Mach's Principle: Explaining the Force We Feel When Changing Velocity

[A.T.]

I might be wrong,

No, it's rather me.

but I have always assumed that's Mach's principle is that interia is caused by the sum of all mass in the universe, near and far,

Yeah, I guess so. But without a quantitative model it is difficult to even envision a test for it. And if Mach & Einstein didn't come up with quantitative model, I will not even try.

 

[bcrowell]

But without a quantitative model it is difficult to even envision a test for it. And if Mach & Einstein didn't come up with quantitative model, I will not even try.

Brans-Dicke gravity is often considered to be more Machian than GR. So if the discussion of Einstein's two-planet example feels like it degenerates into metaphysics because Mach's principle is so vague, I suppose one way to go would be to see what Brans-Dicke gravity predicts about it. The Brans-Dicke theory has an adjustable parameter \omega, and as \omega\rightarrow\infty you recover GR. The latest data from Cassini give \omega > 40000. I wonder whether, for small values of \omega, you'd see the equatorial bulges of the two planets become more equal, and the Sagnac effect become more equal on both planets' surfaces.

Looking at the form of the Brans-Dicke field equations, I'm tempted to say that they probably *cannot* reproduce the result that Einstein expected in his two-planets example. The reason is that the problem has axial symmetry, and the scalar field \phi is real, so I don't see how you \phi could carry any information about the handedness of rotation. But the interaction between g and \phi is complicated and nonlinear, so I could be totally wrong about that.

 

[Naty1]

Brian Greene has some interesting comments regarding Mach and Einstein's theories in THE FABRIC OF THE COSMOS,2004, PAGES 33-38:

...The concept of absolute space left many wondering how absolute space can allow us to identify true accelerated motion (as with the famous spinning bucket of water example), while it cannot provide a way to identify true constant velocity motion. After all, if absolute space really exists, it should provide a benchmark for all motion, not just accelerated motion...Mach argued that in an otherwise empty universe there is no distinction between spinning and not spinning...Without other material-without any benchmarks for comparsion- Mach claimed there would be no way to experience acceleration...Machs suggestion was not a complete theory or description because he never specified how the matter content of the universe would exert the proposed influence...when the dust of relativity finally settled, the question of whether space is something...was transformed in a manner that shattered all previous ways of looking at the universe...

 

[Dmitry67]

Infinite empty universe with few gravitating bodies at one location - is it a realistic GLOBAL solution for GR equations?

 

[bcrowell]

Infinite empty universe with few gravitating bodies at one location - is it a realistic GLOBAL solution for GR equations?

This is another case where it's useful to consider the Brans-Dicke theory as a test theory. I've just read their paper, C. Brans and R. H. Dicke, Physical Review 124 (1961) 925, which IMO is an exceptional piece of scientific writing -- very accessible and physically clear. The case of the nearly empty universe is just an idealization. They start out the paper by doing idealized thought experiments like this and discussing how they appear to violate Mach's principle. Then they develop their field theory and use it to calculate some classic stuff like cosmological solutions and the perihelion rotation of Mercury. The cosmological solutions come out different than in GR, because the scalar field \phi is basically the inverse of the gravitational constant. As the universe expands, \phi gets smaller (because the trace of the stress energy tensor T acts as its source), and G gets bigger. So if you like, you can think of all these thought experiments with nearly empty universes as limiting cases where cosmological expansion has progressed far into the future.

In particular, I now realize that I was wrong about Einstein's two-planet example in the Brans-Dicke theory. They do essentially the same example in their paper, although their thought experiment is phrased a little differently. (A physicist is in a laboratory in a universe that's otherwise empty. He fires a bullet out the window, and the lab rotates in response, which is detectable on a gyroscope.)

Analyzing the two-planet thought experiment in the Brans-Dicke theory, here's what happens. Very little matter is present in the universe, so G is huge. Because of the big G, you get a huge amount of Lense-Thirring frame dragging. Therefore the two planets will synchronize their rotations. This is exactly the Machian result that Einstein wanted, but didn't get in GR.

 

[Al68]

GR predicts that the planet that is really spinning (B) will bulge.

OK, how does GR determine which one is "really spinning"? By the a priori observation of the bulge to be "predicted"?

Yes, my smiley indicated a trap.

The obvious issue is the circular logic.

