2 Local existenceLocal and Global Existence Theorems Local and Global Existence Theorems

1 Introduction

  Many of the mathematical models occurring in physics involve systems of partial differential equations. Only rarely can these equations be solved by explicit formulae. When they cannot, physicists frequently resort to approximations. There is, however, another approach which is complementary. This consists in determining the qualitative behaviour of solutions, without knowing them explicitly. The first and most fundamental step in doing this is to establish the existence of solutions under appropriate circumstances. Unfortunately, this is often hard, and obstructs the way to obtaining more interesting information. It may appear to the outside observer that existence theorems become a goal in themselves to some researchers. It is important to remember that, from a more general point of view, they are only a first step.

The basic partial differential equations of general relativity are Einstein's equations. In general they are coupled to other partial differential equations describing the matter content of spacetime. The Einstein equations are essentially hyperbolic in nature. In other words, the general properties of solutions are similar to those found for the wave equation. It follows that it is reasonable to try to determine a solution by initial data on a spacelike hypersurface. Thus the Cauchy problem is the natural context for existence theorems for the Einstein equations. The Einstein equations are also nonlinear. This means that there is a big difference between the local and global Cauchy problems. A solution evolving from regular data may develop singularities.

A special feature of the Einstein equations is that they are diffeomorphism invariant. If the equations are written down in an arbitrary coordinate system then the solutions of these coordinate equations are not uniquely determined by initial data. Applying a diffeomorphism to one solution gives another solution. If this diffeomorphism is the identity on the chosen Cauchy surface up to first order then the data are left unchanged by this transformation. In order to obtain a system for which uniqueness in the Cauchy problem holds in the straightforward sense 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 given freely. They must satisfy constraint equations. To prove the existence of a solution of the Einstein equations, it is first necessary to prove the existence of a solution of the constraints. The usual method of solving the constraints relies on the theory of elliptic equations.

The local existence theory of solutions of the Einstein equations is rather well understood. Section  2 points out some of the things which are not known. On the other hand, the problem of proving general global existence theorems for the Einstein equations is beyond the reach of the mathematics presently available. To make some progress, it is necessary to concentrate on simplified models. The most common simplifications are to look at solutions with various types of symmetry and solutions for small data. These two approaches are reviewed in sections  3 and  5 respectively. A different approach is to prove the existence of solutions with a prescribed singularity structure. This is discussed in section  6 . Section  7 collects some miscellaneous results which cannot easily be classified. With the motivation that insights about the properties of solutions of the Einstein equations can be obtained from the comparison with Newtonian theory and special relativity, relevant results from those areas are presented in section 4 .

The area of research reviewed in the following relies heavily on the theory of differential equations, particularly that of hyperbolic partial differential equations. For the benefit of readers with little background in differential equations, some general references which the author has found to be useful will be listed. A thorough introduction to ordinary differential equations is given in [90]. A lot of intuition for ordinary differential equations can be obtained from [95]. The article [10] is full of information, in rather compressed form. A classic introductory text on partial differential equations, where hyperbolic equations are well represented, is [101]. Useful texts on hyperbolic equations, some of which explicitly deal with the Einstein equations, are [161, 105Jump To The Next Citation Point In The Article, 127, 118, 159, 102, 68].

An important aspect of existence theorems in general relativity which 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 [122]. An extended discussion of the idea can be found in [59].

2 Local existenceLocal and Global Existence Theorems Local and Global Existence Theorems

image Local and Global Existence Theorems for the Einstein Equations
Alan D. Rendall
© Max-Planck-Gesellschaft. ISSN 1433-8351
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