Theorems on existence and global dynamics for the Einstein equations

DOI10.12942/lrr-2005-6zbMath1316.83008arXivgr-qc/0505133OpenAlexW2170544604WikidataQ42328624 ScholiaQ42328624MaRDI QIDQ5919785

Alan D. Rendall

Publication date: 16 July 2015

Published in: Living Reviews in Relativity (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/gr-qc/0505133

Mathematics Subject Classification ID

Einstein's equations (general structure, canonical formalism, Cauchy problems) (83C05) Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory (83-02) Space-time singularities, cosmic censorship, etc. (83C75) Special relativity (83A05) Exact solutions to problems in general relativity and gravitational theory (83C15)

Related Items

On the nature of Hawking's incompleteness for the Einstein-vacuum equations: the regime of moderately spatially anisotropic initial data ⋮ Stable big bang formation in near-FLRW solutions to the Einstein-scalar field and Einstein-stiff fluid systems ⋮ The maximal development of near-FLRW data for the Einstein-scalar field system with spatial topology \(\mathbb{S}^3\) ⋮ Global hyperbolicity for spacetimes with continuous metrics ⋮ Asymptotic behaviour of the relativistic Boltzmann equation in the Robertson-Walker spacetime ⋮ Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations ⋮ Symmetric hyperbolic systems for a large class of fields in arbitrary dimension ⋮ Singularity theorems for \(C^1\)-Lorentzian metrics ⋮ A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation ⋮ Global solutions of the Vlasov-Poisson-Boltzmann system in a cosmological setting ⋮ On the motion of a compact elastic body ⋮ Lorentzian length spaces ⋮ On the conditions for the breaking of oscillations in a cold plasma ⋮ An approach to stability analyses in general relativity via symplectic geometry ⋮ Strong Cosmic Censorship for Surface-Symmetric Cosmological Spacetimes with Collisionless Matter ⋮ Smooth Gowdy-symmetric generalised Taub–NUT solutions in Einstein–Maxwell theory ⋮ Continuum and discrete initial-boundary value problems and Einstein's field equations

