Reduction of the Einstein equations in 2+1 dimensions to a Hamiltonian system over Teichmüller space

Abstract

In this paper the ADM (Arnowitt, Deser, and Misner) reduction of Einstein’s equations for three-dimensional ‘‘space-times’’ defined on manifolds of the form Σ×R, where Σ is a compact orientable surface, is discussed. When the genus g of Σ is greater than unity it is shown how the Einstein constraint equations can be solved and certain coordinate conditions imposed so as to reduce the dynamics to that of a (time-dependent) Hamiltonian system defined on the 12g−12-dimensional cotangent bundle, T*𝒯(Σ), of the Teichmüller space, 𝒯(Σ), of Σ. The Hamiltonian is only implicitly defined (in terms of the solution of an associated Lichnerowicz equation), but its existence, uniqueness, and smoothness are established by standard analytical methods. Similar results are obtained for the case of genus g=1, where, in fact, the Hamiltonian can be computed explicitly and Hamilton’s equations integrated exactly (as was found previously by Martinec). The results are relevant to the problem of the reduction of the 3+1-dimensional Einstein equations (formulated on circle bundles over Σ×R and with a spacelike Killing field tangent to the fibers of the chosen bundle) and to the recent discussion by Witten of the possible exact solvability of the ‘‘topological dynamics’’ associated with Einstein’s equations in 2+1 dimensions.