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what's important is what we calculate to be the state at the given time "t" = "tau" that we have selected for the end of a given experiment
Previously, I've given two arguments
Quote from: MikeFontenot on 02/11/2022 15:57:00what's important is what we calculate to be the state at the given time "t" = "tau" that we have selected for the end of a given experiment Yes! That's what's important. I did that, and you did not.You did not calculate where the front ship (Fred) would be at tau.
Quote from: MikeFontenot on 09/09/2022 19:26:42Quote from: MikeFontenot on 06/08/2022 23:38:23But some physicists are vague about whether the rapid variations in the current age of the distant person (according to the accelerating person) should be considered to be MEANINGFUL or not. It is possible to show that those rapid age variations (including negative ageing) MUST be considered to be fully real and meaningful.It can be shown to be meaningful to a given observer by showing how to construct an array of clocks which are mutually stationary wrt that observer. Halc then said:It is1) Impossible to keep an array of clocks mutually stationary wrt some accelerating reference worldline. Some of them can be kept stationary if the acceleration schedule is known, but not all of them.
Quote from: MikeFontenot on 06/08/2022 23:38:23But some physicists are vague about whether the rapid variations in the current age of the distant person (according to the accelerating person) should be considered to be MEANINGFUL or not. It is possible to show that those rapid age variations (including negative ageing) MUST be considered to be fully real and meaningful.It can be shown to be meaningful to a given observer by showing how to construct an array of clocks which are mutually stationary wrt that observer.
But some physicists are vague about whether the rapid variations in the current age of the distant person (according to the accelerating person) should be considered to be MEANINGFUL or not. It is possible to show that those rapid age variations (including negative ageing) MUST be considered to be fully real and meaningful.
Question for Halc: All of the above is standard special relativity: it came from Einstein himself. I have introduced some terminology that I believe clarifies some things, but I haven't changed any of Einstein's results. Do you nevertheless want to banish it to the "other theories" forum?
Another question for Halc: Does the fact that I have found an error in Einstein's exponential equation justify banishing this posting to the "other theories" forum?
Have you tried to confirm or refute my proof?
Another question for Halc: Does the fact that I have derived the correct gravitational time dilation equation to replace the incorrect exponential equation justify banishing me to the "other theories" forum?
Einstein said that a clock (and co-located "helper friend (HF)")
Einstein said that a clock whose acceleration "A" is the same as the designated observer's (the "DO's") acceleration
with both of them beginning the acceleration at the same instant with synchronized clocks (and ages) at that instant), and leading (in the direction of the acceleration) the DO by the constant distance "L", will tic faster than the DO's clock, by the ratio exp(L*A).
The "designated observer" (DO) is the person whose conclusion we want to know, about the current reading on each of the clocks in the array.
Einstein clearly regarded the quantity "L" as being constant during the acceleration
Einstein's exponential equation tells the DO, at each instant of his life (after the constant acceleration "A" begins), what the reading on each of the clocks in the array is
I eventually was able to derive the correct gravitational time dilation equation, and it can be used to construct the accelerating array of clocks and HF's that correctly establishes the "NOW at a distance" for the DO, and that makes the DO's conclusions meaningful to him.
I think what might be confusing you (Halc) is that you are probably looking at the scenario through the eyes of a perpetually-inertial observer. (I.e., you are interested in the conclusions of perpetually-inertial people, about what is happening to the separation between the accelerating clocks.) In my analysis, I don't consider the conclusions of any perpetually-inertial observers at all, and don't need or want to.
The acceleration "A" that each clock and their attendants (the DO and the HF's) are undergoing is produced by a rocket attached to each of them, and controlled by an attached accelerometer that always insists on a measured acceleration of exactly "A".
Quote from: MikeFontenot on 16/02/2023 20:33:40(I, Mike Fontenot, said:)Another question for Halc: Does the fact that I have found an error in Einstein's exponential equation justify banishing this posting to the "other theories" forum?(Halc said:)You calling it a fact doesn't make it a fact.
(I, Mike Fontenot, said:)Another question for Halc: Does the fact that I have found an error in Einstein's exponential equation justify banishing this posting to the "other theories" forum?
I told you what I got in the sequence of calculations (for the HF's age at the end of the acceleration) for each of my choices of the acceleration "A" and the duration of the acceleration "tau" (such that the velocity change is the same in each iteration of the sequence). Clearly, in my calculations, the sequence was NOT converging to a finite limit ... the HF's age was clearly going to infinity as tau goes to zero
That results in all cases in the velocity of the two clocks, at the end of the acceleration, being 0.866 ls/s.
and that is inconsistent with the outcome of the twin paradox.
If you think my calculations are wrong, what do YOU get when you do that sequence of calculations?
