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

Discussion in 'Physics & Math' started by Mike_Fontenot, Sep 26, 2021.

1. ### Mike_FontenotRegistered Senior Member

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358
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.

I've also published it on Amazon under the same title. It can be found most easily on Amazon by searching on my full name:

"Michael Leon Fontenot" .

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?

3. ### phytiRegistered Senior Member

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Mike;

If speed=distance/time, then speed=vt/t, and speed= v.
If light speed is a universal constant c, then v is expressed as a fraction of c, or
.5 ls/s=.5c.

'know' definition:
1. hold information in the mind: to have information firmly in the mind or committed to memory
2. be certain about something: to believe firmly in the truth or certainty of something
3. realize something: to be or become aware of something
4.comprehend something: to have a thorough understanding of something through experience or study

Distances and times result from measurements with light signals.

In the 'twin' scenario, we begin with observer A in the rest frame and B the wandering observer. The graphic shows A's perception of events as of t=40. A received a time of t'=10 at t=37.3 which is simultaneous with t=20, and a distance of 20*.866=17.3.
B received a time of t=5.4 at t'=20 which is simultaneous with t'=10.7.
A light cone with its apex at the current position of A and B, is all they know (are aware of). They can assume/speculate beyond their current times with conditional statements.
A can assume, 'IF B doesn't change his velocity he should be at 40*.866=34.6'.
A can't verify that until a signal sent at t=5.4 returns, as in the previous case.
In all cases, A and B can only know the facts after the events.

In the second graphic, A and B follow the same specified acceleration in the U ref. frame. As they accelerate, their t' and green x' axis of simultaneity rotate scissors-like toward the blue light path. Their speed is constant after 4.00 to establish green simultaneity. The common x' axis places A's t1 simultaneous with B's t2.
A thinks the B-clock rate is faster than the A-clock, and
B thinks the A-clock rate is slower than the B-clock.
Both clocks have the same rate and the same accumulated time.
The x' axis of simultaneity is a mathematical convention and does not have an independent existence. It is established with light signals per SR.
Their clock comparison at reunion would depend on the speed profile they follow to reunite.

All the weird effects began with a literal interpretation of the discontinuous motion as shown in the 'twin triangle'. Each segment for B is a different ref. frame. When the direction is reversed, so is the x' axis of simultaneity, a mirror image relative to the ct axis.
The instantaneous reversal defies physical laws, with an application of energy E (from an unknown source) to nullify B's motion, followed by a second application of E to move B in the opposite direction.
In SR the observers motion does not alter a distant clock's rate, but does alter their perception of the distant clock.

Consider the common example of the fast moving anaut. He assumes a pseudo rest frame as he moves toward his destination at constant velocity. His clock runs slower but he is not aware of it, and explains the early arrival of the destination as spatial
contraction. No one on Earth would agree with him. It is his interpretation of the effects of high speed motion.

Last edited: Sep 27, 2021

5. ### Mike_FontenotRegistered Senior Member

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358
I've just published another paper, entitled

"A New Gravitational Time Dilation Equation".

It is available on the viXra repository:

https://vixra.org/abs/2201.0015

and also 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.)

Here is the abstract:

In a previous paper, I showed that the gravitational time dilation equation, which has been accepted since Einstein published it in 1907, is incorrect. It is incorrect because it is inconsistent with the required outcome at the reunion of the twins in the famous twin ‘paradox’ of special relativity. In this paper, I describe a new gravitational time dilation (GTD) equation which IS consistent with the required outcome at the reunion of the twins. And my new GTD equation gives the same instantaneous change of the home twin’s (her) age, according to the traveling twin (him), when he instantaneously changes his velocity, as is given by the CMIF simultaneity method, but without requiring the assumption that the CMIF method requires.

7. ### originHeading towards oblivionValued Senior Member

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11,500
No you didn't. So building on a previous error is not a good start for your new paper.

8. ### Janus58Valued Senior Member

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2,285
Here's the problem; When you assume an instantaneous change of velocity, you are saying that a finite change of velocity occurs in zero time. Since the acceleration value is v/t and t is zero, you are doing a division by zero which is undefined. So of course, if you try and use an undefined value in a time dilation formula you are going to get nonsense for a result. ( It's like the algebra problem where you say a=b, and then later divide by (a-b), and end up by proving 1=2)
All this means is that an actual instantaneous change in velocity is not possible in the real world ( which is what Relativity strives to describe). In real life, you can make reversal of velocity direction time as short as you want, just not zero. And any non-zero duration gives results that are perfectly consistent.

9. ### Mike_FontenotRegistered Senior Member

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358
In both my previous and latest paper, I do a SEQUENCE of calculations where a finite acceleration "A" is applied for a finite amount of time "tau". In all cases, I choose "A" and "tau" such that the rapidity "theta" is equal to 1.317 ls/s, which produces a velocity "v" of 0.866 ls/s (starting from zero velocity in all cases). (The unit of speed is "lightseconds per second, ls/s). "Theta" is always equal to "A" times "tau". The first calculation is for "tau" = 1 second, and "A" = 1.317 ls/s/s. The result is that the change in age of the distant person is 7.513 seconds.

