Absolutely rotationless reference frames?

[Hiero]

So there are no "absolutely motionless" reference frames, but is there a set of reference frames which could be described as "absolutely rotation-less"?

 

[Drakkith]

I believe so. As far as I know, rotation is frame-invariant. If you measure yourself as rotating, then you are indeed rotating. If you measuring yourself as not rotating, then you are not.

 

[Hiero]

The universe then does "prefer" a set of reference frames?
So the angular velocity of a reference frame (or object) is an absolute quantity?

Am I alone in thinking this is quite a profound idea? I have never heard this point be talked about.

 

[Drakkith]

The universe then does "prefer" a set of reference frames?

I don't believe so.

So the angular velocity of a reference frame (or object) is an absolute quantity?

That is my understanding, though I don't know any of the details on how to show this.

Am I alone in thinking this is quite a profound idea? I have never heard this point be talked about.

I find it about as profound as the fact that there are no preferred inertial frames. Why should one be more profound than the other or more profound than any other physical law?

 

[Hiero]

The universe then does "prefer" a set of reference frames?

I don't believe so.

Well I mean how some classical physicists thought about there being an 'absolute reference frame' ("the ether") but found it not to be the case. Well instead of there being a single absolute reference frame ("the ether"), you are telling me there is a set of them (the rotation-less ones).

I can't help but see it as "the ether" but with regards to rotation as opposed to translation.

I find it about as profound as the fact that there are no preferred inertial frames. Why should one be more profound than the other or more profound than any other physical law?

I suppose by "profound" what I really mean is that the idea doesn't sit right with me. The idea of everything being relative and no inertial reference frame being preferred is a comfortable one; the idea of certain frames having some absoluteness to them is almost unsettling.

 

[Bandersnatch]

The idea of everything being relative and no inertial reference frame being preferred is a comfortable one;

Focus on the word 'inertial' here. Rotating frames are not inertial. That's all. Non-rotating frames are as profound as any other kind of non-accelerating ones.

 

[Drakkith]

Well instead of there being a single absolute reference frame ("the ether"), you are telling me there is a set of them (the rotation-less ones).

No, I'm telling you that rotation is invariant. This does not mean that rotating frames are equivalent to an "absolute" or "preferred" inertial reference frame. One of the key things with an absolute inertial frame is that certain physical laws don't hold in another frame. For example, one of the possible consequences is that the speed of light would not be the same in all directions if you were moving with respect to this absolute inertial frame.

I suppose by "profound" what I really mean is that the idea doesn't sit right with me. The idea of everything being relative and no inertial reference frame being preferred is a comfortable one; the idea of certain frames having some absoluteness to them is almost unsettling.

Why? There are plenty of things that are not relative. Mass (invariant), rotation, acceleration, and others.

 

[Hiero]

So angular velocity, as a vector, is an absolute quantity of an object. I just want to make sure I understand this correctly. I suppose all I'm technically saying is there are set of frames which lack any centrifugal forces.

I've just never thought about this very much. It makes me wonder things like;
What is my rotation right now?
Are there places/times when the rotation and the orbit of Earth cancel and I'm rotation-less in space?
Are objects (say on my table) then exerting different tensions to balance centrifugal forces at different times/places (say now compared to a different part of Earth's orbit about the Sun)?

 

[Drakkith]

So angular velocity, as a vector, is an absolute quantity of an object. I just want to make sure I understand this correctly. I suppose all I'm technically saying is there are set of frames which lack any centrifugal forces.

As far as I know, that's right.

Are there places/times when the rotation and the orbit of Earth cancel and I'm rotation-less in space?

Are objects (say on my table) then exerting different tensions to balance centrifugal forces at different times/places (say now compared to a different part of Earth's orbit about the Sun)?

Honestly, I'm not sure how to treat objects in free-fall.

 

[jbriggs444]

What is my rotation right now?

If you are sitting still relative to the earth, you are rotating once every sidereal day.

In the context of Newtonian mechanics, you are also accelerating in a complicated path due to continental drift, the tides, the jack hammer next door, the Earth's rotation about its axis, its orbit around the Sun, the Sun's motion within the milky way, etc, etc.

