Influence of Lorentz Invariation Violation on Arbitrarily Spin Fermions Tunneling Radiation in the Vaidya-Bonner Spacetime

In the spacetime of non-stationary spherical symmetry Vaidya-Bonner black hole, an accurate modification of Hawking tunneling radiation for fermions with arbitrarily spin is researched. Considering a light dispersion relationship derived from string theory, quantum gravitational theory and Rarita-Schwinger Equation in the non-stationary spherical symmetry spacetime, we derive an accurately modified dynamic equation for fermions with arbitrarily spin. By solving the equation, modified tunneling rate of fermions with arbitrarily spin, Hawking temperature and entropy at the event horizon of Vaidya-Bonner black hole are presented. We find the Hawking temperature will increase, but the the entropy will decrease comparing with the case without Lorentz Invariation Violation modification.


Introduction
The theory of Hawking thermal radiation reveals the relationship between gravitational theory, quantum theory, and statistical thermal dynamic mechanics [1]. After the research of Hawking thermal radiation to all kinds of black holes [2], Kraus and Wilczek did some modifications to the Hawking thermal radiation adopting self-gravitational interaction [3]. Hereafter, researchers studied the Hawking tunneling radiation for many types of black holes [4][5][6][7][8][9]. In 2007, Kerner and Mann proposed a semiclassic method to investigate the tunneling radiation of fermions with spin 1/2 [10,11]. In the later research, this semiclassic method is widely used to calculate the tunneling radiation of the other type of particles [12][13][14]. Yang and Lin developed Kerner and Mann's theory and proposed that the Hamilton-Jacobi method is efficient for the tunneling of fermions [15,16]. According to references [15,16], after choosing a suitable Gamma matrix and considering the commutation relation of the Pauli matrix in the Dirac equation, which describes the dynamic of the fermion quite well, the Hamilton-Jacobi equation in the curved space-time can be derived. This result means that the Hamilton-Jacobi equation is also a very important equation in the research of the tunneling theory of fermions. In recent years, the Lorentz light dispersion relationship is generally regarded as a basic relation in modern physics. It seems that both general relativity and quantum mechanics are built on this relationship. However, the research of quantum gravitational theory indicates that the Lorentz relationship should be modified in the high-energy case. Although scientists have not built a successful light dispersion relationship in the high-energy case, current researches are helpful to the development of this theory. People usually estimate that the magnitude of this modification should be in the Plank scale. It is confirmed that both the Dirac equation and the Hamilton-Jacobi equation must be modified if the Lorentz Invariation Violation is considered. In such a case, only an accurate modification can efficiently research fermion tunneling radiation from a black hole, such as the Vaidya-Bonner black hole. In this paper, the most important progress is that we use a new method which is suitable for fermions with an arbitrary spin. We will research the exact modification of tunneling radiation for fermions with an arbitrary spin, considering the Lorentz Invariation Violation.

