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Dynamic Black Holes in a FRW background: Lemaitre transformations

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.