### Chapter 10: Quantum Geometry

George Bernhard Riemann, a nineteenth-century German mathematician,
figured out how to apply geometry to curved spaces. Einstein recognized
that Rienmann’s geometry accurately described the physics of gravity,
and Reinmann’s theories supplied him with the necessary mathematical
foundations to analyze warped space. The curvature of spacetime,
Rienmann found, is expressed mathematically as the distorted distances
between its points. Einstein applied Rienmann’s discovery to the
physical realm and concluded that the gravitational force felt by
an object directly reflects this distortion.

String theory deals in short-distance physics, and Rienmannian geometry
ceases to function at an ultramicroscopic level. This means that,
for string theory to work, physicists must modify both Riemannian
geometry and the general theory of relativity that Einstein derived
from it. A new type of geometry is necessary to decipher tiny Planck-length
scales. Physicists have called this new type of geometry *quantum
geometry*.

Fifteen billion years ago, the universe began with the
big bang. As Hubble discovered, the universe is constantly expanding,
which makes it hard to measure the average density of matter in
the universe. If the average matter density exceeds a so-called *critical
density* of a hundredth of a billionth of a billionth of
a billionth (10–29) of a gram per cubic
centimeter, then a large gravitational force will permeate the cosmos
and reverse the expansion. If the average density is less than the
critical density, the gravitational expansion will be too weak to
do this. (The earth is not a reliable indicator for the average
density of the universe: Matter clumps, and the vast empty spaces
between galaxies bring the average down.)

Conventional wisdom proclaims that the universe began
with a bang from an initial zero-size state. If the universe has
enough mass, it will eventually end with a “crunch” that will reduce
it to a similar state of compression. String theory is required
to help physicists evaluate the extremely compressed early stage;
it has set Planck length as the lower limit on the size of the “Big
Crunch.” It would not make sense to set this same limit for the
point-particle model.

To return to the garden hose analogy for the universe:
strings, unlike point particles, can “lasso” the circular part of
the garden hose. When a string is in this position, it is in a *winding
mode of motion*, which is a possibility that is inherent
to strings. A string in winding mode has a minimum mass that is
determined by the size of the circular dimension it is wrapping
around and the number of times it is wrapped.

Wound-string configurations suggest that a string’s energy
comes from two sources: vibrational motion and winding energy. All
string motion is a combination of sliding and oscillating. Strings’
vibrational movements have energies that are inversely proportional
to the radius of the circle they are wrapping. A small radius, for
example, would confine the string more strictly and would contain
more energy. But the winding mode energies are directly proportional
to the radius. Greene eventually explains what this means: there
is no distinction between geometrically distinct forms. The same
goes for total string energies: there is no distinction between
different sizes for the circular dimension! Through a complicated
chain of explanations, Greene shows that there is absolutely no
way to differentiate between radii that are inversely related to
one another.