Revisiting the Flaws of the Light Clock in Special and General Relativity

[yuiop]

the instantaneous rate of a clock at a single instant, the rate is solely a function of its instantaneous velocity.

Velocity with respect to what?

If your answer is with the prior instant, than it is the acceleration between the two instants that changed the rate! So what causes clock rates to change? Acceleration!

If the answer is different I am happy to await your further explanation.

I think what JesseM meant was that the instantaneous rate of a moving clock relative to the clocks at rest in reference frame S, is solely a function of its instantaneous velocity relative to the clocks at rest reference frame S.

For example, if an clock is moving at constant velocity 0.8c relative to frame S, then its instantaneous clock rate relative to clocks in S is 0.6. A similar clock accelerating at 1,000,000g will also have an instantaneous clock rate of 0.6 relative to clocks at rest in S at the instant its velocity relative to S is 0.8c. The acceleration has no effect at all on its instantaneous relative clock rate. The "instant" is here defined as the time measured simultaneously by synchronised clocks at rest in frame of S.

 

[starthaus]

Look at page 49. It is not a simple "we cannot" statement so perhaps I have misinterpreted it. What do you think of it?

Matheinste.

You are right, he does claim that, nevertheless, on page 260 he treats the problem using hyperbolic motion just fine. Could it be that he considers accelerated motion as part of GR? This is why I only trust math when it comes to physics, I don't trust the literary part.

 

[starthaus]

I think what JesseM meant was that the instantaneous rate of a moving clock relative to the clocks at rest in reference frame S, is solely a function of its instantaneous velocity relative to the clocks at rest reference frame S.

For example, if an clock is moving at constant velocity 0.8c relative to frame S, then its instantaneous clock rate relative to clocks in S is 0.6. A similar clock accelerating at 1,000,000g will also have an instantaneous clock rate of 0.6 relative to clocks at rest in S at the instant its velocity relative to S is 0.8c. The acceleration has no effect at all on its instantaneous relative clock rate. The "instant" is here defined as the time measured simultaneously by synchronised clocks at rest in frame of S.

...all of which is perfectly correct, yet has no bearing on the discussion of the accelerated twins "paradox" where acceleration plays a key role in the final outcome.

 

[Passionflower]

I think what JesseM meant was that the instantaneous rate of a moving clock relative to the clocks at rest in reference frame S, is solely a function of its instantaneous velocity relative to the clocks at rest reference frame S.

You fail to see that a change in velocity is caused by acceleration?

 

[JesseM]

Sure I did, read post 135. If, after reading the paper, you will continue to maintain that the elapsed time differential is not dependent on acceleration, I will provide you with more papers published in peer reviewed journals that contradict your misconception.

I never expressed the "misconception" that "the elapsed time differential is not dependent on acceleration" (in the sense that you can find specific scenarios where the elapsed time is a function of acceleration--I would say that it's impossible to write a general expression for elapsed time as a function of acceleration regardless of the motion of the object, whereas you can write a general expression for elapsed time as a function of v(t)). Here is what you quoted me saying in post 135:

The clock hypothesis says the instantaneous clock rate depends on the instantaneous velocity (so the instantaneous acceleration is irrelevant), which means if you're considering some extended interval of time where the clock is accelerating, the instantaneous velocity will be different at the beginning of the interval than the end, so the clock rate will be different too.

Hmm, did I say anything faintly resembling "the elapsed time differential is not dependent on acceleration" there? Nope, try again to find an error in something I have actually said to you in this thread.

As for your claim about kev's supposed "error":

I looked back over all the posts from p. 6 on (starting with post #81), the only "err, no"'s you wrote in response to actual equations or quantitative statements kev made were when kev posted your equation and said it was wrong, but he thought it was wrong because he misunderstood the scenario you were considering (which you never spelled out), not realizing you were supposing an initial and final acceleration as well as a turnaround. So, looks like no actual physical/mathematical errors, just a misunderstanding of what scenario was being analyzed.

This is not true, look again at posts 97 (last entry) and 124.

Last entry of #97 was:

The proper time that elapses on the traveling twins clock in years is:
d \tau=10*0.6+10*0.6/0.8* \,asinh(0.8/0.6) = 6+8.23959 = 14.2396

...which clearly contradicts your earlier claim that the contribution of the acceleration period is negligible.

But here you seem to be fantasizing about something kev said rather than correcting a real statement of his. I did a search and the only post on this thread where he used the word "negligible" prior to #97 was in #13:

That may be true, but I have shown in the last post, that the time dilation due to acceleration in the twins paradox can be reduced to a negligible error, e.g less than 4 seconds "lost" due to acceleration time dilation compared to 8 years "lost" due to constant velocity.

"Can be reduced" (by making the acceleration brief), not "always reduces to a negligible amount in all problems" (regardless of the length of the acceleration). And a later response by kev to your post #97 in post #112 also shows that this he was talking about choosing a problem where the acceleration was brief:

I was talking about a different scenario where the acceleration phase was much more extreme and took place in seconds. With very extreme acceleration with the acceleration phase period tending to zero, the time dilation due to acceleration becomes negligable. The greater the acceleration is, the more you can ignore it.

So, strike one on finding a real technical error in kev's posts. The next post of yours which you point to is #124 where you say:

The proper time of the traveling twin can be calculated using

d \tau=\frac{T_c}{\gamma}+\frac{c}{a} \, \, asinh(a T_a / c)

Where T_c and T_a are total cruise and acceleration times, then when a is very large and T_a is necessarily brief because the terminal velocity v is reached very quickly, the second term goes to zero

No, it doesn't. Do the exercise I gave you and you'll find how false it is.

This is actually an example of you making a pretty clear physics error--do you not understand that as the acceleration approaches infinity the time needed to change from one cruising velocity to another approaches zero (instantaneous acceleration), which means the proper time elapsed in the acceleration phase approaches zero too? It may not look like the second term approaches 0 from the form above, but if you take into account the physical dependence of T_a on a and v (which is what kev meant above when he said 'T_a is necessarily brief because the terminal velocity v is reached very quickly') and substitute an expression for T_a as a function of a and v into the above equation, you find that the second term does indeed go to zero in the limit as a approaches infinity. Just consider the formula for velocity as a function of coordinate time:

v(T) = aT / sqrt[1 + (aT/c)²]

So if T_a is the total turnaround time, the T_a/2 is the time needed to accelerate from a speed of 0 to the final cruising speed v. So, we have:

v = aT_a / (2*sqrt[1 + (aT_a/2c)²])

So, we have:

v² = a²T_a² / (4 + 4a²T_a²/4c²)

4v² + (v²a²T_a²/c²) = a²T_a²

4v² + (v²a²T_a²/c²) = c²a²T_a²/c²

4v² = T_a²(c²a² - v²a²)/c²

T_a² = 4v²c²/[a²(c² - v²)]

T_a² = 4v²/[a²(1 - v²/c²)]

T_a = 2v/(a*sqrt[1 - v²/c²])

Now, kev actually did make one minor error in the above, if T_a is the total acceleration time then as I said in post #96 the acceleration term should be \frac{2c}{a} \, \, arcsinh(a T_a / 2c), which only differs from his expression by those factors of 2. Anyway, if you substitute in T_a = \frac{2v}{a*\sqrt{ 1 - v^2/c^2}}, you get \frac{2c}{a} \, \, arcsinh(\frac{v}{c*\sqrt{ 1 - v^2/c^2}}). Clearly, this does approach 0 in the limit as a approaches infinity, since a appears only in the denominator of the expression outside.

 

[yuiop]

Conveniently, you left out the fact that Moller, before taking the limit, calculates the exact formula of elapsed proper time:

\tau'_2=\frac{c}{\gamma v} \sinh^{-1}(\frac{g \Delta T}{c}) (eq 158)

where g represents ... acceleration. So, Moller doesn't cut any corners, he derived the exact formula. Selective quoting makes for very bad science, or no science at all.
I have no objection when a true scientist derives an exact formula only to take it to the limit afterwards. I have very strong objections when someone uses the truncated formula as a starting point. This is why I disagree with you and kev.

The equation I gave is also exact:

\tau=\frac{c}{\gamma v} \, sinh^{-1}(\frac{\gamma v}{c})

However, Moller (like me) has cut one corner. He has assumed that the initial or final velocity is zero and so in the general case, his equation is not correct.

For example consider the following case in units of c = 1 lyr/yr.

T is the coordinate time in inertial reference frame S.
The initial velocity of a rocket is zero at time T=0 in S.
The rocket has constant proper acceleration g = 1 lys/yr^2.
The elapsed proper time \tau of the rocket is zero at time T=0 in frame S.

After 1 year in S the elapsed proper time of the rocket is asinh(1) = 0.88137 years.
After 2 years in S the elapsed proper time of the rocket is asinh(2) = 1.44363 years.

The difference in proper times of the rocket, between year 1 and year 2 in S is asinh(2)-asinh(1)=0.56226 years, while the Moller equation predicts 0.88137 years of elapsed proper time for the rocket between year 1 and year 2 in S, because it fails to take account of the fact that the initial velocity of the rocket at the start of the second period is not zero.

The more general equation is:

\Delta \tau =\frac{c}{g}\, asinh\left(\frac{g T_2}{c}\right) - \frac{c}{g}\, asinh\left(\frac{g T_1}{c}\right)

and this only reduces to:

\Delta \tau = \frac{c}{g}\, sinh^{-1}(\frac{g \Delta T}{c})

in the limited case where v=0 at T1 or T2.

This dependency on the inital and final velocities in the general case is clearer in the velocity version of the equation when written as:

\Delta \tau=\frac{c\sqrt{1-v_2^2/c^2}}{v_2 } \, \, asinh\left(\frac{v_2 }{c\sqrt{1-v_2^2/c^2}}\right) - \frac{c \sqrt{1-v_1^2/c^2}}{v_1 } \, \, asinh \left(\frac{v_1 }{c\sqrt{1-v_1^2/c^2}}\right)

When v1 or v2 is zero, the above equation reduces to:

\Delta\tau=\frac{c}{\gamma \Delta v } \, \, asinh(\gamma \, \Delta v/c)

Since you are the one that is always harping on about equations being incorect, beacuase they are not generalised enough, you should state the limiting assumptions you are using, when you are quoting an equation that is only valid for a special case.

 

[yuiop]

I think what JesseM meant was that the instantaneous rate of a moving clock relative to the clocks at rest in reference frame S, is solely a function of its instantaneous velocity relative to the clocks at rest reference frame S.

