zbMATH — the first resource for mathematics

Generalized shear coordinates on the moduli spaces of three-dimensional spacetimes. (English) Zbl 1347.83029
Summary: 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 parametrization of the moduli spaces by the bundle of measured geodesic laminations over Teichmüller space of \(S\) and can be viewed as analytic continuations of the shear coordinates on Teichmüller 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 Teichmüller 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 Teichmüller space, corresponding to different values of the curvature.
See [Commun. Math. Phys. 273, No. 3, 705–754 (2007; Zbl 1148.83016)] for similar results by the same author.
83C80 Analogues of general relativity in lower dimensions
57R57 Applications of global analysis to structures on manifolds
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
53Z05 Applications of differential geometry to physics
83C10 Equations of motion in general relativity and gravitational theory
Full Text: DOI Euclid