by Steve Bryson
We are going to be developing a new view of the world. This new view is founded on the idea that space and time combine to form a four dimensional space that we call spacetime. Before we can do this we need to know what a space is.
The word 'space' is used many different ways in our culture. We speak of empty space, outer space, parking space, and breathing space. If you are from California you may speak of your own personal space and your mind space. The mathematician speaks of abstract spaces while the physicist speaks of configuration spaces. All of these uses of the word 'space' refer to very different things. All of these uses of the word space are completely valid.
Of all the above uses of the word 'space', the use that is closest to the one we shall use is perhaps 'parking space'. It is the place where something happens, namely the parking of a car. It is sometimes empty and sometimes full. Kids can play catch in this space. The space can have properties, like parking regulations, that effect what happens in the space.
In this book, the word space will denote the setting of our world. It is where things happen. People often picture space by picturing an empty container. In this picture, space is, in some sense, what is left over if you remove everything. In the way that I am using the word 'space', this is partly right and partly misleading. It is right in that it shows space to be the background in which things move. It is misleading in that the space that I am referring to is not defined by some container. The space I am talking about is the setting for the entire world. This picture is also misleading because the empty space is seen to be a passive background in which things happen. We shall see that the space in which everything sits can be a very active participant in what happens in that space.
So what are you supposed to picture when I say space? Think this: Space is the place where things are. This is not a complete definition, but it is enough to get started. We will be getting more precise as we go on. The fuller picture will come as we journey into different kinds of spaces.
A simple example of the kind of space that we will be talking about is called for. Allow me to present you with a simple hand held space: The page that these words are written on. If you look above where you are reading you will see that the space contains many many black marks. For contrast, here is some of the space without any marks, some empty space:
You can, if you wish, put more dark markings in this space. If you are ambitious you can even make markings that move around.
Now what about everything else, like your nose or the streetlight outside my window? Are these things in this paper space? No! Only those things on the paper are in the space of this sheet of paper.
How about another example? A different space is the room that you are now in (this may be a very big space indeed if you are outside). Look around you and name some of the things that are in this space. Things can move in this space. You can throw this book across the room, you can observe some creature navigating about, you can even move in this space yourself!
This brings us to the next concept that we must get clear about. What does it mean to move in a space? This is not hard to understand, as we certainly know what it is to move around the room. We need to be precise about this intuition so that we can talk of motion in other spaces. Motion is simply change of position in a space (we will worry about making the concept of position precise later). You move by going from one place to another in a space. If this is too vague for you, it is enough now to think this: motion is going from where you were to where you are.
When you move, you move in a particular direction. How many possible directions can you move in? Infinitely many! You can move forward, sideways, up, and any combination of these three. In the example space of the sheet of paper, a point could also move infinitely many directions, towards the sides of the paper, towards the top, or any combination of the two.
In both the space of your room and the space of the paper there is an infinity of possible directions of motion. Yet your intuition probably says that you have more directions to move in you room than a point has on the sheet of paper. This intuition is correct, and understanding this intuition in a precise way brings us to what is one of the most important property of a space.
In the space of your room, you can choose from an infinity of directions in which to move. Picture yourself now choosing one of those directions. Turn to face in that direction and start to move forward. As you move forward, notice that you are not moving sideways -- that is you are not moving to your right or to your left, at least if you followed my instructions. Notice also that you are not moving up or down. Your nose stays (more or less) at the same altitude from the floor. This should not surprise you as, after all, this is what it means to move forward! But now imagine a direction which is some combination of forward and left. You might point it out by raising your arm in that direction. Now ask yourself "am I moving in the direction of a little bit forward and a little bit left?" You are not moving exactly in that direction, but you are moving a little bit in that direction, as that direction is a combination of forward and left and you are moving forward. Now try to picture other such combinations of forward and sideways or forward and up and down. You are moving a little bit in all these directions if you are moving forward. (Something to think about: are you moving in a direction which is a combination of sideways and up and down only?) However, when you are moving forward you are not moving at all sideways or up and down. Close your eyes, get a good picture of this, and go on.
Now let's pretend we live in the paper space. So that we have a concept of forward in this paper world, let's pretend that we look like a circle with one eye.
When we move in the direction that the eye is pointing, we say that we are going forward. Now when we are going forward, we are not going to our right or left, just like walking around in our room.
Just like in the space of our room, we can choose a direction which is a combination of forward and sideways. When we are moving forward, we are moving a little bit in this new direction. There is one thing about this situation that is different, though, and that is that there is no up and down in the paper world. There is only forwards and sideways.
