# What's the debate about Newton's bucket argument?

If a non-rotating bucket is all there is in the universe, then, initially, all the parts of the bucket are at rest wrt to each other.

But if we want to rotate that bucket with an angular velocity $$\omega$$, then we basically require the different parts of it to have relative acceleration wrt each other. Because if we divide the bottom of the bucket into many concentric rings, then each ring would've an acceleration $$\omega^2 r$$ towards the center, depending on the radius $$r$$ of ring. This means that the rings have relative acceleration wrt to each other. Laws of physics would take different forms for people standing on different rings. Hence, a rotating bucket is a collection of non-inertial frames having relative acceleration.

But non-inertial frames are supposed to detect acceleration in Newtonian physics. So what am I missing?

• In an empty universe (with no gravitational field) the laws of physics will take the SAME form for any observer, standing on any of the different rings. Any ring is an inertial frame. This is because there is no gravitational field and the space curvature is flat. – sintetico Jun 3 '20 at 3:29
• @sintetico But the rings have relative acceleration. In Newtonian physics, the equations are same only for inertial frames – Ryder Rude Jun 3 '20 at 3:57
• yes, they have relative acceleration. the description changes but the equation of motion does not change. You still have $F=ma$, where in this case $F=0$ in ALL reference frames. The water inside the bucket is flat. – sintetico Jun 3 '20 at 4:06
• @sintetico But acceleration is equivalent to gravity. So every frame is increasingly "attracted" to the center, the closer to the center. The water will not be flat. – Rexcirus Jun 12 at 21:39
• Also if F=0 in all frames, then a=0 in all frames, which is not true by hypothesis. – Rexcirus Jun 12 at 21:42

In Newtonian mechanics (and also relativity and quantum mechanics), a hypothetical physicists sitting in the bucket would definitely be able to do an experiment to detect that the bucket is rotating. I'm not sure why that would be mysterious. It should be noted that the velocities of the particles involved (relative to one another) are a fundamental part of the system. You cannot describe a physical situation using only the mass and position of the particles. You need to include their relative velocities. For this reason, a bucket sitting alone in an empty universe is fundamentally different from a rotating bucket sitting alone in an empty universe.

Suppose that instead of talking about the bucket's angular velocity, you talked about its linear velocity. Then it would have indeed been the case that you can't speak of an absolute linear velocity in an empty universe. The paradox is why the same logic doesn't apply to angular velocity, since they're both "velocities".

Of course, within the formulation of Newtonian mechanics, this isn't confusing. Newton's laws tell us unambiguously that there's no such thing as absolute linear velocity, but there is such thing as absolute angular velocity. Newton's bucket argument is really a metaphysical question, asking why it is the case that we have laws that seem to treat angular velocity and linear velocity differently.

• I think angular velocity is define-able because it, by definition, involves the different parts of the bucket having relative acceleration wrt each other. By 'rotating' a body, we implicitly mean for the different parts of the body to have non-zero relative velocity as well as non-zero relative acceleration. This is clearly distinguishable from a non-rotating bucket whose parts are at rest relative to each other. – Ryder Rude Jun 3 '20 at 4:08
• @RyderRude Sure, but you're just implicitly assuming the whole framework of Newtonian physics to make that argument. You've shown that from the usual axioms, you get a certain answer to the bucket question. But the real question is, why those axioms? – knzhou Jun 3 '20 at 4:12
• @RyderRude Of course, I would caution against any response of the form "that's the only logically consistent answer", because that's simply not true. (Indeed, sloppy people have often used the Newton's bucket argument in the exact opposite direction, to say that absolute angular velocity is inherently logically inconsistent!) There are a lot of logically consistent sets of physical laws. We use one that has absolute angular velocity and no absolute linear velocity because it fits with the results of experiments. – knzhou Jun 3 '20 at 4:17
• It doesn't seem absolute to me. It seems definable only because of motion of parts of the bucket relative to other parts of the bucket. Can we define angular velocity for a system of two separated point particles lying along the line x=1 wrt to the axis x=0, assuming these particles are the only things which exist in the universe. If we give consciousness to those two points, how would they detect if they're rotating around x=0 ? – Ryder Rude Jun 3 '20 at 4:27
• Instead of two points, let's make it a linear object connecting the two points. Would the linear object be able to detect its rotation along an axis which is parallel to it? – Ryder Rude Jun 3 '20 at 4:32

Newton thought that there could only be a meniscus on the bucket if the bucket was rotating relative to something. He took it to be a demonstration of the existence of Absolute Space, because his equations were formulated in terms of Absolute Space. Mach may or may not have discussed whether absolute space can be replaced with distant stars (Mach's principle was formulated by Einstein, but this was an exercise in thought, and never made precise).

The problem only exists in Newtonian mechanics, because the formulation depends on an unobservable concept, Absolute Space. Inertial frames are assumed infinite, and in uniform motion relative to each other. It is resolved in general relativity, without reference to either distant stars or Absolute Space, since we can replace Newton's first law with

• An inertial body will locally remain at rest or in uniform motion with respect to other local inertial matter

This can be used to define inertial frames locally.

(As for what you are missing, you appear to be using a relativistic concept of inertial frames, not a Newtonian one, so you don't see the problem).