Exact Modification of Arbitrary Spin Fermion Rarita-Schwinger Equation and Hamilton-Jacobi Equation
In the research of string theory, the authors proposed a relation [17][18][19][20][21]: In the natural unit, P 0 and p are the energy and momentum of the particle with the static mass m, respectively. L is a constant in the magnitude of the Plank scale, which comes from the Lorentz Invariation Violation theory. In Equation (1), α = 1 is adopted in the Liouville-string model. Kruglov obtained a modified Dirac equation considering α = 2 [22]. Therefore, we substitute α = 2 into Equation (1) and get a general Rarita-Schwinger equation in the flat space: where ℏ is the reduced Plank constant, which equals 1 in the natural units. λ is a very small constant. Ψ α 1 ⋯α k is a wave function, where the value of α k corresponds to a different spin. The larger the α k , the higher the spin is. The wave function satisfies following supplementary condition: When k = 0 and Ψ α 1 ⋯α k = Ψ, Equation (2) changes to the Dirac equation for spin 1/2 and condition (3) disappears automatically. When k = 1, Equation (2) describes the dynamic of fermions with spin 3/2 and the condition (3) also disappears automatically. Note that the commutation relation In the curved space-time, the Rarita-Schwinger equation can be rewritten as where D μ = ∂ μ + ði/ℏÞeA μ , λ≪1, and λℏγ t D t γ j D j is a very small term. For fermions with an arbitrary spin, the wave function is where ξ α 1 ⋯α k and S are matrices and the action of the fermion, respectively. The line element of the nonstationary Vaidya-Bonner black hole represented in an advanced Eddington coordinate [23] is given by where v is the Eddington time and MðvÞ and QðvÞ represent the mass and charge of the black hole changes with time, respectively. When QðvÞ = 0, the nonstationary Vaidya-Bonner black hole is reduced to the Vaidya black hole. The electromagnetic four-potential of the Vaidya-Bonner black hole is Corresponding to the line element (7), the inverse metric tensor is where Because the component of the inverse metric tensor g 00 = 0 in the curved space-time of line element (7), so Equations (5) and (6) become In this paper, the range for i and j in the superscript and subscript satisfies i, j = 1, 2, 3. μ and ν in the superscript and subscript are defined as μ, ν = 0, 1, 2, 3. Setting Equation (12) becomes Multiplying Γ ν ð∂ ν S + eA ν Þ in both sides of Equation (14), then Exchanging ν and μ in Equation (15), we get Equations (15) and (16) are equivalent. Considering γ 0 γ 0 = g 00 = 0, firstly adding the left side and the right side 2 Advances in High Energy Physics of Equations (15) and (16), respectively, and then dividing the new equation by 2, finally combining with Equation (13), we obtain Defining Equation (17) changes to Multiplying iλγ ν ð∂ ν S + eA ν Þ at both sides, we get By exchanging μ and ν for Equation (20), then Combining Equations (20), (21), and it is easy to get Equation (23) is a matrix equation. In fact, it is an eigenvalue matrix equation. The condition for the nonsingular solution of this eigenvalue matrix equation requires that the corresponding value of the determinant is zero. Combining Equations (18), (23), and (7), we get Therefore, As λ≪1, oðλ 2 Þ is a high-order term. For accuracy of modification, the term oðλ 2 Þ is kept in Equations (17), (23), and (24). If oðλ 2 Þ is ignored in Equation (17), one cannot obtain a correct result. For the Vaidya-Bonner black hole, only this derivation can get a correct result. In fact, Equation (25) is the dynamic equation describing an arbitrary spin fermion in the Vaidya-Bonner space-time. Moreover, this equation is derived from the Rarita-Schwinger equation in the curved space-time with the Lorentz Invariance Violation. So this equation is a deformation of the Hamilton-Jacobi equation or can be called exactly as the Rarita-Schwinger-Hamilton-Jacobi equation. The first two terms of this equation can be expressed as g μν ð∂ μ S + eA μ Þð∂ ν S + eA ν Þ. Considering Equation (10), one can obtain the first two terms in Equation (25), so Equation (25) can be rewritten as From this equation, we can get the action of the fermion and then study the modified tunneling radiation of fermions. Equation (26) is a highly accurate dynamic equation because the term oðλ 2 Þ is not ignored during the derivation and the Lorentz Invariance Violation is included. If it is not done so, an accurate modification cannot be obtained. Note that the modification of boson Hamilton-Jacobi equation is different from this method [24,25]. This indicates the significance of accurate modification of tunneling for particles with an arbitrary spin. In the following, we will derive the thermal dynamic characteristics at the horizon of the Vaidya-Bonner black hole.

Tunneling Modification for Fermions with Arbitrary Spin in Vaidya-Bonner Black Hole
The Vaidya-Bonner black hole is a charged nonstationary spherical black hole; the line element is shown in Equation (7). The event horizon of the Vaidya-Bonner black hole is determined by the zero supercurved equation From Equations (10) and (27), we find that the event horizon r H satisfies where _ r H = dr H /dv is the change rate of r H with time. Solving Equation (28), we get 3 Advances in High Energy Physics where "+" denotes the event horizon of the Vaidya-Bonner black hole. From Equations (10) and (25) The key to research the tunneling is to get the action S of fermions. For this black hole, the key is the solution of the action in the direction of the radius, so a tortoise coordinate transformation is necessary, where κ is the surface gravity, r H is the event horizon of the black hole, and v 0 is a special moment when the fermion escapes from the event horizon. Both v 0 and κ are constants. From Equations (31) and (32), we have We make a separation of the variable to action S as and set where λ ′ = λ_ r H . For simplification, it is suitable to keep the first-order term of λ in the final results. Multiplying 2κðr − r H Þ to both sides of Equation (37), and taking the limit for the condition r ⟶ r H , we get where ω 0 = eQ/r H . The limit of the coefficient of ð∂R/∂r * Þ 2 is From Equations (33) and (38) Using the residue theorem to solve R in Equation (40), we get From Equation (39), κ in Equation (41) can be obtained as Due to λ ≪ 1, the final result can only retain the λ term. In Equation (41), "+" and "-" represent outgoing and ingoing waves, respectively, from the horizon of the back hole.
So according to the tunneling theory, we get the tunneling rate for fermions with an arbitrary spin in the Vaidya-Bonner space-time.
The κ ′ = κ/ð1 − λmÞ in Equations (42) and (43) is the modified surface gravitational force at the event horizon of the black hole. T H in Equation (43) is Advances in High Energy Physics