I then start over and REPEAT that calculation, but this time with "A" ten times larger, and "tau" ten times smaller, so that "theta" is again 1.317 ls/s. So for that second calculation, "tau = 0.1 second, and "A" = 13.17. The result for this second calculation is that the change in age of the distant person is 6.613 seconds.

I then AGAIN repeat the calculation, this time with "tau" = 0.01 second, and "A" = 131.7. The result for this third calculation is that the change in age of the distant person is 6.518 seconds.

Finally, I AGAIN repeat the calculation, this time with "tau" = 0.001 second, and "A" = 1317. The result for this fourth calculation is that the change in age of the distant person is 6.509 seconds.

It's clear that the above sequence is converging toward a finite value of the age change, as "tau" goes to zero and "A" goes to infinity (with "theta" always equal to 1.317 ls/s). So it is perfectly reasonable to say that in the limit, the age of the distant person increases by about 6.51 seconds when the velocity instantaneously changes from zero to 0.866 ls/s. The age change, 6.51 seconds, is exactly what the CMIF simultaneity method says it should be, and so the new gravitational time dilation (GTD) equation AGREES with the CMIF method, as required.

The above example is described in detail in Section 4 of my paper.

10. ### arfa branecall me arfValued Senior Member

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7,637
Mike, isn't the problem just that you have an instantaneous change from a positive to a negative velocity, relative to the at-home twin?

Suppose the travelling twin lands on a distant planet, stays for a few days, then takes off and heads home? do your equations say there's an instantaneous change somewhere in that senario, that invalidates relativity?

I'm just curious, is all.

11. ### Mike_FontenotRegistered Senior Member

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In my last two papers, the observer whose "viewpoint" I determine (The "AO") has been stationary wrt the distant home twin for a long time, and then, at time "t" = zero, he (and his distant HF (helper friend) accelerate TOWARD the home twin, at a constant "A" ly/y/y for a finite time "tau".

When the existing gravitational time dilation(GTD) equation (which consists of an exponential function) is analyzed, with a sequence of repeated calculations that progressively decrease the time "tau" and increase the acceleration "A", while keeping the total velocity increase at "tau" to be 0.866 ls/s, the result is that the age change of the HF keeps increasing by an INCREASINGLY larger amount on each iteration. That implies that in the limit, as "tau" goes to zero, the age change of the HF goes to INFINITY. That contradicts the CMIF simultaneity method of special relativity, so the existing GTD equation must be incorrect (assuming that the equivalence principle is valid).

Doing the same type of analysis for the new GDT equation that I have proposed, in the limit, as "tau" goes to zero, the age change of the HF rapidly converges to a FINITE value, and that value agrees with the result given by the CMIF simultaneity method. And the new GTD equation gets that result without requiring the assumption that the CMIF method requires.

12. ### arfa branecall me arfValued Senior Member

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Why am I bothering, I ask myself, but here goes

It's well known that introducing an unphysical event (instantly changing a velocity), doesn't show that a theory is invalid or wrong about something else; it isn't possible to change velocity instantly unless you're a beam of light reflecting off a mirror.

Anything with mass has to either come to a complete halt and reverse direction, or accelerate around some point (so they can maintain the same speed if they want to). In either case, there's a change in coordinates to account for.
Otherwise it seems you got nothin'. Sorry.

The idea that tau, the proper coordinate time, can be zero just doesn't really make sense.

Last edited: Jan 7, 2022

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14. ### Mike_FontenotRegistered Senior Member

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In my previous post (and in my last two papers), I discussed the results I got when the time elapsed during the velocity change is FINITE, and the acceleration is FINITE.

In my previous posting, the FINITE quantity "tau" isn't the coordinate time. The coordinate time is "t". The quantity "tau" is the finite duration of the constant finite acceleration "A".

15. ### Mike_FontenotRegistered Senior Member

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358
I'm aware of the Wiki article ... I reference it in my latest paper. But thanks for the Hafele-Keating reference ... I HAVE wanted to see exactly what they did.

16. ### Mike_FontenotRegistered Senior Member

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The H-K experiments certainly sound valid to me. If you look at the last section (Section 9, Conclusions) of my latest paper:

https://vixra.org/abs/2201.0015

you'll see that I agree that the H-K experiments conflict with what the equivalence principle says my special relativity (no gravitational fields) method implies about the equivalent gravitational scenario.

My new special relativity simultaneity method says that, if two separated rockets both start a constant acceleration "A" at the same instant (at "t" = 0), initially the clock in the leading rocket will tic faster than the clock in the trailing rocket by the factor 1 + L A, where "L" is their separation and "A" is the constant acceleration. When converted to a gravitational scenario by the equivalence principle, THAT agrees with H-K. But in my method, that tic rate difference will then decrease with time (whereas H-K says it remains constant), and eventually approach 1.0 as "t" gets arbitrarily large (you can see that in the second plot I give in my latest paper, labeled "Rate Ratio R") ... i.e., the clocks will eventually click at the same rate, which contradicts H-K.

