A singularity theorem in terms of asymptotic expansion

Abstract

We prove a sharp singularity theorem in which the classical focusing hypothesis of Hawking--Penrose theory is replaced by a condition on the asymptotic volume growth of timelike geodesics. More precisely, let $(M,g)$ be a smooth, globally hyperbolic Lorentzian manifold satisfying the strong energy condition, and let $V\subset M$ be a smooth, compact Cauchy hypersurface. We associate to the future-directed timelike geodesics orthogonal to $V$ asymptotic volume-expansion invariants $θ_α$, and show that a uniform lower bound $θ_α\ge c>0$ forces a sharp upper bound on the time separation between $V$ and its chronological past. In particular, $(M,g)$ is past timelike geodesically incomplete. The estimate is sharp and equality is attained by Lorentzian cones. Moreover, in the smooth setting we prove a rigidity theorem showing that equality characterizes these cone models. The argument extends to globally hyperbolic Lorentzian length spaces satisfying the synthetic strong energy condition in the form of the Timelike Measure Contraction Property, TMCP(0,N). In this non-smooth setting we also obtain corresponding raywise and measure-rigidity statements.