Vacuum non-expanding horizons and shear-free null geodesic congruences

TL;DR

This paper explores the geometry of vacuum non-expanding horizons in space-time, developing a formalism that links null geodesic congruences, CR structures, and complex world-lines, enhancing understanding of horizon geometry.

Contribution

It provides a complete geometric description of NEHs, introduces a new cut formalism, and connects null congruences with CR structures via complex world-lines.

Findings

Null geodesic congruences with shear-free property are generated by complex world-lines.

A canonical null tetrad and coordinate system are established for NEHs.

CR structures on horizons are induced by choices of complex world-lines.

Abstract

We investigate the geometry of a particular class of null surfaces in space-time called vacuum Non-Expanding Horizons (NEHs). Using the spin-coefficient equation, we provide a complete description of the horizon geometry, as well as fixing a canonical choice of null tetrad and coordinates on a NEH. By looking for particular classes of null geodesic congruences which live exterior to NEHs but have the special property that their shear vanishes at the intersection with the horizon, a good cut formalism for NEHs is developed which closely mirrors asymptotic theory. In particular, we show that such null geodesic congruences are generated by arbitrary choice of a complex world-line in a complex four dimensional space, each such choice induces a CR structure on the horizon, and a particular world-line (and hence CR structure) may be chosen by transforming to a privileged tetrad frame.