### 2.5 Matter fields

Analogues of the results for the vacuum Einstein equations given in Section 2.2 are known for the
Einstein equations coupled to many types of matter model. These include perfect fluids, elasticity
theory [43], 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 that is uniform in time. In cases where 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. This will be discussed in more detail in Section 2.6. In the case of kinetic or field
theoretic matter models it makes no difference whether the energy density vanishes somewhere or
not.