Mach’s bucket, and a non-puzzle for a non-physicist

Will some physicist please explain what appears as a non-puzzle to me?

This page is part of the author’s topics in physics.

Dear reader, consider this page the equivalent of a message in a bottle, thrown into the cyber-ocean with the hope that it is picked up by some real physicist, who will kindly explain to me where the puzzle is. However, the text is written with a general audience in mind. The puzzle I am talking about is a question that was initially put forth by Newton, who answered it to his own satisfaction; it was then re-examined by Ernst Mach in the 19th C., who disagreed with Newton and gave his own answer. Today — as far as I know — it is assumed that Einstein’s framework gives a satisfactory answer to it. I don’t disagree with any of the previously mentioned gentlemen. I simply don’t understand why there is a question at all. Why have several of the brightest minds, including some contemporary ones, been concerned with something that appears to have a really trivial answer? (You’ll see what I mean after reading.) Will somebody please explain to the author, a scientist but non-physicist? But first, it is I who has to explain what the puzzle is.

It’s the puzzle of the bucket

Take a bucket of water, said Newton. Buckets usually come with a semi-circular handle (see figure, on the left). Bind a rope from the middle of its handle, and hang the rope from somewhere, with the bucket filled with water up to the middle. Now start rotating the bucket around the vertical axis that coincides with the rope, so that the rope becomes twisted, as in the figure. Then let go. What will happen to the surface of the water? Stage 1: In the first moments after you let the system loose the surface of the water will be flat (top figure on the right). As the bucket picks up speed while spinning, its rotating wall will cause the water, reluctantly, to follow it into the rotating motion by friction. Stage 2: When the water rotates somewhat fast, its surface will become concave (bottom figure on the right). Now stop the bucket abruptly! Stage 3: The water will continue rotating for awhile, its surface retaining the concave (parabolic,(*) actually) shape, until finally it comes slowly to a rest, with its surface back to the flat state. That’s Newton’s bucket experiment. Newton thought like this: that the surface of the water becomes concave proves that the water is rotating. But with respect to what is it rotating?

It can’t be rotating with respect to the bucket, because at stage 2 both the water and the bucket were spinning at around the same speed, thus they were stationary with respect to each other, yet the water had a concave surface. When we stopped the bucket at stage 3, the water was rotating with respect to the bucket, but then it slowed down and its surface became gradually flat. So we see that the shape of the surface cannot be predicted by the relative motion between bucket and water: just by looking at what the bucket does (spinning or not) you can’t tell what the surface of the water will look like. So if the water is rotating, said Newton, it must be rotating with respect to absolute space. Thus Newton thought he brought up evidence for the existence of absolute space. That was late 17th century.

Enter Mach. This is the second half of the 19th century, and Ernst Mach does not believe in Newton’s absolute space, but is a relativist (soon to influence Einstein and his relativity, according to some), following in the steps of another great relativist, Gottfried Wilhelm von Leibnitz, who was Newton’s contemporary. Mach thought, what if the bucket were placed away of all things with mass in the universe? What if the bucket were the only thing in the universe? Would it still rotate? But with respect to what? How can one say that something is rotating if there are no landmarks around to verify its rotation? Would the water in the bucket still acquire a concave surface in a bucket-only universe? Okay, water needs gravity in order to stay in the bucket, and since we don’t have gravity in such a universe (call it a “Mach universe” from now on), Mach replaced the bucket-plus-water with two bricks tied together with a rope, and asked whether rotation would tighten the rope or it would stay loose (implying that no rotation is possible in such a universe). But I don’t have a conceptual problem with the bucket-and-water setup. Put iron filings in the bucket instead of water, and a powerful magnet underneath, to simulate gravity. All right, the magnetized iron filings will not form a smooth surface like water does, but either they will be deformed by rotation or not — you get the idea. Besides, adding a magnet is no big deal because, in any setting, we must also assume an observer who can see the result of the experiment and form an opinion. What Mach wanted to exclude was the mass of stars in the rest of the universe, he didn’t care about the insignificant masses of an observer, or a magnet, or the bucket itself. And if you insist that water or some liquid is essential, put a whole Earth-sized planet underneath the bucket! Mach would still not mind; a planet more or less is so insignificant compared to the totality of the universe that it wouldn’t make any difference in Mach’s argument. So I’ll keep talking about the “water in the bucket” experiment, keeping in mind that it might require some technical modifications for it to work, depending on the environment. (This will also save me from having to make another drawing showing bricks and ropes, by the way.)

