The Amelioration of Uncertainty

What’s your connection to the distant stars?

Many have thought motion somehow required the presence of other objects. Just like gravity requires two masses to exist, it seems like motion is meaningless in a universe with just one mass. This suggests inertia itself somehow requires other masses and perhaps results from an interaction with them.

This is bolstered by the otherwise amazing coincidence that the inertia mass, which appears in equation, is numerically the same as the gravitational mass equation. Newton himself knew these were equal to about 1 part in a thousand, but didn’t believe inertia came from particle interactions. Rather he believed inertia was a property of absolute space and would exist even if there were only one mass.

To see why consider Newton’s bucket experiment. Rotate a bucket at a steady angular velocity long enough for the water inside to attain the same speed, and the water will rise up the sides of the bucket.

NewtonsBucket

This has to be caused by the water’s rotation with respect to something, but what? To cut a medium story short, there are two suspects. One is the rotation with respect to the fixed stars and the phenomenon is a byproduct of the water’s interaction with them. Or it’s the result of rotation with respect to absolute space. Newton had good reason to reject the former.

For Newton, the interaction with the fixed stars would be through gravity, but he knew the inverse square law of gravity,

    equation

has a very peculiar property. The net gravitational force exerted by a uniform shell on a mass inside is zero. So if the fixed stars are randomly sprinkled about, their net gravitational effect on the water is zero, and the bucket experiment can’t be a result of the water’s interaction with distant stars.

This property of the inverse square law was very familiar to Newton. After deriving the inverse square law by working backwards from Kepler’s laws, Newton faced a new difficulty. To get the net gravitation attraction between say the earth and moon, you had to add up the attraction between every particle in the earth with every particle in the moon. Newton showed that for the inverse square law, this was the same as a simple interaction between two point masses, one with the earth’s mass and one with the moon’s. The technical fact about uniform shells would have made itself known to anyone working on such a derivation.

An interesting twist to this story occurred in 1925 when Schrödinger[Ref 1] showed how it’s possible to square “relativity” with Newton’s results by reviving the idea that inertia resulted from interactions with the fixed stars. He gave a toy example of a purely relational gravitational force which once the interaction of a local mass with the distant matter was taken into account would reduce to the usual Newtonian gravity plus an “inertia” term F=ma measured with respect to the fixed stars. The toy example had the form:

    equation

It’s clear from V that Schrödinger chose the simplest generalization of Newton’s law of gravity which could have yielded such a result. Unknown to Schrödinger though, this force law has quite a history.

This was the force of interaction between two charged particles in Weber’s Electrodynamics with equation. Weber’s theory was supplanted by Maxwell’s equations, but Weber’s work explained a surprising amount of Electrodynamics. In particular, the discovery that the speed of electromagnetic waves equation is numerically the speed of light, was found by Weber 12 years before Maxwell. But in the end Weber’s theory isn’t correct, so it’s interesting to see what Electrodynamics proper has to say.

Enter Feynman. In 1949, Feynman and his thesis advisor Wheeler published a paper on Electrodynamics, which lead according to his 1965 Nobel lecture directly to the discovery of the Feynman Path Integral. The basic idea is simple and pleasing. Ditch the field concept and work with a delayed action-at-a-distant force. The field emerges as a kind of accounting ledger used to keep track of past motions. In an offhand comment Jaynes (page 3) described the idea thus:

… the Wheeler-Feynman electrodynamics, in which the EM field is not considered a “real” physical entity in itself, but only a kind of information storage device invented by us. That is, the present EM field is a “sufficient statistic” which merely summarizes all the information about past motion of charges that is relevant for predicting their future motion.

Which hints there may be deeper connections between statistics and classical physics than currently known. Feynman ran into one problem though. To make it work, he had to use the average of the advanced and retarded potentials. The retarded potentials have never been a problem since they use past locations of charges, but the advanced potentials depend on future locations.

The usual strategy for dealing with these causality issues is to avoid advanced potentials entirely. Unable to do that easily, Feynman found another solution. In the section “The Paradox of Advanced Actions”, Feynman presents his solution to the problem:

Advanced actions appear to conflict both with experience and with elementary notions of causality. Experience refers not to the simple case of two charges, however, but to a universe containing a very large number of particles. In the limiting case of a universe in which all electromagnetic disturbances are ultimately absorbed it may be shown that the advanced fields combine in such a way as to make it appear – except for the phenomenon of radiative reaction – that each particle generates only the usual and well-verified retarded field. It is only necessary to make the natural postulate that we live in such a completely absorbing universe to escape the apparent contradiction between and advanced potentials and observation.

It’s a plausible assumption given that when looking at the night sky our eyes absorb light some of which was emitted millions of years ago. Here again though is the notion that the local physics we see is warped by our interaction with the fixed stars.