The real question is, what would GR predict in the absence of any a priori knowledge of which one is "really spinning", except that each planet spins relative to the other?

 

[Jonathan Scott]

Analyzing the two-planet thought experiment in the Brans-Dicke theory, here's what happens. Very little matter is present in the universe, so G is huge. Because of the big G, you get a huge amount of Lense-Thirring frame dragging. Therefore the two planets will synchronize their rotations. This is exactly the Machian result that Einstein wanted, but didn't get in GR.

Most of that sounds plausible. However, from what I recall of frame dragging I don't see why it should cause the planets to "synchronize their rotations", because there is no change occurring.

There may be some special case effect here when these are the only masses in the universe, as that affects coordinate systems and other things as well, but certainly in the normal case I don't think frame-dragging changes rotation in this way.

As I understand it (and I'm willing to accept that I may be mistaken), the effect in frame dragging is that a nearby test body's view of local space is influenced by rotation or acceleration of local masses in such a way that this rotation or acceleration is apparently slightly decreased relative to how it would appear to a distant observer.

This means for example that a test body would feel that it wasn't rotating when in fact it was rotating slightly in the same direction as the source body. However, although this induces changes in centripetal and coriolis forces in the object (and of course may induce precession in a gyroscope as in GP-B) I don't think it induces any change in overall rotation (and angular momentum) unless you change the frame dragging effect. For example, if you move the test object closer to the source, then relative to a distant observer it will appear to experience additional forces, but if it stays in the same location I don't see any reason that any angular momentum should be transferred.

 

[bcrowell]

OK, how does GR determine which one is "really spinning"? By the a priori observation of the bulge to be "predicted"?

Yes, my smiley indicated a trap.

The obvious issue is the circular logic.

The real question is, what would GR predict in the absence of any a priori knowledge of which one is "really spinning", except that each planet spins relative to the other?

I don't think Einstein's logic was circular here. Mach's principle predicts that the two planets' equatorial bulges are always equal, and that they must vanish if neither planet is rotating relative to the other. GR predicts that they can be unequal, and that they can be nonvanishing even if neither planet is rotating relative to the other.

(GR also predicts a correlation between the equatorial bulge and other effects like the Sagnac effect and the Lense-Thirring effect, so there are multiple ways of determining each planet's rate of rotation, independent of any observation of the other planet or anything else that may or may not exist in the more distant cosmos.)

 

[bcrowell]

Most of that sounds plausible. However, from what I recall of frame dragging I don't see why it should cause the planets to "synchronize their rotations"[...]

You're right. My interpretation was wrong. Frame dragging's effect on a gyroscope vanishes when the two axes are parallel. I think the Brans-Dicke paper was referring to a case where the gyro's axis was perpendicular to the lab's. Actually I think there's a straightforward argument that the synchronization effect I proposed is impossible in a Machian theory. In a purely Machian theory, observers on an isolated planet can never tell whether their planet rotates or not. Therefore if they start a gyroscope rotating on their planet, with its axis perpendicular to the planet's surface, the gyroscope's environment is perfectly symmetric spatially. If the gyroscope then slows down or speeds up in this spatially symmetric environment, it violates time-reversal symmetry.

So what can we actually say about the Einstein two-planet scenario in Brans-Dicke gravity...?

It seems clear to me that observers on an isolated planet can never do local experiments using mechanical gyroscopes in order to determine their planet's state of rotation. If they could do such experiments using some other effect like the Sagnac effect, then the disagreement between the two would probably violate the equivalence principle. Although Brans-Dicke gravity doesn't obey the strictest form of the equivalence principle, it does obey it in some form, so I suspect that the purely Machian result holds as well in Brans-Dicke gravity in the appropriate limit of a nearly-empty universe: observers can't tell by any local experiment whether they're rotating or not.

This all seems to work out consistently when you consider the equatorial bulges. The gravitational constant is extremely high, so gravity is extremely strong relative to inertia, and the bulges are strongly suppressed.

Adding a second planet would not seem to me to change this prediction. The field \phi reflects the presence of mass in the universe, convoluted with a 1/r distance dependence. In a universe full of matter, this is a big effect, because the amount of matter at distance r grows faster than r. But in the two-planet example, the second planet is not going to have a significant effect on \phi.

I think the ability to get clearcut answers here may be hampered by difficulties in formulating the limits correctly. The Brans-Dicke paper talks about this on the final page, where they can't take the limit of an empty universe because they're using a weak field-approximation.

