The paper extends completeness notions to low-regularity spacetimes.
problem Defining completeness conditions for spacetimes with low-regularity metrics.
method Extending Beem's completeness notions to Lorentzian length spaces and proving relationships between them.
result Equivalence of completeness conditions for globally hyperbolic C1-spacetimes under certain conditions. The study proves timelike Ricci bounds for low regularity spacetimes using optimal transport.
problem Proving timelike Ricci bounds for spacetimes with low regularity.
method Using optimal transport to prove timelike measure-contraction property.
result Timelike curvature-dimension condition holds for C1,1 metrics. Low regularity spacetimes split into simpler structures.
problem Proving splitting theorem for C1 metrics and weights. method Combining elliptic techniques and line-adapted curves.
result Extends Lorentzian splitting theorem to C1 settings. Proves conditions for Cauchy horizons in low-regularity spacetimes.
problem Conditions for the existence of Cauchy horizons in spacetimes with low regularity.
method Analyzes the relationship between complete Cauchy hypersurfaces, almost closed causal curves, and points at infinity.
result Wald's conjecture reformulated as a PDE problem about Cauchy horizons.
No trapped surfaces can form under low-regularity bounds in certain spacetimes.
problem Existence of trapped surfaces in low regularity solutions to Einstein's equations.
method Analyzing the initial data in Besov B2,13/2 norm and extending to H3/2 smallness. result No trapped surfaces can exist initially when the Cauchy data are close to Minkowski spacetime data.
Synthetic proof of Gannon-Lee theorem for spacetimes.
problem Proving incompleteness in globally hyperbolic spacetimes.
method Synthetic null energy condition and synthetically asymptotically regular trappedness condition.
result Generalized classical incompleteness theorem to weighted spacetimes.
New relation between curvature bounds and spacetime inextendibility.
problem Inextendibility of spacetimes under low regularity conditions.
method Synthetic curvature and causal character analysis.
result Low-regularity spacetimes with unbounded curvature.
New relation between curvature bounds and spacetime inextendibility.
problem Inextendibility of spacetimes under low regularity conditions.
method Synthetic curvature and causal character maximizers.
result Low-regularity inextendibility linked to unbounded curvature.
We study the low-regularity (in-)extendibility of spacetimes within the synthetic-geometric framework of Lorentzian length spaces developed in [KS:17]. To this end, we introduce appropriate notions of geodesics and timelike geodesic completeness and prove a general inextendibility result. Our results shed new light on …
Counterexample disproves Borde-Sorkin conjecture on causal continuity of Morse spacetimes.
problem Disproving the Borde-Sorkin conjecture on causal continuity of Morse spacetimes.
method Provided a counterexample with low regularity causal structure and causal bubbling.
result Borde-Sorkin conjecture does not hold for Morse spacetimes with large anisotropy.
We extend the validity of the Penrose singularity theorem to spacetime metrics of regularity C1,1. The proof is based on regularisation techniques, combined with recent results in low regularity causality theory.
We investigate the initial value problem for the Einstein-Euler equations of general relativity under the assumption of Gowdy symmetry on T3, and we construct matter spacetimes with low regularity. These spacetimes admit, both, impulsive gravitational waves in the metric (for instance, Dirac mass curvature singularitie…
In this paper we develop the mathematics required in order to provide a description of the observables for quantum fields on low-regularity spacetimes. In particular we consider the case of a massless scalar field φ on a globally hyperbolic spacetime M with C1,1 metric g. This first entails showing that the …
Stability of timelike Ricci bounds in low-regularity spacetimes.
problem Stability of synthetic timelike Ricci curvature bounds under C0-limits. method Constructing smooth approximations and analyzing limiting behavior via Lorentzian optimal transport.
result Impulsive gravitational waves satisfy synthetic timelike Ricci curvature lower bounds.
New proof shows FLRW spacetimes can't be extended smoothly in certain axisymmetric cases.
problem Proving smooth extension of FLRW spacetimes in specific spacetime classes.
method Extending previous work on spherically symmetric spacetimes to axisymmetric spacetimes.
result Demonstrates C0-inextendibility for FLRW spacetimes in a subclass of axisymmetric spacetimes. Researchers extend the concept of metric spaces to Lorentzian spaces and prove the feasibility of their c-completion.
problem Extending the concept of metric spaces to Lorentzian spaces and proving their c-completion.
method Revisiting Lorentzian metric spaces, constructing c-completion, proving feasibility and endowing with Lorentzian metric space structure.
result The c-completion of Lorentzian metric spaces is feasible and well-suited, completing the original space in a precise sense.