So what do present day Physicists say about all this?

I don’t know. There could be thousands of papers on the subject and I haven’t read one of them. I doubt it’s favorable though. If someone had observed say micro changes in local physics due to the non-uniform distribution of the fixed stars, then we’d probably have heard about it.

Still, it’s a simple old idea that never quite goes away. And it would not only explain why the gravitation mass equation equals the inertial mass equation, but also why the angular velocity of our rotation with respect to the fixed stars equation is the same as the earth’s rotation with respect to inertial frames equation. By the way, the current experimental tolerances [Ref 2,3] for these differences are:

    equation

REFERENCES:
(1) Die Erfüllbarkeit der Relativitätsforderung in der klassischen Mechanik
Annalen der Physik, (4), 77, (1925), 325-336

(2) P.W. Worden and C. W. F. Everitt. Resource letter GI-1: gravity and inertia.
American Journal of Physics, 50:494-500, 1982

(3) L. I. Schiff. Observational basis of Mach’s principle.
Reviews of Modern Physics, 36:510-511, 1964

(4) See especially Relational Mechanics

November 26, 2013
9 comments »
  • November 27, 2013Daniel Lakeland

    nice post and nice to see another topic. The idea that motion requires other masses is interesting. My vague understanding of gauge theory is that it involves measuring space by the presence of other masses.

    As for newton’s rotating bucket experiment. Clearly, the bucket is rotating relative to the earth, and without the gravitational field of the earth, the water would not sit at the bottom of the bucket, and would not crawl up the sides when rotated. An interesting thought experiment would be to take water in a clear bucket with a lid into interstellar space and simply observe it. You couldn’t rotate the bucket without a torque provided by interaction with some mass, such as your space ship. In many ways the distant stars as a newtonian reference frame makes a lot of sense, a sort of statistical average of all the angular momentum in the universe.

    Interestingly, Newton’s laws do not imply conservation of angular momentum without the additional postulate that all forces are central forces. Noether’s theorem says that if the laws of motion are invariant to choice of coordinate system, then there’s a conserved quantity associated with the invariance and this is angular momentum. it seems reasonable to simply postulate the conservation of angular momentum rather than central forces in Newtonian approximations.

  • November 29, 2013Joseph

    Daniel,

    The water would go up the sides in space due to what we’d interpret as centrifugal forces. The shape wouldn’t be the same without gravity though. Interestingly, I saw Elysium over the weekend. The rotating space station didn’t have a contained atmosphere. It was held in place by the rotation. So ships could fly in and land without docking or entering a chamber of some sort.

    That rotation would have to be with respect to a inertial frame though, which in practice means with respect to the fixed stars. You could imagine a station roughly perpendicular to the axis of the earth’s rotation which was at rest with respect to the fixed stars, which would seem to be rotating to an earthbound observer. The atmosphere would not stay in place in this scenario.

    About conservation of angular momentum: in continuum mechanics the conservation of angular momentum really is a different principle as you say. And historically it was definitely seen as such. Physicists tend to think of it as a derivative principle because of their experience with “Classical Mechanics”, which really means “the mechanics of point particles”.

    As you say, from Noether’s Theorem if there is a rotational symmetry in the Lagrangian (which there will be for central forces), it immediately implies conservation of angular momentum.

    Gauge theory originally referred to Electrodynamics. Since only derivatives of the Electric and Magnetic potentials enters in Maxwell’s equations, they are not uniquely specified. It’s like saying if only f’(x) enters into physical equations, then any f(x)+c will do and we have freedom to choose c. Or you might say the answers are invariant with respect to c. In general, Gauge invariance of a lagrangian is enough, by Noether’s Theorem again, to imply some kind of conservation law.

  • November 29, 2013Brendon J. Brewer

    My theoretical physics foo has slipped a little, so apologies if I make any dumb statements.

    “To cut a medium story short, there are two suspects. One is the rotation with respect to the fixed stars and the phenomenon is a byproduct of the water’s interaction with them. Or it’s the result of rotation with respect to absolute space.”

    I’ve always thought Mach’s “principle” was highly suspect, but I don’t think absolute space is the only alternative. Absolute space isn’t a thing in relativity, but it makes the same predictions using the concept of non-inertial reference frames. Then the mystery reduces to “if I define my positions WRT the distant stars, why do I end up with an inertial frame?” Also I thought GR explained why inertial mass = gravitational mass? Maybe I need to study physics again.

  • November 29, 2013Daniel Lakeland

    With the gravitational field of the earth, water in a sealed cylinder will sit at the bottom of the cylinder, and water vapor will be above. Let’s let the axis of the cylinder point towards the center of gravity of the earth so that we can rotate it without problems. In this scenario water rises up the sides and forms a more or less parabolic surface.

