Influence of modified light dispersion theory on fermion tunneling radiation in the Einstein【-逻*辑*与-

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Influence of modified light dispersion theory on fermion tunneling radiation in the Einstein【-逻辑与-

本站小编 Free考研考试/2022-01-02

Jie Zhang^,, Zhie Liu, Bei Sha, Xia Tan, Yuzhen LiuCollege of Physics and Electronic Engineering, Qilu Normal University, Jinan 250300, China

Received:2019-11-28Revised:2020-02-10Accepted:2020-02-10Online:2020-03-26

Fund supported:

National Natural Science Foundation of Chinahttps://doi.org/10.13039/501100001809.11273020
Science Foundation of Sichuan Science and Technology Department.2018JY 0502
Natural Science Foundation of Shandong Provincehttps://doi.org/10.13039/501100007129.ZR2019MA059

Abstract
Based on a light dispersion relationship derived from string theory and quantum gravitational theory, we make an accurate modification to the quantum tunneling radiation rate and black hole temperature at an event horizon in a stationary axial-symmetric Einstein–Maxwell–dilaton–axion black hole. We also analyze our new results and carry out some significant discussions. This work enriches the research content and methods of the frontiers of black hole physics.
Keywords： modified fermion equation;Hawking radiation;fermion tunneling

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Cite this article
Jie Zhang, Zhie Liu, Bei Sha, Xia Tan, Yuzhen Liu. Influence of modified light dispersion theory on fermion tunneling radiation in the Einstein–Maxwell–dilaton–axion black hole^*. Communications in Theoretical Physics, 2020, 72(4): 045402- doi:10.1088/1572-9494/ab7704

1. Introduction

After Hawking demonstrated that black holes have a theoretical thermal radiation in 1974, a series of important scientific research studies have been conducted on both single and double black holes, including theoretical aspects and astronomical observations [1–9]. Although the Hawking radiation spectrum is a pure thermal radiation spectrum, his work promoted other studies of the physics of black holes. In particular, Kraus, Parikh, and Wilczek et al proposed a quantum tunneling radiation theory to research Hawking thermal radiation of black holes [10–17]. Assuming the nonthermal spectrum of Parikh and Wilczek, Zhang et al’s work shows the total information encoded in the correlations among Hawking radiations exactly equals the same amount previously considered lost [18, 19]. In recent years, Kerner and Mann et al investigated the quantum tunneling radiation of fermions of spin 1/2 using a semi-classical method [20, 21]. Following this, authors carried out a series of research studies on the quantum tunneling radiation at the event horizon of a black hole using this semi-classical method (see e.g. [22–24]). With the development of research, some authors began to use the Hamilton–Jacobi method to research the quantum tunneling of fermions [25, 26]. Recent research shows that there is a light dispersion relationship in the string theory and quantum gravitational theory [27–32]. This new relationship indicates that the Lorentz dispersion relationship should be modified in the high energy case, and it is suitable for the correction of Hawking quantum tunneling. Until now, an accurate modification of the tunneling radiation of fermions with arbitrary spin has not been researched thoroughly [33]. In this paper, we make an accurate modification to the quantum tunneling radiation rate of fermions with arbitrary spin at the event horizon in a stationary axial-symmetric Einstein–Maxwell–dilaton–axion black hole, and carry out some necessary discussions around our new results. In the following section, the dynamical equation of fermions with arbitrary spin in the Einstein–Maxwell–dilaton–axion black hole spacetime will be given. In the third section, we will solve and research the above-mentioned equation. Some conclusions are given in section 4.

