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Relativity in Illustrations
By JACOB T. SCHWARTZ, FELIX COOPER
Dover Publications, Inc.Copyright © 1962 New York University
All rights reserved.
WHAT IS TIME?
The question at first seems foolish, because we are so sure we know. However, since Einstein was able to make such interesting discoveries by asking this question seriously and by answering it carefully, we ask again:
WHAT IS TIME?
WHAT IS SPACE?
Let us study the first question first.
WHAT IS TIME?
We think we know, because we seem to feel time passing constantly. Time, we feel at first, is that which passes; that whose passage separates the earlier from the later. What does this mean? It means that our experiences are related to each other as earlier and later—that some things happen first, and others happen afterward—that when the later things happen, we can mostly remember the earlier things that have happened, but that when the earlier things happened, we could not remember the later things which were to happen, but could only guess them.
First we are little and go to school. Then we graduate. Then we work and marry. Then we have children. First they are little and stay in the house. Then they go to school. Then they are big and go away. Then we are old.
Time is like a wire, and we are like beads being pushed along the wire, from earlier to later, without any return. This is what we can feel directly. But not more than this.
If we want to know more about time than what is earlier and what is later; if we want to know how much earlier and how much later, we can no longer rely on our feelings, on our direct perceptions. Our direct perception of time is merely qualitative. Some days seem long, others short. When we are children, hours seem very long, and the years between birthdays seem to be ages. Later, days, weeks, and years seem to vanish in a moment. To understand time, not merely as a qualitative after and before, but as a quantitative THAT MUCH after and THIS MUCH before, we must make use of our physical experience.
Night follows day, and day night; and in each night there is one instant when we see the last star in the Big Dipper at its highest. The intervals between these instants we call a "day." The pendulum swings from right to left, and from left again to right, and we call the intervals between the instants when we see the pendulum at its highest "seconds." The tiny regulator spring in our watch ticks in and out, in and out, and drives the hands of the watch round and round, over and past the marks in the watch face. By adjusting the speed of our watches we can make the hand pass over the successive marks on the watch face at exactly the same instants when the pendulum is at its highest. Then the intervals between the instants when the hand of the watch is over a mark on the watch face are also seconds.
We see from all this how our quantitative idea of time is taken from our physical experience. We arrange our notion of equal intervals of time in such a way as to be able to say of certain simple repetitive physical processes that they repeat themselves in equal intervals of time. When we have arranged our notion of equal intervals of time in this way, we find that many physical happenings have a simple description. The last star in the Big Dipper is at its highest once every twenty-four hours. The pendulum goes from side to side once each second. The flywheel in an engine revolves eight hundred times a minute. A radio wave oscillates seven million times a second, another radio wave oscillates eight million times a second. From the fact that we have arranged our notion of equal intervals of time in such a way that so very many different physical happenings all have a simple description, we know that we have successfully chosen our notion of equal intervals of time in a way appropriate for the understanding of our physical world.
We must remember, however, that this success, like all successes, can turn out to be short of absolute. Above all, we must remember that our quantitative notions of time come from our physical experience, can be made definite ONLY by reference to physical experience, and are SUBJECT TO CHANGE if a reconsideration of the details of our physical experience seems to warrant change.
WHAT IS SPACE?
We have a direct qualitative perception of space also. We see things by moving our eyes and head left or right, up or down. A given object, when looked at, may appear bigger, which we learn to call nearer, or smaller, which we learn to call farther. To this extent, space is seen. In early infancy we learn to move our hands while watching them, and find that certain muscular adjustments bring our hands up or down, left or right, nearer and farther. The fact that things which are on the left "for looking" are also on the left "for reaching," and that things which are nearer "for reaching" are also nearer "for looking," gives us confidence in our space perceptions.
But just as with time, so also, if we wish to arrive at a quantitative notion of space, we must make use of our physical experience, specifically experience with measuring tapes, rulers, with calipers, micrometers, surveyors' transits, magnifying glasses and microscopes, telescopes, etc., of the experience of fitting things together, and finding that sometimes pieces are too big to fit, and that sometimes pieces are too small to reach from one point to another no matter which way you turn them.
In these ways, through looking, reaching, fitting, and measuring, we develop quantitative notions of space. From the fact that our quantitative notions of space and our quantitative notions of time fit together in such a way that many physical happenings have a simple description, we know that we have successfully chosen our notions of space and time in a way appropriate for the understanding of the physical world.
We must remember that this success can turn out to be short of absolute. Our quantitative notions of time and space come from our physical experience, can only be made definite by reference to physical experience, and are subject to change if a reconsideration of the details of our physical experience seems to warrant a change.
