The Analysis of Matter

the body on which the lengths are: the longer length will have to the shorter the ratio . More interesting, however, is the effect as regards time. Suppose that an observer on the body judges two events at and to be simultaneous, and both at time . Then an observer on will judge that they occur at times , where: [Pg 51] and therefore: This is not zero unless ; thus in general events which are simultaneous for one observer are not simultaneous for the other. We cannot therefore regard space and time as independent, as has always been done in the past. Even the order of events in time is not definite: in one system of co-ordinates an event A may precede an event B, while in another B may precede A. This, however, is only possible if the events are so separated that, no matter how we choose our co-ordinates, light starting from either could not reach the place of the other until after the other had occurred. The Lorentz transformation yields the result that: Since and , we have: or, putting , for the distances of the event from the two observers: This result is general—i.e. given any two reference-bodies in uniform relative motion, if is the distance between two events according to one system, the distance according to the other, and if , are the corresponding time-intervals between the events, equation (2) will always hold. Thus represents a physical quantity, independent of the choice of co-ordinates; it is called the square of the "interval" between the two events. There are two cases, according as it is positive or negative. When it is positive, the interval between the events is called "time-like"; when negative, "space-like." In the intermediate case in which it is zero, the events are such that one light-ray can be present at each. In this case, one event might be the seeing of the other. The time-order of two events[Pg 52] will be different in different reference-systems when their interval is space-like, but when it is time-like the time-order is the same in all systems, though the magnitude of the time-interval varies. When the interval between two events is time-like, it is possible for a body to move in such a way as to be present at both events. In that case, the interval is what clocks on that body will show as the time. When the interval between two events is space-like, it is possible for a body to move in such a way that, by its clocks, the two events will be simultaneous; in that case, the interval is what, in relation to that body, appears as their distance. (In these remarks, we are taking the velocity of light as the unit of velocity, which is convenient in relativity theory.) Both these are consequences of the Lorentz transformation. From the first of them it follows that, if two events both happen to me, the time between them as measured by my watch (assuming it to be a good watch) is the "interval" between them, and has still a physical significance. Thus the time that is concerned in psychology is unaffected by relativity, assuming that everything that psychology is concerned with happens, from a physical point of view, in the body of the person whose mental events are being considered. This is an assumption for which grounds will be given at a later stage. It follows from the ambiguity of simultaneity between distant events that we cannot speak unambiguously of "the distance between two bodies at a given time." If the two bodies are in relative motion, a "given time" will be different for the two bodies and different again for other reference-bodies. It follows that such a conception cannot enter into the correct statement of a physical law. On this ground alone, we can conclude that the Newtonian form of the law of gravitation cannot be quite right. Fortunately, Einstein has supplied the necessary correction. [Pg 53] It will be observed that, as a consequence of the Lorentz transformation, the mass of a body will not be the same when it is in motion relatively to the reference-body as when it is at rest relatively to it. The mass of a body is inversely proportional to the acceleration produced in it by a given force, and two reference-bodies in uniform relative motion will give different results for the acceleration of a third body. This is obvious as a consequence of the FitzGerald contraction. The increase of mass with rapid motion was known experimentally before the special theory of relativity had explained it; it is very marked for velocities such as those attained by -particles (electrons) emitted by radio-active bodies, since these velocities may be as great as 99 per cent, of the velocity of light. This change of mass, like the FitzGerald contraction, seemed strange and anomalous until the special theory of relativity explained it. One more point is important as showing how easily what seems axiomatic may be false: it concerns the composition of velocities. Suppose three bodies moving uniformly in the same direction: the velocity of the second relatively to the first is , that of the third relatively to the second is . What is the velocity of the third relatively to the first? One would have thought it must be , but in fact it is: It will be seen that this ; if or , it is , otherwise it is less than . This is an illustration of the way in which the velocity of light plays the part of infinity in relation to material motions. The special theory set itself the task of making the laws of physics the same relatively to any two co-ordinate systems in uniform rectilinear relative motion. There were two sets of equations to be [Pg 54] considered: those of Newtonian dynamics, and Maxwell's equations. The latter are unaltered by a Lorentz transformation, but the former require certain adaptations. These, however, are such as experimental results had already suggested. Thus the solution of the problem in hand was complete, but of course it was obvious from the first that the real problem was more general. There could be no reason for confining ourselves to two co-ordinate systems in uniform rectilinear motion; the problem ought to be solved for any two co-ordinate systems, no matter what the nature of their relative motion. This is the problem which has been solved by the general theory of relativity. [Pg 55] CHAPTER VI THE GENERAL THEORY OF RELATIVITY THE general theory of relativity has a much wider sweep than the special theory, and a greater philosophical interest, apart from the one matter of the substitution of space-time for space and time. The general theory demands an abandonment of all direct relations between distant events, the relations upon which space-time depends being primarily confined to very small regions, and only extended, where they can be extended, by means of integration. All the old apparatus of geometry—straight lines, circles, ellipses, etc.