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...exactly what Iggy tried to do. What makes it worse, is he managed to prove exactly the opposite of what he intended to prove. Funny that you still maintain this fallacy: your post #1 scenario is based on acceleration just the same, the fact that you have been in denial throughout the thread doesn't change the facts.

No, they are incorrect, Iggy thinks that he has created a scenario that proves that acceleration isn't the critical part in creating the paradox, thus agreeing with your original false claim. He also keeps claiming the the debate is about the clock hypothesis, I proved both his claims false. BTW: you can lay off your cowardly practice of negging my posts, I am simply trying to teach you mainstream physics. What Iggy has done is that he has applied the same acceleration value (2g) in a different manner to the two twins: the "carousel twin" has the acceleration normal to the direction of motion , thus having no influence on the speed involved in the evaluation of his elapsed proper time whereas for the other twin, Iggy applied the acceleration in the direction of motion , such that it has direct influence on the speed involved in the evaluation of his elapsed proper time. In other words, Iggy thinks that he has created a counter-example justifying his oft repeated fringe claim that the disparity in the elapsed proper time "doesn't depend on acceleration", when, in reality, he has created a disparity in the speeds of the twins based on applying the same value for the acceleration (2g) in a different manner to the two twins, orthogonal to the motion for the "carousel twin" and tangential to the motion to the other twin. So, he has proven exactly the opposite to his fringe claim, that the disparity in elapsed proper time between the twins, does depend on acceleration. Speed profile is a direct function of acceleration, as shown by numerous examples. Change the acceleration and the results change between the twins.

Then why are you persisting, 100+ posts in the thread? You should be done with it.

You can answer the simple question you have been evading: Acceleration is: A. Important in the twins paradox B. Not important in the twins paradox. Which one is it, Iggy? A or B? Give a straight answer, stop hiding behind inapplicable quotes. Is the claim in the title of this thread TRUE or FALSE?

This is the third time I am pointing to your erroneous claim. Let's cut to the chase: Acceleration is: A. Important in the twins paradox B. Not important in the twins paradox. Is it A or B? Please give a straight answer. Here is post 318 addressing the errors in your scenario. Ok, Here is again the explanation of your misconceptions, replete with your original posts. So, to cut to the chase, please answer this simple question: Acceleration is: A. Important in the twins paradox B. Not important in the twins paradox. Which one it is? A or B? This is not what I am saying, you mangle badly what I've been trying to teach you. First off, it is not "time dilation", it is "elapsed proper time". Two different concepts. Secondly, what I've been trying to teach you is that your claim: "Acceleration is not important in the twins paradox" is FALSE. In reality, acceleration is at the very root of the twins paradox by virtue of the asymmetry it produces in the speed profiles of the twins. So, the claim in your title is false, no matter how many times you have recanted: "Acceleration IS important in the twins paradox". I called it hidden wrt to your scenario, where you were jumping frames. I thought that you finally understood your error. The proper time is definitely dependent on the acceleration profile (see the presence of the variables [math]a,T_a[/math]) , contrary to your unsubstantiated claims.

Not directly, the correct statement is "proper time does not depend directly on acceleration". I already explained to you the dependency on acceleration: differing acceleration profiles->differing speed profiles->differing total elapsed time Are you still claiming that "Acceleration is not important in the twins paradox"? Do you want to discuss the mistakes in your post 297?

But the clock postulate is not in discussion, I am pointing out your misconceptions in terms of how acceleration affects the total elapsed proper time. These are two different subjects you keep co-mingling liberally. You are confusing clock rate (unaffected by acceleration) with total elapsed proper time (affected by acceleration, albeit through the speed profile). On a different note, I addressed your post 297 directly and I pointed out the conceptual errors that you are making in that post. I thought that you wanted to discuss your famed post 297, no?

