B Is Inertia Tied to the Stars According to Mach's Principle?

[timmdeeg]

After rereading your post 33

But it isn't just empty universes that is an issue for Mach. One might oversimplify by saying any universe not matter/energy dominated is primarily determined by boundary conditions. ... Einstein's doubts focused precisely on the importance of boundary conditions in GR. He hoped, initially, that there would be no freedom to choose these, or, at least they would be highly constrained. His Machian doubts precisely focused on this key question of boundary conditions.

Consider any asymptotically flat universe, alternatively any open asymptotic geometry not determined by the matter content. Then the inertial structure (the answer to bucket questions) is not primarily determined by matter. Just because these don't match our universe, is it really correct to say the are edge cases in GR? At best one might claim that our universe as modeled in GR is Machian, but not that GR per se is a Machian theory.

I think to have a better understanding of Einstein's doubts.

To put it as simple as possible could one say the existence of any inertial structure (which requires defined geodesics) is inseparably linked with the rising water question in the bucket, so that there is no need to "show" that (here remembering Max Jammers wording "it could be shown ...)?
Then it should make no difference if we talk about the asymptotically flat Schwarzschild spacetime or about flat Minkowski spacetime. If true whether or not there is a history of matter (to mention Ken G's idea) doesn't make any difference either.

 

[timmdeeg]

Objects don't follow non-geodesic paths because of "inertia". They follow non-geodesic paths because a force is exerted on them. (In GR, gravity is not a force.) An object's "inertia" (more properly it's "inertial mass") determines the amount of force that is necessary to make it deviate from a geodesic by a particular amount, i.e., to give it a particular proper acceleration.

Ok thanks for clarifying.

 

[Ken G]

Einstein hoped he would be able to do away with or highly constrain freedom to choose boundary conditions. If you have no choice in boundary conditions, then the theory is determining the inertial structure rather than something arbitrary. He was very disappointed at how significant boundary conditions remained, and considered this to spoil the idea of GR being truly Machian.

Indeed, and here is where I think Einstein might have been using too specific a version of "Mach's principle." It seems to me the essence of that principle is simply "dynamical history determines the geodesics," as in, the geodesics are not handed to us by some absolute geometry (as Newton thought), but rather, they are part and parcel of the dynamics that involves the devations from them. This is what I would call the "big idea" of looking to the distant stars to understand local inertia, the idea that you need information to know the local geodesics, and that information involves everything that is within your past light cone, including boundaries (after all, what is a "boundary" if not simply a proxy for everything that has gone before that you are choosing not to explicitly include?). So when framed that way, there is no reason to wish to elevate the effects of matter to some higher status than the effects of boundary conditions, it's all about what conditions the geodesics as emergent phenomena rather than dictated by mystery or fiat. In that sense, the ambiguities in the geodesics in an empty universe that foists it all onto the boundary conditions is not a problem for Mach, it is a problem for the universe itself-- a universe in which nothing happens becomes a universe in which it is hard to talk about what will happen, that doesn't sound like a problem.

 

[Torog]

In post N.4 Chronos sent a link to a paper by D. W. Sciama which agrees nicely with my non-academic (a polite way of saying it) world view.

You are correct in that GR is not fully consistent with Mach's principle regarding inertia. Einstein himself acknowledged this fact. The seminal paper; On the Oriicely gin of Inertia, by Sciama: http://articles.adsabs.harvard.edu/...=2&data_type=GIF&type=SCREEN_VIEW&classic=YES, has a nice summar. You may also find this discussion of interest; A Look at the Abandoned Contributions to Cosmology of Dirac, Sciama and Dicke, https://arxiv.org/abs/0708.3518

From Sciama's paper:

“If the rest of the universe determines the inertial frames, it follows that inertia is not an intrinsic property of matter, but arises as a result of the interaction of matter with the rest of the matter in the universe”

and from his calculations:

“This means that the main contribution comes from distant matter – (I) shows that 99 percent of local inertia arises from matter further away than 10⁸ light years.”

I present this as a bit of a poll to understand if any (or all) of the esteemed contributors to this discussion agree with this view.

 

[PeterDonis]

From Sciama's paper

Sciama's paper was using a simplified model of gravity, which was known even at the time to be incorrect; he just used it to illustrate the kind of thing he was talking about. He said he was working on applying his ideas to a more realistic model of gravity, but AFAIK he never completed that or published anything based on it.

from his calculations

Which, as above, should not be taken as a claim about actual numbers, just as an illustration of a general method he was proposing, but which he never actually developed to the point where it could be used with a realistic model of gravity.

 

[Torog]

Sciama's paper was using a simplified model of gravity

The paper is about the origin of Inertia. I suppose, however, with the principle of equivalence you can interchange gravity and inertia at will.

And you are right he never came up with his promised complete theory.

 

[Ken G]

One way to focus the discussion is point out the empirical fact that if we enter a frame that is both distant from any known sources of forces, and where the distant galaxies have no net rotation around us, and no net acceleration in a particular direction, then this is also the frame in which geodesics are consistent with our concept of no net forces. In short, in a frame like that, inertia behaves in the way Newton expected on small scales (say, water in a bucket). This is an empirical fact, and we have two basic approaches for addressing it. The first I would call Newtonian, the second, Machian:
1) (Newtonian) Our local inertia, and the behavior of the distant galaxies, all respect the same intrinsic geometry that is handed to us by fiat. This is the flavor of Newton's first law, and it does indeed require its own law to stipulate the intrinsic character of geodesics. It has the same flavor of when the Greeks thought orbits had to be perfect circles, as that geometry was in some sense "intrinsic" to how motion through space worked.
2) (Machian) Our local inertia is coupled to the behavior of the distant galaxies, the latter (along with any boundary conditions that are proxies for the influences we are choosing not to treat in detail) unfolding dynamically in a unified and self-consistent way to how the geodesics behave. This has the flavor of a dynamical theory rather than an assertion by fiat of some kind of intrinsic character of geodesics.
So framed like that, I cannot see how GR does not take the second perspective. It seems to me the issue basically comes down to whether you put gravity into a modification of Newton's first law, or if you put it into the second law as a particular type of force. I realize that many physicists are attempting to treat gravity as the fourth force, and model gravitons and so forth, which would be fundamentally non-Machian because it would place gravity on the same footing as other forces and thus place it in the second law. But it seems to me that this is not what GR does, so GR should therefore be regarded as Machian. When people say it is technically not Machian, it must therefore be some more specific, and I would argue less useful and less generalizable, version of Mach's principle than the way I framed it above.

 

[PeterDonis]

The paper is about the origin of Inertia.

Yes, but his model for how inertia is produced was that it is due to the interaction of a given piece of matter with all the rest of the matter in the universe, and the interaction he used was a simplified model of gravity.