Change in energy and entropy of the non-spinning black holes

In the present paper, we have derived an expression for change in entropy of non-spinning black holes on the basis of formula ( BH BH s E K R  ) as proposed by Kanak Kumari et al. (2010) and the formula 2 2 ( Mahto et al., 2011) and calculated their values for different test non-spinning black holes.


INTRODUCTION
The laws of black hole mechanics as proposed by Bardeen et al. describe the behaviour of a black hole in close analogy to the laws thermodynamics by relating mass to energy, area to entropy, and surface gravity to temperature [1]. Classically black holes are perfect absorbers, but do not emit anything; their physical temperature is absolute zero [2]. Quantum mechanically, however, there is a possibility that one of a particle production pair in a black hole is able to tunnel the gravitational barrier and escapes the black hole's horizon. Thus, a black hole is not really black; it can radiate or evaporate particles [3]. Dipo Mahto et al. derived an expression for the change in energy and entropy of Non-spinning black holes taking account the first law of the black hole mechanics relating the change in mass M, angular momentum J, horizon area A and charge Q, of a stationary black hole with Einstein's mass-energy equivalence relation [5]. Dipo Mahto

DISCUSSION
The Black hole possesses an event horizon (a one-way membrane) that casually isolates the "inside" of the Black hole from the rest of the universe. The radius of the event horizon of a non-spinning BH given by the Schwarzschild radius in terms of solar mass can be obtained as [7] R s = 2950 (M/M ʘ ) m (1) The energy of non-spinning black holes in terms of radius of event horizon is given as Differentiating eq n (2), we have The change in entropy of different test Non-spinning black holes for their corresponding change in energy is given by the following eqn [5] Putting eq n (4) in the above eqn, we have where  and bh R are the surface gravity & radius of the event horizon of black holes given as [8,10] 1 4M   (8) The term M stands for the mass of black holes. From eqn. (8), it is clear that the surface gravity of black hole is inversely proportional to its mass and the different black holes will have different surface gravity. The role of surface gravity ()  may be seen in the research paper [8][9][10].
The equation (7) can be used to calculate the change in entropy of different test Nonspinning black holes for their corresponding change in the radius of the event horizon.
There are two categories of Black holes classified on the basis of their masses clearly very distinct from each other, with very different masses M ~ 5 20 M ʘ for stellarmass Black holes in X-ray binaries and M ~ 10 6 -10 9.5 M ʘ for super massive black holes in Active Galactic Nuclei [10,11]. The other data in the support of mass of black holes in AGN can seen in references [11][12][13][14][15][16].
On the basis of the data mentioned above regarding the mass of black holes in XRBs and AGN in terms of solar masses, we have calculated energy of black holes and change in entropy in XRBs and AGN for given radius of event horizon of different test Non-spinning black holes listed in the Table1 and 2 respectively. There is negligible change in the radius of the event horizon. So for numerical calculation, the radius of the event horizon (    (Table 1).  (Table 2). . We have calculated the change in entropy of different test nonspinning black holes in X-ray binaries (XRBs) and Active Galactic Nuclei (AGN) and the graphs have been plotted between:

DISCUSSION AND RESULT
(A) the radius of event horizon (R bh ) of different test black holes and their corresponding change in entropy in XRBs (Fig. 1). (B) the radius of event horizon (R bh ) of different test black holes and their corresponding change in entropy in AGN (Fig. 2).
From the observation of the data in the Table 1  The graph plotted between the radius of event horizon (R bh ) of different test black holes & their corresponding change in entropy in XRBs (Fig. 1) has the same nature of the graph plotted between the radius of event horizon (R bh ) of different test black holes and their corresponding entropy in XRBs using the well known relation of entropy

CONCLUSIONS
In the present research paper, we have drawn the following conclusions: