Skew-symmetric endomorphisms in $\mathbb{M}^{1,3}$: A unified canonical form with applications to conformal geometry

Abstract

We derive a canonical form for skew-symmetric endomorphisms $F$ in Lorentzian vector spaces of dimension three and four which covers all non-trivial cases at once. We analyze its invariance group, as well as the connection of this canonical form with duality rotations of two-forms. After reviewing the relation between these endomorphisms and the algebra of conformal Killing vectors of $\mathbb{S}^2$, $\mathrm{CKill}(\mathbb{S}^2)$, we are able to also give a canonical form for an arbitrary element $ \xi \in \mathrm{CKill}(\mathbb{S}^2)$ along with its invariance group. The construction allows us to obtain explicitly the change of basis that transforms any given $F$ into its canonical form. For any non-trivial $ \xi$ we construct, via its canonical form, adapted coordinates that allow us to study its properties in depth. Two applications are worked out: we determine explicitly for which metrics, among a natural class of spaces of constant curvature, a given $\xi$ is a Killing vector and solve all local TT (traceless and transverse) tensors that satisfy the Killing Initial Data equation for $\xi$. In addition to their own interest, the present results will be a basic ingredient for a subsequent generalization to arbitrary dimensions.