Can the Twin Paradox be simplified?

IF the calculations are correct.

 

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Fair enough, and if I'm wrong I will post my error and admission, but it will have to wait until tonight.

 

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Cute trick considering that for there to be .75/1.25 time dilation ratio between Prime and Unprime, prime has to be moving at .8c relative to Unprime. Twice that would be 1.6c.
Putting that aside and assuming >c speeds for both Prime and Triple prime, let's examine were you went wrong:

It is true that, according to Unprime, Prime ages more from the time that Triple prime passes Unprime to the moment that Triple prime catches prime. (Prime is moving slower with respect to unprime than triple prime is and thus show less time dilation over the interval.), Though prime still ages less than Unprime.

It is also true that all three will agree as to what time was on showing on Unprime and Tripleprime's clocks when they passed each other and what time was on Prime and Tripleprime's clocks when Tripleprime passes prime.

However, this does not mean that according to Tripleprime, Prime aged more than he did during the interval between the two passings. Because according to Triple prime, when he passed Unprime, Prime's clock read a later value than it did for Unprime at the same instant, due to the Relativity of Simultaneity. So for instance, if according to Unprime, prime's clock read 0.75c at this moment, then according to Tripleprime, prime's clock will read later than that at the same instant. The amount that Prime ages according to Tripleprime between his passing of Unprime and Prime will be the difference between Prime's age at the time he passes Unprime and prime's age when Triple prime reaches him.

Because of Prime's already advanced age when Tripleprime passes Unprime, this difference will be less than the time that passes for Tripleprime. Tripleprime concludes that Prime aged less than he did during the interval while he was passing between Unprime and prime.

As to the fact that, according to Tripleprime, prime ages more than Unprimeduring the interval, SO WHAT? This just goes back to illustrate the Relativity of Simultaneity again. Prime and Tripleprime simply do not agree as to how old Unprime is when they meet up.

 

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Generally if you're going to examine where someone went wrong you would not concur with their observations and results.

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As to the "what" in your "SO WHAT", I reiterate the following:

 

If you had pursued my exercise with the Red army and the Blue army, you would have found that Blue claims Red is time-dilated, yet Red claims Blue is time-dilated. I suppose you would interpret this to mean, "No absolute conclusions can be drawn," and that, "Acceleration is a necessary causal component of absolute time dilation." If so...

1. Are you claiming it is impossible for Blue to claim Red is time-dilated, and for Red to claim Blue is time-dilated?

2. Are you claiming it is meaningless for Blue to claim Red is time-dilated, and for Red to claim Blue is time-dilated?

 

I have to apologize, Neddy Bate. I truly was going to work through your thought experiment but then Tach was let back on the Island and interjected. I'm not convinced that your red/blue army setup is any different from to single observers watching each other with Doppler effects but I'm not sure. If you let me conclude things with Tach/Janus I'll be happy to move back to our discussion...

 

Ok, let's see the calculations. I'll be waiting.

 

That's fine, take your time.

PS: LOL @ "let back on the Island"

 

Setup: Standard Third-Brother Twin Paradox. Three brothers, Unprime, Prime and Doubleprime. Unprime stays home. Prime passes him at T=T'=0 moving at .8c. At T'=T''=.75 yr, Prime passes Doubleprime moving in the opposite direction, back towards Unprime at a velocity (-.8c) from Unprime's perspective. When Doubleprime reaches Unprime, T''=1.5 yr yet T=2.5 yr. This is all from Tach's link and should be incontrovertible.

Conclusion: Can T'' and T claim that absolute time dilation occurred without acceleration?

No!