You fail to see that a change in velocity is caused by acceleration?

I never said that change in velocity is not caused by acceleration. That is trivially true. What I actually said was:

I think what JesseM meant was that the instantaneous rate of a moving clock relative to the clocks at rest in reference frame S, is solely a function of its instantaneous velocity relative to the clocks at rest reference frame S.

 

[starthaus]

The equation I gave is also exact:

\tau=\frac{c}{\gamma v} \, sinh^{-1}(\frac{\gamma v}{c})

However, Moller (like me) has cut one corner. He has assumed that the initial or final velocity is zero and so in the general case, his equation is not correct.

Of course that the final speed is zero, how else do you expect the rocket to turn around? By jumping frames?
You didn't answer any of my questions relative to realistic cases of acceleration being finite.

 

[starthaus]

But here you seem to be fantasizing about something kev said rather than correcting a real statement of his. I did a search and the only post on this thread where he used the word "negligible" prior to #97 was in #13:

Not true. Look at post #80

 

[starthaus]

you get \frac{2c}{a} \, \, arcsinh(\frac{v}{c*\sqrt{ 1 - v^2/c^2}}). Clearly, this does approach 0 in the limit as a approaches infinity, since a appears only in the denominator of the expression outside.

Not so fast.

v=\frac{at}{\sqrt{1+(at/c)^2}}

so, a->oo implies v->c, meaning \gamma->oo

so you are rushing to conclusions. You have a lot more work to do in order to find the answer.

This is actually an example of you making a pretty clear physics error--do you not understand that as the acceleration approaches infinity the time needed to change from one cruising velocity to another approaches zero (instantaneous acceleration), which means the proper time elapsed in the acceleration phase approaches zero too?

Actually this statement is pretty amusing, speaking of blunders. T_a is the duration the rocket accelerates (see the wiki page), there is no reason whatsoever why T_a should correlate to a.

I see your error, it is a repaet of the error that I flagged above:

T_a = 2v/(a*sqrt[1 - v²/c²])

What happens when a->oo? You are forgetting that , according to your starting point (see the formula for v), v->c so, the limit is undetermined (not that this even the correct way of calculating T_a) . Your math has deserted you today.

 

[JesseM]

Not so fast.

v=\frac{at}{\sqrt{1+(at/c)^2}}

so, a->oo implies v->c, meaning \gamma->oo

That equation only applies during the accelerating phase, but kev was specifically assuming the ship was accelerating up to a fixed cruising speed, so the greater the value of a, the shorter the time t needed to reach this speed. As long as the ship only accelerates up to some fixed v and then coasts, then whatever that v happens to be, the proper time of the accelerating phase will be \frac{2c}{a} \, \, arcsinh(\frac{v}{c*\sqrt{ 1 - v^2/c^2}}), so if we hold v fixed and let a approach infinity, the proper time of the accelerating phase approaches zero. Do you disagree?

Actually this statement is pretty amusing, speaking of blunders. T_a is the duration the rocket accelerates (see the wiki page), there is no reason whatsoever why T_a should correlate to a.

If you are accelerating until you reach some fixed v and then stop, then yes, there is a very good reason T_a should correlate with a. You need to actually think about the physical assumptions of the problem and not forget them when evaluating various equations.

 

[starthaus]

That equation only applies during the accelerating phase, but kev was specifically assuming the time t of the accelerating phase was approaching zero. As long as the ship only accelerates up to some fixed v and then coasts, then whatever that v happens to be, the proper time of the accelerating phase will be \frac{2c}{a} \, \, arcsinh(\frac{v}{c*\sqrt{ 1 - v^2/c^2}}), so if we hold v fixed and let a approach infinity, the proper time of the accelerating phase approaches zero. Do you disagree?

Sure I disagree, you are talking about an unphysical situation, infinite acceleration in zero time. So, what is the resulting cruising value for v?

If you are accelerating until you reach some fixed v and then stop, then yes, there is a very good reason T_a should correlate with a. You need to actually think about the physical assumptions of the problem and not forget them when evaluating various equations.

I was wondering how you will try to get out of your latest two blunders .

 

[JesseM]

I was wondering how you will try to get out of your latest two blunders .

Great, a completely non-substantive response. Do you disagree that the ship is accelerating up to some fixed cruising velocity and then stopping, and that in this case the proper time approaches zero if a approaches infinity? Or are you imagining the ship is accelerating forever, completely contrary to the stated assumptions of the problem?

 

[starthaus]

Great, a completely non-substantive response. Do you disagree that the ship is accelerating up to some fixed cruising velocity and then stopping, and that in this case the proper time approaches zero if a approaches infinity? Or are you imagining the ship is accelerating forever, completely contrary to the stated assumptions of the problem?

You have been given the description of the experiment and the associated math before.
This is a realistic situation, not the unphysical one that you are trying to make up in order to cover your mistakes. The description contains nothing close to the nonsensical infinite acceleration/zero acceleration period you claim.

 

[JesseM]

Sure I disagree, you are talking about an unphysical situation, infinite acceleration in zero time. So, what is the resulting cruising value for v?

No, we are talking about limits as acceleration approaches infinity and time approaches zero. Do you imagine it is somehow less "unphysical" to consider the limit as acceleration approaches infinity but time does not approach zero? In this case the "cruising speed" in both directions would approach the speed of light in this limit, which is not a usual assumption in any textbook version of the twin paradox, even ones that do consider acceleration as quasi-instantaneous. In fact I bet if you looked at a large collection of textbooks on relativity, you would find many that do consider the limit as acceleration goes to infinity, but not a single one that didn't also assume that the time of the acceleration phase was approaching zero so the cruising velocity would have some fixed value. For example, try looking carefully at Moller's textbook where he assumes acceleration approaches infinity, on pages 260, 261 and 262 (278 - 280 of the pdf)--do you think he is not assuming the time approaches zero so the cruising speed is fixed at some value below c?

Once again, the problem is that while you have a decent grasp of the math, your physical intuitions concerning what assumptions are standard in physics discussions and which would be considered bizarre and unusual are totally off. But aside from this, the fact is that kev spelled out the fact that he was assuming a fixed terminal velocity so that increasing a would mean decreasing T_a, so even if you go against every physics textbook that considers the limit as acceleration approaches infinity and treat this as a "strange" assumption to make rather than an utterly commonplace one, you still cannot call kev's comment incorrect on a technical level since in the case he said he was considering, the proper time in the acceleration phase would indeed approach zero. Read again his words that you were reacting to in post #124:

then when a is very large and T_a is necessarily brief because the terminal velocity v is reached very quickly, the second term goes to zero

 

[JesseM]

But here you seem to be fantasizing about something kev said rather than correcting a real statement of his. I did a search and the only post on this thread where he used the word "negligible" prior to #97 was in #13:

Not true. Look at post #80

In post #80 you quote a statement by him made in post #24. If you look at the context of the statement that "I have taken the time dilation due to acceleration into account and it is insignificant compared to the velocity time dilation", it is obvious he is saying this is true for the particular numerical example he presented in that post, he wasn't saying the time dilation in the accelerating phase would always be insignificant in any possible version of the twin paradox.

 

[starthaus]

No, we are talking about limits as acceleration approaches infinity and time approaches zero.

Which is unphysical.
The experiment has been run already, it does not involve any weird accelerations either at takeoff or at landing. The results are known, the time dilation during the acceleration periods is non-neglible:

Hafele, J.; Keating, R. (July 14, 1972). "Around the world atomic clocksbserved relativistic time gains". Science 177 (4044):

Do you imagine it is somehow less "unphysical" to consider the limit as acceleration approaches infinity but time does not approach zero?

You are trying to twist things, what I am telling you is that infinite acceleration is a desperate attempt to save the unphysical scenario that you've created.
So, you haven't answered the question, if a->oo and t->0 what is the crusing speed? Would you please answer the question?

 

[JesseM]

No, we are talking about limits as acceleration approaches infinity and time approaches zero.

Which is unphysical.

But in post #148 you said this was OK when Moller did it:

Conveniently, you left out the fact that Moller, before taking the limit, calculates the exact formula of elapsed proper time:

\tau'_2=\frac{c}{g} sinh^{-1}(\frac{g \Delta T}{c}) (eq 158)

where g represents ... acceleration. So, Moller doesn't cut any corners, he derived the exact formula. Selective quoting makes for very bad science, or no science at all.
I have no objection when a true scientist derives an exact formula only to take it to the limit afterwards. I have very strong objections when someone uses the truncated formula as a starting point. This is why I disagree with you and kev.

This is exactly what I did, I started with an "exact" formula (i.e. one that didn't involve any limits):

d \tau=\frac{T_c}{\gamma}+\frac{2c}{a} \, \, asinh(a T_a / 2c)

...and then I took the limit as a approaches infinity while we keep v fixed. This is the idea kev was talking about too in the post you're objecting to now (although he made the minor error of forgetting to include the factors of 2 in the second term). Is it OK for Moller (and many other physics textbooks) to do this but not for me and kev? Or do you wish to take back what you said above about not having any objection to Moller taking the limit as the acceleration approaches infinity as long as he derived it from a formula not based on limits?

The experiment has been run already, it does not involve any weird accelerations either at takeoff or at landing. The results are known, the time dilation during the acceleration periods is non-neglible

You are attacking fantasy strawmen again, neither I nor kev ever said the elapsed time during the acceleration phase would be negligible in all possible experiments, certainly not in the Hafele-Keating experiment where the planes were accelerating for pretty much the entire trip. The point we're making is just that you can make the elapsed time during acceleration negligible if you consider a scenario where the acceleration is large and brief.

You are trying to twist things, what I am telling you is that infinite acceleration is a desperate attempt to save the unphysical scenario that you've created.

How is it "desperate" when it's exactly the situation that kev was considering, which you said was incorrect on a technical level? He said "when a is very large and Ta is necessarily brief because the terminal velocity v is reached very quickly, the second term goes to zero", the phrase "goes to zero" clearly indicates he was talking about the limit as a gets larger and larger.

So, you haven't answered the question, if a->oo and t->0 what is the crusing speed? Would you please answer the question?

