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Relmuis
2006-Apr-16, 01:42 PM
Suppose that a solid sphere is made of rigid matter, which may be elastically but not plastically deformed. Suppose that the size of this sphere is large enough to make quantum effects unimportant and centrifugal forces small but its mass is small enough to make its self-gravitation negligible. Suppose that the sphere is now forced to rotate at such a rate that particles on its equator attain an appreciable fraction of the speed of light.

Now I wonder:

1. Which shape the former sphere will appear to have, as seen by an observer at rest relative to the sphere but not corotating with it?

2. Which (elastic) deformations the former sphere will actually undergo, as measured by yardstick-carrying observers clinging to its surface or embedded inside an internal system of tunnels?

3. Will it be necessary to do work to maintain its rotation?

antoniseb
2006-Apr-16, 01:50 PM
sphere is large enough to make ... centrifugal forces small ... rotate at such a rate that particles on its equator attain an appreciable fraction of the speed of light.

How big would it need to be to meet this requirement?

Relmuis
2006-Apr-16, 01:58 PM
I am thinking of a "idealized rubber" ball the size of a block of houses.

antoniseb
2006-Apr-16, 02:12 PM
I'm guessing the centrifugal forces on something a couple hundred meters in diameter spinning so the equator is at relativisitic speed would be substantial.

Assuming that a radius of 100 meters, and an equitorial speed of 150,000,000 meters per second, we're talking about spinning around every 4 microseconds. The centripidal acceleration would be crushing to anyone in these tunnels you're talking about. If we are allowed to think about the material, it would also rip the rubber into atoms before one full revolution.

Let me add that you are asking an interesting question, but I want to know the scale we are talking about to make it something we can think about. I think that for centrifugal force not to be a major factor, we'd be looking at a radius of several light years (but that's just a guess).

Ken G
2006-Apr-18, 02:46 AM
We know the answer for small speeds, the sphere becomes an oblate spheroid like the Earth. So I think the question is, as c is approached, does the Lorentz contraction ever catch up and return us to a spherical shape in the stationary frame? antoniseb is right that the overall scale must be important, because the elastic properties of the material do matter. The Lorentz contraction is independent of the material properties, but the tendency to become oblate is not. My guess is, for any material that can be held together, some rotation rate will return it to a spherical shape in the stationary frame, but I really don't know, it is an interesting problem.

Relmuis
2006-Apr-18, 01:46 PM
I feel there are three parameters which govern the shape of the object: the diameter (when stationary), the number of rotations per second (as seen by a stationary observer) and the modulus of elasticity. A fourth parameter, the density (when stationary), would become important if the mass became very large. (For example in the case of a neutron star.)

The stuff which the thing is made off, is supposed to have no breaking point, that is: it will deform elastically only, and return to its original shape when the rotation is stopped. This ensures that any state will have a definite shape associated with it. To simplify the problem, one might let the elasticity modulus go to infinity, ensuring that no amount of force will deform the material; i.e. the "inside" observers will not see local deformations, though they might see global ones.

Intuition, loosely based on special relativity, suggests that in this case the length of the equator will shrink but the length of an equatorial diameter will not. So we would have a circle whose circumference is less than pi times its diameter. This makes me wonder whether there might be a deep "relativistic" reason which makes perfectly rigid matter impossible -- quite apart from the actual inside workings of the kinds of matter we are used to.

trinitree88
2006-Apr-18, 02:00 PM
Einstein spent quite a while on rigid rotating bodies, before finishing GR...might be something useful in his papers from there. It is an interesting idea. Pete.

Ken G
2006-Apr-18, 02:21 PM
I think Einstein did conclude that an infinitely rigid body could not be made to rotate. A real disk that rotates would presumably deform into a helmet shape. What that does to a sphere is not so clear-- did Penrose do some thinking on that?

kzb
2006-Apr-18, 05:56 PM
<<Intuition, loosely based on special relativity, suggests that in this case the length of the equator will shrink but the length of an equatorial diameter will not.>>

The matter along the diameter is moving with respect to the observer, so it should show the Lorentz contraction along its direction of motion. This contraction will be maximal when the diameter is pointing at the observer (highest transverse velocity) and zero when it is perpendicular to the observer.

So what would actually be observed ? The sphere becoming flattened, front-to-back, relative to the observer? This is a difficult question !

<<3. Will it be necessary to do work to maintain its rotation?>>

I don't see why it should

Relmuis
2006-Apr-19, 02:30 PM
Conservation of energy would seem to suggest that it won't be necessary, but I have a gnawing (and perhaps groundless) suspicion that ongoing deformation will be needed to keep the thing spinning. I.E. that the outer shell will tend to rotate at a different rate from the inner core. (As seen by an "internal" observer, that is.) Such a steadily increasing deformation would need a steadily increasing amount of work.