2. Lorentz invariance violation theory and modification to the fermion dynamical equation

It is known that the dynamical equation of fermion, the spin of which is 1/2, can be described by the Dirac equation, and the dynamical equation of fermion, th espin of which is 3/2, can be described by the Rarita–Schwinger equation. In flat spacetime, the dynamical equation describing fermion with arbitrary spin is [34](1)$\begin{eqnarray}\left({\bar{\gamma }}^{\mu }{\partial }_{\mu }+\displaystyle \frac{m}{{\hslash }}\right){{\rm{\Psi }}}_{{\alpha }_{1}\cdots {\alpha }_{k}}=0.\end{eqnarray}$This equation satisfies the condition: ${\bar{\gamma }}^{\mu }{{\rm{\Psi }}}_{\mu {\alpha }_{2}\cdots {\alpha }_{k}}\,={\partial }_{\mu }{{\rm{\Psi }}}_{\ \ {\alpha }_{2}\cdots {\alpha }_{k}}^{\mu }={{\rm{\Psi }}}_{\ \ \mu {\alpha }_{3}\cdots {\alpha }_{k}}^{\mu }=0$. Equation (1) is valid for fermions, the spin of which is a semi-integer in the flat spacetime. As k=0 and ${{\rm{\Psi }}}_{{\alpha }_{1}\cdots {\alpha }_{k}}={\rm{\Psi }}$, equation (1) returns to the Dirac equation. As k=1, equation (1) changes to the Rarita–Schwinger equation in the condition of ${\bar{\gamma }}^{\mu }{{\rm{\Psi }}}_{\mu }={\partial }_{\mu }{{\rm{\Psi }}}^{\mu }$. In equation (1), if we change the usual differential to the covariant differential, and change the usual derivative to the covariant derivative, the Rarita–Schwinger equation in the curved spacetime is derived as(2)$\begin{eqnarray}\left({\gamma }^{\mu }{D}_{\mu }+\displaystyle \frac{m}{{\hslash }}\right){{\rm{\Psi }}}_{{\alpha }_{1}\cdots {\alpha }_{k}}=0.\end{eqnarray}$This equation satisfies the condition(3)$\begin{eqnarray}{\gamma }^{\mu }{{\rm{\Psi }}}_{\mu {\alpha }_{2}\cdots {\alpha }_{k}}={D}_{\mu }{{\rm{\Psi }}}_{\ \ {\alpha }_{2}\cdots {\alpha }_{k}}^{\mu }={{\rm{\Psi }}}_{\ \ \mu {\alpha }_{3}\cdots {\alpha }_{k}}^{\mu }=0,\end{eqnarray}$where the gamma matrix γ^μ satisfies the commutation relation:(4)$\begin{eqnarray}{\gamma }^{\mu }{\gamma }^{\nu }+{\gamma }^{\nu }{\gamma }^{\mu }=2{g}^{\mu \nu }I.\end{eqnarray}$In equation (2),(5)$\begin{eqnarray}{D}_{\mu }={\partial }_{\mu }+{{\rm{\Omega }}}_{\mu }+\displaystyle \frac{{\rm{i}}}{{\hslash }}{{eA}}_{\mu },\end{eqnarray}$where Ω_μ is the spin-connection in the curved spacetime.

Considering the dispersion relationship derived from string theory and quantum gravitational theory, we get [27–32](6)$\begin{eqnarray}{P}_{0}^{2}={p}^{2}+{m}^{2}-{\left({{LP}}_{0}\right)}^{\alpha }{p}^{2},\end{eqnarray}$where m is the static mass of the fermion, P₀ and p are the energy and momentum of the particle in the natural unit, respectively, and L is a constant in the Plank magnitude. Choosing α=2, equation (1) changes to(7)$\begin{eqnarray}[{\bar{\gamma }}^{\mu }{\partial }_{\mu }+\displaystyle \frac{m}{{\hslash }}-\lambda {\hslash }{\bar{\gamma }}^{t}{\partial }_{t}{\bar{\gamma }}^{j}{\partial }_{j}]{{\rm{\Psi }}}_{{\alpha }_{1}\cdots {\alpha }_{k}}=0,\end{eqnarray}$where $\lambda ={\rm{i}}L/{\hslash }$. In the modification term of equation (7), $\lambda \lt \lt 1$, so $\lambda {\hslash }{\bar{\gamma }}^{t}{\partial }_{t}{\bar{\gamma }}^{j}{\partial }_{j}$ is a very small correction term. In the curved spacetime, a general dynamical equation for fermions with arbitrary spin, i.e. the Rarita–Schwinger equation, can be rewritten as(8)$\begin{eqnarray}[{\gamma }^{\mu }{D}_{\mu }+\displaystyle \frac{m}{{\hslash }}-\lambda {\hslash }{\gamma }^{t}{D}_{t}{\gamma }^{j}{D}_{j}]{{\rm{\Psi }}}_{{\alpha }_{1}\cdots {\alpha }_{k}}=0.\end{eqnarray}$The solution of equation (8) depends on specific curved spacetime. In the following, we research equation (8) in terms of the Einstein–Maxwell–dilaton–axion black hole.

