# Chapter 2

## Section 2: Newton’s Bucket and Absolute Space

“There is no coherent sense in which nothingness could be the
subject for the qualities of length, breadth and depth.”

al-Farabi [3]

In 1689 Sir Isaac Newton described an experiment in which he used a simple bucket of water to acutely focus the riddles of space and time. [4] The experiment can easily be repeated. It consists of a bucket partially filled with water and hung by a rope. To perform the experiment we simply fix the bucket in position and twist the rope up enough so that the bucket will unwind when we let it go. Then, we release the bucket. (Figure 2-1)

Figure 2-1 Bucket of water.

Sounds simple right? Well, the implications of what happens next aren’t so simple to make sense of. At first, the bucket begins to spin, but the water remains mostly stationary. Consequently, the surface of the water remains flat and smooth. (Figure 2-2) However, as the bucket continues to pick up speed, its motion is communicated to the water via friction, and the water starts to spin. As it does, the water’s surface takes on a concave shape. (Figure 2-3)

The fact that the surface of the spinning water takes on a concave shape is not what is mysterious. What is mysterious is that observers from all reference frames (inertial or not) will agree that the surface of this water is not flat. This means that acceleration, against all expectations, is not relational. There is no reference frame from which an observer could claim that the water’s surface is flat. Attempts to explain how this could be have forced us to rethink our assumptions about space and time.

Figure 2-2 Bucket begins to spin.

Figure 2-3 The concave shape of spinning water.

The exact reason that this thought experiment is puzzling might appear a bit subtle at first. But if we look close enough we will discover that this experiment is capable of bringing to focus the central enigmas of space and time. You might be tempted to say that there is no mystery at all, that the shape of the water’s surface changes because the bucket is spinning, that is, the points on the bucket’s rim are accelerating (always changing direction); therefore, for the same reason that we feel pressed into the seat of our car when we accelerate, the water is pressed toward the walls of the bucket. Since the only place the pressed water can go is up, the water climbs the walls making the surface of the water assume a concave shape.

This description is fine, but our goal is not to describe how the water takes on the shape it does — our goal is to explain what it means to say that something is accelerating (spinning in an example of accelerating). Accelerating according to what, or who?

Why is it that observers from any imaginable reference frame will agree that the water’s surface is not flat? Shouldn’t acceleration be a relational concept? Shouldn’t we have to mention a comparative reference frame in order to claim that the bucket is spinning? After all, we always have to mention a reference frame when we are describing something’s position, or its velocity. The very meaning of position or velocity comes into focus only when we have a reference frame by which to define them. Shouldn’t acceleration share this property?

If acceleration did depend upon a reference frame, then it would be possible to pick a reference frame from which the acceleration of the water in the bucket would be zero. The surface of the water should look flat from that reference frame. But is there any kind of reference frame from which the surface of the water would appear flat?

It appears the answer to that question is no – and this is the mystery. All positions can be defined as the zero position (the origin) within each reference frame. And there is always a reference frame that will allow us to define any particular velocity as the zero velocity (to define any constant velocity object as not moving). But there appear to be no reference frames that allow us to set a nonzero acceleration to zero. How do we make sense of this?

Some people have argued that this condition reveals a need for a reference frame that is special – a single reference frame that somehow allows us to define acceleration in comparison to it and only it. To secure such a reference frame we must have some rational explanation for how that reference frame has come to be the frame from which all accelerated forms of motion are compared, and we have to be able to explain why quantities like position and velocity are not fixed by this special reference frame.

To entertain the possibility that this special reference frame exists let’s imagine what would happen if we removed all external references? Could an object spin if it was the only material object in the universe? If that object was a bucket of water could the surface of that water be concave? Or, would the very notion of spinning (acceleration) disappear without external references – without something to compare it to?

If we assume that acceleration is a relative quantity, then we can use the relative spinning motion between the bucket and any other object in the universe to explain the concave shape of the water. But without external references the water cannot possess a concave shape. In other words, if acceleration is a relative measure like velocity, then without some exterior object by which to compare it to, the bucket could not meaningfully possess acceleration — just as it cannot meaningfully possess a velocity without comparison.

