# Fermions and vector particles tunnelling from non-rotating weakly isolated horizons

## Abstract

Fermions and vector particles tunnelling from non-rotating weakly isolated horizons is investigated in this paper. By applying the WKB approximation to the Dirac equation and Proca equation, we obtain the emission spectrum and Hawking temperature of fermions and vector particles tunnelling from weakly isolated horizons. We consider the back reaction of emitted particles to the space-time, and get the corrected Hawking radiation spectrum. At last we discuss the information recovery of weakly isolated horizons.

## 1 Introduction

In 1974, Stephen Hawking discovered that black holes can release a black-body radiation due to quantum effects near the event horizon with the help of the Wick Rotation method [1, 2] applied for the gravitational collapse. From Hawking’s discovery, people know that black holes are not the final state of stars, and, with the emission of Hawking radiation, they could lose energy, and shrink. Later on, several different derivations of Hawking radiation relying on quantum field theory were proposed, which further confirms the existence of Hawking radiation [3, 4, 5, 6, 7]. Hawking’s discovery that black holes emit Hawking radiation has greatly stimulated the development of black hole physics, and also provides some hints of underlying quantum gravity.

In this paper, we will investigate fermions and vector particles tunnelling from non-rotating weakly isolated horizons. Weakly isolated horizon is a new, quasi-local framework which was introduced by Ashtekar and his collaborators [45, 46, 47, 48]. Compared with the event horizon, this framework does not need the knowledge of the overall space-time, and only involves quasi-local conditions, so it accords with the practical physical process. In this framework, black holes in equilibrium are described by Weakly Isolated Horizons (WIH). Weakly isolated horizons will cover stationary space-times. So investigations on the tunnelling from weakly isolated horizons are important problems. Reference [49] recovers the semi-classical emission rate in the tunnelling process by applying the Null Geodesic Method of Parikh and Wilczek and Hamilton–Jacibi Ansatz for scalar particles to a weakly isolated horizon. In this paper we investigate the tunnelling effect of fermions and vector particles near non-rotating weakly isolated horizons. We obtain Hawking’s purely thermal spectrum without considering the back reaction of emitted particles to the space-time, which will lead to the information loss puzzle [50, 51, 52, 53]. Then we consider the back reaction of emitted particles to the space-time, and get the corrected Hawking radiation spectrum. At last the information recovery of weakly isolated horizons based on the methods in Refs. [54, 55, 56] is discussed. Another solution for the black hole information loss problem was developed by Cordain, which is founded on the Bohr-like approach to black hole quantum physics [57, 58]. These two solutions are consistent with each other, which has been highlighted in Ref. [59]. Like Ref. [21], we assume that the change of black hole angular momentum due to the spin of the emitted particle is negligible. This is a good approximation for the black hole with mass much larger than the Planck mass.

The remainder of this paper is organized as follows. In Sect. 2, we briefly review the definition of the WIH and the geometry near it. In Sect. 3, we investigate the tunnelling of fermions near non-rotating weakly isolated horizons. In Sect. 4, the tunnelling of vector particles is investigated. In Sect. 5, we discuss the information recovery of weakly isolated horizons. In the last section, discussions and conclusions are given.

In this paper, we take the convenient units of \(k=h=c=G=1\).

## 2 The near horizon geometry of weakly isolated horizons

In this section we briefly review the geometric properties of WIH [45, 46, 47]. Generally speaking, the WIH (\(\Delta \)) is a non-expansion null hypersurface, with almost stationary inner geometry \([\pounds _l, D_a] l^a=0\) on the horizon, where \(l^a\) is the generator of the horizon. \(\pounds \) is Lie derivative, and \(D_a\) is the induced derivative operator on the horizon \(\Delta \).

