*Reference: Einstein’s 1920 Book*

*This paper presents Part III, Chapter 2 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 Possibility of a “Finite” and Yet “Unbounded” Universe

But speculations on the structure of the universe also move in quite another direction. The development of non-Euclidean geometry led to the recognition of the fact, that we can cast doubt on the *infiniteness* of our space without coming into conflict with the laws of thought or with experience (Riemann, Helmholtz). These questions have already been treated in detail and with unsurpassable lucidity by Helmholtz and Poincaré, whereas I can only touch on them briefly here.

*The development of non-Euclidean geometry led
to the recognition of the fact, that we can cast doubt on the **infiniteness** of our space without coming into conflict with the laws of thought
or with experience.*

In the first place, we imagine an
existence in two-dimensional space. Flat beings with flat implements, and in
particular flat rigid measuring-rods, are free to move in a *plane*. For them nothing exists outside
of this plane: that which they observe to happen to themselves and to their
flat “things” is the all-inclusive reality of their plane. In particular, the
constructions of plane Euclidean geometry can be carried out by means of the
rods, e.g. the lattice construction, considered in Section XXIV. In contrast to ours, the universe of
these beings is two-dimensional; but, like ours, it extends to infinity. In
their universe there is room for an infinite number of identical squares made up
of rods, i.e. its volume (surface) is infinite. If these beings say their
universe is “plane,” there is sense in the statement, because they mean that
they can perform the constructions of plane Euclidean geometry with their rods.
In this connection the individual rods always represent the same distance,
independently of their position.

*Let’s consider the reality of flat beings existing in
two-dimensional space. What they observe to happen to themselves and to their
flat “things” in their plane is their reality. To them their space will extend
to infinity. It will be consistent with Euclidian geometry.*

Let us consider now a second
two-dimensional existence, but this time on a spherical surface instead of on a
plane. The flat beings with their measuring-rods and other objects fit exactly
on this surface and they are unable to leave it. Their whole universe of
observation extends exclusively over the surface of the sphere. Are these
beings able to regard the geometry of their universe as being plane geometry
and their rods withal as the realisation of “distance”? They cannot do this.
For if they attempt to realise a straight line, they will obtain a curve, which
we “three-dimensional beings” designate as a great circle, i.e. a
self-contained line of definite finite length, which can be measured up by
means of a measuring-rod. Similarly, this universe has a finite area, that can
be compared with the area of a square constructed with rods. The great charm
resulting from this consideration lies in the recognition of the fact that *the universe of these beings is finite and
yet has no limits.*

*But if that two-dimensional existence is on a spherical surface
instead of on a plane, then their geometry is no longer consistent with the
Euclidean geometry. Their straight line will be a curve. Their universe will
have a finite area. The universe of these beings is finite and yet has no limits.*

But the spherical-surface beings do not need to go on a world-tour in order to perceive that they are not living in a Euclidean universe. They can convince themselves of this on every part of their “world,” provided they do not use too small a piece of it. Starting from a point, they draw “straight lines” (arcs of circles as judged in three-dimensional space) of equal length in all directions. They will call the line joining the free ends of these lines a “circle.” For a plane surface, the ratio of the circumference of a circle to its diameter, both lengths being measured with the same rod, is, according to Euclidean geometry of the plane, equal to a constant value, which is independent of the diameter of the circle. On their spherical surface our flat beings would find for this ratio the value

i.e. a smaller value than ** π**,
the difference being the more considerable, the greater is the radius of the
circle in comparison with the radius

**of the “world-sphere.” By means of this relation the spherical beings can determine the radius of their universe (“world”), even when only a relatively small part of their world-sphere is available for their measurements. But if this part is very small indeed, they will no longer be able to demonstrate that they are on a spherical “world” and not on a Euclidean plane, for a small part of a spherical surface differs only slightly from a piece of a plane of the same size.**

*R**These spherical-surface beings may not know that they are living
on a spherical surface; but if they draw large enough circles, they will find
the ratio of the circumference to its diameter is not constant. It becomes
smaller as the radius increases. Thus, they can discover that they are not
living in a Euclidean universe, and that their universe is finite.*

Thus if the spherical-surface beings are living on a planet of which the solar system occupies only a negligibly small part of the spherical universe, they have no means of determining whether they are living in a finite or in an infinite universe, because the “piece of universe” to which they have access is in both cases practically plane, or Euclidean. It follows directly from this discussion, that for our sphere-beings the circumference of a circle first increases with the radius until the “circumference of the universe” is reached, and that it thenceforward gradually decreases to zero for still further increasing values of the radius. During this process the area of the circle continues to increase more and more, until finally it becomes equal to the total area of the whole “world-sphere.”