Again, I (and kev) assume a fixed cruising speed v and just consider the limit of the exact formula as a approaches infinity while v is kept fixed. It happens to be true that in this limit the time T_a also approaches zero, but T_a is just treated as a function of a and the fixed cruising speed v (this should be clear from my mathematical calculations in post #155), we aren't treating v as an unknown and trying to deduce its value in a double limit as a approaches infinity but T_a approaches 0. Once again this is a completely standard assumption made in any textbook that considers the limit as a approaches infinity in a twin paradox type problem--have you looked at what Moller does on pages 260-262? If so, do you disagree that he, too, is treating v as fixed while he takes the limit as acceleration approaches infinity, not treating v as unknown and trying to find its value in a double limit as acceleration approaches infinity and time approaches 0?

 

[starthaus]

But in post #148 you said this was OK when Moller did it:

In post 148 I showed that you cited selectively. I also showed that Moller first derived the general expression and then calculated the limit. Very different from your/kev hack.

This is exactly what I did, I started with an "exact" formula (i.e. one that didn't involve any limits):

d \tau=\frac{T_c}{\gamma}+\frac{2c}{a} \, \, asinh(a T_a / 2c)

...and then I took the limit as a approaches infinity while we keep v fixed.

You can't do that since v is a function of a. When a goes to infinity, v goes to c. This is unphysical enough to make you stop right here. We've been over this before.

Is it OK for Moller (and many other physics textbooks) to do this but not for me and kev? Or do you wish to take back what you said above about not having any objection to Moller taking the limit as the acceleration approaches infinity as long as he derived it from a formula not based on limits?

This is a red herring to detract from the earlier debate that the time differential is a function of acceleration.

You are attacking fantasy strawmen again, neither I nor kev ever said the elapsed time during the acceleration phase would be negligible in all possible experiments, certainly not in the Hafele-Keating experiment where the planes were accelerating for pretty much the entire trip.

Excellent, then neither of you, faced with experimental counterproof, will continue to claim that the time differential is not a function of acceleration.

Again, I (and kev) assume a fixed cruising speed v and just consider the limit of the exact formula as a approaches infinity while v is kept fixed.

You are using circular logic since the first formula in your "derivation" is:

v=\frac{at}{\sqrt{1+(at/c)^2}}

You can't "assume" v fixed when you are increasing a towards infinity (an unphysical attempt in itself).

I asked you three times to determine v from these conditions. I can only conclude that you realize that you can't do the calculation.

It happens to be true that in this limit the time T_a also approaches zero, but T_a is just treated as a function of a and the fixed cruising speed v (this should be clear from my mathematical calculations in post #155), we aren't treating v as an unknown and trying to deduce its value in a double limit as a approaches infinity but T_a approaches 0.

So, you realize that you are attempting to have a->oo while t->0. What does this tell you about the product at?

Once again this is a completely standard assumption made in any textbook that considers the limit as a approaches infinity in a twin paradox type problem--have you looked at what Moller does on pages 260-262? If so, do you disagree that he, too, is treating v as fixed while he takes the limit as acceleration approaches infinity, not treating v as unknown and trying to find its value in a double limit as acceleration approaches infinity and time approaches 0?

Yes, I can see what he did, It is pretty bad, it doesn't mean that you need to copy it mindlessly, just because he dit it. If you want a realistic scenario, feel free to use the one that describes the Haefele-Keating experiment. It is pretty good, you should give it a try. Once you do that, you should feel a lot more confortable with the role played by acceleration in the difference between the elapsed proper times.


In an earlier post I provided you with three published papers by recognized scientists that explain the role played by acceleration in calculating elapsed time from the PoV of the accelerated twin. Have you read them?

 

[Passionflower]

Excellent, then neither of you, faced with experimental counterproof, will continue to claim that the time differential is not a function of acceleration.

I would not hold my breath.

For the live of me I cannot understand why JesseM and kev are so adamant to keep acceleration out of the picture.

If we take the simple example of a traveling twin going to a place of a distance x away and returning then the total time dilation depends completely on the rate and duration of the accelerations (4 if we assume the start and end at rest relative to each other). In fact we can make a formula without any need for a velocity or 'cruising' time. Here the time dilation is a function of acceleration, duration of acceleration, and total distance.

As an exercise for those who are interested look at the spacetime diagram in the two extremes, the first extreme is a triangle where the accelerations are instant and the other extreme is the smoothest possible curve where the twin starts deceleration half way at each leg of the trip. Express the total time dilation in terms of a and distance x. Can you make a single formula for all stages in-between?

More challenging would be to extend this to an arbitrary number of accelerations in arbitrary directions, e.g. a kind of 'zitterbewegung' formula between two points a distance x away.

 

[starthaus]

I would not hold my breath.

For the live of me I cannot understand why JesseM and kev are so adamant to keep acceleration out of the picture.

It has something to do with never admitting that they are wrong.

As an exercise for those who are interested look at the spacetime diagram in the two extremes, the first extreme is a triangle where the accelerations are instant and the other extreme is the smoothest possible curve where the twin starts deceleration half way at each leg of the trip. Express the total time dilation in terms of a and distance x. Can you make a single formula for all stages in-between?

More challenging would be to extend this to an arbitrary number of accelerations in arbitrary directions, e.g. a kind of 'zitterbewegung' formula between two points a distance x away.

I went ahead and https://www.physicsforums.com/blog.php?b=1954 , such that kev and JesseM can wonder where I "copied" it from. Hint: nowhere, I wrote it up, like the other posts in my blog.

 

[JesseM]

In post 148 I showed that you cited selectively.

You just pointed out that Moller first derives a non-limit formula, but I never said otherwise. I also first gave the non-limit formula d \tau=\frac{T_c}{\gamma}+\frac{2c}{a} \, \, asinh(a T_a / 2c) and then used that make conclusions about the limit as the acceleration approaches infinity (while the cruising speed is kept constant).

I also showed that Moller first derived the general expression and then calculated the limit. Very different from your/kev hack.

How is it "very different"? Did I not first post the general expression above and then calculate the limit?

You can't do that since v is a function of a. When a goes to infinity, v goes to c.

You have to distinguish here between two uses of v--one is the cruising speed which is kept fixed, and one is the variable speed during the acceleration phase which is a function of t and a. I think the distinction was made fairly clear by my equations and discussion, but it might be more precise to distinguish them in the notation as the constant cruising speed v_c and the variable function v(t, a). Here, I'll repost my derivation in post #155 but with the better notation:

if you take into account the physical dependence of T_a on a and v_c (which is what kev meant above when he said 'T_a is necessarily brief because the terminal velocity v is reached very quickly') and substitute an expression for T_a as a function of a and v_c into the above equation, you find that the second term does indeed go to zero in the limit as a approaches infinity. Just consider the formula for velocity as a function of coordinate time:

v(T, a) = aT / sqrt[1 + (aT/c)²]

So if T_a is the total turnaround time, the T_a/2 is the time needed to accelerate from a speed of 0 to the final cruising speed v_c. So, we have:

v_c = aT_a / (2*sqrt[1 + (aT_a/2c)²])

So, we have:

v_c² = a²T_a² / (4 + 4a²T_a²/4c²)

4v_c² + (v_c²a²T_a²/c²) = a²T_a²

4v_c² + (v_c²a²T_a²/c²) = c²a²T_a²/c²

4v_c² = T_a²(c²a² - v_c²a²)/c²

T_a² = 4v_c²c²/[a²(c² - v_c²)]

T_a² = 4v_c²/[a²(1 - v_c²/c²)]

T_a = 2v_c/(a*sqrt[1 - v_c²/c²])

Now, kev actually did make one minor error in the above, if T_a is the total acceleration time then as I said in post #96 the acceleration term should be \frac{2c}{a} \, \, arcsinh(a T_a / 2c), which only differs from his expression by those factors of 2. Anyway, if you substitute in T_a = \frac{2v_c}{a*\sqrt{ 1 - v_c^2/c^2}}, you get \frac{2c}{a} \, \, arcsinh(\frac{v_c}{c*\sqrt{ 1 - v_c^2/c^2}}). Clearly, this does approach 0 in the limit as a approaches infinity, since a appears only in the denominator of the expression outside.

Is it OK for Moller (and many other physics textbooks) to do this but not for me and kev? Or do you wish to take back what you said above about not having any objection to Moller taking the limit as the acceleration approaches infinity as long as he derived it from a formula not based on limits?

This is a red herring to detract from the earlier debate that the time differential is a function of acceleration.

So, I take it you don't actually dispute that Moller does take the limit as acceleration approaches infinity while keeping the final velocity (v_c in my notation, though he doesn't use the same notation) fixed? Do you think he is making an error on the level of theory or mathematics here?

Anyway, you are using a highly equivocal phrase when you say that I am arguing against the position that "the time differential is a function of acceleration". I'm sure you can see that the following two claims are distinct:

1) It is theoretically possible to come up with scenarios where the acceleration is so brief that the total elapsed time (and thus the 'time differential' with another clock) could be calculated very accurately by just adding the elapsed times on the inertial parts of the trip and ignoring the elapsed time during the accelerating phase

2) In all conceivable scenarios, the elapsed time during the accelerating phase can be ignored when calculating the total elapsed time (and thus the time differential)

Of course I have only argued for 1 above, I would never make an argument as boneheaded as 2. Your vague descriptions of my argument allow you to equivocate between 1 and 2 (so, for example, you present the Hafele-Keating experiment as a 'rebuttal' even though it would only be a rebuttal to 2, not 1). I trust you will keep this distinction in mind from now on, and make no further ridiculous suggestions that I have been making argument 2 above.

You are using circular logic since the first formula in your "derivation" is:

v=\frac{at}{\sqrt{1+(at/c)^2}}

You can't "assume" v fixed when you are increasing a towards infinity (an unphysical attempt in itself).

The cruising speed v_c is fixed, v(T, a) during the acceleration phase. The acceleration phase is always defined to end at T_a/2, and we pick the value of T_a/2 in such a way as to ensure that v(T_a/2, a) = v_c.

I asked you three times to determine v from these conditions. I can only conclude that you realize that you can't do the calculation.

If v = at / sqrt[1 + (at/c)²], simply saying that you want a to approach infinity while t approaches 0 does not provide enough information to give a well-defined limit for v without additional information on the relation between a and t. For example, if we added the information at t = 0.6c/a, in this case it would be true that in the limit as a approaches infinity t will approach 0, and we would have v=0.6c/sqrt[1.36]. But if we instead added the information that t = 0.8c/a, then we would have v=0.8c/sqrt[1.64]. So, your question as stated does not give enough information for a value of v. Fortunately my version does provide more information, I want the speed at time t=T_a/2 to equal to the cruising speed v_c, which implies a relation between T_a and a and v_c, namely T_a = 2v_c/(a*sqrt[1 - v_c²/c²]). With this relation, it will be true that v(T_a/2, a) = v_c, even in the limit as a approaches infinity and T_a/2 approaches 0.

