Linear vs rotary motion

[Physicist248]

TL;DR

If linear motion is relative, how come that rotary motion is absolute? And if it is not absolute what is is relative to?

Reference: https://www.physicsforums.com/forums/special-and-general-relativity.70/post-thread

If linear motion is relative, how come that rotary motion is absolute? And if it is not absolute what is is relative to?

 

[Dale]

If linear motion is relative, how come that rotary motion is absolute?

Geometrically it is for the same reason that an angle requires two straight lines but a curved line can make definite angles with itself.

In spacetime speed is represented by an angle between two worldlines. A straight line (representing inertial motion) doesn’t have an angle with respect to itself, but a curved line (representing accelerated motion) does.

 

[Halc]

In spacetime speed is represented by an angle between two worldlines. A straight line (representing inertial motion) doesn’t have an angle with respect to itself, but a curved line (representing accelerated motion) does.

Good example of why proper acceleration is absolute.

If linear motion is relative, how come that rotary motion is absolute?

As for rotation, I suppose it is because the physics of a rotating frame is different than that of an inertial frame (relative to which linear motion is defined). So for example, in a rotating frame, objects at rest tend not to stay at rest. The one frame in which they do (locally) stay at rest is the one frame relative to which the rotation rate is defined. That makes it absolute.

This cannot be done for linear motion. There is no (local) test one can make that determines one's absolute linear motion.

 

[PeterDonis]

how come that rotary motion is absolute?

It isn't. There are absolutes ("invariants" is a better term) associated with rotation, but "rotary motion" itself, at least not with the usual meaning of that term, is not one of them.

if it is not absolute what is is relative to?

Whatever frame of reference you choose.

 

[PeterDonis]

in a rotating frame, objects at rest tend not to stay at rest.

What you mean is that freely falling objects at rest in a rotating frame do not stay at rest. But that's a much more restricted thing. You, sitting at rest relative to Earth as you type your posts, are at rest in a rotating frame, and you stay at rest. You're not freely falling, true, but you're an object at rest in a rotating frame that has no tendency not to stay at rest.

The one frame in which they do (locally) stay at rest is the one frame relative to which the rotation rate is defined.

I'm not sure what you mean by this.

That makes it absolute.

You need to be much more careful here. As I mentioned in post #4, there are invariants associated with rotation, but it takes quite a bit of care and groundwork to properly define them.

 

[Filip Larsen]

As I read it, the OP is effectively asking if Mach's principle holds or not.

 

[Sagittarius A-Star]

As I read it, the OP is effectively asking if Mach's principle holds or not.

Mach described it in the following way:

Source (chapter 2 "Nonrelativistic Machian Theories" , page 109):

https://www.amazon.com/-/he/Machs-Principle-Newtons-Quantum-Einstein/dp/0817638237?tag=pfamazon01-20

Original German text "Mach, Ernst: Die Geschichte und die Wurzel des Satzes von der Erhaltung der Arbeit":

https://www.digitale-sammlungen.de/de/view/bsb11018450?page=54

 

[Halc]

You need to be much more careful here. As I mentioned in post #4, there are invariants associated with rotation, but it takes quite a bit of care and groundwork to properly define them.

Indeed. It takes more coordinates to define a rotating frame (in say Minkowskian spacetime) that it does to define an inertial one. For one, the velocity of the axis (relative to what?) needs to be defined, which makes it relative to one specific inertial frame more than any of the others. This velocity again cannot be determined from inside a box (hence a rotating frame not really being absolute), but the orientation and rotation rate can be determined, which is what is typically meant when stating that rotation is absolute.

So for instance, from inside a lab on Earth somewhere, I can determine where north is, the exact angle, and also the proper rotation period (~23:56). This is a lot easier in zero G, but it can be done under gravity. These things, measured by any observer anywhere. I said 'proper rotation' since Earth spins slower (coordinate rotation rate) as seen by an observer on say Pluto. That makes the coordinate rotation period frame dependent.

