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

I discovered a mistake in my latest paper. I had concluded in that paper that my new method says that as the duration "tau" of the constant acceleration "A" goes to infinity, the tic rate ratio "R" of the separated clocks will go to 1.0 ... i.e., eventually the two separated clocks will tic at the same rate.

That result is correct.

But experimental results many years ago found that two (unaccelerated) vertcally-separated clocks stationary at or near the Earth's surface in the Earth's gravitational field run at slightly different rates, in agreement with the existing gravitational time dilation equation (using the version of the equation for a field that varies with height).

I had concluded in my paper that EITHER (1) those experimental measurements were flawed, or (2) the equivalence principle was incorrect, or (3) I had made a mistake somewhere.

I now know that my results DON'T contradict those experimental measurements. My results (which apply to a special relativity situation (without any gravitational fields)) apply to two clocks that each undergo EXACTLY the same constant acceleration "A" for the entire duration "tau". But in the case of the experimental measurements (conducted in the Earth's gravitational field), the two clocks have slightly different gravitational fields applied to them, because of the inverse-square dependence of the field strength on their distance from the center of the Earth. The clock higher in the field experiences a slightly weaker field.

So my special relativity scenario, and the scenario of the experimental measurements, AREN'T equivalent, and therefore the equivalence principle doesn't apply to them. Therefore the results in my paper DON'T contradict the experimental measurements.

Accordingly, I have revised my paper, on both the viXra repository and on Amazon.

Mike;

Here is a graphical proof for the simple 'twin' scenario. The red curve indicates a constant unit of time for a range of speeds less than c (blue).
All clocks are identical, the straight lines are an approximation to small accelerations.

The universal U frame is observing ship A moving at constant velocity v, while a probe leaves A at Ut=0 at constant velocity w>v, and returns at Ut=1.
The reunion occurs at At<1, the difference in green.
If the motion of B was extended without the reversal, Bt<1, the greater difference in green. The B clock indicates less time than the A clock.
Another approach is to consider A and B reunite at At=tr. For the interval tr, the x distance for B is > the x distance for A. If B moved a greater distance in the same time interval, it had to move faster than A for some part of its trip. SR predicts if a clock moves faster it's rate decreases, accumulating less time.
In summation, any departure from a constant velocity results in time dilation.

I could not see a GR 'twin' scenario. Time dilation is a function of speed in SR and a function of position in GR, two different phenomena.

My suggestion, do more research before your proposal. We have benefited much in history because of inquiring minds.

I have just put a 2nd revised edition of my paper on viXra:

https://vixra.org/abs/2201.0015?ref=13239788

and on Amazon, which you can find there by searching on my full name: Michael Leon Fontenot . (That search will pop up my three papers ... the new one has a yellow/brown cover, as opposed to the green cover of the first one, and the red cover on the second one.)

I had originally computed the age change of the "Helper Friend" (the "HF"), during the constant acceleration "A" (which lasts "tau" units of time), by doing a numerical integration of the Rate Ratio "R(tau)". But I recently discovered that the integral could be done analytically, and it gives a surprisingly simple result: the age change is just

AC(tau) = tau + { L * tanh(A tau)},

where "*" denotes multiplication. The quantity "tanh(A tau)" is just the velocity "v" of the accelerating observer (the "AO") and the "HF", which makes the AC equation similar to the "delta_CADO" equation that I derived many years ago.

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.

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.

In the above, 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.

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.

In 1907, Einstein published a VERY long paper (in several volumes) on his "relativity principle". In volume 2, section 18, page 302, titled "Space and time in a uniformly accelerated reference frame", he investigated how the tic rates compare for two clocks separated by the constant distance L, with both clocks undergoing a constant acceleration "A". He restricted the analysis to very small accelerations (and very small resulting velocities). His result (on page 305) was that the leading clock tics at a rate

R = 1 + L A

faster than the rear clock. Note that that result agrees with my equation, for very small "L" and "A". But he then said:

"From the fact that the choice of the coordinate origin must not affect the relation, one must conclude that, strictly speaking, equation (30) should be replaced by the equation R = exp(L A). Nevertheless, we shall maintain formula (30)."