Once again this is a completely standard assumption made in any textbook that considers the limit as a approaches infinity in a twin paradox type problem--have you looked at what Moller does on pages 260-262? If so, do you disagree that he, too, is treating v as fixed while he takes the limit as acceleration approaches infinity, not treating v as unknown and trying to find its value in a double limit as acceleration approaches infinity and time approaches 0?

Yes, I can see what he did, It is pretty bad,

"Pretty bad" just in the sense that you don't think it's physically realistic, or do you think it's "pretty bad" in the sense that he made an actual error in mathematics or theoretical physics? If you admit there is no error on a mathematical/theoretical level, but claim that there is such an error in my own derivation, what mathematical/theoretical error do you think I made that Moller didn't also make?

it doesn't mean that you need to copy it mindlessly,

Of course I didn't "copy it mindlessly", my actual equations are different than his, and I posted most of those equations before you even mentioned the Moller textbook.

If you want a realistic scenario, feel free to use the one that describes the Haefele-Keating experiment.

I never claimed that the scenario of very fast acceleration was supposed to be "realistic". It is a theoretical example that sacrifices realism for theoretical simplicity, something every physics textbook does routinely, if you have a problem with this sort of thing you have a bizarre attitude that differs from that of the physics community as a whole.

In an earlier post I provided you with three published papers by recognized scientists that explain the role played by acceleration in calculating elapsed time from the PoV of the accelerated twin. Have you read them?

No, because it's irrelevant to this discussion, nothing I have said would imply I disagree with the idea that you can calculate elapsed time in a non-inertial frame of the accelerated twin, and that in this frame the behavior of the two clocks during the accelerating phase would play a crucial role. Perhaps you are trying to imply that I said acceleration is always irrelevant and that this is a counterexample, but this again represents an equivocation between the claims 1 and 2 I listed above, of which only 1 represents anything I have actually argued on this thread.

 

[starthaus]

Anyway, you are using a highly equivocal phrase when you say that I am arguing against the position that "the time differential is a function of acceleration". I'm sure you can see that the following two claims are distinct:

1) It is theoretically possible to come up with scenarios where the acceleration is so brief that the total elapsed time (and thus the 'time differential' with another clock) could be calculated very accurately by just adding the elapsed times on the inertial parts of the trip and ignoring the elapsed time during the accelerating phase

2) In all conceivable scenarios, the elapsed time during the accelerating phase can be ignored when calculating the total elapsed time (and thus the time differential)

Of course I have only argued for 1 above, I would never make an argument as boneheaded as 2.

Good, then we are done. Because scenario 1 has nothing to do with reality whereas scenario
2 is the one encountered in real life.

"Pretty bad" just in the sense that you don't think it's physically realistic, or do you think it's "pretty bad" in the sense that he made an actual error in mathematics or theoretical physics? If you admit there is no error on a mathematical/theoretical level, but claim that there is such an error in my own derivation, what mathematical/theoretical error do you think I made that Moller didn't also make?

Pretty bad as in unrealistic, accelerations in real life are less than 20g even for the fastest rockets/test planes.

Now, kev actually did make one minor error in the above, if T_a is the total acceleration time then as I said in post #96 the acceleration term should be \frac{2c}{a} \, \, arcsinh(a T_a / 2c), which only differs from his expression by those factors of 2. Anyway, if you substitute in T_a = \frac{2v_c}{a*\sqrt{ 1 - v_c^2/c^2}}, you get \frac{2c}{a} \, \, arcsinh(\frac{v_c}{c*\sqrt{ 1 - v_c^2/c^2}}). Clearly, this does approach 0 in the limit as a approaches infinity, since a appears only in the denominator of the expression outside.

Actually, if you took the limit when a->oo of the expression that doesn't mix in the cruising speed (i.e. \frac{2c}{a} \, \, arcsinh(a T_a / 2c),) you would be getting 0 by applying l'Hospital rule. Much cleaner than going through the circular argument about v.

No, because it's irrelevant to this discussion, nothing I have said would imply I disagree with the idea that you can calculate elapsed time in a non-inertial frame of the accelerated twin, and that in this frame the behavior of the two clocks during the accelerating phase would play a crucial role.

Good, then there is nothing further to argue about. Thank you for clarifying your position.

 

[JesseM]

For the live of me I cannot understand why JesseM and kev are so adamant to keep acceleration out of the picture.

This is way too broad a summary of my position, especially given that I already gave you a more nuanced explanation of what aspects of acceleration can be ignored and what aspects are vitally important back in post #120:

I don't know what you mean by "completely ignore", but by "ignore" I only mean that you don't have to consider the contribution that the accelerating phase makes to the total elapsed proper time, since you are treating the acceleration as instantaneous (I think my meaning was fairly clear from the context, especially given my comment 'you don't actually need to consider the acceleration phase when calculating the elapsed proper time'). In that sense the calculation I already gave you in my last post ignores acceleration, though of course acceleration plays a role in that it explains why v₁ on the outbound leg may be different than v₂ on the inbound leg (and since inertial paths are geodesics and geodesics always maximize proper time, it plays a conceptual role in understanding why the inertial twin is always the one who ages more than the one who turns around, similar to the idea that a straight line between two points in Euclidean geometry always has a shorter length than any bent path between the same points).

If you do a search for previous threads where I discussed the twin paradox, you will see that I always emphasize that acceleration is crucial for understanding on a conceptual level why there is no symmetry in each twin's view of the other, and why the twin that accelerates will genuinely have aged less than the one that doesn't. So please don't continue to say that I am trying to "keep acceleration out of the picture" because it just isn't true. Rather, my point is just that when calculating the elapsed time of the non-inertial twin, in a theoretical problem of the kind that routinely appear in textbooks one can just assume that the accelerating phase is extremely brief, so that we don't need to worry about the time elapsed during the accelerating phase when calculating the total elapsed time. And I'm not in any way "adamant" that this is an assumption one should make when thinking about the twin paradox, just saying it's one you can make (and again, is routinely made in textbooks to simplify things), that one isn't making an error on theoretical or mathematical level by doing so. And I wouldn't have needed to spend so much time defending this if starthaus hadn't made a big show of claiming that kev and I are making an "error" when we talk about this scenario.

If we take the simple example of a traveling twin going to a place of a distance x away and returning then the total time dilation depends completely on the rate and duration of the accelerations (4 or 3 if we combine the turn around).

Not if you make the routine assumption that the time of the accelerations is very brief compared to the time of the constant-velocity outbound and inbound phases of the trip. Do you disagree that in this case the accelerations contribute very little to the elapsed time, and that in the limit as the coordinate time of the accelerations approaches zero (instantaneous accelerations) while the velocities of the inertial phases are kept constant, the total elapsed proper time approaches a simple sum of the elapsed proper time in the two inertial phases? If so that is the only point I have been making on this thread, and starthaus has been making a big stink over it for some reason.

 

[Passionflower]

This is way too broad a summary of my position, especially given that I already gave you a more nuanced explanation of what aspects of acceleration can be ignored and what aspects are vitally important back in post #120:

Ok, I accept that.

If you do a search for previous threads where I discussed the twin paradox, you will see that I always emphasize that acceleration is crucial for understanding on a conceptual level why there is no symmetry in each twin's view of the other, and why the twin that accelerates will genuinely have aged less than the one that doesn't. So please don't continue to say that I am trying to "keep acceleration out of the picture" because it just isn't true.

Ok.

Rather, my point is just that when calculating the elapsed time of the non-inertial twin, in a theoretical problem of the kind that routinely appear in textbooks one can just assume that the accelerating phase is extremely brief, so that we don't need to worry about the time elapsed during the accelerating phase when calculating the total elapsed time. And I'm not in any way "adamant" that this is an assumption one should make when thinking about the twin paradox, just saying it's one you can make (and again, is routinely made in textbooks to simplify things), that one isn't making an error on theoretical or mathematical level by doing so.

Ok, extremely brief is possible, but making it zero creates pathological conditions.

Not if you make the routine assumption that the time of the accelerations is very brief compared to the time of the constant-velocity outbound and inbound phases of the trip. Do you disagree that in this case the accelerations contribute very little to the elapsed time, and that in the limit as the coordinate time of the accelerations approaches zero (instantaneous accelerations) while the velocities of the inertial phases are kept constant, the total elapsed proper time approaches a simple sum of the elapsed proper time in the two inertial phases?

Do you understand I have trouble with the statement I highlighted?
Just by making the acceleration period shorter does not make it less important, acceleration is of the essence in the twin experiment. Acceleration sets the clock rate. Accelerating a little but for a long period can have the same effect on gamma as accelerating very much for a very short period.

 

[JesseM]

Good, then we are done. Because scenario 1 has nothing to do with reality whereas scenario
2 is the one encountered in real life.

Good, so I guess you retract your claim that kev and I have made any "error", since of course lack of realism is not an "error" in a theoretical example, the only possible errors in theoretical examples are in mathematics or in steps that violate the basic laws of physics.

Pretty bad as in unrealistic.

Good, no actual error then. By the way, if you complain about "unrealistic" examples you'll need to complain about virtually every example of a spacetime that appears in a textbook which discusses general relativity, they pretty much always contain unrealistic assumptions like the notion that the spacetime is asymptotically flat, or eternally static (as in the external Schwarzschild solution), or that the distribution of matter is completely uniform (as in cosmological solutions), etc.

Actually, if you took the limit when a->oo of the expression that doesn't mix in the cruising speed (i.e. \frac{2c}{a} \, \, arcsinh(a T_a / 2c),) you would be getting 0 by applying l'Hospital rule. Much cleaner than going through the circular argument about v.

You can't take the limit of that expression if you don't know how T_a depends on a. Were you just assuming that T_a is a constant? But then in the limit as a approaches infinity, the final cruising speed will approach c, and you are left with the conclusion that the total elapsed time for the entire trip approaches 0 as well--a much weirder assumption physically than just the assumption that the acceleration time is very small compared to the time spent moving inertially!

No, because it's irrelevant to this discussion, nothing I have said would imply I disagree with the idea that you can calculate elapsed time in a non-inertial frame of the accelerated twin, and that in this frame the behavior of the two clocks during the accelerating phase would play a crucial role.

