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This question was prompted by another question about a paper by Woodward (not mine). IMO Mach's principle is very problematic (?wrong) thinking. Mach was obviously influenced by Leibniz. Empty space solutions in GR would result in a Minkowski metric and would suggest no inertia. Mach's principle seems incompatible with GR. Gravitational waves could also be a problem. I had thought that papers like one by Wolfgang Rindler had more or less marginalised the Mach Principle, but I see lots of Internet discussion of it. Is it correct? Wrong? Is there evidence? (frame dragging experiments)?

Let's use this definition from

In his book The Science of Mechanics (1893), Ernst Mach put forth the idea that it did not make sense to speak of the acceleration of a mass relative to absolute space. Rather, one would do better to speak of acceleration relative to the distant stars. What this implies is that the inertia of a body here is influenced by matter far distant.

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@QGR --yes there are variants. None in particular..the principle itself (or variants) seem to have adherents and enemies of relatively mainstream physicists. The question is argumentative, and that seems to be the most common word used to close questions. Sometimes "argumentative" is a catalyst to a good discussion, but I think in this case degree of belief may coincide with Leibniz' influence on the scientist (Smolin, Barbour). I was wondering more if theoretical or technical, or experimental considerations could (partially) invalidate it rather than philosophical arguments. – Gordon Feb 19 '11 at 17:22
@Gordon: The Stack* sites are not designed for "discussion" and they are not good at "discussion". They are designed for and good at Questions that have Answers. If you are trying to start a discussion you're doing it wrong. And a vague, imprecise question like "Is [something ill-defined] wrong?" is at serious risk of not having Answers which means it does not belong. – dmckee Feb 19 '11 at 17:52
I think of Mach's principle (and the anthropic one) as one of those hotel showers that changes abruptly from being too hot or too cold, but is never quite comfortable. Similarly, these are either trivially right, or trivially wrong, depending on how you define things, but either way not all that useful or interesting. It always comes down to discussions of semantics, or who was right or wrong etc. – user566 Feb 19 '11 at 23:11
Existence of gravitational waves does not require GR. Gravitational waves should be expected in a purely classical field theory. If there is a charge and a field which is produced by the charge and the charge is accelerating, the field experiences wave. This is a purely classical concept. So it is incorrect to claim that gravitational waves contradict Mach: the waves are produced by matter and are matter themselves. The wave is just dynamic field. – Anixx Feb 21 '11 at 5:28
@Anixx: "Anything that spreads energy is material." You can use this definition of "material" if you like, but nobody else defines it that way. By this definition, electromagnetic waves are material, and the term was originally created precisely in order to make the distinction between light and matter. The definition of the term may have evolved somewhat over the years, but not enough to make it include light. – Ben Crowell May 19 '13 at 17:56

10 Answers 10

Mach's principle has influenced Einstein but the final formulation of general relativity as of 1916 clearly invalidates Mach's conjecture. According to Mach's principle, motion - including accelerating and rotating one - may only be defined relatively to other objects. That would imply that there can't exist any gravitational waves.

However, general relativity predicts and experiments confirm that gravitational waves do exist: the relevant observations were awarded by the 1993 physics Nobel prize, too. The waves are vibrations of the space itself. It means that the metric tensor remembers the information about the geometry - and curvature at each point, even in the empty space, something that Mach's principle specifically wanted to prohibit.

Moreover, the perceptions and other effects of acceleration were supposed to be determined by comparisons with distant objects. This simple fact itself violates locality that has become important already in special relativity, and was simply inherited by general relativity.

If you care about history, the new cold relationship is mutual: much like general relativity rejected Mach's principle, Mach rejected general relativity - and already special relativity, in fact. ;-) If you care about sociology, there's been a poll among physicists active in relativity, and a vast majority of them would also say that Mach's principle is invalidated by general relativity.

Some people sometimes say that some effects predicted by general relativity, such as frame-dragging, are "Machian" in character. I think it is very misleading because it tries to make the listeners think that Mach's principle may be made compatible with the observations. It's very questionable what Mach's principle would predict about frame-dragging because Mach's principle has never become any viable candidate for a physical theory. But the idea that frame-dragging is Machian is more ideology and hype than a valid observation. Despite the vagueness about such very detailed effects, Mach's principle has said enough for us to be sure that it's incorrect in all of its forms.

Well, there's a lot of discussion on the Internet about long-dead ideas in physics - and maybe mostly about them. However, the Internet has nothing to do with the current state of physics.

