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 ), 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.

 

[Al68]

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

OK, how does GR determine which one is "really spinning"? By the a priori observation of the bulge to be "predicted"?

Yes, my smiley indicated a trap.

The obvious issue is the circular logic.

The real question is, what would GR predict in the absence of any a priori knowledge of which one is "really spinning", except that each planet spins relative to the other?

 

[Jonathan Scott]

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.

Most of that sounds plausible. However, from what I recall of frame dragging I don't see why it should cause the planets to "synchronize their rotations", because there is no change occurring.

There may be some special case effect here when these are the only masses in the universe, as that affects coordinate systems and other things as well, but certainly in the normal case I don't think frame-dragging changes rotation in this way.

As I understand it (and I'm willing to accept that I may be mistaken), the effect in frame dragging is that a nearby test body's view of local space is influenced by rotation or acceleration of local masses in such a way that this rotation or acceleration is apparently slightly decreased relative to how it would appear to a distant observer.

This means for example that a test body would feel that it wasn't rotating when in fact it was rotating slightly in the same direction as the source body. However, although this induces changes in centripetal and coriolis forces in the object (and of course may induce precession in a gyroscope as in GP-B) I don't think it induces any change in overall rotation (and angular momentum) unless you change the frame dragging effect. For example, if you move the test object closer to the source, then relative to a distant observer it will appear to experience additional forces, but if it stays in the same location I don't see any reason that any angular momentum should be transferred.

 

[bcrowell]

OK, how does GR determine which one is "really spinning"? By the a priori observation of the bulge to be "predicted"?

Yes, my smiley indicated a trap.

The obvious issue is the circular logic.

The real question is, what would GR predict in the absence of any a priori knowledge of which one is "really spinning", except that each planet spins relative to the other?

I don't think Einstein's logic was circular here. Mach's principle predicts that the two planets' equatorial bulges are always equal, and that they must vanish if neither planet is rotating relative to the other. GR predicts that they can be unequal, and that they can be nonvanishing even if neither planet is rotating relative to the other.

(GR also predicts a correlation between the equatorial bulge and other effects like the Sagnac effect and the Lense-Thirring effect, so there are multiple ways of determining each planet's rate of rotation, independent of any observation of the other planet or anything else that may or may not exist in the more distant cosmos.)

 

[bcrowell]

Most of that sounds plausible. However, from what I recall of frame dragging I don't see why it should cause the planets to "synchronize their rotations"[...]

You're right. My interpretation was wrong. Frame dragging's effect on a gyroscope vanishes when the two axes are parallel. I think the Brans-Dicke paper was referring to a case where the gyro's axis was perpendicular to the lab's. Actually I think there's a straightforward argument that the synchronization effect I proposed is impossible in a Machian theory. In a purely Machian theory, observers on an isolated planet can never tell whether their planet rotates or not. Therefore if they start a gyroscope rotating on their planet, with its axis perpendicular to the planet's surface, the gyroscope's environment is perfectly symmetric spatially. If the gyroscope then slows down or speeds up in this spatially symmetric environment, it violates time-reversal symmetry.

So what can we actually say about the Einstein two-planet scenario in Brans-Dicke gravity...?

It seems clear to me that observers on an isolated planet can never do local experiments using mechanical gyroscopes in order to determine their planet's state of rotation. If they could do such experiments using some other effect like the Sagnac effect, then the disagreement between the two would probably violate the equivalence principle. Although Brans-Dicke gravity doesn't obey the strictest form of the equivalence principle, it does obey it in some form, so I suspect that the purely Machian result holds as well in Brans-Dicke gravity in the appropriate limit of a nearly-empty universe: observers can't tell by any local experiment whether they're rotating or not.

This all seems to work out consistently when you consider the equatorial bulges. The gravitational constant is extremely high, so gravity is extremely strong relative to inertia, and the bulges are strongly suppressed.

Adding a second planet would not seem to me to change this prediction. The field \phi reflects the presence of mass in the universe, convoluted with a 1/r distance dependence. In a universe full of matter, this is a big effect, because the amount of matter at distance r grows faster than r. But in the two-planet example, the second planet is not going to have a significant effect on \phi.

