Solving the time-evolution problem in 2 + 1 gravity

Homotopy invariants and time evolution in (2+1)-dimensional gravity

“…The solution of the time-evolution problem of (2 + 1)-dimensional gravity, for spacetimes with the topology Σ g,N × (0, 1) was originally derived [12] from an exact lattice theory [20]. This solution gives a representation of spacetime by means of a polygon embedded in Minkowski space, with identifications of boundary points by elements of Γ ⊂ ISO(2, 1).…”

“…The explicit solution of the time-evolution problem for genus g and N particles was given by one of us (HW) in terms of time-dependent global variables, which define a polygon representation of spacetime. Since these variables evolve in time, they are not observable in the same sense as the homotopy invariants, yet they are what one would intuitively call "observable", in the sense that they can be measured by an observer [12]. Carlip [13] has suggested that the variables of the polygon representation are related to the ISO(2, 1) homotopy invariants, yet it has not been clear, until now, exactly how.…”

“…Euler class, which gives physically acceptable spacetimes: solutions on other sheets represent spacetimes with large "negative mass" singularities, Ω n = Ω − 4πn/G [1], [12].…”

Canonical quantization of (2+1)-dimensional gravity

“…We further require that the wave packet a(K) have support only over P 's which correspond to solutions of the cycle conditions which form a faithful representation of the fundamental group [5], [21] (other sheets of solutions include totally collapsed handles, or curvature singularities with a surplus angle equal to a multiple of 4π).…”

“…The difficulties encountered by Regge et al can be avoided by using these homotopies rather than the invariants derived from them, leaving out a global ISO(2, 1) symmetry. The homotopies can be parametrized in the "polygon representation" of (2 + 1)-dimensional gravity, and the symplectic structure is known (H.W., [21]). Mapping class invariant scattering amplitudes can be computed by the method of images, by summing over all mapping class images of the 'in' state.…”