Now it is time to pass from generalities to details. We want to look at the specific way in which certain events take place in time and space: the when and where of particular happenings. How can this be done conveniently in the pages of a book? [Remark to the mathematically trained reader: the method which we shall use is simply that of graphing the position in space of these events against time. The next few pages are intended to explain the details of this technique. Naturally, the points at issue will seem quite simple to you.] The best way to begin is to use a sort of comic-strip technique, giving a series of pictures of the scene we want to study at successive instants.
Beginning with a series of cartoons showing various typical sequences of events to be studied, we shall, in the next few pages, introduce a number of conventions which will enable us to simplify these cartoons more and more, thereby depicting the events in a more and more convenient way. We will ultimately come to a scheme for representing a sequence of events by a highly simplified sort of chart or diagram, which will have the great advantage of showing everything relevant for our later analysis and of ignoring all that is irrelevant. These charts will be as basic to the reader of what is to follow as blueprints are to the engineer.
We want to look at the specific and detailed order in which certain events take place in time and space.
Consider the showdown in the main street of Snake City between Dead-Eye Dick and Piute Pete as it so often appears (Dick on the left, Pete on the right).
How did this happen? (X marks are bullet holes.) Well, say the witnesses, Pete drew his gun, shot at Dick; Dick drew his gun, shot at Pete. Each was hit and killed by the bullet of the other. But who shot first? Or did they shoot at the same time?
Did Dick shoot first, and Pete a little later, but before being hit by Dick's bullet?
Did Pete shoot first, and Dick later, but before being hit by Pete's bullet?
Did Dick and Pete shoot at the same time?
One can determine which of these possibilities is correct only by careful observation and measurement. The little drama of Pete and Dick is made up of a large number of separate events. Some of these events are: Dick fires his gun; Pete fires his gun; Dick is hit by a bullet; Pete is hit by a bullet; but also: Dick's bullet reaches the halfway mark (front door of saloon) in its flight to Pete; Pete's bullet reaches the three-quarter mark (left saloon window) in its flight to Dick, etc. Which of these events takes place first, which afterward, and which at the same time as a given event can be determined only by careful observation and measurement.
Since we shall have to study quite a number of cases like the case of Pete and Dick, we shall need a system for recording the results of our observations and measurements which is less clumsy than that of drawing a large number of detailed pictures of the scene at successive instants (these pictures, taken together, would clearly show the whole story of a given action).
For this reason, we shall make a number of simplifications in the way in which we draw our pictures.
First of all, we shall not bother to draw the background but shall merely indicate the position of important places by lines.
Second, since all the action takes place along the length of the main street of Snake City, we shall not bother to show how far off the ground anything is, but shall show only where it is to the right or left along Main Street. This means that instead of drawing
When we use these simpler drawings, as we shall always do from now on, we deliberately forget that events, besides happening a certain distance to the right or left along Main Street, also happen a certain distance aboveground and also a certain number of feet closer to the foreground or farther away in the background. That is, we deliberately forget that space is actually three-dimensional and, for the sake of simplicity in description and pictorial representation, pretend that it is ONE-DIMENSIONAL. All our events consequently take place on a "street," or along a "road" or "railroad track," so and so many feet to the left or right. Nothing important is lost by this simplification, but much is gained. The reader, however, must be careful to remember that this simplification is being assumed, or he may become confused.
With this simpler way of drawing, the whole set of pictures given on page 17 appears as
Our simpler way of drawing gives us room to describe the situation not only as it is after whole seconds have elapsed but also as it is after fractions of a second have elapsed. When this is done, many of the labels on Figure 23 become unnecessary.
Since ability to read a chart like Figure 24 will be essential in following the later sections of this book, it is well to pause and comment on this chart.
The first thing to realize is that a given horizontal line represents a given instant in time (one picture in a comic strip, one frame in a reel of motion-picture film), so that the dots on a given horizontal line represent the position of a number of objects at a given instant. For instance, Figure 24 shows that at 2.4 seconds after noon Dick is standing in front of the right jail window; Dick's bullet is a bit to the left of the right saloon window; Pete's bullet is a bit to the right of this same window; and Pete is standing in front of the left post-office window. Figure 24 shows the same sequence of events shown in Figures 23 and 3-8; it is less detailed than Figures 3-8 in that it shows less of the background, etc., but more detailed in that it shows the situation not only after 1, 2, 3, etc. seconds, but also after 1.1, 1.2, 1.3 ... seconds. That is, Figure 24 is able, by omitting those details which we wish to ignore, to show the details which are important to us—the detailed order in which certain events take place in time and space.
