By James Ellias

Abstract: 

The Newton’s Bucket thought experiment has been thought to be evidence of either absolute space or of Mach’s Principle. This paper will show that it is evidence of neither. I show that rotating bodies deform as a result of motion of their parts with respect to one another. It follows that they do not deform as a result of their rotation with respect to absolute space or with respect to the distant stars. 

Motivation: 

1. In his Principia, Newton distinguished relative motion, the change of position of a body with respect to some other body, from true motion, the change of position of a body with respect to “immovable space.” Newton then proposed an experiment attempting to distinguish between true motion and relative motion. I will describe a simplified version of this experiment. 
2. Imagine a bucket filled with water mounted on a motorized turntable. Suppose the motor is activated, causing the bucket to rotate at some constant angular velocity. At first, the bucket will rotate but the water will not. While the bucket is rotating with respect to the water within it, the water’s surface is flat (Fig. 1)

Fig. 1: The bucket is rotating, the water is not. 

3. The bucket will begin to transfer its rotational motion to the water. After a time, the water will rotate at the same rate as the bucket, being at rest with respect to the bucket. When this is the case, the surface of the water is concave: (Fig. 2)

Fig. 2: The water and bucket rotate at the same rate. 

4. Newton argues that the deformation of the water must be caused by the water’s rotation with respect to an absolute space, and not motion with respect to external bodies. He argues this by pointing out that when the bucket is in motion with respect to the water (Fig. 1), no deformation is observed, but that when the water and bucket are at rest with respect to one another (Fig. 2) the deformation is observed. Newton thus argues that this deformation is evidence of an absolute kind of motion, as opposed to relative motion with respect to other bodies. 
5. In 1883, nearly 200 years later, the physicist Ernst Mach interpreted this experiment in a different way. Mach pointed out that the deformation of the water’s surface could be caused, not by an absolute motion, but by rotation with respect to the rest of the matter in the universe. 
6. This led Mach to what was later called “Mach’s Principle,” a general notion that somehow, the properties of bodies are conditioned by their state of motion with respect to distant matter of the universe. Although no particular relationship has yet been found between the properties of bodies and the matter of the universe at large, this thought experiment has been considered evidence for some kind of relationship of that nature, and has motivated much investigation into what the nature of this relationship might be. 
7. In my own conceptualization, rotational motion is not a special kind of motion apart from translational motion, but simply a kind of motion of an extended body’s parts with respect to one another. To demonstrate what I mean by this, consider a rotating wheel; its rotational motion is nothing more than circular motion of the wheels’ atoms with respect to its axle. When I thought about Newton’s Bucket, I realized that the deformation of the rotating body might be entirely accounted for by the motion of the body’s parts with respect to one another. If this were the case, it would refute the arguments of both Newton and Mach. Newton’s argument and Mach’s argument both rely on the assumption that the cause of the deformation can only be understood as being caused by the bucket’s motion to some external existent (whether absolute space, or to the distant stars.) 
8. Newton’s Bucket has been thought of as evidence of either absolute states of motion or of Mach’s principle and has led to a great deal of thought about these two potential implications. If motion of the body’s parts with respect to one another is found to be the cause of the deformation of rotating bodies instead, then these deformations do not constitute evidence of absolute space or of Mach’s principle. Identifying such a cause would close off erroneous lines of inquiry, freeing up time for physicists to pursue other lines of inquiry. 
9. In pursuit of this potential value, I ask the following question: 
10. Question: Is the deformation of an extended rotating body caused by the motion of its parts with respect to one another? 
11. Answer: Yes. 

Argument: 

12. In place of Newton’s bucket, consider the simpler system of two identical spherical bodies connected to one another by a spring, floating in space, quite far away from other bodies so that no external forces act on the system. Suppose that the spheres and spring are stationary with respect to one another and arranged so that the spring is entirely relaxed and at length L. Suppose that the length of the spring can be measured. Then suppose that both balls are simultaneously struck by identical cubes, each moving along a velocity vector which runs through each sphere’s center of mass. Suppose that these velocity vectors are antiparallel and each run perpendicular to the length of the spring (Fig. 3). 

