Finite reduced hydrodynamic equations in the slow-motion approximation to general relativity. Part I. First post-Newtonian equations

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The iteration procedure of Anderson and DeCanio (1975) for obtaining hydrodynamic equations in the slow-motion approximation to general relativity is modified at each iteration step. These improvements keep all expressions needed for the 2 1/2 post-Newtonian approximation (PNA) manifestly finite, but fail to prevent some divergent terms in the third and higher PNAs. In this Part I, the improvements to the iteration scheme are outlined, and the calculation is completed through the second iteration, yielding Newtonian-like hydrodynamic equations in the first PNA. Paper II will complete the calculation through the 2 1/2 PNA, allowing the calculation of the gravitational radiation reaction force, and making explicit the source of the divergences which occur in the higher PNAs.

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