what's the deal with rotating frames of reference

[Yllanes]

Well, actually, there is a point arguing it. You are thinking of a metric tensor. A metric is a function d: (X cross X)->R.

Yes, that's what a metric is, a distance function. The distance between any vector and itself is indeed 0.

We are talking about relativity. In GR and differential geometry metric <-> g. There is no ambiguity. The point is that you don't seem to know what g is. Look at my original post:

You can very well define a metric, through the mass tensor (kinetic energy), as a diffeomorphism between the two.

And you said:

Huh? Metrics always map to the real numbers (a one dimensional space). Diffeomorphisms always map between manifolds of the same dimension. How can a metric be a diffeomorphism?

If you knew what I was talking about, there would have been no ambiguity. I said 'You can define a metric, through the mass tensor'. This is in every book about the mathematics of theoretical mechanics or differential geometry. It is basic that the phase space is a symplectic manifold and so on... That paragraph was not of course aimed at explaining all those concepts, I just wanted to make clear that spacetime is not the only useful manifold in physics.

The distance d makes absolutely no sense here. Can you substract two vectors on different points on a curved manifold? Answer: no you cannot.

Everything goes through g. What is the distance between two points on a curved surface? Hint: if we define the distance as the arc length of a geodesic joining them, the result involves g.

I'm going to drop the whole 'metric of the subspace' thing because it is waste of time. The point is that you can talk about the metric of a rotating disk and it makes no difference in how many dimensions the disk is embedded.

My real problem with what you said is that you implied that it is some special feature of relativity that results in such wacky things as C !=2 pi r. But there's no need to involve relativity.
[example snipped]

Yes, but you are living on a cone, I don't have any need for that. The only thing I said is that on a rotating frame, our space is no longer Euclidean. What you are saying is that other spaces are not Euclidean.

 

[69dodge]

In 3D space, the x coordinate in determined by the distance to the yz plane. The yz plane is not a "thing" in the sense of a material object.

Yes, exactly. So, in the real world, how do you experimentally determine what the distance of an object to the yz plane is?

"point A' is a completely abstract concept. If you're going to allow it as a reference point, why not allow abstract coordinates?

Sorry, I wasn't clear. The cop doesn't think about the point abstractly. He notices that the car moves away from some concrete object at the point.

But nothing in a rotating room moves away from anything else, if the universe is otherwise empty.

 

[Art Vandelay]

Yes, exactly. So, in the real world, how do you experimentally determine what the distance of an object to the yz plane is?

You don't. How do you experimentally determine that something is at rest?

Sorry, I wasn't clear. The cop doesn't think about the point abstractly. He notices that the car moves away from some concrete object at the point.

Strictly speaking, he notices that the distance between them increases. That the car is moving an assumption. For that matter, how does he determine that the distance between them increases? What if this takes place in the middle of the desert, and there are no landmarks? Will the cop be able to know how fast the car's going?

But nothing in a rotating room moves away from anything else, if the universe is otherwise empty.

So then there's no paradox. Nothing is moving away from each other, there's no measureable difference between it and a room at rest, and everything makes sense.

This is in every book about the mathematics of theoretical mechanics or differential geometry.

Where can I find it in Do Carmo's Differential Geometry of Curves and Surfaces?

The distance d makes absolutely no sense here. Can you substract two vectors on different points on a curved manifold? Answer: no you cannot.

Of course not. They're not in the same vector spaces. I already explained this. For d(x,y) to make sense, x and y have to be from the same metric space. You can define d(x,y) where x and y are different points on a manifold, or you can define a different d(x,y) where x and y are both members of the tangent bundle of a particular point.

Is g defined if the two vectors are from different points?

What is the distance between two points on a curved surface? Hint: if we define the distance as the arc length of a geodesic joining them, the result involves g.

How do you define "geodesic"? What if there are no geodesic joining them? Or more than one?

The point is that you can talk about the metric of a rotating disk and it makes no difference in how many dimensions the disk is embedded.

