO by the factor exp(A d). To keep things as simple as possible, we can always let all of the HF's and the AO's ages be the same, immediately before they all start accelerating. Then the ratio of the age of the "behind" HF's age to the AO's age is just 1/exp(A d). And if, instead, another HF is AHEAD of the AO (compared to the direction of the acceleration), then the ratio of that "ahead" HF's age to the AO's age is just exp(A d). (Of course, different "behind" HF's will have different distances "d" to the AO, and likewise for the "ahead" HF's.) So, at some instant T in the AO's life, he computes that the original "behind" HF's current age is T/exp(A d). Or, alternatively, he computes that the "ahead" HF's current age is T exp(A d) The way he SELECTS the HF from among all possible HF's (both ahead and behind him) is such that the chosen HF is momentarily co-located with the home twin (her) at the instant the AO wants to know her current age.

So, if all of the above is correct, that allows the AO to construct an array of (effectively) synchronized clocks and helper observers attached to him, similar to what a perpetually-inertial observer can do, that can put an observer momentarily co-located with the distant twin (her) at the instant in the AO's life when he wants to know her current age. And in both the perpetually-inertial and the accelerated cases, it would be ABSURD for that momentarily co-located observer to observe a large and abrupt change in her age at that instant.

[MikeFontenot (OP)]

Because of the equivalence principle, it is also true that if two clocks that are separated by a fixed distance "d" ly are both accelerated with a constant equal acceleration of "A" ly/y/y, the trailing clock runs slower than the leading clock, by the factor exp(Ad).

You made the same assertion in June, the OP of this topic.
This makes no mention of gravity, so the equivalence principle is irrelevant.

I START with the "gravitational time dilation" equation.  Here's a link to a Wiki article on it:

 https://en.wikipedia.org/wiki/Gravitational_time_dilation

For a CONSTANT, UNIFORM gravitational field, the equation given in the Wiki article reduces to

 T_d(h) = exp(G h),

where T_d is the ratio of the tic rates of the two clocks, G is the strength of the gravitational force, and h is the constant separation of the two clocks.  The clock which is closest to the source of the gravitational field will tic slower than the other clock by the factor T_d.  (I have assumed that the units are such that the speed of light "c" is equal to 1, so for simplicity I don't include "c" in the equation.  I usually use units of years for time and lightyears for distance.

When the equivalence principle is applied to this GR equation, we get the SR equation

  T_d(h) = exp(A h),

where "A" is the constant acceleration that the two clocks are undergoing.  The clock in the rear tics at the slower rate.

Einstein was the first one, I believe, to derive these results, in 1907.
Here is a reference:

 https://einsteinpapers.press.princeton.edu/vol2-trans/315

Start with the bottom of that first page, labeled part V.

[MikeFontenot (OP)]

I've had a new thought that is seeming "air tight" to me:

If some person (the "HF") is momentarily co-located with the home twin (she), the HF cannot possibly witness her age to instantaneously change at that instant, either positively of negatively. That would be an absurdity.

The distant accelerating observer (the "AO") is able to say that the HF always shares his (the AO's) notion of "NOW". Therefore the AO must also conclude that the home twin's age didn't instantaneously change (either positively or negatively) at that instant.

If the above argument is correct, then the CMIF simultaneity method can't be correct.

I now see that the above argument is NOT correct. The fact that it is not correct DOESN'T prove that she never abruptly gets older, however. It may be true that she doesn't abruptly get older, but the above argument doesn't prove that.

I think the only way to prove that she doesn't abruptly get older, or to prove the opposite, is to make use of two DIFFERENT HF's. Immediately before the speed change, she is co-located with HF1. Immediately after the abrupt speed change, she is co-located with HF2. If she DOES get a lot older because of the speed change, HF2 will be far from HF1. If HF2 is close to Hf1, then she DIDN'T get much older. Which one of those outcomes actually happens isn't trivial to determine ... I think it can be done, but I think it will be complicated. It will require, among other things, looking at it from her perspective, and that is complicated by the fact that, according to her, the distances between the various HF's isn't constant. (Those distances ARE constant, according to the HF's and the AO.)

So, I've got a lot of work to do. But I HAVEN'T been able to find a flaw in my original proof that she can't abruptly get younger, according to the AO. So as of now, I think that proof is valid. If so, the CMIF simultaneity method is incorrect. And I also believe that if she can't abruptly get younger, then she can't abruptly get older either ... they either must both be true, or else they must both be false.