I've never understood that one sentence argument he gave. But I DID assume he was right (because he was rarely wrong), until I tried applying his equation to the case of essentially instantaneous velocity changes that are useful in twin "paradox" scenarios in special relativity. Specifically, I worked a series of examples where the separation of the two clocks is always

L = 7.52 ls (lightseconds)

and where the final speed (with the initial speed being zero) is always

v = 0.866 ls/s.

That speed implies a "rapidity" of

theta = atanh(0.866) = 1.317 ls/s.

But

theta = A tau,

where tau is the duration of the acceleration (according to the rear clock). So

A = theta / tau.

I then do a sequence of calculations, each starting at t = 0, with the two clocks reading zero, and with zero acceleration for t < 0.

First, I set the duration tau of the acceleration to 1 second. The acceleration then needs to be

A = theta / tau = 1.317 / 1.0 = 1.317 ls/s/s (that's roughly 40 g's).

For that case, the CURRENT reading on the leading clock (which I'll denote as AC) is

AC = tau R = 2x10sup4 = 20000

(where 10sup4 means "10 raised to the 4th power").

I then start over and work a second case, with ten times the acceleration (13.17 ls/s/s), but with tau ten times smaller (0.1 second). That keeps the final rapidity the same as in the first case, and the final speed is also 0.866, as before. For the second case,

AC = 1.02x10sup42.

So when we made the acceleration an order of magnitude larger, and the duration an order of magnitude smaller, the current reading "AC" on the leading clock got about 38 orders of magnitude larger.

Next, I start over again and work a third case, again increasing the acceleration by a factor of 10, and the decreasing the duration by a factor of 10, so "A" = 131.7 ls/s/s and tau = 0.01 second. Then, AC = 1.27x10sup428. So this time, when we increased "A" by a factor of 10, and decreased tau by a factor of ten, AC got about 380 orders of magnitude larger.

AC is not approaching a finite limit as tau goes to zero and "A" goes to infinity. In each iteration, the change in AC compared to the previous change gets MUCH larger. Clearly, the clock reading is NOT converging to a finite limit. It is going to infinity as tau goes to zero.

So, for the idealization of an instantaneous velocity change, the change of the reading on the front clock is INFINITE. That means that, when the traveling twin instantaneously changes his speed from zero to 0.866 (toward the home twin), the exponential version of the R equation says that the home twin's age becomes infinite. But we know that's not true, because the home twin is entitled to use the time dilation equation for a perpetually-inertial observer, and that equation tells her that for a speed of 0.866 ls/s, the traveler's age is always increasing half as fast as her age is increasing. So when they are reunited, she is twice as old he is, NOT infinitely older than he is, as the exponential form of the gravitational time dilation equation claims. The time dilation equation for a perpetually-inertial observer is the gold standard in special relativity. Therefore the exponential form of the gravitational time dilation equation is incorrect.

The correct gravitational time dilation equation turns out to approximately agree with what Einstein used in his "small acceleration" analysis, for very small accelerations, but differs substantially for larger accelerations. And the correct gravitational time dilation equation agrees with the ages of the twins when they are reunited. It also exactly agrees with the CMIF simultaneity method for the traveler's conclusions about the sudden increase in the home twin's age when the traveler suddenly changes his velocity. The CMIF method provides a practical way to compute the change in the home twin's age when the traveler instantaneously changes his velocity. But it is the new gravitational time dilation equation, and its array of clocks with a common "NOW" moment, that guarantees that the CMIF result is fully meaningful to the traveling twin, and that the CMIF method is the ONLY correct simultaneity method for the traveling twin.

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 above is all correct.

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 ISN'T correct. The designated "AO" (he) (the given accelerating observer whose conclusions about the current age of the "distant person" we want) can indeed calculate the current time on each of the HF's clocks at each instant in his life. And that establishes a "NOW" instant FOR HIM that extends throughout all of space, and tells him the current age of the distant person at that instant in his life (which IS the goal). But it does NOT establish the same "NOW" instant, according to each of the HF's. The "NOW" calculations of the HF's don't agree, either among themselves, or with the AO. But that doesn't matter. All that matters is that it establishes a "NOW" instant for the given AO, and tells him what the current age of the distant person is.

Michael Leon Fontenot

Mike;

The twins example.
Blue lines define a light cone for each observer. A sphere of light containing images converges on each observer at each apex, which has been simplified to circles in an x-y plane. Awareness of events for A and B consists of all events contained within their light cones. Gray lines are for measurement. a green line is an axis of simultaneity.

