Séminaire de Géométrie
Towards a Spacetime Intrinsic Flat ConvergenceAnna Sakovich (Université d'Uppsala)
Tuesday 27 June 2023 10:20 - Tours - 1180 (Bât. E2)
Résumé :
How can we compare spacetimes with stars, black holes and other singularities – for example our Universe – to standard cosmological models that are symmetric and smooth? One can argue that for answering this kind of questions, where the manifolds that one wants to compare are not close in a smooth sense and possibly not even diffeomorphic, it is necessary to rely on notions of distance from metric geometry that are based on comparison of distances rather than metric tensors. The intrinsic flat distance of Sormani and Wenger is an example of such a notion of distance which has been successfully applied to address a number of questions in Riemannian geometry and mathematical general relativity, such as stability of the positive mass theorem and torus rigidity theorems. I will report on our ongoing work with Christina Sormani whose ultimate goal is to develop the analogue of intrinsic flat distance and intrinsic flat convergence for Lorentzian manifolds. One of the difficulties in achieving this goal is that, unlike Riemannian manifolds, Lorentzian manifolds are not natural metric spaces. In this connection, we have recently shown that there is a canonical procedure which allows one to convert spacetimes into metric spaces, in such a way that it is possible to recover the Lorentzian structure back. This motivates a very promising notion of spacetime intrinsic flat distance that I will discuss along with some potential applications.
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