Good, then there is nothing further to argue about.

Glad you see that you now understand you have been attacking a strawman this whole time. In future, if you want to avoid a lot of wasted time, try to avoid leaping to uncharitable conclusions about what people meant, instead if you think someone is saying something clearly erroneous stop and consider if there may be an alternate interpretation of their words, and ask politely for clarification instead of rushing into attack immediately.

 

[JesseM]

Not if you make the routine assumption that the time of the accelerations is very brief compared to the time of the constant-velocity outbound and inbound phases of the trip. Do you disagree that in this case the accelerations contribute very little to the elapsed time, and that in the limit as the coordinate time of the accelerations approaches zero (instantaneous accelerations) while the velocities of the inertial phases are kept constant, the total elapsed proper time approaches a simple sum of the elapsed proper time in the two inertial phases?

Do you understand I have trouble with the statement I highlighted?
Just by making the acceleration period shorter does not make it less important, acceleration is of the essence in the twin experiment. Acceleration sets the clock rate. Accelerating a little but for a long period can have the same effect on gamma as accelerating very much for a very short period.

When I used the word "contribute" above, I was referring to a numerical contribution only, not to the conceptual or causal significance of the acceleration. If we break down the total elapsed proper time into (proper time elapsed in inertial phase) + (proper time elapsed in acceleration phase), then as the coordinate time of the acceleration phase gets very small, (proper time elapsed in acceleration phase) gets very small too, so this term makes very little numerical contribution to the total elapsed proper time. That's all I meant, apologies if the language was ambiguous.

 

[Passionflower]

If we break down the total elapsed proper time into (proper time elapsed in inertial phase) + (proper time elapsed in acceleration phase), then as the coordinate time of the acceleration phase gets very small, (proper time elapsed in acceleration phase) gets very small too...

Up to here I fully agree.

, so this term makes very little numerical contribution to the total elapsed proper time.

The term as in "the elapsed proper time during acceleration" yes. But "this term" and "acceleration" are not synonymous. The rate and duration of this acceleration phase has a very big impact on the second term.

No need for apologies as language is almost always ambiguous.

 

[starthaus]

Good, so I guess you retract your claim that kev and I have made any "error", since of course lack of realism is not an "error" in a theoretical example, the only possible errors in theoretical examples are in mathematics or in steps that violate the basic laws of physics.

Sure you did, your language was ambigous. It took tens of posts for you to admit it by separating the two cases. You were talking about a didactical case, I was talking about the real life case. Your didactical approach does not apply to real life.


.

 

[JesseM]

Sure you did, your language was ambigous. It took tens of posts for you to admit it by separating the two cases.

Presumably you are referring to the "two cases" I distinguished in this part of post #172:

I'm sure you can see that the following two claims are distinct:

1) It is theoretically possible to come up with scenarios where the acceleration is so brief that the total elapsed time (and thus the 'time differential' with another clock) could be calculated very accurately by just adding the elapsed times on the inertial parts of the trip and ignoring the elapsed time during the accelerating phase

2) In all conceivable scenarios, the elapsed time during the accelerating phase can be ignored when calculating the total elapsed time (and thus the time differential)

Of course I have only argued for 1 above, I would never make an argument as boneheaded as 2.

So you really think the problem was that I took "tens of posts" before finally "admitting" that I was not advocating 2? It couldn't be that you just rushed to an uncharitable interpretation and then failed to read my posts carefully enough? Read again what I wrote back in post #155:

I never expressed the "misconception" that "the elapsed time differential is not dependent on acceleration" (in the sense that you can find specific scenarios where the elapsed time is a function of acceleration--I would say that it's impossible to write a general expression for elapsed time as a function of acceleration regardless of the motion of the object, whereas you can write a general expression for elapsed time as a function of v(t)).

And in that post I also responded to statement you made to kev that a certain calculation "clearly contradicts your [kev's] earlier claim that the contribution of the acceleration period is negligible" by quoting his post #13:

That may be true, but I have shown in the last post, that the time dilation due to acceleration in the twins paradox can be reduced to a negligible error, e.g less than 4 seconds "lost" due to acceleration time dilation compared to 8 years "lost" due to constant velocity.

"Can be reduced" (by making the acceleration brief), not "always reduces to a negligible amount in all problems" (regardless of the length of the acceleration). And a later response by kev to your post #97 in post #112 also shows that this he was talking about choosing a problem where the acceleration was brief

The distinction I made there between a claim that time elapsed during the acceleration phase "can be reduced" a negligible amount by an appropriate choice of problem to analyze, vs. the idea that it "always reduces to a negligible amount in all problems" which was a claim kev didn't make, corresponds precisely to the distinction between claims 1) and 2) which I made in the more recent post #172.

But in case you missed that distinction I made in the long #155, I made the same distinction again in the very short post #166:

In post #80 you quote a statement by him made in post #24. If you look at the context of the statement that "I have taken the time dilation due to acceleration into account and it is insignificant compared to the velocity time dilation", it is obvious he is saying this is true for the particular numerical example he presented in that post, he wasn't saying the time dilation in the accelerating phase would always be insignificant in any possible version of the twin paradox.

And, let's see, I also repeated the point again in my next post #168:

You are attacking fantasy strawmen again, neither I nor kev ever said the elapsed time during the acceleration phase would be negligible in all possible experiments, certainly not in the Hafele-Keating experiment where the planes were accelerating for pretty much the entire trip. The point we're making is just that you can make the elapsed time during acceleration negligible if you consider a scenario where the acceleration is large and brief.

So, looks like I have been pretty clear about this distinction all along.

By the way, I also want to point out that from the very beginning when I asked for examples of errors in post #137, I specifically said I wanted clear mathematical or physical errors, not just complaints about unrealistic simplifications of the kind routinely used in textbooks:

Point to any "misconceptions" or "errors" I have made. The only reason for the initial difference between your equation and mind (and kev's) is that we made different physical assumptions, I assumed the only acceleration was at the turnaround while you assumed (or copied your equation from a wikipedia page which assumed) an initial and final acceleration as well. If you think I have made any physical or mathematical errors aside given my physical assumptions please point them out instead of just making vague accusations.

...

I only saw you object on the basis that he made different physical assumptions than you, or that he used "hacks" (simplifications of the type that are routinely used in textbook discussions), or that you interpreted an English statement by him in a silly uncharitable way (like the one below). Again, show me a single clear error in his actual calculations.

So either you initially thought kev and I had made errors on a physical/mathematical level and have changed your mind, or you didn't read that initial request for examples carefully enough to realize that I wasn't just asking for examples where kev and I had made simplifications that would be unrealistic in actual experiments.

 

[Austin0]

For the live of me I cannot understand why JesseM and kev are so adamant to keep acceleration out of the picture.

.

It has something to do with never admitting that they are wrong.
.

From my own direct experience in a number of previous threads I have seen both JesseM and kev recognize and
admit errors of various kinds.
This is absolutely not true wrt certain other well known members whom need not be named.

 

[starthaus]

Presumably you are referring to the "two cases" I distinguished here:

So you really think the problem was that I took "tens of posts" before finally "admitting" that I was not advocating 2?

Precisely. Otherwise, you would have not taken up to post 172 to admit that acceleration plays a key role in the difference in elapsed time. Both Passionflower and I jhave asked you this direct question multiple times but post 172 the first time you made a clear statement of your position on the issue.

So, looks like I have been pretty clear about this distinction all along.

You didn't come clean until post 172. Had you admitted earlier, we would not have spent another 90 posts on the issue. Not that they were wasted posts, I worked out the complete case for asymettric accelerations at Passionflower's suggestion.

 

[starthaus]

I know it is true intuitively, but the maths might get rather involved. Here is my shot at it.

Let us say that rocket B accelerates to 0.8c in one second and then cruises, and exactly 10 years after B departed, A accelerates to 0.8c in one second (as measured by an observer C that remains at rest in the frame that A and B were originally at rest) and now both A and B are rest in new frame that has velocity 0.8c relative to the original frame.

From the accelerating rocket equations of Baez, that I gave earlier, the proper acceleration of a rocket with terminal velocity 0.8c (as measured in the unaccelerated frame) after a time of 1 second (as measured in the unaccelerated frame) is given as:

a = (v/t)/sqrt[1-(v/c)^2] = 0.8*0.6 = 0.48

The proper elapsed time (T) of the rocket during the acceleration phase is given by Baez as:

T = (c/a)*asinh(at/c) = (1/0.48)*asinh(0.48) = 0.9651 seconds.

using units of c=1. Note that the proper elapsed time is not much less than the 1 second measured by the unaccelerated observer C.

During the cruise phase of B's journey the elapsed proper time of B's clock according to C is 10years*0.6 = 6 years so the total elapsed proper time of B's clock is 6years + 0.9651 seconds in C's frame.

The total elapsed proper time of rocket A from the time B took of to the time A joined B in the new rest frame is 10 years + 0.9651s seconds according to C.

In frame C the elapsed time of A is obviously much greater than the elapsed time of B.

Now we look at the times measured by an observer that was at rest in a frame (D) that was always moving with velocity 0.8c relative to frame C. i.e frame D is the final rest frame of observers A and B.

In frame C the elapsed time from B taking off, to A joining B in frame D, was 10 years + 1 second, so in frame D the elapsed time between the two events is (10y + 1s)/0.6 = 16.6667 years + 1.66667 seconds. Other than the initial 1.6666 seconds that rocket B initially took to accelerate to rest in frame D, rocket B has been at rest in frame D for 16.6667 years so the total elapsed proper time of B's clock, according to C is 16.6667 years (plus the 0.9651 seconds of proper time B spent accelerating). The elapsed proper time of rocket A, according to observer D is 10 years (plus the 0.9651 seconds of proper time A spent accelerating).

So in frame D, the time that elapses between B taking off and A joining B in frame D is 10 years + 0.9651 seconds proper time as measured by clock A and 16.6667 years + 0.9651 seconds proper time as measured by clock B. In frame D much more proper time has elapsed on clock B while in the original frame C, much more proper time elapses on clock A.

It can also be seen that I have taken the time dilation due to acceleration into account and it is insignificant compared to the velocity time dilation and an unnecessary complication.

1.Time dilation due to acceleration is not insignificant, it can play a major role in real life, as in https://www.physicsforums.com/blog.php?b=1954 the results of the Haefele-Keating or Vessot experiments.

