[Halc]

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"

Which is zero times NaN which is NaN, not infinity. tau is zero for an instantaneous velocity change.

And my 2nd criticism also still stands.

[MikeFontenot (OP)]

Instantaneous Velocity Changes in the Equivalence Principle Version of the Gravitational Time Dilation Equation - Revised Model (the LGTD Model)

_______________________________________________________

I repeated my previous analysis of the instantaneous increase in the home person's (her) age (according to the accelerating person, AO, him), according to the Equivalence Principle Version of the Gravitational Time Dilation Equation, (the "EPVGTD" equation), and replaced it with the new equation, which I'll call the "Linearized Gravitational Time Dilation Equation", (the "LGTD" equation).  I simply replace the exponential exp(A d) with the quantity (1 + A d).  (This is the same approximation that Einstein used in his 1907 paper).  In what follows below, I'll repeat each affected calculation that I made in my last post, and show the revised calculation.

[...]

[Previous]:

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.

[Revised]:

The "LGTD" equation says that the acceleration A will cause the HF to age faster than the AO by the factor (1 + A d), where d is the constant separation between the AO and the HF.

(Both of the above are for the case where the AO accelerates TOWARD the unaccelerated person (her).)

[...]

[Previous]:

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.

[Revised]:

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

 tau (1 + A d),

because (1 + A d) is the constant rate at which the HF is ageing, during the acceleration, and tau is how long that rate lasts.


[Previous]:

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

  tau [exp(d)] sup {theta / tau}

[Revised]:

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

 tau (1 + [ ( theta d ) / tau ]  =  tau + (theta d)

[...]

It is still true that d = 7.52 lightseconds and theta = 1.317.

Therefore the revised result is that the change in HF's age during the acceleration is equal to

  tau + ( theta d ) = tau + (1.317)(7.52) = tau + 9.904.

So, in the revised model, as tau approaches zero (to give an instantaneous velocity change), the change in the HF's age during the speed change approaches 9.904 seconds from above.  So the HF's age increased by a finite amount, unlike the infinite increase that the EPVGTD equation gave.

Before the instantaneous velocity change, the AO, the HF, and the home twin (she) were all the same age.  She and the HF were co-located.  So after the instantaneous speed change, the AO hasn't aged at all, but the HF is 9.904 seconds older than he was before the speed change, according to the AO.  And since she and the HF have been colocated during the instantaneous speed change, they couldn't have ever differed in age during the speed change ... it would be absurd for either of them to see the other have an age different from their own age at any instant.  So after the instantaneous speed change, the AO must conclude that she and the HF both instantaneously got 9.904 seconds older than they were immediately before the speed change.

By comparison, the CMIF simultaneity method says that the AO will conclude that her age instantaneously increases by 6.51 seconds, so the LGTD and CMIF don't agree.

[MikeFontenot (OP)]

Wait. The HF spends 9.9 proper seconds (years, units, whatever) accelerating to .866c and you assert that he's still co-located with her after that?  Try again.

The speed change is instantaneous.  Neither she nor the HF move during the instantaneous speed change.  And she and the HF are colocated immediately before the instantaneous speed change.

[MikeFontenot (OP)]

I just repeated my previous analysis of instantaneous velocity changes in the "linearized" (LGTD) version of the equivalence principle version of the gravitational time dilation equation, but for the case where the instantaneous velocity change is AWAY FROM the home twin (her). The result is exactly like the previous result, except that she instantaneously gets YOUNGER, not older. (This contradicts my previous possible proof that negative ageing doesn't occur.)

Below, I'll repeat the previous calculations, and show the changes.

[Previous]:

I simply replace the exponential exp(A d) with the quantity (1 + A d).

[New]:

I simply replace the exponential exp(-A d) with the quantity (1 - A d).

[Previous]:

The "LGTD" equation says that the acceleration A will cause the HF to age FASTER than the AO by the factor (1 + A d), where d is the constant separation between the AO and the HF.

