###
2.4
The ADM approach

We next turn to a more traditional approach to classical general
relativity, the conventional metric formalism in the space/time
splitting of Arnowitt, Deser, and Misner
[25]
. As Moncrief
[206]
and Hosoya and Nakao
[159]
have shown, this metric formalism can also be used to give a
full description of the solutions of the vacuum field equations,
at least for spacetimes with the topology
.
We start with the ADM decomposition of the spacetime metric
,
as illustrated in Figure
2
. The action then takes the usual form
^{
2
}
In this section I use standard ADM
notation:
and
refer to the induced metric and scalar
curvature of a time
slice and
is the extrinsic curvature of such a slice,
while the full spacetime metric and curvature are denoted
by
and
.
with canonical momentum
and the momentum and Hamiltonian constraints
To solve the constraints, we can choose the York
time-slicing
[284], in which the mean (extrinsic) curvature is used as a time
coordinate,
. Andersson et al. have shown that this is a good global
coordinate choice for classical solutions of the vacuum field
equations
[20]
. We next select a useful parametrization of the spatial metric
and momentum. Up to a diffeomorphism, any two-metric on
can be written in the form
[1, 121]
where
are a finite-dimensional family of metrics of constant curvature
(
for the two-sphere,
for the torus, and
for spaces of genus
). These standard metrics are labeled by a set of moduli
that parametrize the Riemann moduli space of
. As in Section
2.2, such constant curvature metrics can be described in terms of a
geometric structure - for genus
an
structure - with moduli parametrizing the homomorphisms (10). We can count these just as in Section
2.2
; now, since
is three-dimensional, we find that a constant negative curvature
surface of genus
is described by
parameters.
The corresponding decomposition of the
conjugate momentum is described in
[206]
: Up to a diffeomorphism, the trace-free part of
can be written as a holomorphic quadratic differential
, that is, a transverse traceless tensor with respect to the
covariant derivative compatible with
. The space of such quadratic differentials parametrizes the
cotangent space of the moduli space
[1], and the reduced phase space becomes, essentially, the cotangent
bundle of the moduli space.

With the decomposition of
[206], the momentum constraints
become trivial, while the Hamiltonian constraint becomes an
elliptic differential equation that determines the scale factor
in Equation (31) as a function of
and
,

where
are the momenta conjugate to the moduli,
The theory of elliptic equations ensures that Equation (32) determines a unique scale factor
. The action (29) then simplifies to a “reduced phase space” action, involving
only the physical degrees of freedom,
with a time-dependent Hamiltonian
The classical Poisson brackets can be read off directly from
Equation (34):
Three-dimensional gravity again reduces to a
finite-dimensional system, albeit one with a complicated
time-dependent Hamiltonian. The physical phase space is
parametrized by
, which may be viewed as coordinates for the cotangent bundle of
the moduli space of
. For a surface of genus
, this gives us
degrees of freedom, matching the results of Section
2.2
.

If
, this correspondence can be made more explicit: For
and
, the space (11) of geometric structures is itself a cotangent bundle, whose
base space is the space of hyperbolic structures on
. This follows from the fact that the group
is the cotangent bundle of
. Concretely, in the first-order formalism of Section
2.3, the curvature equation (16) with
implies that
is a flat
connection; and if
is a curve in the space of such flat connections, the tangent
vector
satisfies the torsion equation (17). For
, I know of no such direct correspondence, and the general
relationship between the ADM and first-order solutions seems less
transparent.