 

[yuiop]

OK, how does GR determine which one is "really spinning"? By the a priori observation of the bulge to be "predicted"?

Yes, my smiley indicated a trap.

The obvious issue is the circular logic.

The real question is, what would GR predict in the absence of any a priori knowledge of which one is "really spinning", except that each planet spins relative to the other?

O.K. I was being slightly glib when I "took the bait" in the interests of prompting a conversation on the issues and it certainly seems to have done that. I think the main issue is that GR allows the asymmetric rotation, where as a Machian model probably would not. One of the main hurdles is that we can not remove all the distant stars to carry out a definitive experiment. I think with some ingenuity, we may be able to come up with a thought experiment that might shed some light on the situation. One way of rephrasing the issues that you raise is "how does planet B know it should bulge and how does planet A know it should not bulge?". If I was really honest, I would admit I have a nagging suspicion of circular logic in there somewhere too.

It is also worth noting that if we started in a situation where both planets were non-rotating, it would be impossible to arrive at a situation where only planet B is rotating without any other masses in the universe. The conservation of angular momentum principle, would require some other masses to rotate in the opposite direction to maintain the initial zero angular momentum of the universe.

First attemp at a thought experiment: Consider two flywheels connected by a motor. Both flywheels have the same radius and one flywheel has ten times more mass than the other and there are no other masses in the universe. When the motor is operated, the small flywheel starts rotating faster than the large flywheel rotating in the opposite direction, as required by the conservation of angular momentum. Obviously the small flywheel has less angular inertia than the large flywheel. Does Mach's principle predict the larger mass of the larger flywheel, causes the smaller flywheel to have less inertia and the smaller mass of the smaller flywheel causes the larger flywheel to have less inertia? Does Mach's principle predict that the greater the total mass of the "distant stars" is, the lesser the inertia of objects "here" is? Does that imply in the absence of the distant stars that objects would have near infinite inertia?

There appears to a slight connection between Mach's ideas and the competion that once existed between the Ptolemaic model and the Copernican model. Ptolemy asserted that the Earth was the centre of the universe and produced a complicated set of rules to explain the motion of the planets, while Copernicus showed that the rules were much simpler if the Sun was taken as the centre of the universe. The most significant part of the Copernican revolution was the realisation that any point in the universe could be treated as the stationary centre of the universe and with enough ingenuity the motion of all bodies about that point can be explained. Presumably, if we assume the Earth is stationary and non rotating, Mach's principle should be able to explain the bulge of the Earth, the Coreoilis forces and the Ptolmaic epicycles of the planets, in terms of the effect of the rotating stars, while GR would say emphatically that the Earth is rotating. Is that correct?

 

[D H]

It seems clear to me that observers on an isolated planet can never do local experiments using mechanical gyroscopes in order to determine their planet's state of rotation.

As far as we know, an observer on a real planet can use gyroscopes to determine whether the planet is rotating. Maybe I'm just an old fart who still thinks conjectures in physics should be testable. Conjectures regarding a universe that comprises one and only one planet are not testable.

These discussions of an isolated planet are conjectures based on extrapolating the physics of the universe we know to a universe that is thirty orders of magnitude less massive than and sixty orders of magnitude small than our universe. The one thing we physicists learned (or should have learned) since the early 1900s is that extrapolating the physics we know over dozens of orders of magnitude is not valid.

 

[bcrowell]

As far as we know, an observer on a real planet can use gyroscopes to determine whether the planet is rotating. Maybe I'm just an old fart who still thinks conjectures in physics should be testable. Conjectures regarding a universe that comprises one and only one planet are not testable.

The quote that you were responding to was in the context of the Brans-Dicke theory. Brans-Dicke gravity is testable, and it has passed all observational tests so far (e.g., perihelion rotation of Mercury). There's a cool book called Was Einstein Right? that discusses a lot of the history of the tests of GR and Brans-Dicke as competing theories of gravity in the 1970's. Brans-Dicke gravity has an adjustable parameter \omega that gives GR if you set it to infinity. The lower you set \omega, the more Machian the theory becomes. The current lower limit on \omega is about 40,000, which means that the universe is in this sense not very Machian. So Mach's principle is not just some vague philosophical notion, it's a number that you can measure. In the limit of large \omega, you get GR, which makes certain predictions about the Sagnac effect, frame dragging, etc. In that limit, you can determine your state of rotation and extract the result using GR. In the limit of small \omega, you get a theory where gyroscopes don't work. In the intermediate case, which for all we know really does describe the universe we live in, gyroscopes do precess, but not as much as predicted by GR, and the answer you'll get for your state of rotation will actually be slightly wrong.

