Josep Llosa and Jaume Carot 2009 Class. Quantum Grav. 26 055013 doi:10.1088/0264-9381/26/5/055013
Flat deformation theorem and symmetries in spacetime
Josep Llosa1 and Jaume Carot2
Show affiliationsThe 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 Ψ(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 be written in the extended Kerr–Schild form, namely ηab := agab − 2bk(alb) 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.
- PACS
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02.40.Ky Riemannian geometries
- MSC
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53C21 Methods of Riemannian geometry, including PDE methods; curvature restrictions (See also 58J60)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
- Subjects
- Dates
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Issue 5 (7 March 2009)
Received 11 September 2008, in final form 9 January 2009
Published 17 February 2009
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Flat deformation theorem and symmetries in spacetime
Josep Llosa and Jaume Carot 2009 Class. Quantum Grav. 26 055013
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