New Setup: Fourth-Brother Twin Paradox. Four brothers, Unprime, Prime, Doubleprime and Tripleprime. Unprime stays home. Prime passes him at T=T'=0 moving at .8c. At T'=T''=.75 yr, Prime passes Doubleprime moving in the opposite direction, back towards Unprime at a velocity (-.8c) from Unprime's perspective. Also, Tripleprime passes Unprime at T=T'''=1.25 yr moving at .9756c* in the same direction as Prime. When Doubleprime reaches Unprime, T''=1.5 yr yet T=2.5 yr, as before. However, analyzing things from Prime's perspective it is Unprime that is moving at (-.8c) while Tripleprime is approaching at .8c! (You don't have to do it this way but the conclusion is the same.) This allows us to calculate the Prime's simultaneity of the meeting of Unprime, Tripleprime and himself as
T = T''' = 1.25 yr = T'\(\sqrt{1-(.8)^2} \) = 2.08 yr, with Tripleprime reaching Prime in another 2.08 yrs for a co-location time of T' = 4.166 yrs and T''' = 2.5 yrs.

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RESULT: T = 2.5yrs, T'' = 1.5 yrs and they conclude that T "absolutely" aged less than T'. However, T' = 4.166 yrs, T''' = 2.5 yrs and they conclude that T' "absolutely" aged less than T.

CONCLUSION: Contradictory observations means that no absolute conclusion can be drawn regarding time dilation in scenarios devoid of acceleration. Therefore, my statement that acceleration is a necessary causal component of absolute time dilation stands.



*Despite Janus' claim that moving twice the speed of .8c = 1.6c and a "cute trick" (therefore implying impossible), relativistic addition of velocities would make it V = (.8 + .8)/(1 + .8*.8) = .9756c.

 

Can you define your term, "Contradictory observations" ?

 

Sure. The traditional Twin Paradox offers [apparently] contradictory observations in that each twin claims that the other's clock is running slow. The paradox is resolved when they meet up again due to acceleration.

The "Third Twin" was introduced (certainly by Tach in this thread and perhaps by the original author as well) in an attempt to prove that there can be an absolute determination made devoid of acceleration regarding which twin is actually aging slower. I've shown that the information gained in the Third Twin scenario offers no more information than the original Twin Paradox. In other words, we again have apparently contradictory observations that can only be resolved with acceleration. Under the presumption that the Third Twin scenario offers more information than the original Twin Paradox, I have generated actual contradictory observations which prove that presumption false. Make sense?

 

Good, thank you. This means that you acknowledged that the triple twin scenario, from mainstream physics is correct.

1. "Absolute time dilation"? What is this? Did you make up this term?

2. Mainstream SR teaches you that time dilation occurs anytime there is relative motion. Are you disputing that or are you simply confused about the nature of time dilation in relativity?

3. The twin problem is not exactly a "time dilation" problem, as shown by the example you just reproduced. It is a problem of how SR deals with the computation of elapsed time. By quoting the scenario (remember, you declared it "incontrovertible" yourself) you are using the calculations of elapsed time. To recast it correctly:

\(\tau=T-t_0=2.5-0=2.5\)
\(\tau'=\tau"=\tau \sqrt{ 1-(v/c)^2}=2.5 . 0.6=1.5\)

Both the above equations express "elapsed time" , not "time dilation". I have shown you earlier how elapsed time is computed in mainstream science.

This is just a repeat of the original scenario.

Tripleprime is totally irrelevant, as I will prove below.

Good, nothing new here.

Correct

Err, you just claimed that

So, according to your logic, Tripleprime speed wrt Unprime is -.9756c and its speed wrt Prime is :

\(\frac{0.9756-0.8}{1+0.9756.0.8}\)

since Prime and Tripleprime, according to you, move in the same direction.

Like I said earlier, not that it matter much but I thought I would flag all the gross errors in your scenario. Now, back to the part that really matters:

Tripleprime is moving AWAY from Unprime, so, after the original "meeting" at passing Unprime, he can never "meet" Unprime, ever again. Not that it matter much but I thought I should point out the gross errors.

