Relativistic Variation of Black Hole Temperature with Respect to Velocity in XRBs and AGN

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1. Introduction

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 [1] [2] . But according to general theory of relativity, a black hole is a solution of Einstein’s gravitational field equations in the absence of matter that describes the space time around a gravitationally collapsed star and its gravitational pull is so strong that even light cannot escape from it [3] .

In 1974, Hawking discovered black hole evaporation. Quantum fields on a black hole background space-time radiate thermal spectrum of particles, with a temperature (), where is the surface gravity of the horizon [1] . Hawking (1974, 1975) introduced what is now called Hawking radiation as the effective black body radiation from a black hole in terms of the 4th power of the black hole temperature and the Stefan-Boltzmann constant [4] [5] . Stephan Hawking provided a theoretical argument for its existence in 1974, and sometimes also after the physicist Jacob Bekenstein who predicted that black holes should have a finite, non-zero temperature and entropy and also stated that it is proportional to the black hole area A [6] . Bardeen, Carter and Hawking performed calculations using a semi-classical approximation, putting Bekenstein conjecture on a firm basis. They established that the black hole temperature is proportional to its surface gravity [7] . The surface gravity has the same role in black hole thermodynamics as the temperature in the ordinary thermodynamics. Silva proposed intuitive derivations of the Hawking temperature and the Bekenstein-Hawking entropy of a Schwarzschild black hole [8] . Ved Prakash et al. have discussed the statistical analysis of lifetime and temperature of the black holes existing in X-ray binaries and active galactic nuclei [9] . Mehta et al. derived an expression for the variation of temperature of the black holes with respect to mass and also calculated their values of different test black holes existing only in X-ray binaries [10] . Mahto et al. converted the Hawking temperature in terms of Chandrasekhar limit (M_{ch}) and calculated their values for different test black holes in XRBs and AGN [11] . In the present research work, we have applied the variation of mass with velocity as proposed by Albert Einstein to the Hawking temperature (), where is the reduced Planck constant, c is the speed of light, is the Boltzmann constant, G is the gravitational constant, and M is the mass of the black hole, to obtain the rate of change in temperature with respect to velocity and also calculated their values for super dense stars like black holes existing in XRBs & AGN.

2. Method

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 [12] as given below.

(1)

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:

(2)

With using in natural units, Equation (2) immediately implies that a Schwarzschild black hole in isolation is unstable: it will radiate and so doing loss energy, hence the mass decreases, thus increasing the temperature causing it to radiate with more power leading to runway effect [13] .

Equation (2) can be written as

(3)

Some of the black holes have their spinning velocity from 50% to 99% of the velocity of light [14] and hence the mass of black holes do not remain constant, but the mass will vary with velocity as proposed by Albert Einstein’s special theory of relativity as [15] :

(4)

where is the rest mass and v be the spinning velocity of black holes.

or, (5)

(6)

Since, , hence

, , , and so on (7)

Hence, it is clear that the terms of higher power of in Equation (6) can be neglected and finally, we have

(8)

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

(9)

or

(10)

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 can be neglected and we have

(11)

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, in natural units.

(12)

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.

(13)

or,

(14)

so that

(15)

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.

3. Data in Support of Spinning Velocity of Black Holes

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 [14] . From above it is clear that the super massive black holes existing in AGN have spinning rate from about 50% to 99% of the velocity of light and the massive black holes existing in XRBs may have spinning rate from about 1% to 99% of the velocity of light.

4. Data in Support of Mass of Black Holes

5. Results and Discussion

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, 10, 15, 20 (as mentioned in the Table 1) and super massive, 10^{6}, 10^{7}, 10^{8}, 10^{9}, (as mentioned in the Table 2) black holes in XRBs and AGN respectively and plotted the graphs as in Figures 1-4 with the help of Table 1 and Figures 5-8

Table 1. The rate of change of temperature of black holes w.r.t. spinning velocity in XRBs.

Table 2. The rate of change of temperature of black holes w.r.t. spinning velocity in AGN.

Figure 1. The graph plotted 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 mass M = 5 in XRBs.

Figure 2. The graph plotted 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 mass M = 10 in XRBs.

Figure 3. The graph plotted 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 mass M = 15 in XRBs.

Figure 4. The graph plotted 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 mass M = 20 in XRBs.

Figure 5. The graph plotted 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 mass M = 10^{6} in AGN.

Figure 6. The graph plotted 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 mass M = 10^{7} in AGN.

Figure 7. The graph plotted 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 mass M = 10^{8} in AGN.

Figure 8. The graph plotted 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 mass M = 10^{9} in AGN.

with the help of Table 2. When we observe the graphs plotted in Figures 1-8, we see that all the graphs are straight lines which show that there are uniform variation of the rate of change of temperature of black holes w.r.t. spinning velocity () with corresponding values of spinning velocity of black holes. The straight line also shows that there is a definite relation between the rates of change of temperature of black holes w.r.t. spinning velocity () with corresponding values of spinning velocity of black holes. We also observed that the rate of change of temperature of black holes w.r.t. spinning velocity () with corresponding values of spinning velocity of black holes for each case of the black holes in XRBs (Table 1) starts from the origin, while in the each case of AGN (Table 2), it differs to that of XRBs, because it does not start from origin. This shows that the rate of change of temperature of black holes w.r.t. spinning velocity () with corresponding values of spinning velocity of black holes for each case of the black holes in XRBs starts from the beginning to last, but in AGN, it starts with some fixed values of the rate of change of temperature of black holes so that their gradient will differ for the both cases.

6. Conclusions

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.

Acknowledgements

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.

References

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