Exploring the Mysteries of Rotation in Mathematics

[Shahin.Omar]

Hello!

I am certain that a lot of debate and good debate must have gone in this direction. I do not know where on the net I can get to read about it.

Case 1 (the very usual one)

Telling rotation in absence of a reference frame. How does a person who cannot see any reference point in the sky tell if Earth is rotating?

Case 2

Moons phase locking. Why moons (moons of almost all the planets) are phase locked, and planets closer to Sun rotate much slower?

Case 3

Why planets rotate?

Case 4

How exactly did the celestial bodies turn spherical?

Case 5

WMAP image of the universe, why does it look like any other celestial object, spherical, oblate and having edges?

I know that non-mathematical explanation for these events exist.

Regards
Shahin

 

[HallsofIvy]

Basically, relativity add nothing to those explanations. They involve such relatively low masses and low speeds that relativistic effects are neglegible.

 

[Shahin.Omar]

Basically, relativity add nothing to those explanations. They involve such relatively low masses and low speeds that relativistic effects are neglegible.

Because (without an external reference frame) GR cannot tell if a body is rotating or not, and going by the shape of the universe does it not suggest that it (universe) is rotating?

A rotating universe justifies its expansion.

My basic question really is if rotation is out of scope for GR/Mathematics, I know GR rules out a rotating universe, however even if it were rotating, GR could not tell?

 

[Fightfish]

Hello!

I am certain that a lot of debate and good debate must have gone in this direction. I do not know where on the net I can get to read about it.

Case 1 (the very usual one)

Telling rotation in absence of a reference frame. How does a person who cannot see any reference point in the sky tell if Earth is rotating?

Well, physical evidence of the Earth's rotation include observations of polar flattening and the equatorial budge, and an experiment by Foucault involving the famous Foucault Pendulum demonstrates the existence of the Coriolis effect.

 

[Shahin.Omar]

Well, physical evidence of the Earth's rotation include observations of polar flattening and the equatorial budge, and an experiment by Foucault involving the famous Foucault Pendulum demonstrates the existence of the Coriolis effect.

Thank you for your reply. I knew one can tell rotation, as it causes 'centrifugal force' that would cause weight reduction.

One can tell about rotation (in absence of an external reference frame) by identifying and detecting the effects of rotation. (The same does not apply to linear motion.)

I can see one effect of rotation on universe, exansion of the universe, is there any observational evidence that suggests universe should not be rotating?

Can you please also tell what theory we have for the phase locking of moon, why they lost their angular momentum?

Shahin

 

[ernestpworrel]

Case 1:

I would say that there is no way to tell if the planet is rotating without a reference point to go by.


Case 2:

Moons are phase locked because they are caught inside LaGrange points. Planets closer to the sun would rotate slower because of the imbalance between the sun's gravitational pull and that of other celestial objects.


Case 3:

The reason planets rotate is that they are unevenly influenced by the gravity of other celestial bodies. There may be bodies in space that don't rotate.


Case 4:

I can't exactly explain why stars are spherical, but given that there is no erosin in space I would say that the most likely reason for the spherical shape of solid objects in the universe is meteoric impacts.


Case 5:

I think the WMAP image looks like that because that is how the lens is shaped. I don't think we should assume from those images that the universe is spherical or finite.

 

[Dale]

Telling rotation in absence of a reference frame. How does a person who cannot see any reference point in the sky tell if Earth is rotating?

Use a gyroscope or ring interferometer.

Why planets rotate?

Conservation of angular momentum.

WMAP image of the universe, why does it look like any other celestial object, spherical, oblate and having edges?

It is an Aitoff or Hammer projection. Like one way of drawing a map of the globe on a flat piece of paper. It is not an indication of the shape of the universe.

 

[Shahin.Omar]

ernestpworrel/DaleSpam, thank you for your relies.

Case 1:

I would say that there is no way to tell if the planet is rotating without a reference point to go by.

I think I agree with you and I am hppay to close this case. *I* think Foucault pendulum experiment cannot tell whether the Earth is rotating or it is just the pendulum rotating about stationary Earth's orbit, again perhaps moon is also a bit oblate in shape but it does not rotate. Similarly you would need different tyeps of experiments to conduct if you were to tell if the galaxy we are in is rotating about its axis or not, and despite having done that you would not be able to tell if it is the galaxy rotating about its axis or is this just you rotating about stationary galaxy's orbit?

