# Mach's principle, GR and the nature of space

1. Jun 16, 2013

### epovo

I have been puzzled by this for years, so I would welcome some enlightenment.
It seems that Einstein was enamored with Mach's principle while searching for GR, but in the end GR does not seem compatible with it - or rather has nothing to say about it.
What I mean is that the proverbial spinning skater's arms are pulled out because rotation is absolute even for GR even if the skater is far away from any star. Where as Mach conjectured that it is the presence of the rest of the universe (however far it may be) that confers some meaning to the notion of rotation and therefore produces the pull of their arms, it seems that this idea is now abandoned. I tend to think that Mach's principle came from the notion that empty space is nothingness. This is not the prevalent idea now. Space seems to be a bubbly thing after all, and filled with the Higgs field. Is this the thing against which rotation happens? What is the current thinking about this?
Thank you.

2. Jun 16, 2013

### WannabeNewton

Some of the ideas in Mach's principle are compatible with GR but most aren't. The idea that matter "content" everywhere in the universe affects the local laws of physics is at the heart of GR and probably can't be made more evident than from the effects of frame dragging, the Lense Thirring Precession, and the definition of "locally non-rotating" observers in GR. However as noted there are many aspects of Mach's principle which are not compatible with GR so GR is for the most part non-Machian.

This is one of my most favorite papers on the topic and in the field of gravitation as a whole: http://blogs.epfl.ch/document/19038

3. Jun 16, 2013

### Bill_K

epovo, I can't speak for the current thinking, only my own opinion, and I largely agree with you, only even moreso. Mach's principle was a vague pre-Einsteinian notion, best forgotten! Whatever intuitive appeal it once had, Mach's principle is inconsistent with much of physics, and certainly inconsistent with General Relativity, which gives an absolute definition for the local nonrotating rest frame that's independent of what the "distant stars" are doing.

4. Jun 16, 2013

### WannabeNewton

Bill, while I agree the so mentioned rotation is absolute, how is the definition of locally non-rotating not affected by the matter content in space-time? The family of locally non-rotating observers external to a static spherically symmetric mass is not the same as the family of locally non-rotating observers external to a rotating spherically symmetric mass.

5. Jun 16, 2013

### Staff: Mentor

Does it? In order to specify such a frame, you have to know the metric. In order to know the metric, you have to solve the EFE. In order to solve the EFE, you have to know the stress-energy tensor--or else you have to at least have an ansatz for one. Or, I suppose, you could work backwards, starting with an ansatz for the metric, computing its Einstein tensor, and multiplying that by $8 \pi$ and calling it the stress-energy tensor of your solution. (Which, of course, raises the question of how physically reasonable the resulting SET is, but I don't think we need to open that can of worms here. )

But in any case, you always have *some* specification of the matter-energy content of the spacetime, the SET, associated with your metric, and therefore associated with your definition of locally nonrotating frames. Which, to me, says that GR does *not* give an absolute definition of local nonrotating frames that is independent of what the "distant stars" are doing, because what the distant stars are doing is part of the overall SET of the spacetime, and therefore part of the overall solution from which your definition of local nonrotating frames is derived.

Does this mean that Mach's Principle *is* part of GR after all? IMO it depends on what you think "Mach's Principle" says. I don't really think it's a very important question, since the answer to it doesn't affect any predictions of GR.

6. Jun 16, 2013

### Staff: Mentor

Even more than that, the definition of locally non-rotating observers in an asymptotically flat spacetime (which both of your examples are) is different from that in a spacetime that isn't, such as an FRW spacetime. Asymptotic flatness itself is an assumption about the global matter-energy content of the spacetime--namely, that there isn't any other than the isolated body in the center, whether it's rotating or not.

7. Jun 16, 2013

### Bill_K

Not what Mach said. Here's what Mach said:

The Kerr metric does show that matter is affected by other matter, but you have to be careful what conclusions you draw from that. Orbits around Kerr exhibit "frame dragging", which means for example that even the orbit with angular momentum zero still "goes around" the central body. Closer to the issue is the gyroscopic precession of a Fermi-Walker transported particle (I have a blog post on this) And it happens not just for Kerr - not even just for Schwarzschild - it's a general feature of GR. So to say that rotation of the Kerr object is responsible ignores the other causes.

In place of Kerr, there's another GR solution by Thirring more relevant to the Mach issue. In linearized theory one can calculate the field inside a large, slowly rotating shell, mass M, radius R, angular velocity ω, and find there's inertial dragging inside: Ω ~ Mω/R. But notice what I said: inside a rotating shell. The rotation of the shell is absolute and can be detected and measured. So again this is all very interesting, masses do have an effect on other masses, but it's not what Mach said.

8. Jun 16, 2013

### Bill_K

I'm not sure I see the point. FRW itself is nonrotating, and won't affect the locally nonrotating observers. There are rotating cosmologies, generalizations of FRW in fact, in which the neighbors of every "galaxy" rotate about it, but they're not needed for the observations.

9. Jun 16, 2013

### WannabeNewton

Hi Bill, thanks for the response. Do you remember this problem you helped me with some while back: https://www.physicsforums.com/showthread.php?t=675475?

It was based on this problem from Wald: http://postimg.org/image/6m116zzl9/ [Broken]
In particular that very last sentence. Would you say you agree or disagree with it? This is how I intepreted Mach's principle with relation to what I said above.

Last edited by a moderator: May 6, 2017
10. Jun 16, 2013

### Bill_K

Yeah, this is the Thirring solution. I'd only disagree where he says, "in a manner in accord with Mach's principle." Mach does not deserve credit every time something rotates!

