Hamiltonian formulation of gravitating perfect fluids and the Newtonian limit

The canonical formalism is applied to self-gravitating perfect fluids with particular emphasis on recovering the correct nonrelativistic limit also in (quasi-) Hamiltonian form. We use essentially Lagrangian coordinates by considering the fluid defined by a map from space-time into a three-dimensional material manifold which is equipped with a volume element representing physically the matter (baryon number) density. By eliminating the coordinate freedom in this material space the usual matter conservation and (relativistic) Euler equations are recovered in a (3+1)-dimensional formalism which makes it very easy to compare them to their nonrelativistic limits. By splitting the 3-metric and its canonical momenta into a conformal part and the determinant we arrive at a system of evolution and constraint equations for the gravitational field that also has a well-defined Newtonian limit provided the geometric version of the Newtonian theory is also cast into an analogous (3+1)-dimensional form. Some of the evolution equations of the relativistic theory, however, become additional constraints in the limit which represents the freezing of the gravitational (or radiation) degrees of freedom. We then use this formalism to rederive the first-order post-Newtonian approximation and obtain the standard results in a flexible geometrical form since no gauge or coordinate conditions need be imposed in advance.