Thread: Time Dilation in Objects in Relative Rotation

A rather silly thread has been running about time travel, which raises an interesting (for me) question about relativity that I cannot answer.

If one has, say, a disc, spinning very rapidly relative to its support, do we expect to see a time dilation effect for a clock, atomic vibration, whatever on the disc, as seen by a stationary observer, similar to that we would expect for rapid relative linear motion, or is it quite different? I had the vague idea that the constant acceleration implied by a relative rotation would mean that Special Relativity would not be adequate to account for what happens.

Can any physicist comment?

Not a physicist, but I am fairly sure you are right that SR is not enough to fully calculate this. It probably also relates the the Ehrenfest Paradox, where you try and consider the effects of length contraction in a rigid, spinning disk. I believe this was one of the things that led Einstein to realise that he needed to study differential geometry to describe the curvature of space-time in GR.

Would length contraction of the circumference of a spinning disc violate euclidian geometry or would it just be a perception effect for an observer? Or would the disc physically shrink in diameter by the same principle of length contraction?

The Earth appears to be a sphere.

Originally Posted by Beer w/Straw

The Earth appears to be a sphere.

Oh, do piss off.

Wasn't expecting that, sorry.

How fast would you be travelling if you were on the surface of a pulsar?

Originally Posted by Strange

Not a physicist, but I am fairly sure you are right that SR is not enough to fully calculate this. It probably also relates the the Ehrenfest Paradox, where you try and consider the effects of length contraction in a rigid, spinning disk. I believe this was one of the things that led Einstein to realise that he needed to study differential geometry to describe the curvature of space-time in GR.

Thanks for this. I see from the Wiki article on the Ehrenfest Paradox that someone called Gron is said to mention the "impossibility of synchronising clocks in a rotating reference frame". Which is tantalising, but not an answer.

I bet Markus would be able to answer this, though whether I would be able to understand his explanation is another thing entirely.

Originally Posted by Daecon

Would length contraction of the circumference of a spinning disc violate euclidian geometry or would it just be a perception effect for an observer? Or would the disc physically shrink in diameter by the same principle of length contraction?

As I read the Wiki article on the Ehrenfest Paradox, yes indeed the geometry is non-Euclidean. Suggest you take a look at it: it deals with both the radius and the circumference. The core of the paradox seems to be that the radius should not contract while the circumference should, which leads to a violation of pi (yikes!!)….hence it must non-Euclidean.

Relativistic Rolling Wheel II

Originally Posted by Daecon

Would length contraction of the circumference of a spinning disc violate euclidian geometry or would it just be a perception effect for an observer? Or would the disc physically shrink in diameter by the same principle of length contraction?

The geometry of the disk becomes non-Euclidean (the ration of diameter to circumference is no longer Pi) even though it remains flat. This is a perfect example of "intrinsic curvature", which is an important aspect of the curvature of space-time in GR.

Ah, exchemist beat me to it!

Originally Posted by exchemist

If one has, say, a disc, spinning very rapidly relative to its support, do we expect to see a time dilation effect for a clock, atomic vibration, whatever on the disc, as seen by a stationary observer, similar to that we would expect for rapid relative linear motion, or is it quite different? I had the vague idea that the constant acceleration implied by a relative rotation would mean that Special Relativity would not be adequate to account for what happens.

The time dilation is exactly as given by Special Relativity. Accelerated objects are dealt with in Special Relativity by considering the inertial frames of reference that are at instantaneous rest relative to the object at the given instants.

Originally Posted by KJW

Originally Posted by exchemist

If one has, say, a disc, spinning very rapidly relative to its support, do we expect to see a time dilation effect for a clock, atomic vibration, whatever on the disc, as seen by a stationary observer, similar to that we would expect for rapid relative linear motion, or is it quite different? I had the vague idea that the constant acceleration implied by a relative rotation would mean that Special Relativity would not be adequate to account for what happens.

The time dilation is exactly as given by Special Relativity. Accelerated objects are dealt with in Special Relativity by considering the inertial frames of reference that are at instantaneous rest relative to the object at the given instants.