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@Lubos - well the arxiv is likely a primary source of info dissemination and it is on the internet and I would argue that it is influential and that the internet will become more important, not less. That doesn't minimize all the garbage on it as well. – Gordon Feb 19 '11 at 19:34
It doe depend on the formulation of "Mach's principle"--you can look at the fact that the flat space inside a spinning shell of mass rotates with respect to the flat space outside of the spinning shell of mass as a vindication of a certain version of "Mach's principle". – Jerry Schirmer Feb 19 '11 at 21:33
John, Einstein had thought that Mach's Principle was the way to go because it (also) made the universality encoded in the equivalence principle manifest. The equivalence principle says that all objects will be influenced equally - the same acceleration - by the whatever agent is causing gravity. Mach's principle satisfies the criterion "totally" - it removes any field-like agent. Well, it's going "too far" in this sense. Of course that Einstein was struggling for years to make Mach's principle compatible with the speed limit $c$ - and GR is what eventually came out of it. – Luboš Motl Feb 20 '11 at 8:48
@JerrySchirmer: That in fact invalidates Mach's principle, because in that model, while two forces equivalent to centrifugal and Coriolis force arise, an axial one with no fictitious force counterpart manifests as well. A rotating shell and a stationary shell are still absolutely distinguishable. – Siyuan Ren Feb 6 '12 at 9:59
@Anixx: What does total moment of inertial mean in general relativity? – Siyuan Ren Feb 6 '12 at 10:27

Using Mach's 1893 definition of Mach's principle condemns the discussion to irrelevance. It's like posting on physics.SE with a question titled "How is the emission spectrum of hydrogen determined?," but then saying in the body of the question that we want an answer written in terms of the aether and Newtonian mechanics.

In the 1960's and 70's, there was a golden age of tests of GR, and one of the most active topics was testing GR against alternative theories such as Brans-Dicke gravity. B-D gravity is physically a very well motivated theory. The original paper is available online and is very readable even if you're not a specialist. The idea of B-D gravity is to couple matter to a scalar field $\phi$, which provides a physical mechanism for Mach's idea that an object's inertia comes from the other matter in the universe. B-D gravity is more Machian than GR. Neither GR nor B-D gravity is completely Machian or completely non-Machian. B-D gravity has a dimensionless parameter $\omega$. In the limit $\omega\rightarrow\infty$, B-D gravity reduces to GR. Brans and Dicke committed themselves to the idea that "[...]in any sensible theory $\omega$ must be of the general order of magnitude of unity." This makes the theory falsifiable.

Experiments show that $\omega$ must in fact be quite high. The best current limit comes from the Cassini probe, which requires $\omega \ge 40,000$. Therefore, B-D gravity should be considered as falsified. So the modern, sensible answer to the OP's question is: Mach's principle is false, in the sense that experiments determine the universe to be no more Machian than GR -- which is not very Machian.

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Mach's principle is simply a philosophical ancestor of the equivalence principle: matter tells geometry how to curve, geometry tells matter how to move. So, yes, you can have Minkowski as a solution for the vacuum Einstein equations, but the minute you introduce even the tiniest mass your solution is no longer Minkowski. You might point out that asymptotically the spacetime will still be flat and Minkowski. However, in GR (and in QFT) it is not only the local geometry that matters. Point is that there is no interesting geometry which does not also contain some matter.

Mach's principle has been interpreted in many different ways. I find the following Wikipedia definition to be one I'm most comfortable with:

A very general statement of Mach's principle is "Local physical laws are determined by the large-scale structure of the universe."

Depending on what day of the week it is and whatever interpretation is your favorite for that day, Mach's principle could be "right", "wrong" or "obsolete". But if stated in the simple manner above, then it is nothing more than a restatement of the equivalence principle and the question of its "correctness" is no longer an issue.

