what's the deal with rotating frames of reference

[Yllanes]

As I said before, maybe you are thinking something similar to Mach's principle, which says roughly (and the 'roughly' is the important part):

The inertia of a system is caused by its interaction with the rest of the universe, i.e., every particle in the universe eventually has an effect on every other particle.

According to Mach, if there were a single object in the universe, it would be impossible to determine whether it was rotating.

Einstein was inspired by these ideas when he developed GR, but the theory turned out to be quite anti-Mach. If we have an universe with a rotating bucket of water (Newton's thought experiment) it would feel centrifugal forces, even if it were otherwise completely empty (according to GR).

The real problem with Mach's principle is that it is so vague as to be be meaningless. You could try for a more precise formulation, but until you translate it to mathematical terms it means nothing. And translating it is not an easy problem. For the initiate, one book that discusses this is Gravitation and Inertia, by Ciufolini & Wheeer.

OK, but my question goes to why there should be a special non-rotating frame at all if the space you are floating in consists of a complete nothingness that is incapable of interacting at all with your ship.

You say, if the universe is otherwise completely empty and we can see no stars, what is the room rotating with respect to[FONT=Courier, Monospaced]? [/FONT]The answer is that it is rotating with respect to the metric of spacetime. If you don't think that the metric is a 'real' concept, like a star, walk into a black hole... They are nothing but pure metric, no matter.
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[screw_dog]

If we are able to observe centripetal forces at work, then we know a system is rotating. If we don't observe any such forces then we know it isn't.

Now, why spacetime works like this, nobody knows, at least from a classical view. (QED, as I've mentioned, explains the geodesics rather nicely, but let's not go there.) What is important is that the mathematics of a rotating frame in a non-rotating universe, and the mathematics of a non-rotating frame in a rotating universe are exactly the same, and so there is no way at all to distinguish between them.

I'm sorry, but these two quotes appear to be in direct opposition.

Which is it? Can we tell if the windowless spaceship we are in is spinning?

 

[Yllanes]

I'm sorry, but these two quotes appear to be in direct opposition.

He said a non rotating frame in a rotating universe is no different from a rotating frame in a non rotating universe. You can't tell between those two, so it makes no sense to speak of a rotating universe, but you can tell between a non rotating and a rotating frame in a non rotating universe.

Which is it? Can we tell if the windowless spaceship we are in is spinning?

Yes.

 

[Budric]

I have a question about Newton's laws. Someone mentioned that you start getting ficticious forces in a rotating frame of reference. I keep thinking back to an old video in high school where they showed these two guys sitting at a table and pushing an object (ball or something) and it would not go in a straight line. Then the camera zoomed out and you see they're sitting on a platform that's rotating them and the table.

My question is why can I perform experiments at my table and things don't go in weird directions because the earth is rotating around the sun. Is it because the force is very small (large radius)? Or is it just my desk?

 

[Hellbound]

Budric:

The force is small. You can see this effect with long pendulums (One of the museums here has one, I wanna say the Smithsonian but I could be wrong).

Actually, the effect you see is due to the Earth rotating on it's axis; the effect from it's rotation around the sun would be so small as to be neglible (I think, not sure here).

You see some of this curving effect in weather patterns on the Earth, as well. THe major movements of air masses are, in part, driven by these effects.

The term for this is Coriolis Force. A quick google should bring you all kinds of info on it .

 

[Yllanes]

Actually, the effect you see is due to the Earth rotating on it's axis; the effect from it's rotation around the sun would be so small as to be neglible (I think, not sure here).

Yes. The centrifugal acceleration due to the rotation is of the order of 0.01*cos d (in m/s/s, where d is the latitude). The effect of the rotation around the Sun is of the order of 0.0001 m/s/s. (Compare to gravity, ~10 m/s/s). It's an easy excercise to calculate the effect due to the rotation of the Solar System around the Galaxy (take 8 kpc as the radius of the orbit and 220 km/s as the speed).

My question is why can I perform experiments at my table and things don't go in weird directions because the earth is rotating around the sun. Is it because the force is very small (large radius)? Or is it just my desk?

To further illustrate the point Huntsman made, notice that the Earth needs a whole day to complete a full rotation, so the effect of the rotation will only be significative in processes that take at least this time. The ball across the table takes a second, a hurricane takes more time. Storms in the Northern (Southern) hemisphere rotate counterclockwise (clockwise) for this reason. The other famous example is Foucault's pendulum, which Huntsman also mentioned. If you have a very big pendulum and let it oscillate, the plane of its motion will slowly rotate (we call this effect 'precession'). This was demonstrated by Foucault 150 years ago and was the first clear proof of the rotation of the Earth.