Thanks for the clarification abut the time dilation - which was my question. But I confess to being a bit baffled as to how to apply the idea of inertial reference frames at instantaneous rest, when dealing with an object that is rotating. Would that be something you can explain easily, or is it a huge red herring that we should avoid?

Originally Posted by exchemist

Originally Posted by KJW

Originally Posted by exchemist

If one has, say, a disc, spinning very rapidly relative to its support, do we expect to see a time dilation effect for a clock, atomic vibration, whatever on the disc, as seen by a stationary observer, similar to that we would expect for rapid relative linear motion, or is it quite different? I had the vague idea that the constant acceleration implied by a relative rotation would mean that Special Relativity would not be adequate to account for what happens.

The time dilation is exactly as given by Special Relativity. Accelerated objects are dealt with in Special Relativity by considering the inertial frames of reference that are at instantaneous rest relative to the object at the given instants.

Thanks for the clarification abut the time dilation - which was my question. But I confess to being a bit baffled as to how to apply the idea of inertial reference frames at instantaneous rest, when dealing with an object that is rotating. Would that be something you can explain easily, or is it a huge red herring that we should avoid?

The Lorentz transforms for rotating frames are a (very) late addition to the special relativity [1-3]. They are by no means trivial. The bottom line is that, from these transforms we can conclude that :




The proof is not for the faint of heart.

References

1. R. A. Nelson, "Generalized Lorentz transformation for an accelerated, rotating frame of reference", J.
Math. Phys. 28, 2379 (1987);
2. H.Nikolic, “Relativistic contraction and related effects in non-inertial frames”, Phys.Rev. A, 61, 032109-
032117, (2000)
3. Gron's draft

Originally Posted by xyzt

Originally Posted by exchemist

Originally Posted by KJW

Originally Posted by exchemist

If one has, say, a disc, spinning very rapidly relative to its support, do we expect to see a time dilation effect for a clock, atomic vibration, whatever on the disc, as seen by a stationary observer, similar to that we would expect for rapid relative linear motion, or is it quite different? I had the vague idea that the constant acceleration implied by a relative rotation would mean that Special Relativity would not be adequate to account for what happens.

The time dilation is exactly as given by Special Relativity. Accelerated objects are dealt with in Special Relativity by considering the inertial frames of reference that are at instantaneous rest relative to the object at the given instants.

Thanks for the clarification abut the time dilation - which was my question. But I confess to being a bit baffled as to how to apply the idea of inertial reference frames at instantaneous rest, when dealing with an object that is rotating. Would that be something you can explain easily, or is it a huge red herring that we should avoid?

The Lorentz transforms for rotating frames are a (very) late addition to the special relativity [1-3]. They are by no means trivial. The bottom line is that, from these transforms we can conclude that :




The proof is not for the faint of heart.

References

1. R. A. Nelson, "Generalized Lorentz transformation for an accelerated, rotating frame of reference", J.
Math. Phys. 28, 2379 (1987);
2. H.Nikolic, “Relativistic contraction and related effects in non-inertial frames”, Phys.Rev. A, 61, 032109-
032117, (2000)
3. Gron's draft

OK thanks very much. I will gladly leave the proofs to the guys in long trousers, then.

I've always thought there is something weird about rotation and this discussion strengthens that feeling.

Originally Posted by exchemist

Originally Posted by xyzt

Originally Posted by exchemist

Originally Posted by KJW

Originally Posted by exchemist

If one has, say, a disc, spinning very rapidly relative to its support, do we expect to see a time dilation effect for a clock, atomic vibration, whatever on the disc, as seen by a stationary observer, similar to that we would expect for rapid relative linear motion, or is it quite different? I had the vague idea that the constant acceleration implied by a relative rotation would mean that Special Relativity would not be adequate to account for what happens.

The time dilation is exactly as given by Special Relativity. Accelerated objects are dealt with in Special Relativity by considering the inertial frames of reference that are at instantaneous rest relative to the object at the given instants.

Thanks for the clarification abut the time dilation - which was my question. But I confess to being a bit baffled as to how to apply the idea of inertial reference frames at instantaneous rest, when dealing with an object that is rotating. Would that be something you can explain easily, or is it a huge red herring that we should avoid?

