# Quantum gravity effect on the Hawking radiation of spinning dilaton black hole

## Abstract

The quantum gravity correction to the Hawking temperature of the 2+1 dimensional spinning dilaton black hole is studied by using the Hamilton-Jacobi approach in the context of the Generalized Uncertainty Principle (GUP). It is observed that the modified Hawking temperature of the black hole depends on both black hole and the tunnelling particle properties. Moreover, it is observed that the mass and the angular momentum of the scalar particle have the same effect on the Hawking temperature of the black hole, while the mass and total angular momentum (orbital+spin) of Dirac particle have different effect. Furthermore, the mass and total angular momentum (orbital+spin) of vector boson particle have a similar effect that of Dirac particle. Also, thermodynamical stability and phase transition of the black hole are discussed for scalar, Dirac and vector boson in the context of GUP, respectively. And, it is observed that the scalar particle probes the black hole as stable whereas, as for Dirac and vector boson particles, it might undergoes second-type phase transition to become stable while in the absence of the quantum gravity effect all of these particle probes the black hole as stable.

## 1 Introduction

The establishment of a self-consistent quantum version of gravity is one of the most important problems of modern physics and is still unsolved despite many important attempts. However, the formulation of Quantum Field Theory (QFT) in curved spacetime provide us some important clues about a self-consistent quantum gravity. For instance, the particle creation and the thermal radiation of a black hole are the most spectacular of these clues. In this connection, Hawking, using the QFT in curved spacetime, proved that a black hole can emit particles formed by the quantum fluctuation near its event horizon [1, 2, 3]. Nowadays, various methods have been proposed to calculate the temperature of a black hole known as Hawking temperature. On the other hand, the Hamilton-Jacobi approach is an important version of the tunnelling method of the quantum mechanical point-like particles from a black hole [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. Adopting this approach, Hawking temperature of various black holes were recovered by quantum tunnelling method of the particles across their event horizons. In all these studies, it is seen that the standard Hawking temperature is independent of the properties of a tunneling particle.

Besides the QFT in curved spacetime, there are some important candidate theories of quantum gravity such as the string theory and loop quantum gravity theory [21, 22]. In these theories, unlike QFT, the elementary particles are no longer point-like. Accordingly, it should be a minimal length in order of Planck scale. Due to this new interpretation of the elementary particles, the standard Heisenberg uncertainty principle is modified as generalized uncertainty principle (GUP) [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. Moreover, the relativistic quantum mechanical wave equations such as Klein-Gordon, Dirac and vector boson equations are modified in the context of the GUP. Using these modified equations, Hawking temperature of many black holes was recalculated in the context of Hamilton-Jacobi approach. It was observed that the standard Hawking temperature is modified and it no longer depends only on the black hole but also on the properties of the tunnelling particle [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45]. It is also stated that the tunnelling probability of the particles from a black hole is completely different from each other and thus leads to completely different Hawking temperatures [40, 41, 42, 43, 44]. Moreover, the stability of the (2+1)-dimensional charged rotating Banados-Teitelboim-Zanelli (CR-BTZ) black hole is investigated by using the modified Hawking temperature. And it points out that the black hole may undergo both first and second-type of phase transitions in the context of GUP whereas it undergoes only first phase transition in the absence of GUP [44]. With this motivation, we will investigate the GUP effect on the tunnelling probability of the scalar, Dirac and vector boson particles, respectively, and subsequently on the Hawking temperature of the 2+1-dimensional spinning dilaton black hole. Moreover, using the modified Hawking temperatures, we will analyze thermal stability condition of the black hole in the presence of quantum gravity effect for all of three type particles, respectively.

The paper is organized as follows: In Sect. 2, we introduce the 2+1 dimensional spinning dilaton black hole. In Sect. 3, we investigate the GUP effect on the tunnelling progress of the scalar particle from the black hole, and then, calculate the modified Hawking temperature. In Sects. 4 and 5, we repeat the same procedure for Dirac and vector boson particles, respectively, as we perform for the scalar particle in Sect. 3. In Sects. 6, the modified heat capacity of the black hole is calculated by using the modified Hawking temperatures, and subsequently, stability/instability and phase transition of the black hole are discussed. In the conclusion, the results are summarized.

