Mach's Principle: Explaining the Force We Feel When Changing Velocity

[SW VandeCarr]

Is Mach's principle still the best explanation for the force we feel when changing velocity? If the universe were infinite with infinite homogeneously distributed matter, would Mach's principle still apply?

EDIT: If it's not, how is the force associated with acceleration explained?

 

[turbo]

In his essays on relativity, 1920 and onward, Einstein rejected Mach's principle, calling it "spooky action at a distance". He claimed that inertial effects arose from matter's interaction with the local space in which it embedded. He called space it the "ether" of GR, which none of his contemporaries seemed to like, much.

 

[Al68]

Is Mach's principle still the best explanation for the force we feel when changing velocity? If the universe were infinite with infinite homogeneously distributed matter, would Mach's principle still apply?

EDIT: If it's not, how is the force associated with acceleration explained?

It seems you have it backward. The force we feel is the force applied. A change in velocity is the result of the force, not the other way around.

So the real question is what causes an object's resistance to acceleration (inertial mass), and there is no consensus on this. Mach's principle relates an object's inertial mass to distant masses because the acceleration being resisted is the object's change in velocity relative to those distant masses.

But Mach's principle offers no mechanism or reason for this relationship, so, like gravity, the cause is completely unknown.

 

[SW VandeCarr]

It seems you have it backward. The force we feel is the force applied. A change in velocity is the result of the force, not the other way around.

So the real question is what causes an object's resistance to acceleration (inertial mass), and there is no consensus on this. Mach's principle relates an object's inertial mass to distant masses because the acceleration being resisted is the object's change in velocity relative to those distant masses.

But Mach's principle offers no mechanism or reason for this relationship, so, like gravity, the cause is completely unknown.

Thanks Al68. Since we don't have any good hypotheses regarding this resistance to acceleration, is it safe to say there's no way to make an informed speculation as to whether an infinite universe would be any different than a finite one in this respect?

 

[turbo]

Thanks Al68. Since we don't have any good hypotheses regarding this resistance to acceleration, is it safe to say there's no way to make an informed speculation as to whether an infinite universe would be any different than a finite one in this respect?

In the 1920's, the universe was very poorly understood, but Einstein reasoned that since nothing could propagate faster than the speed of light, inertial effects could not be the result of the interaction of an accelerating or spinning body with ALL the massive bodies in the universe. As he rejected Mach's notion of inertia, he replaced it with the idea that inertial effects arise from matter's interaction with the local space in which it is embedded. In such a case, it matters little whether the universe is finite or infinite in extent.

We can see that, for Newton, space was a physical reality, in spite of the peculiarly indirect manner in which this reality enters our understanding. Ernst Mach, who was the first person after Newton to subject Newtonian mechanics to a deep and searching analysis, understood this quite clearly. He sought to escape the hypothesis of the 'ether of mechanics' by explaining inertia in terms of the immediate interaction between the piece of matter under investigation and all the other matter in the universe. This idea is logically possible, but, as a theory involving action-at-a-distance, it does not today merit serious consideration. We therefore have to consider the mechanical ether which Newton called 'Absolute Space' as some kind of physical reality. The term 'ether', on the other hand must not lead us to understand something similar to ponderable matter, as in the physics of the nineteenth century.

The essay from which this paragraph was excerpted is Chapter One of "The Philosophy of Vacuum" by Saunders and Brown - highly recommended reading.

 

[SW VandeCarr]

As he rejected Mach's notion of inertia, he replaced it with the idea that inertial effects arise from matter's interaction with the local space in which it is embedded. In such a case, it matters little whether the universe is finite or infinite in extent.The essay from which this paragraph was excerpted is Chapter One of "The Philosophy of Vacuum" by Saunders and Brown - highly recommended reading.

Thanks for the recommendation turbo-1. Is there any current theory on this proposed local interaction with the vacuum? I haven't found any in my searches. I've read some discussion on an absolute vacuum in the setting of a rotating finite universe.

