Mach's principle and boundary cases

[happyfunbot]

Okay, so I've read Max Born's book twice and believe I now have a reasonable, high-level understanding of general relativity. The thing I'm now trying to tease out is Mach's principle. One thought experiment that I've been falling asleep thinking about is how the presence of distant masses accounts for centrifugal force, and how our existence in spacetime would be different were we close to the edge of the universe (if such a thing makes sense).

So, if I'm twirling, I feel a centrifugal force that wants to pull pieces of me away from the axis of rotation roughly in proportion to their distance from this axis. Given that I do not have the background in math to grind through the calculations on my own (assuming my question even has a closed solution), is there a qualitative explanation for how the rapidly moving distant masses account for this pull on me in my own frame of reference in which the skies are spinning?

Relatedly, what would I experience were the Earth close to the edge of the universe, with almost all the distant mass located in one half of the sky? I'm not sure this question even makes sense because I do not know what the structure of the universe is (e.g., are all geodetics closed spacelike curves?).

Pardon my newbism. :-)

 

[Bill_K]

Mach's principle historically was a nebulous idea that preceded relativity, and is not really part of current physics. There are solutions of General Relativity that seem to validate the idea, and others that clearly refute it. What constitutes a rotating reference frame can be defined locally without any reference to the distant stars. Just as acceleration is absolute, rotation (or nonrotation) is also absolute.

 

[happyfunbot]

Mach's principle historically was a nebulous idea that preceded relativity, and is not really part of current physics. There are solutions of General Relativity that seem to validate the idea, and others that clearly refute it. What constitutes a rotating reference frame can be defined locally without any reference to the distant stars. Just as acceleration is absolute, rotation (or nonrotation) is also absolute.

How is acceleration absolute? Acceleration can be defined only by reference to an inertial frame: if you have nothing else to which to refer, you're back to the concept of absolute space, which GR rejects.

With respect to linear accelerations, there must be something about the distribution of mass by itself that determines a preferred rest frame for me when I (say) stomp on the accelerator of my car. Why would the same not be true of rotational frames? What determines a preferred orientation of spatial axes?

Referring to one example cited early on in the book, if the universe comprises two balls spinning at different rates about the axis joining them, what *besides* absolute space determines which one, or both, should be experiencing centrifugal force? This is a degenerate example and likely has no relation to actual existence in real spacetime, but it is still useful as a thought experiment. According to GR, the only thing that can determine the preferred rest frame is the distribution of mass itself, which brings me back to Mach's principle.

 

[Mentz114]

How is acceleration absolute? Acceleration can be defined only by reference to an inertial frame: if you have nothing else to which to refer, you're back to the concept of absolute space, which GR rejects.

Any observer who experiences proper acceleration can measure it locally in many ways. It is not relative in the way velocity is. Rotation can be detected also because it is accompanied by proper acceleration and the Sagnac effect to name but two.

... which brings me back to Mach's principle.

Pointlessly. It is not a productive idea IMO. ( as Bill_k has said).

 

[bcrowell]

Mach's principle historically was a nebulous idea that preceded relativity, and is not really part of current physics.

I would not put it that way. There was a time when Mach's principle was nebulous, but it became less nebulous in the 1970's when it became clear that there were physically well motivated theories like Brans-Dicke gravity that were consistent with observation and that were more Machian than GR. Mach's principle then became a concrete, testable thing: does the universe work according to GR, or according to Brans-Dicke gravity? Further observations (mainly solar-system tests) then showed that the universe behaved according to the less Machian theory, GR, not the more Machian one, Brans-Dicke. There is an excellent popular-level account of this in Was Einstein Right? by Clifford Will.

There are solutions of General Relativity that seem to validate the idea, and others that clearly refute it.

Again, I would state it very differently. GR is a theory with a certain level of Machian-ness (Machinity? Machiality?). If it exhibits some Machian characteristics and some non-Machian characteristics, that tells us nothing about whether Mach's principle is true. What tells us the truth or falsity of Mach's principle is observations. Solar-system tests (e.g., from the Cassini probe) tell us that reality is not very Machian.

Pointlessly. It is not a productive idea IMO. ( as Bill_k has said).

Tell that to Brans, Dicke, and Will, who spent their entire (very productive) careers testing it, and ended up by coming up with a concrete yes/no answer.

 

[OnlyMe]

The thing I'm now trying to tease out is Mach's principle. One thought experiment that I've been falling asleep thinking about is how the presence of distant masses accounts for centrifugal force, ...

As mentioned in an earlier post, Mach's Principle is not often referenced in current theory. Einstein both gave it the name "Mach's Principle" and tried to incorporate it into the field equations of general relativity with no success.

Think of Mach's Principle as an attempt to explain the source of inertia and the spinning bucket as both a thought experiment and practical example of inertia, first introduced by Newton. Inertia is an objects resistance to a change in motion, or in other words.., an objects resistance to acceleration. And the spinning bucket even if spinning at a constant velocity, is also constantly accelerating toward the center...

The source of inertia is still an unsolved mystery.

Mach proposed that inertia was the result of the gravitational force of all matter in the universe on individual objects. So it is the gravitational force of those distant masses not the independent velocities, which would be associated with inertia and the centrifugal forces in the bucket experiment.

 

[bcrowell]

Here's a FAQ I've written on this.

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

 

[Mentz114]

Tell that to Brans, Dicke, and Will, who spent their entire (very productive) careers testing it, and ended up by coming up with a concrete yes/no answer.

You must relalise that's not how I meant it. Pace Brans, Dicke and Will. Didn't they came up with very low 'mach index' for our cosmos ?

I don't see how inertia can be caused by 'all the mass' in the unverse.

[Edit] Posted simultaneously with the post above.

 

[bcrowell]

You must relalise that's not how I meant it. Pace Brans, Dicke and Will. Didn't they came up with very low 'mach index' for our cosmos ?

Yes. They came up with a definite answer. That shows that Mach's principle is a real, testable statement.

I don't see how inertia can be caused by 'all the mass' in the unverse.

The 1961 Brans-Dicke paper does a great job of explaining this.

 

[happyfunbot]

This is what I get for reading a book published in 1962. Heh. :-) I have added Will's book to my reading list. Thanks, all.