Are there places/times when the rotation and the orbit of Earth cancel and I'm rotation-less in space?

If, instead of sitting in a chair, you stand on a turntable aligned with the Earth's axis and turn at a rate of once every 23 hours, 56 minutes and a few seconds in a direction opposite the Earth's rotation then you will be free of rotation.

Cancelling your acceleration is trickier and depends on how you account for gravity. In the Newtonian model, you have to identify all gravitating masses, determine the local acceleration of gravity and apply enough counter-balancing force to negate it. In the model of General relativity, gravity is not a force, free fall trajectories are unaccelerated and all you have to do is jump up in the air.

Are objects (say on my table) then exerting different tensions to balance centrifugal forces at different times/places (say now compared to a different part of Earth's orbit about the Sun)?

The Earth rotates at the same rate regardless of whether it is near the sun (January) or far from it (July). Tidal gravity from the Sun theoretically causes stresses in an object on a table on Earth. And those stresses would change slightly based on the distance to the sun. But you'd have to do some monumentally precise work to verify this with experiment. Tidal gravity from the sun most certainly produces measurable stresses in the Earth. The obvious measurable result is the phenomena of spring and neap tides.

I am not certain whether the variation in the Earth's orbital distance from the sun produces measurable seasonal variations in the differential between spring and neap tides.

 

[Dale]

Well I mean how some classical physicists thought about there being an 'absolute reference frame' ("the ether") but found it not to be the case. Well instead of there being a single absolute reference frame ("the ether"), you are telling me there is a set of them (the rotation-less ones).

Yes, these frames are called "inertial" to identify their physically special character. Note that inertial frames do not have any kind of acceleration including rotation, linear acceleration, expansion, etc. So rotation is just one of many forms of non inertial motion which is absolutely identifiable.

I suppose by "profound" what I really mean is that the idea doesn't sit right with me. The idea of everything being relative and no inertial reference frame being preferred is a comfortable one; the idea of certain frames having some absoluteness to them is almost unsettling.

No inertial frame is preferred over any other inertial frame. Any inertial frame is equally distinguishable from any non inertial frame.

I am not sure that we can help how you feel about this stuff, but we can help explain it.

 

[zwierz]

So the angular velocity of a reference frame (or object) is an absolute quantity?

No. There is only angular velocity of one reference frame relative another one. Absolute motion and particularly absolute rotation is nonsense

 

[jbriggs444]

No. There is only angular velocity of one reference frame relative another one. Absolute motion and particularly absolute rotation is nonsense

As @Dale has pointed out, a rotating frame can be distinguished from an inertial frame. It is not a relative measurement. It is "absolute" in the relevant sense.

 

[Dale]

No. There is only angular velocity of one reference frame relative another one. Absolute motion and particularly absolute rotation is nonsense

Rotation can be measured without reference to another reference frame. It is not relative, it is invariant.

 

[zwierz]

has pointed out, a rotating frame can be distinguished from an inertial frame.

what does notion "inertial" do with pure kinematic notion "rotation"? Ok, give me please definition of "rotating frame"

 

[zwierz]

and also give me a definition of angular velocity

 

[Nugatory]

what does notion "inertial" do with pure kinematic notion "rotation"? Ok, give me please definition of "rotating frame"

and also give me a definition of angular velocity

A rotating frame is a frame in which some rotating object is at rest, in the sense that the spatial coordinates assigned by that frame to all parts of that object remain constant.

A rotating object can be recognized in an absolute sense, without relying on any reference frame, by measuring the proper acceleration of each part of that object.

The angular velocity is a parameter that relates the proper acceleration of any given part of the rotating object to its distance from the point of the object that experiences zero proper acceleration.

 

[zwierz]

A rotating frame is a frame in which some rotating object is at rest

So you replaced the word "frame" with the word "object".
Then what is "rotating object"?

by measuring the proper acceleration of each part of that object.

what is a proper acceleration?

 

[Nugatory]

S
what is a proper acceleration?