 

[edpell]

I highly recommend "Gravitation and Inertia" by Ciufolini and Wheeler. To quote from page 4

"A letter of warm thanks he [Einstein] did, however, write to Mach. In it he explained how mass there does indeed influence inertia here, through its influence in the enveloping spacetime geometry. Einstein's geometrodynamics had transmuted Mach's bit of philosophy into a bit of physics, susceptible to calculation, prediction, and test.

Let us bring out the main idea in what we may call poor man's language. Inertia here, in the sense of local inertial frames, that is the grip of spacetime here on mass here, is fully defined by geometry, the curvature, the structure of spacetime here. The geometry here, however, has to fit smoothly to the geometry of the immediate surroundings; those domains, onto their surroundings; and so on, all the way around the great curve of space. Moreover, the geometry in each local region responds in its curvature to the mass in the region. Therefore every bit of momentum-energy, wherever located, makes its influence felt on the geometry of space throughout the whole universe--and felt, thus, on inertia here."

 

[turbo]

Thanks for the recommendation turbo-1. Is there any current theory on this proposed local interaction with the vacuum? I haven't found any in my searches. I've read some discussion on an absolute vacuum in the setting of a rotating finite universe.

Hal Puthoff, Haisch, Rueda et al have explored the nature of the vacuum, most notably with respect to the field of virtual particle pairs of the quantum vacuum. Their work is not mainstream, but was funded in part by NASA's breakthrough propulsion program.

 

[Jonathan Scott]

A key paper on this topic is Dennis Sciama's 1953 paper "On the Origin of Inertia":

http://adsabs.harvard.edu/abs/1953MNRAS.113...34S"

This paper illustrates beautifully how a simplified analogy between gravity and electromagnetism leads directly to inertia.

If gravity does in fact obey a similar scheme, at some level, then one consequence is that the gravitational "constant" G would actually be slightly variable, determined by the distribution of mass in the universe. (This would of course conflict with one of the assumptions of GR, but I find Sciama's basic idea so compelling that I feel that it cannot simply be dismissed on that basis).

 

[Al68]

Thanks Al68. Since we don't have any good hypotheses regarding this resistance to acceleration, is it safe to say there's no way to make an informed speculation as to whether an infinite universe would be any different than a finite one in this respect?

I'd say that's safe to say. As turbo pointed out, inertia can't depend on all the masses in an infinite universe anyway, only those close enough to have an effect, even if we assume such a relationship exists.

 

[yuiop]

http://www.lightandmatter.com/html_books/genrel/ch03/figs/machian-planets.png

The above image is from section 3.5.2 of fellow member bcrowell's web presentation http://www.lightandmatter.com/html_books/genrel/ch03/ch03.html#Section3.5 which nicely describes how our universe is not fully "Machian". One of the two planets is rotating and has an equatorial hydrostatic bulge and General Relativity predicts this bulging of one planet, when both planets appear to be rotating from each other's perspective, will happen even if there are no distant stars, directly contradicting the Machian idea.

 

[heldervelez]

The light and gravitational interactions propagate at 'c' speed.
We are receiving both influences here and now.
Isotropy caracterizes those effects.
If we move in relation to the present background we will introduce a dipole (same as with the CMB).
Inertia is the resultant of this motion induced dipole.
It is irrelevant if the universe is finite or not and it does not envolve action at a distance.

 

[Dmitry67]

http://www.lightandmatter.com/html_books/genrel/ch03/figs/machian-planets.png

The above image is from section 3.5.2 of fellow member bcrowell's web presentation http://www.lightandmatter.com/html_books/genrel/ch03/ch03.html#Section3.5 which nicely describes how our universe is not fully "Machian". One of the two planets is rotating and has an equatorial hydrostatic bulge and General Relativity predicts this bulging of one planet, when both planets appear to be rotating from each other's perspective, will happen even if there are no distant stars, directly contradicting the Machian idea.

Yes, but this example is artificial.
You are assuming flat boundary conditions at infinity, while there are no such solutions for the whole universe.