I think the ability to get clearcut answers here may be hampered by difficulties in formulating the limits correctly. The Brans-Dicke paper talks about this on the final page, where they can't take the limit of an empty universe because they're using a weak field-approximation.

 

[yuiop]

OK, how does GR determine which one is "really spinning"? By the a priori observation of the bulge to be "predicted"?

Yes, my smiley indicated a trap.

The obvious issue is the circular logic.

The real question is, what would GR predict in the absence of any a priori knowledge of which one is "really spinning", except that each planet spins relative to the other?

O.K. I was being slightly glib when I "took the bait" in the interests of prompting a conversation on the issues and it certainly seems to have done that. I think the main issue is that GR allows the assymetric rotation, where as a Machian model probably would not. One of the main hurdles is that we can not remove all the distant stars to carry out a definitive experiment. I think with some ingenuity, we may be able to come up with a thought experiment that might shed some light on the situation. One way of rephrasing the issues that you raise is "how does planet B know it should bulge and how does planet A know it should not bulge?". If I was really honest, I would admit I have a nagging suspicion of circular logic in there somewhere too.

It is also worth noting that if we started in a situation where both planets were non-rotating, it would be impossible to arrive at a situation where only planet B is rotating without any other masses in the universe. The conservation of angular momentum principle, would require some other masses to rotate in the opposite direction to maintain the initial zero angular momentum of the universe.

First attemp at a thought experiment: Consider two flywheels connected by a motor. Both flywheels have the same radius and one flywheel has ten times more mass than the other and there are no other masses in the universe. When the motor is operated, the small flywheel starts rotating faster than the large flywheel rotating in the opposite direction, as required by the conservation of angular momentum. Obviously the small flywheel has less angular inertia than the large flywheel. Does Mach's principle predict the larger mass of the larger flywheel, causes the smaller flywheel to have less inertia and the smaller mass of the smaller flywheel causes the larger flywheel to have less inertia? Does Mach's principle predict that the greater the total mass of the "distant stars" is, the lesser the inertia of objects "here" is? Does that imply in the absence of the distant stars that objects would have near infinite inertia?

There appears to a slight connection between Mach's ideas and the competion that once existed between the Ptolemaic model and the Copernican model. Ptolemy asserted that the Earth was the centre of the universe and produced a complicated set of rules to explain the motion of the planets, while Copernicus showed that the rules were much simpler if the Sun was taken as the centre of the universe. The most significant part of the Copernican revolution was the realisation that any point in the universe could be treated as the stationary centre of the universe and with enough ingenuity the motion of all bodies about that point can be explained. Presumably, if we assume the Earth is stationary and non rotating, Mach's principle should be able to explain the bulge of the Earth, the Coreoilis forces and the Ptolmaic epicycles of the planets, in terms of the effect of the rotating stars, while GR would say emphatically that the Earth is rotating. Is that correct?

 

[D H]

It seems clear to me that observers on an isolated planet can never do local experiments using mechanical gyroscopes in order to determine their planet's state of rotation.

As far as we know, an observer on a real planet can use gyroscopes to determine whether the planet is rotating. Maybe I'm just an old fart who still thinks conjectures in physics should be testable. Conjectures regarding a universe that comprises one and only one planet are not testable.

These discussions of an isolated planet are conjectures based on extrapolating the physics of the universe we know to a universe that is thirty orders of magnitude less massive than and sixty orders of magnitude small than our universe. The one thing we physicists learned (or should have learned) since the early 1900s is that extrapolating the physics we know over dozens of orders of magnitude is not valid.

 

[bcrowell]

As far as we know, an observer on a real planet can use gyroscopes to determine whether the planet is rotating. Maybe I'm just an old fart who still thinks conjectures in physics should be testable. Conjectures regarding a universe that comprises one and only one planet are not testable.

