The Taylor expansion at past time-like infinity

TL;DR

This paper analyzes the initial value problem for conformal vacuum field equations near past time-like infinity, establishing existence, uniqueness, and the determination of fields from free data on a null cone.

Contribution

It provides a rigorous construction of fields satisfying the conformal vacuum equations with prescribed data at past time-like infinity and null infinity, including Taylor expansions and transport equations.

Findings

Existence of smooth fields satisfying conformal vacuum equations near p.

Unique determination of Taylor coefficients from free data.

Matching of fields on the null cone and at p.

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 $p$.