 

[bcrowell]

In his essays on relativity, 1920 and onward, Einstein rejected Mach's principle, calling it "spooky action at a distance".

He used this phrase to refer to quantum correlations between distant, entangled particles. If he also used it to refer to Mach's principle, this is the first I've heard of it.

He claimed that inertial effects arose from matter's interaction with the local space in which it embedded.

I'm not sure here whether you're describing the Machian or anti-Machian view. The Machian view describes inertia as a relationship between matter and *distant* matter.

He called space it the "ether" of GR, which none of his contemporaries seemed to like, much.

I don't think this statement is correct.

In the 1920's, the universe was very poorly understood, but Einstein reasoned that since nothing could propagate faster than the speed of light, inertial effects could not be the result of the interaction of an accelerating or spinning body with ALL the massive bodies in the universe. As he rejected Mach's notion of inertia, he replaced it with the idea that inertial effects arise from matter's interaction with the local space in which it is embedded. In such a case, it matters little whether the universe is finite or infinite in extent.

This doesn't sound quite right to me. First off, I don't think Einstein made an abrupt turn away from Machian ideas at any point in his career. When he initially tried to formulate a relativistic theory of gravity, he was loosely guided by Mach's principle. When he succeeded with GR, it had some Machian properties and some non-Machian properties (describing the vacuum as having dynamics of its own). When Schwarzschild found the Schwarzschild less than a year later, Einstein was disturbed because it was non-Machian, in the sense that there was a gravitational field that wasn't a relationship between two objects. Einstein's popularization of GR, published in 1920, is full of Machian arguments. There was a period of decades after that when various solutions to the field equations were inspected to see whether they had a Machian or anti-Machian character.

I also don't think it's right to associate Mach's principle with superluminal effects. I think Einstein believed GR to be highly Machian when he first published it in 1915, and the theory clearly didn't include superluminal effects. Brans-Dicke gravity is considered by many theorists to be more Machian than GR, and it doesn't include superluminal effects. There is no logical reason why the relationship between distant objects described by Mach's principle has to be an instantaneous relationship; in the Brans-Dicke theory, it isn't.

 

[turbo]

bcrowell, please read Chapter One of The Philosophy of Vacuum. It is Einstein's essay "On the Ether" (1924), which expands upon his 1920 Leiden address regarding the role of space (ether) in relativity. By 1924, Einstein had come to believe that space has properties which are conditioned by local matter, and that gravity and inertial effects are emergent, arising from matter's interaction with the local space in which it is embedded. The essay should clarify the difference between what Einstein himself believed and what others believed about his theories. There is a disconnect.

The book is very expensive for such a small volume, but you should be able to find it in any decent college library.

 

[bcrowell]

bcrowell, please read Chapter One of The Philosophy of Vacuum. It is Einstein's essay "On the Ether" (1924), which expands upon his 1920 Leiden address regarding the role of space (ether) in relativity. By 1924, Einstein had come to believe that space has properties which are conditioned by local matter, and that gravity and inertial effects are emergent, arising from matter's interaction with the local space in which it is embedded. The essay should clarify the difference between what Einstein himself believed and what others believed about his theories. There is a disconnect.

The book is very expensive for such a small volume, but you should be able to find it in any decent college library.

The essay is accessible, except for 2 of the pages, via books.google.com. Your statement that 'He called space it the "ether" of GR' is a complete misrepresentation of the contents of the paper. First, "He called [...] it" implies that this was his general way of referring to it throughout his career, rather than just in this essay. Second, the essay is just setting up a loose analogy between various systems, such as Newtonian gravity and GR, and using "aether" in a general way to talk about those theories' attitudes toward space, and whether space has locally observable properties of its own.

This essay has been a magnet for cranks for a long time. John Baez has the following comments:

Albert Einstein, in his essay On the Aether (1924), made some injudicious comments to the effect that relativity theory could be said to ascribe physical properties to spacetime itself, and in that sense, to involve a kind of "aether". He clearly did not mean the kind of "aether" which had been envisioned by Maxwell and others in the nineteenth century, but his remarks have been seized upon ever since, by various cranks and other ill-informed persons, as evidence that "gtr is an aether theory". Here's a typical claim of this sort:

...the aether is restored in General Relativity see Einstein's 1924 essay "On the Aether". Einstein recanted on his 1905 rejection of the aether since the mutable curved space-geometry is a dynamical object (with shift and lapse fields in ADM formulation), hence an aether.

