### 9.2 The Newtonian limit

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 that encodes those properties that
are desirable on physical grounds (see [136].) 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 [288], 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 [149].) 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 [288]. Asking whether there are families which are times continuously differentiable in their
dependence on is related to the issue of giving a mathematical justification of post-Newtonian
approximations. The approach of [288] has not even been extended to the case , and it would be
desirable to do this. Note however that when is too large, serious restrictions arise [286]. The
latter fact corresponds to the well-known divergent behaviour of higher order post-Newtonian
approximations.
It may be useful for practical projects, for instance those based on numerical calculations, to use hybrid
models in which the equations for self-gravitating 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 [220]. These
results have been put into context and related to the Newtonian limit of the Einstein equations
in [219].

In the case of the Vlasov-Maxwell and Vlasov-Nordström systems the equivalent of the
post-Newtonian approximations have been justified rigorously up to certain orders [40, 39].
Calogero has proved a theorem on the Newtonian limit of the special relativistic Boltzmann
equation [75].