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

TL;DR

This paper develops a coordinate-independent method to classify conformal Killing vectors on locally conformally flat manifolds and applies it to relate certain five-dimensional vacuum metrics to Kerr-de Sitter-like spacetimes.

Contribution

It introduces an algebraic classification algorithm for conformal Killing vectors and links algebraically special metrics to conformal geometry at null infinity.

Findings

Explicit classification for Riemannian case.

Establishes a one-to-one correspondence between certain 5D metrics and Kerr-de Sitter-like class.

Highlights connections between bulk spacetime properties and conformal boundary geometry.

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 $\mathscr{I}$. 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