From spatial to null infinity: Connecting initial data to peeling

The asymptotic structure of space-time is studied by imposing conditions on the asymptotics of the metric. These conditions are weak enough to include large classes of physically relevant isolated space-times, but have a rich enough structure to be able to define important physically meaningful quantities like mass, angular momentum, and gravitational waves. By using a unified expansion of the metric in a neighborhood of spatial infinity that includes a piece of null infinity, we connect the asymptotic expansions of solutions to Einstein's equations in the different asymptotic regimes. Within the class of space-times under consideration, we find a connection between the peeling properties of the Weyl scalars and symmetries of initial data near spatial infinity. In particular, we show that for initial data that to leading order is symmetric under $\mathrm{parity}+\text{time}$ reversal, ${\mathrm{\ensuremath{\Psi}}}_{2}$ has the usual $1/{r}^{3}$ fall-off rate at null infinity. If, in addition, the subleading part of the data is antisymmetric under $\mathrm{parity}+\text{time}$ reversal, then ${\mathrm{\ensuremath{\Psi}}}_{1}$ has the usual $1/{r}^{4}$ fall-off rate at future null infinity.