This talk was part of the Workshop on "Mathematical Relativity: Past, Present, Future" held at the ESI December 4 — 7, 2023.
In General Relativity an “isolated system at a given instant of time” can be modeled as an asymptotically Euclidean initial data set (M,g,K). Such asymptotically Euclidean initial data sets (M,g,K) are characterized by the existence of asymptotic coordinates in which the difference between the Riemannian metric g and the Euclidean metric, the second fundamental form K, and the mass and momentum densities and J decay to 0 suitably fast. As shown by Bartnik using harmonic coordinates this decay ensures the convergence of the (ADM-)energy. However, to obtain convergence of the (BORT-)center of mass one needs to additionally impose the Regge-Teitelboim conditions stipulating stronger decay of the odd parts of g, and J and the even part of K. We will see that, under certain circumstances, harmonic coordinates can be used as a tool in checking whether a given asymptotically Euclidean initial data set admits asymptotically Euclidean coordinates satisfying these Regge-Teitelboim conditions. This allows us to give examples of asymptotically Euclidean initial data sets embedded in Schwarzschild which do not possess any Regge-Teitelboim coordinates. This is joint work with Carla Cederbaum and Jan Metzger.