I Is acceleration absolute or relative - revisited

[Peter Leeves]

Your claim is not , however, asserting that the spacetime geometry changes when you change coordinates.

For any imprecision on my part, I apologise. I agree with your statement above. I hope I was saying that one scenario (rotating bucket) is equivalent to the other (rotating universe) and that both had equal validity and gave the same observational results (due to the acceleration being proper and therefore identical impact in both reference frames).

 

[Peter Leeves]

Thanks for your indicated edits. I prefer your version to my own too.

Spin the (spherical) blob and it flattens due to centrifugal force ; spin the Earth (lots) and the blob elongates (a tiny bit) thanks to frame-dragging. So, not the same... at least for that reasonably well defined scenario.

In the politiest possible way, can I ask why you think spinning the blob flattens it due to centrifugal force, but the Earth spinning would elongate the blob ? The system is symmetrical through all 360° about the the spin axis. I can see no reason for elongation, but I can see reason for identical flattening ? It certainly appears (at first sight anyway) that both scenarios could yield identical results.

 

[Peter Leeves]

This is all about Mach's principle. The question is: does the (inertial) mass m of the water depend on all the other mass M of the universe? Mach believed so; he believed that, whatever m(M) is, the inertial property of it should vanish if M vanishes. I.e. water in the bucket shouldn't become concave in an otherwise empty universe. Newton would disagree; he defined acceleration w.r.t. space.

And Einstein...well, Einstein believed Mach's principle was true, but his own theory of General Relativity is not fully Machian. Yes, inertial properties of a test mass are defined by other masses; they curve spacetime. But in an empty universe a particle can still undergo inertial forces because Minkowski spactime solves the field equations of an empty universe.

In the end, this remains an open question, mostly because neither Mach, Einstein or other people can tell you exactly what m(M) is. Thats's also why nowadays most physicists lost their interest in the topic.

Good grief, I just re-read your post which I failed to understand entirely yesterday, and found that now I can actually follow it all ! Thank you for this post.

 

[Peter Leeves]

This is a different physical situation than spinning the rest of universe while not turning on the motor.

It's more accurate (and makes comprehension easier) when you realize that it's firing of the motor that causes the universe to spin and the new gravitational field to be generated. You then appreciate that when the motor stops, the gravitational field dies away. Also that the gravitational field essentially just produces an equivalent force to the motor, and identical observations.

There’s substantial intuitive appeal to the idea that the effect should be the same ... but that’s not the same as a proof, let alone a proof backed up by observations.

I can't dispute your point at all. All I can do it try and apply logic and reason as best I can. No one seems to have an issue with the equivalence principle (man in lift freefalling towards Earth = man in interstellar space with no external influences). I see the rotating bucket / rotating universe as circular version.

I put this thread in this forum in the hope that someone can either shoot it down (making me happy because my understanding has increased), or confirm it (making me equally happy because my understanding has increased). Even if I come away with no resolution I'm still learning an awful lot, believe me (which has certainly increased my understanding).

 

[PeterDonis]

For any imprecision on my part, I apologise

It's not a matter of imprecision; perhaps my choice of words was a little misleading. I was making the point that spacetime geometry is actually the underlying cause of the shape of the water in the bucket; spacetime geometry is the invariant thing that tells us that it "really" is the bucket that is rotating, not the universe.

I hope I was saying that one scenario (rotating bucket) is equivalent to the other (rotating universe)

And my point is that this is only true as a statement about choices of coordinates for the same spacetime geometry; it is not true as a statement about invariants. As far as invariants are concerned, "rotating bucket in non-rotating universe" is not the same as "rotating universe with non-rotating bucket"; the latter would be a different spacetime geometry from the former (and the former is the spacetime geometry you have been describing).

 

[hmmm27]

heheh... I can't for the life of me now figure out why I thought/wrote the "elongated" bit, though it made sense at the time. Wait... almost had it... darn, slipped away again. Apparently @Dale and @PeroK understood : maybe they can help.

