# Singularities and Conjugate Points in FLRW Spacetimes

## TL;DR

This paper investigates the relationship between singularities and conjugate points in FLRW spacetimes, proving a theorem that links geodesic behavior near singularities with conjugate points using the Raychaudhuri equation.

## Contribution

It establishes a new theorem connecting conjugate points and singularities in FLRW spacetimes under specific conditions, extending understanding of geodesic behavior near singularities.

## Key findings

- Theorem applies to non-comoving, non-spacelike geodesics in FLRW spacetimes with non-negative curvature.
- The theorem holds for scale factors with power law or logarithmic power law behavior near singularities.
- Applicable to a subset of negatively curved FLRW spacetimes under certain conditions.

## Abstract

Conjugate points play an important role in the proofs of the singularity theorems of Hawking and Penrose. We examine the relation between singularities and conjugate points in FLRW spacetimes with a singularity. In particular we prove a theorem that when a non-comoving, non-spacelike geodesic in a singular FLRW spacetime obeys conditions (39) and (40), every point on that geodesic is part of a pair of conjugate points. The proof is based on the Raychaudhuri equation. We find that the theorem is applicable to all non-comoving, non-spacelike geodesics in FLRW spacetimes with non-negative spatial curvature and scale factors that near the singularity have power law behavior or power law behavior times a logarithm. When the spatial curvature is negative, the theorem is applicable to a subset of these spacetimes.