The quote that you were responding to was in the context of the Brans-Dicke theory. Brans-Dicke gravity is testable, and it has passed all observational tests so far (e.g., perihelion rotation of Mercury). There's a cool book called Was Einstein Right? that discusses a lot of the history of the tests of GR and Brans-Dicke as competing theories of gravity in the 1970's. Brans-Dicke gravity has an adjustable parameter \omega that gives GR if you set it to infinity. The lower you set \omega, the more Machian the theory becomes. The current lower limit on \omega is about 40,000, which means that the universe is in this sense not very Machian. So Mach's principle is not just some vague philosophical notion, it's a number that you can measure. In the limit of large \omega, you get GR, which makes certain predictions about the Sagnac effect, frame dragging, etc. In that limit, you can determine your state of rotation and extract the result using GR. In the limit of small \omega, you get a theory where gyroscopes don't work. In the intermediate case, which for all we know really does describe the universe we live in, gyroscopes do precess, but not as much as predicted by GR, and the answer you'll get for your state of rotation will actually be slightly wrong.

These discussions of an isolated planet are conjectures based on extrapolating the physics of the universe we know to a universe that is thirty orders of magnitude less massive than and sixty orders of magnitude small than our universe. The one thing we physicists learned (or should have learned) since the early 1900s is that extrapolating the physics we know over dozens of orders of magnitude is not valid.

Brans-Dicke gravity makes definite predictions. It makes those predictions without having to go to a hypothetical alternative universe that's empty. The discussion of scenarios with empty universes, in the context of Brans-Dicke gravity as the test theory, is simply a way of reasoning about the limiting behavior of the theory compared to GR.

i would actually maintain exactly the opposite of the point of view you're advocating, in the following sense. If GR is the only theory available, then it becomes impossible to design experiments to test GR. Only if you have other theories that make other predictions can you test whether GR is correct. For instance, there was a 2003 experiment by Fomalont and Kopeikin ( http://arxiv.org/abs/astro-ph/0302294 ) that claimed to test Einstein's century-old prediction that low-amplitude disturbances in the gravitational field would propagate at c. Turns out that Fomalont and Kopeikin's experiment doesn't really test this claim, and the reason it can't test it is that there is no competing test theory available that *doesn't* predict propagation at c. In general, without considering competing test theories like Brans-Dicke, or Østvang's quasi-metric relativity, we'd actually be limited to the kind of navel-gazing you were criticizing.

 

[Al68]

I don't think Einstein's logic was circular here. Mach's principle predicts that the two planets' equatorial bulges are always equal, and that they must vanish if neither planet is rotating relative to the other. GR predicts that they can be unequal, and that they can be nonvanishing even if neither planet is rotating relative to the other.

I wasn't referring to Einstein's logic being circular, he used this example to point out the circular logic that already existed.

A simpler example is that we define an inertial reference frame as one in which objects are unaccelerated in the absence of applied forces, (no pseudo-forces are present). Then we "predict" that in an inertial reference frame, objects will not be accelerated without applied forces. This prediction is just the result of assuming the predicted result a priori.

This makes for a very good physical model, but Einstein was unsatisfied with the "epistemological shortcomings" of such models, including his.

A fully Machian theory, Einstein had hoped, would be free of this shortcoming.

 

[Al68]

O.K. I was being slightly glib when I "took the bait" in the interests of prompting a conversation on the issues and it certainly seems to have done that. I think the main issue is that GR allows the assymetric rotation, where as a Machian model probably would not.

Sure GR allows for asymmetric rotation, but there is no "cause" for the bulge in the form of an equation relating the bulge to coordinate rotation that wouldn't equally apply to the non-bulging planet.

Does Mach's principle predict that the greater the total mass of the "distant stars" is, the lesser the inertia of objects "here" is?

I would think a Machian theory would predict the reverse, but I don't think Mach ever got to that point.

I didn't pose the question because I have the answer, I definitely don't.

 

[yuiop]

Does Mach's principle predict that the greater the total mass of the "distant stars" is, the lesser the inertia of objects "here" is? Does that imply in the absence of the distant stars that objects would have near infinite inertia?

