^{1}

^{2}

^{2}

In the present research work, we have proposed a model for the change in the internal energy and enthalpy of the spinning black holes using first law of black holes in the case of spin parameter a*=0.9 and calculated their values in XRBs and AGN.

Classically, the black hole created after the death of red giant star is the perfect absorber like a black body and does not emit anything; their temperature is absolute zero. However, in quantum theory, black holes emit Hawking radiation with a perfect thermal spectrum [

In the present work, we have proposed a model for the change in the internal energy and enthalpy of the spinning black holes using first law of black holes for the case of spin parameter

The change in internal energy and enthalpy of black holes with corresponding change in the radius of the event horizon of black holes is given by [

The change in internal energy and enthalpy of spinning black holes will have different values in compared with that of non-spinning black holes, because the surface gravity of a black hole is given by the Kerr solution [

where

The surface gravity

Wang and Ding-Xiong have shown that the angular velocity

For convenience, let us assume

Putting (3) and (4) in Equation (2) and solving, we have

Putting the above value in Equation (1)

The

Mass of sun

In the present work, we have derived an expression for the change in the internal energy and enthalpy of the spinning black holes taking an account the first law of black hole mechanics for the case of spin parameter

For the angular spin

From the graph plotted between the change in internal energy and enthalpy w.r.t. the change in the event horizon and different values of mass and event horizon in XRBs and AGN, it is obvious that the change in internal energy and enthalpy w.r.t. the change in the event horizon remains the same. The change in the internal energy and enthalpy calculated for each black hole candidates in both categories of spinning black holes XRBs and AGN are exactly the same which are clear from data in

Sl. No. | Mass of black holes (M) in terms of the solar mass ( | R_{bh} = 1475 ( | |
---|---|---|---|

1 | 5 | 7375 | 1.1244 × 10^{−28} |

2 | 6 | 8850 | 1.1244 × 10^{−28} |

3 | 7 | 10,325 | 1.1244 × 10^{−28} |

4 | 8 | 11,800 | 1.1244 × 10^{−28} |

5 | 9 | 13,275 | 1.1244 × 10^{−28} |

6 | 10 | 14,750 | 1.1244 × 10^{−28} |

7 | 11 | 16,225 | 1.1244 × 10^{−28} |

8 | 12 | 17,700 | 1.1244 × 10^{−28} |

9 | 13 | 19,175 | 1.1244 × 10^{−28} |

10 | 14 | 20,650 | 1.1244 × 10^{−28} |

11 | 15 | 22,125 | 1.1244 × 10^{−28} |

12 | 16 | 23,600 | 1.1244 × 10^{−28} |

13 | 17 | 25,075 | 1.1244 × 10^{−28} |

14 | 18 | 26,550 | 1.1244 × 10^{−28} |

15 | 19 | 28,025 | 1.1244 × 10^{−28} |

16 | 20 | 29,500 | 1.1244 × 10^{−28} |

internal energy with change in mass/event horizon of the black holes.

Equation (7) shows that the change in enthalpy and internal energy with change in mass/event horizon of the black holes is directly proportional to the

Sl. No. | Mass of BHs (M) in terms of the solar mass ( | R_{bh} = 1475 × ( | |
---|---|---|---|

1 | 1 × 10^{6} | 1.475 × 10^{9 } | 1.1244 × 10^{−28} |

2 | 2 × 10^{6} | 2.950 × 10^{9} | 1.1244 × 10^{−28} |

3 | 3 × 10^{6} | 4.425 × 10^{9 } | 1.1244 × 10^{−28} |

4 | 4 × 10^{6} | 5.900 × 10^{9} | 1.1244 × 10^{−28} |

5 | 5 × 10^{6} | 7.375 × 10^{9} | 1.1244 × 10^{−28} |

6 | 6 × 10^{6} | 8.850 × 10^{9} | 1.1244 × 10^{−28} |

7 | 7 × 10^{6} | 10.320 × 10^{9} | 1.1244 × 10^{−28} |

8 | 8 × 10^{6} | 11.800 × 10^{9} | 1.1244 × 10^{−28} |

9 | 9 × 10^{6} | 13.270 × 10^{9} | 1.1244 × 10^{−28} |

10 | 1 × 10^{7} | 1.475 × 10^{10 } | 1.1244 × 10^{−28} |

11 | 2 × 10^{7} | 2.950 × 10^{10} | 1.1244 × 10^{−28} |

12 | 3 × 10^{7} | 4.425 × 10^{10 } | 1.1244 × 10^{−28} |

13 | 4 × 10^{7} | 5.900 × 10^{10} | 1.1244 × 10^{−28} |

14 | 5 × 10^{7} | 7.375 × 10^{10} | 1.1244 × 10^{−28} |

15 | 6 × 10^{7} | 8.850 × 10^{10} | 1.1244 × 10^{−28} |

16 | 7 × 10^{7} | 10.320 × 10^{10} | 1.1244 × 10^{−28} |

17 | 8 × 10^{7} | 11.800 × 10^{10} | 1.1244 × 10^{−28} |

18 | 9 × 10^{7} | 13.270 × 10^{10} | 1.1244 × 10^{−28} |

19 | 1 × 10^{8} | 1.475 × 10^{11 } | 1.1244 × 10^{−28} |

20 | 2 × 10^{8} | 2.950 × 10^{11} | 1.1244 × 10^{−28} |

21 | 3 × 10^{8} | 4.425 × 10^{11 } | 1.1244 × 10^{−28} |

22 | 4 × 10^{8} | 5.900 × 10^{11} | 1.1244 × 10^{−28} |

23 | 5 × 10^{8} | 7.375 x 10^{11} | 1.1244 × 10^{−28} |

24 | 6 × 10^{8} | 8.850 × 10^{11} | 1.1244 × 10^{−28} |

25 | 7 × 10^{8} | 10.320 × 10^{11} | 1.1244 × 10^{−28} |

26 | 8 × 10^{8} | 11.800 × 10^{11} | 1.1244 × 10^{−28} |

27 | 9 × 10^{8} | 13.270 × 10^{11} | 1.1244 × 10^{−28} |

28 | 1 × 10^{9} | 1.475 × 10^{12 } | 1.1244 × 10^{−28} |

29 | 2 × 10^{9} | 2.950 × 10^{12} | 1.1244 × 10^{−28} |

30 | 3 × 10^{9} | 4.425 × 10^{12 } | 1.1244 × 10^{−28} |

radius of the event horizon and inversely proportional to mass of the black holes. Hence these two factors mass (M) and the event horizon (R_{bh}) adjust themselves in such a way that they give the constant values for the change in energy (δH) of the black holes in each case.

1) For the angular spin

2) The change in enthalpy and internal energy calculated with the help of above equation as given in the conclusion (1) for each black hole candidate in both categories of XRBs and AGN are exactly the same, showing the constant change in enthalpy and internal energy equal to 1.1244 × 10^{−28} joule.

3) This agrees with the principle of conservation of the enthalpy and internal energy just like the principle of conservation of the energy.

4) The enthalpy and internal energy have the same role as the energy in the case of spinning black holes.

5) The enthalpy and internal energy of spinning black holes are the manifestation of the same thing.

Mahto, D., Ranjan, A. and Kumari, N. (2017) Change in Internal Energy and Enthalpy of Spinning Black Holes in XRBs and AGN Using Spin Parameter a^{*} = 0.9 International Journal of Astro- nomy and Astrophysics, 7, 38-44. https://doi.org/10.4236/ijaa.2017.71004