# Does Mach's Principle Work Both Ways?

1. Jul 18, 2011

### WCOLtd

When you spin around, you look out and you see the stars twirling and your hands sort of are propelled outward (by a thing we call a centrifugal force). Mach's principle supposes that these two things are not a coincidence, that rotation and centrifugal force is defined in relation to all other mass in the universe. In other words, your hands spin because you are rotating in relation to all those distant stars.

However this leads to many questions. Like how does this law operate? Is it based upon the relative ratio's of mass? Does distance between two bodies have an effect?

You may imagine that you have some godlike power to rearrange the universe in a particular way - and for the purpose of this demonstration you choose a configuration of a donut and a small ball directly in the center.

In the donut resides the vast majority of mass (say 99.99999% of all mass in the universe). In the ball only a negligible amount of mass.

The implication of mach's principle is that if you spin the donut around with your godlike powers, you notice a centrifugal bulging effect on the ball as if it's the one being rotated!

The other supposition would be that the donut bulges out, because at rest motion is defined by some arbitrary "resting" state.

The allure of this way of thinking is in a large part influenced by the case of linear motion. There is no true "at rest" state in relation to linear motion, so why should it be any different in relation to rotational motion? Intuition would have us believe that it should be no different. But this isn't anything different than speculation - which is pointless - so the real question is, What are the observable consequences?

In order to be able to get even some kid of indication of what to look for. Lets suppose that there are some determining factors. Intuition would tell me that it has something to do with the ratio of mass of the rotating system and the non rotating one, and distance away from the rotating and non rotating systems should also play a part.

Implications would be that it is harder for an object to fall into a spinning black hole (or neutron star) than it is for it to fall into one which is not rotating. This is because from the perspective of the black hole, it is the object that is rotating around it in the opposite direction, and the object behaves as if it is revolving around it and has to overcome centrifugal forces - even though it's not revolving around the heavy bodies from the perspective of the distant universe.

Mach's principle goes both ways, if a rotating body bulges because it is spinning then it must impart a bit of its perspective on the surrounding mass bodies, that is there is a small component of imparted force on the perpective as if the universe really is rotating around this object.

If Mach's principle does operate in this way, the body falling into a spinning black hole would experience an imparted centrifugal force, because to the perspective of the spinning mass of the black hole, it sees the mass as revolving around and plays into account of reality.

Is this phenomenon observed in science?

2. Jul 18, 2011

### xts

My personal way (disclaimer: I am very far from deep understanging GR) to interpret Mach's principle is that inertial reference frame is defined as such, that the total angular momentum of whole Universe is equal to 0.

Your speculations assume that there exist geometry beyond the Universe - God's inertial frame, in which our Universe may be spinned by Him.

3. Jul 18, 2011

### bcrowell

Staff Emeritus
You seem to be assuming that Mach's principle is true, but it's not true in GR, which is pretty much the only viable theory of gravity we really have right now. If you're asking for a physical mechanism underlying Mach's principle, then you'd have to specify some theory, in which Mach's principle is obeyed, as the context in which to ask the question. For instance, Brans-Dicke gravity with a small ω is fairly Machian, and it has mechanisms involving a scalar field that can be interpreted as changing the strength of gravitational interactions. But Brans-Dicke gravity with a small ω has been falsified by observation, so it doesn't describe our universe.

FAQ: Does Mach's principle really mean anything? Can it be tested empirically?

The short answer is that yes, it really means something, and yes, it can be tested empirically. It turns out that Mach was wrong, and our universe is not very Machian.

Historically, Mach's principle went through a slow process of refinement in which it became more and more well defined. Mach first stated it, in vague philosophical terms, before Einstein became a physicist. When Einstein first developed GR, he wanted it to be Machian, and he was convinced once he'd created the theory that it was indeed very Machian. His big paper on general relativity begins with a Machian thought experiment in which two planets are alone in an otherwise empty universe. If one planet is rotating about an axis that coincides with the line connecting the two planets' centers, then how can we tell which is the rotating one? Einstein claimed that according to GR, the answer would be that there would be no way to tell (because this empty universe would have no external points of reference with which to compare), and therefore the two planets would have to have identical equatorial bulges.

Once people began working on GR, it became clear that GR wasn't anywhere near as Machian as Einstein had hoped. In the two-planet example, GR *does* say that one planet can bulge while the other doesn't. Einstein was upset by the existence of the Schwarzschild metric, because it seemed un-Machian to him that GR could have an exact solution in which the gravitational field of a body could have a meaning even when there was nothing else in the universe for it to be compared with or interact with. Einstein wrote a paper claiming that gravitational waves were a mathematical artifact, because they seemed non-Machian to him; he turned out to be wrong.

During this era, Mach's principle was still poorly defined. That changed in 1961, when Brans and Dicke published a theory of gravity that was an alternative to GR, and that was more Machian. (The paper is very readable for the non-specialist.) At this point, Mach's principle became a concrete, well-defined, testable thing. In the 1970's, various relativists developed an extensive experimental program to test whether our universe behaved according to GR or according to Brans-Dicke gravity. This story has been told in an excellent popular-level book by Will [1993]. Brans-Dicke gravity has an adjustable parameter omega, which measures how non-Machian the universe is. The omega->infinity limit of Brans-Dicke gravity is the same as GR. Brans and Dicke, in their original paper, stated that the only reasonable value for a unitless parameter like this would be somewhere on the order of 1. Solar-system tests then established, over the next few decades, that omega had to be much greater than 1. The best limits to date come from the Cassini space probe, which constrains omega to be greater than 40,000. This is so much larger than 1 that according to Brans and Dicke's own critera, it should be taken as a disproof of Brans-Dicke gravity. We therefore have a definite conclusion about Mach's principle: it is false.

