High School Does Acceleration Affect Time Dilation?

[Adel Makram]

I think they will agree on times (and be slower than your clock if you also are standing at the tangent point.) But I need to sit down with pencil and paper and work out the details. (It's been awhile since I played with this.) I will do this and post my figures. In the mean time, does this scenario address your question?

How do they agree in their time despite that both of them are non-inertial frames with different acceleration?

 

[Ibix]

How do they agree in their time despite that both of them are non-inertial frames with different acceleration?

Because their paths through spacetime are the same "length".

 

[Adel Makram]

Because their paths through spacetime are the same "length".

But I think their metric are different in their respective accelerating frames. And the metric is a function of the acceleration not as simple as in Minkowski metric.

 

[Adel Makram]

But I think their metric are different in their respective accelerating frames. And the metric is a function of the acceleration not as simple as in Minkowski metric.

But this confuses me again because this would mean the proper time is not invariant. I mean in accelerating frame with no gravity, will the metric in front of $dt$ be a function of the acceleration? if yes, then how the proper time attached to the clock at rest in this frame is invariant?

 

[Dale]

But this confuses me again because this would mean the proper time is not invariant.

Proper time is invariant.

if yes, then how the proper time attached to the clock at rest in this frame is invariant?

In the case of the accelerated frame the proper acceleration is in the metric. In the case of an inertial frame the proper acceleration is in the expression of the worldline. Either way the proper time is affected by the proper acceleration.

 

[Raymond Potvin]

Think this through a little more. What if the clocks start out moving and one of them accelerates to rest?

Hi Dale,

I see two possibilities:

If both clocks are at rest side by side and clock A accelerates away from clock B and then decelerates to rest with regard to clock B after a while, then to me, clock A is the one which will have slowed down even if we cannot measure it from a distance, which should show if we accelerate clock A towards clock B and reunite the two clocks again.

If clock B accelerates to rest with regard to clock A after clock A has accelerated away from it, then to me, clock A would still have slowed down with regard to clock B for a while, which should also show if we accelerate clock A towards clock B and reunite the two clocks. Those are circumstances where we know which clock has accelerated, thus which one is actually moving with regard to the other. That's what happens when we send probes for instance, or when we accelerate particles. We also know that an atmospheric muon lives longer than a laboratory one because we know where it started to move and at what speed it has traveled. When we know which clock is traveling, which is the case for practical problems, we can still use relativistic calculations to know how much it has slowed down even if it is no more a relativity problem. Difficult relativity problems seem to be reserved to situations where it is impossible to tell where the motion comes from, thus to useless situations.

 

[Dale]

I see two possibilities

Both of the possibilities you mention are more complicated than the scenario you described in post 25. In post 25 you had only a single acceleration period for a single clock. Consider just that scenario from the reference frame where the inertial clock ends at rest and then consider the same scenario from the reference frame where the accelerating clock ends at rest.

 

[Ibix]

@Raymond Potvin - that wasn't quite what Dale asked. He was asking what happens if two clocks are traveling side by side at 0.6c and one of them accelerates at 1g until it is at rest with respect to you. Compare and contrast what happens if the two clocks are side by side at rest with respect to you and one accelerates at 1g to 0.6c. Which one ticks slowly? Is it always the one that accelerated, which is what you seem to be claiming in #25.

Edit: as Dale already said...

 

[phinds]

Hi Dale,

I see two possibilities:

If both clocks are at rest side by side and clock A accelerates away from clock B and then decelerates to rest with regard to clock B after a while, then to me, clock A is the one which will have slowed down even if we cannot measure it from a distance, which should show if we accelerate clock A towards clock B and reunite the two clocks again.

Again, it will not have "slowed down", it will have produced fewer ticks. That is, it will still be ticking at one second per second but a different number of seconds will have passed for it (fewer in this example) because it took a different path through spacetime.

I keep pointing this out in response to your posts because "slowing down" is seriously misleading and people who believe that clocks run slower in their own reference frames get all confused as to how biological processes could slow down too (as they would have to if "slowing down" were true).

 

[Raymond Potvin]

Again, it will not have "slowed down", it will have produced fewer ticks. That is, it will still be ticking at one second per second but a different number of seconds will have passed for it (fewer in this example) because it took a different path through spacetime.

I keep pointing this out in response to your posts because "slowing down" is seriously misleading and people who believe that clocks run slower in their own reference frames get all confused as to how biological processes could slow down too (as they would have to if "slowing down" were true).

