Equations of hydrodynamics in general relativity via a slow-motion expansion of an integral solution

By means of a formal solution to the Einstein gravitational field equations a slow-motion expansion in inverse powers of the speed of light is developed for the metric tensor. The formal solution, which satisfies the deDonder coordinate conditions and the Trautman outgoing-radiation condition, is in the form of an integral equation which is solved iteratively. A stress-energy tensor appropriate to a perfect fluid is assumed and all orders of the metric needed to obtain the equations of motion and conserved quantities to the 2-1/2 post-Newtonian approximation are found. The results of this method are compared to those previously obtained in another gauge by S. Chandrasekhar. They are shown to be equivalent, but require considerably less labor in their determination. The relation of the fast-motion approximation to the slow-motion approximation is examined.