Cites Work

• Quasi-spherical metrics and prescribed scalar curvature
• Asymptotics of solutions of the Einstein equations with positive cosmological constant
• Generic global solutions of the relativistic Vlasov-Maxwell system of plasma physics
• Asymptotic behaviour of the gravitational field and the nature of singularities in Gowdy spacetimes
• Compressible fluid flow and systems of conservation laws in several space variables
• Neighborhoods of Cauchy horizons in cosmological spacetimes with one Killing field
• On space-times with U(1)\(\times U(1)\) symmetric compact Cauchy surfaces
• Semi-global existence and convergence of solutions of the Robinson-Trautman (2-dimensional Calabi) equation
• On the Cauchy problem for Boltzmann equations: Global existence and weak stability
• Remarques sur l'apparition de singularités dans les écoulements euleriens compressibles. (Remarks on the appearance of singularities in compressible Euler flows)
• The global stability of Minkowski space-time in harmonic gauge
• Existence and structure of past asymptotically simple solutions of Einstein's field equations with positive cosmological constant
• Stationary spherically symmetric models in stellar dynamics
• Formation of singularities in three-dimensional compressible fluids
• The problem of a self-gravitating scalar field
• Remarks on cosmological spacetimes and constant mean curvature surfaces
• Partial differential equations. 4th ed
• The boost problem in general relativity
• The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space. I. Local existence and smoothness properties
• Propagation of moments and regularity for the 3-dimensional Vlasov- Poisson system
• On the global existence and the asymptotic behavior of solutions to the Einstein-Maxwell-Yang-Mills equations
• Smooth static solutions of the Einstein/Yang-Mills equations
• Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data
• Pseudodifferential operators and nonlinear PDE
• On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein's field equations
• Global existence proof for relativistic Boltzmann equation
• Global existence of solutions of the spherically symmetric Vlasov- Einstein system with small initial data
• The Boltzmann equation and its applications
• \(C^\infty\) solutions of hyperbolic non linear equations. Applications in G.R.G
• The instability of naked singularities in the gravitational collapse of a scalar field.
• Global foliations of vacuum spacetimes with \(T^2\) isometry
• Smooth irrotational flows in the large to the Euler-Poisson system in \(\mathbb{R}^{3+1}\)
• On spherically symmetric stellar models in general relativity
• Global existence for the Einstein-Boltzmann equation in the flat Robertson-Walker spacetime
• The initial boundary value problem for Einstein's vacuum field equation
• Global solutions of the relativistic Euler equations
• On the asymptotic behavior of spherically symmetric wave maps
• On the spherical symmetry of static stellar models
• The Newtonian limit for asymptotically flat solutions of the Vlasov- Einstein system
• Cosmic censorship for some spatially homogeneous cosmological models
• Global existence of classical solutions to the Vlasov-Poisson system in a three-dimensional, cosmological setting
• The mathematical theory of dilute gases
• Compactness in Boltzmann's equation via Fourier integral operators and applications. I, II
• On ``asymptotically flat space-times with \(G_2\)-invariant Cauchy surfaces
• On the existence of rotating stars in general relativity
• On the Newtonian limit of general relativity
• The ``Two and one-half dimensional relativistic Vlasov Maxwell system
• Conservation laws for relativistic fluid dynamics
• Gravitational fields near space-like and null infinity
• On the global evolution problem in \(2+1\) gravity
• Existence of constant mean curvature foliations in spacetimes with two-dimensional local symmetry
• Nonlinear stability of homogeneous models in Newtonian cosmology
• The relativistic Vlasov-Maxwell system in two space dimensions: I. II
• Elliptic-hyperbolic systems and the Einstein equations
• Global existence of solutions to the coupled Einstein and Maxwell-Higgs system in the spherical symmetry
• On the topology of vacuum spacetimes
• The weak null condition for Einstein's equations.
• Well-posedness for the linearized motion of a compressible liquid with free surface boundary
• Nonlinear stability of gaseous stars
• Stability and instability of the Cauchy horizon for the spherically symmetric Einstein-Maxwell-scalar field equations