I've already addressed this.

What you are saying is that other spaces are not Euclidean.

No, I'm saying you don't have to go to a non-Euclidean space. The cone is Euclidean.

 

[Yllanes]

Where can I find it in Do Carmo's Differential Geometry of Curves and Surfaces?

You can't, I'm afraid. That book is only about curves and surfaces and I was talking about more general manifolds. You would need a book on theoretical mechanics (Arnold's Mathematical Methods of Classical Mechanics is the classic, there are also several, quite advanced, books by Marsden) or a more advanced book on differential geometry (Spivak's, Choquet-Bruhat's, the list is endless). A very good book on differential geometry and applications in physics (a lot of applications) is Frankel's The Geometry of Physics. Great book, doesn't assume any previous knowledge of differential geometry and is not as painstakingly abstract as others, but goes very far and touches many, many topics. I heartily recommend this to anyone with some mathematical and physical knowledge who wants to go further.

Of course not. They're not in the same vector spaces. I already explained this. For d(x,y) to make sense, x and y have to be from the same metric space. You can define d(x,y) where x and y are different points on a manifold, or you can define a different d(x,y) where x and y are both members of the tangent bundle of a particular point.

Yes. But the tangent space at each point is isomorphic to R^n, so the distance between two vectors there is not very interesting. By the way, so there's no misunderstanding, the tangent bundle is the collection of pairs (p,v), where p is a point and v a vector on that point. In other words, the tangent bundle is roughly the collection of all tangent spaces. Vector fields live on the tangent bundle and 1-form fields live on the cotangent bundle. On a riemannian manifold, there is a diffeomorphism between the two bundles, the metric. But even if we don't have a metric, those spaces still exist (and are manifold of dimension 2n, if n is the dimension of the original manifold).

Is g defined if the two vectors are from different points?

No, in the sense that you cannot do g(u,v), if u and v are on different points, because g is a function of the point. But g is a tensor field, so it is defined everywhere. If u(x) and v(x) are also vector fields, then g(u(x),v(x)) is defined everywhere.

How do you define "geodesic"? What if there are no geodesic joining them? Or more than one?

This is not important for what I was saying. I just wanted to illustrate that the metric is a field, defined continously and differentiably on all the manifold. The distance is no such thing. Sometimes it is easy to talk about the distance between two points, as the arc length of a geodesic joining them.

For example, let's talk about surfaces. You know about that if you have read Do Carmo's. There are several (equivalent) definitions of geodesic on a surface. For instance, a curve is a geodesic if it can be parameterised with acceleration always perpendicular to the tangent plane at each point. Or we can say that a curve is a geodesic if the tangent plane to M is always perpendicular to its osculating plane. Or a curve with extremal arc length (not always a local minimum, as you pointed out earlier). You can also talk about covariant derivatives, Christoffel symbols and geodesic curvature, though I don't remember if that book covered these things. Anyway you define it, if x is a point of M and v a vector on the tangent plane at x, there's always one (and only one) geodesic that passes through x with velocity v. So the concept is well defined.

To summarise, because we have hijacked this thread (sorry, davefoc). The fundamental notion of metric in differential geometry is g, the metric tensor, not the distance. If we want to define some kind of distance, chances are we will do it through g. The metric tensor g is 'useful' for many other things. It is a tensor field, provides a diffemorphism between the tangent and cotangent bundles (=it serves to raise and lower indices), determines the curvature, etc. Einstein's equations say G_mv = k T_mv (k is a unit-dependant constant). T is the amount of energy-momentum and G is a tensor defined through the Riemann tensor which is in turn a function only of the metric tensor. I probably should have been more careful when talking about 'metric', I didn't think it would give raise to misunderstandings. I will say always 'metric tensor' in the future.

We were talking about different things with our 'metric'. Can we give it a rest now? Of course, if you want to ask something about differential geometry, feel free to do it, but let's not keep on arguing about the difference between d <-> g.

