The Taylor expansion at past time-like infinity

Abstract

We study the initial value problem for the conformal field equations with data given on a cone ${\cal N}_p$ with vertex $p$ so that in a suitable conformal extension the point $p$ will represent past time-like infinity $i^-$, the set ${\cal N}_p \setminus \{p\}$ will represent past null infinity ${\cal J}^-$, and the freely prescribed (suitably smooth) data will acquire the meaning of the incoming {\it radiation field} for the prospective vacuum space-time. It is shown that: (i) On some coordinate neighbourhood of $p$ there exist smooth fields which satisfy the conformal vacuum field equations and induce the given data at all orders at $p$. The Taylor coefficients of these fields at $p$ are uniquely determined by the free data. (ii) On ${\cal N}_p$ there exists a unique set of fields which induce the given free data and satisfy the transport equations and the inner constraints induced on ${\cal N}_p$ by the conformal field equations. These fields and the fields which are obtained by restricting the functions considered in (i) to ${\cal N}_p$ coincide at all orders at $p$.