# Quantum tunneling from generalized (2+1) dimensional black holes having Noether symmetry

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## Abstract

We have studied the Hawking radiation from *generalized* rotating and static (2+1)-dimensional BTZ black holes. In this regard, we have benefited from the quantum tunneling approach with WKB approximation and obtained the tunneling rate of outgoing scalar and spinor particles across the horizons. We have also obtained the Hawking temperature at the horizons corresponding to the emission of these particles. It is shown that the tunneling rate and Hawking temperature of generalized (2+1)-dimensional BTZ black holes are different from ordinary (2+1)-dimensional BTZ black holes due to the Noether symmetry. In other words, the Noether symmetry can change the tunneling rate and Hawking temperature of the BTZ black holes. This symmetry may cause the BTZ black holes to avoid evaporation and its breakdown may start the evaporation.

### Keywords

Black Hole Event Horizon Killing Vector Quantum Tunneling Spinor Particle## 1 Introduction

Hawking radiation is a quantum mechanical phenomenon by which the (3+1)-dimensional black holes in the background of classical general relativity can evaporate [1, 2]. This phenomenon has also been considered as quantum tunneling of particles from the horizons of black holes [3, 4, 5]. In this approach, the Klein–Gordon or Dirac wave equations for scalar or spinor particles are solved in the spacetime background of the black holes by using complex path integration techniques and the WKB approximation. This gives the tunneling rate of scalar or spinor particles across the event horizons, as well as the Hawking temperature of the black holes.

On the other hand, (2+1)-dimensional black holes are among the most interesting subjects in lower dimensional gravity. The vacuum solution of (2+1)-dimensional gravity is flat for zero cosmological constant, hence no black hole solution exists [6]. However, (2+1)-dimensional BTZ black hole solutions for a negative cosmological constant were shown to exist by Ba\(\tilde{n}\)ados et al. [7, 8]. These black holes are very similar to (3+1)-dimensional black holes in their thermodynamical properties. Moreover, they have inner and outer horizons, mass, angular momentum and charge. Recently, we have obtained a generalized (2+1)-dimensional BTZ black hole solution by using the Noether symmetry approach [9]. (For a study of Noether symmetry in spherical solutions, in 4D \(F(R)\) and \(F(T)\) gravity; see Refs. [10, 11]). This black hole has three conserved charges as mass \(M\), angular momentum \(J\) and a new conserved charge \(Q_N\) corresponding, respectively, to the invariance of the solution under time translation, rotation, and continuous displacement of the Ricci scalar in the action.

The quantum tunneling for scalar and spinor particles from (2+1)-dimensional charged and/or rotating black holes has been studied in Ref. [12] by using WKB approximation, and the corresponding tunneling rates and Hawking temperatures have been obtained. In this paper, following Ref. [12], we study the Hawking radiation from the above mentioned generalized rotating and static (2+1)-dimensional BTZ black holes to obtain the tunneling rate of outgoing scalar and spinor particles across the horizons and the Hawking temperature at the horizons corresponding to the emission of these particles.

## 2 (2+1)-dimensional BTZ black holes

^{1}The line element for this solution is given by

## 3 Generalized (2+1)-dimensional BTZ black hole

## 4 Quantum tunneling of scalar particles from generalized (2+1)-dimensional BTZ black holes

## 5 Quantum tunneling of fermionic particles from generalized (2+1)-dimensional black holes

## 6 Conclusions

In this paper, we have used the quantum tunneling approach and WKB approximation to calculate the tunneling rate of outgoing scalar and spinor particles across the horizons of rotating and static generalized (2+1)-dimensional BTZ black holes. We have also calculated the Hawking temperature at the horizons corresponding to the emission of these particles. We have shown that the generalized (2+1)-dimensional BTZ black holes, with an extrinsic parameter \(Q_{N}\) as the Noether charge, have a different tunneling rate and Hawking temperature from those of ordinary (2+1)-dimensional BTZ black holes. In other words, the Noether symmetry can change the tunneling rate and Hawking temperature of the BTZ black holes. This is remarkable, because the generalization of ordinary (2+1)-dimensional BTZ black holes by applying the Noether symmetry may cause these black holes to avoid evaporation. When the free parameter \(D_3\) in the generalized action is fixed to a cosmological constant \(\Lambda =-l^{-2}\) as \(D_3=2l^{-2}\), the Noether symmetry is broken, namely \(Q_N=0\), and one recovers the ordinary BTZ black holes, having \(R=-6l^{-2}\). Then the tunneling rate and the Hawking temperature of the generalized (2+1)-dimensional BTZ black holes coincide with those of ordinary (2+1)-dimensional BTZ black holes and they start evaporation.

## Footnotes

- 1.
We use units where \(8 G=1\).

## Notes

### Acknowledgments

This work has been supported financially by Research Institute for Astronomy and Astrophysics of Maragha (RIAAM) under research project NO.1/3252-44.

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