We propose and analyze a new metric that has two conformal factors a(t) and b(t) that combine the expansion of the universe and its effects on the spatial and temporal part of the Schwarzschild metric in isotropic coordinates. We present the solutions, their descriptions and we comment on their shortcomings. In the spatially flat case of an expanding universe, we derive from the proposed metric the special solutions of the field equations for the dust approximation and the McVittie metric. We show that the presence of a black hole does not modify the a(t)αt2/3 law for dust and H = const. for dark energy.
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