U is a 3rd observer of A with no relative velocity and B with v=.866.
Graphic-1:
A's last image of the B-clock shows Bt=11.3, at At=22.6.
B's last image of the A-clock shows At=5.3, at Bt=11.3.
Graphic-2:
After the 180 deg reversal at Bt=20, he continues to receive clock images from A.
B receives A-clock images (1 to 80) during his (1 to 40).

B's motion does NOT alter the A-clock rate. It does alter his perception of the A clock rate, slower out bound, and faster inbound. That is Doppler shift which has nothing to do with aging, which is the accumulation of time.
Any 'time jump' results from misinterpreting the aos as an independently existing thing,
when it is in reality the product of a measurement.

The instantaneous reversal lasts for a time interval of 0, thus produces no effect other than changing direction, and can be physically performed with two observers substituted for B.

The issue here is communication. A and B can speculate as to the state of each other, but until measurements/inquiries are made, they can't know (record as a fact).
Like the box, you can't know what's in it until you look.

Considering the local GPS system requires daily adjustments, any astronomical system of synchronized clocks is fantasy.
Your idea would have been acceptable in the pre-Newton era of instantaneous light speed and universal time.
After thousands of years, all things thinkable are not realizable.

The correct gravitational time dilation equation turns out to approximately agree with what Einstein used in his "small acceleration" analysis, for very small accelerations, but differs substantially for larger accelerations. And the correct gravitational time dilation equation agrees with the ages of the twins when they are reunited. It also exactly agrees with the CMIF simultaneity method for the traveler's conclusions about the sudden increase in the home twin's age when the traveler suddenly changes his velocity. The CMIF method provides a practical way to compute the change in the home twin's age when the traveler instantaneously changes his velocity. But it is the new gravitational time dilation equation, and its array of clocks with a common "NOW" moment, that guarantees that the CMIF result is fully meaningful to the traveling twin, and that the CMIF method is the ONLY correct simultaneity method for the traveling twin.

Are you saying that the full and correct gravitational time dilation equation matches the CMIF method for both the accelerating "away" and the accelerating "toward" scenarios? If so, that is great news, but I must admit I would find that rather surprising. I had never heard of any equivalence between gravitational time dilation and distant simultaneity conventions before.

For the basic twins case that you and I discussed a while ago, when the traveling twin (he) was 20 years old he had said that the stay-home twin (she) was 10 years at that time (for v=+0.866c). Then if he quickly decelerated and stopped (v=0.000c) he would say her age changed to 40 years old. And then if he quickly accelerated toward her (v=-0.866c) he would say her age changed to 70 years old. He could quickly stop again (v=0.000c) and he would say her age changed to 40 years old again (what you call negative aging as it went from 70 to 40). He could then quickly accelerate away again (v=+0.866c) and he would again say her age changed to 10 years old again (what you call negative aging again, as it went from 40 to 10). All of this is the perfectly valid CMIF method of course.

Are you saying you can replicate the negative aging using the full and correct gravitational time dilation equation?

Are you saying that the full and correct gravitational time dilation equation matches the CMIF method for both the accelerating "away" and the accelerating "toward" scenarios? If so, that is great news, but I must admit I would find that rather surprising. I had never heard of any equivalence between gravitational time dilation and distant simultaneity conventions before.

For the basic twins case that you and I discussed a while ago, when the traveling twin (he) was 20 years old he had said that the stay-home twin (she) was 10 years at that time (for v=+0.866c). Then if he quickly decelerated and stopped (v=0.000c) he would say her age changed to 40 years old. And then if he quickly accelerated toward her (v=-0.866c) he would say her age changed to 70 years old. He could quickly stop again (v=0.000c) and he would say her age changed to 40 years old again (what you call negative aging as it went from 70 to 40). He could then quickly accelerate away again (v=+0.866c) and he would again say her age changed to 10 years old again (what you call negative aging again, as it went from 40 to 10). All of this is the perfectly valid CMIF method of course.

Are you saying you can replicate the negative aging using the full and correct gravitational time dilation equation?

Yes, to all your above questions.

Yes, to all your above questions.

Wow! Now that is fascinating! What a great result for you to find, after working so long on this. Are the calculations simple enough that you could post them here? I think it would be fantastic to see the actual 10 years old, then 40 years old, then 70 years old calculations, and the acceleration direction reversed to show the calculations as 70 years old, then 40 years old, then 10 years old. That would be amazing.