Bob
2006-Apr-19, 03:07 PM
The rotating rigid disc is a famous unsolved problem in General Relativity. Einstein discussed it in his 1916 paper. Giants like Eddington and Pauli took a crack at it but didn't work it out.

http://math.ucr.edu/home/baez/physics/Relativity/SR/rigid_disk.html

kzb
2006-Apr-19, 05:49 PM
Relmius and Bob

I've had a look at a few websites including the link Bob gave above. This is from it:

<<The stresses in the rigid disk warp spacetime. This is plausible, even assuming the mass of the disk is negligible. Recall that we had to allow the stress/strain ratio to approach infinity to obtain Born-rigidity. >>

Eric Laithwaite gyroscope drive, anyone? Particularly if work needs to be done to keep it spinning at a constant rate.

Relmuis
2006-Apr-19, 07:01 PM
I think that the work would be re-delivered if the rotation were allowed to stop; if the material is not plastically deformable, that is. If all deformations are elastical, they would all relax and yield back the energy stored in them.

Why do I suspect that ongoing deformation would be necessary? Because (by Special Relativity) the outer parts of the sphere are undergoing more time dilation than the inner parts. If both seem to be rotating at the same rate, as seen by a stationary observer, they would have to rotate at different rates as seen by internal observers. Which means that the material is being deformed. Wound up like the spring of a wristwatch, as it were.

But I am not sure of it, because there might be a deformation of spacetime which precisely counteracts the difference in time-dilation. That's what the website mentioned by Bob seems to suggest, anyway.

trinitree88
2006-Apr-19, 08:58 PM
The rotating rigid disc is a famous unsolved problem in General Relativity. Einstein discussed it in his 1916 paper. Giants like Eddington and Pauli took a crack at it but didn't work it out.

http://math.ucr.edu/home/baez/physics/Relativity/SR/rigid_disk.html

Interesting problem. A Merry-Go-Round, Elevator, and Static Mass are connected more deeply than a cursory examination reveals.Thanks. Pete.

kzb
2006-Apr-24, 06:06 PM
This isn't just an academic question. What about neutron stars. A 20 km diameter neutron star spinning at 38,000 rpm has an equatorial velocity 13% the speed of light.

Now relativistic effects at that speed might not be that obvious at first but would be easily measurable. Time would be going about 1% slower at the equator than at the poles !? What stresses and strains does this set up?

As for black holes.....

biknewb
2006-Apr-24, 06:36 PM
This isn't just an academic question. What about neutron stars. A 20 km diameter neutron star spinning at 38,000 rpm has an equatorial velocity 13% the speed of light.

Now relativistic effects at that speed might not be that obvious at first but would be easily measurable. Time would be going about 1% slower at the equator than at the poles !? What stresses and strains does this set up?

As for black holes.....

That's the reason I'm following this thread. A larger millisecond pulsar might even have the equator going as fast as 25% lightspeed, with the poles and the center at 0%.
My guess would be stress in every way, temporal, material, dimensional.... Can it stand that?

regards

Ken G
2006-Apr-24, 06:42 PM
I think you have to be a GR expert, or do models of pulsars, to know the answer to this one. Hopefully such an expert will happen on this thread and enlighten us, it's a good question.

afterburner
2006-Apr-28, 04:55 PM
would a prefect vacuum be created around the sphere? Would it even reflect light?

I personally think its a silly experiment. With my scewed view of the universe, i say all you're gonna get is a spinning sphere.

Oh and perhaps "dragged" particles creating a "disk" extending from the equator.

Zamise
2006-Apr-28, 06:25 PM
Maybe there would be points partway through the sphere and at the center where gravity and centrifical force would be at equalibrium. The more massive the sphere the more gravity too I'm thinking. So, the faster the sphere spins the closer to the center that balance becomes, the slower the spin the further away it would be from center. Also the angle of an observer standing on an inner or outer surface to resist falling down or out and away would be steeper too as they get closer to the spheres spinning axis. Harder to resist falling down and in or out and away in either direction the nearor they would be to the equator. To an idependent observer already away from it, the gravity from its mass would attract them the same weather they were more toward the equater or spinning axis since the centriphical force would have little to no effect on them. I'm also thinking there might be a 3rd equalibrium that moves in opposite direction in relation to the one closer to the center point depending on the spin and gravity and mass etc. that might help form orbital rings and debree.

