^{1}

^{*}

^{2}

Mahto et al. have shown
*δS*≥0 and
*M*
^{2}
*=a**
*J* by using the first law of the black hole mechanics in the vacuum and Einstein mass-energy equivalence relation specially for spinning black holes. In the present paper, this work is extended to propose a model for the change in mass of the spinning black holes due to corresponding change in the angular momentum for maximum & half spin parameter of black holes (a* = 1 & 1/2) and calculated their values for different test black holes in XRBs and AGN. We have also shown that the change in mass of the spinning black holes due to corresponding change in the angular momentum for maximum spinning rate of black holes (a* = 1) is double to that of the spinning black holes having spinning parameter (
*a** = 1/2).

Classically black holes are perfect absorbers, but do not emit anything; their physical temperature is absolute zero [

In the present work, a model for the change in mass of the spinning black holes due to corresponding change in the angular momentum is proposed for maximum spinning rate of black holes (a* = 1/2 and 1).

The mass (M), angular momentum (J) and spin parameter (a*) of black holes are co-related by the following equation [

M 2 = a ∗ J (1)

where a* is a constant called spin parameter of spinning black holes lying between −1 to +1 including zero of different test black holes [

Now the equation is differentiated , we have

2 M δ M = a ∗ δ J

δ M δ J = a ∗ 2 M (2)

For spinning black holes having maximum spin a ∗ = 1 [

δ M δ J = 1 2 M (3)

The above equation shows that the change in mass of the spinning black holes due to corresponding change in the angular momentum for maximum spinning rate of black holes.

For half spin parameter (a* = 1/2), the Equation (2) becomes [

δ M δ J = 1 4 M (4)

The above equation shows that the change in mass of the spinning black holes due to corresponding change in the angular momentum for half spin parameter 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 20M_{ʘ} for stellar-mass Black holes in X-ray binaries and M ~10^{6} - 10^{9.5}M_{ʘ} for super massive black holes in Active Galactic Nuclei [_{ʘ}) = 1.99 × 10^{30} kg [

On the basis of the data mentioned in the Section 3, the change in mass of different test spinning black holes due to corresponding change in angular momentum are calculated in XRBs and AGN to plot the graphs with the help of Equation (3) & (4) as shown in Figures 1-3 of the Section 4.

To obtain the change in mass of the spinning black holes due to corresponding change in the angular momentum, the equation (1) is differentiated for maximum spinning rate of black holes (a* = 1) and half spin parameter (a* = 1/2) given by the Equation (3) & (4) respectively.

After this, we have calculated the change in mass of the spinning black holes due to corresponding change in the angular momentum for maximum spinning rate of black holes (a* = 1 & a* = 1/2) in XRBs and AGN with the help of (3) & (4).

We have plotted the graph between the change in mass of the spinning black holes w.r.t. the change in the angular momentum for maximum spinning rate of black holes (a* = 1) and spinning parameter (a* = 1/2) ( δ M / δ J ) and corresponding value of the mass of black holes (M). We also have shown that the change in mass of the spinning black holes due to corresponding change in the angular momentum for maximum spinning rate of black holes (a* = 1) is greater than to that of the spinning black holes having spinning parameter (a* = 1/2) in XRBs and shows that the change in mass of the spinning black holes w.r.t. the change in the angular momentum for maximum spinning rate of black holes (a* = 1) is twice times to that of spinning of black holes with spinning parameter (a* = 1/2). This fact is also clear from the comparative study from the

From the graph in the

spinning black holes and corresponding change in their angular momentum in AGN with spin parameter a* = 1 decreases up to a certain value of the mass with the increasing the mass of the spinning black holes and then rapidly increases with a certain value These variations are repeated in periodic manner as shown in

The spinning black holes of mass:

1) (1 × 10^{6}M_{ʘ}, 1 × 10^{7}M_{ʘ}, 1 ×10^{8}M_{ʘ}, 1 × 10^{9}M_{ʘ})―line a

2) (2 × 10^{6}M_{ʘ}, 2 × 10^{7}M_{ʘ}, 2 ×10^{8}M_{ʘ}, 2 × 10^{9}M_{ʘ})―line b

3) (3 × 10^{6}M_{ʘ}, 3 × 10^{7}M_{ʘ}, 3 ×10^{8}M_{ʘ}, 3 × 10^{9}M_{ʘ})―line c

4) (4 × 10^{6}M_{ʘ}, 4 × 10^{7}M_{ʘ}, 4 ×10^{8}M_{ʘ}, 4 × 10^{9}M_{ʘ})―line d

5) (5 × 10^{6}M_{ʘ}, 5 × 10^{7}M_{ʘ}, 5 ×10^{8}M_{ʘ}, 5 × 10^{9}M_{ʘ})―line e

6) (6 × 10^{6}M_{ʘ}, 6 × 10^{7}M_{ʘ}, 6 ×10^{8}M_{ʘ}, 6 × 10^{9}M_{ʘ})―line f

7) (7 × 10^{6}M_{ʘ}, 7 × 10^{7}M_{ʘ}, 7 ×10^{8}M_{ʘ}, 7 × 10^{9}M_{ʘ})―line g

8) (8 × 10^{6}M_{ʘ}, 8 × 10^{7}M_{ʘ}, 8 ×10^{8}M_{ʘ}, 8 × 10^{9}M_{ʘ})―line h

9) (9 × 10^{6}M_{ʘ}, 9 × 10^{7}M_{ʘ}, 9 ×10^{8}M_{ʘ}, 0 × 10^{9}M_{ʘ})―line i

When the graph is plotted for each category in the same graph paper, nine parallel lines are obtained. When the results obtained from our research work is compared with that of the spinning black holes with spin a* = 1/2, we see that the change in mass of the spinning black holes w.r.t. the change in the angular momentum for maximum spinning rate of black holes (a* = 1) is quite different to that of the spinning black holes with spin parameter (a* = 1/2). Hence, we may conclude that the spinning parameters of black holes are mainly responsible for the identification and characterization of black holes.

In the study of present research paper, we can draw the following conclusions:

1) The change in mass of the spinning black holes due to corresponding change in the angular momentum for the black holes of maximum spin is double to that of the spinning black holes of half spin parameter and decreases with increasing the mass of black holes in XRBs.

2) The change in mass of the spinning black holes w.r.t. the change in the angular momentum for the black holes of maximum spin is quite different to that of the spinning black holes with half spin parameter in AGN.

3) The spinning parameter is mainly responsible for the identification and characterization of black holes.

The authors are very grateful to reviewers to find out the errors in the original manuscript and providing the constructive suggestions. This paper is also devoted in memory of great scientist Stephen Hawking (March 14, 2018).

Mahto, D. and Kumari, A. (2018) Change in Mass of Spinning Black Holes w.r.t. the Angular Momentum. Journal of Modern Physics, 9, 1037-1042. https://doi.org/10.4236/jmp.2018.95065