The Earth rotates on its axis. If you watched the Earth from the Moon, for example, you would see the oceans and continents drift past. The surface features of Earth would make it very clear that the Earth is rotating. The same is true for any object. If you were to spin a ball, for example, the texture of the ball’s surface would make it easy to see the ball is rotating. But what if the ball were perfectly smooth? Not just kind-of smooth, but so smooth its surface was absolutely uniform. In that case a rotating ball would look just like a stationary one. So how could you tell if it was rotating? 

This puzzle of rotation is more troublesome than you might think, and it strikes at the heart of some very basic physics. In physics terms, it raises the question of what constitutes an inertial frame of reference. Basically, if you were in an inertial frame of reference, you wouldn’t feel any forces pulling on you. If you were floating freely in deep space, for example, you would be in an inertial frame. Galileo used the idea of inertial frames to explain why objects seem to keep moving even when nothing is acting on it. The motion of an object is relative, Galileo argued, and an object moving at a constant speed is in an inertial frame, just as if it were sitting still.

Newton agreed with Galileo to a point, but he struggled with the idea of rotation. As an example he proposed a theoretical bucket experiment, where a perfectly symmetrical bucket was filled with water. If the bucket is rotating, the water should flare out a bit, making a dip in its surface. If it isn’t rotating the water’s surface would be flat. Since the bucket and water are perfectly symmetrical, why should there be a difference? For Newton the answer was that there must be some universal inertial frame. There is a particular absolute frame of reference against which everything can be said to move. When the bucket rotates, it rotates relative to the cosmic inertial frame, and thus the water flares out a bit.

While this seems perfectly reasonable, it was problematic for Einstein. In Einstein’s relativity there shouldn’t be any absolute frame of reference, since all reference frames can only be observed relative to each other. What would happen, Einstein wondered, if you were floating in deep space and saw the stars rotating around you. Since reference frames are relative, how would you know that you were rotating and the universe wasn’t? To address this issue Einstein invoked what he called Mach’s principle. It was originally proposed by Ernst Mach, and argued that an inertial frame is determined by the overall distribution of matter throughout the cosmos. Since all the stars in the Universe appear to be moving around you, it must be you that is rotating, and not the other way around. This all might seem like theoretical navel gazing, but it has real consequences for astrophysics, particularly for objects like black holes.

A black hole is an object so dense that not even light can escape it. One of the properties of black holes is the no-hair theorem, which means that no matter what shape a mass had before collapsing into a black hole, the resulting black hole (technically its event horizon) will be perfectly spherical. So in a very real sense a black hole is an example of a perfectly smooth sphere, rotating or not. So how do we know whether a black hole is rotating? Without any surface features to observe, we instead have to rely upon something known as momentum.

The momentum of an object is a measure of its motion. In Newtonian physics the linear momentum of an object is the product of its mass and velocity. If an object is rotating, it has an angular momentum determined by its rotational speed and how mass is distributed in an object (moment of inertia). In Einstein’s relativity things are a bit more complicated, but objects still have a linear and angular momentum. What makes these quantities useful is that momentum is conserved, meaning that whatever momentum a system has, it always has. So if a rotating mass collapses into a black hole, the angular momentum it had before collapsing must still exist for the black hole. Since a rotating black hole must be spherical thanks to the no-hair theorem, how do we distinguish between a rotating and non-rotating black hole? It turns out angular momentum twists space.

In general relativity gravity is not a force, but a curvature of spacetime. This means that gravity affects the overall structure of space and time. According to Mach’s principle, the overall structure of space and time must be inertial, and any rotation (angular momentum) must be relative to the global structure of spacetime. This means that angular momentum must twist spacetime near a rotating mass. As a result, any object near a rotating black hole wouldn’t simply fall straight in, but would appear to spiral around the black hole as it falls toward it. This effect is known as frame dragging, and we’ve actually observed it in Earth orbit. Frame dragging is how we can distinguish a rotating from a non-rotating black hole.

So long after a rotating mass collapses into a black hole, and long after all that matter is hidden behind the black hole’s event horizon, the angular momentum of the mass still leaves its mark by twisting space around the black hole.

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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.

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