How Does Mach's Principle Connect Distant Stars to Local Motion?

[stevendaryl]

There are many ways to use Mach's Principle, and I do not see any need to assume anything non-local (that is, involving "faster than light" communication or direct "action at a distance").

Well, at first glance (or first think), Mach's principle is saying that only relative motion is meaningful. So rotation or acceleration of a single object doesn't mean anything--the only meaningful notion is rotation or acceleration relative to some other object (the "fixed stars", for example). But if there is no action-at-a-distance, then how can the rotation or non-rotation of distant stars be relevant here? The answer given by General Relativity is that spacetime itself is an entity that provides a reference for acceleration or rotation. Rotation or acceleration is not measured relative to the distant stars, but relative to the local geodesics determined by the metric tensor. So I would say that GR fails to satisfy Mach's principle.

A way to see this is that Mach's principle would say that without the fixed stars to provide a reference for acceleration and rotation, a single rigid object cannot be said to be rotating. However, there are solutions of GR that consist of a single rotating star in an otherwise empty universe. This is a different solution than a single nonrotating star. So rotation makes a difference, even in the absence of distant stars to provide a reference for rotation.

 

[Jonathan Scott]

I totally agree that if spacetime is assumed to be able to exist without the masses that would define it according to Mach's Principle then any such model doesn't satisfy Mach's Principle, and that includes hypothetical GR solutions where the contents of the universe are anything other than the actual contents.

In my preferred model, the shape and scale of space and time is defined by the wave functions of all masses, and how those propagate is defined by the shape of space and time, so the local observation of a remote value of m/r is defined by the frequency of the wave for m and parallax of the wave fronts for r.

I don't think I can get any further into this without getting into details of speculative personal theories, which aren't allowed here.

 

[stevendaryl]

I totally agree that if spacetime is assumed to be able to exist without the masses that would define it according to Mach's Principle then any such model doesn't satisfy Mach's Principle, and that includes hypothetical GR solutions where the contents of the universe are anything other than the actual contents.

In my preferred model, the shape and scale of space and time is defined by the wave functions of all masses, and how those propagate is defined by the shape of space and time, so the local observation of a remote value of m/r is defined by the frequency of the wave for m and parallax of the wave fronts for r.

I don't think I can get any further into this without getting into details of speculative personal theories, which aren't allowed here.

Fair enough, as long as we agree that it isn't GR that you're talking about, but some possible extension or correction or replacement to GR.

 

[PeterDonis]

I totally agree that if spacetime is assumed to be able to exist without the masses that would define it according to Mach's Principle then any such model doesn't satisfy Mach's Principle, and that includes hypothetical GR solutions where the contents of the universe are anything other than the actual contents.

But even in our actual universe there are regions where there are no masses but only vacuum--yet it is not "the distant stars" that immediately determine which states of motion are inertial and which are not in such regions, but the local geometry of spacetime (at least according to GR). So for the interpretation of Mach's Principle that you appear to be using, even the GR solution that describes our actual universe does not satisfy Mach's Principle. In other words, I don't think the interpretation of Mach's Principle that you appear to be using only requires that spacetime can only exist if there are masses somewhere in it; I think the interpretation you are using requires that there is some invariant meaning to "the distance from here to distant masses" at every event, even events in the middle of a vacuum region. GR does not satisfy that requirement.

 

[Dale]

It is a very attractive gem and I find it hard to believe there isn't something important in it, even through it is difficult to see how it can be consistent with the experimental evidence.

In science experimental evidence beats attractiveness and belief.

Brans-Dicke theory effectively attempts to create a Machian theory by adding a controlled amount of "Machian-ness" to GR, but unfortunately the best fit occurs when the controlled amount is set to zero

To me this is a strong reason to reject Mach's principle, regardless of its philosophical appeal.

