Most textbooks on general relativity discuss the fact that
Newtonian gravitational theory is the limit of general relativity
as the speed of light tends to infinity. It is a non-trivial task
to give a precise mathematical formulation of this statement.
Ehlers systematized extensive earlier work on this problem and
gave a precise definition of the Newtonian limit of general
relativity which encodes those properties which are desirable on
physical grounds (see [67].) Once a definition has been given the question remains whether
this definition is compatible with the Einstein equations in the
sense that there are general families of solutions of the
Einstein equations which have a Newtonian limit in the sense of
the chosen definition. A theorem of this kind was proved
in [142], where the matter content of spacetime was assumed to be a
collisionless gas described by the Vlasov equation. (For another
suggestion as to how this problem could be approached see [76].) The essential mathematical problem is that of a family of
equations depending continuously on a parameter
which are hyperbolic for
and degenerate for
. Because of the singular nature of the limit it is by no means
clear a priori that there are families of solutions which depend
continuously on
. That there is an abundant supply of families of this kind is
the result of [142]. Asking whether there are families which are
k
times continuously differentiable in their dependence on
is related to the issue of giving a mathematical justification
of post-Newtonian approximations. The approach of [142] has not even been extended to the case
k
=1 and it would be desirable to do this. Note however that for
k
too large serious restrictions arise [141]. The latter fact corresponds to the well-known divergent
behaviour of higher order post-Newtonian approximations.