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 transformation of Lorentzian vectors, a kind of spinor correspondence. We express a 1-form generating the contact structure in the twistor space (when it is smooth) explicitly as a form taking line-bundle values. 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.