 

[Art Vandelay]

This is not important for what I was saying. I just wanted to illustrate that the metric is a field, defined continously and differentiably on all the manifold. The distance is no such thing.

Actually, you're now using another term with more than one meaning: field. There's the physics meaning, and the math meaning. It looks like you're using the physics meaning. Anyway, you can't talk about continuity without there being some topology defined. For a metric space, the open ball topology is canonical, which makes d continuous by definition.

For example, let's talk about surfaces. You know about that if you have read Do Carmo's. There are several (equivalent) definitions of geodesic on a surface. For instance, a curve is a geodesic if it can be parameterised with acceleration always perpendicular to the tangent plane at each point. Or we can say that a curve is a geodesic if the tangent plane to M is always perpendicular to its osculating plane. Or a curve with extremal arc length (not always a local minimum, as you pointed out earlier). You can also talk about covariant derivatives, Christoffel symbols and geodesic curvature, though I don't remember if that book covered these things.

Do Carmo takes d to be given, and bases the definitions on that. For instance, the term "extremal arc length" is defined in terms of arc length, and arc length is defined in terms of distance. Acceleration and orthogonality are also tied to distance. So if you want to define distance in terms of geodesics, you must find a way to define geodesics that doesn't rely on a preexisting d, otherwise it's a circular definition.

Anyway you define it, if x is a point of M and v a vector on the tangent plane at x, there's always one (and only one) geodesic that passes through x with velocity v. So the concept is well defined.

Yes, but if you want the distance between point x and point y, you are guaranteed neither existance nor uniqueness of a geodesic passing through both. "The distance between x and y is the arc length of a geodesic from one to the other" therefore is not a valid definition.

 

[Yllanes]

Actually, you're now using another term with more than one meaning: field. There's the physics meaning, and the math meaning. It looks like you're using the physics meaning.

No, I'm not. Field means several things in mathematics. It can be field as in 'the complex field' or it can be a vector/tensor field. Check a book on differential geometry by a pure mathematician and you'll see that they also use it. It's not my fault that sometimes the same word has more than one meaning.

Anyway, you can't talk about continuity without there being some topology defined.

A topological manifold is a space locally homeomorphic to R^n. Of course a topology is defined. A differentiable manifold has even more structure, and a riemannian manifold even more.

For a metric space, the open ball topology is canonical, which makes d continuous by definition

As I said I'm not going to discuss this further, because it's pointless. The distance you are talking about is not a tensor field.

So if you want to define distance in terms of geodesics, you must find a way to define geodesics that doesn't rely on a preexisting d, otherwise it's a circular definition.

Neither of my definitions were circular. Take the first one 'a curve is a geodesic if it can be parameterised with acceleration always perpendicular to the tangent plane at each point'. How does this rely on a distance? I don't recall how Do Carmo does all this. I didn't like that book when I was studying classical differential geometry, so I'm not familiar with it.

Yes, but if you want the distance between point x and point y, you are guaranteed neither existance nor uniqueness of a geodesic passing through both. "The distance between x and y is the arc length of a geodesic from one to the other" therefore is not a valid definition.

Sometimes it is, that's all I said. It was just an example. Take a cylinder. It is isometric to the plane. It's geodesics are the isometric images of straight lines (circles, helices and straight lines). The only geodesic joining two points on a cylinder (and there always is one) is the image of the straight lines joining their inverse images on the plane. Its arc length is just the length of the line, the Euclidean distance between the inverse images of the points. We know they are isometric because they have the same metric tensor. And, again, I don't care for the definition of distance between two points on a manifold, it was just an example.

