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 , is numerically the same as the gravitational mass . 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.
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,
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:
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 . 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 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 equals the inertial mass , but also why the angular velocity of our rotation with respect to the fixed stars is the same as the earth’s rotation with respect to inertial frames . By the way, the current experimental tolerances [Ref 2,3] for these differences are:
(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