266 The Poetry of Physics and The Physics of Poetry
infinite. If the curvature is positive, then the universe curves back upon
itself in the four-dimensional space and hence is closed and finite. On a
clear day one could see the back of one’s head if one stood still for 13.7
billion years. The analogy of a closed and finite space in two dimensions
is a circle, which closes back upon itself to form a one-dimensional
closed space. The analogy in three dimensions is the surface of a sphere,
which closes back upon itself forming a two-dimensional closed space. If
the universe curves back upon itself in the four dimensional space-time
continuum, then it forms a closed three dimensional space, the surface of
a four dimensional sphere. If the universe is closed, then a photon
emitted in a given direction if unimpeded will eventually return to the
point of its origin having traversed a complete hypercircle. This is
exactly analogous to the situation on the surface of a three dimensional
sphere. A creature crawling on the surface of the Earth will eventually
return to the point of its departure having traversed a great circle because
the two-dimensional surface of the Earth curves back on itself.
Einstein showed that the nature of the curvature of the four
dimensional space-time continuum depends on the density of matter
within the universe. If the density of the universe is small, then the
curvature is negative but if the density is great, then the curvature is
positive and the universe is finite and closed. It is difficult to determine
the density of the universe because of our ignorance of intergalactic
space. There is also the question of dark matter and dark energy needed
to explain the acceleration of the rate of expansion, but more of that later.
We can estimate the amount of galactic matter, however. If the space
between the galaxies is void, then the density of the universe is too low
to curve the space-time continuum positively and the universe is open
and infinite. The universe will continue to expand forever because the
forces of gravity will never be strong enough to pull it back together
again.
If the intergalactic medium contains a sufficient amount of matter
then it is possible that the curvature is positive. In order for the universe
to close upon itself the amount of extra galactic material would have
to be equal to 10 times the amount of matter in the galaxies. Because
of the vast expanses of intergalactic space, the density of matter required
to furnish this much mass is only 10-29 grams per cubic centimeter
or one hydrogen atom for every 100 cubic centimeters. This is still a
rather rarefied medium when one considers that the density of a galaxy
is 10-24 grams per cubic centimeter. At the moment the density of