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Absolutely. The right question is not whether it is right or wrong, but whether it is useful or not. Clearly this used to be useful, as a way to converge on something more precise and correct. Is it useful now? I don't see how. – user566 Feb 20 '11 at 3:34
Thank you @Moshe. As for the usefulness of Mach's Principle I think that there is an argument to be made in its favor, but that's for another time. – user346 Feb 20 '11 at 8:34
Dear @Moshe, I would respectfully disagree. In fundamental physics, the ultimate question is always whether XY is right or wrong, not whether it is useful - which is left to managers and perhaps engineers. The observation that Mach's principle is no longer "useful" doesn't mean that we can't answer the question whether it's right or wrong. Yes, we can. – Luboš Motl Feb 20 '11 at 8:52
@Deepak, the variation of Mach's principle you wrote, "Local physical laws are determined by the large-scale structure of the universe," is in no way equivalent to the equivalence principle. Quite on the contrary, the equivalence principle says that the local physical laws in a freely falling frame are completely unaffected by any distant objects - GR is a method to impose locality in a manifest way. Your (or your quote from Wikipedia) version of Mach's principle has undoubtedly been shown invalid, too. In GR, the large structure is determined by local physics and local sources, not vice versa – Luboš Motl Feb 20 '11 at 8:54
@Lubos the proposition->*proof*->*proposition* style of physics that you advocate is not really the universal choice. There are a lot of known factors and also unknown factors which help prop up any great idea. Not all these can assigned an "absolute Truth value". Mach's principle happens to be one of these auxiliary ideas which form the foundation of GR. To attempt to separate Mach's Principle from the historical foundations of General Relativity is a lost cause. It was mixed in with the concrete. – user346 Feb 20 '11 at 9:30

In psychology there is a special effect called the "verbal overshadowing effect". It concerns the phenomenon that describing a previously seen face impairs recognition of this face.

Mach's principle is essentially target of this psychological effect. It is theoretically overshadowed in such a way that most physicists do not relate to the underlying empirical core but to the theoretical context under discussion.

Hermann Bondi and Joseph Samuel have tried to fix this observational core of Machs Principle. (The Lense–Thirring Effect and Mach’s Principle, Physics Letters A 228, 1997, S. 121–126) They called it "Mach0": "The universe, as represented by the average motion of distant galaxies, does not appear to rotate relative to local inertial frames."

This "coincidence" is measured to a very high accuracy: 0.25 milliarcsec/year. (J. Kovalevsky, et. al. "The Hipparcos catalogue as a realisation of the extragalactic reference frame", Astron. Astrophysics. 323, 620 - 633 (1997)

It is just this coincidence that has theoretically to be explained. General Relativity is only a preferred theoretical tool of explaining this coincidence, but all attempts failed. Hence, from a historical point of view this empirical coincidence appears as an ANOMALY - as a fact that cannot be explained in any way by the running paradigm. The Answers given in this website do reflect this epistemological feature in an almost idealized way. MOSHE f.e. has compared it with a hotel shower that changes abruptly from being too hot or too cold, but is never quite comfortable.

As this empirical coincidence refers obviously to the ultimate boundaries of our universe, its solution resp.explanation can possibly not be found within the world, but outside from it.


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This is an interesting argument, but I think it fails to hold water. In inflationary models, the rotation of the universe falls off exponentially with time, so these models provide a very natural explanation of Bondi's Mach0. See Barrow, J. D., Juszkiewicz, R., & Sonoda, D. H., "Universal rotation: how large can it be?," 1985 -- . Even without inflation, a time-dependence is predicted; see Barrow, p. 924, eq 4.8. – Ben Crowell May 30 '13 at 17:37

It is not really a case of Mach principle being wrong. It is something related to general relativity, and frame dragging or Lense-Thirring effect are similar to Mach’s principle. However, these are local laws of physics and Mach’s principle is a global hypothesis. Gravity does tell us that spacetime curvature induces motion of masses, and if these masses are large enough that can in turn change curvature. So gravity or spacetime curvature acts on a mass locally and the same experiences acceleration as seen from another frame. This is a geodesic deviation which is defined by the Riemann curvature. So this tells us the mass which is observed to accelerate is the same mass which responds to the curvature --- the equivalence principle. This also sounds similar to Mach’s principle: Inertia there determines inertia here.

Mach stated the centripetal force on a local rotating frame is equivalent to what happens if the frame is nonrotating and the entire universe rotates around the frame. Kurt Godel worked out a model of a rotating universe and found a very strange time-looping spacetime. If the universe rotates it has a structure which is radically different from Mach’s original conjecture. Further, this universe violates energy conditions. It is not a physical spacetime, even if it comes from the Einstein field equations. This does at least suggests there are some radical departures from Mach’s principle and general relativity. Mach’s principle does not appear to be a consequence of general relativity.

Mach’s principle is rather vague in some respects. The inertia of a body is said to be determined by the inertia (or masses) of all other bodies. Again this sounds like GR, but this extends across the universe. The actual gravitational interaction a particle here has with a galaxy at z = 8 around12.5 billion light years out is miniscule. So the interaction picture appears funny. It might be tempting to say the entire universe is some sort of single quantum wave function, and single masses we observe are entangled subsets. This might sound global and gets away from the $q/r^2$ drop off of gravity. From there one might be tempted to say inertial is inherited this way. However, entanglements do not involve forces, and inertia as an entanglement not defined.