These discussions of an isolated planet are conjectures based on extrapolating the physics of the universe we know to a universe that is thirty orders of magnitude less massive than and sixty orders of magnitude small than our universe. The one thing we physicists learned (or should have learned) since the early 1900s is that extrapolating the physics we know over dozens of orders of magnitude is not valid.

Brans-Dicke gravity makes definite predictions. It makes those predictions without having to go to a hypothetical alternative universe that's empty. The discussion of scenarios with empty universes, in the context of Brans-Dicke gravity as the test theory, is simply a way of reasoning about the limiting behavior of the theory compared to GR.

i would actually maintain exactly the opposite of the point of view you're advocating, in the following sense. If GR is the only theory available, then it becomes impossible to design experiments to test GR. Only if you have other theories that make other predictions can you test whether GR is correct. For instance, there was a 2003 experiment by Fomalont and Kopeikin ( http://arxiv.org/abs/astro-ph/0302294 ) that claimed to test Einstein's century-old prediction that low-amplitude disturbances in the gravitational field would propagate at c. Turns out that Fomalont and Kopeikin's experiment doesn't really test this claim, and the reason it can't test it is that there is no competing test theory available that *doesn't* predict propagation at c. In general, without considering competing test theories like Brans-Dicke, or Østvang's quasi-metric relativity, we'd actually be limited to the kind of navel-gazing you were criticizing.

 

[Al68]

I don't think Einstein's logic was circular here. Mach's principle predicts that the two planets' equatorial bulges are always equal, and that they must vanish if neither planet is rotating relative to the other. GR predicts that they can be unequal, and that they can be nonvanishing even if neither planet is rotating relative to the other.

I wasn't referring to Einstein's logic being circular, he used this example to point out the circular logic that already existed.

A simpler example is that we define an inertial reference frame as one in which objects are unaccelerated in the absence of applied forces, (no pseudo-forces are present). Then we "predict" that in an inertial reference frame, objects will not be accelerated without applied forces. This prediction is just the result of assuming the predicted result a priori.

This makes for a very good physical model, but Einstein was unsatisfied with the "epistemological shortcomings" of such models, including his.

A fully Machian theory, Einstein had hoped, would be free of this shortcoming.

 

[Al68]

O.K. I was being slightly glib when I "took the bait" in the interests of prompting a conversation on the issues and it certainly seems to have done that. I think the main issue is that GR allows the asymmetric rotation, where as a Machian model probably would not.

Sure GR allows for asymmetric rotation, but there is no "cause" for the bulge in the form of an equation relating the bulge to coordinate rotation that wouldn't equally apply to the non-bulging planet.

Does Mach's principle predict that the greater the total mass of the "distant stars" is, the lesser the inertia of objects "here" is?

I would think a Machian theory would predict the reverse, but I don't think Mach ever got to that point.

I didn't pose the question because I have the answer, I definitely don't.

 

[yuiop]

Does Mach's principle predict that the greater the total mass of the "distant stars" is, the lesser the inertia of objects "here" is? Does that imply in the absence of the distant stars that objects would have near infinite inertia?

I would think a Machian theory would predict the reverse, ...

I must admit I too would think Machian theory would predict the reverse, but if that is the case, my little thought experiment indicates Machian theory contradicts conservation of angular momentum.

Perhaps Ben can tell us what Brans-Dicke theory tells about the relationship between total mass of the universe and inertia of individual objects?

 

[Jonathan Scott]

I remember trying to work out what a universe would look like in a specific Machian model if there was only one substantial mass in an otherwise empty universe. I found that relative to the conventional coordinate system, gravitational potential would vary as r and the speed of light as r². I then found that null geodesics were exact circles which passed through the origin. Eventually, I realized that this was effectively flat Euclidean space with the radial coordinate inverted r -> 1/r. So the result was that to an observer moving around in that space, the mass was at infinity in all directions but otherwise space was flat. Weird!

