Study directed completion of spacetimes, focusing on Schwarzschild spacetime.
problem Characterizing directed completions of spacetimes.
method Directed completion of Lorentzian pre-length spaces, focusing on Schwarzschild spacetime.
result Directed completion of Schwarzschild spacetime coincides with future causal completion.
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.
We consider (flat) Cauchy-complete GH spacetimes, i.e., globally hyperbolic flat lorentzian manifolds admitting some Cauchy hypersurface on which the ambient lorentzian metric restricts as a complete riemannian metric. We define a family of such spacetimes - model spacetimes - including four subfamilies: translation sp…
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. Study causal structure of warped spacetimes using novel pre-length spaces.
problem Understanding the causal structure of warped spacetimes.
method Novel notion of Lorentzian pre-length spaces and proof of causal completion as globally hyperbolic pre-length space.
result Causal completion of GRW spacetime is a globally hyperbolic pre-length space under Hausdorff chronological topology.
Geodesic completeness and optimal Sobolev index proven for Minkowski spacetimes.
problem Geodesic completeness and optimal Sobolev index for Minkowski spacetimes.
method Null non-trapping condition and real principal type estimate.
result Optimal Sobolev index proven for asymptotically Minkowski spacetimes.
This paper completes globally hyperbolic conformally flat spacetimes, proving they are topological manifolds.
problem Understanding the structure of spacetimes with specific properties.
method Analyzing globally hyperbolic conformally flat spacetimes, proving their causal completions are topological manifolds.
result Causal completions of globally hyperbolic conformally flat spacetimes are topological manifolds homeomorphic to S x [0, 1].
Geodesic completeness and flow properties of compact Brinkmann spacetimes proven.
problem Geodesic completeness and flow properties of compact Brinkmann spacetimes.
method Proof of geodesic completeness and flow properties of isotropic parallel vector fields in compact Brinkmann spaces.
result Geodesic completeness and flow properties of compact Brinkmann spacetimes proven.
In this paper, under natural geometric and physical assumptions we provide new uniqueness and non-existence results for complete maximal hypersurfaces in spatially open Robertson-Walker spacetimes whose fiber is flat. Moreover, our results are applied to relevant spacetimes as the steady state spacetime, Einstein-de Si…
Study the metric geometry of Cauchy hypersurfaces in spacetimes.
problem Properties of the space of Cauchy hypersurfaces.
method Equipped with a Hausdorff-type metric, studied completeness and local compactness.
result Generalized completeness results for spacetimes.
New rigidity results for specific hypersurfaces in spacetimes.
problem Characterizing maximal hypersurfaces in Generalized Robertson-Walker spacetimes.
method Applying rigidity results under geometric assumptions and the Null Energy Condition.
result New parametric uniqueness and nonexistence results for maximal hypersurfaces.
Study examines two topologies on future causal completion of spacetimes.
problem Characterizing differences between two topologies on future causal completion.
method Systematic examination of the stronger topology τ+ on Geroch-Kronheimer-Penrose future completion IP(X) of spacetimes X. result Complete characterization of the difference in convergence between τ+ and the weaker topology. Criteria for completeness and existence of Cauchy hypersurfaces in spacetimes.
problem Characterizing completeness and existence of Cauchy hypersurfaces in spacetimes.
method Introducing wind Riemannian structures and deriving criteria for completeness and existence of Cauchy hypersurfaces.
result Simple criteria for slices of spacetimes to be Cauchy.
The aim of this paper is to review and complete the study of geodesics on Gödel type spacetimes initiated in [8] and improved in [2] of the References. In particular, we prove some new results on geodesic connectedness and geodesic completeness for these spacetimes.
Study of hypersurfaces in static spacetimes with curvature bounds.
problem Characterizing hypersurfaces in static spacetimes with curvature constraints.
method Mean curvature and gradient estimates for spacelike hypersurfaces.
result Complete CMC hypersurfaces in static spacetimes are maximal under certain curvature conditions.
We provide a simple proof that conformally semi-symmetric spacetimes are actually semi-symmetric. We also present a complete refined classification of the semi-symmetric spacetimes.
Causal spacetimes with Ricci tensor have unique transformations.
problem Understanding transformations in viable causal spacetimes.
method Analyzing Ricci tensor and causal diffeomorphisms.
result Causal diffeomorphisms preserving Ricci tensor are homotheties.