(The above is for the case where the AO accelerates TOWARD the unaccelerated person (her).)

[New]:

The "LGTD" equation says that the acceleration A will cause the HF to age SLOWER than the AO by the factor (1 - A d), where d is the constant separation between the AO and the HF.

(The above is for the case where the AO accelerates AWAY FROM the unaccelerated person (her).)

[...]

[Previous]:

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

tau (1 + A d),

because (1 + A d) is the constant rate at which the HF is ageing, during the acceleration, and tau is how long that rate lasts.

[New]:

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

tau (1 - A d),

because (1 - A d) is the constant rate at which the HF is ageing, during the acceleration, and tau is how long that rate lasts.


[Previous]:

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

tau (1 + [ ( theta d ) / tau ] ) = tau + (theta d)

[New]:

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

tau (1 - [ ( theta d ) / tau ] ) = tau - (theta d)

[both Previous and New]:

It is still true that d = 7.52 lightseconds and theta = 1.317.

[Previous]:

Therefore the revised result is that the change in HF's age during the acceleration is equal to

tau + ( theta d ) = tau + (1.317)(7.52) = tau + 9.904.

[New]:

Therefore the revised result is that the change in HF's age during the acceleration is equal to

tau - ( theta d ) = tau - (1.317)(7.52) = tau - 9.904.

[Previous]

So, in the revised model, as tau approaches zero (to give an instantaneous velocity change), the change in the HF's age during the speed change approaches 9.904 seconds from above. So with an instantaneous velocity change, the HF's age INCREASED instantaneously by a finite amount.

[New]

So, in the revised model, as tau approaches zero (to give an instantaneous velocity change), the change in the HF's age during the speed change approaches -9.904 seconds from above. So with an instantaneous velocity change, the HF's age DECREASED instantaneously by a finite amount.

[Previous]:

Before the instantaneous velocity change, the AO, the HF, and the home twin (she) were all the same age. She and the HF were co-located. So after the instantaneous speed change, the AO hasn't aged at all, but the HF is 9.904 seconds OLDER than he was before the speed change, according to the AO. And since she and the HF have been colocated during the instantaneous speed change, they couldn't have ever differed in age during the speed change ... it would be absurd for either of them to see the other have an age different from their own age at any instant. So after the instantaneous speed change, the AO must conclude that she and the HF both instantaneously got 9.904 seconds OLDER than they were immediately before the speed change.

By comparison, the CMIF simultaneity method says that the AO will conclude that her age instantaneously increases by 6.51 seconds, so the LGTD and CMIF don't agree.

[New]:

Before the instantaneous velocity change, the AO, the HF, and the home twin (she) were all the same age. She and the HF were co-located. So after the instantaneous speed change, the AO hasn't aged at all, but the HF is 9.904 seconds YOUNGER than he was before the speed change, according to the AO. And since she and the HF have been colocated during the instantaneous speed change, they couldn't have ever differed in age during the speed change ... it would be absurd for either of them to see the other have an age different from their own age at any instant. So after the instantaneous speed change, the AO must conclude that she and the HF both instantaneously got 9.904 seconds YOUNGER than they were immediately before the speed change.

By comparison, the CMIF simultaneity method says that the AO will conclude that her age instantaneously decreases by 6.51 seconds, so the LGTD and CMIF don't agree.

[MikeFontenot (OP)]

So, what to make of all these different and contradictory results?

The "EPVGTD Equation" (the one with the exponential), says that, if the AO (he) instantaneously changes his velocity in the direction TOWARD the home time (her), she instantaneously gets INFINITELY older, according to him.  That's nonsense, because it gives incorrect ages for the twins when they are reunited.