The line element of the Einstein–Maxwell–dilaton–axion black hole is [35, 36](9)$\begin{eqnarray}\begin{array}{rcl}{\rm{d}}{s}^{2} & = & {g}_{00}{\rm{d}}{t}^{2}+{g}_{11}{\rm{d}}{r}^{2}+{g}_{22}{\rm{d}}{\theta }^{2}+{g}_{33}{\rm{d}}{\varphi }^{2}+2{g}_{03}{\rm{d}}t{\rm{d}}\varphi \\ & = & -\displaystyle \frac{{\rm{\Delta }}-{a}^{2}{\sin }^{2}\theta }{{\rm{\Sigma }}}{\rm{d}}{t}^{2}+\displaystyle \frac{{\rm{\Sigma }}}{{\rm{\Delta }}}{\rm{d}}{r}^{2}+{\rm{\Sigma }}{\rm{d}}{\theta }^{2}\\ & & +\displaystyle \frac{{\left({r}^{2}+{a}^{2}-2{Dr}\right)}^{2}-{\rm{\Delta }}{a}^{2}{\sin }^{2}\theta }{{\rm{\Sigma }}}{\sin }^{2}\theta {\rm{d}}{\varphi }^{2}\\ & & +\displaystyle \frac{2a{\sin }^{2}\theta }{{\rm{\Sigma }}}[{\rm{\Delta }}-({r}^{2}+{a}^{2}-2{Dr})]{\rm{d}}t{\rm{d}}\varphi ,\end{array}\end{eqnarray}$where(10)$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }} & = & {r}^{2}+{a}^{2}-2{Mr},\\ {\rm{\Sigma }} & = & {r}^{2}-2{Dr}+{a}^{2}{\cos }^{2}\theta ,\end{array}\end{eqnarray}$where M, a and D are the mass, angular momentum of the unit mass and magnetic charge of the black hole. From equations (9) and (10), one can obtain the metric determinant $g=-{{\rm{\Sigma }}}^{2}{\sin }^{2}\theta $, so the non-zero inverse metric tensors ${g}^{\mu \nu }$ are(11)$\begin{eqnarray}\begin{array}{rcl}{g}^{{tt}} & = & -\displaystyle \frac{{\left({r}^{2}+{a}^{2}-2{Dr}\right)}^{2}-{\rm{\Delta }}{a}^{2}{\sin }^{2}\theta }{{\rm{\Delta }}{\rm{\Sigma }}},\\ {g}^{t\varphi } & = & -\displaystyle \frac{a[({r}^{2}+{a}^{2}-2{Dr})-{\rm{\Delta }}]}{{\rm{\Delta }}{\rm{\Sigma }}},\\ {g}^{{rr}} & = & \displaystyle \frac{{\rm{\Delta }}}{{\rm{\Sigma }}},\\ {g}^{\theta \theta } & = & \displaystyle \frac{1}{{\rm{\Sigma }}},\\ {g}^{\varphi \varphi } & = & \displaystyle \frac{{\rm{\Delta }}-{a}^{2}{\sin }^{2}\theta }{{\rm{\Sigma }}{\rm{\Delta }}{\sin }^{2}\theta }.\end{array}\end{eqnarray}$According to the null super-surface equation(12)$\begin{eqnarray}{g}^{\mu \nu }\displaystyle \frac{\partial f}{\partial {x}^{\mu }}\displaystyle \frac{\partial f}{\partial {x}^{\nu }}=0,\end{eqnarray}$the event horizon r_H satisfies the equation(13)$\begin{eqnarray}{\rm{\Delta }}({r}_{H})=0,\end{eqnarray}$i.e. ${r}_{H}=M+\sqrt{{M}^{2}-{a}^{2}}$. Assuming the wave function of equation (8) can be expressed as(14)$\begin{eqnarray}{{\rm{\Psi }}}_{{\alpha }_{1}\cdots {\alpha }_{k}}={\phi }_{{\alpha }_{1}\cdots {\alpha }_{k}}{{\rm{e}}}^{\tfrac{{\rm{i}}}{{\hslash }}S},\end{eqnarray}$and substituting equation (14) into equation (8), keeping the lowest order term of ℏ, we get(15)$\begin{eqnarray}\begin{array}{l}{\rm{i}}{\gamma }^{\mu }({\partial }_{\mu }S+{{eA}}_{\mu }){\phi }_{{\alpha }_{1}\cdots {\alpha }_{k}}+m{\phi }_{{\alpha }_{1}\cdots {\alpha }_{k}}\\ \quad +\,\lambda {\gamma }^{t}({\partial }_{t}S+{{eA}}_{t})({\partial }_{j}S+{{eA}}_{j}){\phi }_{{\alpha }_{1}\cdots {\alpha }_{k}}=0.\end{array}\end{eqnarray}$This equation can be rewritten as(16)$\begin{eqnarray}{{\rm{\Gamma }}}^{\mu }({\partial }_{\mu }S+{{eA}}_{\mu }){\phi }_{{\alpha }_{1}\cdots {\alpha }_{k}}+{m}_{E}{\phi }_{{\alpha }_{1}\cdots {\alpha }_{k}}=0,\end{eqnarray}$where(17)$\begin{eqnarray}\begin{array}{l}{{\rm{\Gamma }}}^{\mu }={\rm{i}}{\gamma }^{\mu }+\lambda ({\partial }_{\nu }S+{{eA}}_{t}){\gamma }^{t}{\gamma }^{u},\\ {m}_{E}=m+\lambda {g}^{{tt}}({\partial }_{t}S+{{eA}}_{t}),\\ {\gamma }^{\mu }({\partial }_{\mu }S+{{eA}}_{\mu })={\gamma }^{t}({\partial }_{t}S+{{eA}}_{t})+{\gamma }^{j}({\partial }_{j}S+{{eA}}_{j}).\end{array}\end{eqnarray}$In equation (16), ${A}_{\mu }=({A}_{t},0,0,{A}_{\varphi })$. Due to the axial symmetry of spacetime, A_r=A_θ=0. Multiplying ${{\rm{\Gamma }}}^{\beta }({\partial }_{\beta }S+{{eA}}_{\beta })$ at both sides of equation (16), equation (16) becomes(18)$\begin{eqnarray}{{\rm{\Gamma }}}^{\beta }({\partial }_{\beta }S+{{eA}}_{\beta }){{\rm{\Gamma }}}^{\mu }({\partial }_{\mu }S+{{eA}}_{\mu }){\phi }_{{\alpha }_{1}\cdots {\alpha }_{k}}-{m}_{E}^{2}{\phi }_{{\alpha }_{1}\cdots {\alpha }_{k}}=0,\end{eqnarray}$where β and μ can be exchanged. An equivalent equation to equation (18) is(19)$\begin{eqnarray}{{\rm{\Gamma }}}^{\mu }({\partial }_{\mu }S+{{eA}}_{\mu }){{\rm{\Gamma }}}^{\beta }({\partial }_{\beta }S+{{eA}}_{\beta }){\phi }_{{\alpha }_{1}\cdots {\alpha }_{k}}-{m}_{E}^{2}{\phi }_{{\alpha }_{1}\cdots {\alpha }_{k}}=0.\end{eqnarray}$Combining equations (17), (18) and (19), we get(20)$\begin{eqnarray}\begin{array}{l}[{g}^{\mu \nu }({\partial }_{\mu }S+{{eA}}_{\mu })({\partial }_{\nu }S+{{eA}}_{\nu })+{m}^{2}\\ \quad +\,2m\lambda {g}^{{tt}}({\partial }_{t}S+{{eA}}_{t})\\ \quad -\,2i\lambda ({\partial }_{t}S+{{eA}}_{t}){g}^{t\beta }({\partial }_{\beta }S+{{eA}}_{\beta })\\ \quad \times \,{\gamma }^{\mu }({\partial }_{\mu }S+{{eA}}_{\mu })+{\lambda }^{2}{\left({g}^{{tt}}\right)}^{2}{\left({\partial }_{t}S+{{eA}}_{t}\right)}^{2}\\ \quad -\,{\lambda }^{2}{\left({\partial }_{t}S+{{eA}}_{t}\right)}^{2}{g}^{t\beta }{g}^{t\mu }({\partial }_{\beta }S+{{eA}}_{\beta })\\ \quad \times \,({\partial }_{\mu }S+{{eA}}_{\mu })]{\phi }_{{\alpha }_{1}\cdots {\alpha }_{k}}=0.