On apparent horizons and the Schwarzschild surface for a uniform fluid sphere in general relativity

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Abstract
The time history of the marginally trapped surfaces, i.e., the apparent horizons for a spherically symmetric nonstatic fluid of uniform density are studied. Generally it is found that apparent horizons may or may not exist dependent upon the choice of arbitrary functions of integration. However, it is shown in this paper that if the metric is conformally flat or if the circumference of the sphere is an increasing function of a radial coordinate, apparent horizons exist if and only if the surface is inside the Schwarzschild surface. Then there exist in fact at least two horizons: The absolute Schwarzschild surface and an apparent horizon in the interior of the fluid matter.