2.4 Matter fields 

Analogues of the results for the vacuum Einstein equations given above are known for the Einstein equations coupled to many types of matter model. These include perfect fluids, elasticity theory, kinetic theory, scalar fields, Maxwell fields, Yang-Mills fields and combinations of these. An important restriction is that the general results for perfect fluids and elasticity apply only to situations where the energy density is uniformly bounded away from zero on the region of interest. In particular they do not apply to cases representing material bodies surrounded by vacuum. In cases where the energy density, while everywhere positive, tends to zero at infinity, a local solution is known to exist, but it is not clear whether a local existence theorem can be obtained which is uniform in time. In cases where there the fluid has a sharp boundary, ignoring the boundary leads to solutions of the Einstein-Euler equations with low differentiability (cf. Section 2.3), while taking it into account explicitly leads to a free boundary problem. For more discussion of this and references see [74]. In the case of kinetic or field theoretic matter models it makes no difference whether the energy density vanishes somewhere or not. There is apparently little in the literature on the initial value problem for the Einstein equations coupled to fermions, e.g. for the Einstein-Dirac system, although there seems no reason to expect special difficulties in that case. One paper related to this question is [13].

Local and global existence theorems for the Einstein equations Alan D. Rendall http://www.livingreviews.org/lrr-1998-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de