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We consider the problem of gravitational collapse of a fluid under the effectof a Schwarzschild black hole (e.g. a primordial one) that suddenly formsinside the fluid. We assume the fluid initially be a uniform dust. Startingfrom this configuration we obtain a class of metrics under some assumptions. Wefind that the metric we obtain includes the dust collapse as a subcase. Afterdiscussing some basic properties of the solution, we discuss the case of dustcollapse in more detail. We find that the radial and tangential pressuresoutside the horizon may take positive or negative values depending on thevalues of the parameters.

A metric for gravitational collapse of dust around a Schwarzschild black hole