Mathematical General Relativity

Abstract

We present a number of open problems within general relativity. After a brief introduction to some technical mathematical issues and the famous singularity theorems, we discuss the cosmic censorship hypothesis and the Penrose inequality, the uniqueness of black hole solutions and the stability of Kerr spacetime and the final state conjecture, critical phenomena and the Einstein-Yang--Mills equations, and a number of other problems in classical general relativity. We then broaden the scope and discuss some mathematical problems motivated by quantum gravity, including AdS/CFT correspondence and problems in higher dimensions and, in particular, the instability of anti-de Sitter spacetime, and in cosmology, including the cosmological constant problem and dark energy, the stability of de Sitter spacetime and cosmological singularities and spikes. Finally, we briefly discuss some problems in numerical relativity and relativistic astrophysics.

References in corpus (154)

• Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant
• Measurements of Omega and Lambda from 42 High-Redshift Supernovae
• The Large N Limit of Superconformal Field Theories and Supergravity
• The Ekpyrotic Universe: Colliding Branes and the Origin of the Hot Big Bang
• Loop Quantum Cosmology: A Status Report
• Black Strings and p-Branes are Unstable
• A rotating black ring in five dimensions
• Black Holes in Higher Dimensions
• Cosmological perturbations
• From Big Crunch to Big Bang
• On weakly turbulent instability of anti-de Sitter space
• Testing General Relativity in Cosmology
• Bouncing Cosmologies: Progress and Problems
• E10 and a "small tension expansion" of M Theory
• Global existence for the Einstein vacuum equations in wave coordinates
• Cosmic Voids in Sloan Digital Sky Survey Data Release 7
• Instability of Ultra-Spinning Black Holes
• Classification of the Weyl Tensor in Higher Dimensions
• The 1965 Penrose singularity theorem
• Black Strings, Low Viscosity Fluids, and Violation of Cosmic Censorship
• Voids in the PSCz Survey and the Updated Zwicky Catalog
• Voids in the 2dF Galaxy Redshift Survey
• Numerical Investigation of Cosmological Singularities
• The linear stability of the Schwarzschild solution to gravitational perturbations
• The global non-linear stability of the Kerr-de Sitter family of black holes
• Numerical Approaches to Spacetime Singularities
• General relativity and cosmic structure formation
• The Bianchi IX attractor
• Is there proof that backreaction of inhomogeneities is irrelevant in cosmology?
• Instability and new phases of higher-dimensional rotating black holes
• Regularity of Wave-Maps in dimension 2+1
• Why the cosmological constant is small and positive
• Beginning inflation in an inhomogeneous universe
• Critical Behaviour in Gravitational Collapse of a Yang-Mills Field
• The past attractor in inhomogeneous cosmology
• The Einstein-Vlasov System/Kinetic Theory
• Analytic description of singularities in Gowdy spacetimes
• Cosmological Solutions in Macroscopic Gravity
• On the Nonlinear Stability of Asymptotically Anti-de Sitter Solutions
• Numerical simulations of generic singuarities
• Lectures on the Cosmological Constant Problem
• Classification of the Weyl Tensor in Higher Dimensions and Applications
• Gravitational Turbulent Instability of Anti-de Sitter Space
• Newtonian Cosmology in Lagrangian Formulation: Foundations and Perturbation Theory
• Geometrical Destabilization of Inflation
• Observational Challenges for the Standard FLRW Model
• Time-periodic solutions in Einstein AdS - massless scalar field system
• Effects of nonlinear inhomogeneity on the cosmic expansion with numerical relativity
• Spacetimes characterized by their scalar curvature invariants
• Initial data engineering
• End Point of Black Ring Instabilities and the Weak Cosmic Censorship Conjecture
• Global Properties of Locally Spatially Homogeneous Cosmological Models with Matter
• Curvature blow up in Bianchi VIII and IX vacuum spacetimes
• Gravitational Collapse and Cosmic Censorship
• Resonant dynamics and the instability of anti-de Sitter spacetime
• Future Global in time einsteinian spacetimes with U(1) isometry group
• Evidence for an Electrifying Violation of Cosmic Censorship
• A comment on AdS collapse of a scalar field in higher dimensions
• Evolution to a smooth universe in an ekpyrotic contracting phase with w > 1
• Asymptotically Flat Initial Data with Prescribed Regularity at Infinity
• The area of horizons and the trapped region
• Gravitational echoes from macroscopic quantum gravity effects
• The global nonlinear stability of Minkowski space for self-gravitating massive fields. The wave-Klein-Gordon model
• End Point of the Ultraspinning Instability and Violation of Cosmic Censorship
• Gravitational wave spectroscopy of binary neutron star merger remnants with mode stacking
• The Stability of the Irrotational Euler-Einstein System with a Positive Cosmological Constant
• Supersymmetric quantum cosmological billiards
• The Generalized Second Law implies a Quantum Singularity Theorem
• Higher order M theory corrections and the Kac-Moody algebra E10
• Numerical simulations of general gravitational singularities
• Oscillatory approach to the singularity in vacuum spacetimes with $T^2$ isometry
• Power-law Inflation in Spacetimes without Symmetry
• Formation of singularities for equivariant 2+1 dimensional wave maps into the two-sphere
• The asymptotic regimes of tilted Bianchi II cosmologies
• Simple Types of Anisotropic Inflation
• Global dynamics of the mixmaster model
• Nonperturbative analysis of the evolution of cosmological perturbations through a nonsingular bounce
• The physics of the relativistic counter-streaming instability that drives mass inflation inside black holes
• Einstein's Universe: Cosmological structure formation in numerical relativity
• Asymptotic expansions close to the singularity in Gowdy spacetimes
• Inflationary models with non-minimally derivative coupling
• Spikes in the Mixmaster regime of G_2 cosmologies
• On Gowdy vacuum spacetimes
• Equivariant self-similar wave maps from Minkowski spacetime into 3-sphere
• A new class of obstructions to the smoothness of null infinity
• Bias and scatter in the Hubble diagram from cosmological large-scale structure
• Singularity theorems based on trapped submanifolds of arbitrary co-dimension
• Quasimodes and a Lower Bound on the Uniform Energy Decay Rate for Kerr-AdS Spacetimes
• Islands of stability and recurrence times in AdS
• The Black Ring is Unstable
• On the uniqueness and global dynamics of AdS spacetimes
• Asymptotic dynamics of the exceptional Bianchi cosmologies
• How does the cosmic large-scale structure bias the Hubble diagram?
• Is AdS stable?
• Gravitational wave memory in $\Lambda$CDM cosmology
• Gravitational wave memory in de Sitter spacetime
• Chasing the non-linear evolution of matter power spectrum with numerical resummation method: solution of closure equations
• A scattering theory construction of dynamical vacuum black holes
• Constant mean curvature foliations in cosmological spacetimes
• The nature of singularity in multidimensional anisotropic Gauss-Bonnet cosmology with a perfect fluid
• Space-time Averages in Macroscopic Gravity and Volume-preserving Coordinates
• ``Nowhere'' differentiable horizons
• Spike Oscillations
• Strong cosmic censorship in vacuum space--times with compact, locally homogeneous Cauchy surfaces
• A proof of Friedman's ergosphere instability for scalar waves
• Geometric inequalities for axially symmetric black holes
• Safely smoothing spacetime: backreaction in relativistic cosmological simulations
• A Reformulation of the Hoop Conjecture
• Mixmaster: Fact and Belief
• A fully anisotropic mechanism for formation of trapped surfaces in vacuum
• Testing the nonlinear stability of Kerr-Newman black holes
• Hawking's singularity theorem for $C^{1,1}$-metrics
• Black rings at large D
• Geometric Horizons
• Identification of black hole horizons using scalar curvature invariants
• Global results for linear waves on expanding Kerr and Schwarzschild de Sitter cosmologies
• Quantum Supersymmetric Bianchi IX Cosmology
• Generic self-similar blowup for equivariant wave maps and Yang-Mills fields in higher dimensions
• Recent developments concerning generic spacelike singularities
• Life, the universe, and everything - 42 fundamental questions
• The Newtonian limit for perfect fluids
• AdS nonlinear instability: moving beyond spherical symmetry
• The Stability of an Isotropic Cosmological Singularity in Higher-Order Gravity
• Naked singularities in Tolman-Bondi-de Sitter collapse
• Conservation laws and flux bounds for gravitational perturbations of the Schwarzschild metric
• Quantum Supersymmetric Cosmology and its Hidden Kac-Moody Structure
• Generating matter inhomogeneities in general relativity
• Geometric inequalities for black holes
• Fermionic Kac-Moody Billiards and Supergravity
• An automatically generated code for relativistic inhomogeneous cosmologies
• Lorentzian manifolds and scalar curvature invariants
• Late-time behaviour of the tilted Bianchi type VIh models
• Large global solutions for energy supercritical nonlinear wave equations on \R^{3+1}
• Averaging in cosmological models using scalars
• Constraining cosmological ultra-large scale structure using numerical relativity
• Averaging Problem in General Relativity, Macroscopic Gravity and Using Einstein's Equations in Cosmology
• The Einstein--null dust system in spherical symmetry with an inner mirror: structure of the maximal development and Cauchy stability
• Inflation followed by Belinskii-Khalatnikov-Lifshitz collapse inside accreting, rotating black holes
• Spacetime Singularities: Recent Developments
• On the existence of global solutions for $T^3$-Gowdy spacetimes with stringy matter
• The Penrose inequality on perturbations of the Schwarzschild exterior
• Bianchi VIII and IX vacuum cosmologies: Almost every solution forms particle horizons and converges to the Mixmaster attractor
• General relativistic density perturbations
• Static Spherically Symmetric Solutions of the SO(5) Einstein Yang-Mills Equations
• On the Penrose inequality along null hypersurfaces
• Non-orthogonally transitive $G_2$ spike solution
• Trapped surfaces in vacuum arising dynamically from mild incoming radiation
• The causal structure of microlocalized Einstein metrics
• Spike statistics
• The instability of anti-de Sitter space-time
• When Do Spacetimes Have Constant Mean Curvature Slices?
• General Bianchi IX dynamics in bouncing braneworld cosmology: homoclinic chaos and the BKL conjecture
• Future complete Einsteinian space times with U(1) symmetry, the unpolarized case
• Nonlinear wave equations

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• Electrically Charged Quark Stars in $4D$ Einstein-Gauss-Bonnet Gravity
• Charged quark stars in $f(R,T)$ gravity
• Singularities, black holes, and cosmic censorship: A tribute to Roger Penrose
• A conjecture about the general cosmological solution of Einstein's equations
• Hunting for gravitational quantum spikes
• Limits on the Inferred Hubble Constant Bias from a Local McVittie Gravitational Field