The problem of the spinning bucket of water…

[rrw4rusty]

Hi,

 

I first searched the web for/and joined over half a dozen science forums because a specific question had bothered me for a long time. That first post is here: http://www.scienceforums.net/forum/showthread.php?t=44928 (I had already belonged to a few science forums so it won’t be my very ‘first’ post everywhere). I thought the question was certainly a easy one: it was basically ‘if motion is relative why isn’t rotation relative’ (I was looking for a cleaver way to explain artificial gravity for my latest sci-fi book).

 

I was dumbfounded by the responses I got—everyone had a different answer! People (scientists, students, forum administrators ) were arguing with each other it! My God hadn’t anyone ever asked this question before (as it turned out, everyone had, starting with Sir Isaac Newton!)?! No doubt someone had the ‘correct’ answer – it would take some searching to tell.

 

Since that post back in Oct 2009 I have plowed through 4 popular (and recommended) quantum physics books to bone up on the latest. It was in the book I’m reading now (The Fabric of the Cosmos: Space, Time, and the Texture of Reality by B. Greene) that I finally got the answer to that post. In fact the book starts out with my very question only its presented as a spinning bucket of water -- and doesn't give the answer until the end of the book.

 

In the 17th century Sir Isaac Newton’s answer was: The absolute position, accelerated motion is not relative… space is an entity.

 

During that same period, Gottfried Wilhelm Leibniz answer was: The relational position, all aspects of motion are relative… space is not an entity.

 

In the 18th century, Ernst Mach’s answer was: The relational position, accelerated motion is relative to the average mass in the universe… space is not an entity. (Fewer matter, less effect).

 

In the 20th century, Albert Einstein’s answer was (at first): The relational position, space-time is an absolute entity… space and time are individually relative.

 

According to Greene, in 1912 Einstein offered the first part of the true and complete answer. In 1965 Dieter Bill and Jeffery Cohen refined Einstein’s answer and, in 1985 German physicists Herbert Pfister and K. Braun completed the solution.

 

The answer is everything I wanted it to be and everything I didn’t want it to be! Next post…

 

Cheers,

Rusty

[rrw4rusty]

Hello again!

 

So, after 300 years the answer to why water climbs up the sides of a spinning bucket and why astronauts can seeming have gravity in a rotating compartment has been worked out mathematically using general relativity. It’s called frame dragging. It seems spacetime is dragged around by spinning objects. This is what makes an object falling into a spinning neutron star spiral around following the twisted contour of spacetime as it falls while from the neutron star’s frame of reference the object is falling straight down toward it.

 

In one way the answer was everything I wanted it to be! I wanted rotation to be relative so that instead of the space ship’s compartment rotating to provide gravity, something else could be happening making it ‘seem’ like the compartment was rotating (in a relative way). Strangely enough, the answer provides for this! If a massive hollow sphere were rotating and a bucket of water standing still (from a distant observer) were placed within that sphere—the water would climb the sides and take on the same concave form it would if the bucket of water were spinning! Ergo my astronauts would have gravity without a spinning compartment! But, of course there’s a catch… ask me how massive that sphere has to be to mask out the universe outside of it and cause the stationary bucket of water to appear to be spinning.

 

According to Greene, an experiment is being planned to detect the ‘frame dragging’ effect of the Earth’s rotation (he gives no further information on this experiment—perhaps someone out there has heard of it). Extremely sensitive gyroscopes will be placed in orbit and aligned at stars and, if the experiment works, that is, if the calculations are correct, after a year the gyroscopes will tilt a tiny bit. How tiny? We barely have the ability to measure the effect!

 

Greene goes on to say that if the sphere were massive enough, that is if the sphere were as missive as the entire universe, it would not matter which was rotating, the sphere or the bucket!

 

But I’m writing science fiction so the shape, the charge, the exotic material of that my hollow sphere is made of and the field placed around it will cause it to drag spacetime around far far more then the mass of my anti-grav sphere would otherwise!

 

Cheers,

Rusty

[insane_alien]

So, after 300 years the answer to why water climbs up the sides of a spinning bucket and why astronauts can seeming have gravity in a rotating compartment has been worked out mathematically using general relativity. It’s called frame dragging.

