16
2
Classical general relativity
in 2+1
dimensions
second step we must factor out (or gauge-fix) these transformations. The
resulting 'reduced phase space' is the space of true degrees of freedom of
the theory [100, 101, 102, 135, 149].
In 3+1 dimensions, we do not know how to carry out such a program:
the constraints are too difficult to solve. In 2+1 dimensions, however, a
further decomposition of the phase space variables noticeably simplifies
the ADM action. For closed 'cosmological' spacetimes, this decomposition,
coupled with a clever choice of time-slicing, will allow us to reduce the
constraints to a single differential equation, and will permit a fairly detailed
description of the reduced phase space. As we shall see, this phase space
is closely related to the Riemann moduli space */T(2) of the surface Z, a
space that has been studied extensively by mathematicians.
We begin with a theorem from Riemann surface theory
[110],
which
states that any metric on a compact surface Z is conformal to a metric of
constant (intrinsic) curvature fe, where k = 1 for the two-sphere, k = 0 for
the torus, and k =
—
1 for any surface of genus g > 2. This result, which is
a version of the uniformization theorem discussed in appendix A, allows
us to write the spatial metric gy as
gij = e
2
%, (2.20)
where gy is a constant curvature metric, unique up to diffeomorphisms
of Z. Moreover, the space of such constant curvature metrics modulo
diffeomorphisms is known to be finite dimensional. This means that we
may express any metric gij in the form
gij =
e
u
fgij,
(2.21)
where / is a diffeomorphism of Z and
gij
is one of a fixed finite-dimensional
family of constant curvature metrics. If Z is open, suitable boundary
conditions are needed for this theorem to hold, and the implications for
(2+l)-dimensional gravity are not yet fully understood; section 6 of this
chapter includes a partial analysis.
The space of constant curvature metrics modulo diffeomorphisms is
known as the moduli space Jf of Z. It has dimension 6g
—
6 if the genus
of Z is g > 1, two if Z is a torus (g = 1), and zero if Z is a sphere
(g = 0). This space of metrics will recur throughout this book, appearing,
for example, in chapter 4 as the space of hyperbolic structures on Z. It
is roughly the same as Wheeler's superspace
[285];
more precisely, it is
'conformal superspace', the space of metrics modulo diffeomorphisms and
Weyl transformations.
Let us denote coordinates on Jf by m
a
, and write gy = gy(m
a
). The
variables conjugate to the m
a
are, roughly speaking, parameters that label
the traceless part of the momentum
%
X
K
More precisely, let V,- denote the