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

[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