 

this isn't frame dragging at all. well, it will play a SMALL part but at the velocities and masses involved the effect will be immeasurable, heck, we're struggling to detect it for the earth. we had to build 4 of the smoothest balls ever made and some of the most precise machinery ever made to get a measurement.

 

the answer to both hose questions is centripetal acceleration.

[swansont]

As insane_alien has already noted that's not frame dragging, and rotation is not relative. It's an accelerating frame, and you can tell if you are rotating — Newton's laws won't seem to hold in a rotating frame.

[Iggy]

...rotation is not relative. It's an accelerating frame, and you can tell if you are rotating...

 

My understanding, which is admittedly tenuous, has always been that rotation is relative. I understand that an observer can tell if they are in an inertial frame by the forces they feel, but whether the cause of the force is acceleration or gravitational is not absolute but relative. If the observer considers themselves at rest then they are gravitational and if they consider the fixed stars at rest then they are inertial.

 

The quote is page 150 of the linked document--Einstein's "The Foundation of the General Theory of Relativity":

 

Does this permit an observer at rest relatively to K' to infer that he is on a "really" accelerated system of reference? The answer is in the negative; for the above-mentioned relation of freely movable masses to K' may be interpreted equally well in the following way. The system of reference K' is unaccelerated, but the space-time territory in question is under the sway of a gravitational field, which generates the accelerated motion of the bodies relatively to K'.

 

This view is made possible for us by the teaching of experience as to the existence of a field of force, namely, the gravitational field, which possesses the remarkable property of imparting the same acceleration to all bodies. The mechanical behaviour of bodies relatively to K' is the same as presents itself to experience in the case of systems which we are wont to regard as "stationary" or as "privileged." Therefore, from the physical standpoint, the assumption readily suggests itself that the systems K and K' may both with equal right be looked upon as "stationary" that is to say, they have an equal title as systems of reference for the physical description of phenomena.

 

http://www.alberteinstein.info/gallery/pdf/CP6Doc30_English_pp146-200.pdf[emphasis mine]

 

I think it can be said that earth rotates relative to the background stars. But, to say which, the earth or the background stars, rotates and which is at rest in an absolute sense would not be consistent with the freedom of coordinate choice in Einstein's gtr. Choosing coordinates where the earth is at rest would result in a gravitational field caused by the rotating background mass indistinguishable form the centrifugal force expected if we choose the background stars to be at rest.

 

There is another quote of Einstein from 1918 that he writes as a dialog between a critic and a relativist--"Dialog about Objections against the Theory of Relativity"--that I think would be relevant to the thread:

 

Critic:

...For example, would anyone get it in his head to actually use the possibility offered by the theory of relativity to relate the motions of the celestial bodies of the solar system to a geocentric coordinate system that on top of that is participating in the rotation of the Earth? Would anyone really be allowed to see this coordinate system as "at rest" and as equally valid, relative to which the fixed stars are tearing around with tremendous speed?...

 

Relativist:

...Nobody will use a coordinate system that is at rest relative to the planet Earth, because that would be impractical. However as a matter of principle such a theory of relativity is equally valid as any other. The situation, that the fixed stars are circling with tremendous velocities, when one bases an examination on such a coordinate system, does not constitute an argument against the admissibility, but merely against the efficiency of this choice of coordinates, nor does the complicated form of the relative to this coordinate system acting gravitational field, which for example would also have the components that correspond to the centrifugal force....

 

http://en.wikisource.org/wiki/Dialog_about_Objections_against_the_Theory_of_Relativity

 

Perhaps I misunderstand your meaning, Swansont. Or, I misunderstand, or don't fully understand, the issue.

[swansont]

Acceleration in a uniform gravitational field is inertial according to GR, and cannot be distinguished from any other linear acceleration. But rotation can't be made to look like that.

 

If you are in a rotating frame and roll a ball in a straight line, it will look like it's curving, in violation of Newton's first law. Or you could observe Foucault pendulum rotating. There's some experiment you could do that would show that you are rotating.

[rrw4rusty]

Seems I must loose faith in books by apparent experts or some self-proclaimed experts on this forum--heck of a choice. Perhaps I can post the pages of the book and see if I've mis-interpreted what I've read or if Greene is just wrong.

