How does the duration of acceleration affect the Twin Paradox?

[PAllen]

Then we both agree that the 'capsule summary" phrase saying that the time differential occurs during the acceleration is really bad, correct?

Personally, saying "accumulate proper time along the paths" is no different than saying "the time on the clocks as they travel" and doesn't off any additional explanation unless you say how you calculate the time on the clocks based on the speed in a frame.

Agree. In my mind, "accumulate proper time" is associated with a notion of line element, and this one notion applies with full generality to SR or GR; whether the line element is:

d tau^2 = d t^2 - (dx^2 + dy^2 + d z^2)/ c^2 [which trivially gives (1/gamma) dt by rearrangement]

or the more general GR metrics, you have one method, one concept.

 

[ghwellsjr]

Agree. In my mind, "accumulate proper time" is associated with a notion of line element, and this one notion applies with full generality to SR or GR; whether the line element is:

d tau^2 = d t^2 - (dx^2 + dy^2 + d z^2)/ c^2 [which trivially gives (1/gamma) dt by rearrangement]

or the more general GR metrics, you have one method, one concept.

But since proper time is the time displayed on any clock, saying "accumulate proper time" is no different than "explaining" the twin paradox by saying "look at the clocks of each twin at the start of the scenario and at the end of the scenario and that will tell you how much each one aged". It doesn't matter what you associate "accumulate proper time" with because it's the problem we're trying to explain, not the explanation of a different problem. I'm trying to at least associate the (dilated) times on the clocks with their speeds relative to a single frame of reference, not to each other.

 

[PAllen]

But since proper time is the time displayed on any clock, saying "accumulate proper time" is no different than "explaining" the twin paradox by saying "look at the clocks of each twin at the start of the scenario and at the end of the scenario and that will tell you how much each one aged". It doesn't matter what you associate "accumulate proper time" with because it's the problem we're trying to explain, not the explanation of a different problem. I'm trying to at least associate the (dilated) times on the clocks with their speeds relative to a single frame of reference, not to each other.

This is really good example of the inherent pitfalls (into which I've frequently slipped) of trying to intuitively express a technical concept. With all positive intent, we start with:

attempt 1) Twin that spends more (coordinate) time at higher speed in some frame ages less. Problem: example in post #24 shows this is wrong.

attempt 2) Twin with higher average speed in some frame ages less. Problem: false by same example

attempt 3) Twin that spends more time at higher gamma ages less. Problem: also false by same example. Twin A had higher gamma 80% of the time but ages more not less.

attempt 4) Twin with higher average gamma ages less. This can also be false. Consider twin A goes at speed with gamma of 3 the whole time. Consider twin B goes at speed with gamma of 1.5 half the time and gamma of 6 half the time. Twin B has higher average gamma (3.75) , but twin B ages more rather than less (assuming coordinate, e.g. t=6 over the paths, A ages 2, B ages 2.5).

attempt 5) Twin with lower average 1/gamma ages less. This is trivially true, but also just says "twin with lower average aging per coordinate time ages less", rather un-profound. We arrive, as the only true statement we can make, one twin ages less and here is how you calculate it.

This underscores my belief that all attempts at a simple rule (other than the formula itself) for which twin ages less run into trouble, even in SR.

 

[R Power]

If acceleration is necessary for differential aging, how can twin's paradox be solved without involving general relativity? i.e solely on the basis of STR.

 

[PAllen]

If acceleration is necessary for differential aging, how can twin's paradox be solved without involving general relativity? i.e solely on the basis of STR.

Erase from your memory the false statement than SR cannot deal with acceleration.

 

[R Power]

Erase from your memory the false statement than SR cannot deal with acceleration.

If so, then STR can't solve twin paradox? bcoz all the articles on internet say that it is acceleration which causes the the traveling twin to age less than the one who stayed on earth.

 

[PAllen]

If so, then STR can't solve twin paradox? bcoz all the articles on internet say that it is acceleration which causes the the traveling twin to age less than the one who stayed on earth.

SR can solve it, did solve it, right from when it was first presented by Einstein. Ignoring gravity (or topologically non-trivial universes), acceleration is necessary to have true twin 'paradox', but it is false to say the acceleration causes the age difference, or that the age difference is localized in any way to a period of acceleration.

 

[R Power]

SR can solve it, did solve it, right from when it was first presented by Einstein. Ignoring gravity (or topologically non-trivial universes), acceleration is necessary to have true twin 'paradox', but it is false to say the acceleration causes the age difference, or that the age difference is localized in any way to a period of acceleration.

Then what causes the differential aging? How does STR solves the paradox. This may be very elementary question but I don't have solid understanding of relativity. Can you explain how STR solves paradox or can you give me good references where I can get my answer?