If this is the case, then the shape of the water inside of the bucket must strictly depend upon the existence of exterior references — even if those references are light years away. This condition becomes quite difficult to explain. What’s more, if acceleration is strictly a relative measure, then the shape of the water in Newton’s bucket must depend upon the observer’s state of acceleration. This is the quintessential property of relative measures — their magnitudes are defined by comparison.

If we assume that acceleration is a relative quantity, then we can use the relative spinning motion between the bucket and any other object in the universe to explain the concave shape of the water. But without external references the water cannot possess a concave shape. In other words, if acceleration is a relative measure like velocity, then without some exterior object by which to compare it to, the bucket could not meaningfully possess acceleration — just as it cannot meaningfully possess a velocity unless that velocity is compared to something else.

If this is the case, then the shape of the water inside of the bucket must strictly depend upon the existence of exterior references — even if those references are light years away. This condition becomes quite difficult to explain. What’s more, if acceleration is strictly a relative measure, then the shape of the water in Newton’s bucket must depend upon the observer’s state of acceleration. This is the quintessential property of relative measures — they all require comparison to gain definition.

For example, if we are floating in deep space and observe a ball flying by, we will define the velocity of that ball (its speed and direction) in relation to us. If we change our velocity by firing our jet pack rockets briefly, then we will observe the ball’s velocity automatically change in response to this action. This is because velocity is strictly a relative measure; it can only be defined in comparison to something else.

If acceleration is also a relative measure and the shape of the surface of the water depends upon how much the bucket is accelerating (spinning), then it follows that if we spin ourselves around the bucket — changing the relative amount of acceleration between us and the bucket — the water should appear to us to take a different shape (climb higher or lower up the walls). When we do this, however, we find that the water’s shape doesn’t depend upon the observer’s state of acceleration at all. (Figure 2-4)

[FIGURE PLACEHOLDER]

Figure 2-4

This means that acceleration cannot be a relative measure; it must gain its meaning from a unique reference frame. Supporting this conclusion is the fact that relative measures like position, time, and velocity have a quality that sets them apart from non-relative measures — you can’t feel them. But you do feel acceleration. This property fundamentally distinguishes acceleration from any relational measure.

By ruling out the possibility that relative frames of acceleration affect the shape of the water’s surface in our spinning bucket, we are left with the condition that a unique reference frame must exist within Nature. This leaves us with a rather big problem – one that requires us to identify what this reference is. How do we do this?

This is a tricky question. Newton tried to answer this question by saying that space itself comprises the ultimate reference frame, which he called ‘absolute space.’ This claim elevates space into a thing, a noun, instead of an abstract idea; but it doesn’t reveal the properties of that noun. Nevertheless, if we briefly put aside our discomfort about not knowing what this absolute space is, we will notice that the existence of an ultimate reference frame would provide us with an intuitive solution to the questions Newton’s bucket raised. An object’s motion becomes a measure of its motion with respect to that reference frame. If absolute space is that reference frame then an accelerating object is accelerating with respect to absolute space. This makes it trivial to understand the shape of the water’s surface in Newton’s bucket.

As the bucket begins to spin, it does so with respect to absolute space; but the water, at first, remains stationary with respect to absolute space, which is why its surface remains flat. As the bucket’s motion increases and is communicated to the water via friction, the water takes on a concave shape because it is spinning with respect to absolute space. Then, when the bucket stops spinning as the rope twists up tightly, the water is still spinning with respect to absolute space — explaining why it retains its concave shape.

So, according to Newton, absolute space itself is the ultimate reference by which motion can be defined. However, when it came to explaining what this absolute space is, Newton didn’t have an answer.

When we ponder Newton’s proposed version of an ultimate reference frame we discover that it has an interesting limitation. It allows us to explain why acceleration is nonrelational — why we can feel it — but at the same time it requires that, contrary to our experience, position, time, and velocity should also be nonrelational. This mismatch needs to be reconciled. Certainly within a framework of continuity an ultimate reference frame should provide an origin for position, time, velocity, acceleration, and any other description of motion through the framework’s dimensions. So how do we explain the fact that acceleration is defined in relation to this ultimate reference frame, while position and velocity are not?

