The top 500 results for "singularity theorems" are:

The interior of the Kerr solution is singular and achronological. The classic singularity theorem by Hawking and Penrose relies on chronology, and thus does not apply to the Kerr solution. An improvement of their theorem by Kriele partially removes the requirement of chronology. However, both of these singularity theorems fail to give any information on the type or location of the incomplete geodesics. Here, using recent results of Minguzzi, we prove a new singularity theorem, specifically designed to apply to black holes, which enables locating null incomplete geodesics within the black hole interior, all the while allowing certain forms of chronology violation in the interior.

Some theorems are proved on the singularities of solutions of differential equations with weighted principal symbols. Bibliography: 5 titles.

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We review the first modern singularity theorem, published by Penrose in 1965. This is the first genuine post-Einsteinian result in general relativity, where the fundamental and fruitful concept of the closed trapped surface was introduced. We include historical remarks, an appraisal of the theorem's impact, and relevant current and future work that belongs to its legacy.

The abstract boundary singularity theorem was first proven by Ashley and Scott. It links the existence of incomplete causal geodesics in strongly causal, maximally extended spacetimes to the existence of abstract boundary essential singularities, i.e., non-removable singular boundary points. We give two generalizations of this theorem: the first to continuous causal curves and the distinguishing condition, the second to locally Lipschitz curves in manifolds such that no inextendible locally Lipschitz curve is totally imprisoned. To do this we extend generalized affine parameters from C¹ curves to locally Lipschitz curves.

We extend the validity of the Penrose singularity theorem to spacetime metrics of regularity ${C}^{\mathrm{1,1}}$. The proof is based on regularization techniques, combined with recent results in low regularity causality theory.

The Raychaudhuri equation and its consequences for chronality are studied in the context of Finsler spacetimes. It is proved that the notable singularity theorems of Lorentzian geometry extend to the Finslerian domain. Indeed, so do the theorems by Hawking, Penrose, Hawking and Penrose, Geroch, Gannon, Tipler and Kriele, and also the Topological Censorship theorem and so on. It is argued that the notable results in causality theory connected to achronal sets, future sets, domains of dependence, limit curve theorems, length functional, Lorentzian distance and geodesic connectedness, extend to the Finslerian domain. Results concerning the spacetime asymptotic structure, horizons differentiability and conformal transformations are also included.

We provide a detailed proof of Hawking's singularity theorem in the regularity class ${{C}^{1,1}}$, i.e., for spacetime metrics possessing locally Lipschitz continuous first derivatives. The proof uses recent results in ${{C}^{1,1}}$-causality theory and is based on regularisation techniques adapted to the causal structure.

We translate Penrose's singularity theorem to a Finsler spacetime. To that end, causal concepts in Lorentzian geometry are extended, including definitions and properties of focal points and trapped surfaces, with careful attention paid to the differences that arise in the Finslerian setting.

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.