Effective null Raychaudhuri equation

TL;DR

This paper investigates how an intrinsic discreteness of spacetime affects null Raychaudhuri's equation, deriving effective expressions that regularize singularities and modify the behavior of null congruences near focal points.

Contribution

It introduces an effective continuous model incorporating spacetime discreteness to modify null Raychaudhuri's equation, providing finite limits and new insights into singularity behavior.

Findings

Finite limiting values for expansion and its rate near focal points

Modified effective equations regularize classical singularities

Non-vanishing limiting cross-sectional area of null congruences

Abstract

The effects on Raychaudhuri's equation of an intrinsically-discrete or particle nature of spacetime are investigated. This is done through the consideration of null congruences emerging from, or converging to, a generic point of spacetime, i.e. in geometric circumstances somehow prototypical of singularity issues. We do this from an effective point of view, that is through a (continuous) description of spacetime modified to embody the existence of an intrinsic discreteness on the small scale, this adding to previous results for non-null congruences. Various expressions for the effective rate of change of expansion are derived. They in particular provide finite values for the limiting effective expansion and its rate of variation when approaching the focal point. Further, this results in a non-vanishing of the limiting cross-sectional area itself of the congruence.