\end{array}\end{eqnarray}$This is a matrix equation, and can be simplified as(21)$\begin{eqnarray}{\rm{i}}\lambda {\gamma }^{\mu }({\partial }_{\mu }S+{{eA}}_{\mu }){\phi }_{{\alpha }_{1}\cdots {\alpha }_{k}}+{m}_{D}{\phi }_{{\alpha }_{1}\cdots {\alpha }_{k}}=0,\end{eqnarray}$where(22)$\begin{eqnarray}\begin{array}{rcl}{m}_{D} & = & \displaystyle \frac{{g}^{\mu \nu }({\partial }_{\nu }S+{{eA}}_{\mu })({\partial }_{\nu }S+{{eA}}_{\nu })+{m}^{2}+({\partial }_{t}S+{{eA}}_{t}){\lambda }^{{\prime} }}{-2\lambda ({\partial }_{t}S+{{eA}}_{t}){g}^{t\beta }({\partial }_{\beta }S+{{eA}}_{\beta })},\\ {\lambda }^{{\prime} } & = & 2m\lambda {g}^{{tt}}+{\lambda }^{2}{\left({g}^{{tt}}\right)}^{2}({\partial }_{t}S+{{eA}}_{t})\\ & & -{\lambda }^{2}({\partial }_{t}S+{{eA}}_{t}){g}^{t\beta }{g}^{t\mu }({\partial }_{\beta }S+{{eA}}_{\beta })({\partial }_{\mu }S+{{eA}}_{\mu }).\end{array}\end{eqnarray}$Multiplying $-{\rm{i}}\lambda {\gamma }^{\mu }({\partial }_{\nu }S+{{eA}}_{\nu })$ at both sides of equation (21), we get(23)$\begin{eqnarray}{\lambda }^{2}{\gamma }^{\mu }{\gamma }^{\nu }({\partial }_{\mu }S+{{eA}}_{\mu })({\partial }_{\nu }S+{{eA}}_{\nu }){\phi }_{{\alpha }_{1}\cdots {\alpha }_{k}}+{m}_{D}^{2}{\phi }_{{\alpha }_{1}\cdots {\alpha }_{k}}=0.\end{eqnarray}$An equivalent equation to equation (23) is(24)$\begin{eqnarray}[{\lambda }^{2}{\gamma }^{\nu }{\gamma }^{\mu }({\partial }_{\nu }S+{{eA}}_{\nu })({\partial }_{\mu }S+{{eA}}_{\mu })+{m}_{D}^{2}]{\phi }_{{\alpha }_{1}\cdots {\alpha }_{k}}=0.\end{eqnarray}$Combining equations (22), (23), (24) and (4), we get(25)$\begin{eqnarray}\begin{array}{l}\left\{{\left[\displaystyle \frac{({g}^{\mu \nu }({\partial }_{\mu }S+{{eA}}_{\mu })({\partial }_{\nu }S+{{eA}}_{\nu })+{m}^{2}+({\partial }_{t}S+{{eA}}_{t}){\lambda }^{{\prime} }}{2({\partial }_{t}S+{{eA}}_{t}){g}^{t\beta }({\partial }_{\beta }S+{{eA}}_{\beta })}\right]}^{2}\right.\\ \quad \left.-\,{\lambda }^{2}{m}^{2}\right\}{\phi }_{{\alpha }_{1}\cdots {\alpha }_{k}}=0.\end{array}\end{eqnarray}$The condition for this matrix equation to have nontrivial solutions is that the value of the determinant of the eigen matrix equation is zero, i.e.