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Looking at older books, I was surprised to see that the general relativity "bible" by Misner, Thorne, and Wheeler is very strongly in favor of Mach's principle, which is treated in section 21.12.

They begin by quickly dismissing an "enormous literature" of "anti-Machian" papers, because

[...] most of them were written before one had anything like the understanding of the initial-value problem that one possess today.

Next, they formulate Mach's principle in a mathematically concrete way:

Specify everywhere the distribution and flow of mass-energy. [...] These initial-value data now known, the remaining, dynamic components of the field equation determine the 4-geometry into the past and the future. In this way, the inertial properties of every test particle are determined everywhere and at all times, giving concrete realization to Mach's principle.

In other words, they say that Mach's principle is just the same thing as being able to compute time evolution given initial value data. This seems to me to be an extremely weak, nearly trivial interpretation of the principle. One could use the same argument to say that charges in a distant galaxy "determine" the motion of charges on Earth, because electromagnetism has a well-defined initial value problem, a statement which is equally trivially true and totally inconsequential.

However, MTW then go on to make their statements more quantitative by invoking the Lense-Thirring effect. My impression has always been that the tiny magnitude of this effect shows that general relativity is only very slightly Machian. However, MTW claim that it is enough to make relativity completely Machian. They show that for a shell of radius $R$ and mass $M$ rotating with angular velocity $\omega$, the frame dragging frequency is $$\omega_{\text{drag}} \sim \frac{M}{R} \omega \tag{21.155}.$$ They then write the total frame-dragging effect of the universe as $$\omega_{\text{drag}} \sim \sum_{\text{stars}} \frac{M_{\text{star}}}{R_{\text{star}}} \omega \sim \frac{M_{\text{universe}}}{R_{\text{universe}}} \omega.$$ Finally, they claim rather vaguely that general relativity only makes sense if the universe is closed, which apparently requires $M_{\text{universe}} \sim R_{\text{universe}}$, giving $$\omega_{\text{drag}} \sim \omega.$$ Hence the Lense-Thirring frame-dragging effect alone makes Mach's principle work. There is a lot of "reasoning by $\sim$" in these pages that I can't really follow.

The book by MTW is now almost 50 years old. How are these two arguments generally viewed in the literature today? Do people accept MTW's formulation of Mach's principle, or their frame-dragging argument?


Edit: here is MTW's argument for $M_{\text{universe}} \sim R_{\text{universe}}$ in full.

Just such a relation of approximate identity between the mass content of the universe and its radius at the phase of maximum expansion is a characteristic feature of the Friedmann model and other simple models of a closed universe.

At phases other than the stage of maximum expansion, $R_{\text{universe}}$ can become arbitrarily small compared to $M_{\text{universe}}$. Then the ratio $M_{\text{universe}}/R_{\text{universe}}$ can depart by powers of ten from unity. Regardless, one has no option but to understand that the effective value of the "sum for inertia" is still unity after all corrections have been made for the dynamics of contraction of expansion, for retardation, etc. Only so can $\omega_{\text{drag}}$ retain its inescapable identity with $\omega$.

I have no idea what they mean by the "effective value", and they really seem to be just assuming the conclusion they want. I really can't follow the reasoning here, but perhaps somebody else can.

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  • $\begingroup$ Is "they don't explain at all why $M_\text{universe} \sim R_\text{universe}$ follows from a closed universe" a proper interpretation of your "which apparently requires"? It kinds seems this assertion is the crucial link in this argument, at least from your presentation of it, so it would be good to be explicit about what their argument for it is (if there is one), however weak. $\endgroup$ – ACuriousMind Feb 28 at 18:45
  • $\begingroup$ @ACuriousMind Added their argument in full; I didn't type it in the original question because I thought it was overly wordy and quite weak, but maybe it'll make more sense to somebody else. $\endgroup$ – knzhou Feb 28 at 18:55
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This section of MTW is not really a coherent argument, it's more like an outline of a research program. They were writing during the 1960's (finished writing in 1970-71). They say, "Much must still be done to spell out the physics behind these equations and to see this physics in action." They were anticipating that these theoretical investigations would be carried out during the 1970's.

A lot of work really was done on this during the 1970's. A nice nontechnical history of this period is given in Will 1986. At least some of what MTW talk about in this section just turns out to be wrong. They make an argument that the universe has to be closed, but in fact the evidence has now converged on the fact that the universe is very nearly spatially flat.

Now that the 1970's are in the past rather than the future, there is a different picture that represents sort of a rough consensus in the literature of what Mach's principle means and what its empirical status is. Basically there were two prominent test theories, which were GR and Brans-Dicke gravity. If you ask a specialist today how Machian these theories are, I think they will say essentially that GR is less Machian, and Brans-Dicke is more Machian, but neither is absolutely Machian or non-Machian. So to the extent that anyone has any well-defined way of describing Mach's principle today, all we can really say is that if Mach's principle is right, the universe should behave more like Brans-Dicke gravity than GR. But actually the experimental work of the 1970's has shown that the universe behaves more like GR than Brans-Dicke (i.e., $\omega\gg 1$), so the universe isn't very Machian.

In other words, they say that Mach's principle is just the same thing as being able to compute time evolution given initial value data.

I don't think this is quite what this passage is saying. I think they have in mind that there should be some restriction on the type of initial data that's required (hence their long justification under point (6)), and they seem to have in mind some strengthening of what people today normally expect GR to predict (hence the requirement of a closed universe). That is, they describe Mach's principle as being able to do more with less.

Will, Was Einstein right? Putting general relativity to the test, Basic Books, 1986

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  • $\begingroup$ Thanks for this very nice answer! Is it possible to say a bit more about what people mean by Machian today, in the mathematical sense? That is, given the 100 modified gravity theories out there, is there a quantity I could compute for each that would roughly measure how Machian people would consider it to be? Or is that kind of question not relevant anymore, i.e. is Machian-ness viewed as an outdated philosophical criterion, when we can just deal with each of these theories mathematically? $\endgroup$ – knzhou Mar 1 at 0:54
  • $\begingroup$ @knzhou: I'm not a specialist, and I suspect that specialists may have many different points of view. But the impression I get is that Mach's principle is still viewed as sort of an important "meta-theory" or aesthetic criterion, although one for which nobody claims to have a satisfying ultimate definition. Most non-GR theories are primarily intended to do something other than embody Mach's principle more nicely. They may be intended to be easier to quantize, or to explain supposed anomalies, or to explain accelerated cosmological expansion without $\Lambda$, whose small value is mysterious. $\endgroup$ – Ben Crowell Mar 1 at 2:12
  • $\begingroup$ Mach's principle has never been formulated clearly and rigorously. There are many authors with different interpretations of what "Mach's principle" should be. So no wonder that GR isn't fully machian! I suggest that the OP consider these references : arxiv.org/abs/physics/0407078 and arxiv.org/abs/1007.3368. Weinberg's book also has a nice discussion on the blurry "Mach's principle". $\endgroup$ – Cham Mar 1 at 2:35

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