Fig. 3

13. Let us say that these collisions are elastic and the cubes move away from the system after impact, no longer affecting it. The system will then rotate at a certain rate. This rotation will be accompanied by a deformation of the spring ΔL. (Fig. 4) This deformation is analogous to the deformation of the water’s surface in Newton’s Bucket.

Fig. 4

14. What causes this deformation? I will now show that it is caused by the motion of the parts of this system with respect to one another. Note that over the course of the following explanation, I will identify the cause of the deformation while only making reference to the relative positions and motions of the parts of the system, not to motion with respect to absolute space, nor to motion with respect to the distant galaxies. 
15. Let us examine in detail the process by which the collision causes the rotation and how the rotation causes the deformation of the spring. Each ball starts in a particular distance relationship with the other (L), with a relative velocity of zero. (Fig. 3)
16. After the collision of the cubes, each ball starts moving away from the other along antiparallel velocity vectors. (Fig. 5)

Fig. 5

17. At first they will each move some small distance Δx₁, in the direction of their respective velocity vectors. (Fig. 6) This will cause the spring to lengthen by ΔL₁. The lengthening of the spring will cause elastic forces F on each sphere, each in the direction of its twin. These elastic forces will increase as the spring’s deformation increases. These forces will cause each sphere to accelerate in the direction of the other sphere. The velocity of each sphere at this moment can be broken into a component which is in the direction of the other sphere V_r and a component which is perpendicular to that direction V_t. 

Fig. 6

18. The force F will reduce V_r over time, it will not reduce V_t. Until Vr is reduced to zero, the spheres will continue to move away from one another, increasing the deformation of the spring to ΔL₂. (Fig. 7) 

Fig. 7

19. After a time, oscillations will subside due to damping forces in the material of the spring, and the elastic force F will have canceled V_r completely, leaving only V_t. (Fig. 8) The spring will have lengthened to L+ΔL, with ΔL being a deformation which causes an elastic force which is just large enough to provide the centripetal force, m(v²/r) on each sphere required to keep each of them in uniform circular motion about an atom in the spring which is equidistant from the two spheres. (This atom is indicated in figure 8 with the crosshair symbol.)  

Fig. 8

Conclusion: 

20. I have thus shown that the deformation of the system is caused by changes in the relative position and motions of the two spheres, not motion with respect to absolute space, or to the distant stars. This deformation then causes a force on each ball in the direction of an atom at the center of the spring which keeps each ball locked in uniform circular motion around this atom. 
21. Note that throughout this explanation, I only made reference to relationships among the parts of the system, not to bodies outside the system or to absolute space. I didn’t even make reference to a coordinate system; I described only positions and states of motion of the parts of the system with respect to one another. 
22. A similar kind of argument can be made in the case of Newton’s Bucket, and all other deformations in a body caused by rotation. Let us return to the original question to make sure I have answered it. 
23. Question: Is the deformation of an extended rotating body caused by the motion of its parts with respect to one another? 
24. Answer: Yes. 
25. In general, when a body begins rotating, some of its parts move away from one another (Figure 5). This motion of the parts away from one another constitutes a deformation of the body (ΔL₁ in Fig.6). These deformations will cause forces between the parts of the body which act to hold the body together (The elastic force F in Fig. 6). These forces will provide the centripetal force which keeps each part locked in uniform circular motion about some one part of the body (the atom indicated by the crosshairs in Fig. 8). This circular motion of the parts is the rotational motion. The cause of the rotating body’s deformation can be explained by reference to the motion of the parts with respect to one another. No reference to absolute space or the distant galaxies is needed to explain the cause of the deformation. 
26. The value which motivated the question has been achieved. Since the deformation of rotating bodies is caused by the motion of the parts of those bodies with respect to one another, this class of phenomena does not count as evidence towards the existence of absolute space, or for Mach’s Principle. Absolute space and Mach’s principle should therefore be rejected, or different inferences should be used to prove their existence.