SOLVING THE TIME-EVOLUTION PROBLEM IN 2+1 GRAVITY

We propose a new formulation of 2 + 1 gravity, with a finite number of observables. The initial surface is represented as a (non-planar) polygon in Minkowski space, with edges that are identified in pairs - any surface which rests on the polygon and is continuous at the identified edges can be thought of as the "initial surface". The time-evolution problem is set in the constrained hamiltonian formalism. As the polygon evolves in time, it carves a polygonal cylinder in Minkowski space, with walls that are identified in pairs. Closed timelike geodesics appear in most solutions with non-trivial topology; they are related to Cauchy horizons in Taub-Nut universes. The total mass (which in the static case is just 2-pi times the Euler number) is computed for various time-dependent solutions. A sphere with N moving particles has a "total mass" (sum of deficit angles)/less than 4-pi. Conversely, for particles on a plane the total mass (deficit angle at infinity) increases with velocity. A sphere with g greater-than-or-equal-to 1 evolving handles can accommodate a total stationary mass up to 2-pi, and no less than - 2-pi(4g - 3). For a disk with g greater-than-or-equal-to 1 wormholes. the bounds are -2-pi(4g - 1) and -2-pi; consequently there are no "geons" in 2 + 1 dimensions.