Weighted Lorentz-Finsler analogues of the Penrose, Hawking, and Hawking-Penrose singularity theorems, plus a weighted Bonnet-Myers theorem, are proved using a new weighted Raychaudhuri equation and a family of ǫ-completeness conditions.
An optimal transport formulation of the Einstein equations of general relativity
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abstract
The goal of the paper is to give an optimal transport formulation of the full Einstein equations of general relativity, linking the (Ricci) curvature of a space-time with the cosmological constant and the energy-momentum tensor. Such an optimal transport formulation is in terms of convexity/concavity properties of the Shannon-Bolzmann entropy along curves of probability measures extremizing suitable optimal transport costs. The result gives a new connection between general relativity and optimal transport; moreover it gives a mathematical reinforcement of the strong link between general relativity and thermodynamics/information theory that emerged in the physics literature of the last years.
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Geometry of weighted Lorentz-Finsler manifolds I: Singularity theorems
Weighted Lorentz-Finsler analogues of the Penrose, Hawking, and Hawking-Penrose singularity theorems, plus a weighted Bonnet-Myers theorem, are proved using a new weighted Raychaudhuri equation and a family of ǫ-completeness conditions.