Prime examples of such situations are
spacetime *singularities* such as the central point of a black hole
or the state of the universe just before the big bang.
These exotic physical structures involve enormous mass scales
(thus requiring general relativity) and extremely small
distance scales (thus requiring quantum mechanics).
Unfortunately, general relativity and quantum mechanics are mutually
incompatible: any calculation which simultaneously uses both of these
tools yields nonsensical answers. The origin of this problem can be
traced to equations which become badly behaved when particles interact
with each other across minute distance scales on the
order of
10
cm
(
10
in)--- the *Planck length*.

String theory solves the deep problem of the
incompatibility of these two fundamental theories
by modifying the properties of general
relativity when it is applied to scales on the order of the
Planck length. String theory is based on the premise that
the elementary constituents of matter are not described correctly when
we model them as point-like objects. Rather,
according to this theory, the elementary ``particles''
are actually tiny closed loops of string with radii approximately
given by the Planck length. Modern accelerators can only probe down
to distance scales around
10
cm
(
10
in) and hence these loops
of string *appear* to be point objects. However, the string theoretic
hypothesis that they are actually tiny loops, changes drastically
the way in which these objects interact on the shortest of distance
scales. This modification is what allows gravity and quantum mechanics
to form a harmonious union.

There is a price to be paid for this solution, however. It turns out
that the equations of string theory are self consistent only if
the universe contains, in
addition to time, *nine* spatial dimensions. As this is in
gross conflict with the perception of three spatial dimensions,
it might seem that string theory must be discarded. This is not true.

A universe with both extended dimensions (two shown) and curled up dimensions (two shown).

A remarkable property of these theories is that the precise size, shape,
number of holes, etc.
of these extra dimensions *determines* properties such as the masses
and electric charges of the elementary `particles'.

Topology is a mathematical concept that embodies those properties of a geometrical space which do not change if the space is stretched, twisted or bent but not torn. A doughnut and a sphere are distinct from the topological viewpoint because there is no way to deform one into the other smoothly, that is, without tearing either object. A doughnut and a teacup, both of which have one hole, can be continuously deformed into each other and hence have the same topology.

General relativity predicts that the
fabric of spacetime will smoothly deform its size and shape in response
to the presence of matter and energy. A familiar manifestation of this
spacetime stretching is the expansion
of the universe. The *topology* of the universe, however,
remains fixed. A long standing question is whether there might be physical
processes which, unlike those familiar from general relativity,
cause the topology of the universe to change. There is a heuristic reason
for suspecting this possibility based on a naive application
of quantum mechanics. Namely, a universal feature of quantum mechanics
is that on the smallest distance scales even the most quiescent systems
undergo `quantum jitter': the value of quantities characterizing
the system fluctuate, sometimes violently, averaging out to their measured
values on larger distance scales. This notion, applied the fabric of spacetime,
yields the image of a frothing, undulating structure on small distance
scales which averages out on larger scales to the smooth geometrical
description of general
relativity. It is conceivable that, behind the veil of quantum jitter,
the fabric of spacetime could momentarily tear and subsequently reconnect
in a manner resulting in a change of the topology of the universe.
Prior to the advent of string theory, the incompatibility of general
relativity and quantum mechanics made it impossible to address
this possibility in a quantitative manner.

There is a well studied mathematical operation called a *flop*
which is a systematic procedure for changing the topology of
a geometrical space in a ``minimal'' manner. It involves singling out
a sphere in the space, continuously shrinking its volume down to zero
(leaving the rest of the space fully intact) and then blowing its
volume back up, but in an orthogonal direction. The point at which the volume
is zero is the singularity which may be considered as a minimal tear.
The result of this operation is a new geometrical space whose topology
is different from the original. The change in topology is not
as drastic as that between a doughnut and a sphere, but nonetheless it
is different.

Mathematically, this is a rigorously defined and well studied operation.
It can, for instance, be applied to the curled up six dimensional
part of spacetime in a theory based on strings. The crucial question is
whether this operation is *physically* realizable. The criterion for
determining this is simple: can this operation be achieved
in a manner
which does not result in any catastrophic physical consequences?
In general relativity the answer to this question is no as the physical
model ceases to make sense at the singular point --- the point at which
the chosen sphere has zero volume. Since string theory differs from
general relativity on short distance scales, it is conceivable that
a different answer might emerge.
At first sight, however, even the equations of string theory appear
difficult to analyze in this context. Only with the tool of mirror
manifolds can this question be addressed.

Although either member of a mirror pair gives rise to the same physical theory, the technical description of a given physical process very often differs drastically between the two constructions. In fact, certain processes which have an extremely complicated, and difficult to analyze, description when one curled up space is used, have a transparent, and easy to analyze, description when the mirror is used.

Recently, the mirror description of the topology changing flop operation discussed above has been analyzed. This results in a remarkable simplification of the string equations governing this process. An analysis of these simplified equations has revealed that there are no catastrophic physical consequences of this topology changing process. In fact, the mirror description makes it clear that such topology changing events are not only physically realizable, but commonplace as well. Thus, using the tool of mirror manifolds, it has been shown that the long suspected possibility of topology changing processes can be explicitly realized in string theory.

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Last modified: March 3, 1995

Brian Greene, greene@hepth.cornell.edu