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2.5 Exact discrete approaches

Discrete approximations to general relativity have existed for decades. In 2+1 dimensions, though, one has the added feature that a discrete description can be exact . This follows from the peculiar nature of the field equations in three dimensions: As discussed above, any vacuum solution can be patched together from finite pieces of constant curvature spacetime, and the dynamics occurs only in the patching.

The “standard” discrete approach to classical general relativity is Regge calculus  [231Jump To The Next Citation Point], 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.  [250252253124274156] . 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 /\ = 0 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.  [266267268270] . 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 M -~ R × S, one can choose a lattice that is simply a 4g -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  [200Jump To The Next Citation Point] . 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  [131Jump To The Next Citation Point] may permit a classical description quite similar to that of Waelbroeck.


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