I think this is the essence of relativity and this was once explained to me by my professor (I did go thrugh G. 't Hooft's Introductory GR available online but could not find my answer). If anyone disagrees please reopen. I am sure there should be some good material available online.

Case 4:
I can't exactly explain why stars are spherical, but given that there is no erosin in space I would say that the most likely reason for the spherical shape of solid objects in the universe is meteoric impacts.

I am happy to close this one as well.

Case 5
It is an Aitoff or Hammer projection. Like one way of drawing a map of the globe on a flat piece of paper. It is not an indication of the shape of the universe.

I am happy to close this one as well.

However I would like some more input on Case 2 and Case 3.

Case 2:
Moons are phase locked because they are caught inside LaGrange points. Planets closer to the sun would rotate slower because of the imbalance between the sun's gravitational pull and that of other celestial objects.

OK, as far as I understand LaGrange points, this tells about a possibility that a small body can be stationary in space.

This does not explain the whole thing. Bodies closer to Sun rotate slowly, bodies farther from Sun rotate much faster, and moons of nearly all the planets are phase locked. This indicates more towards a fact they were rotating in the past but they lost their angular momentum. Somewhere I have read about rotation braking as well.

As it looks to *me*, bodies under large gravitational force of another body gradually loose their angular momentum.

Case 3:
The reason planets rotate is that they are unevenly influenced by the gravity of other celestial bodies. There may be bodies in space that don't rotate.

This one is a bit difficult to appreciate. I have another exlanation that says they planets received their initial angular momentum from the rotating gases they were formed of. There is yet another that says planets started rotating because of meteor fall. Take a look at Saturn and let me know if the above two exlanations justify its rotation.

The question really is how did they start rotating (defying conservation of angular momentum)? Can I have an explanation which is mathematical or testable?

Regards!

 

[sganesh88]

The question really is how did they start rotating (defying conservation of angular momentum)? Can I have an explanation which is mathematical or testable?
Regards!

Calm down. You seem to be in a hurry to close all those 'cases'. Don't spoil the fun.

The start of rotation doesn't defy conservation of angular momentum. Imagine a clay ball going straight with a particular speed. It hits an arm of a straight rigid pole. The arm is so designed such that it can rotate around the pole with a bearing arrangement. Completely inelastic collision. The clay ball sticks to the arm. The arm starts rotating. Where did this angular momentum come from? The thing is that the clay ball which was going straight in fact had an initial angular momentum w.r.t the pole axis. Angular momentum is a mathematical function. R X mv. So rigid and indifferent. It doesn't care whether a particular particle is executing rotation about some axis. As far as there is a reference point, a reference frame and a particle with some speed in a particular direction measure from this frame, angular momentum can be found.
For planets, the centripetal force to capture random gas molecules and make them rotate along with the already swirling ones was provided by gravity.

 

[Dale]

Hi ernest, sorry about this, I was content to let your answers slide, but the OP seems to accept them readily. Unfortunately you are incorrect in pretty much every response.

I would say that there is no way to tell if the planet is rotating without a reference point to go by.

Yes, you can. A sensitive gyroscope or http://en.wikipedia.org/wiki/Ring_laser_gyroscope" has to do with tidal effects, not the LaGrange points.

The reason planets rotate is that they are unevenly influenced by the gravity of other celestial bodies. There may be bodies in space that don't rotate.

No, they rotate due to conservation of angular momentum, not uneven gravity. As a planet or a solar system is formed the initial dust cloud has some overall angular momentum just by chance (it would be an incredible coincidence for it to have 0 angular momentum). That angular momentum must be conserved and results in the angular momentum both of the orbits and the spin of the planets.

I can't exactly explain why stars are spherical, but given that there is no erosin in space I would say that the most likely reason for the spherical shape of solid objects in the universe is meteoric impacts.

That is not relevant for objects like stars or gas giants. The short answer is that a spherical shape is a minimum energy configuration.

