Living Reviews in Relativity

"Theorems on Existence and Global Dynamics
for the Einstein Equations"
by
Alan D. Rendall  

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Abstract
1 Introduction
2 Local Existence
2.1 The constraints
2.2 The vacuum evolution equations
2.3 Questions of differentiability
2.4 New techniques for rough solutions
2.5 Matter fields
2.6 Free boundary problems
3 Global Symmetric Solutions
3.1 Stationary solutions
3.2 Spatially homogeneous solutions
3.3 Spherically symmetric asymptotically flat solutions
3.4 Weak null singularities and Price’s law
3.5 Cylindrically symmetric solutions
3.6 Spatially compact solutions
4 Newtonian Theory and Special Relativity
4.1 Hydrodynamics
4.2 Kinetic theory
4.3 Elasticity theory
5 Global Existence for Small Data
5.1 Stability of de Sitter space
5.2 Stability of Minkowski space
5.3 Stability of the (compactified) Milne model
5.4 Stability of the Bianchi type III form of flat spacetime
6 Prescribed Asymptotics
6.1 Isotropic singularities
6.2 Fuchsian equations
6.3 Asymptotics for a phase of accelerated expansion
7 Asymptotics of Expanding Cosmological Models
7.1 Lessons from homogeneous solutions
7.2 Inhomogeneous solutions with /\ = 0
7.3 Homogeneous models with /\ > 0
7.4 Acceleration due to nonlinear scalar fields
7.5 Other models for cosmic acceleration
7.6 Inhomogeneous spacetimes with accelerated expansion
8 Structure of General Singularities
8.1 Lessons from homogeneous solutions
8.2 Inhomogeneous solutions
8.3 Formation of localized structure
8.4 Cosmic censorship in Gowdy spacetimes
9 Further Results
9.1 Evolution of hyperboloidal data
9.2 The Newtonian limit
9.3 Newtonian cosmology
9.4 The characteristic initial value problem
9.5 The initial boundary value problem
9.6 The geodesic hypothesis
10 Acknowledgements
Open References References