We showed in a previous chapter that the real universe of the Relativists is a four-dimensional continuum—not three-dimensional, as classic science thought—and that in this continuum distances in time and space are relative. The only thing that has a value independent of the conditions of observation—that has an absolute, or at least objective, value—is what we called the “Interval” of events, the synthesis of the spatial and chronological data.
Yet, in spite of its four dimensions, the universe, as we discussed it in connection with the Michelson experiment and the Special Relativity which this discloses, was nevertheless a Euclidean continuum, in which the classical geometry was verified, and light travelled in a straight line. As we have just seen, we have to recant this. The universe not only has four dimensions, but it is not Euclidean.
With what geometry does the universe accord best—or most conveniently, to use the language of Poincaré? Probably that of Riemann. When we take the compasses and draw a small circle on a sheet of paper spread on the table, the radius of the circle is found by the distance between the points of the compasses, and the circle is Euclidean. But if we draw the circle on an egg, the fixed point of the compasses being stuck in the top of the egg, and again get the radius by the distance between the points, the circle we have now drawn is not Euclidean. The proportion of the circumference to the radius as thus defined is smaller than π, just as it is smaller than π when the circle is traced round a massive star.
Well, there is the same difference between the non-Euclidean real universe and a Euclidean continuum as there is between our flat sheet of paper and the surface of the egg, taking into account the fact that these surfaces have only two dimensions while the universe has four.
Two-dimensional space may be flat like the sheet of paper or curved like the surface of the egg. By leaving the sheet of paper flat or rolling it up we can make the geometry of the figures drawn on it correspond with or differ from the Euclidean geometry. In just the same way space with more than two dimensions may or may not be Euclidean.
As a matter of fact, the universe is, as we saw, only approximately Euclidean in those regions which are remote from all heavy masses. It is not Euclidean, but curved or warped in the vicinity of the stars; and the curvature is the greater in proportion as we approach the stars.
Hence the geometry of curved space, as founded by Riemann, seems to be the best adapted to the real universe. It is the one used by Einstein in his calculations.
When we sought to prove, on a previous page, that rays of light fall just as projectiles of the same velocity would, we used the following argument: