# Dirac equation of spin particles and tunneling radiation from a Kinnersly black hole

## Abstract

In curved space-time, the Hamilton–Jacobi equation is a semi-classical particle equation of motion, which plays an important role in the research of black hole physics. In this paper, starting from the Dirac equation of spin 1/2 fermions and the Rarita–Schwinger equation of spin 3/2 fermions, respectively, we derive a Hamilton–Jacobi equation for the non-stationary spherically symmetric gravitational field background. Furthermore, the quantum tunneling of a charged spherically symmetric Kinnersly black hole is investigated by using the Hamilton–Jacobi equation. The result shows that the Hamilton–Jacobi equation is helpful to understand the thermodynamic properties and the radiation characteristics of a black hole.

## 1 Introduction

In 1974, considering quantum effects, Hawking proved that a black hole has thermal radiation [1]. After that, one has carried out a series of research for various types of black hole radiation [2, 3, 4, 5, 6, 7, 8]. About the source of Hawking radiation, a common viewpoint is to believe that, due to a vacuum fluctuation near the event horizon where a virtual particle pair could be created or annihilated, a negative energy particle falls in since there exists a negative energy orbit inside the black hole, however, the positive energy particle left outside is radiated to infinity, which causes Hawking radiation. In 2000, Parikh and Wilczek et al. put forward a quantum tunneling theory to study the thermal radiation of a black hole. By taking into account the background change before and after tunneling radiation, they have carried out modification to the previous tunneling probability [9]. In recent years, a series of significant studies have been made on the tunneling radiation of black holes [3, 10, 11, 12, 13, 14, 15, 16, 17]. Zhang and Zhao et al. have developed this tunneling theory, and they studied the relationship between the tunneling radiation and the black hole entropy, which provided a reasonable explanation for the information loss paradox of a black hole [18, 19]. Further research of Lin and Yang et al. showed that the tunneling rate of the event horizon from a dynamic black hole is not only related to the black hole entropy, but also related to an integral function [20]. Therefore, the information loss paradox of a black hole still needs further research. According to the literature [21, 22, 23], the Hamilton–Jacobi equation in curved space-time is a basic equation describing the dynamic characteristics of all kinds of particles. Applying it to the Hamilton–Jacobi equation, the Dirac equation and the tunneling radiation of a stationary black hole have been studied. However, a black hole in the universe should show dynamic change due to radiation, accretion, merging and other reasons. Therefore, it is of practical significance to study the tunneling radiation characteristics of a dynamic black hole.

For a dynamic black hole, the Dirac equation describing the motion of spin 1/2 and the 3/2 particles is more complex than that of a stationary black hole. In this paper, according to the space-time line element of a Kinnersley black hole, using the advanced Eddington coordinate to represent its dynamic characteristics, we study the tunneling radiation of spin 1/2 and 3/2 particles in this space-time background.

The rest of the paper is organized as follows. Using the Dirac equation for a spin 1/2 particle and the Rarita–Schwinger equation for a spin 3/2 particle, we derived the Hamilton–Jacobi equation in Sect. 2. In Sect. 3, according to the Hamilton–Jacobi equation, the fermion tunneling behavior in a non-stationary Kinnersley black hole is addressed. Section 4 is devoted to our discussion and conclusion.

## 2 Dirac equation of spin particles and the Hamilton–Jacobi equation

*S*is the principal function. The coefficient term can be decomposed into

*C*,

*D*and

*G*are closely related to Eqs. (5)–(12). The expressions are, respectively,

*C*,

*D*and

*G*refer to Eqs. (16)–(18). Taking into account non-trivial solution conditions in Eq. (25), we can obtain

## 3 Hamilton–Jacobi equation and the tunneling from a non-stationary Kinnersly black hole

*j*is a constant that is related to the Killing vector \(( {{\partial / {\partial \varphi }}} )\). After processing the above expression, we can get

*A*is

*a*and the coefficients \(r'_H\), \(\dot{r}_H\), meanwhile setting \(r_H=2M\), we can find that the surface gravity of a dynamic Kinnersley black hole reduces to the case of the Schwarzschild hole. Therefore, Eq. (41) also reduces to the Hawking temperature of a Schwarzschild black hole. Furthermore, it is well known that the relationship between the temperature and the surface gravity is \(T=\kappa /(2 \pi )\), where the \(\kappa \) is the surface gravity of the black hole. Combined with this fact, from Eq. (41) we find the value of \(\alpha \) is equal to \( 1 / ( 2 \kappa )\).

## 4 Conclusion

In this paper, starting from both the Dirac equation of spin 1/2 fermions and the Rarita–Schwinger equation of spin 3/2 fermions, we can obtain a Hamilton–Jacobi equation. Moreover, making use of this equation, we investigate the fermion tunneling rate and the Hawking temperature of a non-stationary Kinnersly black hole. According to our conclusion, the Hamilton–Jacobi equation can be derived from the kinetic equations of arbitrary spin particles. This shows that the Hamilton–Jacobi equation is a basic semi-classical equation, which can be used to study the quantum tunneling behavior of arbitrary spin particles. Furthermore, through the comparison of the previous work, we can obviously see that it is very convenient to study tunneling radiation by using the Hamilton–Jacobi equation. Especially for the fermion tunneling, since there is no need to construct the complex gamma matrix by a tedious calculation, which greatly reduces the workload, it is better to carry out creative research in depth.

## Notes

### Acknowledgements

This work is supported by the Natural Science Foundation of China (Grant No. 11573022)

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