[Halc]

The fact that it is not correct DOESN'T prove that she never abruptly gets older

Couple of issues with that wording.
1) I agree that only if the AO's velocity is discontinuous can her age relative to his worldline also be discontinuous (older, younger, whatever. Distance matters not so long as it is nonzero). Since it requires infinite force for AO's velocity to be discontinuous, it isn't going to happen.
2) Your conclusion " she never abruptly gets older" is worded as if it was some kind of metaphysical fact instead of a relation to something. So say "she never abruptly gets older relative to AO's worldline, or AO's CMIF as you seem to prefer naming it. A statement of simultaneity without a frame reference is either an absolute statement or is meaningless, and given your posts, I'm never quite sure if you're not some kind of solipsistic absolutist.

I think the only way to prove that she doesn't abruptly get older

Technically, everybody is always 'getting older', so I think you need to be more precise in what you mean by 'abruptly'. It's not like we suddenly start doing it when we were not getting older before.
And again, lack of frame reference makes it sound like solipsistic absolutism.

Immediately before the speed change, she is co-located with HF1. Immediately after the abrupt speed change, she is co-located with HF2.

Ouch! Meaningless statements without a frame reference. You're talking I think about immediately after AO's speed change, meaning simultaneous at a distance with that speed change event. Simultaneity at a distance is meaningless without a frame reference. You have to learn to be clear about things, else we're all just guessing.

But I HAVEN'T been able to find a flaw in my original proof that she can't abruptly get younger, according to the AO.

Yay! A frame reference! She can get younger relative to AO, but not abruptly (instantly) due to my reasoning in point 1 above. She might lose 50 years in one of AO's seconds, but that's not abruptly.

[MikeFontenot (OP)]

The fact that it is not correct DOESN'T prove that she never abruptly gets older

Couple of issues with that wording.
[...]
2) Your conclusion " she never abruptly gets older" is worded as if it was some kind of metaphysical fact instead of a relation to something. So say "she never abruptly gets older relative to AO's worldline, or AO's CMIF as you seem to prefer naming it.

You are correct that I should have made it clear that I was referring to the conclusion of the AO.  I probably failed to do that because in my previous post that I was quoting, I had said: "Therefore the AO must also conclude that the home twin's age didn't instantaneously change (either positively or negatively) at that instant."  But I should have repeated that in my new post.  Sorry.

[MikeFontenot (OP)]

Instantaneous Velocity Changes in the Equivalence Principle Version of the Gravitational Time Dilation Equation

When using the CMIF simultaneity method, the analysis is GREATLY simplified by using instantaneous velocity changes, rather than finite accelerations that last for a finite amount of time.  So I decided to try using instantaneous velocity changes in the Equivalence-Principle Version of the Gravitational Time Dilation equation (the "EPVGTD" equation).  The result (assuming I haven't made a mistake somewhere) is unexpected and disturbing.  My analysis found that the age change of the HF, produced by an instantaneous velocity change by the AO and the HF, from zero to 0.866 lightseconds/second (ls/s), directed toward the home twin (her), is INFINITE!

I'll describe my analysis, and perhaps someone can find an error somewhere.

Before the instantaneous velocity change, the AO (he), HF, and the home twin (she) are all mutually stationary.  She and the HF are initially co-located, and the AO (he) is "d" lightseconds away from her and the HF.

I start by considering a constant acceleration "A" ls/s/s that lasts for a very short but finite time of "tau" seconds.  That acceleration over tau seconds causes the rapidity, theta, (which starts at zero) to increase to

  theta = A tau ls/s,

and so we get the following relationship:

  A = theta / tau.

We will need the above relationship shortly.

(Rapidity has a one-to-one relationship to velocity.  Velocity of any object that has mass can never be equal to or greater than the velocity of light in magnitude, but rapidity can vary from -infinity to +infinity.)

We want the velocity, beta, to be 0.866 ls/s after the acceleration.  Rapidity, theta, is related to velocity, beta, by the equation

  theta = arctanh (beta) = (1/2) ln [ (1 + beta) / (1 - beta) ].

("arctanh" just means the inverse of the hyperbolic tangent function.)