But we aren't debating the clock postulate. What we are debating is your fringe claim that acceleration is not what causes the "paradox" in the twins "paradox". To wit, you have created a scenario (post 297) that you think that it supports your fringe claim, see below: OK, let's have a look at your scenario. What you've done is that you have applied the same acceleration value (2g) in a different manner to the two twins: the "carousel twin" has the acceleration normal to the direction of motion , thus having no influence on the speed involved in the evaluation of his elapsed proper time whereas for the other twin, you applied the acceleration in the direction of motion , such that it has direct influence on the speed involved in the evaluation of his elapsed proper time. In other words, you think that you have created a counter-example justifying your oft repeated fringe claim that the disparity in the elapsed proper time "doesn't depend on acceleration", when, in reality, you have created a disparity in the speeds of the twins based on applying the same value for the acceleration (2g) in different manner to the two twins, orthogonal to the motion for the "carousel twin" and tangential to the motion to the other twin. So, you have proven exactly the opposite to your fringe claim, that the disparity in elapsed proper time between the twins, does depend on acceleration. But the results do depend on acceleration. The fact that you are applying the "same acceleration" (correction: the same scalar value of the acceleration) in differing ways to the two twins means that you have proven exactly the opposite, that the outcome of the twins "paradox" does depend on acceleration. Like I said: differing acceleration profiles -> differing speed profiles -> differing total elapsed proper times. So, "Acceleration IS important in the twins paradox". It is what creates the "paradox" in the first place. Markus explained that to you, I explained that to you.

We are looking at your oft repeated misconception that acceleration is not the root of the difference in elapsed proper time. You appeared to had finally gotten it fifty posts ago, now you are regressing. Why? I corrected you several times, gave you the exact mathematical counter-proof to your fringe claim, yet , you persist in your error? Why?

So, you still don't get the point : different acceleration profile->different speed profile->different total elapsed proper time? OK, let's try some very simple example, in the wiki example, make the two twins travel in opposite directions, with acceleration [math]a1[/math] and , respectively , [math]a2[/math] Their total elapsed times will be : [math]\Delta \tau1 = 2 T_c / \sqrt{ 1 + (a1 \ T_a/c)^2 } + 4 c / a1 \ \text{arsinh}( a1 \ T_a/c )[/math] [math]\Delta \tau2 = 2 T_c / \sqrt{ 1 + (a2 \ T_a/c)^2 } + 4 c / a2 \ \text{arsinh}( a2 \ T_a/c )[/math] Depending on the relation between [math]a1[/math] and [math]a2[/math] one gets: [math]\Delta \tau1=\Delta \tau2[/math] [math]\Delta \tau1>\Delta \tau2[/math] [math]\Delta \tau1<\Delta \tau2[/math] You conjecture is nothing but a subcase of the above. Contrary to your insistent misconception that acceleration is at the root of the of the "paradox" in the twins "paradox", quite the opposite, it shows that the acceleration profile determines the relationship between the elapsed proper times.

...which makes the problem into a trivial one. As an aside, your post 297 is sloppy, you can interpret the twin on the carousel experiencing either normal (the trivial case) or tangential (the non-trivial case) acceleration. Do you understand the sequence: difference in acceleration profile -> difference in speed profile -> difference in total elapsed proper time? False, nothing precludes the twin on the carousel to accelerate, coast and decelerate, exactly like the twin that moves linearly. They follow similar acceleration profiles that are both perfectly realistic. I see that you are still stuck on the misconception you espoused in the title of the thread. I see you and md65536 share the same misconception, I thought that you two were over it. Acceleration is the keystone of the "paradox" effect.

Why wouldn't it? Both twins are accelerated, you don't have the monopoly of how they are accelerated. The way Iggy tried to accelerate them created a trivial disparity between them, I fixed that.

I wrote this paragraph (and the next one) for wiki some time ago, I am very familiar with the subject matter. The writeup shows the same answers but in the general form, using symbolic math rather than the particularization that you are showing. The part that you keep missing is that the above is valid for the specific case of only one twin accelerating. If both twins accelerate , the answer depends on the respective acceleration profiles. This is the third time I am pointing this out to you, I am quite sure that you can work out the math all by yourself. If you can't, I'll do it for you, just let me know. Yes, I sensed that. You are talking about normal acceleration. If the angular speed isn't constant (and this is the case for both twins experiencing acceleration), then, you have to deal with the garden variety of tangential acceleration, [math]r \frac{d\omega}{dt}[/math], capisci? Sure, the point is, and always was, that absent acceleration, you do NOT get the "paradox" in the twins "paradox". It is the difference in acceleration profile that determines the difference in the speed profile that determines the difference in the total elapsed proper time. See the two paragraphs I wrote for wiki, the role of acceleration is quite clear.

Simple: you forgot that the rotation is accelerated: [math]r \frac{d \omega}{dt}=2g[/math] .

