Mach versus Einstein

[The Man]

Let’s try to think of it in an engineering sense, a real force when applied to a free substance tends to compresses or elongate that substance due to the transmission of that force through the substance and the inertia of that substance. A fictitious force like gravitational motion or the apparent centrifugal force acts on the entire substance at the same time or on the substance’s reference frame. This, I think, may be the definition of a fictitious force, that when applied or perceived to be applied to a free substance there is no compression or elongation of that substance as there must be with the application of a real force.

 

[Schneibster]

No, the rope will compress the ball. And the rope will stretch. Good try, though, Man.

 

[sol invictus]

Yes, Schneibster is correct. A rotating object for example is stressed more and more by centrifugal force as you move you from the center. So that doesn't work.

There is a mathematical definition that works, which is the one I proposed originally - to ask if the curvature of the metric written using the non-inertial coordinates is zero or not. If it is, the forces are certainly inertial. If it isn't, we're back to my original question.

 

[The Man]

Yes, Schneibster is correct. A rotating object for example is stressed more and more by centrifugal force as you move you from the center. So that doesn't work.

There is a mathematical definition that works, which is the one I proposed originally - to ask if the curvature of the metric written using the non-inertial coordinates is zero or not. If it is, the forces are certainly inertial. If it isn't, we're back to my original question.

No, the rope will compress the ball. And the rope will stretch. Good try, though, Man.

Let's not forget the stipulation of a free substance. A ball attached to a rope is not free and any compression or elongation results from that bound condition. A free ball under a thrust pushing or pulling it would experience compression or elongation. A free ball in the rotating reference frame experiences neither even if it appears as though being pushed or pulled away from the rotational center.

Another way to look at it would be with the free ball and the rotating reference frame initially at rest with regard to some other outside reference frame. As the rotating reference frame begins to rotate anything bound to that reference frame will undergo a compression or elongation. The free ball will remain at rest in relation to the other outside reference frame.

In the now rotating reference frame it is the free ball that appears to accelerate around the center yet still undergoes no compression or elongation due to that acceleration. While objects bound to the rotating reference frame and apparently at rest do undergo elongation or compression.

 

[sol invictus]

It doesn't matter what reference frame you look at it in. If the ball is rotating around its axis it will be distorted. That is a physical statement, and if you have expressed the laws of physics correctly in whatever frame you like that's what you will find.

One particular way to express that is to say that if there is a rotating frame in which the ball is at rest, then it will be distorted (by centrifugal force). Or you can simply say that if it's rotating with respect to an inertial frame, then it will be distorted (by inertia). Same thing.

In all cases the ball is "free" in the sense that no external forces act on it.

 

[The Man]

It doesn't matter what reference frame you look at it in. If the ball is rotating around its axis it will be distorted. That is a physical statement, and if you have expressed the laws of physics correctly in whatever frame you like that's what you will find.

One particular way to express that is to say that if there is a rotating frame in which the ball is at rest, then it will be distorted (by centrifugal force). Or you can simply say that if it's rotating with respect to an inertial frame, then it will be distorted (by inertia). Same thing.

In all cases the ball is "free" in the sense that no external forces act on it.

In the previous example the free ball was not rotating about it’s axis but apparently revolving about the axis of the rotating reference frame (some distance away from the ball) in that reference frame. Sorry if I did not make that condition clearer. In that reference frame the free ball is apparently acting under the imposition of a centripetal force yet is not compressed or elongated by the application of that force. From the point of view of the other outside reference frame the ball is at rest, under no force and again not distorted.

 

[edd]

According to Mach (at least according to the caricature of his position as it's usually presented today, which I referred to before as Straw-Mach ), since the rope was slack initially, and since there are no distant bodies to measure the rotation with respect to, there will never be any tension in the rope, and so it will not snap no matter how long the rockets fire their engines.

There are bodies to measure the motion with respect to.

I'm not sure that this is of significance though.

 

[sol invictus]

There are bodies to measure the motion with respect to.

I'm not sure that this is of significance though.

Well, that's exactly the question. According to Mach that is not only significant but essential. According to Einstein, not at all.

 

[The Man]

Well, that's exactly the question. According to Mach that is not only significant but essential. According to Einstein, not at all.

Wait a second, relative motion is the bases of relativity (Special and General). How can you say that Einstein did not consider it significant and essential to measure motion with respect to some other body?

Perhaps the difference is Mach’s requirement for a distant body reference. Where Einstein knowing that a distant body reference may not always be available asserted that the laws of physics must be the same even when we only have local and somtimes only co-moving body references. Otherwise the laws of physics could be changed just by opening a window and observing some distant body or would be different for a local and distant observer.