*M*is the horizon mass,

*A*is the area of the cross section of WIHs, \(\Omega \) is the angular velocity of the horizon, and \(J=-\frac{1}{8\pi }\oint _S(\omega _a\psi ^a)dS\) is the angular momentum. The expressions of the surface gravity, angular velocity and horizon energy of weakly isolated horizons are given by

*R*is the horizon radius, and is defined as

*A*is the area of cross section of the horizon, so the entropy of WIHs is

*U*,

*X*, \(\omega \), \(\xi _3\) and \(\xi _4 \) near \(\Delta \) is

## 3 Fermions tunnelling from non-rotating weakly isolated horizons

*c*,

*d*,

*e*as elements of the inverse metric of \(h_{ij}\).

*m*is the rest mass, and

*K*is a complex constant. Putting Eq. (27) into Eq. (26), one obtains

*W*(

*r*) yields

*K*is a complex constant.

## 4 Vector particles tunnelling from non-rotating weakly isolated horizons

*I*is the classical action of the trajectory, and is defined as

We get the emission spectrum and Hawking temperature of vector particles tunnelling from weakly isolated horizons. The result is the same as the tunnelling of Dirac particles and scalar particles [49].

## 5 Back reaction to the space-time and the information loss recovery of weakly isolated horizons

In the last two sections, we fix the background of the space-time, and obtain Hawking’s purely thermal spectrum of fermions and vector particles tunnelling from weakly isolated horizons, which will lead to the information loss puzzle [50, 51, 52, 53].

*M*, the mass of the black hole

*M*should reduce to \(M-E_{i}\), and the emission rate should be

*E*is

*E*is considered, we have the corrected spectrum \(\Gamma =\exp (\Delta S)=\exp [-\frac{E}{T}+O(E^2)]\) which deviates from the purely thermal spectrum. We will show that the corrected spectrum leads to the information conservation during the process of black hole evaporation based on the methods in Refs. [54, 55, 56].

*n*particles, we have the following relationship

*M*is the mass of WIHs. The entropy carried out by all the emitted particles is

In this section, we find that there exist correlations between sequential Hawking radiations from the locally defined black holes–weakly isolated horizons, information can be carried out by such correlations, and the entropy of weakly isolated horizons is conserved during the radiation process.

## 6 Discussions and conclusions

In this paper we investigate the tunnelling of fermions and vector particles from a non-rotating weakly isolated horizon. Weakly isolated horizon is a new, quasi-local framework which was introduced by Ashtekar and his collaborators. Compared with the event horizon, this framework does not need the knowledge of the overall space-time, and only involves quasi-local conditions, so it accords with the practical physical process. Weakly isolated horizons not only cover all stationary space-times, but also many non-stationary cases, so the investigation on the Hawking radiation of weakly isolated horizons is a quite significant issue.

We calculate fermions and vector particles tunnelling for non-rotating weakly isolated horizons. Although we discuss non-rotating weakly isolated horizons for simplicity, the shape of the horizon may not have spherical symmetry. In the practical physical process, black holes are often distorted by surrounding matters, so our discussion accords with the practical physical process.

In the discussion of fermions tunnelling, we first follow Kerner and Mann’s methods, and calculate the tunnelling of the Dirac particles with spin-up and spin-down. Then we extend the method to investigate the tunnelling of Dirac particles with arbitrary spin directions, and obtain the expected Hawking temperature.

At last, we consider the back reaction of emitted particles to the space-time, and get the corrected Hawking radiation spectrum. The leading-order term gives the purely thermal spectrum, and when the higher-order term is considered, we have the corrected emission spectrum. We discuss the information recovery of the corrected emission spectrum from weakly isolated horizons based on the methods in Refs. [54, 55, 56]. We find that there exist correlations between sequential Hawking radiations from the locally defined black holes–weakly isolated horizons, information can be carried out by such correlations, and the entropy of weakly isolated horizons is conserved during the radiation process.

## Notes

### Acknowledgements

GRC is supported by National Natural Science Foundation of China (no.11647090). XYC is supported by Natural Science Foundation of Hunan Province (no. 2018JJ3866).

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