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

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. (c) 2022 Elsevier B.V. All rights reserved.