# A new derivation of singularity theorems with weakened energy hypotheses

## TL;DR

This paper introduces a novel derivation of singularity theorems that relax energy condition assumptions, using index form methods instead of Raychaudhuri's equation, and applies to quantum-inspired hypotheses.

## Contribution

It provides a new derivation approach for singularity theorems under weakened energy hypotheses, avoiding Raychaudhuri's equation and incorporating quantum energy considerations.

## Key findings

- Improved singularity theorems with weaker energy conditions
- Application to quantum energy inequalities with quantitative estimates
- Methodological advancement using index form techniques

## Abstract

The original singularity theorems of Penrose and Hawking were proved for matter obeying the Null Energy Condition or Strong Energy Condition respectively. Various authors have proved versions of these results under weakened hypotheses, by considering the Riccati inequality obtained from Raychaudhuri's equation. Here, we give a different derivation that avoids the Raychaudhuri equation but instead makes use of index form methods. We show how our results improve over existing methods and how they can be applied to hypotheses inspired by Quantum Energy Inequalities. In this last case, we make quantitative estimates of the initial conditions required for our singularity theorems to apply.