The Lorentz transforms for rotating frames are a (very) late addition to the special relativity [1-3]. They are by no means trivial. The bottom line is that, from these transforms we can conclude that :




The proof is not for the faint of heart.

References

1. R. A. Nelson, "Generalized Lorentz transformation for an accelerated, rotating frame of reference", J.
Math. Phys. 28, 2379 (1987);
2. H.Nikolic, “Relativistic contraction and related effects in non-inertial frames”, Phys.Rev. A, 61, 032109-
032117, (2000)
3. Gron's draft

OK thanks very much. I will gladly leave the proofs to the guys in long trousers, then.

I've always thought there is something weird about rotation and this discussion strengthens that feeling.

Yes, the bottom line is that the geometry of the rotating disc is not Euclidian , it is Lobachevski.
The other bottom line is that, as Plank pointed out early on, you need to consider the disc spin-up process. As Einstein pointed out immediately, the spin-up process cannot be considered as Born rigid (not possible for all components of the disc to keep their relative distances constant), the problem cannot be treated correctly as a kinematics problem, it is a dynamics problem (the disc deforms and, if spun fast enough, ends up disintegrating).

Originally Posted by exchemist

A rather silly thread has been running about time travel, which raises an interesting (for me) question about relativity that I cannot answer.

If one has, say, a disc, spinning very rapidly relative to its support, do we expect to see a time dilation effect for a clock, atomic vibration, whatever on the disc, as seen by a stationary observer, similar to that we would expect for rapid relative linear motion, or is it quite different? I had the vague idea that the constant acceleration implied by a relative rotation would mean that Special Relativity would not be adequate to account for what happens.

Can any physicist comment?

In terms of pure time dilation we have the clock postulate, which states that the time dilation for such a clock can be calculated just from applying SR to the speed of the clock and that there are no additional effects due to the acceleration it experiences. This has been tested and found to be true by using very high speed centrifuges.

Thus, if you were standing at the center of the disk, the time dilation you would see in the clock at the edge would be the same as that due to it's motion only. However, if you were standing on the disk next to the clock, then your acceleration would effect what you would see happening to a clock at the center of the disk causing it to run fast compared to your own.

Originally Posted by Janus

In terms of pure time dilation we have the clock postulate, which states that the time dilation for such a clock can be calculated just from applying SR to the speed of the clock and that there are no additional effects due to the acceleration it experiences.

How does this relate to the equivalence of acceleration and gravity? If differences in gravity cause time dilation, then shouldn't acceleration also?

Originally Posted by Strange

Originally Posted by Janus

In terms of pure time dilation we have the clock postulate, which states that the time dilation for such a clock can be calculated just from applying SR to the speed of the clock and that there are no additional effects due to the acceleration it experiences.

How does this relate to the equivalence of acceleration and gravity? If differences in gravity cause time dilation, then shouldn't acceleration also?

Yes, it can be proven rigorously that uniform acceleration generates the same exact time dilation effect as a uniform gravitational field. It is not the acceleration directly that generates the effect (otherwise it would contradict "the clock postulate") but indirectly, through speed. Thus, my statement is in perfect agreement with Janus' one. I can post the mathematical proof, it is quite involved.

Originally Posted by xyzt

Yes, it can be proven rigorously that uniform acceleration generates the same exact time dilation effect as a uniform gravitational field. It is not the acceleration directly that generates the effect (otherwise it would contradict "the clock postulate") but indirectly, through speed. Thus, my statement is in perfect agreement with Janus' one. I can post the mathematical proof, it is quite involved.

Ah! Got it! Thanks.

If anyone else wants more detail, see this: http://math.ucr.edu/home/baez/physic.../SR/clock.html

Originally Posted by Strange

Originally Posted by xyzt

Yes, it can be proven rigorously that uniform acceleration generates the same exact time dilation effect as a uniform gravitational field. It is not the acceleration directly that generates the effect (otherwise it would contradict "the clock postulate") but indirectly, through speed. Thus, my statement is in perfect agreement with Janus' one. I can post the mathematical proof, it is quite involved.

