Theorems on Existence and Global Dynamics for the Einstein Equations

Abstract

This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local-in-time Cauchy problem, which is relatively well understood, is surveyed. Global results for solutions with various types of symmetry are discussed. A selection of results from Newtonian theory and special relativity that offer useful comparisons is presented. Treatments of global results in the case of small data and results on constructing spacetimes with prescribed singularity structure are given. A conjectural picture of the asymptotic behaviour of general cosmological solutions of the Einstein equations is built up. Some miscellaneous topics connected with the main theme are collected in a separate section.

The formation of black holes in spherically symmetric gravitational collapse

“…The system (1.1), (1.4), (1.5) is the Einstein-Vlasov system in general coordinates. For an introduction to relativistic kinetic theory and the Einstein-Vlasov system we refer to [1] and [33]. If, for comparison, the matter is to be described as a perfect fluid with density R, four-velocity field U α , and pressure P , then the matter evolution equations are the Euler equations…”

Asymptotic behaviour of the Einstein–Vlasov system with a positive cosmological constant

“…The spatially homogeneous cases of the results of the plane and hyperbolic symmetries in [16] correspond to LRS models of Bianchi type I and LRS of Bianchi III respectively. Surveys of results on the Einstein-Vlasov system in general and other cosmological models can be found in [1,7,8,9,11,12].…”

Axisymmetric evolution of Einstein equations and mass conservation

“…This is called a fully constrained system. On the opposite side we have a system in which we do not solve for the constraints and we solve the evolution equations (24) and (25). This is called a free evolution scheme.…”

“…From the analytical point of view, there exist no results up until now for axially symmetric isolated systems (see the review [25] for results with other kinds of symmetries in cosmologies). One of the purpose of this paper is to initiate the study of this problem.…”