As for the other, more mundane scenario : the center of the Earth's gravitational force is the center of the Earth, no matter where your object is. Equilibrium for an object at the center of the Earth is a spherical shape, which a liquid easily accommodates. If you spin the sphere, the bits closest to the axis experience a slight outward force ; the bits furthest away experience a strong outward force. The outward force is balanced by the gravitational inward force. So, the sphere widens radially near the axial plane ("equator") and shrinks along the axis, which maintains the volume. You might even get a donut-shape out of the deal, since gravity's net force is zero at the center.

That is not going to happen if you spin the Earth, instead. Gravity (imparted by mass) is very much not the same thing as centripetal/fugal force, though a very limited amount of results are similar.

 

[Peter Leeves]

That is not going to happen if you spin the Earth, instead. Gravity (imparted by mass) is very much not the same thing as centripetal/fugal force, though a very limited amount of results are similar.

I would respectfully argue this: "The simplest way to state the equivalence principle is this: inertial mass and gravitational mass are the same thing." I would also add that Centripetal/fugal inertia/force is merely the rotational equivalent of linear inertia/force.

 

[PeterDonis]

No one seems to have an issue with the equivalence principle (man in lift freefalling towards Earth = man in interstellar space with no external influences). I see the rotating bucket / rotating universe as circular version.

No, this is not correct. The equivalence principle is local; it only covers a small patch of spacetime, not the entire universe. And it is not about the global equivalence of different choices of coordinates on the same spacetime geometry; it is about the local equivalence of the same state of motion (i.e., same proper acceleration--zero for free fall, or some fixed nonzero proper acceleration) in different global spacetime geometries (e.g., flat spacetime vs. the curved spacetime geometry around the Earth).

The global equivalence of different choices of coordinates on the same spacetime geometry is called "general covariance", not the equivalence principle. So the bucket thought experiment is an illustration of general covariance, not the EP.

 

[Nugatory]

It's more accurate (and makes comprehension easier) when you realize that it's firing of the motor that causes the universe to spin.

Now wait a moment... are you suggesting that when I switch on a motor to rotate something (like, for example a shaft with a bucket on the end) the motor is actually applying torque to all the rest of the universe causing it to rotate, while the bucket remains still?

You may be thinking of a different question, "does there exist a coordinate system in which the angular velocity of the bucket does not change while the angular velocity of the rest of the universe does?". The answer to that question is clearly yes (and in many problems, such as navigating on the surface of the earth, this coordinate system is more often used). However, the fact that we can use coordinates in which the bucket is rotating or coordiates in which the universe is rotating to describe the same relative motion does not mean that the two situations are otherwise equivalent. In one case, the water in the bucket is following its inertial geodesic path and in the other it is not because of the torque applied by the motor, and an accelerometer measuring proper acceeration will show the difference.

 

[hmmm27]

I would respectfully argue this: "The simplest way to state the equivalence principle is this: inertial mass and gravitational mass are the same thing." Centripetal/fugal inertia/force is merely the rotational equivalent of linear inertia/force.

Yes, and an aircraft driven by a propeller is equivalent to one driven by a rocket motor in that they both produce thrust which allows a properly built craft to fly around. They aren't the same, though and there are very good reasons why there are very few rocket-powered airplanes, and absolutely no propeller-driven spaceplanes.

I like the second sentence, though : no mention of gravity. Take a chair, duct-tape a rocket to the back and light it up. Assuming it flies straight, there's your linear acceleration. Impart the right amount of rotation to the chair, and you can end up with a circular spiral motion that uses the rocket thrust to impart centrifugal force.

[edit:]Impart *exactly* the right amount of rotation, calculated based on the thrust of the rocket, and you end up with the rocket facing directly outwards, always pushing the chair towards the center of a circle, but not getting any closer, ie: exactly the same as swinging the chair around with a rope.