Ah! Got it! Thanks.

If anyone else wants more detail, see this: Does a clock's acceleration affect its timing rate?

Yes, you only need to put this in mathematical terms:

"How do the astronauts describe what is going on? They believe they're accelerating in deep space. The top astronaut reasons "By the time the light from the bottom astronaut reaches me, I'll have picked up some speed, so that I'll be receding from the light at a higher rate than previously as I receive it. So it should be redshifted—and yes, so it is!" The bottom astronaut reasons very similarly: "By the time the light from the top astronaut reaches me, I'll have picked up some speed, so that I'll be approaching the light at a higher rate than previously as I receive it. So it should be blueshifted—and yes, so it is!" As you can see, they both got the right answer, care of the Equivalence Principle. But their analysis only used their speed, not their acceleration as such. So just like our wind chill factor above, applying the Equivalence Principle to the case of the rocket doesn't depend on acceleration per se, but it does depend on the result of acceleration: changing speeds!"

Originally Posted by Strange

Originally Posted by Janus

In terms of pure time dilation we have the clock postulate, which states that the time dilation for such a clock can be calculated just from applying SR to the speed of the clock and that there are no additional effects due to the acceleration it experiences.

How does this relate to the equivalence of acceleration and gravity? If differences in gravity cause time dilation, then shouldn't acceleration also?

Look at it this way: Gravitational time dilation is related to the difference in gravitational potential, IOW, the amount of work needed to move from one height in a gravity field to another.

Now consider a rotating frame. In this frame, the gravitational equivalence is a gravity field that gets stronger as you move away from the axis. Now if you work out the equivalent gravitational potential between the axis and some other point in the frame, the gravitational time dilation works out to being equal to that calculated by someone in a non-rotating frame and just considering the speed that point travels around the axis.

Thus if you had two observers at the axis of the disk, one in the rotating frame of the disk and the other in an inertial frame, both would see the same time dilation for the clock on the edge. In the inertial frame it will be due to the relative motion of the clock. In the rotating frame, the observer could claim no relative motion on the part of the clock, but there would be an equivalent potential gravity difference between the clock and him causing a gravitational time dilation.

So either the disk is rotating and the dilation is motion based, or considered stationary and the dilation is potential based. You don't apply both.

[Damn "like" still isn't working!]

Does this mean that you can account for the (gravitational(*)) time dilation of a satellite (e.g. GPS) in orbit as either due to the difference in gravitational potential or due to its acceleration, and that these are equivalent?

(*) i.e. ignoring the velocity relative to the ground, for the moment

Originally Posted by Strange

[Damn "like" still isn't working!]

Does this mean that you can account for the (gravitational(*)) time dilation of a satellite (e.g. GPS) in orbit as either due to the difference in gravitational potential or due to its acceleration, and that these are equivalent?

(*) i.e. ignoring the velocity relative to the ground, for the moment

it is a combination of both its gravitational potential and the effects of being in a rotating frame. The two effects run counter to each other. In the rotating frame of the satellite alone, it takes energy to move towards the center of the Earth, while by gravitational potential standards it takes energy to go away from the center of the Earth. So if you were to consider the satellite from the rotational frame and not moving, you would have to superimpose these two potentials upon each other to get the net potential.

Luckily, this can be combined into a single formula that works for all orbiting clocks:



with M being the mass of the orbited body and r the radius of the orbit.

i'm selling super-glue, anyone interested?

Originally Posted by Beer w/Straw

The Earth appears to be a sphere.

Beer w/Straw, would you be able to explain how this comment helps me understand the problem I raised?

I was replying to Daecon. Length contraction doesn't happen in your frame. However,xyzt knows better than I do.

The Earth orbits the sun, spins and the Milky galaxy is moving, yet to us it's still a sphere.

Originally Posted by Beer w/Straw

I was replying to Daecon. Length contraction doesn't happen in your frame. However,xyzt knows better than I do.

The Earth orbits the sun, spins and the Milky galaxy is moving, yet to us it's still a sphere.

But how does " The Earth appears to be a sphere" convey any of that?

It doesn't, so far as I can see, unless you add a lot more explanation of what you mean.

