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

TL;DR

This paper presents a unified canonical form for skew-symmetric endomorphisms in Lorentzian spaces, linking them to conformal geometry and applications in constant curvature spaces, with explicit transformations and applications to Killing vectors.

Contribution

It introduces a comprehensive canonical form for skew-symmetric endomorphisms in Lorentzian spaces and connects these forms to conformal Killing vectors and their invariance groups.

Findings

Derived a canonical form for skew-symmetric endomorphisms in Lorentzian spaces.

Connected the canonical form to duality rotations and conformal Killing vectors.

Applied the results to identify metrics admitting specific Killing vectors and solve related tensor equations.

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.