If you want, I can define for you the intrinsic distance between two points. Maybe this is the concept you are always talking about. Let a be a differentiable map a: I=[0,1] -> R^3. We define its length as the integral L(a) = \int_0^1 |a'(t)| dt. Notice that this definition corresponds to the arc length of a curve. Now, if M is a surface and p,q two points on M, we call S_{pq} to the set of all curves a: I -> R^3 such that a(0)=p, a(1)=q and a(I) is contained in M. With all this, we can define the intrinsic distance between two points as the infimum of the L(a) : a in S_{pq}. Is this the definition you are using? The cylinder example works with this definition. If it is, you will agree with me that this distance is not a function of the point, not a field and so it is not the metric we talk about in GR. Notice further that it doesn't act on vectors, but is a function of two points.

 

[SkepDick]

Not to change the subject, but I have a question. Mach's Principle suggests to me that in the early universe, when all the matter was closer togeather than it is today, inertial forces were either greater or smaller. If the strength of inertial forces is, in fact, changing with time, would this have a significant impact on the large-scale evolution of the universe?

 

[Yllanes]

Not to change the subject, but I have a question. Mach's Principle suggests to me that in the early universe, when all the matter was closer togeather than it is today, inertial forces were either greater or smaller. If the strength of inertial forces is, in fact, changing with time, would this have a significant impact on the large-scale evolution of the universe?

Take into account that Mach's principle, in its original form, is too vague to really mean anything solid. Furthermore, if you try to formulate it precisely you may get false results. You can try to get around this by saying, as Einstein did, that it only holds in closed universes. Here's an excerpt from the introduction of Wald's General Relativity, the favourite book of relativity workers:

The second much less precise set of ideas which motivated the formulation of general relativity goes under the name of Mach's principle. In special relativity as in prerelativity notions of spacetime, the structure of spacetime is given once and for all and is unaffected by the material bodies that may be present. In particular,"inertial motion" and "nonrotating" are not influenced by matter in the universe. Mach [...] found this idea unsatisfactory. Rather, Mach felt that all matter in the universe should contribute to the local definition of "nonaccelerating" and "nonrotating"; that in a universe devoid of matter there should be no meaning of these concepts. Einstein accepted this idea and was strongly motivated to seek a theory where, unlike special relativity, the structure of spacetime is influenced by the presence of matter.

The new theory of space, time, and gravitation -general relativity- proposed by Einstein states the following: The intrinsic, observer-independant, properties of spacetime are described by a spacetime metric, as in special relativity. However, the spacetime metric need not have the (flat) form it has in special relativity. [...] Furthermore, the curvature of spacetime is related to the stress-energy-momentum tensor of the matter in spacetime via an equation postulated by Einstein. In this way, the structure of spacetime [...] is related to the matter content of spacetime, in accordance with some (but not all!) of Mach's ideas.

The point is that GR seems to agree sometimes with this principle and sometimes it doesn't. But GR is a much better tool to make predictions than Mach's principle ever was, so the best thing is to concentrate on what GR says. The mass-energy-momentum content is represented by a tensor T, which determines the structure of spacetime. This is what GR says, and it has worked perfectly thus far.

 

[SkepDick]

Thank you for your reply, Yllanes.
I was really thinking about the main idea in Mach's Principle, i.e., that inertia is a property of spacetime (and the distribution of matter) rather than a property of matter. This is a facinating idea because it leaves open the possibility that inertial forces can change with time. If this were the case, there might be (I can't say how) effects on the evolution of the universe which have left traces that are observable today, ruling this idea in or out. I would advise readers that I am approaching the limits of my understanding of these concepts but would like to hear the thoughts of anyone more familiar with them than I.

 

[Art Vandelay]

It's not my fault that sometimes the same word has more than one meaning.

Wasn't saying it was. Just something to be aware of.

Neither of my definitions were circular. Take the first one 'a curve is a geodesic if it can be parameterised with acceleration always perpendicular to the tangent plane at each point'. How does this rely on a distance? I don't recall how Do Carmo does all this.

Do Carmo mainly considers surfaces to be embedded in R^3, and pretty much everything depends on the inherent Euclidean metric of R^3. If R^3 didn't already have a metric, most of the definitions would be meaningless.

