Bousso entropy bound in self - gravitating gas of massless particles

The Bousso entropy bound is investigated in a static spherically symmetric spacetime filled with an ideal gas of massless bosons or fermions. Especially lightsheets generated by spheres are considered. A statistical description of the gas is given. Conditions under which the Bousso bound can be violated are discussed and it is shown that a possible violating region cannot be arbitrarily large and that it is contained inside a sphere of unit Planck radius if the number of independent polarization states gs is small enough. It is also shown that the central temperature must exceed the Planck temperature in order to get a violation of the Bousso bound for gs not too large.