I don't really know, no grav expert, just what I see in my mind, but It is intresting to think about even if I'm totally wrong, and I think this is a reason to explain some of those funny hollow earth theories I've read about, where 2 pendulems could actually have a meeting point in outer space and the center of the earth at the same time if they arnt dead on the axis or equator. Anyway, thats too goofy to get into now, but intresting too.

Dngrsone
2006-Apr-28, 08:00 PM
Would not a massive object such as a neutron star or black hole eventually jump to a toroidal shape at a certain rotation speed?

Relmuis
2006-Apr-29, 02:05 PM
Why would it do that?

By the way, a neutron star would probably behave differently from the object in the proposed thought experiment. Its gravity is not negligible, and it is likely that it may be plastically (as opposed to purely elastically) deformed. I suspect that it isn't even a solid, but more akin to a liquid.

Dngrsone
2006-May-02, 04:22 AM
Why would it do that?

By the way, a neutron star would probably behave differently from the object in the proposed thought experiment. Its gravity is not negligible, and it is likely that it may be plastically (as opposed to purely elastically) deformed. I suspect that it isn't even a solid, but more akin to a liquid.

I agree with the liquid assessment. As for the toroid theory, I'm thinking it's a conservation of energy thing-- the spin imparts enough centrifugal force to push the center of gravity away from teh axis. The resulting shape of the mass would be like a doughnut (or bagel for those averse to sugar).

Of course, this structure would only last as long as there is sufficient spin.

Jens
2006-May-02, 07:24 AM
This is just a wild idea from an ignorant amateur, but what if, although the Lorentz contraction happens, the mass also increases, and so the "natural distance" within the atoms increases, or something like that?

But it is a problem that's vexed me before.

biknewb
2006-May-02, 07:46 AM
I agree with the liquid assessment. As for the toroid theory, I'm thinking it's a conservation of energy thing-- the spin imparts enough centrifugal force to push the center of gravity away from teh axis. The resulting shape of the mass would be like a doughnut (or bagel for those averse to sugar).

Of course, this structure would only last as long as there is sufficient spin.

Just wondering: how is energy conserved this way?
Another: wouldn't the center of gravity of a dougnut still be in its center, in the hole?

regards

Dngrsone
2006-May-03, 04:46 AM
Just wondering: how is energy conserved this way?
Another: wouldn't the center of gravity of a dougnut still be in its center, in the hole?

regards

The effective center of gravity would be in hte center, but the majority of the mass, being driven away from center by centrifugal force, would tend to gravitat toward a spot a certain distance away from center (in a circular path) based on teh rotational speed.

Mind, this is purely conjecture on my part, intuition isn't always a good mathemetician.

Relmuis
2006-May-03, 01:51 PM
In a nonrelativistically rotating sphere, with equal density throughout, the force of gravity points towards the center and is proportional to the distance to the center, but the centrifugal force points away from the axis, and is proportional to the distance to the axis.

Hence, everywhere in the equatorial plane the two forces are pointing in opposite directions, and their sizes are in the same ratio. Their sum must everywhere point to the center or it must everywhere point away from the center. If the former, no torus could be expected to form. If the latter, the particles near the axis will feel less force driving them away, then the particles far from the axis. This must be true even in a plane parallel to the equatorial plane, where only one component of the force of gravity is counteracting the centrifugal force (or, rather, providing a centripetal force).

This suggests that a liquid would fly apart rather than forming a torus, while a solid could not form a torus unless it were stressed beyond its tensile strength near the axis, in which case its tensile strength would also be surpassed further from the axis, so it would shatter, rather than producing a neat hole through the middle.

Therefore, if a torus forms, it must be due to relativistic effects.

I do not mean to say that a pre-existing torus might not be able to maintain itself. That's something I have often wondered about, and about which I have even started another thread.

http://www.bautforum.com/showthread.php?t=27790&highlight=toroidal

biknewb
2006-May-03, 02:11 PM
... while a solid could not form a torus unless it were stressed beyond its tensile strength near the axis,...

A pre-existing torus can exist because it is already separated in the centre. It will fly apart just like the sphere when tensile stress become too great. Before that, it will happily stay a toroid.
And now i'll look at the other thread.

regards

grant hutchison
2006-May-03, 02:26 PM
The situation for fluid bodies held together by gravitation is well worked out in the Newtonian domain, I think.
As you ramp up the rotation speed of such an object, it graduates through a series of ellipsoidal shapes and then starts to "leak" from the equator as Relmuis describes when its self-gravity can no longer supply the necessary centripetal force.
I'm distant from my reference sources at present, but a search on "McLaurin" "Jacobi" and "ellipsoid" should turn up something on the topic.

Grant Hutchison