A Privileged Frame of Reference?

A few decades after Galileo’s lifetime, German philosopher and mathematician Gottfried Wilhelm von Leibniz argued that space doesn’t exist independently of physical objects. Thus, there is no fact about how fast something is really moving, but only facts about how it is moving in relation to other objects. This idea, known as relationalism, seemed to follow naturally from Galilean relativity. According to Galilean relativity, any inertial frame is as good as any other (so far as the laws of motion are concerned). This suggests that there is no privileged frame of reference: no specific frame that is uniquely correct about which objects are really at rest, while other reference frames have it wrong.

One of Leibniz’s contemporaries, the eminent physicist Sir Isaac Newton, disagreed. This was not their only disagreement, by the way. Leibniz, as you may recall from chapter 2, was the guy who invented calculus around the same time that Newton did, prompting them to accuse each other of plagiarism. Newton thought of space itself as a physical substance, a sort of container in which motion takes place. (This view has been called substantivalism.) The speed of an object may be measured from the perspective of any inertial frame; but according to Newton, only one of those reference frames is objectively correct: the reference frame of space itself. The speed and direction of an object’s motion relative to space itself is the object’s true velocity. If an object is at rest relative to space itself, it is really at rest; otherwise, it is really in motion. Or so Newton contended.

In defense of this view, Newton pointed out that although Galilean relativity doesn’t privilege any specific reference frame, it certainly doesn’t consider all reference frames alike. Inertial frames are special. They are the reference frames in which the laws of motion hold. Moreover, all inertial frames agree with each other about many things: not only do they agree about the laws of motion, they also agree about which objects are moving with constant velocity and which are accelerating. In fact, all inertial frames agree about the magnitude and direction of an object’s acceleration. (They may disagree about the speed and direction of the object’s motion, but they still agree on how the speed and/or direction are changing.) For instance, if a falling apple accelerates toward the earth at 9.8 m/s2 from the perspective of one inertial frame, all other inertial frames will agree that the apple is accelerating toward the earth at 9.8 m/s2.

It would be an uncanny coincidence if all of these inertial frames just happened to agree with one another about something that was merely a matter of perspective. Surely those things about which all inertial frames agree must be absolute matters of fact, objective truths about the world. If all inertial frames agree that an object is accelerating, it must really be accelerating as a matter of fact, not merely as a matter of perspective.

Rotation is a form of acceleration: water near the edges of the bucket is constantly changing direction as it moves in a circle.

Furthermore, there are observable differences between objects that are really accelerating and objects that aren’t. To illustrate this fact, Newton described a simple experiment in which a bucket of water hangs from a twisted rope. As the rope begins to unwind, the bucket and the water it contains spin faster and faster. Initially, the surface of the water is flat; but as it spins faster, the water’s surface takes a concave shape, climbing up the sides of the bucket. There is a noticeable difference between water that is rotating and water that isn’t. This difference has nothing to do with the water’s rotation relative to the bucket, as Newton points out:

At first, when the relative motion of the water in the vessel was greatest, it produced no endeavour to recede from the axis; the water showed no tendency to the circumference, nor any ascent towards the sides of the vessel, but remained of a plain surface, and therefore its true circular motion had not yet begun. But afterwards, when the relative motion of the water had decreased, the ascent thereof towards the sides of the vessel proved its endeavour to recede from the axis; and this endeavour showed the real circular motion of the water.The Mathematical Principles of Natural Philosophy (1846), Book 1, in the scholium to the “Definitions” section. The entire text is available here.

In other words, the bucket begins to spin before the water does, so in the non-inertial reference frame of the spinning bucket, the water initially appears to be spinning (even though it isn’t really spinning). Later, when the rotation of the water catches up to that of the bucket, the water is at rest in the bucket’s reference frame; yet its surface is concave (because it is really spinning, objectively, even though it isn’t spinning relative to the bucket). Newton concludes that the rotation of the water must be an absolute, non-relative matter. But if relationalism were true, there could be no fact of the matter about whether the water is really spinning. Therefore, relationalism is incorrect.

Newton’s “bucket argument,” as it is often called, convinced most physicists that relationalism was wrong. Leibniz lost the debate.Later relationalists (notably Ernst Mach) tried to answer the argument by suggesting that the stars—or, more plausibly, the distribution of matter throughout the universe—are somehow responsible for the behavior of water in a spinning bucket. For example, one hypothesis speculated that the stars exert an outward force on anything that rotates relative to them. This could explain why water climbs the walls of the spinning bucket, even if there is no absolute (non-relative) fact of the matter about whether the water is really rotating. As it turns out, such hypotheses were not entirely off the mark: Einstein’s General Theory of Relativity implies that the distribution of matter does play a role in determining which frames of reference are inertial. Nevertheless, as we’ll see, Einstein’s theory still doesn’t justify the sort of relationalism that Leibniz proposed. But as it turns out, Newton wasn’t entirely right either. Even if Newton is right that acceleration is an absolute (non-relative) quantity, this doesn’t imply that velocity must also be absolute. It is tempting to reason as follows: if acceleration is an absolute quantity, then velocity must be absolute too, since the acceleration of an object is just the change in its velocity. That reasoning is mistaken, however. The acceleration of an object can be determined even if there is no uniquely privileged frame of reference. Since all inertial frames agree on the acceleration, the distinction between accelerated (non-inertial) and non-accelerated (inertial) motion can be explained in terms of Newton’s laws regardless of whether substantivalism is correct. In order to determine whether there is a privileged frame of reference, therefore, we’ll have to look beyond Newton’s laws.