I will also say that I suspect Mach had ideas of an ether in space when he suggested this. So he probably had some picture of a rotating frame moving through this medium, and this is somehow equivalent to the whole medium rotating around the frame. So the ether vortex in the case of a rotating world is what generated the centrifugal force on the rotating frame.

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Mach's principle, if interpreted charitably, requires that one include horizons as matter, along with gravitational waves, and light, and all particles. This is required to include black holes, and for consistency requires cosmological horizons too. Once you understand that "matter" means "horizon", the statement that all rotation is relative to distant horizons is just a stunted classical version of the holographic principle, and is sort of vacuously true.

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The Mach's principle in short is that inertia is not absolute, determined by the matter configuration, and there is no other source.

This can be summed up in two statements:

  1. Massive bodies can affect inertia of other objects (inertia is determined by the matter configuration)

  2. The Universe does not rotate and linearly accelerate as a whole (there is no other source of inertia)

The first statement is included in GR and has been proven with the experimental measurement of frame-dragging. In fact inertia even can be screened so people in a rotating (against distant stars) spaceship could not detect the rotation.

The second statement is still not proven (in the rotational part) although verified to a high precision. The linear part simply follows from the conservation of momentum.

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could you elaborate or point me towards a source for your comment on inertia screening? – James Pattarini Jul 20 at 16:51
@James Pattarini frame-dragging – Anixx Jul 20 at 17:00

If you consider that rotation is relative then an observer on a rotating body may regard herself as stationary with the universe rotating about her. In such a view, to avoid exceeding the speed of light, distant objects must have orbits less than c/omega so space must be extremely distorted. Most of the mass in the universe will have a relative velocity close to that of light and will thus have hugely increased mass and therefore enormous gravitational fields. It seems quite possible to me that the resulting net force on the observer will increase linearly with distance from the centre of the frame as do centrifugal forces.

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The honest answer should be no body knows for sure. Einstein was inspired initially by this principle when he was formulating GR. However it gradually became clear that there are solutions to GR field equations which does not confirm to Mach's principle. Einstein was thoroughly disillusioned in his latter years. As a matter of fact GR has some features which are definitely "Machian" but other features which are "non Machian". Whether this principle is right or wrong has been always subject of controversy. The present opinion of most experts seem to be totally negative about the Mach principle.

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Actually, even if we have an ultimate theory of physics, the question of whether Mach's principle is true won't be settled. The main problem is that the principle itself is vague, not that we don't know the physics. – Ted Bunn Feb 19 '11 at 17:05
@Ted: Nevertheless, many great minds have been intrigued by this principle. – user1355 Feb 19 '11 at 17:11
Agreed! There's no doubt that Mach's principle is of historic importance and is fun and salutary to think about, even if it's not sufficiently well-defined to admit of being "true" or "false". – Ted Bunn Feb 19 '11 at 17:29
Dear Gordon, there are no serious physicists who want to test Mach's principle in 2011. Such things were being solved by Einstein in 1911. – Luboš Motl Feb 19 '11 at 17:49
@Luboš: "there are no serious physicists who want to test Mach's principle in 2011. Such things were being solved by Einstein in 1911." This is a mischaracterization of the history of the subject, in that it ignores a major theoretical and experimental program of the 60's and 70's. Einstein "solved" the problem only in the context of GR, and in fact didn't come to the correct anti-Machian interpretation of GR until decades later. It was only ca. 1970 that experiments got good enough to determine whether the universe was actually as non-Machian as predicted by GR. – Ben Crowell Aug 12 '11 at 14:32

Many researchs has been carried out in last dacade on the machs principle. Researchers are trying to develop the theory based on mach's principle which can explain the universe properly. There are some recent papers where it is shown that the mach principle can explain quantum mechanics and also the cosmology without dark matter and dark energy. Therefor is not correct to say that the mach principle is wrong. You can go through these following research papers. They are ofcourse extremely mathematical, but most probably anybody can understand the logic. machian agravity and giant galactic forces. on the wavy mechanics of particles.

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Both papers are by Santanu Das, who appears to be a total crank. No, Mach's principle cannot explain quantum mechanics. – Ben Crowell Jul 25 '13 at 22:34

protected by Qmechanic Jul 26 '13 at 2:55

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