Mach said: Nobody can know by observation whether the surface of the water would become concave in that setting, so we can only speculate about the event. And Mach did speculate: he said, in such a special universe, the water would not become concave, because it would not rotate, having nothing to rotate with respect to. What causes the force that we feel and describe as “centrifugal” is the mass of the fixed stars, i.e., the rest of the universe, he said. The water rotates relative to that, in our universe. But, subtract the stars and everything else (except the apparatus necessary for the experiment) and there will be no rotation, because Newton’s “absolute space” is nonexistent.

By Mach’s reasoning, if we inserted a single star in a Mach universe, there would be the faintest centrifugal force while the water would be rotating with respect to that single star, and the faintest dent on its surface. As we kept adding stars, approaching the mass of our familiar universe, we would likewise approach the observation of the familiar bucket-plus-water experiment.

The reader can find a nice reference to read all the above in Brian Greene’s “The Fabric of the Cosmos”.^[1] In the same book, Greene tells us what the present-day verdict about Newton’s and Mach’s bucket is: according to general relativity, granted, there is no Newton’s absolute space, but there is Einstein’s absolute spacetime, and that is with respect to what the water rotates, and thus acquires its concave surface.

(So, my understanding after all this is that Einstein would agree with Mach: in a specially crafted universe that contains only the experimental apparatus and nothing else, the water surface would remain flat because the absolute spacetime would be absent. At least this is implied by Greene’s book. This does not affect in any way what I am about to describe, but correct me please if my understanding of Einstein’s view is wrong.)

What is the significance of a water bucket anyway?

The water-in-the-bucket experiment is really about inertia and mass. In Newton’s view, because we and everything else are immersed in absolute space, we feel a force when we accelerate (or decelerate) with respect to that space. Note that every object that does not move at constant speed along a straight line experiences acceleration (in the general sense, including deceleration), and so do the water parts, since they move circularly. In Mach’s (and I suppose Einstein’s, and Greene’s, etc.) view, we feel that force because we accelerate with respect to the distant fixed stars (with respect to absolute spacetime, as per Einstein, etc.). Without a frame of reference, in a relativistic universe there can be no acceleration, no force felt, no inertia.

I don’t know if physicists would say the water and the bucket still have mass, even in a Mach universe. Perhaps they do have mass, but without inertia we would be unable to measure their mass in the familiar way, i.e., by how much force they exert on something that accelerates them, because they can’t accelerate. I conclude that if you kicked the bucket (literally, not figuratively!) in such a universe, you’d feel nothing, no force against your foot, otherwise you’d be able to measure its mass (perhaps in a painful way).

Is all this fuss with the absolute-whatevers necessary?

I think not. I think neither distant fixed stars, nor absolute spacetime is necessary for inertia to be observed, and this is what I hope some physicist will kindly either confirm, or reject (and explain the rejection), after reading my explanation. I think the water in the bucket can rotate even in a Mach universe, and will acquire a concave surface. Allow me please to explain my thought: why no reference frame based on distant stars or absolute spacetime is needed for acceleration and inertia.

The claim that I explain below is that the water would rotate with respect to the frame defined by itself and the bucket. But before we can see that, it’s better to consider a situation as simple as possible.

Suppose I am in a Mach universe, standing nowhere, and have in front of me two ball bearings connected with a metal spring. I approach my finger to the ball on the right (as in the figure below), and try to push it. What will happen?

Will the spring be compressed, causing the ball on the right to approach somewhat the ball on the left, as the in following figure?

I think it will. I can find no physical reason to explain how the absence of mass in the rest of the universe will prevent the spring from being compressed. To see this, we can examine what would happen at the biological and molecular level.

First of all, would I be able to move my arm?

I guess so, because the energy required for my muscles to contract and extend is produced by the mitochondria of my cells, the function of which is based on some chemical reactions. Those reactions have nothing to do with stars, nearby or faraway, fixed or moving. My muscles will work, and my finger will approach the rightmost ball. Will the ball be pushed?

Again, I think yes, it will. When my finger is about to touch the ball, at the molecular level, the outermost atoms of the molecules of my finger will approach the outermost atoms of the molecules of the ball. Atoms will approach atoms. This means the electrons of the outermost shells of those atoms will approach the electrons of the outermost shells of the other atoms. Electrons will approach electrons. When the electron shells will come too close, virtual photons will be exchanged probabilistically (at times not determined by any of the laws of physics we are aware of today), and the two atoms will recoil. This is the basic functioning of the electromagnetic force. It is the same force that allows you to sit somewhere now and not pass through objects like a ghost. The electromagnetic force will work, as it always does, independent of distant stars, planets, their alignment, and your horoscope. Why should two electrons care whether the rest of the universe exists?

