^{1}

^{2}

^{2}

Mahto et al. gave an expression for the change in enthalpy and internal energy with corresponding change in the radius of the event horizon of black holes of spin parameter a* = 1 (2013). On the basis of this research work, we have proposed an expression for the change in the internal energy and enthalpy of the spinning black holes to assume spin parameter a* = 1/2 and calculated their values for different test spinning black holes existing in XRBs.

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 [_{BH}) in terms of the radius of the event horizon [_{*} = 1 [

In the present paper, we have proposed an expression for the change in the internal energy and enthalpy of the spinning black holes to assume spin parameter a_{*} = 1/2 on the basis of research work [

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 [

In the case of spinning black holes, the change in internal energy will have different values than that of non-spinning black holes, because in this case, the surface gravity of a black hole is given by the Kerr solution [

where

The radius is smaller in the case of spinning black holes, tending to GM/c^{2} as a_{*} tends to 1 [_{*} = 1/2), then we have,

where

hence

or

we have

or

or

or

or

or

or

or

Putting the Equations ((9) and (14)) into (2) and solving, we have

Putting the above value in Equation (1) becomes

For negligible change in the radius of the event horizon, the radius of the event horizon (R_{bh}) can be used instead of

The above equation shows the change in enthalpy/internal energy of spinning black holes with spin 1/2 and may be used to calculate their values for given mass and corresponding radius of event horizon in XRBs.

The Equation (17) can be written as:

For the spherically symmetric and stationary or Schwarzschild black hole’s mass M, the horizon’s radius is

Using the Equation (18) into the Equation (17) and solving, we have

The standard Bekenstein entropy is given by the following expression [

Using the Equation (20) into the Equation (19) and solving, we have

For constant mass of the black hole, the change in enthalpy/internal energy is directly proportional to the surface area of event horizon (entropy) of black hole. The Equations ((19) and (21)) justify the uniform variation between the change in enthalpy/internal energy and the surface area of event horizon (entropy) of the black hole.

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 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 above regarding the mass of sun and black holes in XRBs, we have calculated change in internal energy and enthalpy of spinning black holes in XRBs for given radius of event horizon and corresponding mass of different test spinning black holes of spin parameter a_{*} = 1/2, listed in the

and then calculated their values of different test spinning black holes existing in XRBs. The above equation is extended in terms of the surface area of event horizon, A and entropy, S of the spinning black hole which shows the dependence of the change in enthalpy/internal energy on the surface area of event horizon and entropy of black hole. To understand clearly the change in enthalpy as well as change in internal energy, the graph has been plotted between the radius of event horizon and corresponding change in the enthalpy/internal energy of spinning black holes with spin (

From the observation of the data in the table, it is clear that the change in enthalpy as well as internal energy of different test spinning black holes increases with increase in corresponding change in values of the mass/radius of the event horizon in XRBs. From the observation of the graph in the

The graph plotted in _{bh}) of different test black holes and their corresponding entropy change w.r.t. the change in event horizon

Sl. No | Mass of black holes (M) | R_{bh} = 1475 (M/M_{ʘ}) (in metre) | Change in enthalpy/internal energy of black holes (dU_{bh} = dH_{bh}) |
---|---|---|---|

1 | 5M_{ʘ} | 7375 | 1.2618 × 10^{−24 } |

2 | 6M_{ʘ} | 8850 | 1.5142 × 10^{−24} |

3 | 7M_{ʘ} | 10,325 | 1.7666 × 10^{−24 } |

4 | 8M_{ʘ} | 11,800 | 2.0189 × 10^{−24 } |

5 | 9M_{ʘ} | 13,275 | 2.2713 × 10^{−24 } |

6 | 10M_{ʘ} | 14,750 | 2.5237 × 10^{−24 } |

7 | 11M_{ʘ} | 16,225 | 2.7760 × 10^{−24 } |

8 | 12M_{ʘ} | 17,700 | 3.0284 × 10^{−24 } |

9 | 13M_{ʘ} | 19,175 | 2.2808 × 10^{−24 } |

10 | 14M_{ʘ} | 20,650 | 3.5332 × 10^{−24 } |

11 | 15M_{ʘ} | 22,125 | 3.7855 × 10^{−24 } |

12 | 16M_{ʘ} | 23,600 | 4.0379 × 10^{−24 } |

13 | 17M_{ʘ} | 25,075 | 4.2903 × 10^{−24 } |

14 | 18M_{ʘ} | 26,550 | 4.5427 × 10^{−24 } |

15 | 19M_{ʘ} | 28,025 | 4.7950 × 10^{−24 } |

16 | 20M_{ʘ} | 29,500 | 5.0474 × 10^{−24 } |

(δS_{bh}/R_{b}_{h}) in XRBs using the relation of entropy change given by

In course of the present research work, we can draw the following conclusions such as:

1) The spinning black holes with spin (

2) There is no change in internal energy entropy and enthalpy in the case of spinning black holes having spin equals to (

3) For max. spin (

4) The change in entropy and enthalpy have the same nature for spinning black holes with half spin parameter.

5) There is uniform variation in the change in enthalpy (dH) and internal energy (dU) with change in surface area of event horizon for different types of spinning black hole candidates in XBRs showing definite relation between the surface area of event horizon and change in enthalpy as well as internal energy.

Dipo Mahto,Amresh Kumar Singh,Niharika Kumari, (2016) Change in Internal Energy and Enthalpy of Spinning Black Hole with Half Spin Parameter in XRBs. International Journal of Astronomy and Astrophysics,06,328-333. doi: 10.4236/ijaa.2016.63027