other particles. Clearly this is not true of our world; for example, the planets do not move in straight lines although they do not suffer any impacts. It is true that if we confine attention to a small region like the interior of Jules Verne's projectile, all the tracks become straight lines for an appropriate observer, or, as we generally say, he detects no field of force. It needs a large region to bring out the differences of geometry. That is not surprising, because we cannot expect to tell whether a surface is flat or curved unless we consider a reasonably large portion of it.
According to Newtonian ideas, at a great distance from all matter beyond the reach of any gravitation, particles would all move uniformly in straight lines. Thus at a great distance from all matter space-time tends to become perfectly flat. This can only be checked by experiment to a certain degree of accuracy, and there is some doubt as to whether it is rigorously true. We shall leave this afterthought to Chapter x, meanwhile assuming with Newton that space-time far enough away from everything is flat, although near matter it is curved. It is this puckering near matter which accounts for its gravitational effects.
Just as we picture different kinds of two-dimensional space as differently curved surfaces in our ordinary space of three-dimensions, so we are now picturing different kinds of four-dimensional space-time as differently curved surfaces in a Euclidean space of five dimensions. This is a picture only[1]. The fifth dimension is neither space nor time nor anything that can be perceived; so far as we know, it is nonsense. I should not describe it as a mathematical fiction, because it is of no great advantage in a mathematical treatment. It is even liable to mislead because it draws distinctions, like the distinction be tween a plane and a roll, which have no meaning. It is, like the notion of a field of force acting in space and time, merely introduced to bolster up Euclidean geometry, when Euclidean geometry has been found inappropriate. The real difference between the various kinds of space-time is that they have
- ↑ A fifth dimension suffices for illustrating the property here considered; but for an exact representation of the geometry of the world, Euclidean space of ten dimensions is required. We may well ask whether there is merit in Euclidean geometry sufficient to justify going to such extremes.
