How to estimate the entropy inside a Schwarzschild black hole

Abstract: We reconsider the interrelation between the Schwarzschild radius and mass of a spherical symmetric non-rotating and uncharged black hole. By applying the relativistic mass-energy equivalence, the area of the horizon is expressed independently of the Schwarzschild radius in terms of an occupation number operator corresponding to the underlying many-body system of the particles inside the black hole. Given the expectation value of the area, the von Neumann entropy based on the matter or radiation inside the black hole is maximized to obtain the equilibrium probability density operator of the underlying ensemble. We apply this approach to an ideal gas of ultra-relativistic particles and to particles of rest mass zero. For both, the entropy is expressed in terms of the horizon area of the black hole. In addition, the quantum fluctuations inside the black hole are expressed in terms of the standard deviation of the horizon area. The thermodynamic concept is finalized by a generalized Planck law of the surface area which is derived at the end.