Generalized shear coordinates on the moduli spaces of three-dimensional spacetimes

Abstract

We introduce coordinates on the moduli spaces of maximal globally hyperbolic constant curvature 3d spacetimes with cusped Cauchy surfaces S. They are derived from the parametrisation of the moduli spaces by the bundle of measured geodesic laminations over Teichm\"uller space of S and can be viewed as analytic continuations of the shear coordinates on Teichm\"uller space. In terms of these coordinates the gravitational symplectic structure takes a particularly simple form, which resembles the Weil-Petersson symplectic structure in shear coordinates, and is closely related to the cotangent bundle of Teichm\"uller space. We then consider the mapping class group action on the moduli spaces and show that it preserves the gravitational symplectic structure. This defines three distinct mapping class group actions on the cotangent bundle of Teichm\"uller space, corresponding to different values of the curvature.