For this particular case, maybe (see below). In general, no. The point was that you made a sweeping comment that isn't true. As explained, the end result for the general case depends on the acceleration profiles of the twins. The acceleration profiles determine the speed profiles. The speed profiles determine the total elapsed proper time. Even this isn't true since the RHS has [math]v=2gt[/math] and the LHS has [math]r \omega=2gt[/math] , so, the two sides are equal provided that the acceleration profile is the same.

No, it doesn't "counteract". The effects are cumulative.

This is incorrect, if the twins exhibit identical acceleration profiles, they will have the same exact total elapsed proper time. The situation gets more complicated if the twins exhibit different acceleration profiles. In this case, their total elapsed can be equal, less than or larger than, all depending on the integrand [math]\sqrt{g_{\mu \nu }dx^{\mu }dx^{\nu }}[/math]. This is definitely untrue. You would need to calculate for each twin [math]\tau =\int_{C}\sqrt{g_{\mu \nu }dx^{\mu }dx^{\nu }}[/math] according to their respective acceleration profiles and you would need to compare their resulting total elapsed proper times. Depending on the acceleration profiles, you would get (if you do the calculation correctly), [math]\tau_1=\tau_2[/math] or [math]\tau_1 < \tau_2[/math] or [math]\tau_1 >\tau_2[/math]

The delivery is purely scientific, there is someone who's an ignorant coward who's negging the posts.

yes, now if the ignorant coward who's been negging my posts on the subject would come forward...

If they are comoving, yes.

For a unique path through spacetime? No. Quite the opposite.

Based on your question, you don't. [math]\tau_{EM}[/math] is the same for all the observers in the universe. [math]\tau_{ME}[/math] is the same for all the observers in the universe. The above does neither require nor imply that [math]\tau_{EM}=\tau_{ME}[/math] . Isotropy would require that but the setup is not isotropic, the path from Moon to Earth is not identical with the path from Earth to Moon. You can make an argument that both Schwarzschild radii [math]r_{sM},r_{sE}[/math] are so small that the two proper times are very close to each other (and very close to the Newtonian value [math]\frac{D}{c}[/math] as I explained early on. On the other hand, in a setup where you would have celestial bodies with a very large gravitational mass, one cannot ignore the Schwarzschild radii and, if there is a big disparity between the Schwarzschild radii, the two proper times will be significantly different and the approximation [math]\frac{D}{c}[/math] will no longer hold. To my best recollection, the EFEs have a solution (Schwarzschild) only for the case when the mass of one celestial body is much larger than the second body.

You don't get , do you? The signal climbs out of the gravitational well of the Earth*, so, BOTH the Earth - bound clock and the Moon bound clock AGREE that it takes the PROPER time [math]\tau_{EM}[/math] for the signal to reach the Moon. Actually, all the observers in the universe agree on [math]\tau_{EM}[/math]. In the reverse direction: The signal climbs out of the gravitational well of the Moon, so, BOTH the Earth - bound clock and the Moon bound clock AGREE that it takes the PROPER time [math]\tau_{ME}[/math] for the signal to reach the Earth. [math]\tau_{ME}[/math] need not be equal to [math]\tau_{EM}[/math]. You need to learn the meaning of proper time, all clocks agree on the elapsed proper time. These are two DIFFERENT directions of the signal, why would you expect them to be the same proper time? Do you understand the notion of invariance? (Hint: I just explained it above). ---------------------------------------------------------------------------------------------------------------------------------- * Things are a little more complicated than that , the signal climbs out of the gravity well of the Earth and falls into the gravity well of the Moon (and vice-versa). This does not affect the symmetry of the problem.

Doesn't matter, you are fixated on the fact that the clocks are placed on different celestial bodies (Earth vs. Moon). Yet GR, teaches you how to do the calculations correctly, so the radar times , despite your nonsense, are the same for both the EME and the MEM trips. Try taking a class, it will cure you from making all these nonsense statements that are clearly incorrect. This is total gibberish, sorry, I just showed you the correct mathematical solution.

Sigh, [math]\tau_{EM}=\int_{r_1}^{r_2}{\frac{dr}{c \sqrt{1-r_{sE}/r}}}[/math] [math]\tau_{ME}=\int_{r_1}^{r_2}{\frac{dr}{c \sqrt{1-r_{sM}/r}}}[/math] [math]\tau_{EME}=\tau_{ME}+\tau_{EM}=\tau_{EM}+\tau_{ME}=\tau_{MEM}[/math]