 

Later today I'll scan them in.

 

Rusty

[insane_alien]

was it a popular science book? because the people who write those, while they may know their stuff in the real world, dumb it down a bit for general consumption.

 

as is inevitable, many things are lost in the translation and it becomes easy to misinterpret.

[Iggy]

Acceleration in a uniform gravitational field is inertial according to GR, and cannot be distinguished from any other linear acceleration. But rotation can't be made to look like that.

 

I don’t think I follow what you mean that freefall in a uniform field can’t be distinguished from any other linear acceleration. Someone in a rocket in deep space who is accelerating relative to the background stars can surely determine that they are accelerating relative to the background stars. I am definitely not arguing that.

 

The first quote in my last post means to me that the acceleration is relative--not absolute. The person in the rocket cannot say that they are absolutely undergoing a change in velocity while the rest of the universe is at rest because it is just as valid, although nowhere near as convenient, for the person to say “I am at rest in a uniform gravitational field that has the background stars accelerating--in freefall.”

 

The person can tell that they are accelerating relative to the bulk mass of the universe, but which is accelerating--the ship or the mass of the universe--is not absolute but relative.

 

I do believe rotation can be made to look like that. The second quote of my last post appears to me to bear that out.

 

If you are in a rotating frame and roll a ball in a straight line, it will look like it's curving, in violation of Newton's first law. Or you could observe Foucault pendulum rotating. There's some experiment you could do that would show that you are rotating.

 

I certainly agree that the Foucault pendulum can reveal rotation relative to the background stars. I’m not arguing that. But, if the earth were at rest while all the mass of the background stars circled around it then the rotating mass should create a gravitational field which moves the Foucault pendulum in the expected way. There is a Flash animation showing this. It shows the pendulum while letting you choose what rotates ‘earth’ or ‘stars’ as a demonstration of Mach's principle:

 

http://www.upscale.utoronto.ca/PVB/Harrison/Flash/ClassMechanics/Foucault/Foucault.html

 

It would appear that GR confirms this at least where the universe is approximated by a shell of rotating mass (which I think is what Rusty was describing):

 

Abstract:

 

Mach's idea of relativity of rotation is confirmed for a shell-type model of the Universe by showing that flat geometry in rotating coordinates, realising correct Coriolis and centrifugal forces, can be continuously connected through a rotating mass shell with not exactly spherical shape and latitude-dependent mass density to an asymptotically Minkowskian outside metric. The corresponding solutions of Einstein's field equations are given to second order in the angular velocity omega but it is plausible that the problem has a solution to any order of omega .

 

http://www.iop.org/EJ/abstract/0264-9381/2/6/015/

 

If my understanding is correct, rotation is truly relative. It would not strictly be correct to say “earth rotates” as much as it would be to say “earth rotates relative to the background stars”. Which of the two is rotating and which is at rest is a coordinate choice--both equally valid--if not incredibly inconvenient.

 

I want to stress that I'm not trying to make an argument--only to explain my understanding which is very tenuous. I'm not a physicist and I defer to the expertise of those here who are.

[rrw4rusty]

was it a popular science book? because the people who write those, while they may know their stuff in the real world, dumb it down a bit for general consumption.

 

as is inevitable, many things are lost in the translation and it becomes easy to misinterpret.

 

See original post.

[swansont]

I don’t think I follow what you mean that freefall in a uniform field can’t be distinguished from any other linear acceleration. Someone in a rocket in deep space who is accelerating relative to the background stars can surely determine that they are accelerating relative to the background stars. I am definitely not arguing that.

 

Freefall is inertial. Standing in a uniform gravitational field is what is indistinguishable from a linear acceleration. This assumes you have no external clues.

 

 

The first quote in my last post means to me that the acceleration is relative--not absolute. The person in the rocket cannot say that they are absolutely undergoing a change in velocity while the rest of the universe is at rest because it is just as valid, although nowhere near as convenient, for the person to say “I am at rest in a uniform gravitational field that has the background stars accelerating--in freefall.”

 

The person can tell that they are accelerating relative to the bulk mass of the universe, but which is accelerating--the ship or the mass of the universe--is not absolute but relative.