Extends Hopf-Rinow theorem to semi-Riemannian spacetimes.
problem Generalizing Hopf-Rinow theorem to compact Lorentzian manifolds.
method Develops null distance for proper cone structures and (n−ν,ν)-spacetimes. result Generalizes Hopf-Rinow theorem to a new class of semi-Riemannian manifolds.
Study on trapped submanifolds in spacetimes, proving rigidity and non-existence results.
problem Rigidity and non-existence of trapped submanifolds in spacetimes.
method Analysis of parabolic spacelike submanifolds with causal mean curvature in orthogonally splitted and globally hyperbolic spacetimes.
result Non-existence of local minima or maxima of certain functions on submanifolds, leading to rigidity results.
Given a globally hyperbolic spacetime endowed with a complete lightlike Killing vector field and a complete Cauchy hypersurface, we characterize the points which can be connected by geodesics. A straightforward consequence is the geodesic connectedness of globally hyperbolic generalized plane waves with a complete Cauc…
Complete set of local gauge invariants for Kerr spacetime identified.
problem Identifying all local gauge invariant quantities for linearized gravity on Kerr spacetime.
method Constructing a complete compatibility complex for the Killing operator and demonstrating equivalence with first compatibility operator.
result Any gauge invariant quantity for linearized gravity on Kerr can be expressed in terms of identified gauge invariants and their derivatives.
New splitting theorem for weighted Finsler spacetimes without Berwald condition.
problem Proving timelike splitting theorems for Finsler spacetimes under weaker conditions.
method Using the p-d'Alembertian and a recently developed strategy. result Established a diffeomorphic splitting for timelike geodesically complete Finsler spacetimes.
Study harmonicity of metrics in generalized Kantowski-Sachs spacetime.
problem Harmonicity of metrics in generalized Kantowski-Sachs spacetime.
method Considered Sasaki, horizontal and complete lifts of generalized Kantowski-Sachs spacetime metrics to tangent bundle and studied their harmonicity.
result Characterized harmonicity properties of lifted metrics.
Study rigidity in Penrose's singularity theorem with weakly trapped surfaces.
problem Understand the global structure of spacetimes with weakly trapped surfaces.
method Show foliation of MOTS generating totally geodesic null hypersurfaces.
result Obtain local or global rigidity results based on assumptions.
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.
We consider restrictions placed by geodesic completeness on spacetimes possessing a null parallel vector field, the so-called Brinkmann spacetimes. This class of spacetimes includes important idealized gravitational wave models in General Relativity, namely the plane-fronted waves with parallel rays, or pp-waves, which…
Study finds unique and non-existent constant mean curvature hypersurfaces in specific spacetimes.
problem Finding unique and non-existent constant mean curvature spacelike hypersurfaces.
method Geometric and physical assumptions applied to Generalized Robertson-Walker spacetimes.
result New uniqueness and non-existence results for complete spacelike hypersurfaces.
Study maximal hypersurfaces in open spacetimes using a maximum principle.
problem Characterize maximal hypersurfaces in open spacetimes.
method Use a generalized maximum principle to analyze hypersurfaces in spatially open Generalized Robertson-Walker spacetimes.
result Provide new uniqueness and non-existence results for complete maximal hypersurfaces in open Robertson-Walker spacetimes.
Study complete space-like stationary surfaces with graphical Gauss image, estimating exceptional values and classifying degenerate surfaces.
problem Estimating exceptional values and classifying degenerate surfaces in Minkowski spacetime.
method Generalizing Fujimoto's theorem, estimating upper bounds, introducing conjugate similarity, and establishing structure theorems.
result Sharp contrast to Bernstein type results for minimal surfaces, estimating upper bounds of exceptional values.