Introduces non-regular spacetime geometry without smooth calculus.
problem Defining gravity without smooth spacetime geometry.
method Discusses non-regular spacetime geometry and curvature without differential calculus.
result Curvature and gravity can be defined without smooth spacetime calculus.
The null distance for Lorentzian manifolds was recently introduced by Sormani and Vega. Under mild assumptions on the time function of the spacetime, the null distance gives rise to an intrinsic, conformally invariant metric that induces the manifold topology. We show when warped products of low regularity and globally…
Derives spacetime regularity under specific curvature conditions.
problem Ensuring smoothness in spacetime models with given curvature constraints.
method General regularity estimate for 4-d spacetimes, using Ricci curvature and Lie derivatives.
result Establishes conditions for smoothness in spacetime models.
On the Geroch-Kronheimer-Penrose future completion IP(X) of a spacetime X, there are two frequently used topologies. We systematically examine τ+, the stronger (metrizable) of them, which is the coarsest causally continuous topology, obtaining a variety of novel results, among them a complete characterization of…
We demonstrate the breakdown of several fundamentals of Lorentzian causality theory in low regularity. Most notably, chronological futures (defined naturally using locally Lipschitz curves) may be non-open, and may differ from the corresponding sets defined via piecewise C1-curves. By refining the notion of a causal…
The maximal analytic Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a twice continuously differentiable metric. In this paper, we prove the stronger statement that it is even inextendible as a Lorentzian manifold with a continuous metric. To capture the obstruction to continuous extens…
Study on unique spacetime extensions in 1+1 dimensions with applications to weak null singularities.
problem Understanding unique spacetime extensions across null boundaries in 1+1 dimensions.
method Analyzing the C0- and C1-structures of continuous spacetime extensions. result Extensions can have the same C0-structure but different C1-structures. The study finds regular null hypersurfaces in a perturbed Schwarzschild black hole exterior.
problem Existence of regular null hypersurfaces in a perturbed Schwarzschild black hole.
method Proof of existence for null hypersurfaces in a perturbed Schwarzschild spacetime.
result Existence of many foliations by regular null hypersurfaces in the exterior region of a perturbed Schwarzschild black hole.
Proves Gannon-Lee theorem for C1 spacetimes.
problem Classical singularity theorems for C1 spacetimes. method Proves theorem for C1 spacetimes, shows geodesic properties. result Gannon-Lee theorem holds for C1 spacetimes. The paper explores uniqueness and non-uniqueness of spacetime extensions in general relativity.
problem Investigating the uniqueness and non-uniqueness of spacetime extensions in general relativity.
method Analyzes the extension of globally hyperbolic Lorentzian manifolds with a focus on low regularities.
result Local uniqueness of anchored extensions for certain regularity classes of extensions.
Study optimal transport on globally hyperbolic spacetimes, focusing on weak Kantorovich potentials' regularity.
problem Investigate regularity of weak Kantorovich potentials on globally hyperbolic spacetimes.
method Apply insights from Riemannian and Lorentzian cases to study π-solutions. result Conclude existence, uniqueness, and structure of optimal transport maps.
We review recent work on the Einstein equations of general relativity when the curvature is defined in a weak sense. Weakly regular spacetimes are constructed, in which impulsive gravitational waves, as well as shock waves, propagate.
The paper extends Hawking's singularity theorem to metrics with Hölder continuity and bounded curvature.
problem Proving singularity theorems for metrics with low regularity.
method Combining elliptic RT-equations for metric regularisation and manifold convolution for curvature refinement.
result Establishes globally hyperbolic and timelike incompleteness for metrics with Hölder continuity and bounded curvature.
Proves properties of maximal hypersurfaces in specific spacetimes.
problem Maximal hypersurfaces in asymptotically AdS spacetimes.
method Uniqueness, existence, and regularity results via mathematical proofs.
result Proves uniqueness, existence, and regularity of maximal hypersurfaces.
Assuming minimal regularity assumptions on the data, we revisit the classical problem of finding isometric immersions into the Minkowski spacetime for hypersurfaces of a Lorentzian manifold. Our approach encompasses metrics having Sobolev regularity and Riemann curvature defined in the distributional sense, only. It ap…
Study baryogenesis in conformally flat spacetimes using causal fermion systems.
problem Understanding baryogenesis in specific spacetimes.
method Analysis of baryogenesis mechanism in conformally flat spacetimes with explicit formula derivation.
result Explicit formula for baryogenesis rate in these spacetimes.
In 2+1 dimensions, all complete spacetimes are cylindrical.
problem Understanding rigidity of Ricci flow spacetimes in (2+1) dimensions. method Analyzing complete and sufficiently regular spacetimes, showing they must be cylindrical.
result Every spatial slice is diffeomorphic to a fixed surface, and the spacetime is isometric to a classical Ricci flow.
