On the convergence of certain expansions at space-like infinity of asymptotically flat, static vacuum solutions
Seminar Room 1, Newton Institute
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.