Proper acceleration is what an accelerometer measures. Because an accelerometer measurement yields the same result in all frames, we don't need any concept of reference frame to talk about proper acceleration.

We can then use measurements of proper acceleration at various points on an object to determine whether the object is rotating, again without any concept of reference frame. That's what Drakkith was getting at above when he said that rotation is frame-invariant.

Now that we have a frame-independent way of determining whether an object is rotating (that is, making a statement about the proper acceleration at various parts of the object) we can consider various reference frames. Some of these have the property that the spatial coordinates of each point on the object are constant, and others do not. By convention we call the ones that do have this property "rotating frames".

 

[gmax137]

To the OP, I too see this as a "profound" subject. At least, it caught my attention when I first came across these ideas.

The contrast between Newton's Bucket and Mach's principle. I'm not sure there is any way to settle the difference, since we cannot do experiments in and otherwise empty universe. Maybe someone here who understands general relativity can say how the ideas work out in that context?

https://en.wikipedia.org/wiki/Bucket_argument

https://en.wikipedia.org/wiki/Mach's_principle

 

[zwierz]

Proper acceleration is what an accelerometer measures.

and how do you deduce formula $\boldsymbol a=\dot{\boldsymbol v}$ from this definition ?

 

[Nugatory]

and how do you deduce formula $\boldsymbol a=\dot{\boldsymbol v}$ from this definition ?

You don't. That's the definition (note that it is a definition, not something that we deduce) of coordinate acceleration, which is an altogether different (and frame-dependent) thing.

Because coordinate acceleration is frame-dependent, if you search hard enough you will always be able to find an inertial frame in which the coordinate acceleration at a point is at least momentarily equal to the proper acceleration... and that can tempt you into thinking that the two are more tightly related than they are.

 

[Dale]

Ok, give me please definition of "rotating frame"

Are you familiar with accelerometers? Particularly the 6 degree of freedom kind?

A rotating reference frame (loosely) is one where 6DF accelerometers at rest show 0 linear proper acceleration but nonzero rotation.

 

[zwierz]

You don't. That's the definition (note that it is a definition, not something that we deduce) of coordinate acceleration, which is an altogether different (and frame-dependent) thing.

Because coordinate acceleration is frame-dependent,

Exactly. By definition, acceleration is relative to a frame and the same is true for angular velocity. Yes, there is a class of frames such that the accelerometer shows the same results relative to all the frames from this class, so called inertial frames. But it is a physical phenomenon. Let us do not mix up physical phenomena and mathematical (kinematical) definitions.

 

[jbriggs444]

Exactly. By definition, acceleration is relative to a frame

Proper acceleration is invariant. It is the same no matter what frame of reference you adopt.

 

[zwierz]

Proper acceleration is invariant.

the thing you are calling "Proper acceleration" is the acceleration relative to an inertial frame

 

[Hiero]

The Earth rotates at the same rate regardless of whether it is near the sun (January) or far from it (July). Tidal gravity...

I'm not concerned with tidal effects. What I was referring to is the change in angular speed of the Earth about the Sun; wouldn't this change the angular velocity of objects on my table thus causing different centrifugal effects? And yes I realize this effect would be very small and unmeasurable I'm just curious as to if I understand correctly, but I think I do now. Thank you for your detailed response, jbriggs.

To the OP, I too see this as a "profound" subject. At least, it caught my attention when I first came across these ideas.

The contrast between Newton's Bucket and Mach's principle. I'm not sure there is any way to settle the difference, since we cannot do experiments in and otherwise empty universe. Maybe someone here who understands general relativity can say how the ideas work out in that context?

https://en.wikipedia.org/wiki/Bucket_argument

https://en.wikipedia.org/wiki/Mach's_principle

Thank you, I feel like you hit the nail on the head as far as what is bothering me: it is Mach's perspective that resonates with me. I know it makes no operational difference, so perhaps my gripe was not really in the spirit of physics, but I would just prefer to think all motion is ultimately relative.

An example of why I prefer that perspective:
What if we spun the entire universe, (whatever that means,) would everything feel centrifugal forces? Even though the stars would appear "fixed" still?
(The above question is rhetorical and not well posed, it's not meant to be answered, it's meant to explain why absolute rotation is an uncomfortable notion for me.)