If there is an ultimate reference frame shouldn’t we be able to use it to define all types of motion? Shouldn’t this reference frame define an absolute position, velocity, acceleration, and so on, for all objects? At first blush we might expect the answers to these questions to be yes, but affirmatively answering these questions contradicts the fact that position, velocity, and so on, are relational. Given this, and the fact that acceleration is nonrelational, what we really need is a structure of spacetime that maps out a unique reference frame for accelerated motion while simultaneously forbidding us to apply that reference frame to position and velocity. To do this we are going to need to have a deeper understanding of what space and time are, and what physical characteristics they possess.

Accelerated frames are somehow different from frames of constant velocity. We can feel acceleration, but we can’t feel velocity. This difference needs to be explained. A model of spacetime that evokes an ultimate reference frame should naturally include a satisfactory explanation for why position, time, and velocity only manifest as relative measures in that model. As we search for our complete map, we need to keep this requirement in mind.

Almost two hundred years after Newton introduced his bucket, and in response to Newton’s inability to reveal properties of this absolute space that caused position and velocity etc. to be relational, Ernst Mach made the bold claim that absolute space doesn’t exist after all. When it came to Newton’s bucket, Mach claimed that a spinning bucket must be spinning compared to some ‘stationary’ external object, otherwise there is no meaning to the word spinning. Since Mach believed it self evident that space is not a physical entity—a belief that was strongly held by his predecessor Leibniz [5] —space could not be that reference.

In this view, space and time become the means by which we make measurements or comparisons between objects, or events, and nothing more. They become bookkeeping devices and not real entities. This implies that in a universe with no fixed references the meaning of a spinning bucket would be as nonsensical as a hungry buffalo nickel. Therefore, if Mach is right, the water in a bucket of an otherwise empty universe will always possess a flat surface. [6]

Whereas Newton held that a bucket in an otherwise empty universe could spin with respect to the ultimate reference, Mach firmly disagreed. In Mach’s framework there was no such thing as an ultimate reference frame. The consequence of this is that all measures of space and time (position, time, velocity, acceleration, jerk, snap, crackle, pop, and so on) lose all meaning in the absence of other objects by which to make comparisons. This follows from the assumption that space and time are strictly relative measures that have no independent physical essence.

With this idea Mach completely redirected the heading of our quest. He put us on a heading that traded one problem for another. First off, Mach’s view fails to account for the obvious difference between acceleration and the true relational measures — namely that we can clearly distinguish accelerated frames from purely relational frames because we can feel acceleration. More importantly, the fact that the shape of the water inside a spinning bucket does not change when we alter our accelerated reference frame means that acceleration cannot be a relational measure. This condition demands the existence of some sort of ultimate reference frame. Newton’s concept of absolute space (which was based on the rules of Euclidean geometry) may not possess the exact character that such a reference frame requires, but in the end there still must be an ultimate reference frame.

Two hundred and sixteen years after Newton introduced his spinning bucket, Albert Einstein published his theory of special relativity (eleven years later, he published general relativity) and profoundly impacted the debate on what space and time actually are. He argued that space and time are somehow united into a single entity called spacetime, which can possess contortions, folds, and warps. He demonstrated that an object’s motion through space affects its motion through time and that its motion through time is directly tied to its motion through space – and the shape of space.

With these revolutionary insights Einstein took the helm and changed our heading once again. He transformed the question of whether space and time are real physical entities to the question of whether spacetime is a real physical entity. Following his intuition, he directed us toward the answer to that question by showing that spacetime can possess different amounts of curvature — a measurable real property. Suddenly we could no longer doubt its physical existence. Spacetime would forevermore be known as a real physical entity, an undeniable noun.

 Is Space an Entity? Accelerated Motion? Stance: Newton Yes Relative only to the ultimate reference frame Absolute Leibniz No All aspects of motion are relative to our choice of reference frame Relational Mach No Relative to average mass distribution in the universe Relational Einstein Space and time are individually relative. Spacetime is an absolute entity with measurable properties. Geometric

Figure 2-5 Historical views of “space.” (Greene 2004, 62)