—is gone. What belongs to analysis situs remains, with certain modifications; and there is a new geometry of geodesics, which has come from Gauss's study of surfaces by way of Riemann's inaugural dissertation. Geometry and physics are no longer distinct, so long as we are not considering the parts of physics which introduce atomicity, such as electrons, protons, and quanta. Perhaps even this exception may not long remain. There are parts of physics which, so far, lie outside the general theory of relativity, but there are no parts of physics to which it is not in some degree relevant. And its importance to philosophy is perhaps even greater than its importance to physics. It has, of course, been seized upon by philosophers of different schools as affording support to their respective nostrums; St. Thomas, Kant, and Hegel are claimed to have anticipated it. But I do not think that any of the philosophers who make these suggestions have taken the trouble to understand the theory. For my part, I do not profess to know exactly what its philosophical consequences will prove to be, but I am convinced that they are far-reaching, and quite different from[Pg 56] what they seem to philosophers who are ignorant of mathematics. In the present chapter, I wish to consider Einstein's theory without any regard to its philosophical implications, simply as a logical system. The system starts by assuming a four-dimensional manifold having a definite order. The form which this assumption takes is somewhat technical: it is assumed that, when we have what might be called an ordinary set of co-ordinates—e.g. those which would naturally be employed in Newtonian astronomy—there are certain transformations of these co-ordinates which are legitimate, and certain others which are not. Those which are legitimate are those which transform infinitesimal distances into infinitesimal distances. This means to say that the transformations must be continuous. Perhaps what is assumed may be stated as follows: Given a set of points , , ,... whose co-ordinates tend towards a limiting set which is the co-ordinates of a point , then in any new legitimate co-ordinate system those points , , ,... must have co-ordinates tending to a limiting set which is the co-ordinates of in the new system. This means that certain relations of order among the co-ordinates represent properties of the points of space-time, and are presupposed in the assignment of co-ordinates. The accurate statement of what is involved can only be made in terms of limits, but the correct meaning is conveyed by saying that neighbouring points must have neighbouring co-ordinates. The exact nature of the ordinal presuppositions of a relativistic co-ordinate system will occupy us in a later chapter; for the present I merely wish to emphasize that the space-time manifold, in the general theory of relativity, has an order which is not arbitrary, and which is reproduced in any legitimate co-ordinate system. This order, it is important to realize, is purely ordinal, and does not involve any metrical elements Nor is it derivable from the metrical relations of points[Pg 57] which are afterwards introduced in the theory—i.e. from "intervals." The points of space-time have, of course, no duration as well as no spatial extension. It is generally assumed that several events may occupy the same point; this is involved in the conception of the intersection of world-lines. I think it may also be assumed that one event may extend over a finite extent of space-time, but on this point the theory is silent, so far as I know. I shall myself, in a later chapter, deal with the construction of points as systems of events, each of which events has a finite extension; this is a subject which has been especially treated by Dr Whitehead, but I shall suggest a method somewhat different from his. So long as we confine ourselves to the theory of relativity, it is not necessary to consider whether events have a finite extension, though I think it is necessary to assume that two events may both occupy the same point of space-time. Even on this, however, there is a certain vagueness in the authoritative expositions, which is due mainly to the large scale of the phenomena with which the theory is principally concerned. Sometimes it would seem as if the whole earth counted as a point; certainly one physical laboratory does so in the practice of writers on relativity. On occasion, Professor Eddington considers an area of square kilometres to be an infinitesimal of the second order. The fact that such a view is appropriate in discussions of relativity makes it unnecessary to be precise as to what is meant by saying that two events occupy the same point, or that two world-lines intersect. For the present I shall assume that this is possible in a strict sense; my reasons will be given in a later chapter. It is assumed that every point of space-time can have four real numbers assigned to it, and conversely that any four real numbers (at any rate within certain limits) are the co-ordinates of a