Conformal reference frames for Lorentzian manifolds

TL;DR

This paper introduces conformal reference frames for Lorentzian manifolds, constructs a Minkowski space example, and explores the celestial transform and contact structures, linking geometry with cosmological causal relations.

Contribution

It defines conformal reference frames for Lorentzian manifolds and constructs a Minkowski space example, advancing geometric understanding of spacetime structures.

Findings

Constructed a conformal compactification for Minkowski space

Defined the celestial transform of Lorentzian vectors

Derived an equation for the flow of time

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.