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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 [288Jump To The Next Citation Point], 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 c, which are hyperbolic for c /= 0 and degenerate for c = 0. 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 c. That there is an abundant supply of families of this kind is the result of [288Jump To The Next Citation Point]. Asking whether there are families which are k times continuously differentiable in their dependence on c 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 k = 1, and it would be desirable to do this. Note however that when k 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 c 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 [4039]. Calogero has proved a theorem on the Newtonian limit of the special relativistic Boltzmann equation [75].


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