Null Geodesics from Ladder Molecules

TL;DR

This paper introduces 'ladder molecules' as a discrete analogue of null geodesics in causal sets, providing a new way to model light-like structures in quantum gravity frameworks, supported by simulations in 2d Minkowski spacetime.

Contribution

It proposes ladder molecules as a novel discrete structure representing null geodesics in causal sets, extending the concept of horismotic relations and demonstrating their density and uniqueness in simulations.

Findings

Ladder molecules are dense in causal sets approximating 2d Minkowski spacetime.

They form a light-cone like grid within the causal set.

Unique ladder molecules exist between linked pairs related by the generalized horismotic relation.

Abstract

We propose a discrete analogue of null geodesics in causal sets that are approximated by a region of 2d Minkowski spacetime, in the spirit of Kronheimer and Penrose's "grids" and "beams" for an abstract causal space. The causal set analogues are "ladder molecules", whose rungs are linked pairs of elements corresponding loosely to Barton et al's horizon bi-atoms. In 2d a ladder molecule traps a ribbon of null geodesics corresponding to a thickened or fuzzed out horizon. The existence of a ladder between linked pairs of elements in turn provides a generalisation of the horismotic relation to causal sets. Simulations of causal sets approximated by a region of 2d Minkowski spacetime show that ladder molecules are fairly dense in the causal set, and provide a light-cone like grid. Moreover, similar to the uniqueness of null geodesics between horismotically related events in the spacetime, in such causal sets there is a unique ladder molecule between any two linked pairs which are related by the generalised horismotic relation.