High School Does Acceleration Affect Time Dilation?

[Dale]

You did not say that the clocks were initially moving,

Yes, I did. Post 29, second sentence.

If you can provide a better answer, please do.

Sure, the best answer is @Ibix's answer $$\tau=\int_0^T\sqrt {1-v^2 (t)/c^2}dt $$

I'm afraid there is no way to solve that kind of problem

With your method there is no way to solve it. The standard method works just fine.

Note that the reason your method appears to work in specific cases is because since you are implicitly assuming v(0)=0 you can then use a(t) to determine v(t) which then allows you to use the standard formula. When v(0) is unknown then your approach fails, precisely because a(t) is not sufficient.

 

[Ibix]

We could almost consider that the actual motion of a body is a remnant of all the accelerations a body has suffered.

The "almost" is the problem. You need an initial velocity. And that's because, as @Dale says, you have $dv/dt$ and you need to integrate to get $v$ to determine elapsed time. If it were acceleration that mattered you would not need to know that original velocity.

 

[Raymond Potvin]

Yes, I did. Post 29, second sentence.

Sorry, I missed it!

Sure, the best answer is @Ibix's answer $$\tau=\int_0^T\sqrt {1-v^2 (t)/c^2}dt $$
With your method there is no way to solve it. The standard method works just fine.

That formula doesn't tell us the direction or the speed the clock that accelerates has with regard to the observer after it has accelerated, so I don't see how it can give a unique answer. Its dilation with regard to that observer depends on its speed with regard to it, which depends on the direction of its acceleration, which is not specified. It may as well get at rest with regard to the observer and suffer no more time dilation, or it may accelerate the other way and suffer more time dilation than before. Maybe I missed something again.

Note that the reason your method appears to work in specific cases is because since you are implicitly assuming v(0)=0 you can then use a(t) to determine v(t) which then allows you to use the standard formula. When v(0) is unknown then your approach fails, precisely because a(t) is not sufficient.

If we don't know if it is the two clocks or the observer that is moving to begin with, then I don't see how the problem can be solved. If we know that it is the two clocks that have accelerated away from the observer, then we know it is the clocks that suffer time dilation. If one of them accelerates away from the observer again, then we know it will suffer more time dilation with regard to the observer than with regard to the other clock since that other clock is already suffering time dilation with regard to the same observer. If it accelerates toward the observer, then we know it will suffer less time dilation with regard to the observer than with regard to the other clock since that other clock is again already suffering time dilation with regard to the same observer. Am I missing something in the way you presented the problem?

 

[timmdeeg]

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.

Do you say his heartbeat slows down and he dies?

 

[Raymond Potvin]

Do you say his heartbeat slows down and he dies?

:0) Hi Timmdeeg,

No he doesn't die, because his atoms also slow down their heartbeat, and without dying, so he won't even notice the difference. Maybe he will die from acceleration though, because at the end, his mass will have increased a lot, so it will take a lot of force to increase its speed, what could crush his heart, and he could bleed to death! :0)

 

[Chris Miller]

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).

I'm not sure how hard you need to stress this. For me, it's more a matter of sloppy writing than understanding. There is (in this 4D universe) no measurement for the rate of time passing. A second is always a second, etc. But your (frame's) second might not equal mine. I read where a clock just a few feet higher off the ground than another ticked slightly faster, that your feet age more slowly than your head. My original question asked whether acceleration alone would slow a clock relative to a non-accelerating observer's, and got the answer: no.

I know that if I'm deep in a gravity well, and you are not, then my clock will appear to you to run slower. Will your clock then appear to run faster to me? If so, the universe must look very strange with planets and galaxies spinning around > c.

 

[phinds]

I'm not sure how hard you need to stress this.

Well, how about for someone who has it totally wrong, as in:

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.

 

[Raymond Potvin]

Well, how about for someone who has it totally wrong, as in:

Stop pushing, my resistance to change has increased so much in that direction that my heart is about to bleed! :0)

 

[phinds]

Stop pushing, my resistance to change has increased so much in that direction that my heart is about to bleed! :0)

Well, you just need to travel faster so the bleeding slows down

 

[Raymond Potvin]

Well, you just need to travel faster so the bleeding slows down

If I succeed, it's going to slow down with regard to you, but not to me, and I will suffer more resistance to change, which might brake my heart definitively. :0)

 

[jbriggs444]

I know that if I'm deep in a gravity well, and you are not, then my clock will appear to you to run slower. Will your clock then appear to run faster to me? If so, the universe must look very strange with planets and galaxies spinning around > c.

Yes, a clock higher in a gravitational field will appear to run fast. Gravitational time dilation is not symmetric.

In special relativity, the speed of light is a global limit. Spacetime is flat and velocities here can compared with velocities there unambiguously. The speed of light is what it is (as long as we are all using inertial frames).

