8 April 2022

**Regge-Teitelboim conditions (and their relationship with harmonic coordinates)**

In General Relativity, an “isolated system at a given instant of time” is 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 Riemannian metric *g* and second fundamental form *K* decay to the Euclidean metric \delta and to 0 suitably fast, respectively. Using harmonic coordinates Bartnik showed that (under suitable integrability conditions on their matter densities) the (ADM-)energy, (ADM-)linear momentum and (ADM-)mass of an asymptotically Euclidean initial data set are well-defined. To study the (ADM-)angular momentum and (BORT-)center of mass, however, one usually assumes the existence of Regge-Teitelboim coordinates on the initial data set (*M,g,K*) in question, i.e. the existence of asymptotically Euclidean coordinates satisfying additional decay assumptions on the odd part of *g* and the even part of *K*. We will show that, under certain circumstances, harmonic coordinates can be used as a tool in checking whether a given asymptotically Euclidean initial data set possesses Regge-Teitelboim coordinates. This allows us to easily give examples of (vacuum) asymptotically Euclidean initial data sets which do not possess any Regge-Teitelboim coordinates. This is joint work with Carla Cederbaum and Jan Metzger.