How to describe a black hole embedded in an expanding universe is an
important topic. Some models about this issue are suggested by assuming that
the metric is a conformal transformation of the Schwarzschild metric or of the
isotropic black hole metric. However, there exists naked singularities in the
two metrics. Recently, it is argued that the singularity in the Sultana-Dyer
space-time is covered by an apparent horizon surface. But we find that such an
apparent horizon does not exist if the null energy condition holds.Comment: 7 pages, 1 figures; typographical errors are corrected; details is
added; conclusin is unchange