In all of the "Twin scenario explained by GR" websites I have ever found, I don't think I've ever seen an actual calculation that I could use to test different directions. It is usually just explained verbally as something like, "When the traveling twin accelerates, it is as if a pseudo-gravitational force arises, and this causes the distant twin to age rapidly" or some such thing. I never even imagined that approach could be used to show the effect that you call negative aging.

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[...]
Are the calculations simple enough that you could post them here?
[...]

If you look back at post #23, it gives this equation:

AC(tau) = tau + { L * tanh(A tau)},

where "AC" is the age change of the helper friend (HF) near the home twin. The function tanh() has the same sign as its argument, so for negative "A" (the case where the acceleration is directed AWAY from the home twin), the quantity in curly brackets is negative. So as tau goes to zero, the age change AC is negative.

But using the CMIF method itself is still the easiest and simplest way to compute the home twin's age (according to the accelerating twin). The most important value of the gravitational time dilation equation is that it allows us to (at least mentally) construct an array of clocks and associated HF's that establish a "NOW" instant, that extends throughout all space, for the given accelerating observer (the AO). And that guarantees that the results of the CMIF simultaneity method are fully meaningful to the AO. And it also guarantees that the CMIF method is the ONLY correct simultaneity method for him. He has no other choice.

This array of clocks does for an accelerating observer what Einstein's array of clocks did for a perpetually-inertial observer: it gives them a "NOW" instant, that extends throughout all space, that is meaningful to them. It's NOT just a convention, and it's not a choice.

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If you look back at post #23, it gives this equation:

AC(tau) = tau + { L * tanh(A tau)},

where "AC" is the age change of the helper friend (HF) near the home twin. The function tanh() has the same sign as its argument, so for negative "A" (the case where the acceleration is directed AWAY from the home twin), the quantity in curly brackets is negative. So as tau goes to zero, the age change AC is negative.

Hmmm. Wouldn't that mean that someone who is simply being stationary someplace in a gravitational field (for example half way up a tall tower on earth) would say that clocks above him tick faster than his own clock, and clocks below him tick backwards? Or at least clocks which are far enough below him would be ticking backwards?

But using the CMIF method itself is still the easiest and simplest way to compute the home twin's age (according to the accelerating twin). The most important value of the gravitational time dilation equation is that it allows us to (at least mentally) construct an array of clocks and associated HF's that establish a "NOW" instant, that extends throughout all space, for the given accelerating observer (the AO). And that guarantees that the results of the CMIF simultaneity method are fully meaningful to the AO. And it also guarantees that the CMIF method is the ONLY correct simultaneity method for him. He has no other choice.

This array of clocks does for an accelerating observer what Einstein's array of clocks did for a perpetually-inertial observer: it gives them a "NOW" instant, that extends throughout all space, that is meaningful to them. It's NOT just a convention, and it's not a choice.

Well, I already had a way to mentally construct such arrays of synchronised clocks. I imagine a very long inertial train (as long as necessary for a given scenario) equipped with its own array of synchronised clocks along the entire length. There can be multiple such trains, all moving at different relative velocities along the same axis. The traveling twin becomes a train hopper, jumping on and off whichever trains he needs to match the scenario. Each time he jumps onto a train, he accepts that the clocks on that particular train are now all displaying the same time simultaneously. He has no other choice either, right?

If you look back at post #23, it gives this equation:

AC(tau) = tau + { L * tanh(A tau)},

where "AC" is the age change of the helper friend (HF) near the home twin. The function tanh() has the same sign as its argument, so for negative "A" (the case where the acceleration is directed AWAY from the home twin), the quantity in curly brackets is negative. So as tau goes to zero, the age change AC is negative.

I realized last night that the above equation (that comes from the new gravitational time dilation equation) is a special case of the "delta_CADO" equation that I derived (and published) more than 20 years ago! The delta_CADO equation is an especially easy way to compute the CMIF results without having to draw a Minkowski diagram (i.e., without drawing the lines of simultaneity for the traveling twin).

Hmmm. Wouldn't that mean that someone who is simply being stationary someplace in a gravitational field (for example half way up a tall tower on earth) would say that clocks above him tick faster than his own clock, and clocks below him tick backwards? Or at least clocks which are far enough below him would be ticking backwards?

