Let us consider a surface of variable curvature—for instance, the surface of any large district with its hills, mountains, and valleys. When we travel in this region, we can proceed in a straight line as long as we are on the level plain. A straight line on a level plain has the remarkable feature of being the shortest distance between two points. It has also this peculiarity, that it is the only one of its kind and its length, whereas we may draw a great number of lines that are not straight uniting the two points, longer than the straight line but all of equal length.
But we have reached the hilly district. It is now impossible for us to follow a straight line from one point to another if there is a hill between them. Whatever path we take, it will be curved. But amongst the various possible paths which lead from one point to the other on the farther side of the hill, there is one—and only one, as a rule—which is shorter than any of the others, as we could prove by means of a tape. This shortest path, the only one of its kind, is what is called the geodetical of the surface covered.
In the same way no vessel can go in a straight line if it is sailing from Lisbon to New York. It must follow a curved path, because the earth is round. But amongst the possible curved paths there is a privileged one which is shorter than the others: the one which follows the direction of the great circle of the earth. In going from Lisbon to New York, though they are nearly in the same latitude, vessels are careful not to head straight westward, in the direction of the parallels. They sail a little to the north-west, so that when they reach New York they come from the north-east, having followed pretty closely a terrestrial great circle. On our globe, as on all spheres, the geodetical, the shortest route between two points, is the arc of a great circle passing through the two points.
Now the “Interval” of two points in the four-dimensional universe precisely represents the geodetical, the minimum path of progress between the two points traced in the universe. Where the universe is curved, the geodetic is a curved line. Where the universe is approximately Euclidean, it is a straight line.
I may be told that it is very difficult to imagine as curved a three-dimensional space, and still more a four-dimensional. I agree. We have already seen that it is difficult enough to imagine four-dimensional space even when it is not curved.
But what does that prove? There are many other things in nature which we cannot visualise or form a mental picture of. The Hertz waves, the X-rays, and the ultra-violet waves exist all the same, though we cannot imagine them, or at least only by giving them a visible form which does not belong to them. It is just one of our human infirmities that we cannot conceive what we cannot picture to ourselves. Hence our tendency to—if one may use an inelegant but expressive word—visualise everything.
Let us therefore return to our geodetics. These we can very well picture to ourselves, because in the universe, in spite of its four dimensions, they are lines of only one dimension, like all other lines that we know.
The existence of geodetics, of shortest-distance lines, will now beautifully explain to us the connection between inertia and weight, which did not appear in the Euclidean world of classic science. Hence the Newtonian distinction between the principle of inertia and the force of gravitation.
We Relativists find this distinction no longer necessary. Material masses, like light, travel in a straight line when they are far from a gravitational field, and in a curved line when they are near gravitational masses. In virtue of symmetry a free material point can only follow a geodetic in the universe.