 

[epepke]

I'm sorry, but these two quotes appear to be in direct opposition.

Which is it? Can we tell if the windowless spaceship we are in is spinning?

We are, after all, two different people. You can tell us apart with the naked eye.

Ceptimus, I think, is speaking from a Newtonian + Special Relativity perspective. In that framework, rotation isn't relative, and you can declare (in that framework) that you are spinning.

I am coming from a General Relativity perspective. In that framework, there is no way at all to tell the difference between the forces that you feel from spinning and the forces that you would feel from a gravitational field of the same shape, which would be a configuration of spacetime exactly the same as if space itself (and the associated inertial paths) were spinning around you. So under GR, you can tell that either you are spinning, or the universe is spinning around you, but you cannot tell which. The point is, under GR, it is equally valid to say that spacetime is rotating as it is to say that you are rotating. The mathematics comes out exactly the same no matter which way you do it. So you cannot say that you are absolutely rotating, just that you're rotating relative to spacetime (or it is rotating relative to you).

The point being that under the GR view, more is relative than in SR or Newtonian/Galilean mechanics. In N/G mechanics, velocity in a straight line is relative, and there is no way to build a local experiment that will tell you your velocity, but times and distances are not relative. SR adds optical experiments as well, and shows times and distances to be relative, though acceleration is not relative (with the cost that now the speed of light is now, in some sense, absolute). GR shows acceleration to be relative, in the sense that there is no way to distinguish acceleration from gravitational effects.

 

[ceptimus]

Objects in the rotating spaceship will tend to move further apart. It's difficult to see how to arrange gravity to do this, unless you allow negative gravity or negative time.
Also the rotating ship will have an axis of rotation, and the apparent forces on the objects in the ship depend on their distance from the axis. This is difficult to mimic with gravity: gravitational forces tend to act in a 3d spherical manner, rather than the more 2d cylindrical distribution that results from rotation.

 

[epepke]

Objects in the rotating spaceship will tend to move further apart. It's difficult to see how to arrange gravity to do this, unless you allow negative gravity or negative time.
Also the rotating ship will have an axis of rotation, and the apparent forces on the objects in the ship depend on their distance from the axis. This is difficult to mimic with gravity: gravitational forces tend to act in a 3d spherical manner, rather than the more 2d cylindrical distribution that results from rotation.

I love these discussions, because I learn more and more about the difficulties in understanding this stuff.

See, within GR, there is no way at all to tell the difference between a rotating spaceship and a non-rotating universe and a non-rotating spaceship in a rotating universe.

I want to be very clear about that: no way at all. I think forgetting that is where people get hung up.

So if you put a spaceship, and you set it spinning, after it's spun up, there is no way at all to tell the difference between two. No way at all. If some aliens with infinite amounts of exotic matter or a warp drive or whatever set spacetime spinning around you, there would be no way at all of telling the difference.

Because there's no way at all, you can't even decide that there is a difference to tell in the first place. You could as easily say that when you span up the spaceship, you were really spinning up spacetime. The cases are in all ways equivalent under GR.

It's whether universe really "is" rotating or not rotating; there is no God with the universe in a little room to tell you the "real" answer. The only answers that exist are measurements from your frame of reference. Nothing else even has any meaning.

Now, it may be emotionally troubling or hard to think about, and one might not be able to think about the universe rotating without drinking a lot of beer first, but GR provides no way whatsoever to tell the difference.

 

[Dark Jaguar]

So the reason that model comes to that conclusion is because the model says a spinning universe would basically drag the stuff in our "non-spinning" space ship off to the sides of the ship, and not cause them to sort of veer clockwise/counterclockwise in relation to the "spin" of the universe? That seems like quite a strong gravitational effect for matter that's so far away it's gravity has no detectible effect on us at all (detectable as in every day experience, which being shoved up against the wall of a spinning spaceship would count as).

I'm fully willing to embrace the idea that centrifugal forces are essentially us being dragged around by the universe spinning around us (or whatever frame of reference you'd like, as they would in fact be identical), but it does seem like they are all a little far away to be having such a strong affect, but only in relation to spinning about. Or, is it more like the "straight paths" are determined by the net result of adding up all those gravitational effects, so the old classical "centrifugal force is caused by everything's straight direction constantly being redirected into a loop" is still in effect, but what is a straight path to take is being determined by the entire universe?