And in any case, he didn't seem to be asking how it appears from the viewpoint of someone rotating with the Earth. Clearly that's not very interesting, the tricky bit being, evidently, how it looks if you are not rotating with it - when, as he rightly surmised, the geometry is non-Euclidean.

See, that's my problem with your posts. You may understand perfectly ( I can't judge) but you don't say enough to make clear the relevance of your contribution. Which is why I told you to piss off in this instance. I had a problem I really wanted help with, and your comment simply failed to contribute at all, so far as I could see. On the contrary, I assumed you were just clogging up the thread with facetious comments.

As I said in the other forum I will now try to take you a tad more seriously, but please, if you can, help by being less Delphic.

He asked if it was only a perception of the observer.

We are not "observers" since we are in the Earth's frame.

Originally Posted by Beer w/Straw

He asked if it was only a perception of the observer.

We are not "observers" since we are in the Earth's frame.

But nobody was talking about the Earth. That was a red herring that you introduced!

The problem concerned a spinning disc which I said in my OP was observed by a stationary observer. That was the context of Daecon's question.

Go and re-read it.

How about you don't bump threads just to pick at me?

Length contraction is length contraction whether whether it's in a straight line or spinning. Daecon asked about perception and I gave a very notable example of something moving really fast but is not perceived to us as being contracted. You couldn't even read that cause your only intent was to pick at me.

Why don't you just ignore me from now on?

Originally Posted by Beer w/Straw

How about you don't bump threads just to pick at me?

Length contraction is length contraction whether whether it's in a straight line or spinning. Daecon asked about perception and I gave a very notable example of something moving really fast but is no perceived to us as being contracted. You couldn't even read that cause your only intent was to pick at me.

Why don't you just ignore me from now on?

That's generally what I've been doing, but now - due to the advice I've received on the other forum that you actually do know what you're talking about - I am trying to make an effort to understand what is behind some of your posts.

It is not picking on you to challenge you on the substance of your contributions. That is what you must expect in a science discussion. In this case it was my thread and so I read all the contributions, looking for the best answer to my question. Please do not adopt the language of victimhood. It makes you sound like Mayflow.

But OK, I'll leave this one there if you want.

Originally Posted by Janus

Originally Posted by Strange

Originally Posted by Janus

In terms of pure time dilation we have the clock postulate, which states that the time dilation for such a clock can be calculated just from applying SR to the speed of the clock and that there are no additional effects due to the acceleration it experiences.

How does this relate to the equivalence of acceleration and gravity? If differences in gravity cause time dilation, then shouldn't acceleration also?

Look at it this way: Gravitational time dilation is related to the difference in gravitational potential, IOW, the amount of work needed to move from one height in a gravity field to another.

Now consider a rotating frame. In this frame, the gravitational equivalence is a gravity field that gets stronger as you move away from the axis. Now if you work out the equivalent gravitational potential between the axis and some other point in the frame, the gravitational time dilation works out to being equal to that calculated by someone in a non-rotating frame and just considering the speed that point travels around the axis.

Thus if you had two observers at the axis of the disk, one in the rotating frame of the disk and the other in an inertial frame, both would see the same time dilation for the clock on the edge. In the inertial frame it will be due to the relative motion of the clock. In the rotating frame, the observer could claim no relative motion on the part of the clock, but there would be an equivalent potential gravity difference between the clock and him causing a gravitational time dilation.

So either the disk is rotating and the dilation is motion based, or considered stationary and the dilation is potential based. You don't apply both.

Does that only apply when the stationary observer and the wheel are in the same plane?

In the sister forum thread SYA produced the following summary table for a solution based on Gron's paper. The wheel has no thickness i.e. x, y and t, z = 0, to avoid Born rigidity issues so the solutions results can be reduced to x and t, y = z = 0 to check that the velocity of the wheels axle remained consistent between each emission event.



Relativistic Rolling Wheel II - Page 3

Once you move the stationary observer outside the plane of the wheel and along the line of the axle (z axis) things change. This particular observation space is Euclidean as the velocity of the plane of the wheel is perpendicular to the axle and all emissions on the rim (at the same time) travel the same distance to the observer (and arrive at the same time). I suppose, gravitation wise, this would remain the case as the observer is not bound gravitationally with the rotating sources being observed.