Anyway thank you for the links, I feel better knowing about Mach's idea.

 

[jbriggs444]

the thing you are calling "Proper acceleration" is the acceleration relative to an inertial frame

And it is still invariant.

 

[jbriggs444]

I'm not concerned with tidal effects. What I was referring to is the change in angular speed of the Earth about the Sun; wouldn't this change the angular velocity of objects on my table thus causing different centrifugal effects?

The Earth rotates at the same rate, once every 23 hours, 56 minutes and a few seconds (in the "absolute" sense that you originally had in mind) regardless of how fast it is revolving around the sun.

 

[Nugatory]

the thing you are calling "Proper acceleration" is the acceleration relative to an inertial frame

No. Proper acceleration is the deviation between the worldline of the object in question and a geodesic tangent to that worldline. This definition has nothing to do with any frame, whether inertial or not.

 

[zwierz]

No. Proper acceleration is the deviation between the worldline of the object in question and a geodesic tangent to that worldline. This definition has nothing to do with any frame, whether inertial or not.

What does your "No" mean? So if I calculate acceleration as $\boldsymbol a=\ddot{\boldsymbol r}$ (here $\boldsymbol r$ is a radius-vector of inertial frame )and if I calculate proper acceleration, do I obtain different results?

 

[Hiero]

The Earth rotates at the same rate, once every 23 hours, 56 minutes and a few seconds (in the "absolute" sense that you originally had in mind) regardless of how fast it is revolving around the sun.

I am talking about the contribution of angular velocity due to Earth's orbit. The angular velocity of the orbit of Earth varies throughout the year, right? And to find the absolute angular velocity of Earth we would need to consider this contribution?

I'm not sure why you say Earth rotates absolutely at that same rate throughout the year... you even said yourself:

In the context of Newtonian mechanics, you are also accelerating in a complicated path due to continental drift, the tides, the jack hammer next door, the Earth's rotation about its axis, its orbit around the Sun, the Sun's motion within the milky way, etc, etc.

 

[Nugatory]

What does your "No" mean?

It means that your statement "the thing you are calling 'Proper acceleration' is the acceleration relative to an inertial frame" is incorrect. The proper acceleration is completely independent of any reference frame.

 

[Dale]

By definition, acceleration is relative to a frame

Not proper acceleration

 

[jbriggs444]

I am talking about the contribution of angular velocity due to Earth's orbit. The angular velocity of the orbit of Earth varies throughout the year, right?

And to find the absolute angular velocity of Earth we would need to consider this contribution?

The motion of the Earth about the sun is completely and utterly irrelevant to the rotation rate of the Earth.

 

[Hiero]

The motion of the Earth about the sun is completely and utterly irrelevant to the rotation rate of the Earth.

See I don't understand this statement. Let me try to elaborate my perspective:

Suppose we had a planet tidally locked in an elliptic orbit about the sun (so on the planet there are no days). As the planet is closer to the sun, it will move through a larger angle per time than when it is far from the sun, hence the planet would rotate at a larger angle per time in order to stay tidally locked, correct?

The same effect should remain if the planet isn't tidally locked (say Earth). This is what I'm referring to. I'm actually not sure if you're just saying this rotation is ignorable (2pi radian per year certainly is not much) or if you're saying it literally has no theoretical effect, in which case I would like to know what is wrong with my above understanding.

 

[Hiero]

Actually, I don't see why the effect should remain if it isn't tidally locked, sorry I'm not sure why I kept thinking that.

Edited to add:
It is the assumption that the length of an (apparent) day is fixed which leads to the conclusion that I had in mind (tidal-locking being a special case of this assumption) but of course this assumption is not true for Earth, it is the sidereal day which is constant in length, not the apparent day.

 

[Dale]

it is Mach's perspective that resonates with me.

Unfortunately, insofar as Mach's principle has been developed into something that can be tested experimentally, it seems that the universe does not follow it. Experiment and observations trump philosophy.