2. You got an "insignificant" value because of the way you contrived your thought experiment.

3. Try calculating what value acceleration you need in order to ramp up to 0.8c in 1s and you'll see how far off from reality your thought experiment is. As a reference, accelerations of 10-20g are the top of the achievable scale.

 

[JesseM]

Precisely. Otherwise, you would have not taken up to post 172 to admit that acceleration plays a key role in the difference in elapsed time.

Uhhh, I just quoted some earlier posts of mine where I clearly said that the acceleration phase would make a significant contribution in experiments where it lasts a significant time, like post #168:

neither I nor kev ever said the elapsed time during the acceleration phase would be negligible in all possible experiments, certainly not in the Hafele-Keating experiment where the planes were accelerating for pretty much the entire trip.

Both Passionflower and I jhave asked you this direct question multiple times but post 172 the first time you made a clear statement of your position on the issue.

Point to a single post where you or Passionflower asked a "direct question" about the time elapsed during accelerations of significant length where I did not say that in this case the proper time during the accelerating phase would be significant. If you're going to bring Passionflower's posts into it, I also made clear in posts to him that even when acceleration is quasi-instantaneous, the acceleration also plays a crucial conceptual role in understanding why one twin ages less, even if the proper time elapsed during the acceleration phase is negligible (see post #120 for example).

 

[starthaus]

Uhhh, I just quoted some earlier posts of mine where I clearly said that the acceleration phase would make a significant contribution, like post #168:

Ok, so it didn't take up to post 172 to admit that acceleration plays a key role in the time dilation, it took you up to post 168.

 

[JesseM]

Ok, so it didn't take up to post 172 to admit that acceleration plays a key role in the time dilation, it took you up to post 168.

Yes, because post #167 was the first post where you brought up the Hafele-Keating experiment as a "rebuttal", before that you gave no clear indication that you were misreading me in such a bizarre way as to think I would disagree that in some experiments the proper time during the accelerating phase would make a large contribution. I note that you didn't answer my request to back up your statement that you and Passionflower had ever asked me "direct questions" about my opinion on whether the accelerating phase would make a significant contribution in cases other than the one that were originally the topic of discussion, namely kev's example where the acceleration becomes very large so the time needed to reach the cruising velocity becomes very small. Why should I randomly volunteer the fact that the acceleration phase can have significant proper time in cases other than the one we were discussing, when it had no relevance to the issue of "errors" in kev's argument, and when you didn't ask me about this? Besides, I think basic reading comprehension would tell you that when in post #155 I quoted kev saying that the time dilation can be reduced to a negligible error, and then commented:

"Can be reduced" (by making the acceleration brief), not "always reduces to a negligible amount in all problems" (regardless of the length of the acceleration).

...the clear implication was that I understand the time dilation during the accelerating phase would not "always reduce to a negligible amount in all problems". But if there was any doubt in your mind that this was my implication, you could have actually done what you misremembered doing and asked a "direct question" about the issue.

 

[starthaus]

Yes, because post #167 was the first post where you brought up the Hafele-Keating experiment as a "rebuttal", before that you gave no clear indication that you were misreading me in such a bizarre way as to think I would disagree that in some experiments the proper time during the accelerating phase would make a large contribution. I note that you didn't answer my request to back up your statement that you and Passionflower had ever asked me "direct questions" about my opinion on whether the accelerating phase would make a significant contribution in cases other than the one that were originally the topic of discussion, namely kev's example where the acceleration becomes very large so the time needed to reach the cruising velocity becomes very small. Why should I randomly volunteer the fact that the acceleration phase can have significant proper time in cases other than the one we were discussing, when it had no relevance to the issue of "errors" in kev's argument, and when you didn't ask me about this? Besides, I think basic reading comprehension would tell you that when in post #155 I quoted kev saying that the time dilation can be reduced to a negligible error, and then commented:

...the clear implication was that I understand the time dilation during the accelerating phase would not "always reduce to a negligible amount in all problems". But if there was any doubt in your mind that this was my implication, you could have actually done what you misremembered doing and asked a "direct question" about the issue.

Both I and passionflower asked you "direct questions" and you kept answering in such a fashion that led us to believe that you negated the role of acceleration. The first time you gave a straight answer to the question was post 168. The first time you gave a complete answer was post 172.

 

[JesseM]

Both I and passionflower asked you "direct questions"

OK so, give an example of a post where you or Passionflower asked such a "direct question" about this issue which I didn't answer (I suspect if you had any you probably would have brought them up by now). As far as I can recall, you didn't ask questions about whether I disputed that the acceleration phase would make a significant contribution to the proper time differential in other examples besides kev's (like the Hafele-Keating experiment), you just jumped to the conclusion I was saying it never would based on your determination to believe I was making some stupid blunder.

and you kept answering in such a fashion that led us to believe that you negated the role of acceleration.

"Negated the role of acceleration" is a totally vague phrase which allows you to equivocate between different meanings--I did say that the acceleration phase didn't contribute significantly to the total proper time differential in kev's example (claim 1 from post 172), but saying we can negate the contribution of the accelerating phase in that example doesn't somehow imply we can negate it in all examples (claim 2 from post 172). Since you brought up my posts to Passionflower, perhaps you didn't see my recent post where I reminded Passionflower that as far back as post #120 I did emphasize that acceleration plays an important role in a conceptual sense, regardless of the actual length of the acceleration phase, so that's another sense in which acceleration cannot be "negated":

I don't know what you mean by "completely ignore", but by "ignore" I only mean that you don't have to consider the contribution that the accelerating phase makes to the total elapsed proper time, since you are treating the acceleration as instantaneous (I think my meaning was fairly clear from the context, especially given my comment 'you don't actually need to consider the acceleration phase when calculating the elapsed proper time'). In that sense the calculation I already gave you in my last post ignores acceleration, though of course acceleration plays a role in that it explains why v₁ on the outbound leg may be different than v₂ on the inbound leg (and since inertial paths are geodesics and geodesics always maximize proper time, it plays a conceptual role in understanding why the inertial twin is always the one who ages more than the one who turns around, similar to the idea that a straight line between two points in Euclidean geometry always has a shorter length than any bent path between the same points).

I would suggest you try to think more carefully and precisely about what your opponents are actually saying and not use vague and over-broad formulations which could have many possible meanings, like accusing people of denying that "the time differential is a function of acceleration" or of believing that "acceleration can be negated". You have a constant tendency to assume everyone you disagree with is stupid/ignorant and to therefore read everything they say in the most uncharitable way (i.e., if an interpretation of someone's words occurs to you that would make them out to be taking a stupid/ignorant position, you immediately jump to the conclusion that this is the correct interpretation and don't consider that there might be others or even ask questions to clarify what they're saying).

The first time you gave a straight answer to the question was post 168.

Straight answer to what question? Still waiting for an example of an earlier post where you asked some question related to this issue and I didn't give a straight answer, seems to me your memory is playing tricks on you.

 

[starthaus]

In that sense the calculation I already gave you in my last post ignores acceleration, though of course acceleration plays a role in that it explains why v₁ on the outbound leg may be different than v₂ on the inbound leg (and since inertial paths are geodesics and geodesics always maximize proper time, it plays a conceptual role in understanding why the inertial twin is always the one who ages more than the one who turns around, similar to the idea that a straight line between two points in Euclidean geometry always has a shorter length than any bent path between the same points).

OK, let's look at post 120 (still 40 posts away from post #80) so we can consider this as your first admission that acceleration plays a role in the differential elapsed time. Yet, you don't come out straight and saying this, instead , you make a claim that acceleration

"explains why v₁ on the outbound leg may be different than v₂ on the inbound leg "

The difference in elapsed time has nothing to do with the difference between the outbound speed v₁ and the inbound speed v₂, it has everything to do with the presence of acceleration during the journey. The fact that the two speeds are different plays no role.

 

[yuiop]

Actually this statement is pretty amusing, speaking of blunders. T_a is the duration the rocket accelerates (see the wiki page), there is no reason whatsoever why T_a should correlate to a.

JesseM has not made a mistake here. Here is a quote from the wikipedia page you mentioned stating the parameters of the scenario: http://en.wikipedia.org/wiki/Twin_p...lt_of_differences_in_twins.27_spacetime_paths

Phase 1: Rocket (with clock K') embarks with constant proper acceleration a during a time Ta as measured by clock K until it reaches some velocity V.

Ta is determined by the terminal velocity V and acceleration a.

You are being your usual unhelpful self by stating that JesseM's equation:

T_a = 2v/(a*sqrt[1 - v²/c²])

is incorrect without saying why or providing a corrected equation. As far as I can tell there is no error in his equation when he has defined T_a as the total round trip time measured in the inertial frame. For a one way trip from zero to terminal velocity v (or vice versa) the time measured in the inertial frame is simply T_a/2 = v/(a*sqrt[1 - v²/c²]).

I see your error, it is a repaet of the error that I flagged above:

T_a = 2v/(a*sqrt[1 - v²/c²])

What happens when a->oo? You are forgetting that , according to your starting point (see the formula for v), v->c so, the limit is undetermined (not that this even the correct way of calculating T_a) . Your math has deserted you today.

Nope, JesseM is right. See below.

, you get \frac{2c}{a} \, \, arcsinh(\frac{v}{c*\sqrt{ 1 - v^2/c^2}}). Clearly, this does approach 0 in the limit as a approaches infinity, since a appears only in the denominator of the expression outside.

Not so fast.

v=\frac{at}{\sqrt{1+(at/c)^2}}

so, a->oo implies v->c, meaning \gamma->oo

so you are rushing to conclusions. You have a lot more work to do in order to find the answer.

Actually, a->oo does not imply v->c although I too initially made that mistaken assumption. Consider the case whan a particle accelerates from zero to 0.8c in zero seconds (or an infinitesimal time interval if you prefer), then that implies infinite acceleration, but the terminal velocity is less than c.

Using the equation I gave earlier for the proper time t of an accelerating particle in terms of terminal velocity v and coordinate time T spent accelerating in the inertial frame:

t = T \sqrt{(1-v^2)}/v \, \, arcsinh(v/\sqrt{(1 - v^2)})

then in the limit as T goes to zero, t also goes to zero, for finite terminal velocity -1<v<1.

 

[starthaus]

As far as I can tell there is no error in his equation when he has defined T_a as the total round trip time measured in the inertial frame.