But it is meaningless to posit (let alone measure) Earth's absolute velocity. We can relate it to another frame. We can say that Earth's rotating frame differs from that of Jupiter by velocity V, orientation O. The rotation periods are not really relative to each other, but rather each relative to not-rotating.

 

[Dale]

As I read it, the OP is effectively asking if Mach's principle holds or not.

I wouldn’t read that into the question. This question can be answered in GR (as I did above) even though GR is non-Machian

 

[PeterDonis]

It takes more coordinates to define a rotating frame (in say Minkowskian spacetime) that it does to define an inertial one

No, it doesn't. Spacetime is a 4-dimensional manifold, so it takes 4 coordinates to define a frame.

he velocity of the axis (relative to what?) needs to be defined

This is not a coordinate, it's a parameter that appears in the metric coefficients. And it's a poor choice of parameter, because...

which makes it relative to one specific inertial frame more than any of the others.

But there are other ways of defining a rotating frame that don't require this.

the orientation and rotation rate can be determined

How? Please be specific. There are plenty of worms in the can you are not opening here.

from inside a lab on Earth somewhere, I can determine where north is, the exact angle, and also the proper rotation period (~23:56).

Again, how? Please be specific.

 

[cianfa72]

I wouldn’t read that into the question. This question can be answered in GR (as I did above) even though GR is non-Machian

In the context of GR, the following invariant physical definition of unaccelerated is taken: a body is unaccelerated (zero proper acceleration) if a physical accelerometer attached to it reads zero.

Speed, by its very definition, is relative to something (you have to look outside the box). Nevertheless we have an intrinsic/invariant definition of inertial traveling as motion that occurs with zero proper acceleration.

 

[Halc]

Again, how? Please be specific.

One of the more precise methods is to use a ring laser gyroscope, which leverages the Sagnac effect to determine rotation rate and axis orientation, all without looking out the window.

Now where the axis actually is (not just its orientation) cannot be determined easily from our lab on Earth, but my prior post didn't make a claim that it could. I suppose it would be an interesting topic to measure the radius of (a perfectly spherical) Earth from a windowless lab, but I can't think of a way. If it's a tall lab and you have really accurate accelerometer at various points up the wall, one could match that curve with Schwarzschild coordinates.

 

[PeterDonis]

One of the more precise methods is to use a ring laser gyroscope, which leverages the Sagnac effect to determine rotation rate and axis orientation, all without looking out the window.

What "rotation rate" does a ring laser gyroscope measure? Relative to what observer?

What "axis orientation" does it measure? Relative to what?

 

[Physicist248]

It isn't. There are absolutes ("invariants" is a better term) associated with rotation, but "rotary motion" itself, at least not with the usual meaning of that term, is not one of them.


Whatever frame of reference you choose.

Let's say I choose the earth as the frame of reference. I.e. the earth is motionless with respect to that frame. Why is it experiencing centrifugal force?

 

[Dale]

Let's say I choose the earth as the frame of reference. I.e. the earth is motionless with respect to that frame. Why is it experiencing centrifugal force?

As I already told you in a previous thread, Centrifugal force is not something that can be experienced. No experiment can ever measure any inertial force. They are only ever inferred from motion.

 

[PeterDonis]

Let's say I choose the earth as the frame of reference. I.e. the earth is motionless with respect to that frame. Why is it experiencing centrifugal force?

It isn't experiencing "centrifugal force". It's experiencing internal forces between its parts because they push on each other. Those forces and the proper accelerations associated with them, are invariant; they don't depend on any choice of reference frame.

In more technical terms, the Earth can be described by a timelike congruence of worldlines, and any such congruence has invariants associated with its kinematic decomposition, which include proper acceleration, expansion, shear, and vorticity. Those invariants are what contain the actual physics for things like "rotary motion" (nonzero vorticity). All of these things are independent of any choice of reference frame (that's why they're called "invariants").

