By Riccardo Benedetti, Francesco Bonsante
ISBN-10: 0821842811
ISBN-13: 9780821842812
The authors strengthen a canonical Wick rotation-rescaling thought in three-d gravity. This comprises: a simultaneous type: this indicates how maximal globally hyperbolic house occasions of arbitrary consistent curvature, which admit a whole Cauchy floor and canonical cosmological time, in addition to complicated projective buildings on arbitrary surfaces, are all diversified materializations of 'more basic' encoding buildings; Canonical geometric correlations: this exhibits how house instances of alternative curvature, that percentage a similar encoding constitution, are on the topic of one another by way of canonical rescalings, and the way they are often remodeled by means of canonical Wick rotations in hyperbolic 3-manifolds, that hold the correct asymptotic projective constitution. either Wick rotations and rescalings act alongside the canonical cosmological time and feature common rescaling features. those correlations are functorial with admire to isomorphisms of the respective geometric different types
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2. Cosmological time We refer to [3] for a general and careful treatment of this matter. 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.
Of compact closed 3-manifolds [60] are partition functions of a QFT pertinent to 3D gravity with Euclidean signature and positive cosmological constant. In fact they can be considered as a countable family of regularizations, obtained by using the quantum groups Uq (sl(2, C)), of the Ponzano-Regge calculus based an the classical unitary group SU2 . ) denotes the Witten-Reshetikhin-Turaev invariant and a complete topological QFT has been developed that embodies these last partition functions (see [59]).
While a geodesic lamination on H can be eventually regarded as a particular lamination on H2 , condition (4) ensures that a measured geodesic lamination on H cannot be extended beyond H. 11. We could include in the picture the degenerate straight convex sets H formed by a single geodesic; in this case the measured lamination consists of a single +∞-weighted leaf (L coincides with the whole of H). This degenerate lamination can be regarded as the “limit” of measured geodesic laminations on non-degenerate convex sets Hn , when Hn tends to a geodesic.
Canonical Wick rotations in 3-dimensional gravity by Riccardo Benedetti, Francesco Bonsante
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