 

[SW VandeCarr]

For my own and others' convenience I'm posting a link on the Brans-Dicke theory with the field equations located at about the middle of the text.

http://www.answers.com/topic/brans-dicke-theory

 

[bcrowell]

Perhaps Ben can tell us what Brans-Dicke theory tells about the relationship between total mass of the universe and inertia of individual objects?

You can interpret the theory either as a theory in which the gravitational constant G varies from point to point, or as one in which inertia varies. There's no way of distinguishing between the two interpretations, but the description that Brans and Dicke use in their paper (after explaining that the two interpretations are equivalent) is that G varies. In their theory, distant masses in the universe create a field \phi, which is essentially 1/G. If there's more mass within your past light cone, G is smaller. In the varying-mass description, if there's more mass within your past light cone, inertial masses in your neighborhood are bigger.

I remember trying to work out what a universe would look like in a specific Machian model if there was only one substantial mass in an otherwise empty universe. I found that relative to the conventional coordinate system, gravitational potential would vary as r and the speed of light as r². I then found that null geodesics were exact circles which passed through the origin. Eventually, I realized that this was effectively flat Euclidean space with the radial coordinate inverted r -> 1/r. So the result was that to an observer moving around in that space, the mass was at infinity in all directions but otherwise space was flat. Weird!

Cool :-) In the Brans-Dicke paper, they work out an approximate metric that's the equivalent of the Schwarzschild metric in their theory. The perihelion precession of Mercury differs from the GR result by a factor of (4+3\omega)/(6+3\omega). This turns out to be independent of the distribution of distant masses in the universe, for reasons that aren't totally clear to me.

The answers.com link that SW VandeCarr posted is, I think, just the same as the Wikipedia article, which doesn't really make even a token effort to explain the physical motivation for the theory. For anyone who has access to journals, I highly recommend the original paper by Brans and Dicke, which is extremely well written, entertaining, and accessible to non-specialists. It's really a shame that we find ourselves in a legal regime where copyright makes scientific knowledge like this inaccessible, fifty years after the paper was originally published.

 

[edpell]

It's really a shame that we find ourselves in a legal regime where copyright makes scientific knowledge like this inaccessible, fifty years after the paper was originally published.

I would guess it was published in Physical Review? I see nothing in copyright law that forbids PR from putting whatever material it owns and wants to place in the public domain into the public domain(?).

 

[yuiop]

... If there's more mass within your past light cone, G is smaller. In the varying-mass description, if there's more mass within your past light cone, inertial masses in your neighborhood are bigger.

The second description would seem to have more of a Machian flavour and that description seems to contradict the prediction of the double flywheel thought experiment I outlined in #40

Consider two flywheels connected by a motor. Both flywheels have the same radius and one flywheel has ten times more mass than the other and there are no other masses in the universe. When the motor is operated, the small flywheel starts rotating faster than the large flywheel rotating in the opposite direction, as required by the conservation of angular momentum. Obviously the small flywheel has less angular inertia than the large flywheel. Does Mach's principle predict the larger mass of the larger flywheel, causes the smaller flywheel to have less inertia and the smaller mass of the smaller flywheel causes the larger flywheel to have less inertia?

 

[bcrowell]

I would guess it was published in Physical Review? I see nothing in copyright law that forbids PR from putting whatever material it owns and wants to place in the public domain into the public domain(?).

Okay, my political views are OT here, but briefly, I think copyright terms should be a decade or two (as they were in the early years of the US), not a century, so I think a PR article from 1961 should be in the public domain.

The second description would seem to have more of a Machian flavour and that description seems to contradict the prediction of the double flywheel thought experiment I outlined in #40

Hmm...the flywheel example is a non-gravitational example. In terms of a theory of gravity, neither description is more Machian than the other, because they both make the same predictions. In Newtonian mechanics, let's say an object with inertial mass m_i and gravitational mass m_g is being accelerated by a gravitational force. For a fixed value of m_g, the acceleration is proportional to G/m_i. So you can increase G or decrease m_i, and either way you observe the same thing.

