Flat deformation theorem and symmetries in spacetime

TL;DR

This paper explores the flat deformation theorem, demonstrating that any Lorentzian analytic metric can be expressed in an extended Kerr-Schild form and that symmetries of the original metric can be preserved in the flat deformed metric.

Contribution

It shows how the flat deformation theorem implies an extended Kerr-Schild form for metrics and connects symmetries of the original metric to those of the flat deformed metric.

Findings

Every Lorentzian analytic metric can be written in an extended Kerr-Schild form.

Symmetries like conformal Killing vectors can be preserved in the flat deformed metric.

The flat deformation leads to a flat metric inheriting the original symmetries.

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 $\eta_{ab} := a g_{ab} - 2 b k_{(a} l_{b)}$ 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.