 

I do believe rotation can be made to look like that. The second quote of my last post appears to me to bear that out.

 

 

 

I certainly agree that the Foucault pendulum can reveal rotation relative to the background stars. I’m not arguing that. But, if the earth were at rest while all the mass of the background stars circled around it then the rotating mass should create a gravitational field which moves the Foucault pendulum in the expected way. There is a Flash animation showing this. It shows the pendulum while letting you choose what rotates ‘earth’ or ‘stars’ as a demonstration of Mach's principle:

 

http://www.upscale.utoronto.ca/PVB/Harrison/Flash/ClassMechanics/Foucault/Foucault.html

 

It would appear that GR confirms this at least where the universe is approximated by a shell of rotating mass (which I think is what Rusty was describing):

 

Mach's principle is a conjecture. It's not actually incorporated in GR; the diurnal rotation of stars would require that they have a speed far in excess of c, and requires the forces involved be essentially instantaneous.

 

The Lense-Thirring effect is very weak, as has already been mentioned.

 

If my understanding is correct, rotation is truly relative. It would not strictly be correct to say “earth rotates” as much as it would be to say “earth rotates relative to the background stars”. Which of the two is rotating and which is at rest is a coordinate choice--both equally valid--if not incredibly inconvenient.

 

I want to stress that I'm not trying to make an argument--only to explain my understanding which is very tenuous. I'm not a physicist and I defer to the expertise of those here who are.

 

You can choose any coordinate system you want. You can use epicycles if that suits your fancy. But mathematical validity is not the same as conforming to physics theories.

[rrw4rusty]

...but at the velocities and masses involved the effect will be immeasurable, heck, we're struggling to detect it for the earth. we had to build 4 of the smoothest balls ever made and some of the most precise machinery ever made to get a measurement.

 

Dear insane_alien,

 

The statement that you make to me above was made first in my post (right from Greene's book). I can appreciate a busy person skimming my post (and it was a long one) but I'm beginning to suspect you've 'skimmed' too much. In another post I'm asked whose book this came from, LOL! Obviously my first post wasn't read at all.

 

Certainly I could be mistaken (in fact, given you personally created the shiniest balls in the history of mankind... most probably I am) but as far as I can tell you are contradicting Einstein, Dieter Bill, Jeffery Cohen, Herbert Pfister and K. Braun -- and of course Brian Greene. You seem to be 'wrong' or at least, one of us is confusing the issue in some way.

 

My goal is to understand the reason... whatever the reason is. Obviously given the different and varied replies my original post generated it has been a disputed question.

 

Cheers,

Rusty

 

EDIT: Oh, I just upgraded to Windows 7 and my scanner is not yet working.

[Cap'n Refsmmat]

I've read The Fabric of the Cosmos, and I recall the bucket example. I think (but I do not have my book handy) that the water in the bucket was merely a metaphor for gravitation. That is, water going up the sides of a bucket is like frame dragging, not caused by it.

 

But I'd have to go and get my book to see, and it's inconveniently several dozen miles away at the moment.

[swansont]

According to Greene, an experiment is being planned to detect the ‘frame dragging’ effect of the Earth’s rotation (he gives no further information on this experiment—perhaps someone out there has heard of it). Extremely sensitive gyroscopes will be placed in orbit and aligned at stars and, if the experiment works, that is, if the calculations are correct, after a year the gyroscopes will tilt a tiny bit. How tiny? We barely have the ability to measure the effect!

 

Greene goes on to say that if the sphere were massive enough, that is if the sphere were as missive as the entire universe, it would not matter which was rotating, the sphere or the bucket!

 

That's the key here, I think. If the mass of the universe were concentrated, you would see significant frame dragging — the rotation of the sphere would cause spacetime to rotate as well.

 

This is not the same as the centripetal acceleration of a rotating object, as in the artificial gravity example you gave.

 

The frame dragging due to the earth would cause a Foucault pendulum to precess 1 degree in 16,000 years. That's how small the effect is.

 

http://en.wikipedia.org/wiki/Lense–Thirring_precession

[rrw4rusty]

I've read The Fabric of the Cosmos, and I recall the bucket example. I think (but I do not have my book handy) that the water in the bucket was merely a metaphor for gravitation. That is, water going up the sides of a bucket is like frame dragging, not caused by it.

