The Analysis of Matter

pre-existing tram-lines. This view of motion, however, is no longer tenable. A moving point is a series of positions in space-time; a later moving point cannot pursue the "same" course, since its time co-ordinate is different, which means that, in another equally legitimate system of co-ordinates, its space co-ordinates also will be different. We think of a tram as performing the same journey every day, because we think of the earth as fixed; but from the sun's point of view, the tram never repeats a former journey. "We cannot step twice into the same rivers," as[Pg 62] Heraclitus says. It is thus obvious that, in place of Euclid's static straight line, we shall have to substitute a movement having some special property defined in terms of space-time, not of space. The movement required is a "geodesic," concerning which we shall have more to say later. In relativity theory, distant space-time points have only such relations as can be obtained by integration from the relations of neighbouring points. Since the distance between two points is always finite, what we call a relation between neighbouring points is not really a relation between points at all, but is a limit, like a velocity. Only the language of the calculus can express accurately what is meant. One might say, speaking pictorially, that the notion of "interval" is concerned with what, at each point, is tending to happen, although we cannot say that this will actually happen, because before any assigned point is reached something may have occurred to cause a diversion. This is, of course, the case with velocity. From the fact that, at a given instant, a body is moving in a given direction with a given velocity, we can infer nothing whatever as to where the body will be at another assigned instant, however near to the first. To infer the path of a body from its velocity, we must know its velocity throughout a finite time. Similarly the formula for interval characterizes each separate point of space-time. To obtain the interval between one point and another, however near together, we must specify a route, and integrate along that route. As we shall see, however, there are routes which may be called "natural"—namely, geodesics. It is only by means of them that the notion of interval can be profitably extended to the relations of points at a finite distance from each other.