It is not hard to check that as time increases,
the modulus (42) moves along a semicircle in the upper half of the complex
plane, with a center on the real axis. Such a curve is a geodesic
in the natural Weil-Petersson (or Poincaré) metric on the torus
moduli space
[159, 135]
. Because of the invariance under the mapping class group (50
), however, the true physical motion in the moduli space of the
torus - the space of physical configurations with the large
diffeomorphisms modded out - is much more complicated; there are
arbitrarily long geodesics, and the flow is, in fact,
ergodic
[93]
.
For spacetimes
with
being a surface of genus
, no explicit metrics analogous to Equation (40
) are known, except for the special case of solutions with
constant moduli. The problem is in part that no simple form such
as Equation (41
) for the “standard” constant curvature metrics exists, and in
part that the ADM Hamiltonian becomes a complicated, nonlocal
function of the moduli. For the case of an asymptotically flat
genus
space, some interesting progress has been made by Krasnov
[172]
; I do not know whether these methods can be extended to the
spatially closed case.
One can write down the holonomies of the
geometric structure for a higher genus surface, of course -
though even there, it is nontrivial to ensure that they represent
spacetimes with
spacelike
genus
slices - but to a physicist, these holonomies in themselves give
fairly little insight into the dynamics. In principle, the ADM
and Chern-Simons approaches might be viewed as complimentary: As
Moncrief has pointed out, one could evaluate the holonomies in
terms of ADM variables in a nice time-slicing, set these equal to
constants, and thereby solve the ADM equations of motion
[207
]
. In practice, though, this approach seems intractable except for
the genus one case. For
, it may be possible to extract a useful physical picture from
the geometrical results of
[55
], which relate holonomies to the structure of the initial
singularity and the asymptotic future geometry, but the
implications have not yet been explored in any depth.
A number of qualitative statements nevertheless
remain possible. The singular behavior of the torus universe
carries over to higher genus: Spacetimes with
expand from a big bang and recollapse in a big crunch, while
those with
expand forever
[200
, 20]
. Moreover, the degeneration of the spatial geometry at the
initial singularity carries over to the higher genus case
[200
, 55
]
. By introducing a global “cosmological time” and exploiting
recent results in two- and three-dimensional geometry, Benedetti
and Guadagnini have shown that when
, a set of parameters describing the initial singularity and a
second set describing the geometry in the asymptotic future
together completely determine the spacetime
[55]
. It seems likely that these two sets are canonically conjugate,
and a better understanding of the symplectic structure could be
useful for quantum gravity.
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