by Steve Bryson
For the purposes of this course, we will think of a space as the background in which things move (like baseballs). We experience walking around in a three-dimensional space. That means that we can go any of three directions (up, forward, or sideways), either one at a time or in any combination. These three directions are special, in that if you go exactly along one, you do not go along any of the others. Think this: If you go forward, you do not go up or sideways (and if you go sideways....). The number of directions (in the space) that have this special property is called the dimension of the space. Remember- the dimension of a space is a number, not a direction. You may wish to entertain yourself by trying to imagine more than three directions with this property. If you succeed, you are not visualizing something properly. Remember that if you move along one direction you are moving along the opposite direction with a negative speed. This is why forwards and backwards are not different special directions.
Following textbooks from grammar school on up, we will name these three special directions x, y, and z. (I think of them as: x is sideways; y is up and down; z is front and back).
All of this is much easier to see if, instead of thinking in three dimensions, we think in terms of two. This is like saying "I will only move forwards and backwards (one dimension) and side to side (another dimension), but I will not move up or down (this is the missing third dimension)". This makes it much easier for me to make diagrams of what I am talking about, as the paper you are holding is two dimensional. Because I am writing these notes (and it is important to understand that there is no better reason), I will choose the x and y directions. I will then represent them with two perpendicular lines called the x axis and the y axis:
In future figures, the axes will be denoted by the letter naming the direction, dropping the word axis: x axis will become x, etc..
What is the significance of these axes? They can be used to indicate the position of an object in our space. The kinds of objects we will be talking about for the near future will be points on the paper. Using the axes, we can indicate the positions of points in the following way. First, call where the two axes meet the center or origin (I will undoubtedly unconsciously interchange them). Then say how far you have to travel along the x axis from the center (in your favorite unit of measurement) until you are directly under the desired point. Finally say how far you have to travel along the y axis from the center until you are exactly beside the point:
Once you know where the center is, you only need two numbers to indicate the position of the point. These numbers are called the coordinates of the point. Of course, it is no coincidence that you need two numbers in a two dimensional space. In fact the simplest way to give the position is in terms of the directions used to define the dimension (though you do not have to do this (why?)).
What would the path of a point look like if it were moving? It would look like a line! It would be a straight line if the point did not turn, and it would look like a wiggly line if it did:
If you are having trouble visualizing where these lines come from, picture the above figure as a map and the lines are the paths of cars (here the cars correspond to the points and the lines correspond to the roads).
We are now well equipped for thinking and visualizing spaces, especially when we can ignore all but two dimensions of the space. We can now, therefore, turn to thinking about adding time to our space.
When we want to find something, we need to know not only where it is, but also when it's there. Thus in addition to the three numbers needed to locate position in our three dimensional world, we need a fourth number--time. We thus need four coordinates to locate something in our world. This gives us the clever idea of treating time as a direction that gives us a fourth dimension! Let's see if this works.
Think of yourself as moving through time, not worrying about whether or not this means anything. Stand still, close your eyes, and feel the motion through time. Are you moving in space? Not if you obeyed my instruction to stand still! So if time is thought of as a direction, it is a direction that you can move along without moving in any of the three spatial directions! It thus defines a fourth dimension! If all this seems too simple to you, you are catching on. It is simple, once we are clear about what we mean by dimension!
Now does this mean anything? What can we mean by saying that we are moving through time? Motion is change in position. If we think of time as a fourth coordinate, then the value of this coordinate certainly changes--in fact we cannot stop it from changing! We cannot stop time, though many have tried! In this sense we are certainly always moving through time. The time coordinate always changes. In this class we are going to combine the observation that we move through time with the observation that the direction we move in time seems to define a fourth dimension. We are going to find that taking this viewpoint leads us to a picture of reality that makes the results of relativity theory very natural. This, to me, says that this idea of motion through time means quite a lot.
Enough philosophy. What we have arrived at is that the space we live in is actually four dimensional. This does not mean that all four dimensions are alike. There is something very special about the time dimension. This is reflected in the fact that you cannot stop moving through time, though you can stand still with respect to the three other (spatial) dimensions. We will never come to understand why time is different (no one does), but we will, by carefully examining this difference, understand many deep facts about our world.
