^{1}

^{2}

^{1}

Stephen Hawking gave a formula for the temperature of black holes as given by . Some of the black holes have their spinning velocity from 50% to 99% of the velocity of light. Due to this velocity, the mass of black holes will vary which cause the variation in the temperature of black holes. In the present research article, we have applied the variation of mass with velocity to obtain the rate of change in temperature of the black holes with respect to velocity. We have also calculated their values for super dense stars like black holes existing in XRBs and AGN and concluded that for super dense stars like black holes of lower velocity as well as the velocity comparable to the velocity of light, the rate of change in temperature with respect to velocity is directly proportional to their velocities. This work will help us to find out the variation in temperature of different black holes spinning with different velocity percentage related to light speed and can be used as the references for other research works.

In classical theory, black holes can only absorb and not emit particles. However, it is shown that quantum mechanical effects cause black holes to create and emit particles like a black body [

In 1974, Hawking discovered black hole evaporation. Quantum fields on a black hole background space-time radiate thermal spectrum of particles, with a temperature (_{ch}) and calculated their values for different test black holes in XRBs and AGN [

Zee presented a nice intuitive derivation of the Hawking temperature on the basis of Schwarzschild solution of Einstein equations for vacuum in his book entitled: Quantum field Theory in a Nutshell [

The above equation for the temperature of black holes was Stephan Hawking’s discovery in 1974, showing that the temperature associated with black holes is inversely proportional to their mass as follows:

With using

Equation (2) can be written as

Some of the black holes have their spinning velocity from 50% to 99% of the velocity of light [

where

or,

Since,

Hence, it is clear that the terms of higher power of

Putting the value of M in Equation (3), we have

or

The terms containing Equation (10) have positive and negative contributions, so the resultant value of the terms in big bracket will be negligible. Also applying the condition of Equation (7), the higher power of

Equation (11) may be designated as Relativistic Hawking temperature of the black holes which is less than Hawking temperature of the black holes. This means that the temperature of black hole decreases with the increase the velocity of black holes. The simplest form of Equation (11) may be written to assume for convenience

The relativistic temperature of different black holes can be calculated with the help of Equation (12) for known rest mass and spinning velocity of the black holes. To obtain the rate of variation of the temperature of the black holes relative to velocity, Equation (12) is differentiated with respect to v.

or,

so that

The above equation shows that for super dense stars like black holes of lower velocity as well as the velocity comparable to the velocity of light, the magnitude of rate of change in temperature with respect to velocity is directly proportional to spinning velocity of black hole.

E.S. Reich demonstrated graphically in his research work that the spinning rate of the super massive black holes begin from about 50% of the speed of light to 99% of the speed of light and there are some super massive black holes spinning at more than 90% of the speed of light. The graph also shows that no super massive black holes spin at rate below than 40% of the speed of light [

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 ^{6} - 10^{9.5} ^{6} ^{9.5}

A black hole has a temperature which is inversely proportional to the mass of black holes as per Hawking temperature formula. A body having mass with a finite temperature radiates energy. Anything that radiates energy is also losing mass according to Einstein’s mass-energy equivalence relation (E = mc^{2}). Looking at the equation, we can see that as the black hole loses mass, the emission of energy from the black hole increases and its temperature increases, and thus the rate of mass loss increases.

In the present work, we have applied the variation of mass with velocity to the Hawking temperature and derived an expression for the relativistic Hawking temperature given by Equation (12). With the help of Equation (12), the rate of change in temperature of black holes with respect to the spinning velocity is obtained by Equation (13). We also calculated their values with the help of Equation (13) for different masses of different massive, 5^{6}, 10^{7}, 10^{8}, 10^{9}, (as mentioned in the

S. No | % Spinning velocity of black holes of velocity of light | Spinning velocity of black holes of velocity of light (m/s) | ||||
---|---|---|---|---|---|---|

For M = 5 | For M = 10 | For M = 15 | For M = 20 | |||

1 | 5% | 0.15 × 108 | 0.60012 × 10^{−25} | 0.30006 × 10^{−25} | 0.20004 × 10^{−25} | 0.15003 × 10^{−25} |

2 | 10% | 0.30 × 108 | 1.20024 × 10^{−25} | 0.60012 × 10^{−25} | 0.40006 × 10^{−25} | 0.30006 × 10^{−25} |

3 | 20% | 0.60 × 108 | 2.40048 × 10^{−25} | 1.20024 × 10^{−25} | 0.80016 × 10^{−25} | 0.60012 × 10^{−25} |

4 | 30% | 0.90 × 108 | 3.60072 × 10^{−25} | 1.80036 × 10^{−25} | 1.20024 × 10^{−25} | 0.90018 × 10^{−25} |

5 | 40% | 1.20 × 108 | 4.80096 × 10^{−25} | 2.40048 × 10^{−25} | 1.60032 × 10^{−25} | 1.20024 × 10^{−25} |

