History of theories of motion; the role of inertia

[Cleonis]

The force measured by the thruster unit is equal to the momentum transferred by it to the ISS per unit proper time,

Naturally I agree that the propulsion from a thruster unit goes back to conservation of momentum: a rocket ejects small amounts of mass at extremely high velocity from its exhaust nozzle, resulting in acceleration of the rocket in the opposite direction.

We can attach that rocket to a larger object, such as ISS, to accelerate it. We can gage the force that is exerted upon ISS independently. For instance, we can use a force gage that works with a coiled spring. Or we can use properties of gas for force calibration. When the volume of a gas is halved its pressure doubles (to a first approximation, more accurate models are available). Bottom line: the way the force gage is calibrated is independent from the operation of the thruster unit.

Assume the thruster unit can vary the nozzle velocity of the exhaust (In other words, it can vary how hard the exhaust is accelerated.) We find that the force that is exerted is proportional to how hard the exhaust is accelerated. We find F=m*a

It's not clear to me how that proportionality could be seen as a matter of definition.

Cleonis

 

[Jonathan Scott]

I think that what I was referring to as the "coupling constant" is what you are referring to as "perhaps a small dimensionless numerical constant", depending on where you think the latter fits in. What I had in mind was that, as I said, the "gravitational force" has to somehow be modeled as a field on Minkowski spacetime, and there should be some coupling constant associated with that field. The value of that coupling constant would affect the numerical value found for G (in the formula you give, I would expect the field coupling constant to affect the value of n).

As a specific example, the value n = 1/2 combined with Marcel Brillouin's radial coordinate R = r - 2Gm/c² (where r is the Schwarzschild radial coordinate) gives a Machian expression for G which is still compatible with GR locally. That is, even if we assume that the global value of G varies in a Machian way, the Schwarzschild solution to the Einstein Field Equations still holds exactly for a single mass with G effectively being a constant. I previously described this result (which I found quite surprising) in more detail in this thread: https://www.physicsforums.com/showthread.php?t=206922"

 

[PeterDonis]

Assume the thruster unit can vary the nozzle velocity of the exhaust (In other words, it can vary how hard the exhaust is accelerated.) We find that the force that is exerted is proportional to how hard the exhaust is accelerated. We find F=m*a

It's not clear to me how that proportionality could be seen as a matter of definition.

In this case you would just do the same momentum balance between the thruster unit itself (more precisely, the nozzle, since that's what the exhaust pushes against) and the exhaust gas. Same result.

Let me try approaching this a different way. If you're surprised that F = ma always seems to hold, you ought to be able to describe a scenario, a way things could be, an alternate set of laws, in which F = ma would not always hold. Can you describe such an alternate set of laws, in which F = ma doesn't always hold, but conservation of energy and momentum still do? In other words, can you describe a way in which it could be possible that the two things I'm saying are connected (F = ma and conservation of momentum) become different?

 

[Cleonis]

Can you describe such an alternate set of laws, in which F = ma doesn't always hold, but conservation of energy and momentum still do?

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.

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.

Cleonis

 

[Cleonis]

Well, Mach's principle is certainly relevant, since if it's true then there is no need to view inertia as an innate property of objects; inertia is just the resultant of effects on the object from external sources.

Then again, also without Mach's principle there is in itself no need to view inertia as an innate property of objects. As far as I can tell those are independent issues. (But maybe you and I have different things in mind when using the expression 'viewing inertia as an innate property of objects'.)

I'm not accustomed to considering Mach's principle. I know different versions of Mach's principle are in circulation. The idea never appealed to me, and since attempts to implement some version of Mach's principle have remained unsuccesful I saw little reason to give Mach's principle any thought.

In the outline of a Machian theory published by Dennis Sciama Maxwell's equations are used to get the exploration going.

I would like to check some things, to see if I understand them correctly.
- In Sciama's exploration each point mass in the Universe is taken as the source of a field. At each point in the Universe the local field that a test mass experiences is a superposition of all fields in the Universe. I will refer to that superposition of fields as 'the resultant field'.
- When the test mass has a uniform velocity with respect to the local resultant field all effects still cancel out. (Hence the resultant field features relativity of inertial motion.)
- When the test mass is accelerating with respect to the local resultant field there is an effect. To accelerate with respect to the local resultant field a force is required and Maxwell's equations (as implemented in Sciama's exploration) imply that the required force is proportional to the acceleration.

Those are the things I would like to check for correctness.

Cleonis

 

[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.