A Matter of Principle

Brian Koberlein

Imagine taking a bucket of water and spinning it. As you rotate the bucket, the water would fling outward a bit, so that the surface of the water is concave. Compare that to a bucket of water that isn’t spinning, so that the water is flat. If both buckets are perfectly smooth and symmetrical, and the water is perfectly calm, the only difference you would see is that one bucket has concave water and the other has flat. But why are they different? You might argue that it’s because of centrifugal force, because that’s what happens when you rotate something. But if motion is relative, then rotation must be relative to something, but what? This idea is known as Mach’s principle, and it isn’t as easy to answer as you might think.

This bucket example is sometimes referred to as Newton’s bucket experiment, because Isaac Newton used it as an example of absolute space. Newton thought that space and time were absolute frames of reference and that all motion could be described relative to that absolute frame. For him, the rotating water bucket was a clear example of this. Even if the bucket were the only thing in the universe, Newton argued, the rotating water would appear concave because of its rotation relative to the absolutes of space and time.

While this became a dominant view, there were still some problems with it. For one thing, although “absolute” space and time seemed intuitively correct, there was no way to determine what that absolute frame might be. The Earth and Sun clearly moved through space, as did the other stars, so an absolute frame wasn’t clear. Even Galileo argued that motion was relative.

In the late 1800s, Ernst Mach argued against Newton’s absolute frame. Mach wasn’t the first to do so, but his argument became the most widely known. Mach thought that in Newton’s “1 bucket in the universe” model, the water wouldn’t appear concave. Without an absolute frame of reference, rotation of a single bucket in the universe had no meaning, since there would be nothing for it to rotate relative to. The reason real water in rotating real buckets appears concave, Mach argued, is that the bucket rotates relative to the distant stars. In principle we can always measure rotation relative to the most distant stars, so they act as a kind of effective absolute frame. For Mach, rotation was relative to the distribution of mass across the universe.

Einstein thought Mach’s argument was incomplete, a kind of philosophical kicking of the can down the road. If motion is truly relative, then how can we know that the distant stars are not themselves rotating relative to an even more distant frame? In Einstein’s model, the global structure of space and time is what determined motion and rotation. So the rotation of an object is relative to the very structure of space and time. Even a single bucket could experience rotation. In general relativity this effect can be observed. When a mass rotates, it twists space and time around it. This is known as frame dragging (or more formally the Lense–Thirring effect), and has been measured experimentally.

While it would seem in the end that Einstein was right, the question of Mach’s principle isn’t completely resolved. The standard solutions of Einstein’s theory assume that distant space isn’t rotating. Theoretically one doesn’t need to make that assumption. In fact Kurt Gödel devised a rotating universe model that agrees with general relativity but specifically violates Mach’s principle. Known as the Gödel universe, it has strange effects like time loops (closed time-like curves). It also doesn’t look anything like the real universe.

What we know is that the universe as a whole doesn’t appear to be rotating, and so Einstein’s assumption does work. That’s the way the universe works, but why that happens to be the case isn’t entirely clear. Mach’s principle remains an interesting concept without a complete solution.