4.5 Closed universes as quotient spaces 71
4.5 Closed universes as quotient spaces
In sections 1 and 2, we encountered two equivalent descriptions of a two-
dimensional surface, as a quotient space H
2
/F and as a geometric space
modeled on H
2
with holonomy group F. We succeeded in generalizing
the latter representation to three dimensions. It is natural to ask whether
the quotient space representation can be extended as well.
In general, it cannot. Consider, for example, the conical spacetime of a
point particle with an irrational deficit angle /?. Rotations by /? are not
periodic, and enough rotations will bring an initial point
XQ
arbitrarily
close to any other point x at the same radius. Consequently, the group
generated by the rotation R(fi) does not act properly discontinuously -
it does not 'separate points' - and the quotient space R
2
/(!?(/?)) is not
well-behaved.^
For cosmological solutions with the topology [0,1] x S, on the other
hand, the situation is more favorable. Mess has shown that if 2 is spacelike,
any such spacetime can be written as a quotient space iV/F, where F is
the holonomy group and (for genus greater than one) N is a region in
the interior of the light cone of Minkowski space
[195].
In principle, this
makes it possible to construct the spacetime [0, l]xl explicitly: one need
merely find a set of coordinates upon which the group F acts nicely and
form the quotient space by identifying appropriate coordinate values.
In practice, this task is already almost unmanageable in two dimensions.
Even for genus two, such coordinate identifications are extraordinarily
difficult to describe explicitly. For genus one, however - that is, for
spacetimes with the spatial topology of a torus - the quotient space
construction allows a complete, simple, and explicit description of all flat
Lorentzian metrics with spacelike hypersurfaces T
2
. We now turn to this
simple case.
We must first determine the possible holonomy groups of the spacetime
M = [0,1] x T
2
. The fundamental group of the torus, and thus of M, is the
abelian group Z©Z, with one generator for each of the two independent
circumferences of T
2
. The holonomy group must therefore be generated
by two commuting Poincare transformations, say (Ai,ai) and (A2,fl2)»
We begin by analyzing the
SO
(2,1) components Ai and
A2.
Any Lorentz
transformation in 2+1 dimensions fixes a vector n, and for Ai and A2 to
commute, they must fix the same vector. This vector may be spacelike,
For the cone, there is a way out: if we remove the line r = 0 from Minkowski space, and
then form the universal covering space (by allowing the polar angle
<\>
to range from —00
to 00), the conical spacetime can be expressed as the quotient of this covering space by
R(P).
Similar constructions have been found in other specific instances, but a systematic
generalization is not known.
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