• Stationary and static stellar dynamic models with axial symmetry
• Isotropic cosmological singularities. III: The Cauchy problem for the inhomogeneous conformal Einstein-Vlasov equations
• Spherically symmetric static solutions of the Einstein equations with elastic matter source
• Solutions of the Einstein constraint equations with apparent horizon boundaries
• Causal geometry of Einstein-vacuum spacetimes with finite curvature flux
• The Darwin approximation of the relativistic Vlasov-Maxwell system
• Isotropic cosmological singularities. II: The Einstein-Vlasov system
• Isotropic cosmological singularities. I: Polytropic perfect fluid spacetimes
• A class of counterexamples to Jeans' theorem for the Vlasov-Einstein system
• Self-gravitating relativistic fluids: The formation of a free phase boundary in the phase transition from soft to hard
• Global existence of generalized solutions of the spherically symmetric Einstein-scalar equations in the large
• The structure and uniqueness of generalized solutions of the spherically symmetric Einstein-scalar equations
• A mathematical theory of gravitational collapse
• Three-manifolds with positive Ricci curvature
• The nature of singularities in plane symmetric scalar field cosmologies
• Examples of naked singularity formation in the gravitational collapse of a scalar field
• The evolution problem in general relativity
• Cosmological models and centre manifold theory
• Kasner-like behaviour for subcritical Einstein-Matter systems
• Global hyperbolicity and completeness
• Local existence and continuation criterion for solutions of the spherically symmetric Einstein-Vlasov-Maxwell system
• Geodesics and the Einstein nonlinear wave system
• The unique limit of the Glimm scheme
• A regularity theorem for solutions of the spherically symmetric Vlasov- Einstein system
• Finite energy solutions of the Yang-Mills equations in \(\mathbb{R}^{3+1}\)
• Self-gravitating relativistic fluids: A two-phase model
• Einstein equations and conformal structure: Existence of anti-de Sitter-type space-times
• Nonlinear instability of double-humped equilibria
• Fuchsian equations in Sobolev spaces and blow-up
• The semigroup generated by \(2 \times 2\) conservation laws
• Self-gravitating relativistic fluids:the continuation and termination of a free phase boundary.
• Global foliations of matter spacetimes with Gowdy symmetry
• Asymptotic stability of the relativistic Maxwellian
• A geometric approach to the Littlewood-Paley theory
• Asymptotically simple solutions of the vacuum Einstein equations in even dimensions
• Global existence of solutions for the relativistic Boltzmann equation with arbitrarily large initial data on a Bianchi type I space-time
• Global aspects of the Cauchy problem in general relativity
• On initial conditions and global existence for accelerating cosmologies from string theory
• A proof of Price's law for the collapse of a self-gravitating scalar field
• An extension principle for the Einstein-Vlasov system in spherical symmetry
• On naked singularities and the collapse of self-gravitating Higgs fields
• The state space and physical interpretation of self-similar spherically symmetric perfect-fluid models
• Asymptotic behavior of homogeneous cosmological models in the presence of a positive cosmological constant
• Manufacture of Gowdy spacetimes with spikes
• Formation of singularities for equivariant (2+1)-dimensional wave maps into the 2-sphere
• Simplified models of electromagnetic and gravitational radiation damping
• The future asymptotics of Bianchi VIII vacuum solutions
• Global existence of spherically symmetric solutions to the coupled Einstein and nonlinear Klein-Gordon system
• Global classical solutions to the spherically symmetric Nordström–Vlasov system
• The surface-symmetric Einstein–Vlasov system with cosmological constant
• The Newtonian limit of the relativistic Boltzmann equation
• Harmonic analysis of linear fields on the nilgeometric cosmological model
• Isotropic cosmological singularities: other matter models
• Existence of CMC and Constant Areal Time Foliations inT²Symmetric Spacetimes with Vlasov Matter
• Asymptotic properties of linear field equations in minkowski space
• Existence and properties of spherically symmetric static fluid bodies with a given equation of state
• The formation of black holes and singularities in spherically symmetric gravitational collapse
• On the regularity of spherically symmetric wave maps
• Theory of Newtonian self-gravitating stationary spherically symmetric systems
• ALL UNIVERSES GREAT AND SMALL
• Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations
• Strong cosmic censorship in polarised Gowdy spacetimes
• Isotropic singularities in cosmological models
• A necessary and sufficient condition for York data to specify an asymptotically flat spacetime
• Qualitative analysis of two-fluid Bianchi cosmologies
• The initial value problem for a class of general relativistic fluid bodies
• On the definition of post-newtonian approximations
• On the StrongL¹Trend to Equilibrium for the Boltzmann Equation
• Cosmic no-hair theorem in homogeneous spacetimes. I. Bianchi models
• Remarks on the abstract form of nonlinear cauchy-kovalevsky theorems
• Blow-up Surfaces for Nonlinear Wave Equations, I
• The Cauchy problem for quasi–linear hyperbolic evolution problems with a singularity in the time
• Well-posedness in Sobolev spaces of the full water wave problem in 3-D
• Black Hole Uniqueness Theorems
• Global smooth solutions to Euler equations for a perfect gas
• Static shells for the Vlasov-Poisson and Vlasov-Einstein systems
• Blow-up surfaces for nonlinear wave equations, II
• Initial-boundary value problem for the spherically symmetric Einstein equations for a perfect fluid
• Growth rates for the linearized motion of fluid interfaces away from equilibrium
• Bounded variation solutions of the spherically symmetric einstein‐scalar field equations
• On the structure of conformal singularities in classical general relativity
• On the structure of conformal singularities in classical general relativity. II Evolution equations and a conjecture of K. P. Tod
• Static solutions of the spherically symmetric Vlasov–Einstein system
• The cosmic no-hair theorem and the non-linear stability of homogeneous Newtonian cosmological models
• Stability of Gaseous Stars in Spherically Symmetric Motions
• Asymptotic states of magnetic Bianchi I cosmologies
• Global dynamics of the mixmaster model
• Existence and non-existence results for global constant mean curvature foliations
• Spatially self-similar spherically symmetric perfect-fluid models
• Future asymptotic expansions of Bianchi VIII vacuum metrics
• Asymptotic dynamics of the exceptional Bianchi cosmologies
• Equivariant wave maps in two space dimensions
• On the EinsteinVlasov system with hyperbolic symmetry
• Local prescribed mean curvature foliations in cosmological spacetimes
• Thoughts on tachyon cosmology
• Global existence and asymptotic behaviour in the future for the Einstein--Vlasov system with positive cosmological constant
• Peeling properties of asymptotically flat solutions to the Einstein vacuum equations
• Asymptotic behaviour of future-complete cosmological spacetimes
• Non-CMC conformal data sets which do not produce solutions of the Einstein constraint equations
• Asymptotic expansions close to the singularity in Gowdy spacetimes
• Global existence problem in T ³ -Gowdy symmetric IIB superstring cosmology
• Static fluid cylinders and their fields: global solutions
• On Gowdy vacuum spacetimes
• On the area of the symmetry orbits in T ² symmetric spacetimes with Vlasov matter
• Dynamical systems approach to relativistic spherically symmetric static perfect fluid models
• Trapped surfaces as boundaries for the constraint equations
• On the area of the symmetry orbits in T ² symmetric spacetimes
• Static, self–gravitating elastic bodies
• Static Axially Symmetric Solutions of Einstein–Yang-Mills-Dilaton Theory
• Conformal “Thin-Sandwich” Data for the Initial-Value Problem of General Relativity
• Curvature blow up in Bianchi VIII and IX vacuum spacetimes
• A class of plane symmetric perfect-fluid cosmologies with a Kasner-like singularity
• Fuchsian analysis of singularities in Gowdy spacetimes beyond analyticity
• Non-tilted Bianchi VII ₀ models - the radiation fluid
• Dynamics of spatially homogeneous locally rotationally symmetric solutions of the Einstein-Vlasov equations
• Asymptotic singular behaviour of Gowdy spacetimes in string theory
• ON ROTATING REGULAR NON-ABELIAN SOLUTIONS
• On the existence of self-similar spherically symmetric wave maps coupled to gravity
• Asymptotic isotropization in inhomogeneous cosmology
• LOCAL EXISTENCE AND CONTINUATION CRITERIA FOR SOLUTIONS OF THE EINSTEIN–VLASOV-SCALAR FIELD SYSTEM WITH SURFACE SYMMETRY
• Structure of solutions near the initial singularity for the surface-symmetric Einstein–Vlasov system
• Scalar fields on SL (2, R ) and H ²   R geometric spacetimes and linear perturbations
• Future asymptotic behaviour of tilted Bianchi models of type IV and VII _h
• The Glimm scheme for perfect fluids on plane-symmetric Gowdy spacetimes
