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.
2.3), while taking it into account explicitly leads to a free
boundary problem. For more discussion of this and references see
. 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 .