 

[Hiero]

Unfortunately, insofar as Mach's principle has been developed into something that can be tested experimentally, it seems that the universe does not follow it. Experiment and observations trump philosophy.

Can you elaborate or refer to such attempts? I had no success in searching it, but I did find this video which has some ideas from this thread. I find the bit at 6:43 in particular to be very interesting.

 

[Drakkith]

Thank you, I feel like you hit the nail on the head as far as what is bothering me: it is Mach's perspective that resonates with me. I know it makes no operational difference, so perhaps my gripe was not really in the spirit of physics, but I would just prefer to think all motion is ultimately relative.

Unfortunately it is not. Translational motion is relative, meaning that different frames will measure different rates of motion. A personal walking down the aisle of a train is moving slowly relative to a person sitting down on the train but is moving very quickly to a person on the ground. Rotation is not relative though. ALL frames will agree that a rotating frame is rotating and ALL frames will agree at what rate it is rotating at.

 

[jbriggs444]

Suppose we had a planet tidally locked in an elliptic orbit about the sun (so on the planet there are no days).
As the planet is closer to the sun, it will move through a larger angle per time than when it is far from the sun, hence the planet would rotate at a larger angle per time in order to stay tidally locked, correct?

How much angular acceleration do you think that tidal locking is able to produce in a satellite such as the Earth orbiting a star such as the Sun?

(2pi radian per year certainly is not much)

2 pi radians per year per year would be astromically huge.

 

[Hiero]

Unfortunately it is not. Translational motion is relative, meaning that different frames will measure different rates of motion. A personal walking down the aisle of a train is moving slowly relative to a person sitting down on the train but is moving very quickly to a person on the ground. Rotation is not relative though. ALL frames will agree that a rotating frame is rotating and ALL frames will agree at what rate it is rotating at.

Perhaps my words did not capture my meaning. Really the idea is not just about reference frames and rotation but also about mass. Mach's idea, summarized through my eyes, is that these centrifugal forces (by which we detect rotation of a frame) only arise by virtue of a mass's rotation relative to the greater mass distribution of the universe (as opposed to it's rotation relative to some absolute background space).

In other words, I prefer to think that, if we somehow keep Earth perfectly rotation-less, and then somehow begin rotating the rest of the mass in the universe, that Earth will experience centrifugal forces as if it were rotating.

Obviously this is not a viable experiment, I am just emphasizing that there are two distinct physical principles here (with different predictions) and it is unclear to me why either should be rejected. This is why I'm interested in the experiments that @Dale spoke of.

 

[Drakkith]

Obviously this is not a viable experiment, I am just emphasizing that there are two distinct physical principles here (with different predictions) and it is unclear to me why either should be rejected. This is why I'm interested in the experiments that @Dale spoke of.

I don't have any experiments for you, but perhaps you should ask yourself why they should be accepted instead of rejected? What reason would we have to accept that the universe is rotating? What does it mean to rotate the entire universe? Are all objects revolving around a central axis somewhere? That's problematic and probably violates the cosmological principle, but I admit I'm not certain.

 

[Hiero]

I don't have any experiments for you, but perhaps you should ask yourself why they should be accepted instead of rejected?

Well I'm not accepting it nor advocating it as a superior choice. I just think it is an idea worthy of consideration, which I personally prefer (and perhaps I will change that preference when I learn more).

What reason would we have to accept that the universe is rotating? What does it mean to rotate the entire universe? Are all objects revolving around a central axis somewhere? That's problematic and probably violates the cosmological principle, but I admit I'm not certain.

I don't mean to say the universe is rotating, just that if it were rotating on the whole, we wouldn't know. See the 4th minute of this video (particularly the point at about 5:00).

What does it mean to rotate the entire universe? Are all objects revolving around a central axis somewhere?

No, I don't mean the whole universe rotating together like some rigid body. Maybe it can be stated like this: 'the frames in which the universe has a net angular momentum of zero are the frames in which no centrifugal forces are felt.' This statement is still very fuzzy to me. Is the angular momentum of the entire universe even calculable (even in theory I mean)? Any time I have to consider "the entire universe" I feel like I've left the realm of physics.