What would you apply in this case?

Originally Posted by Laurieag

Originally Posted by Janus

Originally Posted by Strange

Originally Posted by Janus

In terms of pure time dilation we have the clock postulate, which states that the time dilation for such a clock can be calculated just from applying SR to the speed of the clock and that there are no additional effects due to the acceleration it experiences.

How does this relate to the equivalence of acceleration and gravity? If differences in gravity cause time dilation, then shouldn't acceleration also?

Look at it this way: Gravitational time dilation is related to the difference in gravitational potential, IOW, the amount of work needed to move from one height in a gravity field to another.

Now consider a rotating frame. In this frame, the gravitational equivalence is a gravity field that gets stronger as you move away from the axis. Now if you work out the equivalent gravitational potential between the axis and some other point in the frame, the gravitational time dilation works out to being equal to that calculated by someone in a non-rotating frame and just considering the speed that point travels around the axis.

Thus if you had two observers at the axis of the disk, one in the rotating frame of the disk and the other in an inertial frame, both would see the same time dilation for the clock on the edge. In the inertial frame it will be due to the relative motion of the clock. In the rotating frame, the observer could claim no relative motion on the part of the clock, but there would be an equivalent potential gravity difference between the clock and him causing a gravitational time dilation.

So either the disk is rotating and the dilation is motion based, or considered stationary and the dilation is potential based. You don't apply both.

Does that only apply when the stationary observer and the wheel are in the same plane?

In the sister forum thread SYA produced the following summary table for a solution based on Gron's paper. The wheel has no thickness i.e. x, y and t, z = 0, to avoid Born rigidity issues so the solutions results can be reduced to x and t, y = z = 0 to check that the velocity of the wheels axle remained consistent between each emission event.



Relativistic Rolling Wheel II - Page 3

Once you move the stationary observer outside the plane of the wheel and along the line of the axle (z axis) things change. This particular observation space is Euclidean as the velocity of the plane of the wheel is perpendicular to the axle and all emissions on the rim (at the same time) travel the same distance to the observer (and arrive at the same time). I suppose, gravitation wise, this would remain the case as the observer is not bound gravitationally with the rotating sources being observed.

What would you apply in this case?

The thread tries to deal (rather badly) with the geometry of rolling, not rotating objects. Nothing to do with the subject of this thread: time dilation in rotating frames. We already sorted out the issue.

What would you apply in this case?

None of the stuff applies.

Originally Posted by Janus

Originally Posted by Strange

[Damn "like" still isn't working!]

Does this mean that you can account for the (gravitational(*)) time dilation of a satellite (e.g. GPS) in orbit as either due to the difference in gravitational potential or due to its acceleration, and that these are equivalent?

(*) i.e. ignoring the velocity relative to the ground, for the moment

it is a combination of both its gravitational potential and the effects of being in a rotating frame. The two effects run counter to each other. In the rotating frame of the satellite alone, it takes energy to move towards the center of the Earth, while by gravitational potential standards it takes energy to go away from the center of the Earth. So if you were to consider the satellite from the rotational frame and not moving, you would have to superimpose these two potentials upon each other to get the net potential.

Luckily, this can be combined into a single formula that works for all orbiting clocks:



with M being the mass of the orbited body and r the radius of the orbit.

Janus, this and your previous one are very good, for me at least. Just what I was looking for.

I should have thanked you earlier but forgot, and now the latest additional remarks have reminded me. Pity the "likes" are still out of action…...

Originally Posted by xyzt

None of the stuff applies.

Thanks, that stationary observer (at 90 degrees to the plane of rotation) sees the same thing if the frame is either rotating or rolling.

Originally Posted by Laurieag

Originally Posted by xyzt

None of the stuff applies.

Thanks, that stationary observer (at 90 degrees to the plane of rotation) sees the same thing if the frame is either rotating or rolling.

Not necessarily, if he is "glued" to the disc , he observes one thing, if he isn't, he observes a different thing.