Covariant classification of conformal Killing vectors of locally conformally flat $n$-manifolds with an application to Kerr-de Sitter

Abstract

We obtain a coordinate independent algorithm to determine the class of conformal Killing vectors of a locally conformally flat $n$-metric $\gamma$ of signature $(r,s)$ modulo conformal transformations of $\gamma$. This is done in terms of endomorphisms in the pseudo-orthogonal Lie algebra $\mathfrak{o}(r+1,s+1)$ up to conjugation of the its group $O(r+1,s+1)$. The explicit classification is worked out in full for the Riemannian $\gamma$ case ($r=0,s=n$). As an application of this result, we prove that the set of five dimensional, $(\Lambda >0)$-vacuum, algebraically special metrics with non-degenerate optical matrix, previously studied by Bernardi de Freitas, Godazgar and Reall, is in one-to-one correspondence with the metrics in the Kerr-de Sitter-like class. This class exists in all dimensions and its defining properties involve only properties at $\mathcal{I}$. The equivalence between two seemingly unrelated classes of metrics points towards interesting connections between the algebraically special type of the bulk spacetime and the conformal geometry at null infinity

25 pages, 3 figures

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Cited by in corpus (2)

• On the uniqueness of the Kerr-(A)dS metric as a type II(D) solution in six dimensions
• New characterization of Robertson-Walker geometries involving a single timelike curve