If you want, I can define for you the intrinsic distance between two points. Maybe this is the concept you are always talking about. Let a be a differentiable map a: I=[0,1] -> R^3.

This is particular type of metric. It takes the space, embeds it in a space that already has a metric, and then derives a new metric from the parent space. It reminds me of the time Homer Simpsons says "I carved this spoon myself. From a bigger spoon." If you're talking about a completely abstract manifold, then this option is not available to you. (And, of course, it assumes that the space is path connected.)

Is this the definition you are using?

It's an example of such a metric. It's not the only possibility. In general, a metric is any function d(x,y)X cross X)->R where
d(x,y) = 0 iff x=y
d(x,y) > 0 otherwise
d(x,y) = d(y,x)
d(x,y) + d(y,z) >= d(x,z)

The properties of a manifold are largely determined by this metric. You can put a bunch of points on an open hemisphere and another bunch on a plane, for instance, and define metrics on each such that there's a homeomorphism that preserves the metrics. So it's not really how you label the points, but how you define the metric that determines the surface. You can label the points as if they're on a hemisphere, but if you define a flat metric, the set's going to be a plane, for all intents and purposes. Conversely, you can (locally) parameterize any two dimensional manifold with two variables, making it seem like a plane, but it's the metric that's going to differentiate it from an actual plane. If, as in your example, you give a cylinder a flat metric, then it's going to be (locally) identical to a plane, for all practical purposes. You could take the exact same set of points, give it a different metric, and have it (locally) be, for all practical purposes, the same as a sphere. Or a torus. Or a Klein bottle. Or a completely abstract space.

 

[69dodge]

So then there's no paradox. Nothing is moving away from each other, there's no measureable difference between it and a room at rest, and everything makes sense.

If centrifugal force is present, you can measure it with, e.g., a spring scale, even though nothing is moving relative to anything else.

So the question is, if this room is the only thing in the universe, and nothing in the room is moving relative to anything else, might centrifugal force still be present?

"Paradox" is probably too strong a word---if centrifugal force is present, then it just is, and that's that---but it is an interesting question, I think. The naive view is that rotation causes centrifugal force, rather than "rotation" just being another name for the existence of centrifugal force. On that view, we need a definition for "rotation" other than "the existence of centrifugal force", and we can take "rotation with respect to everything else in the universe" as that definition. But what if there isn't anything else in the universe to rotate with respect to?

 

[Art Vandelay]

How can a spring scale measure force?

 

[Art Vandelay]

Another question: suppose we were to that since everything is relative, we can consider the Earth to be at rest and the stars to be orbiting around us. Then a star 4 light years away is moving about 25 light years each day, or more than 9000 times the speed of light. Does this contradict the principle that nothing can travel faster than light?

 

[davefoc]

If centrifugal force is present, you can measure it with, e.g., a spring scale, even though nothing is moving relative to anything else.

So the question is, if this room is the only thing in the universe, and nothing in the room is moving relative to anything else, might centrifugal force still be present?

My own cut at this which I put forth a bit in the post before I left on a little vacation to Death Valley is that space controls the motion of an object moving without forces applied to it. It is the line drawn by an object moving without external forces that serves as our fundamental method for distinguishing a rotating frame from a non-rotating frame.

If the universe is rotating and we are synchronized with that rotation of course we will not sense any rotation. But if one part of the universe was rotating with respect to the space we are in we will see the path of an object moving without external forces in the part of the universe we aren't in as moving in a curve in the universe we are in.

This is true not only of objects but of light beams as well. It seems like space not only controls the path of objects but of the light that passes through it.

I have also considered how a line that is formed by stretching a string between two points in the part of the universe which is rotating differently than the part we are in. Would it be curved like the light beam or does the method of stretching a string to define a line produce a line which is straight even if it is viewed from an alternate universe which is rotating at a different rate from the observing section of the universe?