If you're going to say my calculations are wrong, you need to show what the correct calculations are.
The results of my calculations, in Section 3 of the above referenced paper, are given in a table near the end of Section 3.
The exponential GTD equation says that the leading clock tics faster by the factor exp(A*d). And so the change in the reading on the leading clock, during a time interval "tau", is just tau * exp(A*d).
In Einstein's 1907 paper, https://einsteinpapers.press.princeton.edu/vol2-trans/319 , he gives his exponential gravitational time dilation equation. Actually, he was working a special relativity problem (with accelerations and no gravitation) because he knew how to do that, and he was hoping the result would give him (via the equivalence principle) some help in his search for a gravitational theory. So Einstein's equation was actually a time dilation equation for accelerating clocks that are separated by a fixed distance. According to Einstein, for a pair of accelerating clocks separated by the distance "L" in the direction of the acceleration, the leading clock tics faster than the trailing clock by the factor R = exp(A*L),where "A" is the acceleration. In the iterations described below, I will limit myself to the case where "A" is constant during the acceleration. I will show that the exponential time dilation equation is incorrect. (I suspect that Einstein, and also physicists who came along later, didn't ever notice that the exponential equation is incorrect, because they never used it in the nonlinear range where its argument is large ... they only used it for very small arguments, where it is very nearly linear.)
Suppose that the two clocks are initially inertial (unaccelerated) at time "t" = 0, and both read zero at that instant. Then, for t > 0, both clocks undergo a constant acceleration "A" (as determined by accelerometers attached to each of them, which control a rocket attached to each of them, so as to achieve the specified acceleration). Their separation remains constant at "L" during the acceleration. In all of my calculations below, I chose L = 7.520. Let "tau" be the duration of the acceleration.Therefore the reading on the leading clock, when the trailing clock reads "tau", is given byAC = tau * R = tau * exp(A*L) .
So we get the following table: tau AC 1.0 2.0 * 10^4 0.1 1.0 * 10^42 0.01 1.0 * 10^428Clearly, this iteration is NOT approaching a finite value for the leading clock's reading, as tau goes to zero.
So when we use this method to determine by how much the home twin's (her) age increases when the traveling twin (he) instantaneously changes his velocity by 0.866 when he reverses course at the turnaround it tells us that the home twin gets INFINITELY older, which is not true.
From the time dilation equation for an inertial observer (which the home twin IS), we KNOW that both she and he have a finite age at their reunion. Therefore the exponential gravitational time dilation equation CAN'T be correct.
[...] you have a self-contradiction going on. If the lead clock 'tics faster' as you put it, it will finish its tau-seconds of acceleration first, meaning there will be a period of time where one ship is accelerating and not the other.
Quote from: MikeFontenot on 28/02/2023 16:55:03(I, Mike_Fontenot, said:) The leading clock stops accelerating when that clock reads tau * exp(L*A), and the trailing clock reads "tau". (And now I'm adding:)It is sometimes simpler to just assume that the constant acceleration goes on forever, and we just choose to assess the situation at any instant that we want. But for now, we can stick to the idea that they stop accelerating when the trailing clock reads tau, and the leading clock reads tau * exp(L*A). (And in that last sentence, I am specifying a SINGLE instant in their accelerating reference frame ... they both agree about that.) (Then, Halc says:)If they start at the same time and the rear clock tics slower (thousands of times slower according to your post), then when the lead clock reaches tau, the rear clock will will not yet be at tau. You cannot have it both ways. Do they stop at the same time or is there time dilation going on?(end of Halc's statement).
(I, Mike_Fontenot, said:) The leading clock stops accelerating when that clock reads tau * exp(L*A), and the trailing clock reads "tau". (And now I'm adding:)It is sometimes simpler to just assume that the constant acceleration goes on forever, and we just choose to assess the situation at any instant that we want. But for now, we can stick to the idea that they stop accelerating when the trailing clock reads tau, and the leading clock reads tau * exp(L*A). (And in that last sentence, I am specifying a SINGLE instant in their accelerating reference frame ... they both agree about that.)
The leading observer does NOT consider the stopping time to be tau on HIS watch But he DOES expect the trailing clock to read tau when they stop
and who then simultaneously start accelerating (according to their accelerometers) at a constant "A" ls/s/s...In each of the cases, the acceleration "A" and the duration "tau" of the acceleration are chosen such that the product "A tau" is constant for all cases at 1.317 ls/s. That results in all cases in the velocity of the two clocks, at the end of the acceleration, being 0.866 ls/s.
A "NOW-at-a-distance" instant, according to these two accelerating observers, consists of any arbitrary instant "t" in the life of the trailing observer, and the corresponding instant t * exp(L*A)