Thermal Correction to the Kinnersley Black Hole in a Lorentz-Violating Dirac Field Theory

According to Lorentz-violating theory, the dynamical equation of Dirac particles in the Kinnersley black hole with variably accelerated linear motion is modified. The Hawking quantum tunneling radiation characteristics of Kinnersley black hole are obtained by solving the modified equation. The expression of the Hawking temperature of Kinnersley black hole has been updated.


INTRODUCTION
During past years, people have made many meaningful researches on the quantum tunneling radiation of black holes [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. These researches have enriched us with knowledge about the thermodynamic evolution of black holes. Using the Hawking quantum tunneling radiation theory, Kraws et al calculated the Hawking temperature and entropy of black hole. The research of quantum tunneling radiation for black hole has been increased greatly by the semiclassical Hamilton-Jacobi method [8,9]. Kerner and Mann used the semiclassical theory to study the quantum tunneling radiation characteristic of Dirac field particles in black hole [16,17]. They divided the fermion spin into up and down and decomposed the Dirac equations into two groups, then obtained the tunneling rate of Dirac particles at the event horizon and the Hawking temperature of black hole.
In 2009, Lin and Yang proposed a new method to study the quantum tunneling radiation of black hole. They transformed the Dirac equation in curved space-time into a matrix equation by using the semiclassical approximation theory, and then the resulting matrix equation was further converted to the Hamilton-Jacobi equation for Dirac particles in curved space-time by using the commutation relation of gamma matrices. Finally they derived the Hawking quantum tunneling rate of fermions and other important physical quantities of black hole based on the Hamilton-Jacobi equation [11,14,[18][19][20][21]. Their work showed that the Hamilton-Jacobi equation and its Hamilton principal function S can be applied to the study of quantum tunneling radiation of fermions in curved space-time. The developing Hamilton-Jacobi method can effectively solve the problems related to fermion tunneling radiation and unify the expressions of quantum tunneling radiation theory in curved space-time.
The study on modifications to the Hawking radiation of black holes can help for understanding the information loss paradox [22][23][24][25][26]. Banerjee and Majhi extended the beyond semiclassical approximation to include all quantum corrections [27][28][29][30][31][32]. Lin and Yang investigate tunnelling of charged black holes based on Klein-Gordon scalar particle theory to derive corrections to the tunnelling rate and temperature in Reissner-Nordstr€ om space-time and Reissner Nerdstr€ om-de Sitter space-time, respectively [33]. Beyond semiclassical theory and semiclassical theory are both modified theories, for which the former can give more accurate correction to the quantum tunneling rate of black hole.
With research of string theory and quantum gravity theory, people have realized that the Lorentz relation should be modified at high energy, which will lead to the Dirac equation in curved space-time to be modified. The application of Lorentz-violating theory in curved space-time is a Frontier subject worthy of attention [34][35][36][37][38][39][40]. Cruz et al introduced the Lorentz symmetry violating term in the scalar field Lagrangian, namely the Lorentzviolating scalar field theory, to research thermal corrections to the Casimir energy [41]. The Dirac particle action and Dirac equation with Lorentz symmetry violating in flat space-time are introduced in [42]. The influence of Lorentz violating theory on the quantum tunneling radiation of Dirac particles in the Vaidya black hole is investigated in [43].
However, for the stationary and dynamic axisymmetric black holes, the effect of Lorentz violating theory on their quantum tunneling radiation has not been studied. In this paper, Lorentz violating theory is considered to correct the quantum tunneling rate, the temperature and entropy of a dynamic Kinnersley black hole with variably accelerated linear motion, resulting in some new conclusion.
In the second section, we introduce the modification of Lorentz violating Dirac field theory to Dirac particle dynamics equation in the Kinnersley space-time. The third section studies the thermal and entropy corrections to Kinnersley black hole in a Lorentz violating Dirac field theory. The last section gives a discussion on the results obtained in this paper.