In particular, note that for this solution the precession rate is much smaller than the rotation rate of the sphere, Ω << ω. Mach's Principle would have us believe that Ω = ω.

11. Jun 16, 2013

### WannabeNewton

Right so I'm wondering in what way Wald is saying this effect is in accordance with Mach's principle? It just seems like a very loose association based on the idea that the central observer precesses due to the spinning of the thin shell which changes which family of observers are locally non-rotating. In other words, when saying such effects are "in a manner in accord with Mach's principle", how seriously is Wald interpreting the statements of Mach and is such a loose interpretation even valid? As you say it cannot be true word for word since the effect is not of the same order as the rotation of the shell itself!

P.S. he does it again in a related (albeit much easier) problem later on in the text: http://s18.postimg.org/bb2fqvf09/mach_2.png

Last edited: Jun 16, 2013
12. Jun 16, 2013

### Staff: Mentor

Sure it does. It makes them nonrotating in the same sense that "FRW itself is nonrotating". If the metric were not FRW but something different, the definition of locally nonrotating observers would also be different.

Meaning, they don't match observations. Agreed. But that just proves my point: if the metric were different, observations, including the observations that determine our definition of "locally nonrotating observers" would be different.

Once again, I'm not saying this means Mach's Principle is correct; I think that depends on what you mean by "Mach's Principle". I'm just saying that it makes no sense to me to say that the definition of anything that is determined by the metric, including the definition of "locally nonrotating observers", is "absolute", independent of the rest of the matter in the universe, since the metric is not absolute--different solutions of the EFE lead to different metrics.

13. Jun 16, 2013

### WannabeNewton

But in what sense is the world "absolute" being used here? If we go with Wald's definition that locally non-rotating observers in an axisymmetric stationary space-time are ones who follow orbits of $\nabla^{a} t$ (in the coordinates adapted to the killing vector fields) then all observers at a given event will unambiguously agree on who is locally non-rotating and who isn't because they can measure their own angular momentum $L = u^{a}\psi_{a}$ (where $\psi^{a}$ is the axial killing vector field) and if $L = 0$ then they know they have $u^{a} = \nabla^{a}t/(-\nabla^{b}t\nabla_{b}t)^{1/2}$ and if $L \neq 0$ then they know they aren't following such an orbit. However it is certainly true that the matter content affects the local standard of non-rotation since $\nabla^{a}t$ depends on the metric which itself depends on the matter content so different matter content will yield different families of locally non-rotating observers (ZAMOs in this case). For example in Schwarzschild space-time the ZAMOs will necessarily be the static observers who follow an orbit of the time-like killing vector field $\xi^{a}$ whereas in Kerr space-time the ZAMOs and the static observers will not be the same.

I'm assuming Wald is loosely interpreting Mach's principle to mean the second part regarding the matter content (based on the way he states it in the two exercises I linked above) whereas Bill's quote from Mach is saying something much different (at least from the way I interpret it) and something that isn't really in accord with the fact that observers can unambiguously agree on who is locally non-rotating, within the framework of GR.

Last edited: Jun 16, 2013
14. Jun 17, 2013

### pervect

Staff Emeritus
My position isn't quite as extreme as Bill K's, but it's close.

I'm sure Mach's principle was inspirational, but I haven't seen much actually useful come directly from it - by useful, I mean experimental predictions based on the principle itself. I'll agree that historically it was important in inspiring GR (but that's not the standard by which I"m attempting to judge it, I'm attempting to find specific experimental predictions from the principle itself).

On the other hand, I haven't looked carefully, either!

The topic is apparently interesting enough that Julian Babour had a conference about it, which inspired a book. Mach's Principle: From Newton's Bucket to Quantum Gravity (Einstein Studies). Which, I might add, I haven't read.

Without a better grounding in the literature on this particular topic, I can't say more.

I suppose I can say a bit about my opinion on its usefulness in introductory GR courses. That opinion would be that it is not a good topic for introductory courses.

15. Jun 17, 2013

### yenchin

16. Jun 17, 2013

### Staff: Mentor

That is my big problem with Mach's principle: what does it mean experimentally? You talk about the equivalence principle or the principle of relativity and you can get pretty concrete testable statements, but Mach's principle usually reduces to untestable statements about how an "otherwise empty" universe would behave.

As a result, I don't usually care too much about Mach's principle. It isn't sufficiently well-defined to determine if it is correct or not.

17. Jun 17, 2013

### Bill_K

Here's another quote:

Nothing vague about that! The local nonrotating frame is the frame in which the distant stars do not rotate. Period, end of sentence.

And thus both the Thomas precession and the Lense-Thirring effect already violate Mach's Principle.

18. Jun 17, 2013

### Staff: Mentor

In which case Mach's principle is experimentally falsified, and I care about it even less!

I still think that is vague, mostly because there are no fixed stars. They all move relative to each other. But that is the most concrete statement I have seen. I think the "fixed stars" ambiguity can be resolved.

Last edited: Jun 17, 2013
19. Jun 17, 2013

### WannabeNewton

Why does the second sentence follow from the first necessarily? The non-vanishing of the induced coordinate angular velocity of the locally non-rotating observer which would make the distant stars look like they were rotating about him/her from his/her viewpoint, according to Mach?

Last edited: Jun 17, 2013
20. Jun 17, 2013

### Staff: Mentor

For extra fun, in at least one book by prominent relativists (Cuifolini and Wheeler's Gravitation and Inertia), exactly the opposite claim is made: effects like Thomas precession and Lense-Thirring *support* Mach's principle, because they show that the rotation of nearby matter *does* have an effect on inertia. Obviously they are using a different definition of "Mach's Principle" than Mach himself appeared to use!