*It may be difficult for them to discover this if they have
access to only a very small part of their universe.*

Perhaps the reader will wonder why we
have placed our “beings” on a sphere rather than on another closed surface. But
this choice has its justification in the fact that, of all closed surfaces, the
sphere is unique in possessing the property that all points on it are
equivalent. I admit that the ratio of the circumference ** c** of a circle to its
radius

**depends on**

*r***, but for a given value of**

*r***it is the same for all points of the “world-sphere”; in other words, the “world-sphere” is a “surface of constant curvature.”**

*r**In this example, we assume the “world-sphere” to be a “surface
of constant curvature because this makes all points equivalent.*

To this two-dimensional
sphere-universe there is a three-dimensional analogy, namely, the
three-dimensional spherical space which was discovered by Riemann. Its points
are likewise all equivalent. It possesses a finite volume, which is determined
by its “radius” (** 2π^{2}R^{3}**). Is it possible to imagine a
spherical space? To imagine a space means nothing else than that we imagine an
epitome of our “space” experience, i.e. of experience that we can have in the movement
of “rigid” bodies. In this sense we can imagine a spherical space.

*Riemann discovered a three-dimensional analogy of a spherical
space. That means an object moving in a straight line in any direction will
ultimately reach its starting point.*

Suppose we draw lines or stretch
strings in all directions from a point, and mark off from each of these the
distance ** r** with a measuring-rod. All the free end-points of these
lengths lie on a spherical surface. We can specially measure up the area (

**) of this surface by means of a square made up of measuring-rods. If the universe is Euclidean, then**

*F***; if it is spherical, then**

*F = 4πr*^{2}**is always less than**

*F***. With increasing values of**

*4πr*^{2}**,**

*r***increases from zero up to a maximum value which is determined by the “world-radius,” but for still further increasing values of**

*F***, the area gradually diminishes to zero. At first, the straight lines which radiate from the starting point diverge farther and farther from one another, but later they approach each other, and finally they run together again at a “counter-point” to the starting point. Under such conditions they have traversed the whole spherical space. It is easily seen that the three-dimensional spherical space is quite analogous to the two-dimensional spherical surface. It is finite (i.e. of finite volume), and has no bounds.**

*r**It is easily seen that the three-dimensional spherical space is
quite analogous to the two-dimensional spherical surface. It is finite (***i.e. of finite volume), and has
no bounds.**

It may be mentioned that there is yet another kind of curved space: “elliptical space.” It can be regarded as a curved space in which the two “counter-points” are identical (indistinguishable from each other). An elliptical universe can thus be considered to some extent as a curved universe possessing central symmetry.

*We can have the curved space to be “elliptical” in form too. This shall look like a galaxy.*

It follows from what has been said, that closed spaces without limits are conceivable. From amongst these, the spherical space (and the elliptical) excels in its simplicity, since all points on it are equivalent. As a result of this discussion, a most interesting question arises for astronomers and physicists, and that is whether the universe in which we live is infinite, or whether it is finite in the manner of the spherical universe. Our experience is far from being sufficient to enable us to answer this question. But the general theory of relativity permits of our answering it with a moderate degree of certainty, and in this connection the difficulty mentioned in Section XXX finds its solution.

*Thus, close spaces without limits are conceivable. From amongst
these, the spherical space (and the elliptical) excels in its simplicity, since
all points on it are equivalent.*

.

*We as three-dimensional beings live in a material universe. We find it hard to imagine the fourth dimension of inertia in which the substance of variable durations **has existence**. This fourth dimension is not some abstract “time”. It is a dimension that accounts for variable duration characteristic (inertia) of substance. *

*The fourth dimension is INERTIA and not some abstract notion of
TIME.*

*.*