† Corresponding author. E-mail:
The surface gravity of Schwarzschild black hole can be quantized from the test particle moving around different energy states analog to the Bohr’s atomic model. We have quantized the Hawking temperature and entropy of Schwarzschild black hole from quantization of surface gravity. We also have shown that the change of entropy reduces to zero when the boundary shrinks to very small size.
Black holes are one of the most fascinating objects in physics, which have been concerned to exist in our Universe. There are good observational evidences that black holes exist on scales from a solar mass in stellar binary systems up to million of solar masses in the center of galaxies and quasars. The theoretical studies of such objects, in particular the effects on their environment are of great importance to black hole candidate. In recent years, considerable attention has been concentrated on the study of the quantization of black holes. Black holes, however, are not objects of direct observing and so the quantization of black holes have remained esoteric, and there still exists no satisfactory solution yet. The quantization of black hole can be performed by considering the way of electron’s motion inside the atom from quantum theory like Bohr’s atomic model. It is therefore becomes one of the most important issues in physics.[1]
The unification of the general relativity (GR) with the quantum mechanics (QM) is one of the unsolved problems of the theoretical physics. The concern of the general relativity and gravitation depends on large scale structure in a fully classical ambit but the concern of QM depends on the small atomic or sub-atomic scale and remarkable understanding of the fundamental interactions.[2–4] The incongruity between the two theories comes from the couple of the quantum mechanics[3] with the classical one in modern physics in which the general relativity is embedded. For very small scale quantum properties of vacuum break the scale invariance of the classical approach. For the Planck-scale black hole quantum property of vacuum can be prevented by quantum effect due to smallest mass forming a black hole. Recently, Chiarelli and Cerruti et al. have shown the way in passing from quantum to classical mechanics which is the decoherence of QM induced by fluctuations.[5–7] In this approach, the vacuum fluctuations is considered as never end and there is a certain ground state in which the mass of the black hole is non-zero produce the quantum decoherence, breaking the QM on large scale.[8–9]
The quantization of the horizon area of black holes was first discovered by Bekenstein.[10–12] The horizon area of a nonextremal black hole was discovered by Chirstodoulou et al.[13–14] Analyzing this work Bekenstein pointed out that reversible transformation of such type of horizon have an adiabatic nature. Of course, in accordance with the corresponding principle the quantization of an adiabatic invariant is perfectly natural. Later, Mukhanov[15] and Kogan[16] have discussed the quantization of black hole. Specially, Kogan was initiated this problem within string theory.
On the other side, the entropic framework developed by Verlinde[17] and He and Ma,[18] propose new way for quantizing gravity beyond classical physics. The quantization of gravity can be used to quantize the black hole. Within the framework of general relativity, gravitoelectromagnetism (GEM) has been discussed by many authors[19–34] in which the electric charge and the electric field of Maxwell electromagnetic theory play the role of the mass of the test particle and the gravitational acceleration, respectively. Following GEM it is possible to split the upper bound of energy without the quantization effects on energy level splitting in atoms and molecules[35] due to the hypothetical nature of the gravitipole. Considering GEM a new method has recently been developed in which the black hole energy can be quantized from the test particle moving around different circular orbits,[36–37] which is analog to the Bohr’s atomic model.
In this research, we have study Hawking radiation and Hawking temperature from quantization of surface gravity rather than using gravitational energy quantization method. The work studied here has the intention to contribute the development of the quantization of black hole by using thermodynamics law
The remainder of this paper is as follows: In Sec.
The line element of the Schwarzschild metric in terms of a spherical coordinates
The time-like geodesic equation corresponding to metric (
We consider a test particle of mass m orbiting around the black hole at the radius
The quantization formula of surface gravity can be performed from Eq. (
The Hawking temperature for each of the energy state can be derived from dividing both sides of Eq. (
In this section, we quantize the entropy from quantization of surface gravity of Schwarzschild black hole. The quantize formula of the surface area for the higher states can be written with the help of Eq. (
The main task of the work is to develop entropy quantize formula. We consider the initial mass and radius of the Schwarzschild black hole as M0 and
If A0 denotes the surface area of the ground state then we have
In Ref. [36], we have developed the hawking radiation from energy quantized method. In this research, the same results have been shown by using quantization of surface gravity rather than energy and is more interesting and relevant in quantum theory. Our present work stringily supports the results of Bekenstein[10] that a black hole entropy is proportional to the horizon area of black hole. The present work shows that the different energy labels of black hole in the nature can be performed in the same way as that for the electron occupy different energy labels outside the atom like quantum theory,[41] however, it suggests a new idea to unify gravity with quantum theory.
We have shown some interesting properties of black holes from quantization of entropy, which was suggested in a previous work to unify the black hole entropy formula as entropic framework.[17,45] It indeed offers new perspectives on quantum properties of gravity. This suggests a way to unify gravity with quantum theory.
[1] | |
[2] | |
[3] | |
[4] | |
[5] | |
[6] | |
[7] | |
[8] | |
[9] | |
[10] | |
[11] | |
[12] | |
[13] | |
[14] | |
[15] | |
[16] | |
[17] | |
[18] | |
[19] | |
[20] | |
[21] | |
[22] | |
[23] | |
[24] | |
[25] | |
[26] | |
[27] | |
[28] | |
[29] | |
[30] | |
[31] | |
[32] | |
[33] | |
[34] | |
[35] | |
[36] | |
[37] | |
[38] | |
[39] | |
[40] | |
[41] | |
[42] | |
[43] | |
[44] | |
[45] |