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.
Highlights
Systems of partial differential equations are of central importance in physics
If the proof of a local existence theorem is examined closely it is generally possible to give a continuation criterion. This is a statement that if a solution on a finite time interval is such that a certain quantity constructed from the solution is bounded on that interval, the solution can be extended to a longer time interval. (In applying this to the Einstein equations we need to worry about introducing an appropriate time coordinate.) If it can be shown that the relevant quantity is bounded on any finite time interval where a solution exists, global existence follows
Since black holes are apparently incompatible with this symmetry, one may hope to prove geodesic completeness of solutions under appropriate assumptions. (It would be interesting to have a theorem making the statement about black holes precise.) A proof of geodesic completeness has been achieved for the Einstein vacuum equations and for the source-free Einstein–Maxwell equations in [34], building on global existence theorems for wave maps [83, 82]
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 as 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 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 [178]. An extended discussion of the idea can be found in [85]
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