6 | 50% | 1.50 × 108 | 6.00120 × 10^{−25} | 3.30006 × 10^{−25} | 2.200060 × 10^{−25} | 1.50030 × 10^{−25} |

7 | 60% | 1.80 × 108 | 7.20144 × 10^{−25} | 3.60072 × 10^{−25} | 2.40048 × 10^{−25} | 1.80036 × 10^{−25} |

8 | 70% | 2.10 × 108 | 8.40168 × 10^{−25} | 4.20084 × 10^{−25} | 2.80038 × 10^{−25} | 2.10042 × 10^{−25} |

9 | 80% | 2.40 × 108 | 9.60192 × 10^{−25} | 4.80096 × 10^{−25} | 3.20064 × 10^{−25} | 2.40048 × 10^{−25} |

10 | 90% | 2.70 × 108 | 10.80216 × 10^{−25} | 5.40108 × 10^{−25} | 3.60072 × 10^{−25} | 2.70054 × 10^{−25} |

11 | 99% | 2.97 × 108 | 12.00236 × 10^{−25} | 6.00120 × 10^{−25} | 4.30080 × 10^{−25} | 3.00060 × 10^{−25} |

S. No | % Spinning velocity of black holes of velocity of light | Spinning velocity of black holes of velocity of light (m/s) | ||||
---|---|---|---|---|---|---|

For M = 106 | For M = 107 | For M = 108 | For M = 109 | |||

1 | 50% | 1.50 × 108 | 3.000067 × 10^{−30} | 3.000067 × 10^{−31} | 3.000067 × 10^{−32} | 3.000067 × 10^{−33} |

2 | 55% | 1.65 × 108 | 3.300739 × 10^{−30} | 3.300739 × 10^{−31} | 3.300739 × 10^{−32} | 3.300739 × 10^{−33} |

3 | 60% | 1.80 × 108 | 3.600806 × 10^{−30} | 3.600806 × 10^{−31} | 3.600806 × 10^{−32} | 3.600806 × 10^{−33} |

4 | 65% | 1.95 × 108 | 3.900873 × 10^{−30} | 3.900873 × 10^{−31} | 3.900873 × 10^{−32} | 3.900873 × 10^{−33} |

5 | 70% | 2.10 × 108 | 4.200941 × 10^{−30} | 4.200941 × 10^{−31} | 4.200941 × 10^{−32} | 4.200941 × 10^{−33} |

6 | 75% | 2.25 × 108 | 4.501008 × 10^{−30} | 4.501008 × 10^{−31} | 4.501008 × 10^{−32} | 4.501008 × 10^{−33} |

7 | 80% | 2.40 × 108 | 4.801075 × 10^{−30} | 4.801075 × 10^{−31} | 4.801075 × 10^{−32} | 4.801075 × 10^{−33} |

8 | 85% | 2.55 × 108 | 5.101142 × 10^{−30} | 5.101142 × 10^{−31} | 5.101142 × 10^{−32} | 5.101142 × 10^{−33} |

9 | 90% | 2.70 × 108 | 5.401209 × 10^{−30} | 5.401209 × 10^{−31} | 5.401209 × 10^{−32} | 5.401209 × 10^{−33} |

10 | 95% | 2.85 × 108 | 5.701277 × 10^{−30} | 5.701277 × 10^{−31} | 5.701277 × 10^{−32} | 5.701277 × 10^{−33} |

11 | 99% | 2.97 × 108 | 5.941330 × 10^{−30} | 5.941330 × 10^{−31} | 5.941330 × 10^{−32} | 5.941330 × 10^{−33} |

with the help of

In the study of present research paper, we conclude that:

1) The relativistic Hawking temperature of the black hole is less than to that of Hawking temperature.

2) The black hole of lower velocity as well as the velocity comparable to the velocity of light, the rate of change in temperature with respect to velocity is directly proportional to its spinning velocity.

3) All the graphs between % spinning velocity of black holes of velocity of light and the rate of change of temperature of black holes w.r.t. spinning velocity for different masses of black holes either for XRBs or AGN are in straight line showing that there is a definite relation and uniform variation between the rate of change in temperature with respect to velocity and spinning velocity.

4) The straight line graph gives the validity of relation between the rate of change in temperature with respect to velocity and its spinning velocity.

Authors are highly grateful and obliged to the reviewers specially Najat M.R. Al- Ubaidi who made some excellent unbelievable corrections and editors for pointing out the technical errors in the original manuscript and providing valuable suggestions to make it better.

Mahto, D., Ranjan, A. and Singh, K.M. (2017) Relativistic Variation of Black Hole Temperature with Re- spect to Velocity in XRBs and AGN. International Journal of Astronomy and Astrophysics, 7, 1-10. https://doi.org/10.4236/ijaa.2017.71001