Synthetic framework for null hypersurfaces in non-smooth spacetimes.
problem Analyzing null hypersurfaces in non-smooth spacetimes.
method Develops synthetic null hypersurfaces using optimal transport and Lorentzian geometry.
result Synthetic null energy condition stabilizes under convergence and applies to low-regularity spacetimes.
Novel approach to wave equations near null infinity in flat spacetimes.
problem Analyzing regularity and decay of wave equations near null infinity in asymptotically flat spacetimes.
method Microlocal analysis in a compactified spacetime with corners, focusing on edge-type wave operators.
result Microlocal regularity propagates across null infinity via radial sets, leading to new estimates for wave equations.
Establishes a Lorentzian Lasry-Lions regularization theorem for functions on globally hyperbolic spacetimes.
problem Optimal transport with C1,1 regularizing pairs method Local semiconcavity and future-directed timelike superdifferentials
result Derives C1,1 regularizing pairs for optimal transport under general assumptions The paper proves smoothness of event horizons in Kerr spacetime perturbations.
problem Smoothness of event horizons in Kerr spacetime perturbations.
method Proof of smoothness using stability of slowly rotating Kerr spacetimes.
result Smooth null hypersurfaces of event horizons are proven for Kerr spacetime perturbations.
Study of Dirac fields on Kerr spacetimes using peeling method.
problem Understanding decay and regularity of Dirac fields on Kerr spacetimes.
method Penrose conformal compactification and geometric energy estimates.
result Optimal initial data spaces for peeling of Dirac fields on Kerr spacetimes.
New approach confirms Kruskal-Szekeres extension for Schwarzschild spacetime.
problem Confirming the Kruskal-Szekeres extension for Schwarzschild spacetime.
method Reformulating the problem as an ODE and showing the ODE admits a solution if and only if the horizon is non-degenerate.
result Photon surfaces approaching the Killing horizon must necessarily cross it.
We consider polyhedra and 4-polytopes in Minkowski spacetime - in particular, null polyhedra with zero volume, and 4-polytopes that have such polyhedra as their hyperfaces. We present the basic properties of several classes of null-faced 4-polytopes: 4-simplices, "tetrahedral diamonds" and 4-parallelotopes. We propose …
The paper confirms the existence of 5D regular static vacuum solutions with multiple black holes and Kasner asymptotics.
problem Existence of 5D regular static vacuum solutions with multiple black holes.
method Construction of specific examples with different horizon topologies and analysis of spacetime properties.
result Existence of 5D vacuum solitons with Kasner asymptotics and regular static space-periodic spacetimes.
Warped-product black hole spacetimes are C0-inextendible.
problem Future C0-inextendibility of warped-product black hole spacetimes method Adapting Sbierski's proof for a broad class of warped-product black hole spacetimes
result Future C0-inextendibility established for spacetimes with a static exterior region We investigate the local regularity of pointed spacetimes, that is, time-oriented Lorentzian manifolds in which a point and a future-oriented, unit timelike vector (an observer) are selected. Our main result covers the class of Einstein vacuum spacetimes. Under curvature and injectivity bounds only, we establish the ex…
Given a regular curve in Minkowski spacetime, we describe necessary and sufficient conditions for this curve to admit a family of pairwise-disjoint crooked planes. Using this criterion, we describe crooked foliations along orbit curves of one-parameter groups of Lorentzian isometries.
It is shown that the initial singularities in spatially compact spacetimes with spherical, plane or hyperbolic symmetry admitting a compact constant mean curvature hypersurface are crushing singularities when the matter content of spacetime is described by the Vlasov equation (collisionless matter) or the wave equation…
One of the central difficulties of settling the L2-bounded curvature conjecture for the Einstein -Vacuum equations is to be able to control the causal structure of spacetimes with such limited regularity. In this paper we show how to circumvent this difficulty by showing that the geometry of null hypersurfaces of En…
Characterizes photon surfaces in static spacetimes, proving uniqueness.
problem Understanding photon surfaces in static spacetimes of arbitrary dimension.
method Complete characterization and new insights into spacetime geometry.
result Proves uniqueness of certain electrostatic spacetimes.
Investigates geometric properties of Bardeen black hole spacetime.
problem Examines curvature properties of Bardeen black hole spacetime.
method Analyzes pseudosymmetry, pseudosymmetric Weyl curvature, and recurrent structures.
result Bardeen spacetime is a manifold of pseudosymmetry Weyl curvature and satisfies special recurrent like structure.