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 ${\mathcal{N}}_p$ with vertex $p$ so that in a suitable conformal extension the point $p$ will represent past time-like infinity $i^{-}$, the set ${\mathcal{N}}_p\setminus \{p\}$ will represent past null infinity ${\mathcal{J}}^{-}$, 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 ${\mathcal{N}}_p$ there exists a unique set of fields which induce the given free data and satisfy the transport equations and the inner constraints induced on ${\mathcal{N}}_p$ by the conformal field equations. These fields and the fields which are obtained by restricting the functions considered in (i) to ${\mathcal{N}}_p$ coincide at all orders at $p$.

40 pages

References in corpus (2)

• The many ways of the characteristic Cauchy problem
• Solutions of the vacuum Einstein equations with initial data on past null infinity

Cited by in corpus (5)

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• Graviton scattering in self-dual radiative space-times
• Fully pseudospectral solution of the conformally invariant wave equation near the cylinder at spacelike infinity
• On the smoothness of the critical sets of the cylinder at spatial infinity in vacuum spacetimes