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It is suggested a metric ansatz to describe local matter in an expandinguniverse, hence interpolating between the Schwarzschild metric at small spatialscales and the FLRW metric at large spatial scales. This is acomplishedmaintaining space-time free of singularities except for the Schwarzschild masspole at the origin as opposed to metrics already considered in the literaturewith the same purpose, namely the McVittie metric. The modified Newton law isanalyzed and the static orbit solutions computed. It is concluded that theeffects of expantion in the solar system are negligible, however depending onthe metric parameter value, at galactic scales there is a significant deviationfrom the General Relativity Newton law which may contribute to dark mattereffects allowing for a flattening of galaxy rotation curves.

A Locally Anisotropic Metric for Matter in an Expanding Universe: I. The Ansatz and the Modified Newton Law