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 nontrivial 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 that encodes those properties that are desirable on
physical grounds (see [98].) 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 that have a Newtonian limit in the sense of
the chosen definition. A theorem of this kind was proved
in [205], 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 [109].) 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 that depend continuously on
. That there is an abundant supply of families of this kind is
the result of [205]. 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 postNewtonian approximations. The approach of [205] has not even been extended to the case
k
=1, and it would be desirable to do this. Note however that when
k
is too large, serious restrictions arise [203]. The latter fact corresponds to the wellknown divergent
behaviour of higher order postNewtonian approximations.
It may be useful for practical projects, for instance those
based on numerical calculations, to use hybrid models in which
the equations for selfgravitating Newtonian matter are modified
by terms representing radiation damping. If we expand in terms of
the parameter
as above then at some stage radiation damping terms should play
a role. The hybrid models are obtained by truncating these
expansions in a certain way. The kind of expansion that has just
been mentioned can also be done, at least formally, in the case
of the Maxwell equations. In that case a theorem on global
existence and asymptotic behaviour for one of the hybrid models
has been proved in [160]. These results have been put into context and related to the
Newtonian limit of the Einstein equations in [159].

Theorems on Existence and Global Dynamics for the
Einstein Equations
Alan D. Rendall
http://www.livingreviews.org/lrr20026
© MaxPlanckGesellschaft. ISSN 14338351
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