 

[Physicist248]

I wouldn’t read that into the question. This question can be answered in GR (as I did above) even though GR is non-Machian

"GR is non-Machian" Hmm. In layman's terms, does the mass in the large universe cause the local centrifugal force (according to GE)?

 

[Physicist248]

As I already told you in a previous thread, Centrifugal force is not something that can be experienced. No experiment can ever measure any inertial force. They are only ever inferred from motion.

And how do you think inertial navigation systems work?

 

[Dale]

"GR is non-Machian" Hmm. In layman's terms, does the mass in the large universe cause the local centrifugal force (according to GE)?

No. The local centrifugal force does not have a cause. It is not a physical thing subject to cause and effect. It is purely math. It can be introduced or removed simply by changing the math.

And how do you think inertial navigation systems work?

They work by measuring proper acceleration and rotation, not centrifugal or other fictitious forces.

 

[Physicist248]

What "rotation rate" does a ring laser gyroscope measure? Relative to what observer?

What "axis orientation" does it measure? Relative to what?

Yeah, that was basically my question: relative to what observer?

 

[Physicist248]

No. The local centrifugal force does not have a cause. It is not a physical thing subject to cause and effect. It is purely math. It can be introduced or removed simply by changing the math.

They work by measuring proper acceleration and rotation, not centrifugal or other fictitious forces.

How do you change the math of F = ma?

 

[Dale]

How do you change the math of F = ma?

By changing $\vec a$. The acceleration is different in different frames.

 

[PeterDonis]

"GR is non-Machian"

That's a matter of some debate in the literature. The argument for it being non-Machian is that there are solutions of the Einstein Field Equation in which the spacetime geometry is not solely determined by the stress-energy content--the simplest such example being flat Minkowski spacetime, which has no stress-energy anywhere but still has a definite spacetime geometry.

The counter-argument for why GR is "Machian", or at least as Machian as it needs to be, is that the solutions I referred to above, where the spacetime geometry is not solely determined by stress-energy content, are all idealizations that are only useful for describing isolated systems (for example, asymptotically flat solutions describing single planets or stars), and when we look at solutions that describe the universe as a whole (such as the FRW solutions used in cosmology), they do have spacetime geometry that is entirely determined by stress-energy content--you don't have to add any extra conditions like asymptotic flatness.

 

[PeterDonis]

that was basically my question: relative to what observer?

You're the one that should be answering that question, since you're the one that made the claims about the ring laser gyroscope. If you can't back up those claims, you shouldn't have made them.

 

[Dale]

By changing $\vec a$. The acceleration is different in different frames.

To explain this further, let’s consider a simple mass-spring accelerometer attached radially on a spinning wheel. The mass is deflected outward, which (by Hookes law) means an inward, or centripetal, proper acceleration.

The external inertial observer sees the measured proper acceleration. Since the proper acceleration matches the coordinate acceleration they infer that there is no centrifugal force.

Meanwhile a co rotating observer sees that the spring has the same proper elongation as before, a real centripetal force. But I this frame there is no acceleration. So to prevent that they must introduce a fictitious centrifugal force to cancel the centripetal force.

Note that the centrifugal force is associated with the analysis of the co-rotating observer. It is not associated with the physical scenario since it does not exist for one of the observers observing that same physical scenario.

 

[cianfa72]

To explain this further, let’s consider a simple mass-spring accelerometer attached radially on a spinning wheel. The mass is deflected outward, which (by Hookes law) means an inward, or centripetal, proper acceleration.

Note that this centripetal force (inward proper acceleration) is given by the spring acting on the mass (not the other way around). The force entering F=ma is the force acting "from the outside/external" to the body (here the outside is the spring).

Meanwhile a co rotating observer sees that they mass has the same proper elongation as before, a centripetal force. But I this frame there is no acceleration to prevent that they must introduce a fictitious centrifugal force.