No, I don't think it means that. When you are talking about a constant gravitational field, existing for all time, none of the clocks run backwards. But note that in the new gravitational time dilation equation (in contrast to the old GTD equation), the tic rates of those clocks aren't constant in the new equation ... they vary with time. And after an infinite time, the rates of all the clocks are equal.

Well, I already had a way to mentally construct such arrays of synchronised clocks. I imagine a very long inertial train (as long as necessary for a given scenario) equipped with its own array of synchronised clocks along the entire length. There can be multiple such trains, all moving at different relative velocities along the same axis. The traveling twin becomes a train hopper, jumping on and off whichever trains he needs to match the scenario. Each time he jumps onto a train, he accepts that the clocks on that particular train are now all displaying the same time simultaneously. He has no other choice either, right?

Since the new GTD equation agrees with the CMIF simultaneity method, he doesn't have any other choice but to agree with CMIF. But the really important thing that the new GTD equation does, is that it implies that ONE array of clocks can be set up for the lifetime of the given observer, which establishes a "NOW" instant, that extends throughout all space, for him. It works as well for him as the array of synchronized clocks work for a perpetually-inertial observer. No frame jumping for the accelerated observer is required ... He is always using the same set of clocks to establish his spatially-unlimited "NOW". Those clocks aren't synchronized, but he knows how to CALCULATE the current reading of each of those clocks, and that establishes his spatially-unlimited "NOW".

But that doesn't mean that the tools of CMIF can't be used. Since the new GTD equation agrees with CMIF simultaneity, any of the CMIF tools can be used, and often they are the easiest way to bet an answer. The importance of the GTD equation is that it GUARANTEES that the CMIF results are fully meaningful. And it guarantees that the CMIF method is NOT just one choice among many ... it's the law!

If you look back at post #23, it gives this equation:

AC(tau) = tau + { L * tanh(A tau)},

where "AC" is the age change of the helper friend (HF) near the home twin.

I realized last night that the above equation (when tau is zero, and "A" is infinite, and the product tau*A is finite) is just a special case (when the initial velocity is zero) of the "delta-CADO" equation that I derived (and published) more than 20 years ago! The delta-CADO equation is an especially easy way to determine the CMIF results, without having to draw a Minkowski diagram with lines of simultaneity for the traveling twin.

No, I don't think it means that. When you are talking about a constant gravitational field, existing for all time, none of the clocks run backwards.

I don't understand how you can agree, as you did in post #30, that you can use the gravity equation to calculate that the traveler says the stay-home twin goes from 10 to 40 to 70 to 40 to 10 years old, and then also say that none of the clocks run backwards in a constant gravitational field.

Consider only the change from 40 to 10 years old, and only that one acceleration. Clearly you are saying that he calculates her clock to run backward due to him experiencing that one pseudo-gravity effect, right? Granted it only happens for a short period of time, but it does happen, and you are saying you can calculate it using only that pseudo-gravity, (and presumably also the distance between the two twins). It seems to me that you must also be saying that is the case for a real gravitational field as well.

But note that in the new gravitational time dilation equation (in contrast to the old GTD equation), the tic rates of those clocks aren't constant in the new equation ... they vary with time. And after an infinite time, the rates of all the clocks are equal.

I don't think that's right. Consider our GPS system. The GPS satellite clocks run faster than ground clocks by about 45 microseconds per day, due only to the difference in location in the earths gravitational field, (neglecting time dilation due to the relative speed between the clocks). The faster rate of the higher clocks due to gravity alone does not change over time, and the rate will never match the rate of ground clocks, no matter how long the GPS system is in existence.

I think you must have something wrong in your calculations, or else I am misunderstanding your statements.

Since the new GTD equation agrees with the CMIF simultaneity method, he doesn't have any other choice but to agree with CMIF. But the really important thing that the new GTD equation does, is that it implies that ONE array of clocks can be set up for the lifetime of the given observer, which establishes a "NOW" instant, that extends throughout all space, for him. It works as well for him as the array of synchronized clocks work for a perpetually-inertial observer. No frame jumping for the accelerated observer is required ... He is always using the same set of clocks to establish his spatially-unlimited "NOW". Those clocks aren't synchronized, but he knows how to CALCULATE the current reading of each of those clocks, and that establishes his spatially-unlimited "NOW".

