By Riccardo Benedetti, Francesco Bonsante

The authors increase a canonical Wick rotation-rescaling conception in three-dimensional gravity. This contains: a simultaneous category: this exhibits how maximal globally hyperbolic house occasions of arbitrary consistent curvature, which admit a whole Cauchy floor and canonical cosmological time, in addition to advanced projective constructions on arbitrary surfaces, are all various materializations of 'more primary' encoding buildings; Canonical geometric correlations: this indicates how area occasions of other curvature, that percentage a comparable encoding constitution, are concerning one another through canonical rescalings, and the way they are often reworked by way of canonical Wick rotations in hyperbolic 3-manifolds, that hold definitely the right asymptotic projective constitution. either Wick rotations and rescalings act alongside the canonical cosmological time and feature common rescaling services. those correlations are functorial with appreciate to isomorphisms of the respective geometric different types

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Here we limit ourselves to recalling the main features of this notion. Let M be any spacetime. The cosmological function of M τ : M → (0, +∞] − is deﬁned as follows: let C (q) be the set of past-directed causal curves in M that start at q ∈ M . Then τ (q) = sup{L(c)| c ∈ C − (q)} , where L(c) denotes the Lorentzian length of c. In general, the cosmological function can be very degenerate; for example, on the Minkowski space τ is the +∞-constant function. We say that τ is regular if: (1) τ (q) is ﬁnite valued for every q ∈ M ; (2) τ → 0 along every past-directed inextensible causal curve.

The plane P (id) could be regarded as the set of projective classes of timelike vectors of (sl(2, R), ηId ). The isometry I : P (id) → H2 extends to a linear isometry I¯ : sl(2, R) → X0 that is P SL(2, R) equivariant, where P SL(2, R) acts on sl(2, R) via the adjoint representation and on X0 via the canonical isomorphism P SL(2, R) ∼ = SO(2, 1). If l is an oriented geodesic of H2 = P (id) whose end-points are projective classes of null vectors x− , x+ , then the inﬁnitesimal generator of positive translations along l, say X(l) ∈ sl(2, R) is sent by I¯ to the unit spacelike vector v ∈ X0 orthogonal to l such that x− , x+ , v form a positive basis of X0 .

Moreover, each of our concrete models is real analytic and the isometry groups is made by analytic automorphisms. e. g. [56] or Chapter B of [15] for more details). We recall that X denotes the model (real analytic) manifold, G is a group of analytic automorphisms of X (which possibly preserve the orientation). A special atlas has charts with values onto open sets of X, and any change of charts is given by the restriction to each connected component of its domain of deﬁnition of some element g ∈ G.