I think the WMAP image looks like that because that is how the lens is shaped. I don't think we should assume from those images that the universe is spherical or finite.

You are correct, you should not assume anything about the shape or size of the universe from those images, but it is not due to the lens, it is simply a projection.

 

[D H]

Case 1 (the very usual one)
Telling rotation in absence of a reference frame. How does a person who cannot see any reference point in the sky tell if Earth is rotating?

I would say that there is no way to tell if the planet is rotating without a reference point to go by.

As already mentioned, both a Foucault pendulum and ring laser gyro falsify this claim. There are frames in which the Sagnac effect is null. The equations of motion take on a simpler form in such frames compared to frames where the Sagnac effect is observed. These frames have one thing in common: The remote stars, and particular the very remote quasars are stationary. In short, they are non-rotating frames.

Case 2
Moons phase locking. Why moons (moons of almost all the planets) are phase locked, and planets closer to Sun rotate much slower?

Moons are phase locked because they are caught inside LaGrange points. Planets closer to the sun would rotate slower because of the imbalance between the sun's gravitational pull and that of other celestial objects.

Nonsense. Google the terms you use.

The reasons moons become phase locked is because of gravity gradient torques. Averaged over the span of an orbit these torques act to reduce the moons rotation rates.

Case 3:
The reason planets rotate is that they are unevenly influenced by the gravity of other celestial bodies. There may be bodies in space that don't rotate.

The reason planets rotate is because they were rotating when they formed and because angular momentum is a conserved quantity.

Case 4:
I can't exactly explain why stars are spherical, but given that there is no erosin in space I would say that the most likely reason for the spherical shape of solid objects in the universe is meteoric impacts.

More nonsense. Stars are spherical because of self gravitation.

 

[sganesh88]

As already mentioned, both a Foucault pendulum and ring laser gyro falsify this claim. There are frames in which the Sagnac effect is null. The equations of motion take on a simpler form in such frames compared to frames where the Sagnac effect is observed. These frames have one thing in common: The remote stars, and particular the very remote quasars are stationary. In short, they are non-rotating frames.

@Dalespam and DH
Just because the laws take on a simpler form doesn't mean its the only acceptable frame right. This isn't a matter of consensus. Do you guys mean to say rotation is absolute?

 

[D H]

Just because the laws take on a simpler form doesn't mean its the only acceptable frame right. This isn't a matter of consensus.

Correct. There is nothing wrong with using a rotating or accelerating frame.

Do you guys mean to say rotation is absolute?

In the sense that one can measure whether a frame is rotating with respect to an inertial frame, yes.

 

[sganesh88]

In the sense that one can measure whether a frame is rotating with respect to an inertial frame, yes.

So its absolute rotation w.r.t an inertial frame. Not a 'stand alone' absolute rotation. A frame detecting its own motion is meaningless.

 

[D H]

So its absolute rotation w.r.t an inertial frame. Not a 'stand alone' absolute rotation. A frame detecting its own motion is meaningless.

There is a big difference between motion (i.e., velocity) and acceleration. If you are in an isolated spacecraft (or a windowless elevator car), there is no way to tell from experiments conducted inside the vehicle how fast the vehicle is moving or how it is oriented. There are however experiments that will tell you whether it is accelerating or rotating. In that sense, acceleration and rotation are absolute.

 

[sganesh88]

There are however experiments that will tell you whether it is accelerating or rotating. In that sense, acceleration and rotation are absolute.

Ok. Let me be clear in my argument. Please correct me if i go wrong somewhere.
What i have understood is that position,displacement, velocity, acceleration etc are all quantities that can be inferred by the observer in a frame about other bodies. An observer detecting his own motion seems ridiculous to me(by motion i mean all the parameters associated. Position, velocity, acceleration, jerk, everything). It shatters the concept of a reference frame.
Its similar to saying that me along with the train is moving forward seeing the whole platform moving back. Just an accumulation of experiences that makes me think "No kidding. The platform can't move back".
When you're saying he can detect his frame's rotation, you mean w.r.t an inertial frame. But that is a narrow minded thinking i would say. Like a person "deciding" that he is an eccentric viewing the society going around monotonously.

I m sorry if my line of thinking is absurdly wrong. Would be happy if you could correct me.