 

[epepke]

Or, is it more like the "straight paths" are determined by the net result of adding up all those gravitational effects, so the old classical "centrifugal force is caused by everything's straight direction constantly being redirected into a loop" is still in effect, but what is a straight path to take is being determined by the entire universe?

That's exactly correct, as near as I can tell.

The "straight paths" are called geodesics, and they're determined by the geometry of spacetime itself.

Theres a minor wrinkle, though. Minor conceptually, though it makes the mathematics huge. The geodesics are in spacetime, not just in space. It's important, because you might think that something that isn't moving in your frame of reference isn't going to start moving, just because some lines get bent. But it is moving: it's moving through time. Its odometer might not be changing, but its clock is.

 

[Dark Jaguar]

Ah, there's an interesting wrinkle! But I think I'll stick with fully wrapping myself around geodesics first. It's just great to finally get a handle on the GR equivilant of centrifugal force.

I'll add it's nice because this way centrifugal forces aren't any "special thing" that has to have it's own solution any more to me. It's now just a natural consequence and just a particular arrangement of things in GR as I had understood it before, but just hadn't properly been applying to that situation.

 

[Art Vandelay]

You could say that there is another metric at work, we are no longer in Euclidean space, but in a hyperbolic geometry (notice that we haven't talked about a graviation, just acceleration in special relativity).

Well, a metric is a function on the entire space, and here we'll talking about a three-dimensional subspace, so that's not exactly an appropriate term. Also, we're not in a different space, we're in a different frame of reference.

Exercise: Would the angles of a triangle sum more or less than 180º on this platform?

The term "triangle" is not well-defined.

If space is a completely empty void then would it be possible to tell the difference between a rotating and a non-rotating frame of reference?

Of course not. In order for you to tell the difference between different frames of reference, you must exist. And if you exist, then space is not void (it has you in it).

I am coming from a General Relativity perspective. In that framework, there is no way at all to tell the difference between the forces that you feel from spinning and the forces that you would feel from a gravitational field of the same shape, which would be a configuration of spacetime exactly the same as if space itself (and the associated inertial paths) were spinning around you.

Wait- are you saying that gravity normally works like a rotating reference frame, or are saying that it's theoretically possible for there to be a configuration of gravity that acts like a rotating reference frame? I don't think you've been quite clear about this.

GR shows acceleration to be relative, in the sense that there is no way to distinguish acceleration from gravitational effects.

Locally.

You say, if the universe is otherwise completely empty and we can see no stars, what is the room rotating with respect to[FONT=Courier, Monospaced]? [/FONT]The answer is that it is rotating with respect to the metric of spacetime.

I think that a more direct answer is that it's rotating with respect to itself. If you're in a room, then there isn't just one object in the universe. There's you, the ceiling, the floor, the four walls, etc. All of these are moving with respect to each other.

If you don't think that the metric is a 'real' concept, like a star, walk into a black hole... They are nothing but pure metric, no matter.

Are you saying that blacks holes are not made up of matter?

My question is why can I perform experiments at my table and things don't go in weird directions because the earth is rotating around the sun.

The Earth is not rotating around the sun. The Earth is revolving around the sun, which isn't quite the same thing. If you look at a rotating table, the ball's route is taking up a significant portion of the table, so the rotation of the table is significant. The table doesn't take up a significant portion of the Earth, so the Earth's rotation doesn't cause significant effects. Hurricanes do take up a significant portion of the Earth, so the Earth's rotation affects them, but the Earth's revolution doesn't. To be affected by the Earth's revolution, something would have to be not merely a significant portion of the Earth, but of the Earth's orbit.

Also, the orbit of the Earth involves gravity, which obviously counterbalances the centrifugal force (otherwise, it wouldn't be an orbit). Locally, the Earth's frame is actually inertial (ignoring the Earth's gravity). You only see the effect of the sun in tidal forces.

The force is small. You can see this effect with long pendulums (One of the museums here has one, I wanna say the Smithsonian but I could be wrong).

Actually, I don't think you'd really see it at all. As I said, the rotation and the sun's gravity cancel each other out.

Actually, the effect you see is due to the Earth rotating on it's axis; the effect from it's rotation around the sun would be so small as to be neglible (I think, not sure here).

Its.

 

[epepke]

Wait- are you saying that gravity normally works like a rotating reference frame, or are saying that it's theoretically possible for there to be a configuration of gravity that acts like a rotating reference frame? I don't think you've been quite clear about this.

It doesn't matter, at all, how gravity normally works. GR is based on the equivalence principle. The effects of acceleration (or, as I said, more accurately, resistance to acceleration) are gravity.

And GR has solutions for every possible configuration. That's why the equations are so big and hard to solve.