I'm left with the following situation: my new simultaneity method, in a special relativity situation, agrees perfectly with the "Co-Moving Inertial Frames" (CMIF) method, without requiring the assumption required by the CMIF method. And as a bonus, it gives a way to construct a distributed array of clocks that can define a "NOW" moment for an accelerating observer, just as Einstein did for a perpetually-inertial observer. That's a wonderful thing, that's never been done before. But ODDLY, when that method is converted to a GR result via the equivalence principle, it conflicts with the H-K measurements. I don't know why.

17. ### Neddy BateValued Senior Member

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2,274
Hi Mike,

I haven't looked at your latest work, but the last time we discussed this, Halc and I were asking you to consider Bell's spaceships which I believe are analogous to your two accelerating points. Back then, I believe you were claiming the distance between your two accelerating points remained constant in their mutual rest frame, which is not correct under SR if you start with the premise that they also remain a constant distance apart in the "lab frame" that measures both of their coordinate accelerations as constant and equal. In that case, each of the two accelerating points should measure the distance between them as increasing, and each of them do not agree on simultaneity.

https://math.ucr.edu/home/baez/physics/Relativity/SR/BellSpaceships/spaceship_puzzle.html

Unless you have changed your arrangement since then, I believe this shows that you have made some serious mistakes in your assumptions. The lower half of that page does show a way to keep the distance constant in the accelerating-pair-frame, and it does mention the equivalence principle. I would think that you would need to take that approach.

Last edited: Jan 12, 2022
18. ### Mike_FontenotRegistered Senior Member

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I'll take a look. Thanks. But I think the equivalence principle is on my side. In the GR case, the two clocks are by definition mutually stationary and a fixed distance apart. So I believe that the equivalence principle guarantees that in the equivalent SR case, the two clocks are likewise mutually stationary and a fixed distance apart.

19. ### Mike_FontenotRegistered Senior Member

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358
Here's where you're going wrong: In the first paragraph of the "Bell's Spaceship" write-up, it says (the red coloring is mine)

"Bell considered two rocket ships connected by a string, with both having the same acceleration in the inertial 'lab frame', with one ship trailing the other and both moving along one line. The ships start out at rest in the lab. Their accelerations in the lab frame are required always to be equal, ..."

I'm NOT using "the lab (inertial) frame". I'm using the frame of the two mutually-stationary accelerating observers. In THEIR frame, their distance apart is constant. And they agree that their accelerations are constant and exactly equal.

20. ### Neddy BateValued Senior Member

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The second half of the "Bell's Spaceship" write up deals with what you describe. Namely: Two mutually-stationary accelerating observers, such that in their one mutual frame, their distance apart is constant. The difference is, in that case, they do not remain a constant distance apart in the lab frame. I think that must mean that they do not share the same proper acceleration in terms of g-forces, at least not simultaneously in the lab frame.

Refer to the two different red and blue graphs. The upper graph is the one where they remain the same distance apart in the lab frame, (and they do not share one mutual frame together because each says the other is moving away). It is the lower graph where they remain the same distance apart in their own frame. So, if you are assuming all of that in your paper, then you should be dealing with the motions shown in the lower graph. Perhaps you already are (I don't know), but if not then you would have to change it so that you are.

I doubt it would make much difference, though, for other reasons I provided when we discussed this in the past. But if you are intent on doing the analysis this way, then you should be careful to make sure you are using the motions shown in the lower graph.

Last edited: Jan 12, 2022
21. ### Mike_FontenotRegistered Senior Member

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Yes. That is correct. That's what I'm doing.

True, but I don't care about the lab frame viewpoint.

True, the lab frame says their accelerations are different, but I don't care about the lab frame viewpoint. The two accelerating observers are carrying accelerometers that each show the exact same constant acceleration.

22. ### Neddy BateValued Senior Member

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2,274
Okay, good.

I think you might be using the lab frame when you consider the age of your third observer whom never had any proper acceleration. When you are done with your "GR" analysis, and you then claim the results must also apply to a pure SR scenario by invoking the equivalence principle, does your SR equivalent scenario have the motions of the lower blue and red graph according to her? I think it should, because the motions of the upper blue and red graph do not match your other assumptions.

Yes, that does seem to be the case for the lower blue and red graph, but not the upper graph.

23. ### phytiRegistered Senior Member

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612
Mike;

The red lines indicate td, B-time projected onto the common ct axis.
Assume the physically impossible instantaneous reversal made by B at t2'.
A's history contains all events within the blue light cone that terminates at t2.
The B reversal is not included.
B's history contains all events within the blue light cone that terminates at t2'.
A's current age is not included.
B cannot be aware of t3 until sending a signal at t2' and getting a return, which establishes the green x' axis of simultaneity.
The jump forward for the A clock that you claim, can be avoided with a curved transition from outbound to inbound.

Here are some equations for GR time.
http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html