On the other hand, when he instantaneously changes his velocity in the direction AWAY from her, the EPVGTD equation says that her age doesn't change instantaneously.  While it's not certain that that result itself is incorrect, it seems to result in an inconsistency at the reunion. The EPVGTD equation says that, with zero acceleration, he and all the HF's age at the same rate.  That seems to require that on the outbound and inbound legs, his conclusion about the correspondence between his and her ages must be the same.  And clearly, on the OUTBOUND leg, he MUST say she is ageing SLOWER than he is, by the factor gamma.  So he must say that, on the INBOUND leg, she is ageing slower by the factor gamma.   But in that case, their conclusions about the correspondence between their ages at the reunion won't be consistent: she says she is the OLDER, but he says she is the YOUNGER. So his conclusions won't match her conclusions at the reunion, which is impossible since they are colocated then and they MUST agree about the correspondence between their ages then.

So much for the EPVGTD equation. What about the LGTD equation? The linearized equation (the LGTD equation) gives results that are qualitatively similar to the CMIF simultaneity method: her age instantaneously changes, according to him, during his instantaneous velocity change (instantaneously increasing when his momentarily infinite acceleration is TOWARD her, and instantaneously decreasing when his momentarily infinite acceleration is AWAY FROM her).  But the AMOUNT of the instantaneous change is greater than CMIF says it should be.  It is interesting that the amount of the instantaneous age changes would be exactly the same for CMIF and LGTD if the linearized equation multiplied the distance "d" by the velocity "v", rather than by the rapidity "theta".  But, in determining the velocity effect obtained by integrating the acceleration "A", it IS necessary to use the rapidity "theta", not the velocity "v", as the variable of integration. (Taylor and Wheeler go over this in detail).

WHY does the EPVGTD equation fail so miserably in this example? Isn't the GTD equation a well-established result in general relativity?  And the equivalence principle is certainly well-established.  Is the GTD equation WRONG?

And WHY goes the LGTD work better than the EPVGTD, at least qualitatively?  The LGTD should be a justified approximation of the EPVGTD only when the argument (A d) is small, and an infinite "A" (even though it lasts only an infinitesimal time) certainly isn't small!  The LGTD equation shouldn't give results that are even qualitatively correct, but it does. Why?

[Halc]

The answers to your problems have already been posted, and perhaps you should read them rather than dismiss them. I will repeat them, attempting to flesh it out for additional clarity.

The HF spends 9.9 proper seconds (years, units, whatever) accelerating to .866c and you assert that he's still co-located with her after that?  Try again.

The speed change is instantaneous.

Mostly meaningless statement. OK, AO's speed change is instantaneous relative to AO or to the UO (unaccelerated observer, or 'she'), but not relative to HF.

A statement of simultaneity without a frame reference is either an absolute statement or is meaningless

Your statement lacked a reference, and what is going on would become much more clear to you if you would include them. A meaningful statement would be that the speed change is instantaneous relative to AO. It is not instantaneous relative to HF for instance since by definition HF and AO are always stationary in their proper frame, and HF takes t=9.9 to complete the acceleration.

Neither she nor the HF move during the instantaneous speed change.

Again no references, so again meaningless. To reword:
Relative to AO's ARF (accelerated reference frame), HF does not move because his proper separation from AO would change if he did move. Likewise AO does not move relative to HF's ARF (the same ARF). Relative to that ARF, the UO very much does move since she moves to a point halfway (3.76) between AO and HF and ages 6.51 in the process. So HF and UO do not remain colocated.
Relative to HF there is t=9.9 of constant proper acceleration away from AO and UO after which AO is still 7.52 behind and UO is 3.76 behind.
Relative to UO's inertial frame, UO is stationary and nobody goes anywhere during the AO's instantaneous acceleration event.  But HF commences acceleration away from AO, so over time he departs company from UO.

So things are quite different depending on the point of view. AO and HF always share the same ARF but different potentials in it, so their time does not remain synced.

So, what to make of all these different and contradictory results?

The problems in this post are mostly using correct equations in inappropriate situations.

The "EPVGTD Equation" (the one with the exponential), says that, if the AO (he) instantaneously changes his velocity in the direction TOWARD the home time (her), she instantaneously gets INFINITELY older, according to him.