 

[PeterDonis]

Impart the right amount of rotation to the chair, and you can end up with a circular motion that uses the rocket thrust to impart centrifugal force.

No, the centrifugal force in this case would not be coming from the rocket. The force of the rocket (pushing on you through the chair) would be pushing you towards the center of rotation, not away from it.

Centrifugal force would appear if you chose coordinates in which you were at rest and the rest of the universe was rotating; it would be required (on the Newtonian view of "force") to balance the force exerted on you by the rocket and keep you at rest.

 

[Dale]

heheh... I can't for the life of me now figure out why I thought/wrote the "elongated" bit, though it made sense at the time. Wait... almost had it... darn, slipped away again. Apparently @Dale and @pbuk understood : maybe they can help.

I didn't worry about it. Elongation is just negative flattening. I don't know the sign, but the amount will be dramatically smaller in the "spinning shell" case than in the "spinning blob" case. Frame dragging is tiny!

 

[Peter Leeves]

Hello Dale, yes I think I've got it straight in my mind now that proper acceleration is invariant (and never relative). Also it's demonstrated by an accelerometer and is true in any coordinate system (I'm now wondering if "coordinate system" has the same meaning in physics as "reference frame" - but that's by-the-by).

Given your next paragraph, do we need to specify whether proper acceleration is invarient in any inertial coordinate system ? Or does it still remain true in a non-inertial coordinate system, hence no need to specify which ?

Focusing on the scenario where the surface is curved, you can describe that in inertial coordinates or in (non inertial) coordinates where the bucket is stationary. In the inertial coordinates the bucket is undergoing coordinate acceleration (the universe is not) and in the co-moving coordinates the bucket is not undergoing coordinate acceleration (the universe is). So coordinate acceleration is relative.

I follow this. Co-ordinate acceleration is relative. Proper acceleration is invariant.

Here you are using the word “rotating” without specifying if you are talking about “proper” or “coordinate”. That is likely the source of the confusion.

I was using the word "rotating" with respect to "proper" (not coordinate).

 

[PeterDonis]

do we need to specify whether proper acceleration is invarient in any inertial coordinate system ? Or does it still remain true in a non-inertial coordinate system, hence no need to specify which ?

"Invariant" means "the same in any coordinate system, period". It is not limited to inertial coordinate systems.

 

[Dale]

It certainly appears (at first sight anyway) that both scenarios would yield identical results.

No, the results are very different. The spinning blob (proper rotation) scenario will produce a very large distortion in the blob shape. The spinning shell (again, proper rotation) scenario will produce a very small distortion in the blob shape.

Of course, in either case you can choose either object as your reference frame. So you can choose coordinates where the shell is fixed regardless of which is undergoing proper rotation, and similarly for the blob. While you can choose either object as your reference frame, the distortion of the shape depends on the invariant facts (which is experiencing proper rotation), not on the relative facts (which is experiencing coordinate rotation).

 

[Peter Leeves]

spacetime geometry is the invariant thing that tells us that it "really" is the bucket that is rotating, not the universe.

You say "it really is the bucket that is rotating". Intutition is telling me this shouldn't be so. It seems directly analagous to saying somehow you can distinguish whether the man really is freefalling in an enclosed lift towards earth, or the man really is in interstellar space. Assuming the observations are identical in both scenarios (emphasis on "assuming" because that's still in question) then surely they would be equivalent and equally valid perspectives, and no preference as to which is "really" rotating ?

 

[Dale]

You say "it really is the bucket that is rotating". Intutition is telling me this shouldn't be so. It seems directly analagous to saying somehow you can distinguish whether the man really is freefalling in an enclosed lift towards earth, or the man really is in interstellar space.

It really is the bucket that is (proper) rotating.

A better analogy would be to have two observers with a relative (coordinate) acceleration between them. You really can identify which of the two observers is (proper) accelerating. The physics (the outcome of any experiment) depends on the proper acceleration, not the coordinate acceleration.