 

[PAllen]

Then what causes the differential aging? How does STR solves the paradox. This may be very elementary question but I don't have solid understanding of relativity. Can you explain how STR solves paradox or can you give me good references where I can get my answer?

Start here:

http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html

 

[harrylin]

[..] I'm trying to at least associate the (dilated) times on the clocks with their speeds relative to a single frame of reference, not to each other.

That is also how Einstein (1905) analysed the difference in clock readings and how Langevin (1911) discussed the difference in ages. I find that the clearest way of presenting it.

 

[harrylin]

Then what causes the differential aging? How does STR solves the paradox. This may be very elementary question but I don't have solid understanding of relativity. Can you explain how STR solves paradox or can you give me good references where I can get my answer?

- Done in post #22; it's not really considered paradoxical in SR.
Note also that Einstein already gave a correct but one-sided analysis in his 1905 paper:
section 4 of http://www.fourmilab.ch/etexts/einstein/specrel/www/

 

[PAllen]

That is also how Einstein (1905) analysed the difference in clock readings and how Langevin (1911) discussed the difference in ages. I find that the clearest way of presenting it.

And this I have no problem with at all. This is what I've been saying: pick a frame, accumulate dilated time (=proper time) along different paths. Compare. Result same no matter what frame you use for the analysis. Where I have a problem is attempts to state some other simple rule to predict or explain this - all such that I've ever seen I consider simply wrong.

 

[ghwellsjr]

Then what causes the differential aging? How does STR solves the paradox. This may be very elementary question but I don't have solid understanding of relativity. Can you explain how STR solves paradox or can you give me good references where I can get my answer?

Since you asked, I will show you how STR solves the twin paradox. We'll use the scenario that PAllen presented in post #24:

Consider that, while neither twin is ever inertial (due to continuous changes in direction), twin A is always moving at speed .4c in this chosen inertial frame. Suppose twin B is moving .1c for 80% of the coordinate time between separate and meet up, and at .99999c for 20% of the coordinate time.

And I'm going to use the process I described in post #16:

But this thread and all the discussion up to this point has been about the Twin Paradox where they start out together, separate, and come back together and I'm saying that if you agree to ignore gravity, then you can analyze the scenario in any single inertial Frame of Reference and the "time spent at a higher speed" is defined uniquely in that FoR and the "higher speed" is defined uniquely in that FoR and we're talking about the coordinate time of each body in that FoR and we apply Einstein's time dilation formula to convert coordinate time into proper time for each body and then we see how much proper time has accumulated for each body as it travels at different speeds according to the FoR for whatever coordinate time intervals from the time they separated until the time they reunite and we get the amount that each one aged and subtract them and we have the differential aging and no time disappeared or needs to be accounted for.

I know that's a mouthful but it's really very simple to analyze using Einstein's formula to get the proper time interval, τ, (tau, the time interval on a clock) as a function of its speed, β, (beta, the speed as a fraction of the speed of light), and the coordinate time interval, t, as specified in the Frame of Reference:

τ = t√(1-β²)

First we analyze Twin A who travels at 0.4c for 100% of the time:

τ_A = 100%√(1-0.4²) = 100%√(1-0.16) = 100%√(0.84) = 100%(0.9165) = 91.65%

Now we analyze the first part of Twin B's trip at 0.1c for 80% of the time:

τ_B1 = 80%√(1-0.1²) = 80%√(1-0.01) = 80%√(0.99) = 80%(0.995) = 79.6%

And the last part of Twin B's trip at 0.99999c for 20% of the time:

τ_B2 = 20%√(1-0.99999²) = 20%√(1-0.99998) = 20%√(0.00002) = 20%(0.00447) = 0.09%

Finally we add the two parts of Twin B's trip to get the total time:

τ_B = τ_B1 + τ_B2 = 79.6% + 0.09% = 79.69%

So we see that Twin B with 79.69% of the coordinate time of the scenario ages less than Twin A with 91.65% of the coordinate time.

 

[ghwellsjr]

This underscores my belief that all attempts at a simple rule (other than the formula itself) for which twin ages less run into trouble, even in SR.

Consider the previous post as attempt 6), the only one I presented.

I made the statement "time spent at a higher speed" specifically regarding a variant of the Twin Paradox to show that acceleration is not the cause of one twin aging less. In that variant, both twins experience exactly the same acceleration but one twin spends very little time at the higher speed and most of the time at the slower (rest) speed while the other one spends practically all of time at the higher speed.

Here's exactly what I said and I continue to stand by it:

I'm glad you mentioned both bodies experiencing acceleration because we can have another variant of the Twin Paradox in which both of them accelerate exactly the same except that one returns home immediately while the other one continues far away from home before matching the acceleration of his twin and returning home a lot later. This clearly shows that it's not the acceleration that causes the differential aging but rather time spent at the relatively higher speed that causes the differential aging.