Hi Phinds,

I prefer to call a cat a cat: if something ages less because of motion, then I prefer to look for a physical phenomenon. If particles' frequencies go down when we accelerate them, then I prefer to attribute this dilation to the time their components take to produce those frequencies. Of course it doesn't work if we don't know they are the ones that have accelerated with regard to the detector, but if we do, it seems to work. To me, if a twin ages less than the other, it is because his metabolism slows down during the time he is traveling with regard to his brother. If a clock records less time, it is because its atoms' frequencies slow down.

More generally, if light takes more time between the mirrors of the moving light clock, it means that the molecules of the mirrors take more time to reflect it, that the bonding between the atoms of those molecules also take more time to be executed, and so on for the components of those atoms. To me, this is the only way the laws of physics can stay the same for all observers on inertial motion, and also the only way to explain the null result of the MM experiment.

 

[Ibix]

If particles' frequencies go down when we accelerate them, then I prefer to attribute this dilation to the time their components take to produce those frequencies. Of course it doesn't work if we don't know they are the ones that have accelerated with regard to the detector, but if we do, it seems to work.

Contradicting the principle of relativity doesn't seem to me like a good way to go about understanding the theory of relativity.

 

[Raymond Potvin]

Both of the possibilities you mention are more complicated than the scenario you described in post 25. In post 25 you had only a single acceleration period for a single clock. Consider just that scenario from the reference frame where the inertial clock ends at rest and then consider the same scenario from the reference frame where the accelerating clock ends at rest.

I meant «at rest with regard to the other clock», not at rest with regard to a third observer as Ibix pointed out. The way the two clocks move with regard to one another does not depend on the way they move with regard to another observer, but the way they accelerate still does even if it is a bit more complicated to illustrate. There is more possibilities then, but if we study everyone of them, we should be able to use acceleration to tell which clock has slowed down with regard to each observer. If acceleration was not determinant, I think we couldn't tell which twin has aged less.

 

[Ibix]

If acceleration was not determinant, I think we couldn't tell which twin has aged less.

Not true. Pick a frame. Write down the speed $v(t)$ of one of the twins at all times $t$ between the first (t=0) and second (t=T) meetings of the twins as measured in that frame. Evaluate $$\tau=\int_0^T\sqrt {1-v^2 (t)/c^2}dt $$This is the age of that twin at their second meeting. Repeat for the second twin. You have your answer with no mention of acceleration. Note that since I only asked for speed not velocity the acceleration cannot, in general, be inferred from $v (t)$. Jambaugh's circular track for which $v$ is a constant but there is always acceleration is an extreme example of this.

 

[Dale]

if we study everyone of them, we should be able to use acceleration to tell which clock has slowed down with regard to each observer

Good luck with that. It won't work, but going through the exercise will be valuable for you. You will find that you need to know the velocity, not just the acceleration

 

[Raymond Potvin]

Good luck with that. It won't work, but going through the exercise will be valuable for you

I used two clocks only because the problem was easier to describe, which would be the case if, in the problem you asked me to solve, you could pick only one possibility where you think acceleration is not determinant.

 

[Raymond Potvin]

Not true. Pick a frame. Write down the speed $v(t)$ of one of the twins at all times $t$ between the first (t=0) and second (t=T) meetings of the twins as measured in that frame. Evaluate $$\tau=\int_0^T\sqrt {1-v^2 (t)/c^2}dt $$This is the age of that twin at their second meeting. Repeat for the second twin. You have your answer with no mention of acceleration. Note that since I only asked for speed not velocity the acceleration cannot, in general, be inferred from $v (t)$. Jambaugh's circular track for which $v$ is a constant but there is always acceleration is an extreme example of this.

Hi Ibix,

Since motion is relative, the relative speed would be the same for both twins if we could not tell which one has accelerated. If we start the experiment with both twins side by side in space for example, the twin that accelerates knows he does, and the twin that does not accelerate also knows he doesn't, so if the one that knows he has accelerated gets back to his twin later on, he knows he will have aged less, and he knows how much if he knows how much he has accelerated and how long the roundtrip took.

 

[Ibix]

It's trivial to set up situations where both twins undergo the same accelerations but end up different ages. A variant on Jambaugh's circular tracks will do it.

 

[Raymond Potvin]

Contradicting the principle of relativity doesn't seem to me like a good way to go about understanding the theory of relativity.

The relativity principle is about not knowing that we are moving, thus when we know and we need to calculate time dilation, it is easier not to refer to it.

 

[Bartolomeo]

It is possible completely rule out accelerations the following way. FIRST twin is at rest, the SECOND approaches him. When they meet, they synchronize clocks, their clocks show 0. Twins recede from each other and the SECOND meets another - THIRD twin who flies towards the FIRST. When they meet, they synchronize clocks. Let's say their clocks show 3. Then that THIRD twin meets FIRST and they compare clock readings again. THIRD clock will show less time.

