*a*and

*k*, and decreases with the increase in electric

*e*and magnetic charges

*g*.

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- Chinese Physics C
- Vol. 44, Issue 1, (2020)

Abstract

Keywords

1. Introduction

A black hole (BH) is a physical object that has the ability to absorb all types of energy from its surroundings as a result of its strong gravitational pull. According to the theory of general relativity, a BH attracts all types of particles that interact with the event horizon. Hawking (1974) described that a BH acts as a black body and emits particles in the form of radiation through its horizon Considering quantum effects in the background of curved spacetime, this radiation is known as Hawking radiation [

According to the tunneling phenomenon, particles are permitted to follow classically forbidden trajectories, by starting from outside the horizon to infinity [

To calculate the imaginary part of the classical action, there are two main approaches, i.e., null geodesic technique and Hamilton-Jacobi strategy. The first one proposed by Parikh and his colleagues [

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By incorporating generalized uncertainty principle (GUP) effects, it is conceivable to discuss quantum-corrected thermodynamical properties of BH [

where

Javed et al. [

2. Pair of charged accelerated black holes involving rotation and nut parameters

Universally, the NUT parameter is associated with the twisting behavior of the surrounding spacetime or with the gravito-magnetic monopole parameter of the central mass. The precise physical significance of the NUT parameter might not be investigated. In contrast, the higher dimensional origin of the Kerr-NUT-(anti) de-Sitter BH and its physical significance was studied [

The exact understanding of the NUT parameter is conceivable when a static Schwarzschild mass is submerged in a stationary source-free electromagnetic universe [

The line-element of this BH can be defined as follows [

where

where

We observe that
*C*-metric is implied.

The line-element (2) can be rewritten in the following form

where the metric functions *Z*, *B*, *C*, *D,* and *F* are given as follows

The electromagnetic potential for this pair of BHs is defined as

The horizons are obtained from
*r* has real roots, i.e.,

Here,

The angular velocity of BH at outer horizon is given as follows

To study the corrected tunneling rate for vector bosons through the pair of BHs horizon, we consider the Lagrangian equation with quantum gravity effects. Considering a spacetime with a potential due to both the electric and magnetic field, the behavior of massive (spin-1) vector is described by the wave equation. The GUP modified Lagrangian equation with vector field

The modified wave equation for massive vector bosons can be defined as follows

where

and

where

Applying the WKB approximation [

where

and (for

Using the variables separation technique, the particle's action is defined as follows

where *J* and *E* denote the angular momentum and particles energy, respectively, and from the above Eqs. (26)–(29), we obtain a non-trivial matrix equation, i.e.,

where

where

where

and

After applying Taylor's series, the functions

Using the above relations in Eq. (48), we consider that the leading wave equation has two poles at

where

The surface gravity

The corrected tunneling probability

We calculate the corrected Hawking temperature by comparing the tunneling probability with the Boltzmann factor, i.e.,

by considering only first order quantum corrections, we can write

where the semi-classical Hawking temperature

The corrected tunneling probability depends on
*E*,

The corrected temperature of vector bosons given in Eq. (56) reduced to the temperature of fermion particles in Eq. (4.20) for

where

For

Here, it is important to mention that the value of the corrected Hawking temperature is smaller than the original temperature, and BH stops radiating when the mass of the BH reaches its minimal value

2.1. Graphical analysis of

This subsection is devoted to the study of the graphical behavior of Hawking temperature

Figure 1.(color online)

Figure 2.(color online)

Figure 3.(color online)

3. Conclusion and discussion

In this study, we have investigated the quantum gravity effects for vector bosons from charged accelerating rotating BH with the NUT parameter. By considering the GUP effects, we employed the modified Lagrangian equation incorporating quantum effects describing the motion of spin-1 particles. Subsequently, by applying the Hamilton-Jacobi technique, we have calculated the tunneling probabilities of vector bosons. Moreover, we analyzed the corrected Hawking temperatures of these BHs. We concluded that the modified tunneling probabilities are not just dependent on the BHs properties, but also on the properties of emitted vector bosons, i.e., energy, potential, surface gravity, particle charge, and total angular momentum. Moreover, it is important to note that the modified tunneling probabilities as well as the Hawking temperature depend on the quantum particles, which contribute gravitational radiation in form of massive particle (BH's energy carrier) tunneling.

When the quantum gravity effects are neglected, i.e.,

In our analysis, we have found that the quantum corrections decelerate the increase in temperature during the radiation process. This correction causes the radiation to cease at some specific temperature, leaving the remnant mass. The remnant mass is obtained at the specific condition

Here, it is important to mention that the value of the corrected Hawking temperature is smaller than the original Hawking temperature, and the BH stops radiating, when its mass reaches the minimal value

The results from the graphical analysis of corrected Hawking temperatures with respect to the horizon for the given BH are summarized as follows:

● For accelerating and rotating BH with NUT parameter, the

● Corrected temperature

● The

● The

● In our analysis, we considered the value

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Wajiha Javed, Riasat Ali, Rimsha Babar, Ali Övgün. Tunneling of massive vector particles under influence of quantum gravity *[J]. Chinese Physics C, 2020, 44(1):

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