This claim is misleading, to say the least. What Einstein really meant was that the aether which had been overthrown by str (and thus was incompatible with gtr, which incorporates str) involved a a specific "preferred frame of reference" in the classical field theory, whereas the field equation of gtr involves no "prior geometry" (such as the euclidean geometry of "space" which has assumed by Maxwell and his contemporaries), much less any "preferred frame". Nonetheless, gtr does not quite say there is "nothing" in "empty space"; in general there will be gravitational waves running about, and these carry (very tiny) amounts of energy, which gravitate. So in this sense, a very different kind of "aether" in the very weak sense of there being "something there" in a vacuum (namely nonlocalizable gravitational field energy, metric properties of "space" in a 3+1 decomposition, etc.), could be said to enter into gtr. In modern quantum field theories, of course, there are still more "things which are there" in a vacuum, but again these do not constitute an "aether" in the nineteenth century sense in which this word was used as a technical term.

Einstein was criticizing people who claimed, in effect, that the classical notion of the aether was such nonsense that people like Maxwell should have known better. He was saying that the problem with the classical aether was not ontological, merely that it is inconsistent with observation and experiment; hence the need for str.

Many years ago, Andrei Sakharov (yes, that Sakharov!) proposed to interpret gtr in terms of something like "stresses" on spacetime as something like a material. This is discussed in Chapter 17 of MTW, but here too, ill-informed readers of that theory have badly misunderstood the meaning of Sakharov's work.

This has been discussed before on PF: https://www.physicsforums.com/archive/index.php/t-4021.html A PF member provided a translation of the opening of the essay in that thread.

Hal Puthoff, Haisch, Rueda et al have explored the nature of the vacuum, most notably with respect to the field of virtual particle pairs of the quantum vacuum. Their work is not mainstream, but was funded in part by NASA's breakthrough propulsion program.

I'm glad that you were more up front here about disclosing the crank character of what you were referring to. Puthoff is known for his research in parapsychology, http://www.parapsych.org/members/h_puthoff.html , and for example was a participant in Ingo Swann's demonstration of his ability to psychically visit the planet Jupiter: http://en.wikipedia.org/wiki/Ingo_Swann

 

[turbo]

I am so gratified to have such guidance in determining whether or not Einstein's words should be taken at face value. Surely Baez has a place in the Pantheon of eminent physicists.

If I had known about the "crank" designation earlier, I could have dismissed Dennis Sciama's paper out-of-hand. (Chapter 6. The Physical Significance of the Vacuum State of a Quantum Field) You know, the one in which he equates vacuum fluctuations with a Lorentz-invariant ether. Especially "cranky" is section 9 - Do Zero-Point Fluctuations Produce a Gravitational Field?

I should have known Sciama was a crank. After all, he earned his PhD under Dirac with a dissertation on Mach's Principle and the origin of inertia. And Sciama himself supervised a whole raft of cranks as they earned their PhDs, including Hawking, Ellis, Rees, and Carter.

/sarcasm

There is still plenty of work to be done in fundamental physics, and we do not know all there is to know about inertia or gravitation or how they arise. We have mathematical representations of these effects that are quite predictive, at least on some scales, but we should not confuse them with reality. The map is not the territory.

 

[Dmitry67]

BTW I am really surprised that nobody had mentioned the Goedels solution, as it had been designed as a contre-example to Mach principle:

http://en.wikipedia.org/wiki/Gödel_metric

Some have interpreted the Gödel universe as a counterexample to Einstein's hopes that general relativity should exhibit some kind of Mach principle, citing the fact that the matter is rotating (world lines twisting about each other) in a manner sufficient to pick out a preferred direction, although with no distinguished axis of rotation.