 

[Art Vandelay]

It doesn't matter, at all, how gravity normally works. GR is based on the equivalence principle. The effects of acceleration (or, as I said, more accurately, resistance to acceleration) are gravity.

No, it's not. Gravity is the curvature of space time. Acceleration is the curvature of a path. The priciple of equivalence says that spacetime is locally indistinguishable from a flat spacetime. Come to think of it, it's a rather vacuous statement. The actual spacetime can be locally approximated by an uncurved spacetime. I.e., it's differentiable.

 

[epepke]

No, it's not. Gravity is the curvature of space time. Acceleration is the curvature of a path. The priciple of equivalence says that spacetime is locally indistinguishable from a flat spacetime. Come to think of it, it's a rather vacuous statement. The actual spacetime can be locally approximated by an uncurved spacetime. I.e., it's differentiable.

Art, I can only answer questions. I cannot cure your mental disorder, whatever it may be.

 

[Yllanes]

Well, a metric is a function on the entire space, and here we'll talking about a three-dimensional subspace, so that's not exactly an appropriate term. Also, we're not in a different space, we're in a different frame of reference.

That doesn't make sense. Of course you can talk about the metric of a rotating disk, or a cylinder, or a sphere, why wouldn't you? And you can talk about the spatial part of the metric of spacetime, what do you think we are doing when we talk about cosmology and the different universes (open,closed, flat)?

We can define a metric in a very general class of manifolds. And a manifold is a very general concept. For example, the configuration space (the set of all posible positions of a system of particles) in classical mechanics is a manifold. The space of all velocities at all points is its tangent bundle and the phase space (momenta and positions) its cotangent bundle. You can very well define a metric, through the mass tensor (kinetic energy), as a diffeomorphism between the two.

The term "triangle" is not well-defined.

Why not? It has three sides, which are straight lines. The straight lines are not the ones you would find on a plane, but that doesn't matter.

Are you saying that blacks holes are not made up of matter?

In a sense they aren't. Once they are formed, they are a singularity and a metric. They are not matter in the sense stars are. Black holes are pure gravitation, one of the reasons theoretical physicists like them so much. Once you have a star, you need an astrophysicist to tell you the equation of state. With black holes there's only geometry to worry about.

Actually, I don't think you'd really see it at all. As I said, the rotation and the sun's gravity cancel each other out.

I don't know what you are saying here. Foucault's pendulum is an observable non inertial effect of the rotation of the Earth around its axis. It doesn't take a significant part of the surface, by the way, it just takes time, which, as I said before, is the important thing.

 

[Hellbound]

I don't know what you are saying here. Foucault's pendulum is an observable non inertial effect of the rotation of the Earth around its axis. It doesn't take a significant part of the surface, by the way, it just takes time, which, as I said before, is the important thing.

I think he responded before reading farther in my post, where I pointed out that the effect was due to the Earth's rotation, rather than it's orbit around the Sun.

 

[Yllanes]

I think he responded before reading farther in my post, where I pointed out that the effect was due to the Earth's rotation, rather than it's orbit around the Sun.

That's what I thought, but he also said: 'Actually, I don't think you'd really see it at all. As I said, the rotation and the sun's gravity cancel each other out'. He made a point of distinguishing between 'revolving' and 'rotating'.

And, anyway, the effect of the orbit around the Sun is minute, but that doesn't mean it is cancelled by gravity.

 

[davefoc]

Well, I am almost off on a little mini-vacation but I wanted to take a moment to post my incomplete thoughts about all this.

I thought Ceptimus did a good job of expressing the reason that if you just think about Newton's laws of motion then it becomes obvious that you don't need any special property of space to figure out when something is rotating and when it is not.

But then I go back to the notion of the merry-go-round rotating, but rotating relative to what? If the universe was rotating at the same rate as the rotating merry-go-round we would judge the merry-go-round as not rotating. It seems difficult to reconcile the two views.

I think now that the reason for the apparent conflict is that we tend to assume that the notion of a straight line is absolute. An object without any forces acting on it moves in a straight line.

But it is actually the universe that controls our ability to judge what is straight. If the universe rotated differently than however it happens to be rotating now we wouldn't know it because we would judge the path of an object without external forces on it as straight in that other hypothetical universe. But in fact the straight in the hypotheical universe would not be the same as straight in the current universe.

I was hoping to think this out a little better before I posted, but I am about to leave so I decided to post now. I think it is very possible that what I have said above as already been said, and I just didn't quite understand it. I also think that maybe I am just wrong here and still don't quite get it.