The “standard” discrete approach to classical
general relativity is Regge calculus
[231], initially developed for (3+1)-dimensional gravity but
extendible to arbitrary dimensions. Classical Regge calculus in
2+1 dimensions was investigated by Roček and Williams
[235], who showed that it gave exact results for point particle
scattering. Regge calculus will be discussed further in
Section
3.6
.
The first discrete formulation designed
explicitly for 2+1 dimensions was developed by ’t Hooft et
al.
[250, 252, 253, 124, 274, 156]
. This approach has been used mainly to understand point particle
dynamics, but recent progress has allowed a general description
of topologically nontrivial compact spaces
[169]
. ’t Hooft’s Hamiltonian lattice model is based on the metric
formalism, and starts with a piecewise flat Cauchy surface
tessellated by flat polygons, each carrying an associated frame.
The Einstein field equations with
then imply that edges of polygons move at constant velocities
and that edge lengths may change, subject to a set of consistency
conditions. One obtains a dynamical description parametrized by a
set of lengths and rapidities, which turn out to be canonically
conjugate. Complications occur when an edge shrinks to zero
length or collides with a vertex, but these are completely
understood. The resulting structure can be simulated on a
computer, providing a powerful method for visualizing classical
evolution in 2+1 dimensions.
A related first-order Hamiltonian lattice model
has been studied by Waelbroeck et al.
[266, 267, 268, 270]
. This model is a discretized version of the first-order
formalism of Section
2.3, with triads assigned to faces of a two-dimensional lattice and
Lorentz transformations assigned to edges. The model has an
extensive gauge freedom available in the choice of lattice. In
particular, for a spacetime
, one can choose a lattice that is simply a
-sided polygon with edges identified; the resulting spacetime can
be visualized as a polygonal tube cut out of Minkowski spacetime,
with corners lying on straight worldlines and edges identified
pairwise. This reproduces the quotient space picture discussed by
Mess in the context of geometric structures
[200
]
. With a different gauge choice, Waelbroeck’s model is
classically equivalent to ’t Hooft’s
[271], but the two models are related by a nonlocal change of
variables, and may not be equivalent quantum mechanically.
Much of the recent work on lattice formulations
of (2+1)-dimensional gravity have centered on spin foams and on
random triangulations, both inherently quantum mechanical. These
will be discussed below in Section
3.6
. It is worth noting here, though, that recent work on
diffeomorphisms in spin foam models
[131]
may permit a classical description quite similar to that of
Waelbroeck.
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