Geometrisch definierte asymptotische Koordinaten in der Allgemeinen Relativitätstheorie

01/10/2020 to 30/09/2023

The main objective of this project is to study the relation of geometric, asymptotic foliations of an asymptotically flat manifold to the asymptotic structure itself. Of main interest are physical and geometric invariants such as mass and center of mass. We wish to contruct geometrically defined asymptotic "center of mass" coordinate systems to study the condition of being asymptotically flat from a purely invariant point of view. We intend to compare these coordinate systems to other choices of asymptotic coordinate systems, such as harmonic coordinates or CMC-coordinates. The important new aspect is that we consider not only Riemannian manifolds but more genereally initial data sets in General Relativity and wish to construct coordinate systems that do not depend on the choice of initial data set in a give spacetime. A further objective is to investigate the Regge-Teitelboim asymptotic parity conditions for asymptotic coordinates in asymptotically flat initial data sets and study their generality with respect to coordinate changes respecting the asymptotic struture. In particular, we are interested in constructing or otherwise proving existence of examples of inititial data sets that do not satisfy these conditions in any asymptotically flat coordinate system. We attempt to develop methods that are robust enough to allow being carried over to the asymptotically hyperbolic setting.