    Now remove the earth and the sun, put the cylinder in interstellar space. Water will form some other shape. Interactions with the material of the cylinder will significantly affect things. Let’s suppose it’s highly hydrophobic. Depending on the volume of the fluid, the shape will be different. Let’s suppose it’s sufficient that the fluid doesn’t form a ball floating in the middle of the cylinder. I suspect you’ll wind up with something like a cylinder with two convex meniscus caps. Now rotate the cylinder. What happens to the shape of the slug of water?

    It’s not at all clear to me that it will be different. The only potential energy that enters in here is the surface energy, and the static meniscus already minimizes that. If we assume there is a steady state, then one important question is whether that steady state has a constant angular velocity or whether there is shear in the fluid. let’s suppose there’s a constant angular velocity, then we can change into a rotating coordinate system where we are back to the static problem, and so the shape won’t change…. if it’s not a constant angular velocity, then there is no such coordinate system. Also, due to shear related dissipation (heating) you will have to constantly add external energy to keep things going. in this case it’s pretty obvious that you’re interacting with whatever external power source is involved, not the far off stars.

  • November 30, 2013Joseph

    Brendon,

    No I don’t think absolute space is the only alternative either. Of course Einstein was explicitly motivated by Mach critique of mechanics, but his solution doesn’t appear to be in complete agreement with it. There were attempts at truly relational theories of electrodynamics, which presumably would have accorded with Mach better, but they seem to have failed. Here’s one:

    http://en.wikipedia.org/wiki/Emission_theory

    But don’t read this discussion as an attempt to explain a physical theory or to expound a viewpoint. It’s more like groundwork needed if one wanted to create similar but slightly different theories in order to suggest interesting experiments. Nothing more.

  • November 30, 2013Joseph

    Daniel,

    Ah, I see what you’re getting at. You really are an engineer! All that’s needed to illustrate the point is that something (a fluid in this case) experiences a centrifugal force. It’s irrelevant to the point how the rotation was created. My comments about the shape of the water were under the assumption the water’s already rotating with constant angular velocity somehow and the shape is stable.

    If you consider the atmosphere in that spoked-wheel space station in the movie Elysium, it’s held in place by the rotation. Even if it was outside the solar system the rotation would still hold the atmosphere in place just the same.

    The torque needed to create the rotation could have been provided by booster rockets thereby avoiding any counter-torques. Once rotating it would continue to do so for a very long time, just like the earth has been spinning in space for a very long time. Getting the atmosphere to rotate with the station could have been achieved in any number of ways.

    But this is getting way far away from the point. It’s the mere existence of centrifugal forces when rotating with respect the fixed stars that’s relevant for the post. So forgetting about the fluid atmosphere for a moment, and look at the individuals on the outer ring. If the rotation is right they’ll experience 1g of force against the ring. This is just the old idea of using rotation to simulate gravity in space. That rotation has to be rotation with respect to the fixed stars though, not the earth.

  • November 30, 2013Joseph

    Incidentally, I met a very senior Military NCO a few years ago named Mach. I asked him if he was any relation to the big Mach and he claimed to be a direct descendant. He knew all about Mach’s relation to Einstein which is pretty unusual for an NCO who probably never went to college.

    He also mentioned that this made him a distant relative of Marilyn vos Savant the high IQ lady. I though for sure he was pulling my leg, but then I looked it up:

    http://en.wikipedia.org/wiki/Marilyn_vos_Savant#Biography

    Sure enough her maiden name was indeed Mach and she was related to the big one.

  • December 1, 2013Daniel Lakeland

    I guess what I was trying to get at is that it’s not irrelevant how the thing comes to be rotating if you want to argue that motion requires other masses by which to measure it. I like the idea that averaging over the motion of the far off stars is like averaging over all the finite angular momentum in the universe. Relative to that average, if you rotate some body in some way it must by conservation of angular momentum, cause the rest of the universe to rotate the other way. Of course, I don’t believe you interact directly with the far off stars, so I would say that how you come to be rotating is very important, because whatever reaction mass you interacted with must have the other chunk of angular momentum that ensures the average stays constant.

    You were referring to the question of why does the person in the spaceship (or the water molecules or whatever) experience centrifugal force when they are rotating relative to the far off stars. And why do they not experience centrifugal force when someone watching them rotates themselves?

    It’s an interesting idea that eventually every tiny quantity of angular momentum thrown off by one object must be absorbed by something else and so that something else *will* eventually feel some kind of torque (or if not that something else, then some other something else). The fields we invent do become an enormous accounting convenience.


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