Aristotle's Space-Time: A primer on space-time theories (pt 1)

Before we look at special relativity, it will be helpful to look at the historical theories of space and time which it replaced. With these in the background it will become easier to see what assumptions are being made in any of these theories, and how the picture of space and time change with each new theory.

In order to start, we need to learn how to read a "space-time diagram". This is really just a fancy word for a graph, specifically which plots a spatial dimension against a temporal dimension. We're all familiar with this concept, even if it's been a few years since our high school physics class. In the graph below (fig 1) time is plotted on the x-axis, and distance (or position) is plotted on the y-axis. The little triangle points represent the movement of a particle as a function of time.

 Figure 1: A distance-time plot as you might see in your physics textbook.

Figure 1: A distance-time plot as you might see in your physics textbook.

In general, the way to read graphs like the one above is to imagine that each step along the x-axis (time) is like a snapshot of the state of the system in question at that moment. We represent a dynamic system (i.e. one that changes) by imagining a series of static systems progressing in time. This is precisely how we'll represent space and time in the coming discussions. However, physicists are creative folks, and not content to plot space and time as we typically do, with time on the x-axis, they flip the axes, so we think of time as going "up" and space as going "left" and "right" (fig 2).

 Figure 2: The same plot as above, but with the x and y-axes exchanged. The object is moving to the left and speeding up (since it covers more horizontal distance per vertical tick). Labels also flipped for dramatic effect.

Figure 2: The same plot as above, but with the x and y-axes exchanged. The object is moving to the left and speeding up (since it covers more horizontal distance per vertical tick). Labels also flipped for dramatic effect.

This plot will be our most fundamental tool for examining theories of space and time. I have a bit more to say about such diagrams, and how to think of them, but it will be useful to think about them in the context of an actual theory.

Aristotelian Space-Time

For obvious reasons we will consider theories of space-time which are limited to two dimensions: One spatial, and one temporal. The universe in which we will be examining these theories consists, in its spatial extent, in a straight line. That's it. Organisms in this universe have two choices when they want to move: left or right. If it helps, you might conceptualize such creatures as tiny ants on a clothesline, pacing back and forth, but with no other options.*

 Figure 3: The universe.

Figure 3: The universe.

Let's meet two of these creatures, Anselm and Baruch. They both live somewhere in the universe, and we can represent that by drawing a point on the line.

 Figure 4: The universe, now with Anselm and Baruch.

Figure 4: The universe, now with Anselm and Baruch.

How do we know where to draw these points? How far apart are they? How might we measure in order to know if we've got the drawing right? It is exactly these questions it turns out, which motivate a theory of space-time! That is, how we answer these questions, what assumptions we make, will differentiate Aristotle's space-time from Galileo's, and Galileo's from Einstein's.

For Aristotelian space-time we pick an origin and a standard length measure. With these two things we can impose a coordinate system on our space (fig 5). The important thing to notice about our treatment of space is that the origin is fixed. There is, on this picture, a correct answer to the question 'who is closer to the center of the universe?' If the origin is located as it is in figure 5, then the answer is Anselm. **

Things get more interesting when we start talking about how things change in time. Figure 5 only shows the positions of Anselm and Baruch at a given instant. By looking at this picture there is no way to tell if they are moving, and if they are, in which direction and at what speed?

 Figure 5: The universe with an origin and a coordinate system.

Figure 5: The universe with an origin and a coordinate system.

 Figure 6: 1 second later

Figure 6: 1 second later

One typical way that we understand the motion of objects is by seeing their position change in time. But this requires animation, and isn't suited for drawings (and blog posts). Instead, we will return to our space-time plots from above.

Consider figure 6, which is much like figure 5 but Anselm's position is different. Suppose I told you that this figure was a snapshot taken 1 second after figure 5. We now have much more information about what's going on. We know that Baruch is (virtually) stationary, and that Anselm is moving toward Baruch (to the left) at roughly one unit per second.

 Figure 7: Consecutive snapshots of our universe.

Figure 7: Consecutive snapshots of our universe.

If we take more of these snapshots, each 1 second apart (fig 7), we can begin to see a clearer picture of how Anselm is moving. We can imagine, rather than taking snapshots, we just let the position of our objects continuously record as we scroll upwards (fig 8). This upwards scrolling is what allows us to represent time in a drawing! And here we finally have it, a space-time diagram:

 Figure 8: B stands still while A moves to the left.

Figure 8: B stands still while A moves to the left.

A few comments are in order before moving on. First, the objects A and B (i.e. our friends Anselm and Baruch) should be thought of as represented by the entirety of the (magenta and green, respectively) lines. We will call these 'world-lines', and they mark out the series of events which together constitute the object. Second, events are the points on the worldline---they are the instantaneous 'happenings' which strung together make up a persistent object. While this is perhaps a rather odd and foreign way to think about objects, you will find you quickly adapt.

This picture of space-time has some intuitive features which will no longer hold in the coming theories of space-time. To end we'll look at one in particular. Notice that in figure 8 there is no way to read this diagram as saying Anselm is at rest while Baruch is moving.*** There is an objective fact of the matter about who is moving, and more precisely, at what speed, and in which direction. This seems entirely reasonable! When I stand on the sidewalk and see cars drive past, it is as clear as day that they are the ones moving, not me.

What is surprising is that our ability to make such distinctions in Aristotelian space-time relies wholly on the fact that we posited that there was a single, fixed, objective origin. On this picture of space-time it is always possible to determine the single, unique, position of any event, as well as the speed and direction of any object. This is made possible by the fact that everyone must agree on the origin.

As we'll see next time, even though this intuitively seems plausible, there are equally plausible considerations against there being such an origin. We will then begin to see what, if anything, we can say about objects and events in such a space-time devoid of a center. This will be Galilean space-time.


*You might wonder, "What happens when the ant gets to the end of the line (that is, goes all the way left or right)?" Great question! We're going to ignore it for this series of posts, but there's no reason why it wouldn't make a great topic itself.

**The assumption that is being made in this case is that the space (i.e. the one dimensional line) has some sort of real, physical existence. The idea is that we could, in principle, determine every object's true position relative to the origin if only we knew where it was. This commitment is at the heart of the Aristotelian picture.

***The distance between them is shrinking so at least one of them must be moving!