Change in entropy of the spinning black holes

Aims: To derive an expression for change in entropy of spinning black holes on the basis of the model for the energy of spinning black holes


INTRODUCTION
Classically, the black holes are perfect absorbers and do not emit anything; their temperature is absolute zero. However, in quantum theory black holes emit Hawking radiation with a perfect thermal spectrum. This allows a consistent interpretation of the laws of black hole mechanics as physically corresponding to the ordinary laws of thermodynamics [1]. The laws of black hole mechanics as proposed by Bardeen

DISCUSSION
The Black hole possesses an event horizon (a oneway membrane) that casually isolates the "inside" of the Black hole from the rest of the universe. The radius of the event horizon of a nonspinning BH given by the Schwarzschild radius in terms of solar mass can be obtained as [7] ( ) In the case of spinning black holes, the equation can be written as The energy of spinning black holes in terms of radius of event horizon is given as [3] '.

ILCPA Volume 32
The change in entropy of different test Nonspinning black holes for their corresponding change in energy is given by the following eqn [ Putting eq n (4) in the above eqn, we have The term M stands for the mass of black holes. From eqn. (9), 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 ( ) κ may be seen in the research paper [8,10].
The equation (8) can be used to calculate the change in entropy of different test spinning black holes for their corresponding change in the radius of the event horizon.

DATA IN THE SUPPORT OF MASS OF BLACK HOLES
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 stellar -mass black holes in Xray binaries and M ~ 10 6 -10 9.5 M ʘ for super massive black holes in active galactic nuclei [10,11]. The mass of black holes never be greater than 5x10 9 M ʘ [17]. The other data in the support of mass of black holes in AGN can seen in references [1116].
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 change in entropy of spinning black holes in XRBs and AGN for given radius of event horizon of different test spinning black holes listed in the Table 1 & 2 respectively. There is negligible change in the radius of the event horizon due to change in entropy. Hence in the numerical calculation, the radius of the event horizon ( bhs R ) can be used instead of bhs R δ [5].   (Table 1).  (Table 2).

DISCUSSION AND RESULT
In present work, we have derived the formula for the change in entropy of spinning (i) the radius of event horizon ( bhs R ) of different test spinning black holes and their corresponding entropy change in XRBs (Fig. 1).
(ii) the radius of event horizon ( bhs R ) of different test spinning black holes and their corresponding entropy change in AGN (Fig. 2).
From the observation of the data in the Table 1  .
When we compared the present work with that of nonspinning black holes , we see that the change in energy and entropy of spinning black holes are the exactly same to the nonspinning black holes, while the event horizon of spinning black holes is half to that of the nonspinning black holes of the same mass. This shows that the change in energy and entropy are mainly dependent on the mass and independent of the event horizon of black holes.

CONCLUSIONS
In the present research paper, we have drawn the following conclusions: (i) The calculated values for change in entropy show that 0 S δ ≥ for each spinning black holes in XRBs and AGN. This result is good agreement with the second law of thermodynamics.
(ii) Larger the mass/ radius of event horizon, greater is the change in entropy of spinning black holes and viceversa. (iv) Mahto model for energy of the spinning black hole is justified.
(v) The change in energy and entropy are mainly dependent on the mass and independent of the event horizon of black holes.