 

[bcrowell]

I should have known Sciama was a crank. After all, he earned his PhD under Dirac with a dissertation on Mach's Principle and the origin of inertia. And Sciama himself supervised a whole raft of cranks as they earned their PhDs, including Hawking, Ellis, Rees, and Carter.

Baez did not criticize Sciama as a crank. I did not criticize Sciama as a crank. Baez criticized crank interpretations of the Einstein essay. I agree with his criticism. I also criticized Puthoff as a crank.

 

[bcrowell]

BTW I am really surprised that nobody had mentioned the Goedels solution, as it had been designed as a contre-example to Mach principle:

Interesting! There's a clear similarity with Einstein's example of the two planets. I don't fully understand the discussion of Machian issues in the WP article, and it doesn't seem to reach a clear conclusion. In general, I would say that GR has lots of problematic solutions, such as solutions with CTCs or naked singularities, and if any of these solutions could be shown to be potential occurrences within our own universe, then GR would be in big trouble as a classical field theory that's supposed to be able to make predictions. However, the indications so far are that chronology protection and cosmic censorship do hold (in some form), and therefore I'm not sure how much attention to pay to solutions that violate them when I'm trying to interpret the physical meaning of GR.

 

[Al68]

http://www.lightandmatter.com/html_books/genrel/ch03/figs/machian-planets.png

The above image is from section 3.5.2 of fellow member bcrowell's web presentation http://www.lightandmatter.com/html_books/genrel/ch03/ch03.html#Section3.5 which nicely describes how our universe is not fully "Machian". One of the two planets is rotating and has an equatorial hydrostatic bulge and General Relativity predicts this bulging of one planet, when both planets appear to be rotating from each other's perspective, will happen even if there are no distant stars, directly contradicting the Machian idea.

In the absence of any distant masses, which planet does GR predict will bulge?

 

[yuiop]

In the absence of any distant masses, which planet does GR predict will bulge?

GR predicts that the planet that is really spinning (B) will bulge.

What Mach predicts is more difficult to specify, because Mach never put his ideas in a mathematical form. However, it is probably safe to say Mach's ideas are more fully relativistic than GR. From that point view, an observer on planet B could consider planet B as stationary and to him it will look like it is planet A that is spinning and in the fully relativistic Machian interpretation planet A should bulge too. In other words in the Machian interpretation it should be impossible to tell which planet is really spinning, just as it is impossible to tell which observer is really moving in Special Relativity. Of course, your little smile tels me that you may have set up a trap for me (or I am just paranoid :tongue2:), but I do not mind. I am prepared to discuss the possibilities and I might learn something

 

[D H]

What planet is "really" spinning? Without any remote stars, how do you even go about defining an inertial frame?

I am not convinced by these toy universe arguments. They are not scientific in the sense that they are not testable. There is no way to construct a universe that comprises only two planets, one spinning and the other not.

Discarding the fact that Mach's principle is not particularly well-specified, is there any test that would falsify Mach's principle in our real universe?

 

[yuiop]

What planet is "really" spinning? Without any remote stars, how do you even go about defining an inertial frame?
...

You could define an inertial frame as one in which you do not feel any forces acting upon you. Such a definition does not care about the distant stars and can be measured by accelerometers. In the two planets example, an observer on B feels forces acting upon him, so hs claim that he is a stationary inertial frame is not valid. There are clear differences between what observers on A and B would feel and measure. As for falsifying Mach, that would be very difficult, as with no clear definition of his ideas it is difficult to know what to falsify.

 

[Jonathan Scott]

I'd say that the most obvious Machian interpretation would be that in the two-planet universe both planets would be spinning in opposite directions relative to the frame in which their average spin (in some sense) is zero.

Even in the Machian interpretation, there is a local concept of a local inertial frame of reference (non-rotating non-accelerating space), but it is effectively a field defined by the distribution and motion of all of the mass in the universe.

 

[D H]

You could define an inertial frame as one in which you do not feel any forces acting upon you. Such a definition does not care about the distant stars and can be measured by accelerometers.