 

[Peter Leeves]

Now wait a moment... are you suggesting that when I switch on a motor to rotate something (like, for example a shaft with a bucket on the end) the motor is actually applying torque to all the rest of the universe causing it to rotate, while the bucket remains still?

Spot on. In the sense that it is equivalent (and will show identical observations and therefore be an equally valid perspective). This is the very principle that Einstein used in a lecture in 1918 when he was asked to resolve the Twin Paradox. I'm not smart enough to think this stuff up on my own. I just saw that you could apply the same answer, but rotationally, to the Newton's Bucket scenario to explain why the water would still become concave if you rotated the universe and the bucket/water remained stationary.

Don't get stuck on the size of the torque necessary to rotate the entire universe with all it's mass. It seems that would take ridiculous amounts of energy. Not at all. The torque required to rotate the entire universe to produce identical observations on the bucket/water system, is precisely equivalent to that generated by the motor (not that much).

You may be thinking of a different question

No, I'm not thinking of a different question.

 

[Dale]

This is the very principle that Einstein used in a lecture in 1918 when he was asked to resolve the Twin Paradox.

Note that the principle you are referring to in Einstein’s 1918 lecture was based on coordinate acceleration, not proper acceleration.

One problem with applying the same strategy to a rotational problem is that rotating coordinates very quickly lose any semblance of physical meaning. Even with a rotation rate of just 1 Hz, the moon is moving at v>c and the rotating coordinate time no longer describes anything that could be measured with a clock. Talking about the rotation of the universe as a coordinate transform is non-trivial if you intend for the coordinates to have a physical interpretation.

Usually, when physicists talk about a rotating universe this is not what we mean. What we mean is that we locally make sure that we are not (proper) rotating and then we look at the distant stars and galaxies. If those are rotating around us then we say the universe is rotating. This is not a coordinate effect.

 

[Peter Leeves]

but the amount will be dramatically smaller in the "spinning shell" case than in the "spinning blob" case. Frame dragging is tiny!

I agree it will be dramatically smaller - when you're only considering the inluence caused by the mass of the Earth rotating. But that's not the case I've described. It's the frame dragging due to the rest of the entire mass of the visible universe rotating (I choose those words very carefully) that influences the "stationary blob"of water, to produce identical squishing of the blob.

Why did I specify "the entire visibile universe" ? Because gravity works at the speed of light in a vacuum. Anything beyond that can't gravitationally influence the water blob. Every bit of mass within that sphere does influence every other bit of mass in that sphere.

 

[Dale]

I agree it will be dramatically smaller - when you're only considering the inluence caused by the mass of the Earth rotating. But that's not the case I've described. It's the frame dragging due to the rest of the entire mass of the visible universe rotating (I choose those words very carefully) that influences the "stationary blob"of water, to produce identical squishing of the blob.

The mass of the Earth is easier to analyze because you can simply place an accelerometer on it and because you can use rotating coordinates that cover both the blob and the Earth. But the qualitative conclusion does not change if you go to the universe as a whole. This is, IMO, a way that Mach's principle fails although I am sure there are other opinions (Mach's principle is too loosely defined to have definitive answers).

Suppose that we have a situation where an observer/blob looks out and sees that the distant stars and galaxies are rotating around the observer/blob. It is possible that the universe is rotating (observer has no proper rotation) or it is possible that the observer is rotating (observer has proper rotation). The blob will be greatly distorted if the blob is rotating and the blob will barely distort if the universe is rotating. They are physically different scenarios with physically different outcomes, even though the relative motion is the same.

 

[PeterDonis]

You say "it really is the bucket that is rotating". Intutition is telling me this shouldn't be so.

Your intuition is faulty and needs to be retrained. Go back and read my posts about spacetime geometry again, carefully.

It seems directly analagous to saying somehow you can distinguish whether the man really is freefalling in an enclosed lift towards earth, or the man really is in interstellar space.

It isn't. Go read my post about what the equivalence principle does and doesn't say again, carefully.

Assuming the observations are identical in both scenarios (emphasis on "assuming" because that's still in question)

It shouldn't be. Changing your choice of coordinates doesn't change the results of any observations or measurements. That has never been a point at issue in this discussion.

then surely they would be equivalent and equally valid perspectives, and no preference as to which is "really" rotating ?

Go back and read my post about what "equivalent and equally valid perspectives" is and is not asserting again, carefully.

 

[PeterDonis]

This is the very principle that Einstein used in a lecture in 1918 when he was asked to resolve the Twin Paradox.

No, it isn't.

Einstein said that when the traveling twin fires his rocket to turn around, this can be viewed as creating a gravitational field in which the stay-at-home twin is at a much higher altitude than the traveling twin, and this accounts for the stay-at-home twin's much greater elapsed time during the period when the field is present (i.e., when the traveling twin is firing his rocket).

Einstein did not say that the traveling twin firing his rocket was applying a force to the stay-at-home twin. This is obviously false since (a) the stay-at-home twin is in free fall the whole time, and (b) the traveling twin's rocket can't apply a force to the stay-at-home twin, that would violate causality.

I'm not smart enough to think this stuff up on my own.

Unfortunately, you are also apparently not well informed enough to correctly interpret what you are reading.

 

[PeterDonis]

It's the frame dragging due to the rest of the entire mass of the visible universe rotating (I choose those words very carefully) that influences the "stationary blob"of water, to produce identical squishing of the blob.

And this is wrong. In the spacetime geometry in question, there is no frame dragging. Frame dragging is not a coordinate effect; it is an effect of the spacetime geometry, and only certain kinds of spacetime geometries have it.

You really need to take several steps back at this point. As far as I can tell, pretty much everything you think you know about this topic is wrong.

 

[Peter Leeves]

This is, IMO, a way that Mach's principle fails although I am sure there are other opinions (Mach's principle is too loosely defined to have definitive answers).

I think I read yesterday that Einstein agreed with Mach (I'm not suggesting Einstein was infallible, but he did get quite a bit right). Sorry, I should stick to physics.

I hope you agree gravity travels at the speed of light in a vacuum and every piece of mass within our visible universe necessarily influences every other piece of mass. At big distances this is imperceptible (the distortion of space-time would be tiny) but nonetheless there. Considering there's quite a lot of mass in the universe (at varying distances from the bucket) it could nevertheless add up to have some influence when totalled (put Jupiter in Mars' location and see what happens on Earth, lol). It must have some influence, if you agree with the first sentence.

Suppose that we have a situation where an observer/blob looks out and sees that the distant stars and galaxies are rotating around the observer/blob. It is possible that the universe is rotating (observer has no proper rotation) or it is possible that the observer is rotating (observer has proper rotation). The blob will be greatly distorted if the blob is rotating and the blob will barely distort if the universe is rotating. They are physically different scenarios with physically different outcomes, even though the relative motion is the same.

Well described and I follow your description. For completeness, it's also possible the blob is proper rotating and the universe is proper rotating. In this case, the blob would squish mostly due to it's own proper rotation but also must be influenced to some lesser degree (due to gravitational frame dragging) by the universe's proper rotation. If the rotations were coincident I suppose the blob would be squished a tiny bit more. If opposite, then a tiny bit less (frame dragging countering the centripetal effect). Any other direction could have a small net disorting effect (depending on net distribution of the universe's mass).

In the bucket/water scenario (prior to releasing the bucket) the shape of the water is accurately determined by the shape of the bucket, the Earth's gravity sucking the water to the bottom of the bucket, the location of the centre of Earth's gravity at Earth's centre (hence slight curvature of the water surface due to it's radial distance), the proper rotation of the Earth and resulting centripetal forces plus ... ta da ... the tiny net influence of the remaining mass in the visible universe and any proper rotation of the universe (if any). [There's a couple other influences like surface tension and atmospheric pressure but I think we can say irrelevant to this case].