• Intermediate inflation and the slow-roll approximation
• GLOBAL CHARACTERISTIC PROBLEM FOR EINSTEIN VACUUM EQUATIONS WITH SMALL INITIAL DATA: (I) THE INITIAL DATA CONSTRAINTS
• Timelike self-similar spherically symmetric perfect-fluid models
• Cosmological spacetimes not covered by a constant mean curvature slicing
• Dynamics of spatially homogeneous solutions of the Einstein-Vlasov equations which are locally rotationally symmetric
• Asymptotic self-similarity breaking at late times in cosmology
• On local and global aspects of the Cauchy problem in general relativity
• Bianchi II magnetic cosmologies
• Asymptotic behavior in polarized T2-symmetric vacuum space–times
• Relativistic elasticity
• On a wave map equation arising in general relativity
• Asymptotic states of magnetic Bianchi VI cosmologies
• Asymptotic behaviour in polarized and half-polarized U (1) symmetric vacuum spacetimes
• Fuchsian analysis of S2 $times$ S1 and S3 Gowdy spacetimes
• Well‐posedness for the linearized motion of an incompressible liquid with free surface boundary
• On the existence of global solutions for T ³ -Gowdy spacetimes with stringy matter
• On future asymptotics of polarized Gowdy   ³ -models
• Global properties of higher-dimensional cosmological spacetimes
• Accelerated cosmological expansion due to a scalar field whose potential has a positive lower bound
• On future geodesic completeness for the EinsteinVlasov system with hyperbolic symmetry
• Asymptotic behaviour of the Einstein–Vlasov system with a positive cosmological constant
• Global prescribed mean curvature foliations in cosmological space–times. I
• Existence of initial data satisfying the constraints for the spherically symmetric Einstein–Vlasov–Maxwell system
• Crushing singularities in spacetimes with spherical, plane and hyperbolic symmetry
• Conservation laws for the relativistic P-system
• Constant mean curvature solutions of the Einstein constraint equations on closed manifolds
• Existence of maximal hypersurfaces in some spherically symmetric spacetimes
• The blow-up problem for exponential nonlinearities
• The spectrum of adiabatic stellar oscillations
• Cosmological solutions of the Vlasov-Einstein system with spherical, plane, and hyperbolic symmetry
• Global properties of locally spatially homogeneous cosmological models with matter
• Hyperbolic reductions for Einstein's equations
• A set of nonconstant mean curvature solutions of the Einstein constraint equations on closed manifolds
• Regularity of the Gain Term and Strong $L^1 $ Convergence to Equilibrium for the Relativistic Boltzmann Equation
• The Einstein flow, the sigma-constant and the geometrization of 3-manifolds
• Analytic description of singularities in Gowdy spacetimes
• Dynamics of magnetic Bianchi VI ₀ cosmologies
• Compact support of spherically symmetric equilibria in non-relativistic and relativistic galactic dynamics
• The initial singularity in solutions of the Einstein–Vlasov system of Bianchi type I
• COSMOLOGICAL SINGULARITIES, EINSTEIN BILLIARDS AND LORENTZIAN KAC–MOODY ALGEBRAS
• Global existence and future asymptotic behaviour for solutions of the Einstein–Vlasov-scalar field system with surface symmetry
• Spherically symmetric spacetimes with a trapped surface
• Relativistic elastostatics: I. Bodies in rigid rotation
• Dynamical Systems in Cosmology
• Stability of spherically symmetric wave maps
• Solutions in the large for nonlinear hyperbolic systems of equations
• Mixmaster Universe
• Algebraic expansions for curvature-coupled scalar field models
• Critical phenomena in gravitational collapse
• Computational cosmology: from the early universe to the large scale structure
• Numerical approaches to spacetime singularities
• Brane-world gravity
• The Einstein-Vlasov system/kinetic theory
• On stationary vacuum solutions to the Einstein equations
• Static solutions of the Vlasov-Einstein system
• On the structure of solutions to the static vacuum Einstein equations
• The asymptotic regimes of tilted Bianchi II cosmologies
• Quiescent cosmological singularities
• Scalar curvature deformation and a gluing construction for the Einstein constraint equations
• Equivariant self-similar wave maps from Minkowski spacetime into 3-sphere
• Isotropic steady states in galactic dynamics
• Proof of a generalized Geroch conjecture for the hyperbolic Ernst equation
• The Bianchi IX attractor
• On the stability of the Kerr metric
• The Vlasov-Poisson system with radiation damping
• Future global in time Einsteinian spacetimes with \(U(1)\) isometry group
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item