It is clear that the acceleration or curvature of path which fig ures as gravitation cannot be an acceleration or curvature in ordi nary three-dimensional space. Before the apple starts to fall from the tree there is neither acceleration nor curvature. and yet the apple is undoubtedly acted on by gravitation. Moreover, this three-dimensional space is, as we have seen, different for different observers—it is a subjective and not an objective conception, and the gravitation resulting from such a curvature could not conform to the relativity condition. Einstein was accordingly led to sup pose that gravitation arose from curvature in the four-dimensional space, or continuum, in which time formed the fourth dimension. This continuum, as has been seen, is objective and if the path of the particle can also be made objective, the resulting gravitation will conform to the relativity principle. The path of the particle in the continuum is, however, simply its "world line," which we have already had under discussion. This world line is determined by natural laws, and if these are to be objective the specification of the world line must also be objective. There is, however, only one specification of world lines in the continuum which is objec tive in the sense that the same specification will give the same world lines to observers moving with different velocities. It is that every world line must be so drawn as to represent the short est path between any two points on it. Mathematically, lines which satisfy this condition are known as geodesics. Thus Ein stein was led to suppose that world lines must be geodesics in the four-dimensional continuum.
Consider for a moment a page of this volume as presenting a two-dimensional analogy of the continuum. The shortest distance between any two points is of course the straight line joining them, so that the geodesics are simply straight lines. These possess no curvature of path, and if they formed a true analogy to the geo desics in the continuum there could clearly be no explanation of gravitation of the type we have been contemplating. There is, however, another type of two-dimensional surface. It is repre sented by the surface of a solid body such as a sphere—say the earth. On the earth's surface the geodesics are the great circles; every mariner or aviator who desires to sail the shortest course be tween two points sails along a great circle. To take a definite in stance, the shortest course from Panama to Ceylon is not along the parallel of lat. (about 9°N.) which joins them—the aviator who wishes to fly the shortest course between the two countries will fly north-east from Panama, he will pass over England and finally reach Ceylon from the north-west. The reader may rapidly verify this by stretching a thread tightly over the surface of an ordinary geographical globe. Let him now trace out the course on an ordinary Mercator chart, and it will be found to appear very curved indeed—the course of the aviator will look surprisingly like that of a comet describing an orbit under the attraction of a sun situated somewhere near the middle of the Sahara.
The reader who performs these simple experiments will under stand how Einstein was led to suppose that gravitation could be explained by a curvature inherent in the continuum. The world lines of particles are geodesics but the space itself, so to speak, provides the curvature. The curvature of path is thrust upon the particle by the nature of the continuum, but we, who until recently have been unaware even of the existence of the continuum, have been tempted to ascribe it to the action of a special agency which we have invented ad hoc and called "gravitation." According to Einstein's view of the nature of gravitation, it is no more accurate to say that the earth attracts the moon than to say that the pockets of an uneven billiard table attract or repel the balls.
Perhaps this train of thought may seem artificial. If so, the reason is that we have not been able to explore the other pos sibilities which have branched off our main line of thought. In point of fact, Einstein found himself practically limited to the conclusion we have stated. Not only so, but the actual type and degree of curvature in the continuum prove to be uniquely fixed in terms of the masses of the gravitating bodies. Thus Einstein, knowing the mass of the sun, found himself in a posi tion to predict absolutely what the motion of the perihelion of Mercury must be. It was found to be 42.9" a century, a figure which agreed with observation to well within the limits of error of the observations. The motions of the other planets as pre dicted by the theory of relativity, have also been found to agree with those observed to within the errors of observation. This latter test is not a very stringent one, since the departures from the motion predicted by the Newtonian law are too small to admit of very precise measurement.