No, it says undefined, not infinite. This is correct since the inputs are acceleration and duration instead of a celerity change. The former do not define a unique change in celerity.

The linearized equation (the LGTD equation) gives results that are qualitatively similar to the CMIF simultaneity method: her age instantaneously changes, according to him, during his instantaneous velocity change

This is what I mean by using it inappropriately. The LGTD equation is for comparing two locations at constant proper separation in an accelerated reference frame, and she (UO) does not do this, so the equation simply does not apply. It only works for HF who does maintain this constant proper separation per you specification. He’s at the front end of a rigid ship with AO at the rear, accelerating past UO.  After t=6.51 of UO, she sees the midpoint of the ship go by at .866c and the abrupt cessation of acceleration of the thing.

UO is inertial, and thus the equation appropriate for her (relative to AO) is the CMIF maths just like we do with the twins scenario.

WHY does the EPVGTD equation fail so miserably in this example?

It was spot on actually, but only applicable to what HF does relative to AO.

[MikeFontenot (OP)]

The complete title should be:
  "An Inconsistency Between the Gravitational Time Dilation Equation and the Twin Paradox"

_______________________________________________

I've pulled together the material I've posted recently on the "A possible proof" thread about the gravitational time dilation equation being inconsistent with the outcome at the reunion of the twin paradox, and I've put it on the viXra repository:

http://viXra.org/abs/2109.0076?ref=12745236

Here is the title and abstract:

"An Inconsistency Between the Gravitational Time Dilation Equation and the Twin Paradox"

Abstract:

It is shown in this monograph that the Gravitational Time Dilation Equation, together with the well-known Equivalence Principle relating gravitation and acceleration, produce results that contradict the required outcome at the reunion of the twins in the famous twin ‘paradox’. The Equivalence Principle Version of the Gravitational Time Dilation Equation (the “EPVGTD” equation) produces results that say that, when the traveling twin (he) instantaneously changes his velocity, in the direction TOWARD the distant home twin (her), that he will conclude that her age instantaneously becomes INFINITE. It is well known that, according to her, at their reunion, she will be older than him, but both of their ages will be FINITE. The twins clearly MUST be in agreement about their respective ages at the reunion, because they are co-located there.


<amazon sales plug removed>

When my manuscript comes up, you can click on "Look Inside", and on the back cover, there is a small photo of yours truly, impersonating an old codger.

The remaining question that I'd like the answer to, is "Has the gravitational time dilation equation been experimentally confirmed for arguments of the exponential function that are large enough to test the non-linear portion of the exponential curve?"  I.e., is it possible that the "modified linearized Gravitational Time Dilation equation (the MLGTD equation, in Section 7 of my monograph) is the correct GDT equation, and that the exponential version is incorrect?

[MikeFontenot (OP)]

Anybody is free to use any coordinate system they want, and if they both choose the same coordinate system, then they'll agree about their respective ages at all times. So for instance, when I'm in the car traveling to Grandma's house, I'm moving relative to Grandma, but both of us probably choose the coordinate system of the ground below us and hence are not in disagreement about our respective ages even while spatially separated.

At any given instant "tau" in the
life of a given INERTIAL observer (he), it's clear that there is just a
single answer to the question "How old is that particular distant person
(she) right now (at the given time "tau" in the life of the inertial
observer): it is what the particular "Helper Friend" (HF) who happens to be momentarily co-located with the distant person (she), says it is, at the instant when he is age "tau". The only way there could be any other
allowable answer is if the synchronization of the clocks isn't valid,
and that is impossible if the velocity of light in that inertial
reference frame is equal to the universal constant "c".

My argument above is that, IF those clocks are synchronized (according to the given observer), then he can't help but conclude that the current age of that distant person IS completely meaningful TO HIM.  And the only way that those clocks AREN'T synchronized according to him, is if the velocity of light in his inertial reference frame ISN'T equal to the universal constant "c".  But the fundamental assumption of special relativity IS that light will be measured in all inertial reference frames to have the value "c".  Therefore, FOR any given inertial observer (he), the current age of a distant person is completely meaningful to him.

