Statistical Approach to the Bekenstein-Hawking Entropy

We consider a Schwarzschild black hole type in this work whose particles, only those that lies on its surface, the event horizon ( r + ), contributes to the entropy and we found it by using the canonical ensemble. We don’t consider any interaction between this particles, but the inner energy.


Introduction
A black hole of mass M and raius r + , with not charge and rotation is considerated in this work. The surface of the black hole is composed by a number N fixes of indistinguishable particles of the same mass m 0 which are restricted to move on the black hole surface and does not interacts one with another.
In [4] was calculated the entropy via thermodynamic consideration, taking in account the work made by the net force on the event horizon at constant temperature. Others consideration are made to find the entropy in [3,6,2]. In [1] is considerated that the main contribution to the entropy is given by thermally excited 'invisible' modes propagating in the close vicinity of the horizon, which is true. In [1] is considerated a dynamical origin of the entropy, and in [5] is considerated that black hole entropy counts only those states of a black hole that can influence the outside, which is true too. Here our goal is to find this magnitude from the statistical point of view, considering the canonical ensemble. In the first part we found the entropy without consider the internal energy, after this we consider the internal energy of the superficial mass of the black hole, which is considerated to be constant.

Partition function 2.1 With low interacting energy
The canonical partition function for indistinguishable particles that describe this system where is the cannonical partition function for distinguishable particles, and the Hamiltonian.
We consider that the potential energy U (r) is negligible and only the particles on the event horizon contributes to the entropy.
Using spherical coordinates and considering that the particles are restricted to move on the black hole surface(f = 2), only the components p θ and p φ contribute to the total momentum p of each particle, that is, Then, the Eq. 4 can be write as, Doing the calculations we get, where A is the area of the event horizon and T the temperature. The Eq. 7 is the partition function of N particles.
Then, the partition function, Eq. 1 , can be write as The free energy, F , is equal to Now we can determine the entropy of this system, where c 0 is a constant igul to 2N k B .
Entropy reach its maximum value when the temperature does. This temperature corresponds to the Hawking temperature, then we get The terms Ac 3 /4 GN and m 0 /4π 2 M can be approximate to e Ac 3 /4 GN and e m0/4π 2 M respectively, then the entropy give The term m = N m 0 corresponds to the superficial mass of the black hole which is much less than that the corresponding black hole mass M , then we can neglected this term, and we get the Bekenstein-Hawking entropy,

Considering the internal energy
Considering the Eq. 4, and that the internal energy, U , of the superficial mass of the black hole does not depend on | r j − r i |, where r i and r j are the position vectors of each particles (U = N m 0 c 2 ), the partition function Z is, and then Eq. 1 can be write as, Now we can calculate the free energy and get and the entropy which correspond to the same results as Eq. 10 . This means that the constant internal energy in the potential energy could not be considerated in the calculus of the black hole entropy. The Eq. 17 show the dependence of the entropy from temperature T , area A and number of particles N and we see that entropy increase as temperatura does, which we hope.

Conclusions
The Bekenstein-Hawking entropy come from the particles that lies on the surface of the black hole, on the event horizon. This results is due to the hight density of the inner surfaces of this type of object. Particles in the inner of the black hole has low kinetic energy, that reduces the temperature. On the other hand it seems that both, internal and interaction energy does not affect the Bekenstein-Hawking entropy.