 

[sol invictus]

Wait a second, relative motion is the bases of relativity (Special and General). How can you say that Einstein did not consider it significant and essential to measure motion with respect to some other body?

In GR motion is relative but acceleration is not. Mach wanted both to be relative.

Otherwise the laws of physics could be changed just by opening a window and observing some distant body or would be different for a local and distant observer.

Huh? Mach wasn't an idiot - his idea has nothing to do with observation. He just believed that the existence of those distant bodies was what defined the rest frame for acceleration. It's not a crazy idea, but it's incorrect according to GR.

 

[edd]

Well, that's exactly the question. According to Mach that is not only significant but essential. According to Einstein, not at all.

Following on from my earlier thoughts I'm not sure there's a meaningful way to come up with an experiment that is sufficiently clear cut.

What I was trying to hint at is not that we have bodies to measure with respect to, but that in your thought experiment, despite the initial lack of such bodies you do end up with them when the system begins to rotate. If you start off with your rocket system non-rotating with the rope slack and then spin it up, the rocket exhaust provides a reference against which you can measure the rotation. More generally, with any non-rotating initial system, if you spin it up then something must have the corresponding 'negative' angular momentum to balance the 'positive' angular momentum of the thing you've postulated. Does this exhaust provide what's needed in the same way a (on average on large scales) uniform distribution of matter would in Mach's view? I do find it unsatisfying that Mach's Principle is always talked about in terms of a fixed distribution of background objects with little discussion of how dependent the whole deal is on that distribution.

If you magic the system into existence rotating, but without any external objects then it's not clear what should be going on - there's several lines of thought that, combined, essentially lead me to think that it's an ill-posed thought experiment in some way. But I'm not too sure. I'd have to think about it, but I don't think it's as simple a thought experiment as it seems on the surface.

edit: I suppose really what I'm trying to say is that I find Mach's Principle alone unsatisfying as it's not any kind of complete theory. GR of course is. So it only makes sense to me to set up situations in GR and ask what happens in order to gain insight into whether there's anything behind Mach's Principle.

 

[The Man]

In GR motion is relative but acceleration is not. Mach wanted both to be relative.



Huh? Mach wasn't an idiot - his idea has nothing to do with observation. He just believed that the existence of those distant bodies was what defined the rest frame for acceleration. It's not a crazy idea, but it's incorrect according to GR.

How does a distant body accelerating with you define a rest frame for your acceleration? How can a distant body define anything if it can not be observed? If someone is asserting an absolute yet indefinable reference frame to measure acceleration against, then I would call that crazy. If the results of such an assertion are that the laws of physics are different in your reference then in some absolute reference frame that may be indefinable and unobservable to you, then I would call that idiotic. Certainly the statement of Mach’s principle as “Mass there influences inertia here” is consistent with General Relativity. However Mach’s own statements are so vague that people make many of their assertions and call them “Mach’s principle”.


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

"[The] investigator must feel the need of... knowledge of the immediate connections, say, of the masses of the universe. There will hover before him as an ideal insight into the principles of the whole matter, from which accelerated and inertial motions will result in the same way."

[2]Mach, Ernst (1960). The Science of Mechanics; a Critical and Historical Account of its Development. LaSalle, IL: Open Court Pub. Co.. LCCN 60010179. This is a reprint of the English translation by Thomas H. McCormack (first published in 1906) with a new introduction by Karl Menger

The fact is that Einstein developed General Relativity based on his interpretation of “Mach’s principle” and did not perceive a conflict.

"it... turns out that inertia originates in a kind of interaction between bodies, quite in the sense of your considerations on Newton's pail experiment... If one rotates [a heavy shell of matter] relative to the fixed stars about an axis going through its center, a Coriolis force arises in the interior of the shell; that is, the plane of a Foucault pendulum is dragged around (with a practically unmeasurably small angular velocity)."[3]

A. Einstein, letter to Ernst Mach, Zurich, 25 June 1923, in Misner, Charles; Thorne, Kip S.; and Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 0-7167-0344-0.

So I think that when people make an assertion they claim to represent “Mach’s Principle” saying that principle is somehow in conflict with General Relativity then it is simply their interpretation of “Mach’s Principle” that is in conflict with Einstein’s interpretation. Since Mach’s statements upon which those interpretations of that principle are based are so vague, I do not see any great significance in that people’s different interpretations would be in conflict.