Now, the recoiling atoms will propagate their action to nearby atoms, and those to nearby atoms, and so on, throughout the mass of the impacting bodies (the ball and myself). This is described theoretically through phonons, which are entities that I don’t pretend to have understood, but I don’t think I am missing something essential here. The point is, the impact will be propagated through the two bodies that collided. Assume that my mass is substantially larger than the mass of the ball (contrary to what that wimpy hand in the figure above suggests). When I say “my mass is larger” I mean I have many more particles (e.g., molecules) than the ball. Given this, my molecules will take longer to react as a whole to the impact than the fewer molecules of the ball, and this is why a more massive body moves back (recoils) slower after its impact with a less massive body, which recoils faster. So the molecules of the ball (i.e., the ball itself) will tend to move as a whole to the opposite direction from the one I will move, only faster than me.

Now, there is a spring there. Will the spring be compressed?

Of course it will, and for the same reason for which the ball moved. I don’t need to repeat the explanation in detail, it’s again the electromagnetic force that causes the molecules of the body of the spring to acquire such positions in space that give to the spring its compressed shape. Stars have nothing to do with this.

But the job of a spring is to extend as soon as it is compressed, so the spring will push the leftmost ball further to the left. If I stop exerting any force, the system will come to the original equilibrium (spring extended) after a short vibration, the duration of which depends on how sturdy the spring is: if the spring is very pliable, the vibration will last a bit longer.

But suppose I continue exerting the force on the right. I continue acting with a constant force F. What will happen then?

My guess is that the spring will remain compressed (again, after a brief vibration, but coming to a new equilibrium in a more compressed state), as in the figure below.

I know, there is no wall on the left pushing against the leftmost ball and preventing it from moving, and in a Mach universe there is really nothing for the ball to move “into”.^(*) But my force F is applied constantly. The spring would extend back to its original shape if I stopped applying the force. As long as the force acts, the spring will remain compressed. This happens also in our familiar universe, and has nothing to do with stars. The difference is that in our universe we’ll see the whole system accelerating with respect to its surroundings, because there are surroundings and we can register and measure its acceleration. In a Mach universe there are no surroundings to use as landmark objects, so we won’t be able to observe any acceleration, but this doesn’t mean that the spring will not be compressed! The compression of the spring is a molecular phenomenon, independent of external, landmark objects.

This is the important conclusion from all the above: that the apparatus of the two balls will be compressed with respect to itself; that is, with respect to a frame of reference defined by the apparatus itself, and such a frame of reference can always be defined, since we assume that the apparatus occupies some space, however little space a Mach universe can provide.

Exactly the same reasoning can be applied to the water bucket system: a frame of reference can be defined by the system itself, and the water can be observed rotating with respect to that frame of reference. Indeed, the thought experiment with the balls connected by a spring that was described above is directly applicable to the water bucket experiment, because we can abstract the water molecules by little balls (for the purposes of this discussion), and the role of a spring is played by the intermolecular forces that keep water molecules at a certain distance from each other. It’s again the electromagnetic force that prevents water molecules from coming too close to each other. (There aren’t many forces in nature, there’s only four of them: strong, weak, electromagnetic, and gravity, and of the four, only the electromagnetic force is relevant in keeping molecules from colliding.) When a molecule is “pushed” through the electromagnetic force (the “spring”), it likewise “pushes” its neighboring molecules, and so on.

It remains to be convinced that the water will acquire a concave surface in the Mach universe.

First comes the question: will the bucket, with the rope twisted and then let free, rotate with respect to the rope? (Remember, there is a force pulling it “down”: if we use iron filings, it’s the magnetic pull of the magnet, and if we use water, then it’s the gravitational pull of the planet; I use “water” in this text to mean either of the two setups.) Yes, the bucket will rotate, because the forces that cause the twisted string to unwind are entirely local (considering the pulling force as a local one, too) and do not depend on the faraway stars. They are forces developed due to the molecular arrangement of the strings that form the twisted rope. (Need I mention it again? It’s the electromagnetic force.) “With respect to the rope” means that while the rope is unwinding slowly, the bucket rotates faster with respect to it.

Next, consider a body on a rotating plane (figure, below). Will the body tend to be pushed to the sides of the rotating plane?