(26)$\begin{eqnarray}\begin{array}{l}{g}^{\mu \nu }({\partial }_{\mu }S+{{eA}}_{\mu })({\partial }_{\nu }S+{{eA}}_{\nu })+{m}^{2}\\ \quad -\,2m\lambda ({\partial }_{t}S+{{eA}}_{t})[{g}^{t\beta }({\partial }_{\beta }S+{{eA}}_{\beta })-{g}^{{tt}}]\\ \quad -\,{\lambda }^{2}({\partial }_{t}S+{{eA}}_{t})[{\left({g}^{{tt}}\right)}^{2}{\left({\partial }_{t}S+{{eA}}_{t}\right)}^{2}\\ \quad +{\left({g}^{t\varphi }\right)}^{2}{\left({\partial }_{\varphi }S+{{eA}}_{\varphi }\right)}^{2}-{\left({g}^{{tt}}\right)}^{2}]=0,\end{array}\end{eqnarray}$or(27)$\begin{eqnarray}\begin{array}{l}{g}^{{tt}}{\left({\partial }_{t}S+{{eA}}_{t}\right)}^{2}+2{g}^{t\varphi }({\partial }_{t}S+{{eA}}_{t})({\partial }_{\varphi }S+{{eA}}_{\varphi })\\ \quad +\,{g}^{{rr}}{\left({\partial }_{r}S\right)}^{2}+{g}^{\theta \theta }{\left({\partial }_{\theta }S\right)}^{2}\\ \quad +\,{g}^{\varphi \varphi }{\left({\partial }_{\varphi }S+{{eA}}_{\varphi }\right)}^{2}+{m}^{2}-2m\lambda ({\partial }_{t}S+{{eA}}_{t})\\ \quad \times [{g}^{{tt}}({\partial }_{t}S+{{eA}}_{t})-{g}^{{tt}}+{g}^{t\varphi }({\partial }_{\varphi }S+{{eA}}_{\varphi })]\\ \quad -\,{\lambda }^{2}({\partial }_{t}S+{{eA}}_{t})[{\left({g}^{{tt}}\right)}^{2}{\left({\partial }_{t}S+{{eA}}_{t}\right)}^{2}\\ \quad +\,{\left({g}^{t\varphi }\right)}^{2}{\left({\partial }_{\varphi }S+{{eA}}_{\varphi }\right)}^{2}-{\left({g}^{{tt}}\right)}^{2}]=0.\end{array}\end{eqnarray}$This is the dynamical equation of fermions with arbitrary spin in the Einstein–Maxwell–dilaton–axion black hole spacetime. It is an accurate dynamical equation after modification. If the modification term λ is ignored, it returns to the Hamilton–Jacobi equation. Therefore, we called equation (27) the Rarita–Schwinger–Hamilton–Jacobi equation. We find that the dynamical equation of fermions with different spin is decided by the line element of spacetime if their wave functions adopt the WKB approximation, named after Wentzel, Kramers, and Brillouin. Moreover, the Hamilton–Jacobi equation can be derived not only from the Rarita–Schwinger equation or the Dirac equation but also from the Klein–Gordon equation [37]. In the following, we will research the quantum tunneling radiation and the corrections to some quantities of the Einstein–Maxwell–dilaton–axion black hole by solving equation (27).