Note that this fictitious centrifugal force is a property of the frame, i.e. it isn't associate to any kind of physical interaction (it hasn't a cause).

 

[Halc]

What "rotation rate" does a ring laser gyroscope measure? Relative to what observer?

What "axis orientation" does it measure? Relative to what?

It measures a proper rotation rate, which is observer invariant.

The axis orientation is also proper, effectively 'thataway', which again is observer invariant.
Sure, coordinate orientation differs from one to the next, but the device outputs proper orientation, not numbers, similar to how a compass needle points north without relying on how many degrees clockwise north is from where the N is on the compass face.


And yes, my super precise accelerometer method can determine how far away the axis is (assuming we know we're on a planet), and can distinguish gravity from linear acceleration, all presuming the lab has some finite spatial extent.
I'd have to brush up on my tex to describe that more precisely

 

[PeterDonis]

It measures a proper rotation rate, which is observer invariant.

But you still have to define what observer (or worldline, if you prefer), you are using to determine the "proper rotation rate". That term has no meaning by itself--you have to specify a worldline, because the 4-velocity of the worldline appears in the formula for the proper rotation rate.

The axis orientation is also proper, effectively 'thataway', which again is observer invariant.

Same comment here: "thataway" has no meaning by itself, you have to specify a worldline. Which worldline?

And yes, my super precise accelerometer method can determine how far away the axis is (assuming we know we're on a planet), and can distinguish gravity from linear acceleration, all presuming the lab has some finite spatial extent.
I'd have to brush up on my tex to describe that more precisely

Then you need to brush up and do it. So far all I see here is handwaving.

 

[pervect]

TL;DR: If linear motion is relative, how come that rotary motion is absolute? And if it is not absolute what is is relative to?

Reference: https://www.physicsforums.com/forums/special-and-general-relativity.70/post-thread

If linear motion is relative, how come that rotary motion is absolute? And if it is not absolute what is is relative to?

If you do an experiment in a small sealed box where you only look at things inside the box, the laws of physics according to special relativity don not allow you to tell whether you are moving or not. If your question is about general relativity, you might be able to detect motion relative to a nearby mass , which would be explained by the curvature of space-time (in General relativity) , or "gravitation", in Newtonian theories.

But, if you do experiments in the same small sealed box and the box is rotating, you can tell that it's rotating, for instance with a laser gyro.

So the answer is that in the flat space-time of special relativity, there's no way experimentally to detect one's motion.

As to "why", that's a difficult question to answer. Are you talking from the background of Newtonian theory, the background of Special Relativity, or some other theory? If it's a personal theory, we probably can't help you.

This is the special and general relativity forum, so one might assume that you are asking for an answer in that context. But it's worth pointing out that, with the exception of light, Newtonian theory predicts the same lack of ability to detect absolute motion and the ability to detect rotation.

In the context of special relativity, we might say that absolute motion can't be detected in flat space-time because of the "boost symmetries" of the Minkowskii metric. This is a specific sort of symmetry present in flat space-time, that more or less says all states of linear motion leave the metric (and hence physical observables) unchanged. I'm not sure if you'll find that helpful. This symmetry is not present in the curved spacetime near massive bodies in GR, leaving the possibility of detecting motion relative to them available in GR.

Rotation in general doesn't have such a symmetry, so it can be detected.

To addres the question of light in Newtonian theory, it was at one time hypotesized that one could detect abolute motoin relative to the theories of the time (the ether), but experimetns such as Michelson and Morely failed to show any such epxeirmental effects. This was influential in the acceptance and development of Special relativity, which explained the null result.

It remains the case that Newtonian mechanics predicts that one cannot detect absolute motion via physical experiments not involving light. So it's difficult to see why you'd expect to be able to detect such motion experimentally in any case.

The inability to detect absolute motion in Newtonian theory is commonly called "Gallilean relativity", by the way - a keyword that might help you do some of your own reading and research.