Using the traveler's ONE array of clocks, what is the time on his array clock which is located next to his twin sister at the moment when he says that she changes from 40 to 10 years old? Is the answer 40 at the time she is 40, and 10 at the time she is 10? If so, then his ONE clock array could simply be HER clock array.

But that doesn't mean that the tools of CMIF can't be used. Since the new GTD equation agrees with CMIF simultaneity, any of the CMIF tools can be used, and often they are the easiest way to bet an answer. The importance of the GTD equation is that it GUARANTEES that the CMIF results are fully meaningful. And it guarantees that the CMIF method is NOT just one choice among many ... it's the law!

It is the law in my train-hopping method as well. Otherwise the train hopping traveler could jump onto a train, and even though everyone else on the train would tell him that all of the clocks on the train are synchronised, he would be allowed to say that they are not, but only for himself, because he is somehow different than everybody else. That would be nonsense.

I don't understand how you can agree, as you did in post #30, that you can use the gravity equation to calculate that the traveler says the stay-home twin goes from 10 to 40 to 70 to 40 to 10 years old, and then also say that none of the clocks run backwards in a constant gravitational field ...

I'm not sure you've understood me correctly about the role that the gravitational time dilation equation plays in my analysis. The original gravitational time dilation equation (which I contend is incorrect, if the equivalence principle is correct) says that two stationary clocks, separated by distance "L" in the direction of a gravitational field "g" that does not vary along the direction of the field, will tic at constant rates that are different from one another. Specifically, the clock farther from the source of the field will tic faster than the other one, by the factor exp( L g ).

Now, I don't actually have any interest in gravitational fields and in general relativity. But I AM interested in special relativity, with no gravitational fields but with accelerations instead. So I use the equivalence principle (the "EP") to convert the gravitational scenario into an equivalent acceleration scenario. The equivalence principle says that all I have to do is replace the "g" in the gravitational time dilation equation with an acceleration "A", and I get an equation that tells me that if two clocks are separated by the constant distance "L", the leading clock will tic faster than the trailing clock by the factor exp( L A ).

That was a wonderful equation for me, because it would allow me to set up an array of clocks for an accelerating observer that would effectively give him a "NOW" extending throughout all space, similar to what was done so effectively by Einstein for a perpetually-inertial observer.

So I did some calculations with that equation. Specifically, I took the case where there has been no acceleration in the past, and the two clocks are synchronized. I arbitrarily choose a separation of the clocks of 7.52 ls. Then suddenly the two separated clocks simultaneously accelerate with a constant acceleration "A" for some period of time "tau" (as measured on the trailing clock). First, I do a case where the duration tau of the acceleration is 1.0 second, and the acceleration is 1.317 ls/s/s (about 40 g's). That produces a rapidity of 1.317 ls/s, which corresponds to a speed of 0.866 ls/s. I evaluate the tic factor exp(L A), and multiply it by tau to get the total change in the reading of the leading clock. The answer is 2 times 10 raised to the 4th power.

Then, I repeat that experiment, but with tau decreased by a factor of 10, and "A" increased by a factor of 10. So for the 2nd experiment, tau is o.1 second and "A" is 13.17 ls/s/s. Note that, by design, the final rapidity is 1.317 like before, and the speed is 0.866 like before. I calculate the new (tau)(exp[L A]), and the result is 1.02 times 10 raised to the 42nd power. So when we reduced tau one order of magnitude, and increased "A" by one order of magnitude, the reading on the leading clock is about 38 orders of magnitude greater than it was in the first calculation!

Then, I repeat the experiment again, but with tau = 0.01 and "A" 131.7 ls/s/s. The new (tau)(exp[L A]) is 1.27 times 10 raised to the 428 power. So when we reduced tau one order of magnitude, and increased "A" by one order of magnitude, the reading on the leading clock is almost 400 orders of magnitude greater than it was in the second calculation!

Clearly, this sequence is NOT going to converge to a finite value. The limit of the sequence, as tau goes to zero, and "A" goes to infinity (such that the rapidity and the speed at the end of the acceleration), is INFINITY. But that is inconsistent with the outcome of the twin paradox, where the home twin's age is FINITE. So the exponential equation we've been using can't be correct.