This describes a point that is moving inertially. It does not describe a reference frame. A reference frame comprises an origin plus orientation.

Our best estimate of what comprises an inertial frame of reference, the Second Realization of the International Celestial Reference Frame (see http://www.iers.org/MainDisp.csl?pid=46-1100252) has a very strong Machian flavor to it: It is based on long-term observations of over 3400 quasars. The ICRF2 was developed as a joint project of the International Astronomical Union (IAU), the International Earth Rotation and Reference System Service (IERS) and the International VLBI Service for Geodesy and Astrometry (IVS).

In the two planets example, an observer on B feels forces acting upon him, so hs claim that he is a stationary inertial frame is not valid. There are clear differences between what observers on A and B would feel and measure.

Two objections:
(1) With respect to what is the rotating planet is rotating?
(2) This discussion is not scientific. It is metaphysics at best.

As for falsifying Mach, that would be very difficult, as with no clear definition of his ideas it is difficult to know what to falsify.

I agree. Mach's Principle is not particularly well-defined.

 

[yuiop]

Two objections:
(1) With respect to what is the rotating planet is rotating?
(2) This discussion is not scientific. It is metaphysics at best.
...

The Kerr metric which is a solution of GR, is the description of a rotating mass in an otherwise empty universe. Your objection "With respect to what is the rotating planet is rotating?" applies equally to the Kerr metric. Do you disagree with the Kerr solution? Do you think the Kerr metric is unscientific?

The Schwarzschild metric is the description of a non rotating mass in an otherwise empty universe. How do we know the mass in this metric is not rotating? The Kerr and Schwarzschild solutions are distinct and not just a transformation from one point of view to another. A ideal point particle orbiting an Schwarzschild mass is not equivalent to a stationary point particle in the Kerr metric. There is a sense in the two metrics that rotation is defined with respect to spacetime itself, rather than to physical reference objects.

 

[A.T.]

In the absence of any distant masses, which planet does GR predict will bulge?

GR predicts that the planet that is really spinning (B) will bulge.

What Mach predicts is more difficult to specify,

Wouldn't Mach predict no bulge on both planets in an otherwise empty universe, because all inertia is caused by distant masses?

 

[A.T.]

You could define an inertial frame as one in which you do not feel any forces acting upon you. Such a definition does not care about the distant stars and can be measured by accelerometers.

This describes a point that is moving inertially. It does not describe a reference frame. A reference frame comprises an origin plus orientation.

You could add ring laser gyroscopes to the accelerometers. But you still don't know if distant stars are causing the Sagnac effect or if it is completely local.

 

[yuiop]

Wouldn't Mach predict no bulge on both planets in an otherwise empty universe, because all inertia is caused by distant masses?

I might be wrong, but I have always assumed that's Mach's principle is that interia is caused by the sum of all mass in the universe, near and far, and that he simply referred to the "the distant stars" as they represented the majority of mass in the universe. In the toy two planet universe, the inertia of one planet would be due to the presense of the other planet and vice versa.

Is Mach's principle still the best explanation for the force we feel when changing velocity? If the universe were infinite with infinite homogeneously distributed matter, would Mach's principle still apply?

Eintein would say that mass beyond the visible horizon of a particle would have no effect on the particle, because changes in gravity are transmitted at the speed of light. If Mach's principle implies that distant stars have an instantaneous cause and effect on matter nearby to us, then matter beyond the visible horizon would have an influence on us. However it is not clear, in the absense of a clear definition of Mach's principle, whether he was suggesting instantaneous action at a distance.

What we do know is that Einstein tried to incorporate Mach's principle, as he understood it, into GR and ulltimately was unable to do so and using Mach's ideas as a guiding principle hampered and slowed down his search for a general theory of relativity. Presumably Einstein had a better idea of what Mach had in mind, as they were friends and private discussions on the matter. It is also probable that because they were friends, that Einstein did not publish a thorough demolition job on Mach's ideas and publicly discredit his ideas.