At first I thought it would be curved like the light beam, but now I'm not so sure.

When I first started thinking along this line, I thought that I might have begun to understand what is meant by the GR term, bending of space. It seems that what is meant by the bending of space is exactly this idea that space is modified in such a way that light and objects moving without external forces take a different path than they would in another part of space.

Of course, all of this leans in the direction of something that seems like it must be true to me. Space has structure.

 

[Yllanes]

Do Carmo mainly considers surfaces to be embedded in R^3, and pretty much everything depends on the inherent Euclidean metric of R^3. If R^3 didn't already have a metric, most of the definitions would be meaningless.

Which is why all you are saying doesn't make sense for general manifolds. We want intrinsic properties, not something that depends on the embedding.

This is particular type of metric. It takes the space, embeds it in a space that already has a metric, and then derives a new metric from the parent space. It reminds me of the time Homer Simpsons says "I carved this spoon myself. From a bigger spoon." If you're talking about a completely abstract manifold, then this option is not available to you. (And, of course, it assumes that the space is path connected.)

This is not a metric in the relativity sense, it is a distance. In relativity we don't call the distance a 'metric'. Of course it is not for any manifold, it is valid for surfaces, which is what Do Carmo was talking about. I didn't claim it was available for general manifolds, I didn't use it at all and I don't use it in my everyday life (hint: my everyday life has very much to do with differential geometry).

It's an example of such a metric. It's not the only possibility.

You are really tiresome. I was asking you if that's what you are using, I was not claiming it is the only possible distance.

In general, a metric is any function d(x,y)X cross X)->R where
d(x,y) = 0 iff x=y
d(x,y) > 0 otherwise
d(x,y) = d(y,x)
d(x,y) + d(y,z) >= d(x,z)

Do you think I do not know that? But the metric we talk about in relativity is not this. This time I'm really finished with this. When we talk about metric in relativity we mean g. Have you read any relativity books? The same goes
for differential geometry in general manifolds.

 

[Yllanes]

"Paradox" is probably too strong a word---if centrifugal force is present, then it just is, and that's that---but it is an interesting question, I think. The naive view is that rotation causes centrifugal force, rather than "rotation" just being another name for the existence of centrifugal force. On that view, we need a definition for "rotation" other than "the existence of centrifugal force", and we can take "rotation with respect to everything else in the universe" as that definition. But what if there isn't anything else in the universe to rotate with respect to?

As I said before, there always is a metric. The room rotates with respect to the metric. In GR rotating with respect to the metric is as real as rotating with respect to the stars.

Let's consider a simpler experiment: a glass of water spining. On Earth, we know that the water will be pulled to the sides of the glass. Imagine the glass is now located in an otherwise empty universe. GR predicts that the water will be pulled to the sides (which seems anti-Machian) just the same. And again, the glass rotates with respect to the metric.

Another different concept, to show once more that GR makes predictions and is not just vague philosophy. Consider the same glass of water, without rotation this time. Now we are in our universe. Imagine we rotate stars, galaxies, planets... around the bucket. Newtonian physics says that the water would remain flat. In GR the water would get curved. This is calling 'frame-dragging' and it has been experimentally verified (Google for Gravity Probe B).

When I first started thinking along this line, I thought that I might have begun to understand what is meant by the GR term, bending of space. It seems that what is meant by the bending of space is exactly this idea that space is modified in such a way that light and objects moving without external forces take a different path than they would in another part of space.

Yes, that's pretty much the basic meaning.

 

[69dodge]

As I said before, there always is a metric. The room rotates with respect to the metric. In GR rotating with respect to the metric is as real as rotating with respect to the stars.

It seems to me there is some difference: I can tell whether I'm rotating with respect to the stars by looking at the stars, and then based on that, I can predict whether I will measure a nonzero centrifugal force. But if there are no stars in the universe, I can't tell whether I'm rotating with respect to the metric except by measuring the centrifugal force and seeing whether it's zero. So GR doesn't really make any prediction in that case. All it says is, if centrifugal force is present, then centrifugal force is present. Yes, of course, but will centrifugal force be present?

