2.1 The
topological
setting 11
topology. (Recall that the Euler number of an orientable genus g surface
is 2
—
2g.) S~ and E
+
need not be connected, however, so topology
change is not completely ruled out. Instead, we obtain an interesting set
of selection rules - for instance, a genus g surface can evolve into two
surfaces of genus gi and g2 only if 2
—
2g = (2
—
2gi) + (2
—
2g2), i.e.,
gi +g2 = g +
1.
We shall discuss quantum mechanical topology change in
chapter 9; for now, let me simply note that a similar selection rule occurs
in the path integral formalism.
We now turn to the more difficult problem, the question of which
three-manifolds admit flat Lorentzian metrics. A complete answer to this
question is not known, but a number of useful results can be found in
the mathematics literature. For example, closed three-manifolds with flat
Lorentzian metrics are understood fairly well [18, 113, 131]. All such
manifolds are geodesically complete, making them interesting candidates
for singularity-free spacetimes, and their fundamental groups can be de-
scribed explicitly. Unfortunately, though, closed Lorentzian manifolds
always contain closed timelike curves, and thus have limited value as
models in classical general relativity.
Less is known about noncompact three-manifolds with flat Lorentzian
metrics. A number of interesting examples are given in [103] and
[104];
in particular, it is shown that any handlebody (that is, any 'solid genus g
surface') can be given a geodesically complete flat Lorentzian metric. The
resulting geometries are fairly bizarre - for instance, it is unlikely that
they allow any time-slicing - and they could potentially serve as coun-
terexamples for a number of plausible claims about (2+l)-dimensional
gravity. These spacetimes have not yet been studied by physicists in any
detail.
For our purposes, the most important result is a theorem due to Mess
[195],
Suppose that M is a compact three-dimensional manifold with a
flat, time-orientable Lorentzian metric and a purely spacelike boundary.
Then M necessarily has the topology
M*[0,l]xS, (2.2)
where H is a closed surface homeomorphic to one of the boundary com-
ponents of M. This means that for spatially closed three-dimensional uni-
verses, topology change is forbidden by the field equations - the topology
of spacetime is completely fixed by that of an initial spacelike slice. This
powerful result greatly simplifies the study of classical (2+l)-dimensional
cosmology, allowing us to ignore many of the more complicated space-
time topologies. Moreover, we shall see below that if 2 is any surface
other than the two-sphere, a manifold with the topology (2.2) actually
admits a large family of flat Lorentzian metrics, which can be described
in considerable detail.