# Classical and quantum strong energy inequalities and the Hawking singularity theorem

## TL;DR

This paper develops weaker energy conditions, including quantum energy inequalities, to extend Hawking's singularity theorem to scenarios where classical and quantum fields violate the strong energy condition.

## Contribution

It introduces lower bounds on the effective energy density for classical and quantum scalar fields, enabling singularity theorems under less restrictive energy conditions.

## Key findings

- Derived lower bounds for classical scalar fields with mass and nonminimal coupling.
- Established state-dependent quantum energy inequalities for Hadamard states.
- Applied these bounds to prove Hawking-type singularity theorems with weaker energy conditions.

## Abstract

Hawking's singularity theorem concerns matter obeying the strong energy condition (SEC), which means that all observers experience a non-negative effective energy density (EED). The SEC ensures the timelike convergence property. However, for both classical and quantum fields, violations of the SEC can be observed even in the simplest of cases, like the Klein-Gordon field. Therefore there is a need to develop theorems with weaker restrictions, namely energy conditions averaged over an entire geodesic and weighted local averages of energy densities such as quantum energy inequalities (QEIs). We present lower bounds of the EED for both classical and quantum scalar fields allowing nonzero mass and nonminimal coupling to the scalar curvature. In the quantum case these bounds take the form of a set of state-dependent QEIs valid for the class of Hadamard states. We also discuss how these lower bounds are applied to prove Hawking-type singularity theorems asserting that, along with sufficient initial contraction, the spacetime is future timelike geodesically incomplete.