 

[Drakkith]

I don't mean to say the universe is rotating, just that if it were rotating on the whole, we wouldn't know. See the 4th minute of this video (particularly the point at about 5:00).

I don't agree with this video. Specifically the statement that you only know you're rotating because you can reference the entire universe around you. A hypothetical society brought up inside of a very large, rotating, sealed environment would have little trouble developing the same physical laws we have despite the fact that they cannot observe the rest of the universe.

Note that pop-sci videos often present you with nice, pretty pictures and diagrams that make you feel like you understand what they're talking about. This is often not the case. We spend a great deal of time here at PF helping people get past misunderstandings brought on by these videos. You'd be amazed at how much nonsense is spread through these videos and other mediums.

No, I don't mean the whole universe rotating together like some rigid body. Maybe it can be stated like this: 'the frames in which the universe has a net angular momentum of zero are the frames in which no centrifugal forces are felt.' This statement is still very fuzzy to me. Is the angular momentum of the entire universe even calculable (even in theory I mean)?

No, because we cannot see the entire universe and will never be able to. But even if we could, this criteria doesn't work. All frames would measure the same total angular momentum. Besides, now you're talking about the angular momentum of objects within the universe. That may not be the same thing as saying the universe as a whole is rotating. Hence why I asked what it even means to say that the universe is rotating. Without a proper definition or description, is it useful to prefer it over the non-rotating description?

Any time I have to consider "the entire universe" I feel like I've left the realm of physics.

Indeed. It's always problematic to try to make statements about the universe and great care should be taken when doing so.

 

[Hiero]

I don't agree with this video. Specifically the statement that you only know you're rotating because you can reference the entire universe around you. A hypothetical society brought up inside of a very large, rotating, sealed environment would have little trouble developing the same physical laws we have despite the fact that they cannot observe the rest of the universe.

To be honest I'm not sure what you're talking about.

Note that pop-sci videos often present you with nice, pretty pictures and diagrams that make you feel like you understand what they're talking about. This is often not the case. We spend a great deal of time here at PF helping people get past misunderstandings brought on by these videos. You'd be amazed at how much nonsense is spread through these videos and other mediums.

Yes I'm sure, but I found that video after this thread was created. At any rate these ideas are in my mind somehow, so I will contemplate it all the same.

No, because we cannot see the entire universe and will never be able to.

This is a good point, but it could be restricted to the local (observable) universe.

But even if we could, this criteria doesn't work. All frames would measure the same total angular momentum.

What do you mean? Angular momentum is a quantity which depends on the coordinates with which it is measure with respect to. It depends not only on the location but also rotation of the coordinates, right? In other words, coordinate frames which are rotating with respect to each other will measure different angular momentum of the same object, right? So then I'm simply proposing that those frames which give zero angular momentum of the observable universe also give zero centrifugal force (i.e. are said to be absolutely rotation-less).
I am sure something is wrong with my statement, but in my mind it is how I understand mach's principle.

Besides, now you're talking about the angular momentum of objects within the universe.

Well of course. Like I said, it's not just about rotating frames, it's also about the mass distribution. Rotating mass gives angular momentum, hence angular momentum captures both aspects.

That may not be the same thing as saying the universe as a whole is rotating.

I just want to be clear; I don't mean 'if the universe as a whole is rotating', I mean 'if the universe on the whole is rotating,' i.e. there is a net angular momentum.
For example, the angular momentum of Earth would contribute to the angular momentum of the universe, even though the Earth is not rotating about some universally central axis.

Without a proper definition or description, is it useful to prefer it over the non-rotating description?

Oh it's certainly not useful in any respect. I do not prefer it as a physicist I prefer it as a human being. As a physicist I seek a more precise formulation, which I do not seem to be making progress towards.

 

[Dale]

Can you elaborate or refer to such attempts?

Here is a decent introduction of what I am referring to:

http://www.scholarpedia.org/article/Jordan-Brans-Dicke_Theory

 

[Drakkith]

To be honest I'm not sure what you're talking about.

