is that, as observers change their relative motion, their time axes take slightly different directions, so that what is purely space or purely time for the one becomes space with a small component in the time direction, or time with a small component in the space direction, for the other. This it will be seen explains fully why observers in relative motion can differ about space and time measurements. We should not be at all surprised if the two observers of the figure reported different values for horizontals and verticals; we should realize that what was vertical for one had become partly horizontal for the other. It is just so, says Einstein, with his observers of time and space who are in relative motion to one another; what one sees as space the other sees as partly time, because their time axes do not run quite parallel.
The natural question here, of course, is “Well, where are their time axes?” If you know what to look for, of course, you ought to be able to perceive them in just the way you perceive ordinary time intervals—with the reservation that they are imaginary, after all, just like your space axes, and that you must only expect to see them in imagination. If you look for a fourth axis in Euclidean three-space to represent your time axis, you will of course not find it. But you will by all means agree with me that your time runs in a definite direction; and this it is that defines your time axis. Einstein adds that if you and I are in relative motion, my time does not run in quite the same direction as yours.
How shall we prove it? Well, how would we prove it if he told us that our space axes did not run in precisely the same direction? Of course we could not proceed through direct measures upon the axes themselves; we know these are imaginary. What we should do would be to strike out, each of us, a very long line indeed in what seemed the true horizontal direction; and we should hope that if we made them long enough, and measured them accurately enough, we should be able to detect any divergence that might exist. This is precisely what we must do with our time axes if we wish to verify Einstein’s statement that they are not precisely parallel; and what better evidence could we demand of the truth of this statement than the evidence already presented—that when we measure our respective time components between two events, we get different results?
What It All Leads To
The preceding chapters have been compiled and written with a view to putting the reader in a state of mind and in a state of informedness which shall enable him to derive profit from the reading of the actual competing essays which make up the balance of the book. For this purpose it has been profitable to take up in detail the preliminaries of the Special Theory of Relativity, and to allow the General Theory to go by default, in spite of the fact that it is the latter which constitutes Einstein’s contribution of importance to science. The reason for this is precisely the same as that for taking up Euclidean geometry and mastering it before proceeding to the study of Newtonian mechanics. The fundamental ideas of the two theories, while by no means identical, are in general terms the same; and the conditions surrounding their application to the Special Theory are so very much simpler than those which confront us when we apply them to the more general case, that this may be taken as the controlling factor in a popular presentation. We cannot omit the General Theory from consideration, of course; but we can omit it from our preliminary discussion, and leave its development to the complete essays which follow, and which in almost every case give it the larger half of their space which its larger content demands. In the process of the slow and difficult preparation of the lay mind for the assimilation of an altogether new set of fundamental ideas, it is altogether desirable to give the Special Theory, with its simpler applications of these ideas, a place out of proportion to its importance in Einstein’s completed structure; and this we have therefore done.
The Special Theory, postulating the relativity of uniform motion and deducing the consequences of that relativity, is often referred to as a “special case” of the General Theory, in which this restriction of uniformity is removed. This is not strictly speaking correct. The General Theory, when we have formulated it, will call our attention to something which we really knew all the time, but to which we chose not to give heed—that in the regions of space to which we have access, uniform motion does not exist. All bodies in these regions are under the gravitational influence of the other bodies therein, and this influence leads to accelerated motion. Nothing in our universe can possibly travel at uniform velocity; the interference of the rest of the bodies in the universe prevents this.
Obviously, we ought not to apply the term “special case” to a case that never occurs. Nevertheless, this case is of extreme value to us in our mental processes. Many of the motions with which we are concerned are so nearly at constant velocity that we find it convenient to treat them as though they were uniform, either ignoring the resulting error or correcting for it at the end of our work. In many other cases we are able to learn what actually occurs under accelerated motion by considering what would have occurred under uniform motion were such a thing possible. Science is full of complications which we unravel in this fashion. The physicist deals with gas pressures by assuming temperatures to be constant, though he knows temperature never is constant; and in turn he deals with temperatures by assuming pressures to be constant. After this, he is able to predict what will happen when, as in nature, pressures and temperatures are varying simultaneously. By using as a channel of attack the artificially simple case that never occurs, we get a grip on the complex case that gives us a true picture of the phenomenon. And because in actual nature we can come as close as we please to this artificial case, by supposing the variable factor to approach constancy, so when we assume it to be absolutely constant we speak of the result as the limiting case. This situation does not occur, but is the limiting case for those that do occur.
When, in the matter of motion, we abandon the artificial, limiting case of uniform velocity and look into the general, natural one of unrestricted motion, we find that the structure which we have built up to deal with the limiting case provides us with many of the necessary ideas and viewpoints. This is what we expect—in it lies the value of the limiting case. We shall see that the relativity of time and space, established for the limiting case, holds good in the general one. We shall see that the idea of the four-dimensional space-time continuum as representing the external world persists, forming the whole background of the General Theory much more definitely than in the Special Theory. Incidentally we shall see that the greater generality of the case under consideration will demand a greater degree of generality in the geometry of this continuum, a non-Euclideanism of a much more whole-hearted type than that of the Special Theory. But all the revisions of fundamental concepts which we have been at such pains to make for the sake of the Special Theory will remain with us in the General. With this we may consider our preliminary background as established, and give our attention to the essayists, who will try to take us more deeply into the subject than we have yet gone, without losing us in its intricacies.