But what about a non-inertial observer?  In particular, what about a
given observer who is undergoing a constant acceleration?  What does HE say the current age of a distant person is?  It turns out to be possible for such an accelerating observer to rely on an array of clocks and associated "helper friends" (HF's) to give him the answer.  Unlike in
the inertial case, those clocks DON'T run at the same rate.  But the
ratio of the rates of those clocks can be CALCULATED by the given
observer.  And if he (and the HF's) are initially stationary and
unaccelerated, they can start out with synchronized clocks (and ages).
Then, if they all fire their identical rockets at the same instant, they
can each CALCULATE the current reading of each of the other clocks, at each instant in their lives.  The calculations of each of the HF's all
agree.  So, at any instant in their lives during that acceleration, they
each share the same "NOW" instant with all of the other HF's.  That
means that the given observer (he), at any instant "tau" in his life,
can obtain the current age "T" of some distant person (her), by asking
the HF, who happens to be momentarily co-located with her at that NOW instant, what her age is then.

[MikeFontenot (OP)]

In the above post, I said that the given accelerating observer (he) (abbreviated, the "AO"), at each instant of his life, can CALCULATE the current reading on each of the HF's clocks.  What IS that calculation?

Let t = 0  be the reading on his clock at the instant that the constant acceleration "A" begins, and let all the HFs' clocks also read zero at that instant.  Thereafter, he and all of the HFs are accelerating at "A" ls/s/s, and the ratio R of any given HF's clock rate to his (the observer's (he) whose conclusions we are seeking) clock rate is

  R(t)  =  [ 1 +-  L  A  sech^2 (A t) ],

where L is the constant distance between him and the given HF, and sech() is the hyperbolic secant (which is the reciprocal of cosh(), the hyperbolic cosine).  The "^2" after the sech indicates the square of the sech.  The "+-" in the above equation means that the second term is ADDED to 1 for the HF's who are LEADING the accelerating observer, and the second term is SUBTRACTED from 1 for the HF's who are TRAILING the accelerating observer.  For brevity, I'll just take the case where the HF of interest is a leading HF.

The limit of R(t), as "t" goes to zero, is 1 + L A.  The limit of R(t), as "t" goes to infinity, is 1.0  So R(t) starts out at some positive number greater than 1, and then approaches 1.0 as t goes to infinity. So eventually, all the clocks essentially tic at the same rate, but early in the acceleration, the ratio of the tic rates varies significantly with time.

The current reading of the HF's clock (the "Age Change" or "AC"), when the AO's clock reads "tau", is

  AC(tau)  =  integral, from zero to tau, of { R(t) dt }

                 =  tau  +  L tanh( A tau ).

The above result depends on the fact that

  sech^2(u) = d{tanh(u)} / d{u}.

As tau goes to zero, AC goes to zero.  As tau goes to infinity, AC goes to tau + L, which goes to infinity, approaching a slope of 1.0 from above.

So there you have it.  That's the calculation that defines "NOW" for the AO and all of the HF's, and makes simultaneity at a distance a meaningful concept for them.  Simultaneity at a distance is not a choice.

[MikeFontenot (OP)]

So there you have it.  That's the calculation that defines "NOW" for the AO and all of the HF's, and makes simultaneity at a distance a meaningful concept for them.  Simultaneity at a distance is not a choice.

But what does the above say about the current age of the home twin (she), according to the traveling twin (he), for each instant in his life on his trip? The answer is that the above equations give the same results as the Co-Moving-Inertial-Frames (CMIF) simultaneity method. That is very fortuitous, because the CMIF method is relatively easy to use. The value of the array of clocks discussed above (which establish a "NOW" moment for the accelerating observer that extends throughout all space) is that they GUARANTEE that the CMIF results are fully meaningful to the traveler, and that the CMIF method is the ONLY correct simultaneity method for him. He has no other choice.