So, according to you, a rocket would decelerate from +0.8c to 0 , turn around and accelerate to -0.8c in zero time, right?

Actually, a-&gt;oo does not imply v-&gt;c although I too initially made that mistaken assumption. Consider the case whan a particle accelerates from zero to 0.8c in zero seconds

I considered it for exactly zero seconds and...I discarded it as unphysical.

You never answered the question I asked you: what is the necessary acceleration to reach 0.8c in 1s in your original scenario? Could you answer , please?

 

[yuiop]

So, according to you, a rocket would decelerate from +0.8c to 0 , turn around and accelerate to -0.8c in zero time, right?

You asked jesseM what would happen if the particle/rocket had infinite acceleration. I was answering that question. I recon infinite acceleration would be sufficient to accelerate a particle from +0.8c to -0.8c. Wouldn't you recon that was sufficient?

You never answered the question I asked you: what is the necessary acceleration to reach 0.8c in 1s in your original scenario? Could you answer , please?

That would be trivial to calculate, but I can not be bothered. The acceleration would be huge, but that does not make it in principle impossible. Your objections are petty as JesseM has pointed out. This forum of full of hypothetical questions about rockets traveling at 0.9c plus, when in practice our best rockets have only achieved less than 0.0001c. That does not mean a rocket could not in principle travel at 0.9c plus, although I expect there would be some practical difficulties. JesseM also gave the example of the barn and pole paradox. A man cannot run at 0.9c, but that does not mean we can not discuss the problem. You could criticize just about every thread in this forum, if we are not allowed to discuss hypothetical situations that have not been achieved in reality. Cut it out and grow up.

I could also point out that particles in particle accelerators are subjected to huge accelerations. If you want to change the the words "twins" and "rockets" to particles then that is fine.

The PF FAQ gives the example of muons in a cyclotron accelerated to 1,000,000,000,000,000,000 times the acceleration of gravity. That discredits your argument that we cannot discuss accelerations of greater than the practical limit of 10 or 20g.

Your objection also means that we are not allowed to discuss distances of the order of light years, because in practice actual manned missions have never gone beyond the Moon and we are barely out the back gate in terms of space exploration. Do you agree that despite the fact that no manned mission has ever gone much further than the Moon, that we could in principle go further?

Are you just deliberately trying to be disruptive with your inane, petty and juvenile objections to everything?

 

[starthaus]

You asked jesseM what would happen if the particle/rocket had infinite acceleration.

No, I asked you how realistic is such a thing. How realistic is it?

That would be trivial to calculate, but I can not be bothered.

I suggest you do, so you can learn how unphysical is the scenario you proposed at post #24.

Your objection also means that we are not allowed to discuss distances of the order of light years, because in practice actual manned missions have never gone beyond the Moon and we are barely out the back gate in terms of space exploration. Do you agree that despite the fact that no manned mission has ever gone much further than the Moon, that we could in principle go further?

Of course we can. As a matter of fact, the experiments that verified the twins paradox were quite "local". Yet, the experimenters, in constructing the theory of their experiments, applied math correctly, without hacks in the style of instantaneous turnaround (see Vessot) and they considered the effects of acceleration as non-negligible (see Vessot, Haefele-Keating). In a word, they did the physics the right way, they did not try to reduce it to a cartoon exercise.

Are you just deliberately trying to be disruptive with your inane, petty and juvenile objections to everything?

You are back to your ad-hominems, please keep it civil.

 

[JesseM]

OK, let's look at post 120 (still 40 posts away from post #80) so we can consider this as your first admission that acceleration plays a role in the differential elapsed time.

It is of course silly to portray this as an "admission" when I volunteered it unprompted, and there was nothing in my previous posts to suggest I would think otherwise. You may have fantasized I had been saying otherwise, but that's just because you aren't very good at reading what people actually say rather than over-generalizing narrow statements about specific situations into ridiculously broad claims like "acceleration doesn't matter" which you can then ridicule. If you think anything I said prior to #120 suggests I was making some broad claim like that, rather than simply a narrow claim about the meaning of the clock hypothesis or the fact that the acceleration phase can be left out of a calculation of total elapsed proper time if it is very brief, then please provide a quote (my posts on this thread begin with #81 so there aren't too many to read through).

Also, if you do a search for previous posts by me that use the words "twin" and "acceleration", you can see I usually emphasize that in a conceptual sense, acceleration does play a crucial role since the twin that accelerates to turn around will always have aged less than the twin that moves inertially--for example:

https://www.physicsforums.com/showthread.php?p=2692235#post2692235
https://www.physicsforums.com/showthread.php?p=2338031#post2338031
https://www.physicsforums.com/showthread.php?p=2253255#post2253255

Yet, you don't come out straight and saying this, instead , you make a claim that acceleration

"explains why v₁ on the outbound leg may be different than v₂ on the inbound leg "

The difference in elapsed time has nothing to do with the difference between the outbound speed v₁ and the inbound speed v₂, it has everything to do with the presence of acceleration during the journey. The fact that the two speeds are different plays no role.

It does help explain the difference in a situation where the acceleration lasts a very brief time compared to the time spent moving inertially (for example, if the acceleration lasts a few days but the inertial legs last years), which is the type of situation I had been discussing with Passionflower, because in that case the total elapsed time for the traveling twin will be approximately equal to T_1 * \sqrt{1 - v_1^2/c^2} + T_2 * \sqrt{1 - v_2^2/c^2} while the total elapsed time for the inertial twin is approximately equal to T_1 * \sqrt{1 - v_0^2/c^2} + T_2 * \sqrt{1 - v_0^2/c^2}, where v₀ is the speed of the inertial twin in the frame you're using and T₁ and T₂ are the times of the two inertial legs of the traveling twin's journey in this frame with speeds v₁ and v₂ respectively (so the sum of T₁ and T₂ is about equal to the total time between the twins departing and reuniting in this frame). As long as you don't use a frame where the two twins are moving at different speeds in the same direction during the outbound leg, then in order for the traveling twin to catch up with the inertial twin again after they've been moving apart, the speed v₂ of the traveling twin must be greater than the inertial twin's speed v₀ throughout the return leg of the trip, so the elapsed time of the traveling twin during this phase of the trip, T_2 * \sqrt{1 - v_1^2/c^2}, will be less than the elapsed time of the inertial twin T_2 * \sqrt{1 - v_0^2/c^2} (if you did use a frame where they were moving at different speeds in the same direction during the outbound leg, then in that case the traveling twin's speed during the outbound leg would have to be higher, so the traveling twin's elapsed proper time during the outbound leg would be smaller). Of course this is not sufficient to explain why the total elapsed time for the traveling twin is less (since v₀ may be greater than v₁ during the outbound leg), nor is it a general argument that covers cases where the acceleration is not brief compared to the inertial legs, but it does help to understand conceptually why you can analyze the twin paradox from different frames and still get the conclusion that the twin that changes velocities has aged less in total, especially when paired with some specific numerical example as I did in post 63 from this thread for example (stevmg found this type of explanation helpful on this thread as you can see from post #51 here, and I think this approach to explaining things has been helpful to others in the past).

 

[starthaus]

It is of course silly to portray this as an "admission" when I volunteered it unprompted, and there was nothing in my previous posts to suggest I would think otherwise.

So, you admit that your claim in post 120 about the time differential being due to the difference between v_1 and v_2 is incorrect?

You may have fantasized I had been saying otherwise, but that's just because you aren't very good at reading what people actually say rather than over-generalizing narrow statements about specific situations into ridiculously broad claims like "acceleration doesn't matter" which you can then ridicule.

You have not admitted the role of acceleration until post 172.

If you think anything I said prior to #120 suggests I was making some broad claim like that, rather than simply a narrow claim about the meaning of the clock hypothesis or the fact that the acceleration phase can be left out of a calculation of total elapsed proper time if it is very brief, then please provide a quote (my posts on this thread begin with #81 so there aren't too many to read through).

I simply think that your post 120 clearly reflects your misconception that the difference between the speeds v_1 and v_2 is the root of the time difference. I cited your claim exactly.

It does help explain the difference in a situation where the acceleration lasts a very brief time compared to the time spent moving inertially (for example, if the acceleration lasts a few days but the inertial legs last years), which is the type of situation I had been discussing with Passionflower, because in that case the total elapsed time for the traveling twin will be approximately equal to T_1 * \sqrt{1 - v_1^2/c^2} + T_2 * \sqrt{1 - v_2^2/c^2} while the total elapsed time for the inertial twin is approximately equal to T_1 * \sqrt{1 - v_0^2/c^2} + T_2 * \sqrt{1 - v_0^2/c^2}, where v₀ is the speed of the inertial twin in the frame you're using and T₁ and T₂ are the times of the two inertial legs of the traveling twin's journey in this frame with speeds v₁ and v₂ respectively (so the sum of T₁ and T₂ is about equal to the total time between the twins departing and reuniting in this frame). As long as you don't use a frame where the two twins are moving at different speeds in the same direction during the outbound leg, then in order for the traveling twin to catch up with the inertial twin again after they've been moving apart, the speed v₂ of the traveling twin must be greater than the inertial twin's speed v₀ throughout the return leg of the trip, so the elapsed time of the traveling twin during this phase of the trip, T_2 * \sqrt{1 - v_1^2/c^2}, will be less than the elapsed time of the inertial twin T_2 * \sqrt{1 - v_0^2/c^2} (if you did use a frame where they were moving at different speeds in the same direction during the outbound leg, then in that case the traveling twin's speed during the outbound leg would have to be higher, so the traveling twin's elapsed proper time during the outbound leg would be smaller). Of course this is not sufficient to explain why the total elapsed time for the traveling twin is less (since v₀ may be greater than v₁ during the outbound leg), nor is it a general argument that covers cases where the acceleration is not brief compared to the inertial legs, but it does help to understand conceptually why you can analyze the twin paradox from different frames and still get the conclusion that the twin that changes velocities has aged less in total, especially when paired with some specific numerical example as I did in post 63 from this thread for example (stevmg found this type of explanation helpful on this thread as you can see from post #51 here, and I think this approach to explaining things has been helpful to others in the past).

Note that you are still insisting on dealing with the limiting cases of zero acceleration period and that none of your formulas includes the terms expressed in acceleration. All your formulas deal with the extreme cases where acceleration is not relevant.

 

[JesseM]

So, you admit that your claim in post 120 about the time differential being due to the difference between v_1 and v_2 is incorrect?

