# General relativity, Mach's principle and rotation

During the Gravity Probe B press conference, Kip Thorne gave the following statement:

Suppose that the entire universe were rotating rigidly instead of being non-rotating [...]. How would we know? Well, we would know because we have our gyroscopes here and we see that the stars move relative to the gyroscopes, but that's not the way it works. In general relativity, the angular momentum associated with all that motion of all the mass in the universe would drag space into a rotational motion near the Earth and the gyroscope would move right with the rest of the universe. And so it would not be possible to have the gyro fail to point at the fixed stars unless you have the Earth nearby spinning and influencing the pointing direction of the gyroscope more strongly than the rest of the universe is. This is called Mach's principle: that inertia here is determined by the rotational state of the rest of the mass in the universe.

So, if we consider a bucket of water spinning on top a planet that is not rotating, the water will assume a parabolic shape. Now, taking what Thorne said into account, does that mean if the entire universe starts to rotate in the exact same way as it was seen from the bucket's perspective in the first case, will the water assume the exact same shape? In other words, will the bucket behave precisely in the same way, with the same angular speed, acceleration and so on?

• Do you mean: by coincidence, after the bucket has a given angular velocity, all the stars rotate together? So that from the bucket rotating frame, that stars would not be rotating. Is it the question? Feb 1 at 1:31
• Consider the universe is shaped as the 3-sphere. To make it easier to imagine, let’s drop one dimension. Consider a flat universe as a surface of a sphere. Imagine we are in a spaceship flying straight. We can always orient our example for the spaceship to be flying along the equator of the sphere. This is equivalent to the spaceship being at rest while the sphere is rotating. Therefore, at least in this particular setup, the rotation of the universe is equivalent to a linear motion within the universe, which doesn’t affect anything per the relativity principle. Feb 1 at 16:35

I will give historical information

This historical information will pretty much have the effect of unquestioning the question.

What we know today under the name 'Mach's principle' was not proposed by Ernst Mach. It was first proposed as a principle by Einstein. From the start Einstein referred to the notion as 'Mach's Principle'. For context: historians of science describe that Einstein wasn't the only one at the time to do so, and that Einstein wasn't the first to refer to the notion as 'Mach's principle.

whatever the case, it is clear that Einstein referring to the notion as 'Mach's principle' made other physicist think that Ernst Mach had proposed it. But that is not the case.

Ernst Mach adhered to a very austere philosophy of physics. Mach advocated that the bar for accepting something as scientific fact or scientific principle should be very high. For instance, Ernst Mach was against regarding the existence of atoms as certain. Mach was aware of the circumstantial evidence, of course, but he argued that on the existing evidence the existence of atoms should not be declared scientifically certain.

In his book about newtonian mechanics Ernst Mach had discussed the observation that the inertial coordinate system of the solar system does not rotate with respect to the fixed stars. Mach declared that it was out of bound for science to speculate why that coincidence is there. Science should use it, of course, but should not speculate about the origin of it.

In the following I will refer to the notion as: Einstein's Mach's principle

Einstein used an interpretation of Einstein's Mach's principle along the following lines: the inertia of a Universe is generated by the combination of all matter in the Universe. That is, Einstein preferred to regard inertia as a particular instance of gravitational effect.

Einstein expected that GR embodies that interpretation of Einstein's Mach's principle as follows: GR applied to cosmology will not allow solutions that describe a Universe devoid of matter. Einstein expected: in order for a Universe to exist at all there must be matter in it.

Historicans of physics describe: a few years after the publication of GR the astronomer Willem de Sitter found a cosmological solution that describes a Universe with no matter in it.

Historians of physics describe that for a prolonged time Einstein made great efforts to show that de Sitter's solution was invalid.

It took a while, but in the end Einstein acknowledged: the solution that de Sitter had found was a valid solution.

Historians of physics describe that in the following years ceased to mention Einstein's Mach's principle altogether. By contrast, prior to the de Sitter solution Einstein had repeatedly asserted that Einstein's Mach's principle was a foundation of GR. Einstein never published an explicit retraction of Einstein's Mach's principle, the evidence is that he ceased mentioning it.

Many years later (1954) the physicist Felix Pirani, when interviewing Einstein, asked about Mach's principle, to which Einstein replied: "Really, we should not be talking about Mach's principle anymore."

von dem Machschen Prinzip sollte man eigentlich überhaupt nicht mehr sprechen.

(Quoted in Mathematische Probleme in der Einstein – de Sitter Kontroverse, author: Stefan Röhle. Röhle gives the Einstein biography by Pais as source.)

The fact that Einstein soon abandoned Einstein's Mach's principle does not necessarily mean that other physicists should abandon the notion too, but still: it is relevant.

There are multiple interpretations/formulations of Einstein's Mach's principle in circulation. The question is not so much whether GR embodies Einstein's Mach's principle. Rather the question is: which version/interpretation of Einstein's Mach's principle, if any, is GR compliant with?

• This an extremely informative answer! And what you pointed out should always be taken into consideration when talking about Mach's principle. However, in a certain sense, my question was indeed that, that is, is the interpretation of Mach's principle which I described through the bucket experiment in my question compatible with GR? Or, in other words, would GR predict that both scenarios will give the same results? If this is still not quite clear, then I would like to know what is the most broad/general/strong interpretation that is compatible with GR? This would, I think, provide the answer. Feb 8 at 21:19
• @WordP I will try to expand my answer. However, there is this to consider: by conscious design stackexchange has a policy of specifically encouraging questions that are answerable with facts. See also: on meta.stackexchange: What types of questions should I avoid asking?, and this 2010 blog post on stackoverflow: Good Subjective, Bad Subjective If can write something to my satisfaction I will post that, but I'm not in a position to promise. Feb 10 at 1:09

If you're asking what would happen in the real world, in an experiment, then nobody knows, because it is not possible to make the whole universe rotate around the bucket in a way that is different from just putting the bucket in rotation w.r.t. Earth and then changing the frame of reference to that of the bucket.

In other words, Mach's idea, while interesting theoretically, and stimulating thinking and development of theories, is extremely hard to study and check or disprove experimentally. It's on the borderline of unfalsifiable.

• While you are correct, I guess my question is about what modern physics would predict in this hypothetical scenario Jan 31 at 22:31
• @WordP I recommend editing the question to make that clear. Jan 31 at 22:48