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Flat deformation theorem and symmetries in spacetime
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The 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 (c,F, x) = 0, such that the deformed metric η = cg − F2 is semi-Riemannian and flat. In this paper we first show that the above result implies that every (Lorentzian analytic) metric g may bewritten in the extendedKerr-Schild form, namely ηab := agab−2bk(a lb) where η is flat and ka, la are two null covectors such that kala = −1; next we show how the symmetries of g are connected to those of η, 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 η 'inherits' that symmetry.
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LLOSA, Josep and CAROT, Jaume. Flat deformation theorem and symmetries in spacetime. Classical and Quantum Gravity. 2009. Vol. 26, num. 5, pags. 055013-055032. ISSN 0264-9381. [consulted: 22 of October of 2026]. Available at: https://hdl.handle.net/2445/171250