Study characterizes 2-Killing vector fields on complex spacetimes.
problem Characterize 2-Killing vector fields on multiply twisted product spacetimes. method Determine nonlinear differential equations, find twisted functions, provide solutions, and construct examples.
result Completely describe 2-Killing vector fields and twisted functions on multiply twisted product spacetimes. We study proper, isometric actions of nonsolvable discrete groups Gamma on the 3-dimensional Minkowski space R^{2,1} as limits of actions on the 3-dimensional anti-de Sitter space AdS^3. To each such action is associated a deformation of a hyperbolic surface group Gamma_0 inside O(2,1). When Gamma_0 is convex cocompact…
A Margulis spacetime is a complete affine 3-manifold M with nonsolvable fundamental group. Associated to every Margulis spacetime is a noncompact complete hyperbolic surface S. We show that every Margulis spacetime is orientable, even though S may be nonorientable. We classify Margulis spacetimes when S is homeomorphic…
We study global aspects of complete, non-singular asymptotically locally AdS spacetimes solving the vacuum Einstein equations whose conformal infinity is an arbitrary globally stationary spacetime. It is proved that any such solution which is asymptotically stationary to the past and future is itself globally stationar…
We formulate the concept of time machine structure for spacetimes exhibiting a compactely constructed region with closed timelike curves. After reviewing essential properties of the pseudo Schwarzschild spacetime introduced by A. Ori, we present an analysis of its geodesics analogous to the one conducted in the case of…
New findings on how conformal rescalings affect spacetime metrics.
problem Understanding how conformal rescalings impact spacetime metrics.
method Analyzing the null curvature condition and causal structure.
result Proving constraints on conformal rescalings in vacuum and non-vacuum spacetimes.
In order to apply variational methods to the action functional for geodesics of a stationary spacetime, some hypotheses, useful to obtain classical Palais-Smale condition, are commonly used: pseudo-coercivity, bounds on certain coefficients of the metric, etc. We prove that these technical assumptions admit a natural i…
The paper classifies critical closed RW spacetimes using elliptic functions.
problem Classifying critical closed Robertson-Walker spacetimes.
method Study of critical RW spacetimes via volume-preserving variations and action functionals.
result Complete classification of critical RW spacetimes with explicit solutions.
Researchers extend parametrization of Margulis spacetimes using strip deformations.
problem Parametrize Margulis spacetimes with decorated horoballs.
method Use strip deformations to parametrize complete finite-area hyperbolic surfaces with spikes decorated with horoballs.
result Generalized parametrization of Margulis spacetimes with photons.
Algorithm counts Killing vectors in 3D spacetime.
problem Counting Killing vectors in 3D spacetime.
method Algorithm based on Ricci tensor and differential invariants.
result Complete classification of 4 Killing vector spacetimes.
Study generalizes submetry concept for spacetimes, linking to curvature and foliations.
problem Understanding submetry in non-positive signature spacetimes.
method Developed Lorentzian submetries and established their equivalence to semi-Riemannian submersions.
result Lorentzian submetries correspond to locally C^{1,1} semi-Riemannian submersions under completeness assumptions.
Study connects stationary spacetimes with pre-Randers metrics using Fermat's principle.
problem Understanding causal structures in stationary spacetimes.
method Relativistic Fermat's principle linking stationary spacetimes and pre-Randers metrics.
result Description of causal ladder in terms of pre-Randers metric elements.
The chapter explores globally hyperbolic spacetimes using topology and functional analysis.
problem Understanding global hyperbolicity in spacetimes.
method Foundational tools from order theory and topology, geometric analysis, and connections to physics.
result A connection between global hyperbolicity and geodesic completeness of space-like surfaces.
This study proves energy bounds in specific AdS spacetimes.
problem Proving positive energy theorems in asymptotically locally AdS spacetimes.
method Derived positive energy theorem for spacetimes with compact, Einstein cross-sections.
result First complete proofs of BPS inequalities in AdS and locally AdS spacetimes.
Study finds conditions for stationary spacelike surfaces in a generalized Robertson-Walker spacetime.
problem Characterize complete stationary spacelike surfaces in a generalized Robertson-Walker spacetime.
method Assume a natural inequality involving curvature and warping function, use parabolicity and superharmonic function properties.
result Non-compact complete stationary spacelike surfaces are totally geodesic under given conditions.
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.
Study proves existence of MOTTs in de Sitter spacetime.
problem Proving existence of marginally outer trapped tubes in de Sitter spacetime.
method Combining results from spacetimes satisfying null convergence condition and properties of CMC surfaces in S3.
result Existence of complete MOTTs with CMC sections in de Sitter spacetime.
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.
Higher-dimensional spacetimes have well-behaved boundaries.
problem Understanding boundaries of higher-dimensional spacetimes.
method Analyzing (n+1)-dimensional Myers-Perry metrics at spacelike infinity. result Optimal conformal completion at spacelike infinity for Cn−3,1 differentiability class.