# GMR

## Seminar

### On the convergence of certain expansions at space-like infinity of asymptotically flat, static vacuum solutions

Seminar Room 1, Newton Institute

#### Abstract

We study asymptotically flat, static solutions to Einstein's vacuum field equations. It is known that under weak fall-off conditions at space-like infinity these solutions admit real analytic conformal extensions in which space-like infinity is represented by a regular point i. Furthermore, the expansion of the field at i is determined uniquely by the sequence of the (suitably defined) multi-poles and a given sequence of multi-poles determines via the conformal static field equations a unique formal expansion of the field at $i$. We address the question of the convergence of a related type of expansion in a particular conformal gauge. If the conformal fields are extended near i holomorphically into the complex domain they induce certain `null data' on the complex null cone at i. The expansion coefficients of these null data, which are related in a 1:1 fashion to the multi-pole moments, also determine a unique formal expansion of the conformal fields. We give estimates on the null data under which these formal expansions define analytic solutions to the static field equations.