The Geometry of Gravitational Radiation

Abstract: We consider 4-dimensional asymptotically flat vacuum spacetimes near future null infinity endowed with the most general allowable Carroll geometry. We show that the near-boundary radial expansion (to a certain order) can be organised in terms of connections that can be obtained by gauging the conformal Carroll algebra. The only non-vanishing curvatures in this gauging procedure are those that are associated with the special conformal generators and we will refer to these as the $K$-curvatures. The vanishing of these $K$-curvatures defines an asymptotic vacuum spacetime and we use this to construct a boundary (i.e. Carroll) covariant expression for the vacuum (soft) shear in terms of two boundary Carroll scalar fields. The $K$-curvatures transform in a hierarchical fashion into one another under Carroll boosts. This leads to a classification of 4 types of spacetimes: vacuum, strongly and weakly non-radiative, and radiative spacetimes. It is shown that the $K$-curvatures correspond to 5 of the 10 Weyl tensor components at leading order in their $1/r$ expansion. We furthermore observe that one of the $K$-curvatures is equal to the recently found Carroll boost anomaly. Finally, we show that the Bondi loss equations for the boundary energy-momentum-news complex can be cast into a form involving another energy-momentum tensor with vanishing energy flux that is traceless and whose non-conservation is entirely captured by the $K$-curvatures. The BMS currents can be obtained by contracting this latter energy-momentum tensor with a Carroll conformal Killing vector.