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

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 o(r + 1, s + 1) up to conjugation of the 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, (? > 0)-vacuum, algebraically special metrics with nondegenerate optical matrix, analyzed in [G. Bernardi de Freitas et al., Commun. Math. Phys. 340, 291 (2015).] is in one-to-one correspondence with the metrics in the Kerr- de Sitter-like class. This class [M. Mars et al., Classical Quantum Gravity 34, 095010 (2017).; M. Mars and C. Peon-Nieto, Phys. Rev. D 105, 044027 (2022).] exists in all dimensions, and its defining properties involve only properties at 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.

Automated summary

This paper presents a coordinate independent algorithm for determining the class of conformal Killing vectors in locally conformally flat n-metric gamma. The algorithm is based on endomorphisms in the pseudo-orthogonal Lie algebra and the group O. The explicit classification is provided for the Riemannian gamma case. As an application, the paper shows the one-to-one correspondence between a set of five-dimensional vacuum metrics analyzed in a previous study and the metrics in the Kerr-de Sitter-like class. The equivalence between these seemingly unrelated metric classes suggests interesting connections between the algebraically special spacetime and the conformal geometry.