Tripleprime moves at the speed 0.9756 in the same direction as Prime (who moves at 0.8). According to you , Prime has a head start wrt Tripleprime of 1.25, so , the correct equation that you should be using in order to find out when Tripleprime catches Prime is:

\(v""T=v'T+v'.1.25\)

\(T=\frac{1.25v'}{v"'-v'}=5.69\)

all measured in Unprime's frame of reference.
Expressed in Prime's frame of reference the above interval will be (see the formulas for calculating elapsed time):

\(T'=T \sqrt{1-0.64}=3.412\)

Likewise, Tripleprime elapsed proper time is:

\(T'"=T \sqrt{1-0.9756^2}\)

So, Tripletime shows the least amount of elapsed time (he's traveling at the highest speed) followed by Prime (higher elapsed time) and followed by Unprime (the highest elapsed time). So, your introduction of Tripleprime did not change anything.

Not that it matters much but I thought I'd point out the gross errors.

Of course, all of the above is nullified by the earlier errors pointed out along the way.

Conclusion: The above is total rubbish since the whole derivation is marred by elementary errors. The introduction of the fourth "twin" does not change the conclusion of the original scenario (there is differential aging in the absence of acceleration): acceleration plays no role.
The only thing that the fourth "twin" occasioned is a series of elementary errors.

 

Well, Tach, I was curious how you might respond...

Absolute time dilation refers to that which cannot be denied by a change in measuring frame. Absolute time dilation is able to be determined, for example, when the twins meet up again after one of them accelerates and they are co-located for a second time. You apparently understood this term when I made the following statement:

Well...the calculations are correct.

This is incorrect, this would give Tripleprime a velocity of .0986c. We're switching to Prime's frame, which necessitates a shift of -.8c. You got the negative on the numerator correct but neglected it in the denominator. The correct relative velocity of Tripleprime from Prime's perspective is

\(\frac{0.9756-0.8}{1+0.9756*(-0.8)} = .8c\)

as I said. My math has no errors, "gross" or otherwise.

Now, will you adhere to our agreement?

 

In mainstream physics this is called "elapsed proper time". You have been shown (several times , already) how to calculate is correctly. It is time you started learning the correct terms and the correct math that goes with them.

Yes, this was a mistake on my part. Nevertheless, it does not affect the outcome of the problem in any fashion, you have a lot of other unrelated gross mistakes to deal with. So deal with them.

 

My math had no errors, and you're straying from the point in an effort to renege on our agreement. Perhaps you could answer this question: what does the Third Brother scenario illustrate which the original Twin Paradox does not?

 

The only thing you did right was to take Prime's speed wrt Unprime as 0.8c. The rest is total rubbish.
I have pointed them out to you , so you are simply denying them by assertion. Deal with them.

For the mainstream science: that you get the same elapsed time differentials in the absence of acceleration. But this has been explained to you several times throughout this thread.

 

OK, I'll take the time to respond to your entire post on the condition that you adhere to our agreement. Refresher for you:

Are you going to claim that, with proper calculations, the conclusions drawn by Unprime and Doubleprime will be in agreement with those drawn by Prime and Tripleprime? If so, what conclusions would those be, specifically?

 

Your calculations of elapsed time have been already shown to be wrong. All that is left is for you to honor your promise and admit that you have a set of nonsensical answers as in, for example:

You obviously seem to believe that T = T''' . Even worse, you conclude that 1.25=2.08.

You have been already given the calculations for T>T'>T"', look them up .

 

I'll get to the math in a bit. I want to make sure that there's a point before I spend the time, however. Please answer the following question:

Are you going to claim that, with proper calculations, the conclusions drawn by Unprime and Doubleprime will be in agreement with those drawn by Prime and Tripleprime? If so, what conclusions would those be, specifically?

 

Please do so. I will be very curious how you explain away this monumental fumble:

You obviously seem to believe that T = T''' . Even worse, you conclude that 1.25=2.08.

This is about you explaining (or, even better) admitting your gross fumbles. Once you do that we can discuss more advanced subjects.