 

[D H]

sganesh88, to measure your position, velocity, orientation you do need some external reference. To measure acceleration or rotation you don't. The external reference is in a sense built into space-time. Google Mach's Principle.

 

[Dale]

What i have understood is that position,displacement, velocity, acceleration etc are all quantities that can be inferred by the observer in a frame about other bodies.

It should be clear from Newton's second law that acceleration is fundamentally different than position and velocity. The equation f = m a = m x'' is second-order in position. So it should be no surprise that position and its first derivative (velocity) are relative and its second derivative (acceleration) and higher are absolute.

Acceleration (rotation is a type of acceleration) can be measured with a variety of physical devices, this is the basis of http://en.wikipedia.org/wiki/Inertial_guidance_system" .

 

[sganesh88]

It should be clear from Newton's second law that acceleration is fundamentally different than position and velocity. The equation f = m a = m x'' is second-order in position. So it should be no surprise that position and its first derivative (velocity) are relative and its second derivative (acceleration) and higher are absolute.

I think all that results due to the very way we define an inertial frame. Going by that line of reasoning, we could also say that the distance between two points,difference in velocities between any two particles and a host of other quantities as absolutes. All these are absolutes w.r.t an inertial frame. They are not universal absolutes.
I am just skeptical about the special status you guys attribute to the "acceleration" factor.

 

[sganesh88]

Whats the role of GR in this? Doesn't it say that both inertial and non-inertial frames are equivalent? That should mean that the results of the equipment that supposedly detect "your" rotation could be attributed to the motion of some other object.
For example the detectors on the under side of the mouse could say to the CPU that the mouse is moving. We could create the same result by moving the mouse pad.
Does a statement "the ball is rotating" cut off from reference to any frame make sense? Nature wouldn't give preference to acceleration this way i feel. :-O

 

[mikeph]

You can measure a force, so you can measure acceleration.

A rotating observer is not valid to claim they are at rest and something else is rotating, because they will feel the centrifugal force which will make no sense to them if they think they are at rest. Why are your arms flying out if you're at rest?

 

[D H]

Whats the role of GR in this? Doesn't it say that both inertial and non-inertial frames are equivalent?

A theory that makes a claim contradicted by observation is a non-viable theory. The observation falsifies the theory. If Einstein had said that inertial and non-inertial frames are equivalent his theory would have been dead on arrival. Fortunately, GR does *not* say that. It instead says that inertial and non-inertial frames are equally valid. There is a big difference between “equivalent” and “equally valid”. The space-time metric in a rotating frame takes on a quite different form that the space-time metric in a non-rotating frame.

That should mean that the results of the equipment that supposedly detect "your" rotation could be attributed to the motion of some other object.

In this case that other object is apparently all of the mass in the universe. I said apparently because physicists do not know why inertia arises. If you want to explain how the collective mass of the universe results in inertia, have at it. Without that explanation, we need to resort to axiomatic explanations. The equivalence principle and the existence of non-rotating frames are axiomatic in general relativity. General relativity retains the concept of the “fixed stars.”

 

[Dale]

All these are absolutes w.r.t an inertial frame.

That is a meaningless sentence. In a physics context "absolute" means that all reference frames agree on the value and "relative" means that different reference frames disagree on the value. So saying "absolute w.r.t. an inertial frame" means that it is a value that all reference frames agree on but only a single inertial frame agrees. It is self-contradictory.

All reference frames (inertial and non-inertial) agree on what is called the proper acceleration, which is the Minkowski norm of the acceleration four-vector. Therefore it is absolute. That is the acceleration which is measured by an accelerometer or a ring interferometer.

 

[Shahin.Omar]

sganesh88/DaleSpam/D H/MikeyW, thank you for your replies!

 

[D H]

In any case what exeriment do you propose if I wanted to know, as part of solar system Earth is rotating about Sun's axis or not?

Again what do you suggest if I wanted to know whather the galaxy I am in is rotating about its axis or not?

Whoa there! You appear to be confusing rotation with revolution. These are quite distinct concepts.

So you cannot tell if a body is rotating or not, the success is very limited.