I guess we could say, GR says that centrifugal force might be present and Mach says that it definitely won't be. Does that sound basically right to you?

Let's consider a simpler experiment: a glass of water spining. On Earth, we know that the water will be pulled to the sides of the glass. Imagine the glass is now located in an otherwise empty universe. GR predicts that the water will be pulled to the sides (which seems anti-Machian) just the same. And again, the glass rotates with respect to the metric.

Is there any way to tell whether the glass is rotating, in order to predict whether the water will be pulled to the sides, besides just seeing whether the water is in fact pulled to the sides?

Another different concept, to show once more that GR makes predictions and is not just vague philosophy. Consider the same glass of water, without rotation this time. Now we are in our universe. Imagine we rotate stars, galaxies, planets... around the bucket. Newtonian physics says that the water would remain flat. In GR the water would get curved. This is calling 'frame-dragging' and it has been experimentally verified (Google for Gravity Probe B).

They're not finished with the data analysis yet for Gravity Probe B. But I think there have been previous experiments that verified frame dragging?

I'm still not sure exactly what GR predicts. According to GR, is there any reason why all the matter in our universe isn't rotating with respect to the metric, just as a single room in an empty universe might be rotating? That is, would it be consistent with GR if we did experience centrifugal force while not rotating with respect to the stars? I don't see why not. But in fact we don't. So maybe Mach was right.

 

[69dodge]

How can a spring scale measure force?

Huh? Force is exactly what spring scales measure.

Just use it in the normal manner to weigh something. If it shows that the weight is nonzero, then your room in an empty universe is not at rest.

 

[Yllanes]

I guess we could say, GR says that centrifugal force might be present and Mach says that it definitely won't be. Does that sound basically right to you?

Actually, GR says that the centrifugal force will be present. It makes a definite prediction. The scenario is a glass of water in an otherwise empty (asymptotically Minkowskian) spacetime. As I said, it rotates with respect to the metric. If you don't believe me, notice that the metric is a field ijust like the electromagnetic field. Only that now, instead of a vector, we have a rank (0,2) tensor at each point. Gravitational waves (provided they exist) are a good indication that the geometry is as good a field as any.

I'm still not sure exactly what GR predicts. According to GR, is there any reason why all the matter in our universe isn't rotating with respect to the metric, just as a single room in an empty universe might be rotating?

Yes, there are observational bounds on the rotation of the universe.

Imagine only SR at play. A rotation of the Universe could be detected by measuring redshift of light from distant sources, looking in a direction perpendicular to the axis of rotation and comparing it with sources on the axis. The redshift would not be the usual (since there is no radial velocity) but a cuadrupolar redshift, due to time dilation at the sources. Roughly the same happens in GR.

In short: a rotating universe is theoretically possible in GR, well defined, and different from a nonrotating one. However, there are strong observational bounds (isotropy) on the rotation velocity.

 

[davefoc]

Is there any way to tell whether the glass is rotating, in order to predict whether the water will be pulled to the sides, besides just seeing whether the water is in fact pulled to the sides?

That is a very good way of putting the question that I was trying to pose at the beginning of this thread.

I don't know if it is a scientifically accepted answer, but right now I think my thoughts about space having a structure that allows a body to move through it with a constant speed on a particular path without external forces is a plausible answer.

In the case of your pail, it is only necessary to determine a frame of reference in which light beams trace a straight path (determined say by a stretched string) and fix your pail with respect to that frame of reference to know that the water in your pail will remain flat. Alternatively to the light beams floating objects could also be used.

I don't know that I am not stating something that is so obvious that it isn't worth stating and if that is the case then I apologize.

Here is an article in Wikipedia on a somewhat related subject:
http://en.wikipedia.org/wiki/Aether_and_general_relativity