 

[Jonathan Scott]

But even in our actual universe there are regions where there are no masses but only vacuum--yet it is not "the distant stars" that immediately determine which states of motion are inertial and which are not in such regions, but the local geometry of spacetime (at least according to GR). So for the interpretation of Mach's Principle that you appear to be using, even the GR solution that describes our actual universe does not satisfy Mach's Principle. In other words, I don't think the interpretation of Mach's Principle that you appear to be using only requires that spacetime can only exist if there are masses somewhere in it; I think the interpretation you are using requires that there is some invariant meaning to "the distance from here to distant masses" at every event, even events in the middle of a vacuum region. GR does not satisfy that requirement.

The scale and shape of space and time is determined by the sum of m/r which is generally dominated by distant masses everywhere in the observable universe (except when very close to compact objects), regardless of how empty it is locally.

GR requires additional boundary conditions to match it to the actual universe, and it does not give a reason for the actual value of G. For the actual universe, the sum of Gm/rc^2 is of order 1, which is very Machian and suggests a relationship between G and the distribution of mass in the universe, which might for example turn out to be related to such boundary conditions via additional physical constraints on GR or something similar.

 

[vanhees71]

In GR the full energy-momentum-stress tensor is universally coupled to the gravitational field, not only mass, and gravity is not simply described by a potential which goes like $1/r$ as in Newtonian gravity theory. I don't think that the original version of Mach's principle can be accommodated with the local field-theory picture that's provided by Einstein's theory. Of course, Mach's principle is vague enough so that you can reformulate to match in some sense GR, but what's the merit of this?

 

[Jonathan Scott]

In science experimental evidence beats attractiveness and belief.

It isn't that Machian theories can't match the experiments (specifically to give the same PPN $\beta$ as GR and to give the appearance of constant $G$) but rather that none of the ways of doing so seems particularly natural at present.

GR is neat but it is only part of a theory, requiring additional constraints and parameters. Machian theories also naturally extend to the scale of the universe and explain inertia, mass and the value of G as relative effects with no additional parameters. So in many ways the Machian theories seem to give more value for less complexity. However, the simplest Machian theories are too simple in some ways theoretically and also conflict with experiment, so one needs to fill in some more detail. These problems are only at the post-Newtonian level, relating for example to non-linearity due to the gravitational effect of potential energy. Although one can arbitrarily choose parameters to match experiment, this undermines the compelling simplicity of the Machian theories.

 

[vanhees71]

How do you come to the conclusion that "GR is neat but it is only part of a theory, requiring additional constraints and parameters"? I don't know any example, where GR fails to predict observations. So where does it need additions?

 

[Jonathan Scott]

How do you come to the conclusion that "GR is neat but it is only part of a theory, requiring additional constraints and parameters"? I don't know any example, where GR fails to predict observations. So where does it need additions?

Firstly, GR solutions require assumed boundary conditions, even at the scale of the universe.
Secondly, in GR the constant G is effectively an arbitrary parameter which is matched to experimental observations.

 

[vanhees71]

Any field theory includes appropriate initial and boundary conditions, and of course $G$ is a parameter which is mathced by experimental observations. So what?

 

[Jonathan Scott]

Any field theory includes appropriate initial and boundary conditions, and of course $G$ is a parameter which is mathced by experimental observations. So what?

In basic Machian theories, $G$ is effectively an abbreviated way of representing the relative distribution of mass in the universe, rather than being an arbitrary parameter. For boundary conditions, I can't be so specific without getting into speculative details, but I feel that without additional rules for the boundary conditions GR has exotic but implausible solutions which simply don't arise in Machian theories because the boundary conditions are simpler.

The simplest Machian theories clearly diverge from GR in one important aspect; they don't quite lead to black holes, in that the metric factor for the time dilation is not of the form $\sqrt{1-2Gm/rc^2}$ but rather typically approximately $1/(1+Gm/nrc^2)^n$ where $n$ depends on the specific theory (and this expression is not necessarily in Schwarzschild coordinates).

 

[Dale]

For boundary conditions, I can't be so specific without getting into speculative details,

This seems like a good point and a good reason to close the thread.

IMO, your defense of Mach's principle sounds like other people's defense of the luminiferous aether. If there arises any compelling scientific evidence to support it then we can open a new thread at that time.