Twistor Theory on a Finite Graph

Abstract

We show how the description of a shear-free ray congruence in Minkowski space as an evolving family of semi-conformal mappings can naturally be formulated on a finite graph. For this, we introduce the notion of holomorphic function on a graph. On a regular coloured graph of degree three, we recover the space-time picture. In the spirit of twistor theory, where a light ray is the more fundamental object from which space-time points should be derived, the line graph, whose points are the edges of the original graph, should be considered as the basic object. The Penrose twistor correspondence is discussed in this context.