LORENTZ'S VIOLATING THEORY AND DIRAC EQUATION IN DYNAMICAL CURVED SPACE-TIME
According to Hamilton principle, the dynamic equation of Dirac particle in flat space-time can be acquired from the action of Dirac particle in flat space-time [41][42][43]. Adding the Lorentz symmetry violating term into the action and using Hamilton principle, we can obtain the Dirac equation of Lorentz symmetry violating in a flat space-time. It only needs to pay attention to two points for generalizing the particle dynamics equation from the flat space-time to the curved space-time: one is to generalize the gamma matrix c μ from the flat space-time to the curved spacetime, with different curved space-time having different c μ ; the other is to generalize the ordinary derivative to the covariant derivative related to connection. Therefore, in the Kinnersley curved space-time, the dynamics equation of spin 1/2 fermion with Lorentz symmetry violating can be expressed as [42].
where Ψ is the wave function. For Dirac particles, the wave function Ψ and the action S are linked by where ψ 0 is a column matrix. For Dirac particles the nonstationary Kinnersley black hole where v is the advanced Eddington coordinate. The covariant derivative D μ in Eq. 1 is defined by where Γ αβ μ is the connection in Riemannian geometry, and Π αβ is expressed as i Z Γ αβ μ Π αβ is the spin connection term that characterizes the spinor covariant derivative in curved space-time. In Eq.1, a, b and c are all small quantities that satisfy a, b, c ≪ m, where m is particle mass. The Gamma matrices c μ or c ] meet the following anticommutation relation: The 4D ether-like field vector u α is not a constant vector in curved space-time, but it must satisfy the following condition We substitute Eq. 2 into Eq. 1, and keep the terms associated with the lowest order of Z, then Eq. 1 is reduced to Considering Eq. 6, the following equality holds: Then the dynamics Eq. 9 becomes Taking square for both sides of Eq. 11, and omitting the b m c 5 term, where O is a high order small quantity. Then we use Eq. 6 to simplify Eq. 12, resulting in Obviously,Ψ ≠ 0, making it necessary that In this equation S is the action, also called Hamiltonian principal function. In the process of derive this equation, we ignore the term b m c 5 for two reasons. One is that b is a small quantity; the other is that the term b m c 5 in the square bracket of Eq. 11 only indicates the correction in quantity and thus can be ignored. However, the terms other than b m c 5 in the square bracket of Eq. 11 are related to the metric tensor or ether-like field vector, thus can not be ignored. In fact, Eq. 14 is completely equivalent to Eq. 1, since Eq. 14 is also the dynamic equation of Dirac particles. Eq. 14 is a new form of modified Hamilton-Jacobi equation about the action S of Dirac particles. Starting from Eq. 14, we can conveniently study the tunneling radiation characteristics of fermion with mass m in the curved non-stationary Kinnersley black hole. This is an innovation that has not been reported yet.
For a Dirac particle with mass of m and charge q, its dynamic equation is very complicated. Firstly, Eq. 4 must be modified to Substituting Eq. 15 into Eq. 1 and using the same method as deriving Eq. 14, we can get the dynamic equation of Dirac particle with mass m and charge q as follows During the derivation of Eqs 14, 16, we ignored higher order quantities in terms of a and c. From the point of view of mathematics and physics, this is a reasonable and effective approximation. In the next section, we will study the tunneling radiation of Dirac particles in the non-stationary Kinnersley black hole according to Eq. 16.

THERMAL CORRECTION TO THE KINNERSLEY BLACK HOLE IN THE LORENTZ-VIOLATING THEORY
According to Kinnersley's research on the metric of accelerating black holes, the space-time line element of a linearly moving black hole with variable acceleration described by the advanced Eddington coordinate v is [44].
f −a k sin θ, where a k a k (v) is the acceleration of the Kinnersley black hole, θ and ϕ are spherical coordinates, and M is the mass of the black hole. The north pole θ 0 of the black hole always points in the direction of acceleration. Therefore, the covariant and contravariant metric tensors are where Obviously, this black hole is uncharged. According to the zero hypersurface equation The event horizon of the black hole should satisfy the following equation: Note that A μ is not the electromagnetic potential of the Kinnersley black hole itself, but the electromagnetic potential of the cosmological space around the moving non-charged Kinnersley black hole. According to Eqs 17, 19, 20, we construct the ether-like field vector u μ such that u v c v g 00 √ , u r c r g 01 g 00 √ , u θ c θ g 02 g 00 √ , u ϕ c ϕ g 33 √ where c v , c r , c θ , c ϕ are all constants. It is easy to show that u v u v c 2 v , u r u r c v c r , u θ u θ c v c θ , u ϕ u ϕ c 2 ϕ and thus condition (8) is met. Substituting Eq. 26 into Eq. 25 will result in This is a time-dependent equation. To solve this equation, general tortoise coordinate transformation must be performed, i.e.
Therefore, we have z zr Then, the following variable separation is performed for S and let where ω is the particle energy, p θ is a component of the particle generalized momentum in the θ direction, and the constant j is a component of the particle generalized moment in the ϕ direction. Substituting Eqs 29-31, 33-35 into Eq. 27, and considering the special condition of r → r H , θ → θ 0 , v → v 0 , we can get the dynamic equation of Dirac particles at the event horizon of the black hole as follows where and further studying the thermodynamic evolution of black holes. It is necessary to further note that the key to solve Dirac-Hamilton-Jacobi equation is to construct the etherlike field vector correctly. The specific form of u μ must be selected according to the characteristics of the curved spacetime to be investigated, so as to ensure the validity of the derivation. The reference time v 0 and the reference angle θ 0 in Eq. 28 are arbitrarily selected, so the results derived from the general tortoise coordinate transformation are of universal significance. In addition, the entropy S of black hole is closely related to Hawking temperature of black hole.
Using the change of Bekenstein-Hawking entropy ΔS BH to express the tunneling rate will give Γ ∼ e ΔSBH . Therefore, the entropy of the black hole should also be corrected. If the Lorentz symmetric violating is not considered, the results in this paper will return to the uncorrected cases that have been known ubiquitously.

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.