 

[A.T.]

I might be wrong,

No, it's rather me.

but I have always assumed that's Mach's principle is that interia is caused by the sum of all mass in the universe, near and far,

Yeah, I guess so. But without a quantitative model it is difficult to even envision a test for it. And if Mach & Einstein didn't come up with quantitative model, I will not even try.

 

[bcrowell]

But without a quantitative model it is difficult to even envision a test for it. And if Mach & Einstein didn't come up with quantitative model, I will not even try.

Brans-Dicke gravity is often considered to be more Machian than GR. So if the discussion of Einstein's two-planet example feels like it degenerates into metaphysics because Mach's principle is so vague, I suppose one way to go would be to see what Brans-Dicke gravity predicts about it. The Brans-Dicke theory has an adjustable parameter $\omega$, and as $\omega\rightarrow\infty$ you recover GR. The latest data from Cassini give $\omega > 40000$. I wonder whether, for small values of $\omega$, you'd see the equatorial bulges of the two planets become more equal, and the Sagnac effect become more equal on both planets' surfaces.

Looking at the form of the Brans-Dicke field equations, I'm tempted to say that they probably *cannot* reproduce the result that Einstein expected in his two-planets example. The reason is that the problem has axial symmetry, and the scalar field $\phi$ is real, so I don't see how you $\phi$ could carry any information about the handedness of rotation. But the interaction between g and $\phi$ is complicated and nonlinear, so I could be totally wrong about that.

 

[Naty1]

Brian Greene has some interesting comments regarding Mach and Einstein's theories in THE FABRIC OF THE COSMOS,2004, PAGES 33-38:

...The concept of absolute space left many wondering how absolute space can allow us to identify true accelerated motion (as with the famous spinning bucket of water example), while it cannot provide a way to identify true constant velocity motion. After all, if absolute space really exists, it should provide a benchmark for all motion, not just accelerated motion...Mach argued that in an otherwise empty universe there is no distinction between spinning and not spinning...Without other material-without any benchmarks for comparsion- Mach claimed there would be no way to experience acceleration...Machs suggestion was not a complete theory or description because he never specified how the matter content of the universe would exert the proposed influence...when the dust of relativity finally settled, the question of whether space is something...was transformed in a manner that shattered all previous ways of looking at the universe...

 

[Dmitry67]

Infinite empty universe with few gravitating bodies at one location - is it a realistic GLOBAL solution for GR equations?

 

[bcrowell]

Infinite empty universe with few gravitating bodies at one location - is it a realistic GLOBAL solution for GR equations?

This is another case where it's useful to consider the Brans-Dicke theory as a test theory. I've just read their paper, C. Brans and R. H. Dicke, Physical Review 124 (1961) 925, which IMO is an exceptional piece of scientific writing -- very accessible and physically clear. The case of the nearly empty universe is just an idealization. They start out the paper by doing idealized thought experiments like this and discussing how they appear to violate Mach's principle. Then they develop their field theory and use it to calculate some classic stuff like cosmological solutions and the perihelion rotation of Mercury. The cosmological solutions come out different than in GR, because the scalar field $\phi$ is basically the inverse of the gravitational constant. As the universe expands, $\phi$ gets smaller (because the trace of the stress energy tensor T acts as its source), and G gets bigger. So if you like, you can think of all these thought experiments with nearly empty universes as limiting cases where cosmological expansion has progressed far into the future.

In particular, I now realize that I was wrong about Einstein's two-planet example in the Brans-Dicke theory. They do essentially the same example in their paper, although their thought experiment is phrased a little differently. (A physicist is in a laboratory in a universe that's otherwise empty. He fires a bullet out the window, and the lab rotates in response, which is detectable on a gyroscope.)

Analyzing the two-planet thought experiment in the Brans-Dicke theory, here's what happens. Very little matter is present in the universe, so G is huge. Because of the big G, you get a huge amount of Lense-Thirring frame dragging. Therefore the two planets will synchronize their rotations. This is exactly the Machian result that Einstein wanted, but didn't get in GR.