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Rotation
#1
Rotation
Here's a question that's been bugging me for a long time. It's actually been rather hard to formulate it. What is the fundamental property that defines rotation from translation? That may be a stupid question but here's the reasoning:

Let us assume two reference frames O1 x1 y1 z1 and O2 x2 y2 z2. Axes z1 and z2 are parallel to each other and points O1 and O2 remain at a constant distance. Let's assume then that the second frame is spinning at a constant angular velocity about z2 axis relative to the first frame.

If we view the problem exclusively from the point of kinematics there should be no way to tell which frame is rotating: you can take either one of them as an inertial frame (non-rotating) then the other one would be non-inertial (either just spinning or orbiting & spinning).

But if we bring dynamics into play things have to get different. Let us assume that each of the above frames is fixed to a body with a z-axis symmetry. Then the spinning body must experience centrifugal forces. This begs an assumption that centrifucal forces may serve as a clue to differentiate rotation from translation on a fundamental level. But these forces themselves are 'fictitious' (introduced to account for normal accelerations in a rotating frame) so this reasoning leads to circular logic. We can also introduce another mass moving across the spinning body to use Coriolis force as a sign of rotation but Coriolis force is also a 'fictitious' one so that's no better.

A simple illustration would be shape of Earth: we can formally take a fixed reference frame associated with Earth as an inertial one yet there will still be centrifugal forces acting upon Earth and flattening it slightly and Coriolis forces acting upon objects moving upon Earth's surface.

So, long story short: what is the fundamental way to tell if something is rotating?
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#2
RE: Rotation
I have a hard time visualizing that, a geometric representation for the reference frame and its transformation of the spin would help. But if I understand it correctly - wouldn't a gyroscope be able to determine that, in the case of Earth?

If you're in the reference system itself, I don't see how you would be able to tell without some outside comparison, which is kinda the point, right? I'm likely not imaginative and creative enough, and a gyroscope seems too easy an answer, TBH.
"The first principle is that you must not fool yourself — and you are the easiest person to fool." - Richard P. Feynman
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#3
RE: Rotation
This is actually a matter for general relativity.

The basic problem is that acceleration is equivalent to gravity and therefore to a curved spacetime. So, it *is* possible to find a frame that works for a rotating object. But that locally inertial frame simply doesn't work globally. The Coriolis forces are seen as gravitational in that frame.

Technically, the metric tensor is diagonal in 'nice' reference frames. In rotating ones (like in rotating black holes), there are off-diagonal components that represent frame dragging.

One way of thinking about this is with latitude and longitude on the Earth. Close to any point (other than the poles), they give a nearly Euclidean description of the local geometry. That corresponds to a locally inertial frame. Geodesics locally look flat.

But, if there are mountains and valleys, small distances on the coordinate grid can correspond to large distances in the actual geometry. And if the mountains and valleys have some 'texture' to them, the locally Euclidean aspect is destroyed. That is (sort of) what happens in rotating frames.

That said, we generally look for frames that are close to inertial on large segments of spacetime. The 'rotating' ones simply cannot be extended to large regions.

Think of it like this. Suppose you take a frame in which the Earth is not rotating. A geosynchronous satellite then hovers above a single point while satellites above and below will move. THAT shows you have a frame that is not inertial when extended to those sattelites.
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#4
RE: Rotation
(October 13, 2020 at 4:47 pm)Smaug Wrote: Here's a question that's been bugging me for a long time. It's actually been rather hard to formulate it. What is the fundamental property that defines rotation from translation? That may be a stupid question but here's the reasoning:

Let us assume two reference frames O1 x1 y1 z1 and O2 x2 y2 z2. Axes z1 and z2 are parallel to each other and points O1 and O2 remain at a constant distance. Let's assume then that the second frame is spinning at a constant angular velocity about z2 axis relative to the first frame.

If we view the problem exclusively from the point of kinematics there should be no way to tell which frame is rotating: you can take either one of them as an inertial frame (non-rotating) then the other one would be non-inertial (either just spinning or orbiting & spinning).

But if we bring dynamics into play things have to get different. Let us assume that each of the above frames is fixed to a body with a z-axis symmetry. Then the spinning body must experience centrifugal forces. This begs an assumption that centrifucal forces may serve as a clue to differentiate rotation from translation on a fundamental level. But these forces themselves are 'fictitious' (introduced to account for normal accelerations in a rotating frame) so this reasoning leads to circular logic. We can also introduce another mass moving across the spinning body to use Coriolis force as a sign of rotation but Coriolis force is also a 'fictitious' one so that's no better.