## 2 2+1 dimensional spinning dilaton black hole

*f*(

*r*),

*H*(

*r*) and

*R*(

*r*) are;

*M*and

*J*are the cosmological constant, the mass and angular momentum of the black hole, respectively [47]. The angular velocity of the horizon is determined as follows:

## 3 Tunneling of scalar particle from spinning dilaton black hole

*C*is a complex constant, and

*E*,

*j*and

*W*(

*r*)=\(W_{0}(r)+\alpha W_{1}(r)\) are the particle’s energy, angular momentum, and radial trajectory, respectively. After some calculations, the radial trajectory of the tunneling scalar particle \(W_{\pm }(r)\) is written as

*E*is total energy of the tunnelling particle, and \(T_{H}\) is Hawking temperature. Then, the modified Hawking temperature of the scalar particle, \(T_{H}^{KG}\), is obtained as follows

## 4 Tunneling of massive Dirac particle from spinning dilaton black hole

*r*. To proceed the tunneling probability of a massive Dirac particle from the black hole, we use the following ansatz for the modified wave function;

## 5 Tunneling of massive vector boson from spinning dilaton black hole

## 6 Quantum gravity correction to the Stability of the black hole

*J*, in term of mass,

*M*, and modified Hawking temperature, \(T_{H}\), of the black hole is given as follows:

*j*, \(\hbar \),

*J*, and

*m*. According to Eq. (35), the modified heat capacity of the spinning dilatonic black hole is always positive, hence, the black hole always stable according to the tunnelling process of a scalar particle in the context of GUP (red line in Fig. 1). In this case, there is no any phase transition. On the other hand, according to the tunnelling process of Dirac particle, the modified heat capacity given in Eq. (36) is negative for the region \(0<r_{h}<0.2\) while it is positive for the region \(0.2<r_{h}\). Therefore, the black hole is unstable in the region \(0<r_{h}<0.2\) while it is stable in the region \(0.2<r_{h}\). The modified heat capacity diverges at point \(r_{h}=0.2\), hence this point corresponds to the phase transition point known as second-type phase transition (red line in Fig. 2). Similarly, in view of the tunneling process of vector boson particle, the modified heat capacity given in Eq. (37) is negative for the region \(0<r_{h}<1.13\) while it is positive for the region \(1.13<r_{h}\) (red line in Fig. 3). Therefore, the stability/instability and phase transition situations of the black hole according to the tunnelling of vector boson particle are similar that of the Dirac particle.

## 7 Concluding remarks

In the absence of GUP effect, Hawking temperatures of the three different types of particles are the same and depend only on the properties of the black hole, i.e. they are not related to the particles properties.

In contrast to the standard results, our results demonstrate that the modified Hawking temperature depends not only on the black hole properties but also on GUP parameter, \(\alpha \), and hence on the properties of the tunneling particles. Furthermore, it is observed that the modified Hawking temperature is lower than that of the standard one.

In the presence of the GUP effect, tunnelling processes of the three different particles are completely different from each other, and hence their Hawking temperatures are completely different, as well.

According to Eq. (15), the modified Hawking temperature of the black hole is decreases via angular momentum of the scalar particle. However, according to Eq. (24), the total angular momentum (orbital+spin) of Dirac particle has an increasing effect on the Hawking temperature of the black hole. Moreover, the total angular momentum (orbital+spin) of the vector boson particle has a similar impact that of Dirac particle (see Eq. (33)). This case indicates that the total angular momentum (orbital+spin)-spacetime geometry interaction depends on the particle type in the presence of quantum gravity correction term. On the other hand, in the absence of this term, all types of particles interact with spacetime geometry in the same way.

In the case of the all three particles, the modified Hawking temperature decreases with mass of the tunneling particle (see Eqs. (15), (24) and (33)).

Thermodynamical local stability of the black hole is analyzed by using the modified Hawking temperatures of scalar, Dirac and vector boson particles for special values of \(\varLambda \), \(\alpha \),

*j*, \(\hbar \),*J*, and*m*.The black hole is always locally stable according to the scalar particle tunnelling in the presence of quantum gravity effect. Therefore, we can say that tunnelling of a scalar particle does not affect the local stability of the black hole (red line in Fig. 1).

On the other hand, according to the tunneling process of both Dirac and vector boson particles, the modified heat capacities given in Eqs. (36) and (37 diverge. Hence, the black hole may undergoes second-type of phase transitions in order to become stable in the presence of the quantum gravity effect (red lines in Figs. 2, 3). Also, for same values of \(\varLambda \), \(\alpha \),

*j*, \(\hbar \),*J*, and*m*, the unstable region in the context of vector boson particle tunneling is wider than that of Dirac particle. This shows that, in the context of tunneling of the Dirac particle, the black hole may undergo stable earlier than that of the vector particle particle.In the absence of GUP effect, all of the three modified heat capacities reduce to standard one (Eq. (38)), and in this situation, the black hole is always stable (blue lines in Figs. 1, 2, 3).

## Notes

### Acknowledgements

This work was supported by Research Fund of the Akdeniz University (Project No: FDK-2017-2867) and the Scientific and Technological Research Council of Turkey (TUBITAK Project No: 116F329).

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