From spatial to null infinity: Connecting initial data to peeling

TL;DR

This paper explores the connection between initial data near spatial infinity and the asymptotic behavior of gravitational fields at null infinity, linking peeling properties of Weyl scalars to initial data symmetries.

Contribution

It introduces a unified expansion approach that connects spatial and null infinity, revealing how initial data symmetries influence the peeling behavior of Weyl scalars.

Findings

Parity + time reversal symmetry leads to standard $ ext{1/r}^3$ fall-off of $oldsymbol{ extPsi_2}$

Antisymmetry in subleading data results in $ ext{1/r}^4$ fall-off of $oldsymbol{ extPsi_1}$

Unified expansion connects asymptotics of Einstein solutions across regimes

Abstract

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 parity + time reversal, $\Psi_2$ has the usual $1/r^3$ fall-off rate at null infinity. If, in addition, the subleading part of the data is antisymmetric under parity + time reversal, then $\Psi_1$ has the usual $1/r^4$ fall-off rate at future null infinity.