So velocity = 0.866 corresponds to a rapidity of about 1.317 ls/s.

The "EPVGTD" equation says that the acceleration A will cause the HF to age faster than the AO by the factor exp(A d), where d is the constant separation between the AO and the HF.

Note that the argument in the exponential exp(A d) can be separated like this:

  exp(A d) = [exp(d)] sup A,

where "sup A" means "raise the quantity exp(d) to the power "A" ".  The rationale for doing that is because the quantity exp(d) won't change as we make the acceleration greater and greater, and the duration of the acceleration shorter and shorter.  That will make the production of the table below easier.

The change in the age of the HF, caused by an acceleration "A" that lasts "tau" seconds is just

  tau [exp(d)] sup A,

because [exp(d)] sup A is the constant rate at which the HF is ageing, during the acceleration, and tau is how long that rate lasts.

But we earlier found that A = theta / tau, so we get

  tau [exp(d)] sup {theta / tau}

for the change in the age of the HF due to the short acceleration.  So we have an expression for the change in the age of the HF that is a function of only the single variable tau ... all other quantities in the equation (d and theta) are fixed.  We can now use that equation to create a table that shows the change in the age of the HF, as a function of the duration of the acceleration (while keeping the area under the acceleration curve constant).

In order to make the table as easy to produce as possible, I chose the arbitrary value of the distance "d" to be such that

  exp(d theta) = 20000.

Therefore we need

  ln[ exp (d theta) ] = d theta = ln (20000) = 9.903,

and since theta = 1.317, d = 7.52 lightseconds.

If we were creating this table for the CMIF simultaneity method, we would find that as the duration of the acceleration decreases (with a corresponding increase in the magnitude of the acceleration, so that the product remains the same), the amount of ageing by the HF approaches a finite limit.  I.e., in CMIF, eventually it makes essentially no difference in the age of the HF when we halve the duration of the acceleration, and make the acceleration twice as great.

But here is what I got for the EPVGTD simultaneity method:

(in the table, "10sup4" means "10 raised to the 4th power".)

  tau     |    (tau) (2000)sup(1/tau)

____________________________

  1.0     |     2x10sup4 = 20000

  0.5     |     2x10sup8

  0.4     |     2.26x10sup10

  0.3     |     6.3x10sup13

  0.2     |     0.64x10sup21

  0.1     |     1.02x10sup42

  0.01   |     1.27x10sup428

  0.001 |     ? (My calculator overflowed at 10sup500)

Clearly, for the EPVGTD simultaneity method, the HF's age goes to infinity as the acceleration interval goes to zero.  That seems like an absurd answer to me.  And it is radically different from what happens with CMIF simultaneity, where the HF's age quickly approaches a finite limit as tau goes to zero.

[Halc]

The result (assuming I haven't made a mistake somewhere) is unexpected and disturbing.

If the result is nonsense, maybe that assumption is best not made. I always assume the mistake is with me if I get inconsistent results.

My analysis found that the age change of the HF, produced by an instantaneous velocity change by the AO and the HF, from zero to 0.866 lightseconds/second (ls/s), directed toward the home twin (her), is INFINITE!

The equation you're using gives the rate of age change relative to AO, but you are interpreting that as an age change, not a rate of age change. Simple as that. The difference in velocity (.866 in this case) is irrelevant since the same answer is computed from any abrupt positive change in velocity.

If you insist on using this formula for an abrupt change in velocity, do it as a limit as the time to change velocity by .866c approaches zero. But don't plug in zero to something that just yields NaN.

I stand unconvinced that it will yield a correct value since the formula is a local one being used in a non-local way.

[MikeFontenot (OP)]

The equation you're using gives the rate of age change relative to AO, but you are interpreting that as an age change, not a rate of age change.

No, I DID compute the HF's total age change during the acceleration.  If you look back at my post, you'll see that I said:

"The "EPVGTD" equation says that the acceleration A will cause the HF to age faster than the AO by the factor exp(A d), where d is the constant separation between the AO and the HF."

So there, I WAS talking about the rate.  But then I said:

"The change in the age of the HF, caused by an acceleration "A" that lasts "tau" seconds is just

  tau [exp(d)] sup A"

So there I AM talking about the total age change of the HF:  I multiplied the rate by the amount of time "tau" that he was ageing at that rate.