In case if we consider the case (motion of twins) from any arbitrary chosen frame (as @Ibix proposed), the paradox simply turns into effect. Two twins move side by side. One if them suddenly stops. It is clear his clock now will tick faster than moving one. His (stopped) clock will tick at the same rate as any synchronized clock of that frame, in which motion takes place. Thus, it will be ticking faster than moving one. Then he suddenly starts (or passes clock readings to third brother, who passes by) and catches up moving one. It is easy to calculate, that while he overtakes moving one, his clock will show gamma times less time.

 

[Dale]

I used two clocks only because the problem was easier to describe, which would be the case if, in the problem you asked me to solve, you could pick only one possibility where you think acceleration is not determinant.

I already did that. Two clocks moving initially at the same velocity, the one on the right accelerates to the right. The one in the right may tick faster or it may tick slower, dependent on the initial velocity. The acceleration alone does not determine it, the velocity (in an inertial frame) does.

 

[phinds]

To me, if a twin ages less than the other, it is because his metabolism slows down during the time he is traveling with regard to his brother. If a clock records less time, it is because its atoms' frequencies slow down.

Well, if you insist, as you clearly do, on being completely and totally wrong, then there's nothing else I can do for you.

 

[Dale]

Since motion is relative, the relative speed would be the same for both twins if we could not tell which one has accelerated.

The v in @Ibix's formula (post 43) is the velocity relative to an inertial frame, not the relative speed. Your statement "motion is relative" is an incorrect statement of the first postulate.

 

[Raymond Potvin]

It's trivial to set up situations where both twins undergo the same accelerations but end up different ages. A variant on Jambaugh's circular tracks will do it.

Nevertheless, if the twins know their accelerations and the time spent between each of them, they also know how much they have aged since the beginning, so they can compare their calculations at the end to know which one is younger instead of looking at their clocks. All they need is a precise accelerometer and a precise clock.

 

[Dale]

instead of looking at their clocks. All they need is a precise accelerometer and a precise clock

Hahaha!

 

[Ibix]

All they need is a precise accelerometer and a precise clock.

Really? If I've got clock readings from both, why do I need the accelerometer to calculate their age? Why not just use the clock reading?

 

[Raymond Potvin]

I already did that. Two clocks moving initially at the same velocity, the one on the right accelerates to the right. The one in the right may tick faster or it may tick slower, dependent on the initial velocity. The acceleration alone does not determine it, the velocity (in an inertial frame) does.

You did not say that the clocks were initially moving, so I didn't know you added a third observer. Now I know, and of course, the accelerating clock can either decelerate or accelerate with regard to that observer, whereas it can only accelerate away from the other clock. I'm afraid there is no way to solve that kind of problem, unless maybe we put your observer side by side with the clocks in the beginning, and we accelerate it away from the clocks. This way, we know that his clock will be suffering time dilation with regard to the two other clocks, so if one of those clocks accelerates until it gets at rest with regard to him, we know it will suffer time dilation with regard to its twin clock and no more time dilation with regard to the observer's one, whereas if it accelerates in the other direction, we know it will suffer less time dilation with regard to its twin clock than with regard to the observer's one. This way, it seems to work, but it means that the accelerations need to have the same origin in space, thus in time also. If you can provide a better answer, please do.

 

[Ibix]

That's just a really long-winded way of admitting that you need to know the velocity, not just the acceleration.

 

[Raymond Potvin]

Hahaha!

Thanks for making me laughing! It's good for health! :0)

 

[Ibix]

The relativity principle is about not knowing that we are moving, thus when we know and we need to calculate time dilation, it is easier not to refer to it.

That'll go badly wrong if you try it in non-flat spacetime.

If you were saying that it's always possible to analyse any experiment in any frame, I'd agree with you. Sometimes it is easier to pick a frame and work entirely in its coordinates. But you're stating that as "knowing that you are moving", which is a really silly way to think of it when, by your own admission, you can't know that.

Moving in some reference frame you can know. Moving, you can't know. And thinking that you kind-of-can will come back to bite you if you approach GR like that.

 

[Raymond Potvin]

That's just a really long-winded way of admitting that you need to know the velocity, not just the acceleration.

What I said is that knowing the velocity with regard to the third observer did not seem to help solve the problem, and I added that, to solve it, we may have to know where that velocity came from. It looks as if bodies' inertial motions could have a common origin even if we cannot identify it. We could almost consider that the actual motion of a body is a remnant of all the accelerations a body has suffered.