Exploring Mach's Principle: Connecting Local and Distant Events

[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.