I would think a Machian theory would predict the reverse, ...

I must admit I too would think Machian theory would predict the reverse, but if that is the case, my little thought experiment indicates Machian theory contradicts conservation of angular momentum.

Perhaps Ben can tell us what Brans-Dicke theory tells about the relationship between total mass of the universe and inertia of individual objects?

 

[Jonathan Scott]

I remember trying to work out what a universe would look like in a specific Machian model if there was only one substantial mass in an otherwise empty universe. I found that relative to the conventional coordinate system, gravitational potential would vary as r and the speed of light as r². I then found that null geodesics were exact circles which passed through the origin. Eventually, I realized that this was effectively flat Euclidean space with the radial coordinate inverted r -> 1/r. So the result was that to an observer moving around in that space, the mass was at infinity in all directions but otherwise space was flat. Weird!

 

[SW VandeCarr]

For my own and others' convenience I'm posting a link on the Brans-Dicke theory with the field equations located at about the middle of the text.

http://www.answers.com/topic/brans-dicke-theory

 

[bcrowell]

Perhaps Ben can tell us what Brans-Dicke theory tells about the relationship between total mass of the universe and inertia of individual objects?

You can interpret the theory either as a theory in which the gravitational constant G varies from point to point, or as one in which inertia varies. There's no way of distinguishing between the two interpretations, but the description that Brans and Dicke use in their paper (after explaining that the two interpretations are equivalent) is that G varies. In their theory, distant masses in the universe create a field \phi, which is essentially 1/G. If there's more mass within your past light cone, G is smaller. In the varying-mass description, if there's more mass within your past light cone, inertial masses in your neighborhood are bigger.

I remember trying to work out what a universe would look like in a specific Machian model if there was only one substantial mass in an otherwise empty universe. I found that relative to the conventional coordinate system, gravitational potential would vary as r and the speed of light as r². I then found that null geodesics were exact circles which passed through the origin. Eventually, I realized that this was effectively flat Euclidean space with the radial coordinate inverted r -> 1/r. So the result was that to an observer moving around in that space, the mass was at infinity in all directions but otherwise space was flat. Weird!

Cool :-) In the Brans-Dicke paper, they work out an approximate metric that's the equivalent of the Schwarzschild metric in their theory. The perihelion precession of Mercury differs from the GR result by a factor of (4+3\omega)/(6+3\omega). This turns out to be independent of the distribution of distant masses in the universe, for reasons that aren't totally clear to me.

The answers.com link that SW VandeCarr posted is, I think, just the same as the Wikipedia article, which doesn't really make even a token effort to explain the physical motivation for the theory. For anyone who has access to journals, I highly recommend the original paper by Brans and Dicke, which is extremely well written, entertaining, and accessible to non-specialists. It's really a shame that we find ourselves in a legal regime where copyright makes scientific knowledge like this inaccessible, fifty years after the paper was originally published.

 

[edpell]

It's really a shame that we find ourselves in a legal regime where copyright makes scientific knowledge like this inaccessible, fifty years after the paper was originally published.

I would guess it was published in Physical Review? I see nothing in copyright law that forbids PR from putting whatever material it owns and wants to place in the public domain into the public domain(?).

 

[yuiop]

... If there's more mass within your past light cone, G is smaller. In the varying-mass description, if there's more mass within your past light cone, inertial masses in your neighborhood are bigger.

The second description would seem to have more of a Machian flavour and that description seems to contradict the prediction of the double flywheel thought experiment I outlined in #40

Consider two flywheels connected by a motor. Both flywheels have the same radius and one flywheel has ten times more mass than the other and there are no other masses in the universe. When the motor is operated, the small flywheel starts rotating faster than the large flywheel rotating in the opposite direction, as required by the conservation of angular momentum. Obviously the small flywheel has less angular inertia than the large flywheel. Does Mach's principle predict the larger mass of the larger flywheel, causes the smaller flywheel to have less inertia and the smaller mass of the smaller flywheel causes the larger flywheel to have less inertia?