I've already messed up a couple of times trying to make qualitative predictions about the results of the Brans-Dicke theory, but I'll go ahead and give this a shot. I think the value of 1/G is sort of determined by an average over your entire past light cone of all the matter (just the surface of the light cone, not the interior). That is, 1/G is the scalar field \phi, and \phi is governed by a wave equation that propagates at c, so the value of 1/G at a particular point samples the wavelets that came out from all those masses on your past light cone. These wavelets have an amplitude that falls off like 1/r. For two extended bodies (as opposed to two point masses), I think the 1/G experienced by each one would be dominated by itself. That is, the 1/G of each atom in object A would be mainly determined by the other atoms in object A lying on its own past light cone.

In your flywheel example, I can imagine two possible ways to apply this:

(1) It's a nongravitational example, so the value of G doesn't matter.

(2) The Brans-Dicke theory is a theory of gravity, and within that theory of gravity, you can also interpret \phi as a local scaling of inertial mass. If you want to extend it to be more than a theory of gravity, then you should use this interpretation. Therefore the more massive flywheel has more inertia, both because it has more atoms and because each of those atoms has more atoms nearby on its past light cone.

Either way, it seems to me that the flywheel with more atoms is also the one with the greater inertia.

 

[Jonathan Scott]

In a Mach's principle universe with few masses, odd things happen to space and time (where the details depend on the specific theory), so you have to be careful about coordinate systems and you may not even be able to find one where space looks anything like Euclidean.

 

[bcrowell]

In a Mach's principle universe with few masses, odd things happen to space and time (where the details depend on the specific theory), so you have to be careful about coordinate systems and you may not even be able to find one where space looks anything like Euclidean.

Yeah, I think you're right. For instance, on the final page of the Brans-Dicke paper, they come back to a thought experiment that they posed near the beginning, which is one of these artificial thought experiments involving a universe that's empty except for a few isolated objects. They show that as the universe gets less dense, the trend of the result is in the direction you expect from Mach's principle, but they say they can't evaluate the limit as the density goes to zero, because it violates the weak field approximations they're using.

 

[edpell]

In a Mach's principle universe with few masses, odd things happen to space and time (where the details depend on the specific theory), so you have to be careful about coordinate systems and you may not even be able to find one where space looks anything like Euclidean.

These are new and interesting idea to me. Can you tell us more? Thanks.

 

[Jonathan Scott]

These are new and interesting idea to me. Can you tell us more? Thanks.

Sorry, not really - I just discovered this myself by taking some Machian theories, of which Brans-Dicke theory is an example, and trying to work out what happens in the limit. In the one case where I could get an exact answer (with a single point mass in a specific toy theory based on Sciama's "Origin of Inertia" paper) it turned out to be equivalent to flat space described in an inside-out coordinate system.

 

[yuiop]

Okay, my political views are OT here, but briefly, I think copyright terms should be a decade or two (as they were in the early years of the US), not a century, so I think a PR article from 1961 should be in the public domain.

Children and grandchildren of authors feel they have the right to an income because they had famous ancestors. They have successfully lobbied to extend the copyright period, because of the undue hardship it would cause them if they had to go out and work for a living like the rest of us.

Actually my pet peeve, is that papers relating to research that has been funded by public money should be freely available to the public, but I digress...

... That is, the 1/G of each atom in object A would be mainly determined by the other atoms in object A lying on its own past light cone.

In your flywheel example, I can imagine two possible ways to apply this:

(1) It's a nongravitational example, so the value of G doesn't matter.
...

The general impression I get from the little information that is available about Mach's principle, is that he meant non-gravitational inertia of objects here is determined by the mass of distant stars. If Brans-Dicke theory does not imply that, then it does not seem truly Machian to me.

Some possible interpretation of Mach's principle:
1)An object that is at rest with the majority of mass, is truly at rest and only has velocity or momentum if it is moving relative to the majority of mass.
2)An object that is at rest in the zero momentum frame of the universe, is truly at rest and only has velocity or momentum if it is moving relative to the zero momentum frame.
3)The amount of energy required to accelerate an initially inertial object is determined by the distribution of mass in the universe.
4)A planet rotating with respect to the "fixed" distant stars is physically identical to a stationary planet being orbited by the distant stars. (Fully relativistic idea.)

1) and 2) imply a preferred frame and is similar to the idea that some people (mostly lay persons) have that the CMB defines a preferred frame.

Actually it would be interested to know how other people interpret what Mach had in mind for his "principle".

I think one clear difference between GR and a Machian model is that GR can define a spinning gravitational object in an otherwise empty universe (the Kerr metric) while the Machian principle would probably not allow such a concept.

