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?