Mach’s Principle

What is relative? What is absolute? Is there anything absolute, or is it just relative to “something else”?

Rotation is True Motion! Is it?

The theoretical physicist Ernst Mach was pondering about those questions, thinking about Newton’s concept of an “absolute space” that made things easy. But are things really that easy?

Newton was thinking about his experiment with a bucket of water hanging on a cord in a closed room. When twisting the rope, the bucket would rotate and form a curved – parabolic – water surface. Without this rotation, the water surface would remain flat. So his question was: relative to what does the water bucket need to rotate such that it forms a parabolic surface? Under which conditions does it remain flat?

Newton’s answer to this question was the “absolute space”, the cosmos, the universe, that one absolute entity that is all around and infinite. The formation of the parabolic surface is thus an indication of “true motion”, as it can be measured and seen independently of whatever else is in the room or however the room is formed. Even without seeing the outside of the room, the parabolic surface of the water can always been measured regardless of any information about the outside. Thus, it must be an absolute quantity indication motion in absolute space.

Is rotation relative motion? Relative to what?

Ernst Mach questioned the need, and thus existence, of the concept of an absolute space, by wondering what would happen if the bucket on the cord was actually at rest and the entire universe would rotate around it? Would we detect a flat or parabolic surface in this case? And if the latter was the case, then how could we ever know whether the bucket or the universe was rotating, as both situations where fully equivalent from the observational point of view?

What if there was only one object in the universe? Mach argued that it could not have a velocity, because according to the theory of relativity, you need at least two objects before you can measure their velocity relative to each other.
Taking this thought experiment a step further, if an object was alone in the universe, and it had no velocity, it could not have a measurable mass, because mass varies with velocity.
Mach concluded that inertial mass only exists because the universe contains multiple objects. When a gyroscope is spinning, it resists being pushed around because it is interacting with the Earth, the stars, and distant galaxies. If those objects didn’t exist, the gyroscope would have no inertia.

https://en.wikipedia.org/wiki/Woodward_effect#Mach’s_principle

Mach speculated that the concept of “absolute space” is just unnecessary, and rotation is just “relative” motion as well. Spinning the idea further, he wondered how “locally” the universe would need to be to “define” rotation. What if just a subregion of the unverse was rotating, would this be sufficient to “define” local rotation? He envisioned that it is the distribution of matter in the universe that causes a local water surface to bulge: If a sufficiently large region of matter in the universe was rotating around a certain center, it would define the rotational reference there, and any water bucket that was in rotation relative to this matter distribution would show a bulge. Any water bucket rotating at the same speed as this matter distribution, would remain flat. Thus, in different parts of the universe there can be different kinds of “rotational rest frames”, just depending on the more or less local matter distribution, but always without an “absolute space” as rotation is always relative to this local distribution.

Note that this rotational effect cannot be due to Newtonian gravity since in the center of a – homogenous – matter distribution, all Newtonian gravitational forces are in equilibrium and thus cancel out. Neither can tidal effects explain it as we consider a very small-scale effect within e.g. a (rotating) cluster of galaxies. It is rather the large-scale (still local, on a universal scale) distribution of matter that defines “relative to what” rotation “happens”.

Mach’s ideas where contemporary to the invention of the theory of General Relativity by Einstein. It inspired Einstein immensely as it sounded so convincing, and particularly it gave rise to an concept to get rid of the concept of “absolute space” which Einstein despised already in his theory of Special Relativity. So when Einstein conceived a theory of gravity, he intended to built it in a way to incorporate Mach’s principle. And he succeeded in this – almost.

Gravito-Magnetism, Frame Dragging & The Lense-Thirring Effect

Einstein’s theory of gravitation comes with the Lense-Thirring Effect: A rotating mass induces a rotation in its nearby objects. This property of general relativity is also called gravito-magnetism, in style of electro-magnetism where a changing electrical current induces a magnetic field and vice versa. Due to gravito-magnetism a satellite around Earth will feel the rotation of Earth itself, a mechanism also known as frame dragging. This is not possible in Newtonian gravity.

Also on Earth we feel the rotation of the Sun that we are orbiting, which results in an angular shift at which we can observe the Sun. Remarkably, this angular shift is of the same magnitude as the angular shift due to the finite speed of gravity emitted by the Sun, resulting in a cancellation of both effects such that we see the Sun at just the position in the sky where we would expect to see it also in Newtonian gravity.

So far, Einstein’s gravity comes well with effects that perfectly fits Mach’s principle. However, to Einstein’s surprise, it turned out that his theory of General Relativity is actually violating Mach’s principle. However, it does so in an way that neither Newton nor Mach could have anticipated.

General Relativity violates Mach’s Principle

Gravitational waves are an ultimate consequence of Einstein’s theory of general relativity, caused by the fact (or rather, core assumption of the theory) that no information can be transported faster than “speed of causality” (which is identical to the speed of light in vacuum, assuming photons have zero rest mass). Therefore changes in the gravitational field can not be felt in immediately, but it takes time for a changes to travel through space to reach a certain destination. Pure space-time without matter thus can transport and contain energy as well, which in turn has a mass equivalent and influences masses around. An extreme case are the so-called Brill Waves that describes a theoretical setup where black holes can be formed from pure gravitational energy. The inertio-gravitational field is thus determined even in a completely matter-free scenario – which contradicts Mach’s principle saying that matter is required to define a frame of reference.