Strong Quantum energy inequality and the Hawking singularity theorem

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Singularities, boundary points of spacetime beyond which no extension is possible, continue to intrigue both mathematicians and physicists since they are places where our current understanding of physical law breaks down. The question of whether they exist in physical situations is still an open one. Fifty years ago, Hawking and Penrose developed the first general model independent singularity theorems. These theorems showed that singularities have to exist in any spacetime that satisfies certain properties. Some of these properties are mild assumptions but others, called energy conditions, depend on matter content and are more problematic. For both classical and quantum fields, violations of these conditions can be observed in some of the simplest of cases. Therefore there is a need to develop theorems with weaker restrictions, namely energy conditions averaged over an entire geodesic and quantum inequalities, weighted local averages of energy densities. In this work we investigate the strong energy condition in the presence of both classical and quantum non-minimally coupled scalar fields and derive bounds in each case. In the quantum case these bounds take the form of a set of state-dependent quantum energy inequalities valid for the class of Hadamard states. Finally, we show how the quantum inequalities derived can be used as an assumption to a modified Hawking singularity theorem.