Espousing personal theories violates the rules of this forum. You have been given two very specific ways to determine whether a body is rotating, a Foucault pendulum and a ring laser gyro.

Case 2: (tidal locking) Can you please explain a little bit more? As I can see GGT does not seem to explain the loss of angular momentum.

Just because you don't understand something does not mean it isn't true.

=======

Your eagerness to close your cases for nonsense reasons combined with the five rather disparate questions raised in the opening post combined with some of your early statements ("I know that non-mathematical explanation for these events exist" in post #1; "A rotating universe justifies its expansion" in post #3) makes me strongly think that you have an ulterior motive here. This forum is not the place for expounding personal theories of physics, particularly non-mathematical and non-sensical personal theories.

 

[Shahin.Omar]

OK, let me remove whatever you have problem with!

Case 1: Rotation

You can measure a force, so you can measure acceleration.
A rotating observer is not valid to claim they are at rest and something else is rotating, because they will feel the centrifugal force which will make no sense to them if they think they are at rest. Why are your arms flying out if you're at rest?

So can you tell if it is only Foucault pendulum rotating about the stationary Earth's axis, or is it the Earth rotating about itx axis, the effect should be the same? I consider this a genuine question from relativistic point of view.

Case 2:

The reasons moons become phase locked is because of gravity gradient torques. Averaged over the span of an orbit these torques act to reduce the moons rotation rates.

Can you please explain a little bit more?

Case 3:

The reason planets rotate is because they were rotating when they formed and because angular momentum is a conserved quantity.

This to me is still like passing the buck on. This does not explain why the gases they were formed of were rotating.

Case 4: Dropped.

Case 5: Dropped.

Are you happy now?

If you have any roblem with any statement, just let me know I will edit correct it. I am not interested in expounding any personal theory here. I am just trying to verify my understanding of contemporary physics. Trust me on this!
Cheers!

 

[D H]

So can you tell if it is only Foucault pendulum rotating about the stationary Earth's axis, or is it the Earth rotating about itx axis, the effect should be the same? I consider this a genuine question from relativistic point of view.

It isn't a genuine question from relativistic point of view. It is a kooky question. Relativity distinguishes between rotating and non-rotating frames via the apparent motion of the "fixed stars", which is the same concept as used in Newtonian mechanics.

You appear to be intentionally ignoring the Sagnac effect and ring laser gyros.

Case 2: Can you please explain a little bit more?

http://en.wikipedia.org/wiki/Tidal_locking#Mechanism
http://books.google.com/books?id=aU6vcy5L8GAC&pg=PA160#v=onepage&q=&f=false

Case 3: This to me is still like passing the buck on. This does not explain why the gases they were formed of were rotating.

You are asking me to write a book! There are plenty of articles and books on this subject aimed at everyone from the lay reader to experts in the field. Here are a couple:
http://books.google.com/books?id=CukAzsJHQaQC
http://iau.hq.eso.org/science/meetings/past/symposia/155/

 

[Dale]

This to me is still like passing the buck on. This does not explain why the gases they were formed of were rotating.

This is simply a matter of probability. Consider a gas cloud composed of a specific number of particles (trillions and trillions and trillions). Now, suppose that those perticles are moving around randomly. What is the probability that it has no measurable angular momentum? It is vanishingly small. Indeed, if we did discover a star with no angular momentum we would be surprised. The answer to your question is simple statistics.

 

[jambaugh]

I came upon the phase locking answer differently. I was thinking about a satellite with a boom (long arm to induce stability in orbit through tidal forces). If you put a flywheel in a satellite and began spinning it up the torque on the flywheel will require an equal opposite torque on the rest of the satellite. But with a boom arm one could use the tidal force to oppose this, essentially an off alignment boom would get a tidal torque which is transferred to the flywheel. Over time the wheel spins faster and faster and then we let the boom swing back toward the planet.

This puzzled me because total angular momentum must be conserved. The satellite will not induce a similar opposite torque on the planet (replace the planet with a point mass and the effect on the satellite is the same). It dawned on me that the change in angular momentum of the flywheel must reflect a change in orbital angular momentum. The orbit must change.

