Matt Strassler 11/14/11
One of the most confusing things about the Big Bang is that it involves an expanding universe. Any reasonable person, hearing about the Big Bang, will imagine something that he or she has seen expanding: a cloud of smoke exploding outward, or a balloon expanding as it is filled with air. This is very natural. And having imagined this, the reasonable person will ask, “But what is the universe expanding into?”
This reasonable though seemingly paradoxical question is simply the wrong question. It is a consequence of having imagined the wrong thing. My goal here is to set your thinking straight.
Let’s go back and look at Figure 3 from the Worlds of 1 Spatial Dimension article, the relevant part of which is shown at the top of Figure 1 below. Notice there are two very different representations of the aeolian line (the dimension of possible wind-directions, which include the directions north, south-east, west-north-west, etc.) One representation is as a line segment whose left end is the same as its right end. The other is as a loop drawn in a plane. Now wait a second, you might ask. These look different. The loop surrounds an area, so it has an inside and an outside. The line segment doesn’t seem to have this. So how can these represent the same thing?
Ah! This is indeed a crucial question, and the answer is vital to understanding spaces. The two representations — the loop, and the line-segment with its two ends matched — do represent the same one-dimensional aeolian line. The area that the loop surrounds is a property of the representation that we have chosen, not a property of the aeolian line itself! We must never confuse properties of pictures that we use in visualizing a space with properties of the space itself!!! That is very easy to do, but it is crucial not to do it.
As another example, it looks as though the doughnut (torus) drawn in Figure 1 has an inside and an outside. But it doesn’t. Just as a circle can be represented as a line segment with the left end and right end being the same point, a torus can be represented as a rectangle whose top and bottom edges are the same and whose left and right edges are the same. (To see that this is true, take a piece of paper. Attach the top edge to the bottom edge. You will now have a cylinder. Now you have to use your imagination to bend the left and right ends of the cylinder around so that they touch: but you will quickly see this will give you a torus.) There’s no inside or outside for our rectangle with matched edges, so there’s no inside or outside for a cylinder or for a torus. In other words, what is essential about a space is what you would learn about it if you traveled within it. To see the circle has an outside or an inside, you would have to travel across it: but if your circle is the aeolian line, that’s impossible. You cannot ask the wind to cut across the circle from north to south-east! It can only go around the circle, via east or via west and south. The only thing intrinsic to the aeolian line is the line itself!
And similarly, you cannot ask the tight-rope walker, from Figure 6 of the Worlds of 1 Spatial Dimension article, to cross from one side of the circular high-wire to the other. The only safe motion for the walker is to go around the circle one way or the other. So there is no way for the wind, or the walker, to find out whether there is or isn’t an interior or exterior to the circle.
This conceptual point is actually really important in understanding the expanding universe. If you are like most people, you’ve probably wondered (as I did when I was young), “What is it expanding into?” Well, in asking this question you are making the same mistake as for the circle: you are confusing something expanding with a representationof something expanding.
For example, you might imagine an expanding balloon. A balloon looks to us as though it is growing into the larger three-dimensional space in which it sits. But if you were an ant on the balloon, you would not know anything about some rumored interior or exterior; all you would know is that the space on which you can walk has become larger. In fact you might (as an ant cartographer) represent the space as a disk whose edge is all one point (for instance, the point where the balloon is being inflated). You wouldn’t think about inside and outside; all you’d know is that the distance between the yellow and blue points (and indeed between all pairs of points on the balloon) is growing.
Another two-dimensional space is the surface of the earth itself. Suppose you woke up one morning and the earth’s surface had doubled in size. You wouldn’t know whether the earth looked different to an observer out in space, or whether the diameter of the earth had grown. All you’d know is that when you walked to work or drove to get groceries it would take longer than it used to.
That the interior or exterior of the balloon is a property of the representation of an expanding space, and not of the space itself, is more obvious in the case of an infinite plane. It is possible for an infinite plane to expand, though it is not expanding into anything. It fills exactly the same space after it expands as before; but the distances between objects on the space (for instance, the dots shown in the figure) has grown. The plane is intrinsically growing in size; it isn’t inside a larger space, and so it obviously does not — cannot meaningfully — expand into that larger space. It just expands, period.
So it is with the universe. Like the plane just described, the space of the universe simply expands. There’s no way to see this from the outside; there is no outside. But you can tell the universe is expanding from within the plane itself: the distance between all of the big objects in the universe (in particular, between the galaxies, the universe’s great cities of stars) grows and grows as the universe expands. Over time, it takes longer and longer to go from one big galaxy to the next. That’s what the Big Bang did: it took small regions of space and made them huge. It wasn’t an explosion; it’s not like a bomb going off. It’s an expansion of space itself.