# Charged fermion tunneling from electrically and magnetically charged rotating black hole in de Sitter space

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## Abstract

Thermal radiation of electrically charged fermions from a rotating black hole with electric and magnetic charges in de Sitter space is considered. The tunneling probabilities for outgoing and incoming particles are obtained and the Hawking temperature is calculated. The relation for the classical action for the particles in the black hole’s background is also found.

### Keywords

Black Hole Event Horizon Dirac Equation Magnetic Charge Black String## 1 Introduction

Hawking radiation has attracted a lot of attention since it was proposed [1]. Different methods have been applied [2] to its study. The semi-classical tunneling approach that was proposed by Kraus and Wilczek [3, 4] has gained considerable interest recently. It was shown that the Hawking temperature is defined by the imaginary part of the emitted particle’s action for the classically forbidden region near the horizon. To calculate it two methods were proposed. The first one is the so-called null-geodesic method proposed by Parikh and Wilczek [5] which is based on the fact that the imaginary part of the action is caused by the integration of the radial momentum \(p_\mathrm{r}\) for the emitted particles. The second method is based on the relativistic Hamilton–Jacobi equation, and the imaginary part of the action can be obtained after integration of that equation [6]. This second approach can be treated as an extension of the complex path method proposed by Padmanabhan et al. [7, 8, 9].

The tunneling approach based on the Hamilton–Jacobi equation at first was applied to the emission of scalar particles. Then it was successfully applied to the vast area of well-known and exotic spacetimes, in particular the Kerr and Kerr–Newman ones [10, 11], the Taub-NUT spacetime [12] the Gödel spacetime [13], BTZ black holes [14], and dynamical black holes [15]. A review of the tunneling method is considered in Ref. [16], where further references can be found.

The tunneling approach was also successfully applied to the tunneling of fermions. In their seminal work, Kerner and Mann used the Dirac equation instead of the Hamilton–Jacobi one to obtain the temperature of the emitted fermions and showed that for a chosen type of spacetime the temperature of the emitted fermions would be the same as the temperature of scalar particles [17]. That method was later applied to different kinds of black hole spacetimes, including the Reissner–Nordström one [18], the Kerr–Newman one [19, 20], dilatonic black holes [21], BTZ black holes [22], black holes in Hořava–Lifshitz gravity [23, 24], accelerating and rotating black holes [25, 26], and rotating black strings [27].

In our work, we consider the Kerr–Newman–de Sitter black hole which carries both electric and magnetic charges. Using the Kerner–Mann procedure, we consider the emission of charged spin \(1/2\) particles. We show that in the presence of electric and magnetic charges of the black hole the variables of the Dirac equation can also be separated and as a consequence the temperature can be found. We also find the quasiclassical action of an emitted particle.

## 2 Charged spin \(1/2\) particle tunneling from Kerr–Newman–de Sitter black hole

It should be stressed that the coordinate \(\theta \) in our decompositions in the vicinity of the horizon point \(r_+\) is not fixed similarly as in Ref. [20]. We also remark that in order to get the black hole’s temperature in [12], the polar angle \(\theta \) was fixed in the near-horizon metric. This was done for the following reason: if the coordinate \(\theta \) is fixed the equations of motion can easily be integrated, especially in the null-geodesic approach. It was noted that the resulting temperature would not depend on the chosen angle \(\theta \), so it can take an arbitrary value [12]. A similar situation was considered in Ref. [10], but there, to simplify the equations for the null geodesics, an additional transformation of the coordinates was performed.

## 3 Action for the emitted particles

## 4 Conclusions

In this paper, we considered charged fermion tunneling from the electrically and magnetically charged Kerr–Newman–de Sitter black hole. Using the Kerner–Mann approach [20], we successfully recovered the black hole’s temperature. It was shown that similarly to the case when the black hole carries only an electric charge, inclusion of an additional magnetic charge does not spoil the separability of the Dirac equation in the vicinity of the horizons. So the relations for the temperature are obtained in the same manner and take almost the same form as in the case of an electrically charged black hole. We also note that for the temperature an explicit dependence on the magnetic charge is hidden in the definition of the radii of the horizons.

We also obtained relations for the radial and angular parts of the action. Those relations might be helpful if one tries to find corrections to the spectrum of emitted particles. Here we have an explicit dependence on the electric as well as magnetic charges, so these terms might have a different influence on the spectrum of the emitted particles.

Another issue that still remains open is taking into account higher orders of the WKB corrections. This problem is connected with the calculation of terms caused by the spin connection. These terms might affect the separability and tractability of the Dirac equation and this problem requires additionally careful consideration.

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