 

But I'd have to go and get my book to see, and it's inconveniently several dozen miles away at the moment.

 

Oh, that could be.

 

I mentioned what was said on another forum where I made the same posts... one reply said:

 

"Before I read your post, and looked up the original papers, I probably would have said the same thing. Centripetal acceleration (i.e., the force in a non-rotating frame which induces circular motion) isn't the issue here. The issue is whether the centrifugal force (that is, the outward force observed in a rotating frame) is relative or absolute.

 

It is very interesting to me that the answer appears to be that Mach was right, and it's absolute."

 

I'm afraid this only confused me further. :-\ The terms and ideas are too new to me -- I still have to look up everything an then figure out how/if it applies (I thought Mach said it was relative but... what are we talking about?).

 

Too much, too fast. Time for a nap LOL.

 

Rusty

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Consecutive posts merged

Originally Posted by Iggy

If my understanding is correct, rotation is truly relative. It would not strictly be correct to say “earth rotates” as much as it would be to say “earth rotates relative to the background stars”. Which of the two is rotating and which is at rest is a coordinate choice--both equally valid--if not incredibly inconvenient.

 

Iggy,

I find this outlook hard to believe... if in some coordinate or frame of reference stars and galaxies were rotating around Earth they would be surpassing the speed of light to say the very least. Now... I want to stress that I'm not trying to make an argument--only to explain my understanding which is very tenuous. I'm not a physicist and I sometimes (LOL) defer to the expertise of those here who are.

 

;-)

 

Rusty

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Merged post follows:

Consecutive posts merged

I've read The Fabric of the Cosmos, and I recall the bucket example. I think (but I do not have my book handy) that the water in the bucket was merely a metaphor for gravitation. That is, water going up the sides of a bucket is like frame dragging, not caused by it.

 

But I'd have to go and get my book to see, and it's inconveniently several dozen miles away at the moment.

 

No, absolutely not a metaphor for gravitation. It sure reads like 'it is', not 'it is like'. I wish you could go get it... I should have a working scanner later today.

 

Rusty

[Iggy]

It's not actually incorporated in GR; the diurnal rotation of stars would require that they have a speed far in excess of c...

 

Iggy,

I find this outlook hard to believe... if in some coordinate or frame of reference stars and galaxies were rotating around Earth they would be surpassing the speed of light to say the very least.

 

Yes. It would be like the ergosphere of a rotating black hole. Speeds are, and must be, faster than c because spacetime is being dragged around with the mass.

 

I looked for a paper this morning supporting the Machian view of the relativity of rotation with GR and this is the first I found:

 

Cosmological Vorticity Perturbations, Gravitomagnetism, and Mach’s Principle

 

The axes of gyroscopes experimentally define non-rotating frames. But what physical cause governs the time-evolution of gyroscope axes ? Starting from an unperturbed, spatially flat FRW cosmology, we consider cosmological vorticity perturbations (i.e. vector perturbations, rotational perturbations) at the linear level. We ask: Will cosmological rotational perturbations drag the axis of a gyroscope relative to the directions (geodesics) to galaxies beyond the rotational perturbation? We cast the laws of Gravitomagnetism into a form showing clearly the close correspondence with the laws of ordinary magnetism.

 

Our results are:

1) The dragging of a gyroscope axis by rotational perturbations beyond the˙H

radius (H = Hubble constant) is exponentially suppressed.

2) If the perturbation is a homogeneous rotation inside a radius significantly larger than the ˙H radius, then the dragging of the gyroscope axis by the rotational perturbation is exact for any equation of state for cosmological matter.

3) The time-evolution of a gyroscope axis exactly follows a specific average of the matter inside the ˙Hradius for any equation of state.

 

In this precise sense Mach’s Principle follows from cosmology with Einstein Gravity.

 

http://arxiv.org/PS_cache/gr-qc/pdf/0201/0201095v1.pdf

 

I think Swansont is correct that Mach's principle is a conjecture. The paper I was just reading sounds sure of itself, but I'm sure other authors have found different results and would disagree. Maybe "rotation is relative" and "rotation is not relative" are both conjectures, and mostly philosophical.