C. Brans and R. H. Dicke, Physical Review 124 (1961) 925

Will, Was Einstein Right?, 1993

4. Jul 18, 2011

### WCOLtd

Yes I understand all that. The conventional notion is that the universe cannot spin because there is no reference for which to judge that rotation. The notion I bring up is that inertia is defined in terms of relation to all other mass, and that the universe does rotate from the perspective of a spinning object, and that mass, - in that reference frame - imparts it's perspective on reality. What does this mean in practical terms? all mass bodies recive a centrifugal force - however slight - propelling them away from any spinning object.

Yes, I understand that was not a perfect anology but that was necessary to demonstrate the concept.

5. Jul 18, 2011

### WCOLtd

Thanks for the detailed response, I will read C. Brans and R.H. Dicke.

My intuition makes me feel like Mach's principle can't be proven false because mach's principle, as I understand it, is so ambigious that a person can mold it so that it fits into just about any model for how the universe operates. As you mentioned a more precise definition for Mach's principle exists. I'll read that paper, as you might have guessed I'm not well versed in GR or Brans-Dicke theory, so I guess I'll have to take your word for it.

It doesn't really constitute and understanding having to take someone's word for something.

If someone asks if Mach's principle is right or wrong I'll have to say wrong. Then when they ask me to explain I'll have to tell them "it's wrong because someone smarter than me said so".

6. Jul 18, 2011

### bcrowell

Staff Emeritus
That would be sad :-) There's a good popular-level treatment of all this in Was Einstein Right, by Will.

7. Jul 19, 2011

### xts

@ Bcrowell

Thanks!

I see I was totally wrong (good for me, I gave the disclaimer...)

So one more naive question before I get my copy of Will's book..

As I understand your post, Mach's principle is not true in the meaning it is not necessary for GR and that there exist valid GR solutions leading to spinning universes. So it is not a fundamental principle behind GR nor it is its implication.

So now we should change its proud title from 'principle' to 'law', and ask another question: is Mach's law true as an experimantal law applying to our Universe? Is the Universe spinning? Or rather - what are observational limits on Universe angular momentum or rotational speed? If the lower limit is non-zero: what is the rotation axis direction?
Intuition says me that even marginal rotation would have to lead to observable anisotropies, e.g. in galaxy distribution, background microwave radiation, etc., or observed galaxy axises should be somehow aligned.

Last edited: Jul 19, 2011
8. Jul 19, 2011

### bcrowell

Staff Emeritus
Rotation of the universe is just one example of something that can happen in a non-Machian theory but can't happen in a Machian theory. We have a FAQ entry about rotation of the universe: https://www.physicsforums.com/showthread.php?t=506988 [Broken]

Last edited by a moderator: May 5, 2017
9. Jul 19, 2011

### xts

Good FAQ!
If I may suggest small improvement: both Hawking's articles cited there are available on-line free. It could be worth to provide URLs as for other cited articles - not everone googles for references ;)

[ returnig to reading Hawking's article ]

Thanks again for directing there!
XTS

Last edited by a moderator: May 5, 2017
10. Jul 19, 2011

### Staff: Mentor

Can you elaborate on how omega measures the non-Machianness (if that's a word) of the universe? Or is there a short treatment of it somewhere online? (I don't have Will's book, but I have put it on my list of books to get.)

I should confess here that I've always been confused by claims that GR is not Machian. It seems to me that such claims are based on the observation that in GR, Newton's gravitational constant G is a constant (whereas in Brans-Dicke theory, for example, the "effective" G is not a constant by varies with the scalar potential). But it seems to me that G is just a conversion factor between units of mass and units of length (more precisely, the conversion factor is G/c^2, or G/c^4 if you're converting between energy and length); it appears in the equations at all only because we commonly measure mass (or energy) in different units than length, just as c arises in SR only because we commonly measure length and time in different units. If we use "geometric" units throughout, G never appears at all.

I understand that G can also be viewed as a "coupling constant", telling us how much spacetime curvature is produced by a given amount of stress-energy. But in that case I would expect the coupling constant to be determined by the small-scale details of how gravity works at the point I'm interested in (for example, by a local quantum gravity theory), not by the distribution of mass in distant portions of the universe. The latter is already fully accounted for by the Einstein Field Equation, so if "Machian" is supposed to mean that distant matter affects local motion, the GR *is* Machian. (IIRC, Cuifolini and Wheeler, in their book *Gravitation and Inertia*, take this viewpoint.)

But the Schwarzschild solution does assume asymptotic flatness, which has to have some physical reason. I believe Einstein also said that, if the universe were closed, the issue he raised would go away, because, for example, the local Schwarzschild solution around a black hole would merge, at some very large r-coordinate, into the general closed background spacetime of the universe. The only requirement to make asymptotic flatness work for the local solution would be that the overall mass distribution of the closed universe would be isotropic, so that the local region where the black hole is could be viewed as being inside a uniform spherical shell of matter at a very large radius (and inside such a shell, the metric, which would be the asymptotic metric of the local black hole solution, is flat).

I'm not sure, but it seems to me that the same logic ought to apply to the open FRW cosmological models as well as the closed one; as far as I know, none of the FRW models require any assumption of asymptotic flatness as a boundary condition, only the general assumptions of homogeneity and isotropy. In which case a local Schwarzschild solution patched to any global FRW solution could still be Machian.