We start by trying to visualize our four dimensional world. There is immediately a problem. This world is four dimensional and we only know how to visualize in (at most) three dimensions. What to do? We will have to somehow reduce the number of dimensions that we are trying to visualize!
The space part of spacetime is three dimensional. It is a fortunate fact that the three spatial directions are completely equivalent. If we know one, we know the others. We can thus forget one of the spatial directions (let's drop the z direction), and reduce our problem to trying to visualize a three dimensional spacetime, with a time direction and two spatial directions. We have succeeded in making the world possible to visualize. However, not everybody finds it easy to visualize things in three dimensions. But wait! We still have a two dimensional spatial part, and these two spatial directions are equivalent. For your sake and for the sake of my drawings let's drop another spatial direction (the y direction). We are now left with a two dimensional spacetime with the time direction, denoted by t, and the x direction:
This is a spacetime diagram. We will be seeing a lot of this.
This diagram allows us to see motion in the x and t directions only. This is the same as saying that I will only move sideways (in the x direction) and, unavoidably, in the time direction.
What is a point in our spacetime diagram? It is something with two coordinates, a value for x and a value for t. So it is something that happened in a particular place at a particular time. This is called an event. Events are the points on a spacetime diagram. Examples of events are: a bat hitting a ball, a hand clap, or a firecracker going off. Note that these examples are not exactly events, as they take a (very short) amount of time to occur, but they are the closest that everyday things come.
What does an object in the real world look like on our spacetime diagram? Since we only have one spatial dimension, think about a very thin rod. Idealize this rod as a one dimensional object. Put the end of the rod at the center of the diagram. As time goes by, the rod will sweep out an area on the spacetime diagram, for it is not moving in space but is moving in time. Thus an object in the real world looks like a collection of events on our spacetime diagram:
The rod, which we idealize as a one dimensional object in space is (unavoidably) a two dimensional object in spacetime. You can think of this in several ways. I think of the rod as 'sweeping out' a two dimensional object in spacetime. It is perhaps more accurate to think of the rod as fundamentally two dimensional and we experience seeing one dimensional 'slices' of the rod at any one time (see the above diagram).
For the rest of these notes, I am only going to think about objects that are like points in our three dimensional space. Though we say that these points are zero dimensional objects in three-space, like the rod they 'sweep out' one dimensional lines on our spacetime diagram. Try to visualize the appropriate spacetime diagram before you go on.
We can now think about how motion of our point looks on a spacetime diagram. Let's imagine our point starting at the center (so that the x coordinate and time coordinate are zero) and moving steadily to the right. After a certain time, say three seconds, the point will be off to the right. This is an event in the upper right section of the diagram. After a later time we will have another event with the point further to the upper right. Further, all positions of the point in spacetime will form a line of events. Because the motion is steady, this line will be straight. Because the point is moving towards the right the line will be tilted towards the right:
What would happen if the point were moving towards the left? What if it started somewhere besides the center? Try to see the appropriate diagrams before moving on.
The line swept out by our point is called the world line of the point. What would the world line of the point look like if it were moving slower or faster? It would be like this:
Summary:
It is natural, though strange at first, to think of time as a fourth dimension. It is not equivalent to the three spatial dimensions. One way this shows up is that we cannot stop moving through time.
What would happen if the point were moving towards the left? What if it started somewhere besides the center? Try to see the appropriate diagrams before moving on.
On a spacetime diagram, an object has one more dimension than we are used to. A point (zero dimension) in three-space looks like a (one dimensional) line in spacetime. A rod (one dimensional) in three-space looks like a (two dimensional) area in spacetime.
A steadily moving point (in three-space) looks like a tilted straight line on a spacetime diagram. The faster the motion the greater the tilt.
