Abstract
This mini course will introduce the basic techniques of optimal transportation in smooth Lorentzian geometry. Starting from the existence of causal couplings and optimal causal couplings, we will discuss dynamical optimal couplings and their structure (i.e. the Mather graph Theorem). Further topics include the Monge problem in Lorentzian geometry and Kantorovich duality. Conditions for the existence of dual solutions will be discussed as well. The last part of the course will be devoted to the characterization of the Einstein field equations via optimal transportation.
Schedule
- Tue 04.03.2025 13.15-14.45 Location: Seminarraum 1 Oskar-Morgenstern-Platz 1, Ground floor
- Thu 06.03.2025 15.00-16.30 Location: Seminarraum 1 Oskar-Morgenstern-Platz 1, Ground floor
- Tue 11.03.2025 13.15-14.45 Location: Seminarraum 1 Oskar-Morgenstern-Platz 1, Ground floor
- Thu 13.03.2025 15.00-16.30 Location: Seminarraum 1 Oskar-Morgenstern-Platz 1, Ground floor
- Mon 17.03.2025 09.45-11.15 Location: Seminarraum 6 Oskar-Morgenstern-Platz 1, 1. floor
- Thu 20.03.2025 15.00-16.30 Location: Seminarraum 6 Oskar-Morgenstern-Platz 1, 1. floor
Abstract
Synthetic methods have profoundly transformed Riemannian geometry in recent decades, extending core concepts and results beyond smooth manifolds. More precisely, bounds on sectional and Ricci curvature, using triangle comparison and optimal transport respectively, have proven to be so robust as to be independent of any differentiable structure. Recently, the foundations for an analogous synthetic Lorentzian geometry have been laid based on the core notion of Lorentzian length spaces. This talk explains the basics of this new geometry, outlines initial results, and explores potential applications in general relativity and discrete approaches to quantum gravity.
Abstract
Consider the trapped photon region in the domain of outer communication of sub-extremal Kerr spacetime. Cederbaum and Jahns (Gen.Relativ.Gravit, 51, 79, 2019) proved that the canonical projection of this trapped photon region in the (co-)tangent bundle is a five-dimensional submanifold of topology \(SO(3)\times \mathbb{R}^2\). By adapting the latter’s methodology, we generalize this result to two stationary axisymmetric classes of spacetimes admitting black hole horizon, namely Kerr- Newman spacetime and Kerr-Sen spacetime. The former is a solution to Einstein-Maxwell equations, while the latter is a solution to Einstein-Maxwell-Dilaton-Axion equations. These include both sub-extremal and extremal cases. The result has potential applications in various areas of mathematical relativity, like black hole uniqueness theorems, black hole dynamical stability, gravitational lensing, black hole shadows, and cosmic censorship conjecture. (This work is in collaboration with Cederbaum).
