*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.*

*The heading below is linked to the original materials.*

.

## 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 connection^{1} 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.*

.

## FINAL COMMENTS

*The final comments are pretty much the comments above.*

*.*