Flat deformation theorem and symmetries in spacetime

Abstract

The \emph{flat deformation theorem} states that given a semi-Riemannian analytic metric $g$ on a manifold, locally there always exists a two-form $F$, a scalar function $c$, and an arbitrarily prescribed scalar constraint depending on the point $x$ of the manifold and on $F$ and $c$, say $\Psi (c,F,x)=0$, such that the \emph{deformed metric} $\eta =cg-\epsilon F^2$ is semi-Riemannian and flat. In this paper we first show that the above result implies that every (Lorentzian analytic) metric $g$ may be written in the \emph{extended Kerr-Schild form}, namely ${\eta }_{ab}:=a{g}_{ab}-2b{k}_{(a}{l}_{b)}$ where $\eta$ is flat and $k_a,l_a$ are two null covectors such that $k_a{l}^a=-1$; next we show how the symmetries of $g$ are connected to those of $\eta$, more precisely; we show that if the original metric $g$ admits a Conformal Killing vector (including Killing vectors and homotheties), then the deformation may be carried out in a way such that the flat deformed metric $\eta$ `inherits' that symmetry.

30 pages, 0 figures