Hawking's singularity theorem under a bounded integral norm of Ricci curvature

Abstract

In classical general relativity, the timelike convergence condition is Ric(v,v)≥0 for every timelike vector v. Hawking's singularity theorem is that the timelike convergence condition with a positive future convergence entails the future timelike incompleteness.But recent astronomical observations show that the timelike convergence condition does not hold. We prove a singularity theorem using an integral norm of Ricci curvature instead of Ric(v,v)≥0.