Conformal reference frames for Lorentzian manifolds

Abstract

We define a conformal reference frame, i.e., a special projection of the six-dimensional sky bundle of a Lorentzian manifold (or the five-dimensional twistor space) to a three-dimensional manifold. We construct an example, a conformal compactification, for Minkowski space. Based on the complex structure on the skies, we define the celestial transform of Lorentzian vectors, a kind of spinor correspondence. We express a 1-form generating the contact structure explicitly as a (line bundle)-valued form. We prove a theorem on the projection of this 1-form to the fiberwise normal bundle of a reference frame; its corollary is an equation for the flow of time. The Appendix is less mathematical than the main body and discusses the causal relation in context of the FLRW cosmology and its natural conformal reference frame.