Release the bucket and the tension in the string causes the universe to rotatationally accelerate (bucket/water stationary co-ordinate system). Frame dragging by the visible universe's mass increases significantly, distorting the water such that it mimics the centripetal force.

Dale, at this stage (your patience has been very much appreciated) are you able to say agree, disagree or indeterminate ? I think I should probably drop this now for everyone's sanity, lol.

 

[PeterDonis]

I think I read yesterday

Where? Please give a reference.

I hope you agree gravity travels at the speed of light in a vacuum

With appropriate qualifications, yes. But see below.

every piece of mass within our visible universe necessarily influences every other piece of mass

Yes, but this doesn't mean what you think it means. For example, it does not mean there is "frame dragging" on a bucket that is rotating relative to the rest of the universe.

The way that the rest of the matter in the universe influences the bucket is by determining the spacetime geometry.

Release the bucket and the tension in the string causes the universe to rotatationally accelerate (bucket/water stationary co-ordinate system). Frame dragging by the visible universe's mass increases significantly, distorting the water such that it mimics the centripetal force.

All of this is wrong--not about what happens to the water in the bucket, but about why it happens. There is no "frame dragging" by the universe's mass in this scenario.

 

[Dale]

I think I read yesterday that Einstein agreed with Mach

Well, philosophically he definitely wanted to agree with Mach, but his actual theory does not. Of course, it is a little difficult to exactly pin down what Mach’s principle means in an empirical sense.

It must have some influence, if you agree with the first sentence.

Certainly, it does have some influence. But for equivalence it must not merely have some influence, it must have the exact same influence. It does not.

Frame dragging by the visible universe's mass increases significantly, distorting the water such that it mimics the centripetal force.

No. What you are describing here is not frame dragging. It is those Christoffel symbols.

Dale, at this stage (your patience has been very much appreciated) are you able to say agree, disagree or indeterminate ?

Sorry, what is the question at this stage?

 

[Peter Leeves]

Sorry, what is the question at this stage?

Will a PROPER rotating bucket in a static universe give identical observation (water pushed out towards the circumference) as a static bucket in a PROPER rotating universe, as the acceleration is invariant in both cases ? Or in the PROPER rotating universe would the water remain as-is since there's nothing to push the water outwards ?"

 

[PeterDonis]

Will a rotating bucket in a static universe give identical observation (water pushed out towards the circumference) as a static bucket in a rotating universe

It depends on what you mean by "a static bucket in a rotating universe". That phrase can mean one of two things:

(1) You have the same spacetime geometry, you've just changed coordinates. This is the case I think we have been discussing. In this case, all observations are the same since the spacetime geometry is the same (as well as the relative motion of bucket and rest of universe). You're just describing the same observations using a different choice of coordinates. However, as I have already pointed out, the phrase "static bucket in a rotating universe" is not really a good one to describe this case, since there is an invariant sense in which the bucket is rotating and the universe is not.

(2) You have a different spacetime geometry, one in which there is an invariant sense in which the universe is rotating and the bucket is not. In this case, observations will be different because the spacetime geometry is different. Taken literally, this is what "a static bucket in a rotating universe" describes, but I don't think it's what you intended to describe in your earlier posts in this thread.

These answers were already given in an earlier post, which I asked you to go back and read carefully. Apparently you didn't.

 

[Dale]

Will a rotating bucket in a static universe give identical observation (water pushed out towards the circumference) as a static bucket in a rotating universe

Disagree, as I described in detail in post 81.

as the acceleration is invariant in both cases ?

These are two different situations, not one situation described from two frames. The (proper) accelerations are different in the two different spacetimes.