3. Quantum tunneling radiation of fermions with arbitrary spin at the event horizon of the Einstein–Maxwell–dilaton–axion black hole

The line element (9) indicates that there are killing vectors ∂_t and ∂_φ in this curved spacetime. Hence, the action of fermions with arbitrary spin in equation (27) can be separated as(28)$\begin{eqnarray}S=-\omega t+R(r)+{\rm{\Theta }}(\theta )+n\varphi ,\end{eqnarray}$where ω is the energy of the particle, and n is the general angular momentum in the φ direction. Substituting equations (10), (11) and (28) into the modified fermion dynamical equation (27), equation (27) near the event horizon of the black hole can be further expressed as(29)$\begin{eqnarray}\begin{array}{l}{{\rm{\Delta }}}^{2}{\left(\displaystyle \frac{{\rm{d}}R}{{\rm{d}}r}\right)}^{2}-{\left[({r}^{2}+{a}^{2}-2{Dr})\omega -{an}-{{eA}}_{t}\right]}^{2}\\ \quad -\,2\lambda m(\omega -{{eA}}_{t})[{\left({r}^{2}+{a}^{2}-2{Dr}\right)}^{2}\\ (\omega -{{eA}}_{t})+{\left({r}^{2}+{a}^{2}-2{Dr}\right)}^{2}\\ \quad -\,a(n+{{eA}}_{\varphi })({r}^{2}+{a}^{2}-2{Dr})]-{\lambda }^{2}(\omega -{{eA}}_{t})\\ \left[{\left(\omega -{{eA}}_{t}\right)}^{2}{\left({r}^{2}+{a}^{2}-2{Dr}\right)}^{4}\right.\\ \quad -\,{\left({r}^{2}+{a}^{2}-2{Dr}\right)}^{4}+{n}^{2}{\left({r}^{2}+{a}^{2}-2{Dr}\right)}^{2}]=0.\end{array}\end{eqnarray}$For simplication and clearance, one can ignore the term $\unicode{x000F8}({\lambda }^{2})$, but we keep it in this paper. At the event horizon of the black hole, there must be a radial derivative of general angular momentum as follows(30)$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\partial S}{\partial r}{| }_{r\to {r}_{H}}=\displaystyle \frac{{\rm{d}}R}{{\rm{d}}r}{| }_{r\to {r}_{H}}=\pm \displaystyle \frac{1}{{\rm{\Delta }}{| }_{r\to {r}_{H}}}\{[({r}^{2}+{a}^{2}\\ \quad {\left.-2{Dr})\omega -{an}-{{eA}}_{t}\right]}^{2}+2\lambda m(\omega -{{eA}}_{t})\\ ({r}^{2}+{a}^{2}-2{Dr})[({r}^{2}+{a}^{2}-2{Dr})(\omega -{{eA}}_{t})\\ \quad +\,({r}^{2}+{a}^{2}-2{Dr})-{na}]+{\lambda }^{2}(\omega -{{eA}}_{t})\\ {\left({r}^{2}+{a}^{2}-2{Dr}\right)}^{2}[{\left(\omega -{{eA}}_{t}\right)}^{2}{\left({r}^{2}+{a}^{2}-2{Dr}\right)}^{2}\\ \quad {\left.-{\left({r}^{2}+{a}^{2}-2{Dr}\right)}^{2}+{n}^{2}]\right\}}^{1/2}\end{array}\end{eqnarray}$where±corresponds to an outgoing or incoming wave solution at the horizon of the Einstein–Maxwell–dilaton–axion black hole. Here, ${\rm{\Delta }}{| }_{r\to {r}_{H}}=0$, so we can assume the event horizon of the black hole as a singular point. Adopting the integral from the inner part to the outer part by using the residue theorem, the fermion action S_± at the event horizon is(31)$\begin{eqnarray}\begin{array}{rcl}{S}_{\pm } & = & \pm \displaystyle \frac{{\rm{i}}\pi }{{{\rm{\Delta }}}^{{\prime} }({r}_{H})}({r}_{H}^{2}+{a}^{2}-2{{Dr}}_{H})(\omega -e{{\rm{\Phi }}}_{0}\\ & & -n{\rm{\Omega }})(1+\lambda {A}_{0}+{\lambda }^{2}{B}_{0})+{S}^{{\prime} },\end{array}\end{eqnarray}$where(32)$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Delta }}}^{{\prime} }({r}_{H}) & = & {\left.\displaystyle \frac{{\rm{d}}{\rm{\Delta }}}{{\rm{d}}r}\right|}_{r={r}_{H}}=2{r}_{H}-2M,\\ {{\rm{\Phi }}}_{0} & = & \displaystyle \frac{{A}_{t}}{{r}_{H}^{2}+{a}^{2}-2{{Dr}}_{H}},\\ {\rm{\Omega }} & = & \displaystyle \frac{a}{{r}_{H}^{2}+{a}^{2}-2{{Dr}}_{H}},\\ {A}_{0} & = & 2m(\omega -{{eA}}_{t})({r}_{H}^{2}+{a}^{2}-2{{Dr}}_{H})\\ & & \times [(\omega -{{eA}}_{t})({r}_{H}^{2}+{a}^{2}-2{{Dr}}_{H})\\ & & +({r}_{H}^{2}+{a}^{2}-2{{Dr}}_{H})-{na}]\\ & & \times {\left[({r}_{H}^{2}+{a}^{2}-2{{Dr}}_{H})(\omega -{{eA}}_{t})-{an}+{{aeA}}_{t}\right]}^{-2},\\ {B}_{0} & = & (\omega -{{eA}}_{t}){\left({r}_{H}^{2}+{a}^{2}-2{{Dr}}_{H}\right)}^{2}\\ & & \times [{\left(\omega -{{eA}}_{t}\right)}^{2}{\left({r}_{H}^{2}+{a}^{2}-2{{Dr}}_{H}\right)}^{2}\\ & & -{\left({r}_{H}^{2}+{a}^{2}-2{{Dr}}_{H}\right)}^{2}+{n}^{2}]\\ & & \times {\left[({r}_{H}^{2}+{a}^{2}-2{{Dr}}_{H})(\omega -{{eA}}_{t})-{an}+{{aeA}}_{t}\right]}^{-2}.\end{array}\end{eqnarray}$Here, ${S}^{{\prime} }$ in equation (31) is an integral term besides the integral in the radial direction, which will naturally disappear in the calculation of the tunneling rate.