We are now going to look at what the statement "motion is relative" looks like in terms of spacetime diagrams. Let us imagine two spatial points. For the sake of brevity, I will call them me and you, taking advantage of the appropriate pronouns. Let's say that I am standing still, and you are steadily moving right:
• (me standing still ) •»» (you moving right )
What will this look like in a spacetime diagram? Since I am watching you move, I will put myself in the center of my diagram. As I am not moving in space, my world line will stay at x coordinate = 0, so my world line will coincide with my time axis. Your world line will be tilted towards the right:
Now I decided that it is you who is moving. This is perfectly natural, since I am sitting watching you move along. Perfectly natural, that is, if you are me. However, you are not me (by definition), and so to you it would be natural to say that you are standing still and I am moving towards the left:
««• (me moving left ) • (you standing still )
The situation has not changed! Physically, these two situations are the same. We have only shifted our point of view! What does the spacetime diagram look like for this point of view? First, it is only natural that you would put yourself at the center of the diagram. I would be to the left of you. As I am, relative to you, moving steadily left, my world line will be a straight line tilted towards the left:
As a final exercise in the relativity of motion, think about the situation where you are steadily moving relative to me towards the right as above, and a third point, named her, is steadily moving towards the right relative to me faster than you. Then in my point of view your world line is tilted towards the right and her world line is tilted more than yours to the right. You, however, see your world line as vertical, mine as tilted towards the left and hers tilted towards the right, but less than I saw hers tilted:
my point of view: • (me ) •»» (you ) •»»»» (her ) --- your point of view: ««• (me ) • (you ) •»» (her )
Try to draw the spacetime diagram from her point of view.
Thus we see that the tilt of world lines that we see will depend on our own state of motion. To get a handle on the above situation, translate it into statements about who is moving what direction relative to and faster than whom.
What if she were a light ray? Then the statement "light travels at a constant velocity" would imply that no matter whose point of view you took the world line of the light would be tilted the same. This contradicts the analysis above. Resolving this conflict will give us the Special Theory of Relativity! This is what we do next time.
Let us look a little more closely at the above analysis of the relativity of motion. When we were making a spacetime diagram from my point of view, we had my world line coinciding with my time axis. This is a natural thing to do--it is the same as saying that I am always at my own position (my x coordinate is zero relative to myself). It also says that I measure my own time. After all, if I carry a watch with me, I am perfectly justified in using the ticks of this watch to measure my time. So taking my time axis to be along my world line is natural for several reasons.
Now from your point of view, for the same reasons given above (with you replacing me), your time axis will coincide with your world line. But your time axis is tilted relative to mine--We do not have the same time axis and so we will not measure time the same!!! In fact, I can ask what your time axis will look like from my point of view. In my spacetime diagram your time axis is given by your world line:
Our time axes do not (and cannot) coincide so long as you are moving relative to me. This means that you and I will measure time differently! This bears saying again.
Because you and I each carry our own time axes with us, and because when you move relative to me your time axis rotates relative to mine, we will measure time differently.
This is the insight behind so-called time dilation, or the fact that if you move relative to me we measure time differently. (Caution--It is not true that we measure the time of all things differently, only those that are in relative motion to you and in a different motion relative to me.)
At this point, we have only found that if you are moving relative to me we will measure time differently. We do not know enough to estimate the nature of this difference (slower, faster, whatever). Getting actual quantitative results will require more thinking. (In particular, you do not just try to draw equally spaced timing marks along both axes and somehow project. This will give an incorrect answer). I will now give an outline of what we will do in the next session in order to do this.
We know that your motion makes your time axis rotate from my point of view. We need to know what happens to the x axis--Does it rotate by the same amount, not rotate at all, or something else.
We will answer this question about the x axis by appealing to our demand that the speed of light be a constant from all points of view. We will find that if the x axis rotates backwards from what we would expect then we have the speed of light remaining constant from all points of view.
This strange behavior of the x axis will bring us to seeing motion as a rotation of our coordinate axes which is different from rotations we are used to in three-space. Just as rotation in three-space preserves the distance between two points (this distance is the same no matter how you rotate in three-space), this new kind of rotation defines a new kind of distance which is preserved under these new rotations.
This new kind of distance will allow us to make quantitative estimates of how we observe space and time differently if we are in relative motion (though we will observe spacetime to be the same), and will give us the Lorentz transformation (the equations that tell us how our views of space and time are related).