Just think of the experiments we have done here on Earth to show that it's rotating. A simple experiment is to create a large pendulum and watch as it precesses over time.

To start, a small experiment can be done in which a pendulum is mounted on a rotatable base and rotated. Assign a frame of reference (which includes a coordinate system) to the base of the pendulum such that no point of the base moving. The pendulum's path in this coordinate system, the arc it makes as it move, changes over time, the result of the precession of the pendulum. This indicates that the frame of reference is rotating and all other frames of reference will agree with that.

Now, if the world isn't rotating, a large pendulum, whose base is stationary with respect to the ground, shouldn't have any precession. The arc the pendulum follows should remain the same over time. But if the world is indeed rotating, then the pendulum will precess, indicating that the ground itself is rotating as is everything attached to the ground. It doesn't matter if the scientists performing the experiments can see the rest of the universe or not.

Interestingly, this is similar to a scenario used by Einstein. He posited that an observer, unable to observe anything other than the room he was in, would not be able to tell if he was on the ground or if he was on an accelerating spaceship since all physical laws would be the same in either frame. See here: https://en.wikipedia.org/wiki/Introduction_to_general_relativity#Gravity_and_acceleration

Yes I'm sure, but I found that video after this thread was created. At any rate these ideas are in my mind somehow, so I will contemplate it all the same.

Honestly I wouldn't expect you to stop.

This is a good point, but it could be restricted to the local (observable) universe.

Sure, but that doesn't get you anywhere. See below.

What do you mean? Angular momentum is a quantity which depends on the coordinates with which it is measure with respect to. It depends not only on the location but also rotation of the coordinates, right? In other words, coordinate frames which are rotating with respect to each other will measure different angular momentum of the same object, right? So then I'm simply proposing that those frames which give zero angular momentum of the observable universe also give zero centrifugal force (i.e. are said to be absolutely rotation-less).
I am sure something is wrong with my statement, but in my mind it is how I understand mach's principle.

No, angular momentum is frame invariant. Imagine two frames which are rotating about the same axis at the same rate. Even though no points in either frame are moving with respect to each other, experiments can be done to show that the frames are rotating and measure how much angular momentum they have.

I just want to be clear; I don't mean 'if the universe as a whole is rotating', I mean 'if the universe on the whole is rotating,' i.e. there is a net angular momentum.
For example, the angular momentum of Earth would contribute to the angular momentum of the universe, even though the Earth is not rotating about some universally central axis.

We can already observe matter rotating about some axis. We can talk about the net angular momentum of whole regions of space. But, to me, this says nothing about whether the universe is rotating. It only says something about the distribution of matter within the universe and how it is moving. This doesn't seem to match up with the usual uses of the phrase "rotating universe" that I've seen, but perhaps I haven't looked into this topic enough.

 

[Hiero]

No, angular momentum is frame invariant.

This isn't how I understand angular momentum. In my eyes, it is simply defined as the sum of the cross product of position and momentum for all mass elements in the system. If we change frames (maybe change the location of the origin, maybe have the new frame move at a constant speed relative to the old, maybe give the frame a constant rotational speed) then we will generally find different angular momenta for the same system, because the position and momentum vectors will change with the frames.

Perhaps you're speaking of general relativity's definition of angular momentum (if that's a thing) or something else I know nothing of, but that is my basic understanding of the topic.

 

[Drakkith]

This isn't how I understand angular momentum. In my eyes, it is simply defined as the sum of the cross product of position and momentum for all mass elements in the system. If we change frames (maybe change the location of the origin, maybe have the new frame move at a constant speed relative to the old, maybe give the frame a constant rotational speed) then we will generally find different angular momenta for the same system, because the position and momentum vectors will change with the frames.

Hmmm. I admit this is getting a bit outside my area of expertise, so I think I'll have to let someone else answer that.

Perhaps you're speaking of general relativity's definition of angular momentum (if that's a thing) or something else I know nothing of, but that is my basic understanding of the topic.

I wasn't, but you did inspire me to look up more information on the topic and it seems that angular momentum isn't always conserved in curved spacetime. So that's probably important if you want to talk about angular momentum in cosmological scales.