  Since the conformal transformations of metric do not change its causal
structure, we use these transformations to embed the Lemaitre metrics into the
FRW background. In our approach, conformal transformation is in agreement with
the universe expansion regimes. Indeed, we use the Lemaitre metrics because the
horizon singularity is eliminated in these metrics. For our solutions, there is
an event horizon while its physical radii is increasing as a function of the
universe expansion provided suitable metrics for investigating the effects of
the universe expansion on the Black Holes. In addition, the physical and
mathematical properties of the introduced metrics have well-defined behavior on
the event horizon. Moreover, some physical and mathematical properties of the
introduced metrics have been addressed.