This puzzled me further because as I saw it at first no extra force is being applied to the satellite to change its orbit. Then I realized this given the tidal effect the force due to gravity will not be parallel to the line from satellite center of mass through the planet's center. There is a lateral component effecting a change in the orbit. (See picture)

With orbiting planets or moons there will be a similar effect. The rotation of the orbiting body is the "flywheel" in question and it is being slowed by the tidal effect. This because the tide make the body deform (bulge) but the rotation pulls this bulge off line with the center of orbit. It can't re-deform fast enough to always be in line. Through this mechanism rotational angular momentum gets transferred to orbital angular momentum. A similar (actually reverse I think) effect causes elliptic orbits to eventually become circular.

Ultimately given enough time all orbiting bodies will become tidally locked. But the effect is faster the stronger the tidal force i.e. the closer the orbiting body is to the center of orbit. Hence Earth's moon has already become locked. Mercury is almost there being in a 2 to one orbit-rotation resonance.

The Earth will eventually become tidally locked with the Moon too. Ultimately the Earth-Moon system would become tidally locked with the Sun either by the Moon merging with the Earth or (with very low probability) achieving a Lagrange point orbit. However the Sun should go red giant and engulf the Earth long before this can actually occur.

 

[sganesh88]

There is a big difference between “equivalent” and “equally valid”.

Whats the difference? My understanding of "equivalent" is that while inertial and non inertial observers tries to make sense of the motion of an object, both of them make use of the same set of laws. Nature's laws don't vary with the reference frames. Just like Einstein made all the inertial frames equivalent "again" after the michelson-morley debacle, i thought he made ALL the reference frames equivalent.

All these are absolutes w.r.t an inertial frame.

That was a mistake. I meant absolute w.r.t all inertial frames. Not "an inertial frame"

 

[D H]

Whats the difference? My understanding of "equivalent" is that while inertial and non inertial observers tries to make sense of the motion of an object, both of them make use of the same set of laws. Nature's laws don't vary with the reference frames. Just like Einstein made all the inertial frames equivalent "again" after the michelson-morley debacle, i thought he made ALL the reference frames equivalent.

You are mistaken. Nature's laws do in a sense vary with reference frames. There are no fictitious forces in an inertial frame. You can't even begin to describe the laws of motion in a rotating frame unless and until you know the frame's angular velocity and angular acceleration with respect to inertial.

 

[Dale]

we could also say that the distance between two points,difference in velocities between any two particles and a host of other quantities as absolutes. All these are absolutes w.r.t an inertial frame.

That was a mistake. I meant absolute w.r.t all inertial frames. Not "an inertial frame"

Neither the distance between two points nor the difference in velocities between two particles are frame invariant. Different frames will disagree and therefore they are both relative quantities. Things which are absolute and which are relative are well understood and have a coherent mathematical framework (Minkowski geometry) where the relative things are the components of four-vectors and the absolute things are the Minkowski norm of four-vectors. It is completely unambiguous.

 

[sganesh88]

Neither the distance between two points nor the difference in velocities between two particles are frame invariant. Different frames will disagree and therefore they are both relative quantities.

At speeds<<c, they are frame invariants (all inertial frames will agree). Check it out. For relativistic speeds, even acceleration ceases to be absolute- going by the definition of dv/dt.

 

[Dale]

For relativistic speeds, even acceleration ceases to be absolute- going by the definition of dv/dt.

No, proper acceleration is absolute, even at relativistic speeds. I think you are confusing coordinate acceleration which (obviously) depends on your coordinate system with proper acceleration which is frame invariant and which is the acceleration measured by accelerometers. Proper acceleration is frame invariant, it is the Minkowski norm of the four-acceleration, all reference frames agree on it, it is coordinate independent, it can be measured using accelerometers and without reference to any external object.

 

[Shahin.Omar]

I came upon the phase locking answer differently.

Thanks for explaining it as things should be explained without throwing your weight around, you know what I mean.

"Sagnac effect and ring laser gyros", I have not understood it very clearly, however if you rotate the gyros around stationary Earth's orbit, or you rotate the source of light around stationary Earth's orbit, in one of which situations, different from the original situation you will get the same result? And is not the use of source of light akin to using external reference point, given that I have not understood it very clearly, sorry?

Cheers!