3.3 The torus universe 53
the solution (3.70):
For A > 0, the solutions display a 'big bang' or 'big crunch' singularity: as
T increases from an initial value of 2^/A, the universe collapses, reaching
a final singularity of zero area at T — oo. (The time-reversed solution
represents an expanding universe.) For A negative, on the other hand, the
area reaches a minimum value: the universe 'bounces'.
Note that as T approaches infinity,
%2
also goes to zero. This indicates
that as the universe approaches a big bang or big crunch, the spatial
geometry also becomes singular: the parallelogram of figure 3.3 collapses
to a line. Similar - but typically much more complex - singular behavior
of the spatial geometry is known to occur for (2+l)-dimensional universes
with more complicated topologies; a good deal of the mathematics is
understood, in a fairly technical form, but this behavior has not yet been
analyzed in the physics literature
[195].
So far, I have been rather cavalier in my treatment of the symmetries
of these solutions. By construction, the solutions are invariant under
the 'small' spacetime diffeomorphisms, that is, the diffeomorphisms that
can be built from infinitesimal transformations. The torus also admits
'large' diffeomorphisms, however, generated by Dehn twists around the
circumferences. These act nontrivially on the moduli
T
and the momenta
p.
As described in appendix A, the group of such large diffeomorphisms
of the torus (modulo 'small' diffeomorphisms) has two generators,
S
:
T
-• —, p -> T
2
p
(3.73)
This means that not all moduli are geometrically distinct. Rather, the
independent geometries are characterized by values of
T
that range over
a single fundamental region, such as the 'keyhole' region shown in fig-
ures 3.4 and A.6. Equation (3.71) for the trajectory is therefore somewhat
misleading; projected back to a single fundamental region, a portion of
a typical trajectory looks like the path shown in figure 3.4. The resulting
motion is quite complicated: there are arbitrarily long closed geodesies,
and the geodesic flow is actually ergodic [80, 150]. Note, though, that the
evolution of the physical geometry of space is not ergodic. The full geom-
etry depends on both the modulus and the area, and the time-dependence
(3.72) of the area is simple and well-behaved.
The role of large diffeomorphisms is a subtle one, but it will be im-
portant later when we discuss quantization. For the torus, the upper