Canonical Wick rotations in 3-dimensional gravity

We develop a canonical Wick rotation-rescaling theory in 3-dimensional gravity. This includes (a) A simultaneous classification: this shows how generic maximal globally hyperbolic spacetimes of arbitrary constant curvature, which admit a complete Cauchy surface, as well as complex projective structures on arbitrary surfaces, are all different materializations of “more fundamental” encoding structures. (b) Canonical geometric correlations: this shows how spacetimes of different curvature, that share a same encoding structure, are related to each other by canonical rescalings, and how they can be transformed by canonical Wick rotations in hyperbolic 3-manifolds, that carry the appropriate asymptotic projective structure. Both Wick rotations and rescalings act along the canonical cosmological time and have universal rescaling functions. These correlations are functorial with respect to isomorphisms of the respective geometric categories. In particular, the theory applies to spacetimes with compact Cauchy surfaces. By Mess classification, for every fixed genus g ≥ 2 of a Cauchy surface S, and for any fixed value of the curvature, these spacetimes are parametrized by pairs (F, λ) ∈ Tg ×MLg, where Tg is the Theichmüller space of hyperbolic structures on S, λ is a measured geodesic lamination on F . On the other hand, Tg ×MLg is also Thurston’s parameter space of complex projective structures on S. The WR-rescaling theory provides, in particular, a transparent geometric explanation of this (a priori amazing) coincidence of parameter spaces, and contains a wide generalization of Mess classification. These general spacetimes of constant curvature are eventually encoded by straight convex sets H in H, equipped with suitably defined measured geodesic laminations λ, possibly invariant under the proper action of some discrete subgroup of PSL(2,R). We specifically analyze the remarkable subsectors of the theory made by ML(H)-spacetimes (H = H), and by QD-spacetimes (associated to H consisting of one geodesic line) that are generated by quadratic differentials on Riemann surfaces. In particular, these incorporate the spacetimes with compact Cauchy surface of genus g ≥ 2, and of genus g = 1 respectively. We analyze broken T -symmetry of AdS ML(H)spacetimes and relationship with earthquake theory, beyond the case of compact Cauchy surface. WR-rescaling does apply on the ends of geometrically finite hyperbolic 3manifolds, that hence realize concrete interactions of their globally hyperbolic ending spacetimes of constant curvature. WR-rescaling also provides further “classical amplitudes” of these interactions, beyond the volume of the hyperbolic convex cores.