"Due to" is a pretty imprecise phrase, but I explained above why I think that, in the specific situation I was talking about with Passionflower (where the acceleration period was negligible), it is helpful conceptually to think about the fact that the velocity of the traveling twin must be greater than that of the inertial twin on at least one of the two legs of the trip, regardless of what frame you use to define velocities. What's more if you look at my wording in post 120, what I said was that "the calculation I already gave you in my last post ignores acceleration ... though of course acceleration plays a role in that it explains why v₁ on the outbound leg may be different than v₂ on the inbound leg". In other words, here I was primarily referring to the relevance of acceleration to the calculation for elapsed time which used the formula T_1 * \sqrt{1 - v_1^2/c^2} + T_2 * \sqrt{1 - v_2^2/c^2}, not making a more general claim that the only conceptual significance of acceleration is that it allows the velocities to be different. In fact right after that sentence I immediately went on to discuss the broader conceptual significance of acceleration, saying "and since inertial paths are geodesics and geodesics always maximize proper time, it plays a conceptual role in understanding why the inertial twin is always the one who ages more than the one who turns around, similar to the idea that a straight line between two points in Euclidean geometry always has a shorter length than any bent path between the same points". In this comment, the "conceptual role" played by acceleration had nothing to do with different velocities, but rather to do with the fact that inertial paths are geodesics and geodesics always maximize the proper time in SR.

You have not admitted the role of acceleration until post 172.

Once again you use overly-broad phrases which carelessly lump together quite different notions. By "the role of acceleration", you could either be talking about A) the fact that the elapsed proper time during the acceleration phase must be taken into account to get an accurate the total elapsed proper time (which is true in experiments with long non-inertial phases like the Hafele-Keating experiment, but not true in cases where the time of the acceleration phase is negligible), or B) the fact that acceleration plays an important conceptual role in understanding why the non-inertial clock elapses less than the inertial one (true in all possible experiments involving a pair of clocks that are separated and later reunited, where one clock moves inertially between the separation and the reunion). If you are talking about B), I didn't actually talk about the conceptual importance of acceleration in post #172 at all, in fact I didn't bring up the conceptual importance of acceleration in a post to you until post #184 (because you never gave any indication that you wanted to discuss this topic), where I directed you to read my post #120 to Passionflower which did discuss the conceptual importance.

On the other hand if you are talking about A), then you have a short memory, since in post #185 you already admitted I had made clear this is true in some experiments earlier:

Ok, so it didn't take up to post 172 to admit that acceleration plays a key role in the time dilation, it took you up to post 168.

And of course in my reply in post #186 I pointed out that I had already made this pretty damn clear in post #155:

Besides, I think basic reading comprehension would tell you that when in post #155 I quoted kev saying that the time dilation can be reduced to a negligible error, and then commented:

"Can be reduced" (by making the acceleration brief), not "always reduces to a negligible amount in all problems" (regardless of the length of the acceleration).

...the clear implication was that I understand the time dilation during the accelerating phase would not "always reduce to a negligible amount in all problems".

...which you didn't give any substantive response to. I also pointed out in post #186 that the reason I didn't say anything more direct about experiments with significant non-inertial periods until post 168 was that I didn't even realize you were concerned with such cases, given that our discussion until then had been about what happens in kev's example where the acceleration is large and brief:

Yes, because post #167 was the first post where you brought up the Hafele-Keating experiment as a "rebuttal", before that you gave no clear indication that you were misreading me in such a bizarre way as to think I would disagree that in some experiments the proper time during the accelerating phase would make a large contribution.

You didn't give any substantive response to this either, you just made the false claim that 'Both I and passionflower asked you "direct questions" and you kept answering in such a fashion that led us to believe that you negated the role of acceleration.' I guess you now tacitly admit this was a false memory on your part, since you didn't even attempt to find an example of an earlier post where you or Passionflower had asked me any "direct question" about experiments where the non-inertial phase lasted a non-negligible time (or about the conceptual importance of acceleration, i.e. interpretation B above).

I simply think that your post 120 clearly reflects your misconception that the difference between the speeds v_1 and v_2 is the root of the time difference.

No, post 120 said that the conceptual importance of acceleration is that geodesics maximize proper time, and the inertial path is a geodesic in SR while the path with an acceleration is not. But of course it is also true that acceleration's relevance to the calculation T_1 * \sqrt{1 - v_1^2 / c^2} + T_2 * \sqrt{1 - v_2^2 /c^2} is that it explains why v₁ and v₂ can in general be different, whereas the corresponding calculation of the inertial twin's elapsed time would involve only a single velocity (but as I said above in #194, the fact that one formula involves two velocities v₁ and v₂ and the other only involves a single velocity v₀ is not sufficient to prove that one formula will yield a smaller total elapsed time, even if you add some reasoning to show that either v₁ or v₂ must be greater than v₀. Like I said though, I think if you add the latter observation it is helpful in understanding how it can be true that different frames all agree the non-inertial twin elapsed a smaller time, especially when combined with some specific scenario analyzed from the perspective of two different frames as an illustration).

Note that you are still insisting on dealing with the limiting cases of zero acceleration period

No I'm not:

It does help explain the difference in a situation where the acceleration lasts a very brief time compared to the time spent moving inertially (for example, if the acceleration lasts a few days but the inertial legs last years), which is the type of situation I had been discussing with Passionflower, because in that case the total elapsed time for the traveling twin will be approximately equal to T_1 * \sqrt{1 - v_1^2/c^2} + T_2 * \sqrt{1 - v_2^2/c^2}

And of course, the reason I am "insisting on" dealing with a case where the acceleration is small enough to be ignorable in an approximate calculation is because that is "the type of situation I had been discussing with Passionflower", so if you insist on this increasingly-desperate attempt to show I said something wrong in my previous posts, you'll have to deal with the case I was actually discussing at the time.

 

[yuiop]

Almost forgot this question!

O.K, you are right that this is how things would look from D's point of view, but that is just D's point of view and D is not a final arbitrator.

Yu are right D is not the final arbiter just an exception to agreement , on the other hand what frame is the final arbiter if there is not frame independent agreement?

There is no final arbiter for the elapsed times on A and D's clocks relative to each other when they never started and finished in the same place.

However, for the first leg of the journey, when B is going away from A, D sees A coming towards him at 0.5c (same as in the final leg) but B is going away from him at (0.8-0.5)/(1-0.8*0.5)=0.92857c so the time dilation of B's clock on the outward journey is much greater than the time dilation of A's clock, according to D. The first leg of the experiment took 14 years of A's proper time in D's frame. This equates to 14/sqrt(1-0.5^2)=16.1658 years by D's clock. D calculates that the proper time of B's clock on the outward journey "during the same time" is 16.1658*sqrt(1-0.92857^2)=6 years. Overall, D says 20 years of proper time elapsed on A's clock and 12 years of proper time elapsed on the combined times of B and C and all is good.

Well maybe not quite 100% just yet. .

How exactly does B take 16.1658 years on D's clock to travel approx. 6.9 ly at 0.92867c ?

Here is one way to picture it. Imagine a 8 ly long rod that extends from where B starts to where B finishes in A's frame. In D's frame this rod is moving at 0.5c to the right and B is chasing after it. In 16.166 yrs the end of the rod B is chasing moves 16.166*0.5=8.66 lys. Add this to the head start the end of the rod had and the total distance traveled by B in D's frame is 8.66+6.93=15.56 lys. This equates to a velocity of 15.56/16.166=0.928c.

You can do the calculations more formally, by considering the two events (A,B) and (B,C) in A's frame in x,t terms as (x1,t1) = (0,0) and (x2,t2) = (8,10) and converting those coordinates of the two events to the point of view of D using the Lorentz transformations. Just let me know if you need that clarifying.

 

[yuiop]

Here is one way to picture it. Imagine a 8 ly long rod that extends from where B starts to where B finishes in A's frame. In D's frame this rod is moving at 0.5c to the right and B is chasing after it. In 16.166 yrs the end of the rod B is chasing moves 16.166*0.5=8.66 lys. Add this to the head start the end of the rod had and the total distance traveled by B in D's frame is 8.66+6.93=15.56 lys. This equates to a velocity of 15.56/16.166=0.928c.

You can do the calculations more formally, by considering the two events (A,B) and (B,C) in A's frame in x,t terms as (x1,t1) = (0,0) and (x2,t2) = (8,10) and converting those coordinates of the two events to the point of view of D using the Lorentz transformations. Just let me know if you need that clarifying.

Mentz has just reminded me another thread about a program he created that allows you construct scenarios in a space time diagram and transform from one frame to another. Might be fun to try out Mentz's program and set up the scenario we were discussing and see if it adds up. See https://www.physicsforums.com/showpost.php?p=2864155&postcount=15

 

[Austin0]

Almost forgot this question!


There is no final arbiter for the elapsed times on A and D's clocks relative to each other when they never started and finished in the same place.



Here is one way to picture it. Imagine a 8 ly long rod at rest in frame A extending from B starts to where B finishes in A's frame. in D's frame this rod is moving at 0.5c to the right and B is chasing after it. In 16.166 yrs the end of the rod B is chasing moves 16.166*0.5=8.66 lys. Add this to the head start the end of the rod had and the total distance traveled by B in D's frame is 8.66+6.93=15.56 lys. This equates to a velocity of 15.56/16.166=0.928c.

You can do the calculations more formally, by considering the two events (A,B) and (B,C) in A's frame in x,t terms as (x1,t1) = (0,0) and (x2,t2) = (8,10) and converting those coordinates of the two events to the point of view of D using the Lorentz transformations. Just let me know if you need that clarifying.

I can see my bad habit of hasty mental appraisals may have struck again. :-)
I should tattoo on my forhead DO THE MATH before speaking.
In any case I need to refresh myself with this scenario as I have lost track with all the irrelevant dust in the air of this thread

 

[Passionflower]

You never answered the question I asked you: what is the necessary acceleration to reach 0.8c in 1s in your original scenario? Could you answer , please?

Hmm, out of curiosity I checked it, when we have:

\alpha = 329,356,000 m/s^2

\tau = 1 s

I get a coordinate velocity of 0.8 and a coordinate acceleration of 7488461 m/s² about 2.3% or the proper acceleration.

It is rather high but on the other hand if we consider the maximum acceleration from QT perspective then we should get to somewhere in the 5.56 x 10⁵¹ m/s² region.

Plenty of playroom it seems