Conformal reference frames for Lorentzian manifolds

I. V. Maresin

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

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.

Keywords: Lorentzian manifold, sky, null geodesic, twistor, contact geometry, line bundle, spinor, conformal symmetry, light cone, Penrose compactification.

Received: 05.04.2016

DOI: 10.4213/tmf9202

 English version: Theoretical and Mathematical Physics, 2017, 191:2, 682–691

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