This is the essential difference between the paper space and the space of your room. In your room when you were moving forward there were two directions that you were not moving in: sideways and up and down. In the paper space, when you move forward, there is only one direction that you do not move in: sideways. This is because the space of your room is a three dimensional space and the space of the paper is a two dimensional space. The idea here is that a space has some special directions. These directions are special in that if you move along one of these directions you do not move along any of the other directions. The dimension of a space is the number of these special directions. Here is the definition of the dimension of a space:
The dimension of a space is the number of directions in that space so that if you move along one of these special directions you do not move along any of the other special directions.
There is one subtlety here. The dimension of a space counts the number of directions that are special in the way described above. In the space of your room, that number is three and in the space of the paper that number is two. This just means that, in the space of your room, there are three special directions. Nothing says which three directions they are. After all, you choose what forward is. Once you choose what direction forward will be, then the other two special directions are fixed. They are sideways (to your left and right) and up and down. It is important to understand that you choose the three special directions by choosing one of them. If you are lying in bed, the your forward would be towards the ceiling, your sideways is to your left and right, and your up and down is along the length of the bed. To say a space is three dimensional is to say that there three special directions of motion--not which three directions they are.
Why do we care about the dimension of a space? The real answer is that by understanding the dimension of a space we get deep understanding of how things are in that space. This is very abstract, however. Is there not a more tangible excuse for understanding dimension than this?
Of course there is! We need to know the number of dimensions of a space in order to locate objects in that space. Consider yourself in the space of your room. Holding very still, picture a point in space about two feet in front of your nose. How would you tell your cleaning person, who will be working tomorrow while you are away, where that spot in space is so that he can make sure that it is not dirty? Think about this problem a little before you go on. There are actually many ways to specify where the spot is in space. The most natural way is to say that the spot is so far from the floor, so far from this wall, and so far from this side wall. Notice that you needed three numbers to tell where this point is. You may specify the point by saying something like "It is two feet in front of me." This seems to only use two numbers (distance and direction). Actually, however, for this to work you need to specify your position in space as well, and that will take three numbers. You can be very clever and try to come up with other ways to specify the position of that point in this space. If you do this correctly, you will find that you will always need three numbers!
This is no coincidence. The need for three numbers in the space of your room is intimately bound with the fact that your room is three dimensional. This means that there are three directions that do not effect each other -- these directions are independent. The dimension of a space measures the number of independent directions in that space. If you try to indicate the position of a point in a three dimensional space with only two numbers, you are fixing only two of these independent directions. This leaves no information about the third, and the position of the point can be anywhere along that third direction. Topology, a very pretty and pictorial branch of mathematics, concerns itself in part with this extra freedom of position in abstract spaces. For our purposes, however, we are going to demand that we specify the position of an object in space completely. This means that in a three dimensional space we will need three numbers.
What about in the two dimensional space of this paper? Let's start over again in the context of this example. How would specify the position of the following dot in this paper space?
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The most obvious way is to say that it is such a distance from the top of the page and such a distance from the left side. Or it could have been from the bottom and from the right side, or any combination of these. The main thing to notice here is that two numbers do it. There are other ways to specify the position of that point, and you may wish to amuse yourself by trying to picture them. All of them will take at least two numbers to do the job (of course, you can invent ways that take more than two numbers, by using each dimension more than once). This is because the space of the paper is two dimensional.
When you are specifying the position of a point in a way that uses the smallest number of numbers, these numbers are called the coordinates of the point. Thus in the space of the paper you will need two coordinates to specify a position. In the space of your room you will need three coordinates to specify the position of a point.
How does the motion of objects in space appear in terms of coordinates? Very simple! The coordinates of something that is moving in a space change as the something moves. Thus motion is simply the changing of coordinates.
Coordinates are tremendously important if you are going to attempt to make a precise description of nature. Physicists wish to describe objects in the universe, measure their motion and by so doing come to some understanding of how the universe is put together. In order to do this, they must use coordinates to make precise, simple statements about what the objects in the universe are doing. Without this, it would be very hard to see patterns in the behavior of objects, and without patterns it would be very hard to understand how the universe works.
But wait! How can coordinates be so important? After all, the coordinates we use to describe our spaces are entirely made up by us, and we could as easily as not have made up another set of coordinates for our description! This is a very important point to understand. We impose our own coordinate systems on nature. This does not mean that there is no independent nature out there that we are observing (ignoring for now the current controversies around quantum mechanics). It does mean that if we measure something, then for it to be actually part of nature, as opposed to some artifact of our measuring process, we had better find that this measurement has the same value in all coordinate systems. It is a wonderful and somewhat surprising fact that there are things in nature that satisfy this. Relativity theory is about finding those aspects of spacetime that are independent of what coordinates we use to measure them.