Twistor theory on a finite graph

TL;DR

This paper extends twistor theory concepts to finite graphs by defining holomorphic functions on graphs and relating graph structures to space-time and light rays, offering a discrete analogue of continuous twistor geometry.

Contribution

It introduces the notion of holomorphic functions on graphs and formulates twistor theory concepts within a finite graph framework, connecting graph theory with space-time geometry.

Findings

Holomorphic functions on graphs can model shear-free ray congruences.

Regular coloured graphs of degree three recover space-time structures.

The line graph captures the fundamental role of light rays in twistor theory.

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.