 

[sol invictus]

What I was trying to hint at is not that we have bodies to measure with respect to, but that in your thought experiment, despite the initial lack of such bodies you do end up with them when the system begins to rotate. If you start off with your rocket system non-rotating with the rope slack and then spin it up, the rocket exhaust provides a reference against which you can measure the rotation. More generally, with any non-rotating initial system, if you spin it up then something must have the corresponding 'negative' angular momentum to balance the 'positive' angular momentum of the thing you've postulated. Does this exhaust provide what's needed in the same way a (on average on large scales) uniform distribution of matter would in Mach's view?

It's a nice idea, but it doesn't work that way in GR. The force in the rope is totally independent of what happens to the rocket exhaust. For example there could be a big sphere of mass out there, and someone could grab the exhaust and smear it all over that sphere. In that case it would have precisely zero effect on the rockets.

Or you could just set the gravitational constant to zero, in which case the exhaust again has precisely zero effect.

How does a distant body accelerating with you define a rest frame for your acceleration? How can a distant body define anything if it can not be observed?

Ask Mach. But it could presumably be observed by its effect.

If someone is asserting an absolute yet indefinable reference frame to measure acceleration against, then I would call that crazy.

I don't know what you mean by indefinable, but otherwise that sounds exactly like what GR does.

Certainly the statement of Mach’s principle as “Mass there influences inertia here” is consistent with General Relativity.

Influences, OK, but not determines. That's the whole point of this thread.

The fact is that Einstein developed General Relativity based on his interpretation of “Mach’s principle” and did not perceive a conflict.

He seems to have meant different things by it at different times. I presented a clear version above it here, which may or may not have been Mach's actual view, and argued that GR refutes that version.

However in some ways GR is closer to Mach than it is to Newton; in other ways, it's closer to Newton. Really, it's just different than either.

So I think that when people make an assertion they claim to represent “Mach’s Principle” saying that principle is somehow in conflict with General Relativity then it is simply their interpretation of “Mach’s Principle” that is in conflict with Einstein’s interpretation. Since Mach’s statements upon which those interpretations of that principle are based are so vague, I do not see any great significance in that people’s different interpretations would be in conflict.

I agree, particularly since Einstein himself wasn't very consistent on this. That's why I made clear what I meant by Mach's principle.

Incidentally, I thought Mach died in 1916. Why would E. write a letter to him in 1923? Anyway, the scenario he mentions in that letter is a specific case and is more complicated than the simple example above. It doesn't demonstrate the form of M's principle I gave to be correct; on the contrary, it's actually more evidence against it.

 

[69dodge]

It's a nice idea, but it doesn't work that way in GR. The force in the rope is totally independent of what happens to the rocket exhaust. For example there could be a big sphere of mass out there, and someone could grab the exhaust and smear it all over that sphere. In that case it would have precisely zero effect on the rockets.

I don't follow. Aren't you just transferring the angular momentum of the exhaust to the big sphere? Something is still rotating the opposite way of the rockets.

Or you could just set the gravitational constant to zero, in which case the exhaust again has precisely zero effect.

It seems to me that this would be a bigger change in the theory than you think. If the whole theory is based on inertial mass being essentially the same thing as gravitational mass, how can you eliminate one but keep the other without radically altering the character of the theory? Would it be general relativity anymore? Everyone agrees that in special relativity there's a big difference between inertial and accelerated reference frames.

 

[The Man]

Incidentally, I thought Mach died in 1916. Why would E. write a letter to him in 1923? Anyway, the scenario he mentions in that letter is a specific case and is more complicated than the simple example above. It doesn't demonstrate the form of M's principle I gave to be correct; on the contrary, it's actually more evidence against it.

True, Mach did die, February 19th 1916; I think Einstein wrote the letter to express his interpretation of Mach’s principle and the conformance to that interpretation of his General Relativity, published around the time of Mach’s death (1915/16).

I would have to agree that I do not believe the interpretation of Mach’s principle that you presented is consistent with Einstein’s interpretation of Mach’s principle or GR. Of course I certainly do not consider myself an expert in either GR or Mach’s principle.

 

[sol invictus]

I don't follow. Aren't you just transferring the angular momentum of the exhaust to the big sphere? Something is still rotating the opposite way of the rockets.

Well, you could have started with the sphere rotating the other way, and have the exhaust just barely cancel that. Anyway, it's silly to argue this way - we can calculate how much effect the exhaust would have on the rockets, and it's negligible.

It seems to me that this would be a bigger change in the theory than you think. If the whole theory is based on inertial mass being essentially the same thing as gravitational mass, how can you eliminate one but keep the other without radically altering the character of the theory?

