7.2 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 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.

Local and Global Existence Theorems for the Einstein Equations Alan D. Rendall http://www.livingreviews.org/lrr-2000-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de