Abstract
AbstractIn classical general relativity, the timelike convergence condition is Ric(v, v) ≥ 0 for every timelike vector v. But recent astronomical observations show that the timelike convergence condition does not hold. In this paper, we obtain an explicit upper bound of the volume form and the distance to a focal point (such as Hubble distance) along timelike geodesic with integrals of the negative part of Ricci curvature, which entails singularity theorems.
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