...
I think the 1/G experienced by each one would be dominated by itself. That is, the 1/G of each atom in object A would be mainly determined by the other atoms in object A lying on its own past light cone.
...
... Therefore the more massive flywheel has more inertia, both because it has more atoms and because each of those atoms has more atoms nearby on its past light cone.

Either way, it seems to me that the flywheel with more atoms is also the one with the greater inertia.

O.K. I understand what you are saying here and it sort of makes sense. If we stay with methods you have outlined here we can analyse them a new thought experiment. This time we have two connected flywheels of equal mass but different radii. Flywheel A has 10 times the radius of flywheel B and the mass of both flywheels is concentrated at the rims. Now when the motor connecting the two flywheels is started the larger flywheel has the greater moment of angular inertia and spins slower than the smaller flywheel using conservation of angular momentum. Using the inertia is proportional to the 1/r distribution of mass Machian idea, the larger flywheel should have less inertia.

Here is yet another modified form of the experiment. This time there are 3 flywheels of equal mass and radius in an otherwise empty universe. The two outer flywheels are connected by a common axle and rotate in the same direction, effectively as one large flywheel (call this assembly A), while the flywheel in the middle (B) spins in the opposite direction around the axle of the double flywheel. The distribution of mass for the flywheel A from its centre of mass is identical to the distribution of mass from the centre of mass of flywheel B because the locations of the two centres of mass are superimposed. Again, the larger flywheel A has a greater angular inertia than flywheel B, despite the distribution of mass in their universe (including there own mass) is identical from the point of view of either flywheel.


Now I have a further question about the Machian idea. Let us say we have a spinning planet in a universe with a single atom somewhere near the visible horizon of the planet's equatorial plane. The single atom provides a reference point so that we can have a unequivocal notion of rotation. In the rest frame of the atom the planet is rotating at say 1 rpm. According to the fully relativistic Machian concept the planet can be considered to be at rest and the atom is orbiting around the planet at many times the speed of light. In another interpretation of the Machian idea, we could possibly go further and say that since the planet represents the majority of mass, the planet IS at rest. Is the gravitational influence of the rapidly orbiting atom causing the planet to bulge and therefore having an anti-gravitational effect on the planet? Are the Coreolis effects and centrifugal effects experienced on the planet caused by the frame dragging of the single atom? Presumably GR would not allow such an interpretation.

 

[bcrowell]

Is the gravitational influence of the rapidly orbiting causing the planet to bulge and is therefore having an anti-gravitational effect on the planet? Are the Coreolis effects and centrifugal effects experienced on the planet caused by the frame dragging of the single atom? Presumably GR would not allow such an interpretation.

The Brans-Dicke paper poses a very similar thought experiment on the first page, and then claims to more or less resolve it on the last page. However, they don't claim to be able to figure out the limit in which the rest of the universe is empty.

Re the flywheels with unequal radius, again it's probably hard to say for sure in the context of the Brans-Dicke theory, since the limit of an empty universe is apparently intractable.

 

[Al68]

Some possible interpretation of Mach's principle:
1)An object that is at rest with the majority of mass, is truly at rest and only has velocity or momentum if it is moving relative to the majority of mass.
2)An object that is at rest in the zero momentum frame of the universe, is truly at rest and only has velocity or momentum if it is moving relative to the zero momentum frame.
3)The amount of energy required to accelerate an initially inertial object is determined by the distribution of mass in the universe.
4)A planet rotating with respect to the "fixed" distant stars is physically identical to a stationary planet being orbited by the distant stars. (Fully relativistic idea.)

I think number 3 is the only viable interpretation. 1 and 2 are fine, but would constitute only a semantical difference from GR.

4 is problematic because it wouldn't allow for multiple rotating bodies.

My personal thought is that Mach was thinking along the lines of 3, that in addition to normal gravity, mass produces an "inertial field" that provides a resistance to acceleration.

Of course that's problematic, too, if we assume any kind of inverse square law would apply.

But we could suppose that even if the "strength" of the field varies with distance from its source, it's effect is independent of its strength above a certain minimum.

 

[bcrowell]

For anyone who's interested, I've written up a description of the Brans-Dicke theory, the experimental tests it's been subjected to, and what I think that tells us about Mach's principle: http://www.lightandmatter.com/html_books/genrel/ch07/ch07.html#Section7.3