By setting G=0, for example. Or simply by considering relatively light objects, such as these rockets.

Everyone agrees that in special relativity there's a big difference between inertial and accelerated reference frames.

Then the conversation is over, because SR is a limit of GR. And in fact, almost all the time, SR plus Newtonian gravity and the other three forces is good enough. I chose this example partly because that's true for it.

 

[edd]

It's a nice idea, but it doesn't work that way in GR. The force in the rope is totally independent of what happens to the rocket exhaust. For example there could be a big sphere of mass out there, and someone could grab the exhaust and smear it all over that sphere. In that case it would have precisely zero effect on the rockets.

Or you could just set the gravitational constant to zero, in which case the exhaust again has precisely zero effect.

Right, I'm totally with GR all the way - to make that clear. I think this represents an issue with how the 'naive' setup of the more traditional Machian version is set up though.

 

[69dodge]

Well, you could have started with the sphere rotating the other way, and have the exhaust just barely cancel that.

Isn't that the question, though? Could I have?

You're saying that GR allows the total angular momentum of the universe to be nonzero?

Or simply by considering relatively light objects, such as these rockets.

Relatively light, compared to what? They're the only things in the universe.

And in fact, almost all the time, SR plus Newtonian gravity and the other three forces is good enough.

Yes, but also in fact there's a lot more stuff in the universe than a pair of rockets. Doesn't rotation relative to that stuff correspond well with "absolute" rotation? Is this merely a coincidence, according to GR?

I just get the feeling that there's some constant somewhere which isn't really constant, but rather depends on the total amount of stuff in the universe. And you're using, in the hypothetical case of rockets in an empty universe, the same value for it that was experimentally determined in our actual, stuff-filled, universe. If that makes any sense.

 

[sol invictus]

Isn't that the question, though? Could I have?

Yes, the theory allows it, there's nothing wrong with it.

You're saying that GR allows the total angular momentum of the universe to be nonzero?

Yes. As you may be aware, there is a subtlety in GR - in a sense, energy, momentum, and angular momentum are all zero no matter what the configuration of matter is. However there is a way to define them so they can take non-zero values. So in a word, yes.

Relatively light, compared to what? They're the only things in the universe.

GR has a dimensionful parameter, usually denoted G. In the Newtonian limit it is Newton's constant. Roughly speaking the gravitational effects of a mass are Gm/rc^2, where m is the mass and r is the distance. So long as that combination never gets to be of order 1, gravity is weak. So that's what I mean when I say light.

Yes, but also in fact there's a lot more stuff in the universe than a pair of rockets. Doesn't rotation relative to that stuff correspond well with "absolute" rotation? Is this merely a coincidence, according to GR?

That's an interesting question. First of all I'm not sure how true that is. The earth is rotating around the milky way, and the milky way is rotating around a galaxy cluster, and our cluster is rotating around others... but I'll think about that a little more.

I just get the feeling that there's some constant somewhere which isn't really constant, but rather depends on the total amount of stuff in the universe. And you're using, in the hypothetical case of rockets in an empty universe, the same value for it that was experimentally determined in our actual, stuff-filled, universe. If that makes any sense.

It makes sense, but you have to remember that GR is a mathematical theory which allows us to answer hypothetical questions. Of course it might be wrong, and so the answers might be wrong, but we can at least determine what they are. One thing GR says is that there will be tension in that rope, regardless of the fact that the rotation takes place in an empty universe.

 

[The Man]

It makes sense, but you have to remember that GR is a mathematical theory which allows us to answer hypothetical questions. Of course it might be wrong, and so the answers might be wrong, but we can at least determine what they are. One thing GR says is that there will be tension in that rope, regardless of the fact that the rotation takes place in an empty universe.

Correct, or we should specify in an otherwise empty universe, since the rotating body makes the universe not entirely empty. I think this is the problem with the entirely distant body interpretation of Mach’s principle. I think Mach’s assertion of “the immediate connections, say, of the masses of the universe” refers to the local influences of both distant and local bodies. In other words the generally flat space-time, from a significantly distant body, in which a non-gravitational acceleration could be represented by GR or the very locally curved space-time that moves with a single body accelerating alone in the universe. Since any curvature moves with the accelerating body that body is basically moving through flat space-time. I do not think this would even present a problem when considering the frame dragging of a rotating black hole alone in the universe but I’m not sure. I think that the best interpretation of Mach’s principle, that I have seen so far, is that without mass (or energy) there is no space (or space-time).