Microcanonical approach to the entropy of a quantum eld in a Kerr

It is shown that that the area law for the entropy of a quantum eld in a Kerr space-time is due to the quantum statistics with the maximal entropy principle. The entropies for one particle, a Boltzmann gas, a quantum mechanical gas obeying Bose-Einstein or Fermi-Dirac statistics, and a quantum eld in the Kerr space-time are calculated using the microcanonical ensemble approach and the brick wall method. The area law holds only when the system follows the quantum statistics and it satisses the maximal entropy principle of a given energy.