A number of features of the chart in Figure 24 are worth emphasizing. A period of time in which an object is not moving is represented in this chart by a vertical sequence of dots. Thus, for instance, from 1 second after noon on, Dick is continually in front of the right jail window; the dots representing his position at successive intervals of 1/10 second occur one under the other, all on the vertical line upon which all events occurring in front of the right jail window are recorded. These dots consequently form a vertical sequence. In contrast, a period of time in which an object is moving is represented in the chart by a sequence of dots forming a slanted line. For example: at 2 seconds after noon, Pete's bullet, just having been fired, is in front of the left post-office window. One tenth of a second later, it has moved 1/5 of the distance to the right saloon window; 5/10 of a second later, it has reached this window; and so forth. The dots representing the position of this bullet at successive intervals of 1/10 second consequently appear not along a vertical but along a slanted line.
The chart in Figure 24 consequently shows that between noon and 1 second after noon Dick is walking to the right and Pete to the left (they are walking toward each other). Then both stop moving; Dick fires, at 2 seconds after noon Pete fires; Dick's bullet and Pete's bullet pass each other at 2.5 seconds after noon, etc. The reader should examine the chart on page 20 carefully, and be sure that he sees how all the facts just stated are represented there. Without a clear understanding of how such charts are to be read, it is useless to go on.
The position of an object at any given instant is, naturally enough, represented by a single point in our chart. Conversely, a point in our chart represents both the time and the position in (one-dimensional) space of a single event. The position in (one-dimensional) space of the event which the point represents governs how far to the right or left in the chart the point appears; the time of the event governs how far up or down in the chart the point appears. Since many successive instants are shown all at once in the chart, the successive positions of an object are represented by a succession of points.
The succession of dots showing the position of a given object at successive instants traces out a path in our chart—conveniently, though purely figuratively, called the "path through time" of the object. A stationary object as well as a moving object has such a "path"; the stationary object a vertical "path," the moving object a slanted "path." The reader should avoid confusing this sort of figurative "path" in a chart of events in space and time with the ordinary notion of path through space. In what follows we shall often speak of the "path" of an object in this sense, i.e., in the sense of the path in a diagram representing the successive positions of the object at a large number of instants. The reader, forewarned, should avoid confusion as to what is meant.
Notice that we are putting all events which we determine to have occurred at the same time on the same horizontal line, and putting all events which we determine to have occurred at the same place on the same vertical line. Remembering these two facts, we realize that the horizontal and vertical lines are just guide lines which we might as well leave out, except for a few which we may want to keep for the sake of orientation. Moreover, instead of drawing a more and more closely spaced series of dots showing the successive position of, say, Pete's bullet, at successive intervals of time, first at intervals of 1/10 second, then at intervals of 1/100 second, etc., it is simpler and more informative to let these more and more closely spaced dots "run into each other," and to draw a continuous line which will then show the position of Pete's bullet at every instant of time. With these new stipulations, our picture appears as
This is the way we shall draw our simplified "film-strips" from now on. It is a good idea to look back over pages 10, 11, 17, 19, 20, and 23 to see how these simplified drawings have been evolved, and to be sure of what they mean.
The explanation of the way in which Figure 24 is to be read is particularly relevant. Figure 25 is to be read in exactly the same way, except that, whereas Figure 24 shows the situation only at intervals of 1/10 second, the present diagram shows the situation also at, say, 2.15 seconds or 2.1583 seconds after noon. The reader, before going on, should be sure that he can read charts like Figure 25, and that he understands exactly why Figures 3–8, 21, 23, 24, and 25 all give schematic representations of the same sequence of events in various systems of drawing.
It is to be noted that our diagrams show all at once the situation at a whole range of different instants in time; and that the "path" of an object, as it appears in such a diagram, shows the position of the object at a range of different instants, as is explained on page 22.
The continual occurrence of two-dimensional charts should not lead the reader to forget that our objects are moving, like trains along a railroad track, in only one dimension. The charts are two-dimensional only because an extra dimension is needed to show the time at which events occur in our one-dimensional space. Dick's bullet, in our chart, is moving in one dimension, from left to right, and not moving along a slanted line in two dimensions. In the same way, a film actress who walks from left to right is represented on the actual celluloid film by a series of images which lie along a slant in the celluloid strip; but the actress is not walking at an angle down the celluloid but merely walking, at a certain rate, from left to right on the stage.
Excerpted from Relativity in Illustrations by JACOB T. SCHWARTZ, FELIX COOPER. Copyright © 1962 New York University. Excerpted by permission of Dover Publications, Inc..
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