Coordinates are messy

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.