The upper bound on the entropy of self-gravitating radiation systems

The entropy of a self-gravitating radiation system (sgrs) confined to a spherical box with a central black hole is calculated by using the equation of state of radiation near a black-hole horizon, which was derived by Li and Liu. The upper bound on the total entropy of the system including the central black hole is exactly equal to the entropy of the black hole with the same mass. A reasonable mode of the gravitational collapse of ansgrs is given, and a possible explanation for the origin of black-hole entropy is provided.