Reliability of perturbation theory in general relativity

Abstract

The relation between perturbation theory and exact solutions in general relativity is tackled by investigating the existence and properties of smooth one-parameter families of solutions. On the one hand, the coefficients of the Taylor expansion (in the parameter) of any given smooth family of solutions necessarily satisfy the hierarchy of perturbation equations. On the other hand, it is the converse question (does any solution of the perturbation equations come from Taylor expanding some family of exact solutions ?) which is of importance for the mathematical justification of the use of perturbation theory. This converse question is called the one of the ‘‘reliability’’ of perturbation theory. Using, and completing, recent results on the characteristic initial value problem, the local reliability of perturbation theory for general relativity in vacuum is proven very generally. This result is then generalized to the Einstein–Yang–Mills equations (and therefore, in particular, to the Einstein–Maxwell ones). These local results are then partially extended to global ones by: (i) proving the existence of semiglobal vacuum space-times (respectively, Einstein–Yang–Mills solutions) which are stationary before some retarded time u0, and radiative after u0, and which admit a smooth conformal structure at future null infinity; and (ii) constructing smooth one-parameter families of such solutions whose Taylor expansions are of the ‘‘multipolar post-Minkowskian’’ type which has been recently used in perturbation analyses of radiative space-times.