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

Abstract

We show the existence of families of orthonormal, future directed bases which allow to cast every skew-symmetric endomorphism of $\mathbb{M}^{1,n}$ ($\mathrm{SkewEnd}(\mathbb{M}^{1,n})$) in a single canonical form depending on a minimal number of parameters. This canonical form is shared by every pair of elements in $\mathrm{SkewEnd}(\mathbb{M}^{1,n})$ differing by an orthochronous Lorentz transformation, i.e. it defines the orbits of the orthochronous Lorentz group under the adjoint action on its algebra. Using this form, we obtain the quotient topology of $\mathrm{SkewEnd}(\mathbb{M}^{1,n})/O^+(1,n)$. From known relations between $\mathrm{SkewEnd}(\mathbb{M}^{1,n})$ and the conformal Killing vector fields (CKVFs) of the sphere $\mathbb{S}^n$, a canonical form for CKVFs follows immediately. This form is used to find adapted coordinates to an arbitrary CKVF that covers all cases at the same time. We do the calculation for even $n$ and obtain the case of odd $n$ as a consequence. Finally, we employ the adapted coordinates to obtain a wide class of TT-tensors for $n=3$, which provide Cauchy data at conformally flat null infinity $\mathscr{I}$. Specifically, this class of data is characterized for generating $Λ>0$-vacuum spacetimes with two-symmetries, one of which axial, admitting a conformally flat $\mathscr{I}$. The class of data is infinite dimensional, depending on two arbitrary functions of one variable as well as a number of constants. Moreover, it contains the data for the Kerr-de Sitter spacetime, which we explicitly identify within.