In general relativity, the speed of light is a local limit. Spacetime is not flat and comparing velocities here to velocities there is ambiguous. There is no such thing as an global inertial frame and the velocity that you get when you measure something far away depends on the coordinate system you use.

As we speak, you are sitting in a gravitational potential well. Does the universe look strange to you?

 

[timmdeeg]

No he doesn't die, because his atoms also slow down their heartbeat,

So if atoms slow down consequently molecules will slow down too. If water molecules slow down due to cooling the water freezes. So in case of our twin, the water in his glass will freeze, right? Not because of cooling in this case but because of the slowing down of the water molecules due to less aging.

 

[jbriggs444]

So if atoms slow down consequently molecules will slow down too. If water molecules slow down due to cooling the water freezes. So in case of our twin, the water in his glass will freeze, right? Not because of cooling in this case but because of the slowing down of the water molecules due to less aging.

The fact that you are, right now, moving very rapidly in the rest frame of a particle in an accelerator at CERN does not cause your body to freeze instantly.

 

[Raymond Potvin]

So if atoms slow down consequently molecules will slow down too. If water molecules slow down due to cooling the water freezes. So in case of our twin, the water in his glass will freeze, right? Not because of cooling in this case but because of the slowing down of the water molecules due to less aging.

If our own time dilation is unobservable, then any phenomenon that is happening inside us because of that is also unobservable. Length contraction is unobservable for instance, but without it, our data would be unexplainable. If our atoms' frequencies slow down with regard to an observer, then our molecules must slow down, our metabolism must slow down, everything we do must slow down. To me, that's the only way to explain relativistic aging anyway, otherwise we might as well invoke magic to explain the data.

 

[Raymond Potvin]

The fact that you are, right now, moving very rapidly in the rest frame of a particle in an accelerator at CERN does not cause your body to freeze instantly.

Hi Jbriggs,

The fact that we know the particle is actually accelerating with regard to us means that we know it is actually getting speed with regard to us, and we know it is not us that is getting speed because we know we are not accelerating, so we also know that it is the particle that is suffering more and more time dilation.

 

[A.T.]

The fact that we know the particle is actually accelerating with regard to us means that we know it is actually getting speed with regard to us,

Not if it goes in circles. Centripetal acceleration doesn't increase speed.

 

[timmdeeg]

The fact that you are, right now, moving very rapidly in the rest frame of a particle in an accelerator at CERN does not cause your body to freeze instantly.

I know, I was interested in the answer. I wasn't sure about the meaning of "slowing down" as Raymond Potvin understands it.

 

[phinds]

I know, I was interested in the answer. I wasn't sure about the meaning of "slowing down" as Raymond Potvin understands it.

Yes, but he DOESN'T understand it. He has not yet grasped differential aging due to different world-lines.

 

[timmdeeg]

If our own time dilation is unobservable, then any phenomenon that is happening inside us because of that is also unobservable.

It is unobservable for us, because in our frame nothing slows down which is at rest with us.

 

[phinds]

It is unobservable for us, because in our frame nothing slows down which is at rest with us.

Actually, it's not that this "slowing down" that he thinks is happening is "unobservable", it's that it IS NOT HAPPENING. In our rest frame there is no "slowing down", no "time dilation", no nuttin'. @Raymond Potvin you really need to come to grips with differential aging and world lines and stop with this nonsense about "slowing down".

 

[Bartolomeo]

Actually, it's not that this "slowing down" that he thinks is happening is "unobservable", it's that it IS NOT HAPPENING. In our rest frame there is no "slowing down", no "time dilation", no nuttin'. @Raymond Potvin you really need to come to grips with differential aging and world lines and stop with this nonsense about "slowing down".

Note: rotating absorber in centrifuge dilates himself. That's why observer on the rim of rotating disc sees, that that clock in the center ticks faster. Two rotating observers on the opposite sides of rotating disc will not see any dilation (though they are in relative motion) since they dilate themselves on the same magnitude. It is experimentally proven fact. Two inertial observers, that momentarily coincide with those rotating observers on the rim, will not see any dilation of each other clock.
An observer, who moves in reference frame of another twin, will see that other twin's clock is ticking faster since his own clock dilates - R. Feynman, Feynman Lectures, Relativistic Effects in radiation.

 

[PeterDonis]

Two inertial observers, that momentarily coincide with those rotating observers on the rim, will not see any dilation of each other clock.

Yes, they will, because they are moving relative to each other.

It is experimentally proven fact.

What experiment are you thinking of? Please give a reference.

An observer, who moves in reference frame of another twin, will see that other twin's clock is ticking faster since his own clock dilates

You are misinterpreting this and misapplying it as well.

 

[PeterDonis]

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