Energy Quantization from Reissner-Nordstrӧm Anti-de Sitter Black Hole

Energy Quantization from Reissner-Nordstrӧm Anti-de Sitter Black Hole

Jakir Hossain and M. Ilias Hossain
Department of Mathematics, Rajshahi University, Rajshahi – 6205, Bangladesh

DOI: https://doi.org/10.51584/IJRIAS.2023.8802

Received: 01 August 2023; Accepted: 06 August 2023; Published: 03 September 2023

ABSTRACT

In this work we address the study of Bekenstein-Hawking entropy of Reissner-Nordstrӧm anti-de Sitter (RNdS) black hole using energy quantization method like Bohr’s atomic model. We have quantize the energy of the test particle orbiting around RNAdS black hole and the change of entropy between two nearby states approaches to zero for large quantum numbers. 

Keywords: Energy Quantization, RNAdS black hole, Circular orbit.

INTRODUCTION 

Hawking radiation is black-body radiation that is expected to be released by black holes as a result of quantum effects near the event horizon. Hawking radiation is named after Stephen Hawking, who provided a theoretical argument for its existence in 1974, [1] and after Jacob Bekenstein in 1972, who forecasted that black holes should have a finite, non-zero temperature and entropy [2]. Hawking’s visited Moscow in 1973 where the Soviet physicists Yakov Zeldovich and Alexei Starobinsky showed him that, consistent with the quantum mechanical uncertainty principle, rotating black holes should create and emit particles [3]. Hawking radiation reduces the mass and energy of black holes and as such it is also known as black hole evaporation. Owing to this, black holes that lose more mass than they gain through other means are expected to reduce in size and ultimately disappear. Micro black holes are predicted to be larger net emitters of radiation than larger black holes and should get smaller and disappear faster. In September 2010, a signal that is closely related to black hole Hawking radiation was claimed to have been observed in a laboratory experiment involving optical light pulses. However, the results remain unverified and debatable [4, 5]. Other projects have been launched to look for this radiation within the framework of analog gravity. In June 2008, NASA launched the Fermi space telescope, which is searching for the terminal gamma-ray flashes expected from evaporating primordial black holes. In the event that speculative large extra dimension theories are correct, CERN’s Large Hadron Collider may be able to create micro black holes and observe their evaporation [6, 7, 8, 9]. In Refes. [10, 11] many research works on the thermal radiation of black holes have been made by Kraus and Wilezek [12, 13]; Parikh and Wilezek [14]; Hemming and Keski-Vakkuri [15]. It seems that an initially pure quantum state, by collapsing to a black hole and then evaporating completely. When the black hole has evaporated down to the Planck size, quantum fluctuations dominate and the semi classical calculations would no longer be valid, as spacetime is subject to violent quantum fluctuations on
this scale.

A conjectured de Sitter / conformal field theory (CFT) correspondence [16] defined in a manner analogous to the very successful AdS/CFT correspondence [17, 18] has been proposed recently that there is dual between a quantum gravity on a de Sitter space and a Euclidean conformal field on a boundary of the de Sitter space. Many works on Hawking radiation from massive uncharged particle tunneling [19, 20] and charged particle tunneling [19, 20, 21] from black hole was first proposed by Zhang and Zhao. Accomplishment this work, a few researches have been carried out as charged particle tunneling [22, 23]; Ali [24]; Wu and Jiang [23, 25] and for rotating black hole [26] and using this method Hawking radiation of Kerr-NUT black hole [27] and the charged black hole with a global monopole [28] have been developed but the quantization of black hole by the method of energy quantization developed not yet. The quantization of black hole or its gravity is very limited and therefore there is ample scope for further study. Recently we have investigated the quantization of black hole [29, 30] using the energy quantization method. We will use the same method to investigate the quantization of black hole for RNAdS black hole.

The rest of the composition is planned as follows: Firstly in section 2 we have discussed Reissner-Nordstrӧm anti-de Sitter(RNAdS) Spacetime and secondly in section 3 we have calculated the Lagrangian and canonical momentum of a test particle of the RNAdS metric according to our previous work [31]. In  Section 4, we have designed the effective possibility of radial motion using the Lagrangian and canonical momentum. Here we have also quantized the angular momentum of the test particle moving around the black hole while in section 5 and in section 6 we have established by quantizing the energy of the RNAdS black hole gravity. Finally in section 7, we present our concluding remarks.