But is that exponential equation also wrong when it is used as a gravitational equation, rather than as an acceleration equation? I don't know. Maybe it works fine for a constant (in time) field that varies as the inverse square of the distance from the center of the earth. But if you do the scenario I've described for a gravitational scenario rather than an acceleration scenario, would it give the same infinite result? And would that infinite result obviously be wrong? What would the analog of the twin paradox outcome be in the gravitational case? I don't know, and I don't really care.

What I DID is develop the linearized equation that Einstein used when he was trying to figure out how to do the acceleration problem. His equation was only guaranteed to be good for very small arguments, but I noticed that for large arguments it gave results that weren't off by very much ... it almost agrees with the CMIF simultaneity. That lead me to slightly modify his linearized equation in a way that produced perfect agreement with CMIF simultaneity. But that simple modification turned out to be non-causal, and I had to make it more complicated to make it causal.

"I don't understand how you can agree, as you did in post #30, that you can use the gravity equation to calculate that the traveler says the stay-home twin goes from 10 to 40 to 70 to 40 to 10 years old, and then also say that none of the clocks run backwards in a constant gravitational field ..."

In the above, I don't understand why you are trying to compare a scenario in special relativity that has periods of no acceleration and then acceleration, with a scenario in general relativity that has a gravitational field that doesn't change with time. Those two scenarios aren't linked by the equivalence principle.

I don't understand how you can agree, as you did in post #30, that you can use the gravity equation to calculate that the traveler says the stay-home twin goes from 10 to 40 to 70 to 40 to 10 years old, and then also say that none of the clocks run backwards in a constant gravitational field.

I just re-read your words above, and I think they indicate that you HAVE misunderstood what the equivalence principle says. In the first part of your sentence, you are talking about the special relativity scenario, which contains lots of instantaneous velocity changes by the traveler (requiring infinite accelerations, lasting for only infinitesimal times), and no velocity changes by the home twin. But in the second part of your sentence, you are talking about a CONSTANT gravitational field. For the equivalence principle to apply, the two scenarios (gravitational vs accelerational) have to be the same, with the only difference being gravitational fields versus accelerations. So to use the equivalence principle, the gravitational scenario would have to have essentially infinite gravitational fields, existing for only an infinitesimal time, acting on the traveler, but with NO gravitational fields acting on the home twin. In that gravitational scenario, the home twin WOULD instantaneously get younger, according to the traveling twin. But that's not the gravitational scenario you've been talking about.

I just re-read your words above, and I think they indicate that you HAVE misunderstood what the equivalence principle says. In the first part of your sentence, you are talking about the special relativity scenario, which contains lots of instantaneous velocity changes by the traveler (requiring infinite accelerations, lasting for only infinitesimal times), and no velocity changes by the home twin. But in the second part of your sentence, you are talking about a CONSTANT gravitational field. For the equivalence principle to apply, the two scenarios (gravitational vs accelerational) have to be the same, with the only difference being gravitational fields versus accelerations. So to use the equivalence principle, the gravitational scenario would have to have essentially infinite gravitational fields, existing for only an infinitesimal time, acting on the traveler, but with NO gravitational fields acting on the home twin. In that gravitational scenario, the home twin WOULD instantaneously get younger, according to the traveling twin. But that's not the gravitational scenario you've been talking about.

I would not expect any clocks to run backwards regardless of their location in a constant gravitational field, so I think you are correct about that. As for what would happen if an actual gravitational field were to suddenly appear out of nowhere, I don't think there is any answer to that, since it is physically impossible.

But one thing that is important to keep in mind is that in the purely SR scenario, her clock never runs backwards according to her, and she never gets younger. It is only his reckoning of the time her clock currently shows, and his reckoning of what her current age is that changes, and only because his line of simultaneity changes.

So, if you did want to go so far as to say that if an actual gravitational field were to suddenly appear out of nowhere, someone somewhere would get younger, I think you would have to qualify it by saying that it only happens according to someone else located somewhere else, and only because their simultaneity line changes. But as I said, I don't think there is any answer to that, since it is physically impossible for an actual gravitational field to suddenly appear out of nowhere.

Regardless of all that, if you have found a way to use gravitational equations to obtain the same results that we get from SR (i.e. the CMIF method), then I think that is great. However, I do not claim to fully understand exactly what you have done mathematically for the gravitational approach, so I will not say much else.

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