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I was reading up on Mach’s principle and the historical relation it had with the development of General Relativity. As I understand it, one formulation of Mach’s principle states that for all motion to be relative (even rotation/acceleration) the global mass-energy of the universe must somehow be affecting the motion of an inertial observer to create an apparent “pseudo-force”. This is in contrast to the view that the pseudo-forces seen in Accelerated frames are an indicator that the observer in that frame of reference is really in an absolute state of motion.

So to uphold this idea that “all motion is relative” Mach needed some sort of underlying mechanism to explain how the two situations could be equivalent, but whatever that underlying cause was was a mystery.

There’s an anecdote on Wikipedia that sums up the idea behind this pretty well:

“You are standing in a field looking at the stars. Your arms are resting freely at your side, and you see that the distant stars are not moving. Now start spinning. The stars are whirling around you and your arms are pulled away from your body. Why should your arms be pulled away when the stars are whirling? Why should they be dangling freely when the stars don't move?“

So my question is, in the actual mathematical framework is this reproduced? Would the stars whirling around a stationary observer somehow make their arms be pulled away, thus making the two situations, one where the observer is spinning and the stars are stationary and the other way around physically equivalent?

I’ve heard that there are frame-dragging effects in general relativity that reproduce something like this, but I’m unaware of whether or not this makes all forms of motion truly relative. I know that Einstein desired a theory such as this, but was it achieved with General Relativity and is it feasible?

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General relativity is a local theory. It only defines motion directly with respect to local reference matter.

It is known that prior to developing general relativity, Einstein thought very deeply about what he termed Mach’s principle, but controversy has surrounded the question as to whether general relativity actually incorporates the principle, perhaps largely because it was never given clear expression. If the principle merely means that we can only talk of acceleration relative to other matter, then that is clearly the case in general relativity. However, the referenced matter is always local to the matter under consideration and, generally, discussion of Mach’s principle seems to invoke a suggestion that rotation only makes sense in the context of the distribution of matter in the universe as a whole.

The origins of the discussion lie in Newton’s rotating bucket argument. Newton had observed experimentally that a concave meniscus forms in a spinning bucket hung from a rope, as the water starts to rotate with the bucket. He argued that in the absence of absolute space it would not make sense to say that the water in the bucket is rotating, and therefore that no concave meniscus would form in its surface.

Mach appears to suggest that the answer lies in the motion of the water relative to distant stars. This idea is certainly not directly expressed in the assumptions of general relativity, which is essentially a local theory. According to the restatement, N1*, of Newton’s first law as a local law, we need consider only the motion of the particles of water relative to each other:

  • N1*: An inertial body will locally remain at rest or in uniform motion with respect to other local inertial matter.

The local structure of spacetime is determined from the interactions of particles locally. An inertial frame is one in which inertial particles can maintain a state of rest with respect to each other while transmitting no net force — in effect this is the situation when there is no meniscus signifying rotation of the water.

On the other hand a converse argument can be presented. If spacetime is divided into local, and overlapping, regions, each described in inertial coordinates, then no rotation is possible in the global structure resulting by conjoining the regions. Thus we cannot say that the frame of the non-rotating bucket is determined from the “frame of the fixed stars”, but rather must say that the “frame of the fixed stars” is determined from local structures. In other words, Mach’s principle is a consequence, not an underlying assumption in general relativity.

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  • $\begingroup$ Thank you for your answer. What i’m wondering in particular though is, in General Relativity, would the case where the stars are spinning and not the observer be enough to create a pseudo-force experienced by the observer? Like what happens here in this video at 6:48 youtu.be/cPEwkMHRjZU $\endgroup$ – Thatpotatoisaspy Mar 18 at 8:50
  • $\begingroup$ If that is the case, then could you say the situation where the stars are spinning is physically equivalent? $\endgroup$ – Thatpotatoisaspy Mar 18 at 8:51
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    $\begingroup$ @Thatpotatoisaspy, the argument is to show that the stars can only appear to be spinning when viewed from a non-inertial frame. The pseudo-forces are entirely a local phenomenon, dependent only on the choice of a local frame. $\endgroup$ – Charles Francis Mar 18 at 8:59
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    $\begingroup$ I have added a link to an answer with more about inertial frames. Yes, inertial forces ("pseudo-forces") including gravity are all treated in the same way. And yes, the general principle of relativity states that local laws of physics are the same in all reference frames, not just inertial frames. General relativity dispenses with absolute space, using only the concept of reference frames. $\endgroup$ – Charles Francis Mar 18 at 11:11
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    $\begingroup$ @Umaxo “All systems of reference are equivalent with respect to the formulation of the fundamental laws of physics.” Basically, the general principle of relativity, which GR is based on assumes all motion is relative, even non-inertial. That is also what Charles Francis and the wikipedia page on the general principle of relativity states. $\endgroup$ – Thatpotatoisaspy Mar 18 at 13:55
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In the equivalence where you are spinning, your arms (and their radiations) simply 'try' to follow tangent lines (restrained by your shoulders and arms).

In the equivalence where the exterior universe is spinning, the relativistic mass of distant objects is vastly increased and your arms, being off center, are drawn toward the closest part of that distant 'shell'.

In either case, there is an equivalent gradient in the vacuum density (curvature of space). ;)

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  • $\begingroup$ Not being expert on GR, this answer seems very suspitious. Serious sources say, that a "rotating universe" is an issue for GR: en.wikipedia.org/wiki/G%C3%B6del_metric. Rotation of universe is input into GR models not an effect explained by GR. $\endgroup$ – F. Jatpil Mar 18 at 11:50
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Rotation (as a type of an accelerated movement) has to be somehow related to the universe because there is nothing outside universe. With my very poor understanding of general relativity I believe that GR does not solve the problem "what is rotating and what is not", because also in GR rotating-universe models exist: https://en.wikipedia.org/wiki/G%C3%B6del_metric. With respect to what a rotating universe rotates? Well, wrt some idealization that does not exist in reality. My point of wiev: if the whole universe would be compound of two (and not 10^80) point like particles (one positron, one electron) then these particles would necesarilly fall on each other because they constitue the whole universe and there is no an objective frame in which they could rotate. So my answer is: no, GR introduces concept of reference frame without explaining it, so not all motion is relative. "Rotating or not" is an observed but unexplained feature of the universe.

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