These difficulties and obscurities disappear in great part when we consider space, or space-time, from the Einsteinian standpoint of General Relativity.
What is the meaning of the sentence, “The universe is infinite”? From either the Einsteinian, the Newtonian, or the Pragmatist point of view it means: If I go straight ahead, going on eternally, I shall never get back to my starting-point.
Is it possible? Newton is compelled to say yes, because in his view space stretches out indefinitely, independent of the bodies that occupy part of it, whether the number of the stars is or is not limited.
But Einstein says no. For the Relativist the universe is not necessarily infinite. Is it therefore limited, fenced in by some sort of railings? No. It is not limited.
A thing may be unlimited without being infinite. For instance, a man who moves on the surface of the earth may travel over it indefinitely in every direction without ever reaching a limit. The surface of the earth, thus regarded, or the surface of any sphere whatsoever, is therefore both finite and unlimited. Well, we have only to apply to space of three dimensions what we find in two-dimensional space (a spherical surface), to see how the universe may be at one and the same time finite and unlimited.
We saw that, in consequence of gravitation, the Einsteinian universe is not Euclidean, but curved. It is, as we said, difficult, if not impossible, to visualise a curvature of space. But the difficulty exists only for our imagination, which is restricted by our life of sense, not for our reason, which goes farther and higher. It is one of the commonest of errors to suppose that the wings of the imagination are more powerful than those of reason. If one wants proof of the contrary, one has only to compare what the most poetic of ancient thinkers made of the starry heavens with what modern science tells about the universe.
Here is the way to approach our problem. Let us not notice for the moment the rather irregular distribution of stars in our stellar system, and take it as fairly homogeneous. What is the condition required for this distribution of the stars under the influence of gravitation to remain stable? Calculation gives us this reply: The curvature of space must be constant, and such that space is bent like a spherical surface.
Rays of light from the stars may travel eternally, indefinitely, round this unlimited, yet finite, universe. If the cosmos is spherical in this way, we can even imagine the rays which emanate from a star—the sun, for instance—crossing the universe and converging at the diametrically opposite point of it.
In such case we might expect to see stars at opposite points in the heavens, of which one would be the image, the spectre, the “double” of the other—in the sense which the ancient Egyptians gave to the word. Properly speaking, this “double” would represent, not the generating star as it is, but as it was at the time when it emitted the rays which form the double, or millions of years earlier.