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 or late-time asymptotics 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.
Highlights
Systems of partial differential equations are of central importance in physics
It is known that the Hs developments cannot shrink with increasing s [87], and so the existence of a C∞ solution is obtained for C∞ data. It appears that the Hs spaces with s sufficiently large are the only spaces containing the space of smooth functions for which it has been proved that the Einstein equations are locally solvable
Solutions of the Einstein equations with cylindrical symmetry that are asymptotically flat in all directions allowed by the symmetry represent an interesting variation on asymptotic flatness
Summary
Systems of partial differential equations are of central importance in physics. Only the simplest of these equations can be solved by explicit formulae. In order to obtain a system for which uniqueness in the Cauchy problem holds in the straightforward sense that it does for the wave equation, some coordinate or gauge fixing must be carried out Another special feature of the Einstein equations is that initial data cannot be prescribed freely. A different approach is to prove the existence of solutions with a prescribed singularity structure or late-time asymptotics An important aspect of existence theorems in general relativity that one should be aware of is their relation to the cosmic censorship hypothesis This point of view was introduced in an influential paper by Moncrief and Eardley [248]. A collection of relevant equations together with the background necessary to understand the notation can be found in [301]
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