REISSNER-NORDSTRӧM ANTI-DE SITTER SPACETIME

We have used the canonical formulation [32, 33, 34, 35] to quantize the RNAdS black hole gravity for this
section. With the negative cosmological constant ( ), the metric of the Reissner-Nordstrӧm anti-de
Sitter black hole is given by [25, 36]

where M being the mass, l is the cosmological radius , Q the total charge with respect to the static de Sitter space are defined such that  . At large , the metric (1) tends to the  space limit. It is seen that the explicit  case is obtained by setting  while the explicit Reissner-Nordstrӧm case is obtained by taking the limit . For the simplicity we can rewrite the metric given in Eq. (1) of the following form

The characteristic polynomial of RNAdS spacetime can be written as . Solving this equation and using  , since , therefore  , thus the position of the black  hole horizon [25] can be obtained as

For minisuper space  and neglecting higher power of , the first approximated position of horizon can be simplified from Eq. (3) of the following form

Using Eq. (4), into Eq. (2), the metric equation can be taken of the following form

If we let , then the above equation becomes as

Now let us consider a test particle of mass m orbiting along the circular geodesics in the equatorial plane  around RNAdS black hole. Thus the above Eq. (7) as described in [37, 38] can becomes in the following form

LAGRANGIAN AND CANONICAL MOMENTUM

Now we assume that the black hole mass is larger than the Planck mass so that the Compton radius , where rh is the radius of the RNAdS black hole. In this circumstances the quantum fluctuations of the black hole disregards [39]. In terms of the metric components , the Lagrangian of the test particle can be written as

For static and spherically charged RNAdS black hole, there exist two constants of motion for the test particles, associated with two Killing vectors in terms of energy E, angular momentum L and the canonical momenta components ,  can be written in the following form

RADIAL MOTION AND EFFECTIVE POTENTIAL FOR RNADS LINE ELEMENT

The radial motion of a geodesic can be written as

the four vectors  can be expressed in the magnitude of the energy-momentum. Inserting Eqs. (9)-(10) and Eq. (7) into Eq. (11), we have

In Refs. [31, 40], the energy and momentum of the test particle per unit rest mass  are respectively and setting these are into Eq. (13), we have

For the time-like particle orbit the velocity is expressed in terms of two parameters energy and angular momentum. Thus we have the effective potential  for the radial motion from Eq. (15) of the following form

The radial acceleration of the test particle can be obtained by taking the derivative of Eq. (16) with respect to the proper time and then equating to zero, we obtain

Now our prime task in this section is to find the radius of the circular orbit namely . For this purpose, we can be written the above equation of the following form

We observe that the roots of the above equation is real only when and for the smallest stable orbit the square root on the right-hand side of Eq. (20) dissolves. Therefore, we must have

For large and largest stable circular orbits the conditions and are holds respectively.

QUANTIZATION OF ANGULAR MOMENTUM OF THE TEST PARTICLE

According to Wilson [41] and Sommerfeld [42] idea, angular momentum can be quantized as a periodic function of time and with the help of quantize energy for the reason that it is closely related to the quantize angular momentum of the orbiting test particle. In order to quantize angular momentum  with the help of  canonical momentum L conjugate to the angular variable of the form

 Since L is a constant of motion, thus   gives the quantization condition for the angular momentum of the following form

With the help of compton radius  and equation Eq. (23), the Eq. (21) can be written as

By using Eq. (24) into Eq. (20), we get the radius of the different stable circular orbit of the particle corresponds to of the form

In the limiting case  when which agrees with the result given in Ref. [40], where rs=2M  represents the Schwarzschild radius. The position of the next higher circular orbit  can be obtained from Eq. (25) by replacing n0 with n1 = n0+1 of the form

By using the Eq. (24) into the right side of the above equation can be approximated of the form

Progressing in this way, the radius of the next higher circular orbit  of the test particle with the help of Eq. (32) can be obtained from Eq. (25) of the form

Here we have, . For large quantum number we can say that the two nearby states coincide for RNAdS black hole.

ENERGY QUANTIZATION FOR RNADS BLACK HOLE

With the help of angular momentum we want to quantize the energy of the orbiting test particle in this section. For zero velocity at r = R the Eq. (15) gives as

Therefore, Eq. (37) becomes as

Using Eq. (23) into Eq.(26) we have

From the Eq. (45), we obtains the corresponding th label energy of the form

When , we observe that the radius of the RNAdS black hole approaches to the radius of the RNdS black hole and which has been developed in [30]. For RNAdS black hole, the Eq. (47) can be written as

We have which indicates for a larger circular orbit the change of energy between two nearby states approaches to zero.

CONCLUDING REMARKS

In this paper, we have presented the change of entropy for two nearby circular orbit around Reissner-Nordstrӧm Anti-de Sitter Black Hole by Energy Quantization process. We have also found the different energy labels of RNAdS black hole in nature can be performed in the same way as that for the electron signal inside the atom like Bohr’s quantum theory and leads to the results on quantization of black hole [43]. For further study, we may extend the quantum relation of Reissner-Nordstrӧm Anti-de Sitter black hole and their temperature can be evaluated in agreement with the Hawking temperature. The results we have obtained agree with the results of RN black hole [29] when cosmological constant goes to infinity. Again when and the RNdS black hole reduced to Schwarzschild black hole [40].

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