History of theories of motion; the role of inertia

[PeterDonis]

I would like to check some things, to see if I understand them correctly.

Check post #24; I posted a summary there of how Machian theory handles these things. The basic idea is that, if a test body is in free fall, the total "resultant field" on it must be zero, because it feels no force. When there's no massive body nearby, the resultant field is entirely due to distant matter in the universe, and that turns out as you say: bodies in uniform motion are the ones for which the resultant field cancels out and no force is felt.

When there *is* a massive body nearby, it turns out that for the total resultant field to cancel out, so the test body is in free fall, the test body must accelerate towards the massive body. If we want to attribute that acceleration to the gravity of the nearby massive body alone, then we can write down the equation for the total resultant field to be zero, rearrange some terms, and find that the acceleration of the test body is

$$a = - \frac{G M}{r^2}$$

where M is the mass of the nearby massive body, r is the distance to it, and G is an "effective" gravitational constant which arises from the effects of the rest of the mass in the universe.

The derivation of the above in Sciama's paper does, as you say, use Maxwell's equations instead of the "correct" tensor equations for the gravitational interaction. As far as I know, nobody has ever actually done this type of analysis using a tensor field for gravity. Sciama's paper references a second paper he planned to publish that would do that, but apparently he never actually did.

 

[PeterDonis]

That's an interesting angle. F=m*a is a linear law. Would any other law be compatible with the principle of relativity of inertial motion? The very essence of the galilean and Lorentz transformations is that they are linear transformations. Intuitively I'd say the principle of relativity of inertial motion rules out any non-linear law, leaving F=m*a as the only possibility.

That's kind of what I was getting at. I'm wondering, though, whether it would be possible to gin up a theory that had the rest mass m be a scalar function instead of just a constant for each object. If it was a scalar function it would still be invariant (i.e., the same in all frames), and it might be possible to maintain the principle of relativity.

(Actually, now that I think about it, something like this happens with bound systems in a potential well--I think Jonathan Scott already mentioned that in this thread. A bound system can have a smaller rest mass, when seen "from the outside", than the sum of the rest masses of its parts when the parts are separated, because the bound parts are in a potential well, so the negative potential energy associated with the binding cancels out some of the rest mass.)

Tying in F=m*a with the conservation principles is interesting. It's not explanation in terms of a deeper theory, but the interconnection is nice.

The conservation laws, if not exactly a "deeper theory", certainly would seem to count as deeper principles.