According to the tunneling theory of a black hole, the tunneling rate is(33)$\begin{eqnarray}\begin{array}{rcl}{\rm{\Gamma }} & = & \exp [-2(\mathrm{Im}{S}_{+}-\mathrm{Im}{S}_{-})]\\ & = & \exp \left(-\displaystyle \frac{\omega -e{{\rm{\Phi }}}_{0}-n{{\rm{\Omega }}}_{0}}{{T}_{{\rm{H}}}^{{\prime} }}\right),\end{array}\end{eqnarray}$where ${T}_{{\rm{H}}}^{{\prime} }$ is the Hawking temperature after modification. Therefore, the Hawking temperature corresponding to the tunneling radiation of fermions with arbitrary spin at the event horizon of the black hole is(34)$\begin{eqnarray}\begin{array}{ccc}{T}_{{\rm{H}}}^{{\rm{{\prime} }}} & = & \displaystyle \frac{{r}_{H}-M}{\left.2\pi ({r}_{H}^{2}+{a}^{2}-2{{Dr}}_{H}\right)}(1-\lambda {A}_{0}-{\lambda }^{2}{B}_{0}+\cdots )\\ & = & {T}_{0}(1-\lambda {A}_{0}-{\lambda }^{2}{B}_{0}+\cdots ).\end{array}\end{eqnarray}$Due to ${T}_{{\rm{H}}}=\tfrac{\kappa }{2\pi }$, the surface gravitation at the event horizon of this black hole is(35)$\begin{eqnarray}\kappa =\displaystyle \frac{{r}_{H}-M}{{r}_{H}^{2}+{a}^{2}-2{{Dr}}_{H}}(1-\lambda {A}_{0}-{\lambda }^{2}{B}_{0}+\cdots ).\end{eqnarray}$Obviously, κ is a constant, which is consistent with the zeroth law of black hole thermodynamics. According to the zeroth law of thermodynamics of a black hole, surface gravitation κ of a stationary black hole is a constant. Equations (33), (34) and (35) are the new results we get from the accurate modification of fermion quantum tunneling and by adopting the Lorentz invariance violation theory in the Einstein–Maxwell–dilaton–axion black hole spacetime. Another important physical quantity in the thermodynamics of a black hole is entropy. The modification to the Hawking temperature must change the entropy of the black hole. According to the first law of thermodynamics of a black hole,(36)$\begin{eqnarray}{\rm{d}}M=T{\rm{d}}S+V{\rm{d}}J+U{\rm{d}}{Q}^{{\prime} },\end{eqnarray}$where V and U are the rotation angular velocity and electric potential of the black hole, respectively. Denoting the entropy without modification as S₀, and temperature without modification as T₀, they have a relationship(37)$\begin{eqnarray}{\rm{d}}{S}_{0}=\displaystyle \frac{{\rm{d}}M-V{\rm{d}}J-U{\rm{d}}{Q}^{{\prime} }}{{T}_{0}}.\end{eqnarray}$From equation (34), the entropy after accurate modification at the event horizon of a black hole is(38)$\begin{eqnarray}{S}_{H}={S}_{0}(1+\lambda {A}_{0}+{\lambda }^{2}{B}_{0}+\cdots ).\end{eqnarray}$These results show that the tunneling rate, surface gravitational force, Hawking temperature and entropy should all be modified by a term related to the parameters λ and λ². Although it is a minor modification term, it is still valuable for further research.

4. Conclusion

According to the deformed dispersion relationship, we accurately modify the tunneling radiation for fermions with arbitrary spin in the Einstein–Maxwell–dilaton–axion black hole; then, we obtain a series of new results, such as the temperature, surface gravitation and entropy of the black hole. If the λ and λ² terms are ignored, the results can return to those without modification. If a and D are zero in equations (34), (35) and (38), we will go back to the case of the Schwarschild black hole. This supplementary information means that our results are correct by other means [12, 14]. Our research includes fermions with arbitrary semi-integer spin, such as 1/2, 3/2, ⋯. For the deformed dispersion relationship, we have just chosen the parameter α=2 during the modification to the Rarita–Schwinger equation in our research. In fact, more general relationships are also needed. In this paper, we researched a representative stationary axial-symmetry black hole; more types of black hole spacetime should be investigated in the future. Our research is based on current theoretical results, but new experiments and self-consistent theories may provide support to some extent for our work. All the above topics are significant and beneficial for theoretical physics and astrophysics.