The cube shown on the round table, above, is pushed to the circumference of the table because of a so-called “fictitious centrifugal force”. That is, if our frame of reference is the table, then it appears as if a force is acting on the cube (green arrow) that pushes it toward the circumference. Alternatively, if we fix our frame of reference to a non-rotating one with respect to the table, then there is no force acting on the cube. The latter simply tends to retain its state of non-rotation, and when the table rotates it forces the cube to go in a circle; the combined result of the cube tending to retain its non-rotation and of it being forced to move along a tangent to its circular path is that it follows a route that brings it closer to the circumference (unless friction holds it in place).

This is all elementary, high school physics, and it rests on our ability to consider a fixed frame of reference with respect to which the table rotates, if we want to avoid fictitious centrifugal forces. But we do have such a frame of reference in our water-bucket experiment. Recall that we have a non-rotating or slowly-unwinding rope, and that the bucket rotates faster relative to the rope.

Now, one might think, Mach might want to get rid of the rope, if he were alive to counter this argument. He might request the existence of a bucket with water only, or of two bricks tied by a rope, as indeed was one of his formulations of the problem.

But this is not possible. Something must set the system in rotating motion (or attempt to set it in rotating motion, if Mach insisted it wouldn’t rotate). There must be some mechanism: a twisted rope, an observer’s finger stirring the water (or pushing the bricks), something anyway. Magic alone will do nothing to a mere bucket with water.

It is not possible to have a pure body (e.g., a bucket with water only) without any apparatus to rotate it, and ask whether the pure body can rotate in an empty universe. Mach can’t have the pie whole and eat it.

Essentially the argument above says that the entire system prior to the rotation defines a fixed frame of reference. After the rotation, the part of the apparatus that causes the rotation and doesn’t rotate (there must be some) remains immobile in that frame of reference. The rest of the system rotates with respect to the fixed frame. Neither distant stars, nor absolute spacetime is needed.

As for the final conclusion about the shape of the water surface, it’s easy. Think of water molecules as bodies like the cube on the table, in the figure above. Consider one such molecule, closest to the bucket wall, and on the top surface of the water (golden ball in figure, below).

The green arrow in the figure above shows the direction of motion of the molecule due to the rotation of the bucket (as in the cube-on-table case, in an earlier figure). However, motion along that direction is impossible, because the molecule is prohibited by the molecules of the bucket wall (shown in solid brown, on the left). The only way of moving is along the direction of the red arrow. As soon as this molecule moves along the red arrow, its location is emptied and can be occupied by other molecules, and so on. This causes some of the molecules close to the wall to move up, leaving an empty space that becomes progressively larger as we move toward the center, which gives the overall concave shape to the surface of the water. The situation involving real molecules is much more complex than the ideal metallic balls shown in the figure above, but, roughly, the principle is the same.^(*)

The above concludes the argument by which I understand that the water will acquire a concave surface after rotating in a Mach universe, without needing to resort to notions such as distant stars or absolute spacetime to explain why. If there is any error in my reasoning, or if you have other suggestions, corrections, or comments, please feel free to contact me and let me know.

Post Scriptum, Summer 2012

After some experimental confirmation regarding the existence of Higgs boson and the ensuing discussion, it is now my understanding that, for inertia to exist, the system must be immersed in a Higgs field. Roughly speaking, the more massive an object is, the harder it is for it to change its state of motion as it makes its way through the Higgs field. An elephant has more particles than a bee. It’s harder for the elephant to change his state of motion than the bee, because many more particles from the elephant’s body than those of the bee need to negotiate with the ubiquitous Higgs field. At least that’s my naïve understanding. Now, how does this change any of the things stated earlier on this page? I’m still trying to figure that out.

What I do understand is that Einstein’s absolute space-time is not the reason for the existence of inertia. The true reason is the existence of Higgs field, which permeates our familiar universe and its space-time.

Footnotes: (Clicking on the footnote caret (^) brings back to the text)

1. (^) Thanks to Phil Jilbert, from the UK, for pointing this out.
2. (^) Of course the apparatus of the two balls and the spring will be pushed away from me. How far it will go, or how far I can keep on pushing it, depends on the size and properties of the Mach universe. But for this thought experiment to work there is no requirement that the apparatus must move far from the observer, or stay close — no fixed frame of reference is required here; we only need to observe whether the spring will be compressed or not.
3. (^) Note that the slope of the bucket wall is inconsequential: even with a vertical wall (or even with one reclining in the opposite direction) real molecules will find the way to “glide” on top of each other, unlike balls in the macro-world that can be hindered by friction.

References:

[1] Greene, Brian R. (2004). The fabric of the cosmos: space, time, and the texture of reality. New York: Knopf.

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