Abstract
The Newtonian limit of general relativity is by no means as straightforward as is commonly assumed. In particular, the correct limit of the Bianchi identities must be taken to the second (non-linear) order. Furthermore Newtonian cosmology does not have a well-posed initial value formulation, while relativistic cosmology does. We show in this paper that the c-4 approximation of general relativity, although non-linear, provides a non-standard version of Newtonian theory which is in fact completely equivalent to the Heckmann–Schucking version of Newtonian cosmology. The next approximation (order c-6), when the limit is taken in a particular way, gives rise to a closed and self-consistent post-Newtonian cosmological theory which has a well-posed initial value problem. This seems to be a suitable, if somewhat complicated, theory for cosmological and astrophysical problems
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