Abstract
The original singularity theorems of Penrose and Hawking were proved for matter obeying the null energy condition or strong energy condition, respectively. Various authors have proved versions of these results under weakened hypotheses, by considering the Riccati inequality obtained from Raychaudhuri’s equation. Here, we give a different derivation that avoids the Raychaudhuri equation but instead makes use of index form methods. We show how our results improve over existing methods and how they can be applied to hypotheses inspired by quantum energy inequalities. In this last case, we make quantitative estimates of the initial conditions required for our singularity theorems to apply.
Highlights
A central question in gravitational physics is to determine conditions under which singularities arise either as the endpoint of gravitational collapse or at the origin of an expanding universe
The main goal of this paper is to show how the index form methods described in propositions 2.2 and 2.7 can be used to prove singularity theorems with weaker energy conditions than the strong energy condition (SEC) or null energy condition (NEC), using much simpler arguments than those used in existing literature
Given an expansion rate of the universe, and other conditions, drawn from actual cosmological data, would one be able to conclude that the universe is necessarily past timelike geodesically incomplete? To do this we must consider the time-reverse of the analysis presented above, so the question is whether the extrinsic curvature of surfaces of constant cosmic time—the Hubble parameter—is sufficiently positive; that is, whether the expansion rate is sufficiently large
Summary
A central question in gravitational physics is to determine conditions under which singularities arise either as the endpoint of gravitational collapse or at the origin of an expanding universe. We will point out a more direct method for obtaining such results, which avoids the use of the Raychaudhuri equation and Riccati inequalities Instead it is based on the study of the index form, which arises as the the second variational derivative of the length functional about a geodesic. For simplicity and as a proof of principle, we have concerned ourselves solely with singularity theorems for globally hyperbolic spacetimes This fits well with the fact that QEIs are typically derived under the assumption of globally hyperbolicity, and the stability of this condition under small metric perturbations gives some confidence in the relevance of our results for semi-classical gravity. A consequence of our conventions is that timelike and null convergence conditions take the form RμνUμUν 0 for timelike or null vectors Uμ respectively
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