Einstein 1920: The Structure of Space According to the General Theory of Relativity

Reference: Einstein’s 1920 Book

This paper presents Part III, Chapter 3 from the book RELATIVITY: THE SPECIAL AND GENERAL THEORY by A. EINSTEIN. The contents are from the original publication of this book by Henry Holt and Company, New York (1920).

The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding.  Feedback on these comments is appreciated.

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The Structure of Space According to the General Theory of Relativity

According to the general theory of relativity, the geometrical properties of space are not independent, but they are determined by matter. Thus we can draw conclusions about the geometrical structure of the universe only if we base our considerations on the state of the matter as being something that is known. We know from experience that, for a suitably chosen co-ordinate system, the velocities of the stars are small as compared with the velocity of transmission of light. We can thus as a rough approximation arrive at a conclusion as to the nature of the universe as a whole, if we treat the matter as being at rest.

Matter is a substance of infinite inertia. It occurs at the upper end of the dimension of inertia. The geometrical properties of space are not only dependent on matter, but also determined by the inertia of substance. This is because space is the extent of substance. As the inertia of the substance increases, its velocity decreases. This appears as a “whirlpool” of substance. At the center of the “whirlpool” we have substance as solid matter. Because of the “whirlpool” phenomenon, the matter at the center spins about an axis.

This whirlpool is elliptical that is almost flat as visible in the shape of a galaxy. The same whirlpool model applies to the solar system. It is very likely that this model applies also to the atom and to the whole universe too.

We already know from our previous discussion that the behaviour of measuring-rods and clocks is influenced by gravitational fields, i.e. by the distribution of matter. This in itself is sufficient to exclude the possibility of the exact validity of Euclidean geometry in our universe. But it is conceivable that our universe differs only slightly from a Euclidean one, and this notion seems all the more probable, since calculations show that the metrics of surrounding space is influenced only to an exceedingly small extent by masses even of the magnitude of our sun. We might imagine that, as regards geometry, our universe behaves analogously to a surface which is irregularly curved in its individual parts, but which nowhere departs appreciably from a plane: something like the rippled surface of a lake. Such a universe might fittingly be called a quasi-Euclidean universe. As regards its space it would be infinite. But calculation shows that in a quasi-Euclidean universe the average density of matter would necessarily be nil. Thus such a universe could not be inhabited by matter everywhere; it would present to us that unsatisfactory picture which we portrayed in Section XXX.

Einstein is operating on the model of matter and void. He is considering the dimension of inertia only indirectly. It is true that inertia takes a big jump at the interface between “void” and matter. It increases very slowly in the “void” domain, and again quite slowly in the material domain.

If we are to have in the universe an average density of matter which differs from zero, however small may be that difference, then the universe cannot be quasi-Euclidean. On the contrary, the results of calculation indicate that if matter be distributed uniformly, the universe would necessarily be spherical (or elliptical). Since in reality the detailed distribution of matter is not uniform, the real universe will deviate in individual parts from the spherical, i.e. the universe will be quasi-spherical. But it will be necessarily finite. In fact, the theory supplies us with a simple connection1 between the space-expanse of the universe and the average density of matter in it.

Each atom is an elliptical whirlpool at the center of which the nucleus exists. That means a solid body is made if an infinite number of atomic whirlpools. But that solid body itself forms the center of a much larger whirlpool. This same model then scales up to solar and star systems, the galaxies, and finally the universe. The universe shall exist much farther outwards than its relatively solid core.

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