Twistor Space Origins of the Newman-Penrose Map

TL;DR

This paper presents a geometrical, twistor space-based reformulation of the Newman-Penrose map, linking Einstein solutions and Maxwell equations, enhancing theoretical understanding and invariance properties.

Contribution

It introduces a purely geometrical, twistor space formulation of the Newman-Penrose map, emphasizing invariance and potential for theoretical insights.

Findings

Reformulation of the Newman-Penrose map in twistor space

Enhanced invariance under spacetime and twistor transformations

Potential for deeper theoretical understanding

Abstract

Recently, we introduced the "Newman-Penrose map," a novel correspondence between a certain class of solutions of Einstein's equations and self-dual solutions of the vacuum Maxwell equations, which we showed was closely related to the classical double copy. Here, we give an alternative definition of this correspondence in terms of quantities that are defined naturally on twistor space, and a shear-free null geodesic congruence on Minkowski space whose twistorial character is articulated by the Kerr theorem. The advantage of this reformulation is that it is purely geometrical in nature, being manifestly invariant under both spacetime diffeomorphisms and projective transformations on twistor space. While the original formulation of the map may be more convenient for most explicit calculations, the twistorial formulation we present here may be of greater theoretical utility.