A simple illustration would be shape of Earth: we can formally take a fixed reference frame associated with Earth as an inertial one yet there will still be centrifugal forces acting upon Earth and flattening it slightly and Coriolis forces acting upon objects moving upon Earth's surface.

So, long story short: what is the fundamental way to tell if something is rotating?


Yes, by the forces/acceleration with respect to frame they experience besides those which can be accounted by other stuff?
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#5
RE: Rotation
(October 13, 2020 at 4:47 pm)Smaug Wrote: Here's a question that's been bugging me for a long time. It's actually been rather hard to formulate it. What is the fundamental property that defines rotation from translation? That may be a stupid question but here's the reasoning:

Let us assume two reference frames O1 x1 y1 z1 and O2 x2 y2 z2. Axes z1 and z2 are parallel to each other and points O1 and O2 remain at a constant distance. Let's assume then that the second frame is spinning at a constant angular velocity about z2 axis relative to the first frame.

If we view the problem exclusively from the point of kinematics there should be no way to tell which frame is rotating: you can take either one of them as an inertial frame (non-rotating) then the other one would be non-inertial (either just spinning or orbiting & spinning).

But if we bring dynamics into play things have to get different. Let us assume that each of the above frames is fixed to a body with a z-axis symmetry. Then the spinning body must experience centrifugal forces. This begs an assumption that centrifucal forces may serve as a clue to differentiate rotation from translation on a fundamental level. But these forces themselves are 'fictitious' (introduced to account for normal accelerations in a rotating frame) so this reasoning leads to circular logic. We can also introduce another mass moving across the spinning body to use Coriolis force as a sign of rotation but Coriolis force is also a 'fictitious' one so that's no better.

A simple illustration would be shape of Earth: we can formally take a fixed reference frame associated with Earth as an inertial one yet there will still be centrifugal forces acting upon Earth and flattening it slightly and Coriolis forces acting upon objects moving upon Earth's surface.

So, long story short: what is the fundamental way to tell if something is rotating?

There is also a misunderstanding of what it means to be an inertial frame. It is literally one in which bodies move in straight lines except for forces that arise from gravitational or electrical effects (well, we can include strong and weak forces if you wish).

So, if you take the Earth as stationary rotation), the motion of a satellite is NOT given by F=ma where F is determined by the masses around gravitationally. It is precisely that the Coriolis force is NOT associated with anything else that makes it 'fictitious'. That we *can* find frames where F=ma is what says that inertial frames exit.
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#6
RE: Rotation
(October 13, 2020 at 4:47 pm)Smaug Wrote: Here's a question that's been bugging me for a long time. It's actually been rather hard to formulate it. What is the fundamental property that defines rotation from translation? That may be a stupid question but here's the reasoning:

Let us assume two reference frames O1 x1 y1 z1 and O2 x2 y2 z2. Axes z1 and z2 are parallel to each other and points O1 and O2 remain at a constant distance. Let's assume then that the second frame is spinning at a constant angular velocity about z2 axis relative to the first frame.

If we view the problem exclusively from the point of kinematics there should be no way to tell which frame is rotating: you can take either one of them as an inertial frame (non-rotating) then the other one would be non-inertial (either just spinning or orbiting & spinning).

But if we bring dynamics into play things have to get different. Let us assume that each of the above frames is fixed to a body with a z-axis symmetry. Then the spinning body must experience centrifugal forces. This begs an assumption that centrifucal forces may serve as a clue to differentiate rotation from translation on a fundamental level. But these forces themselves are 'fictitious' (introduced to account for normal accelerations in a rotating frame) so this reasoning leads to circular logic. We can also introduce another mass moving across the spinning body to use Coriolis force as a sign of rotation but Coriolis force is also a 'fictitious' one so that's no better.

A simple illustration would be shape of Earth: we can formally take a fixed reference frame associated with Earth as an inertial one yet there will still be centrifugal forces acting upon Earth and flattening it slightly and Coriolis forces acting upon objects moving upon Earth's surface.

So, long story short: what is the fundamental way to tell if something is rotating?

In Newtonian Mechanics and Special Relativity, any non-inertial frame will require fictitious forces to explain the laws of motion.  As you state, the Coriolis and Centrifugal forces are evidence of rotation.

In General Relativity things may be a bit different, but a rotating reference frame has different mathematics.  The universe, for instance, must not be rotating or we wouldn't have a flat spacetime.
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