1. Introduction
Black holes are a special type of celestial bodies existing in the Universe, including static black holes, stationary black holes and dynamic black holes. The curved space-time metric associated with a black hole possess is a solution to Einstein’s field equation. Hawking found the black hole’s thermal radiation, called Hawking radiation, by studying quantum effects near the black hole’s event horizon [1, 2]. Hawking radiation effectively linked gravity theory, quantum theory and thermodynamic statistics physics, and inspired researchers to study the evolution of black hole thermodynamics [3–6]. A theory that can really explain Hawking radiation is the quantum tunneling radiation theory, namely the event horizon of the black hole is taken as a barrier and due to the quantum tunneling effect, virtual particles inside the horizon have a certain probability to go through this barrier and be converted to real particles, then these particles are radiated out of the black hole. Literature [7–20] used the quantum tunneling radiation method to study Hawking temperature and entropy of the black hole. The semi-classical theory proposed in literature [12, 13] can deduce the Hamilton–Jacobi equation in curved space-time from the scalar field equation in curved space-time. Kerner and Mann et al studied the tunneling radiation of Dirac particles using the semi-classical theory [21–23]. Lin and Yang proposed a new method to study the quantum tunneling radiation of fermions [24–29]. Their method can also be used to study the quantum tunneling radiation of bosons. The results obtained in [24–27] show that the Hamilton–Jacobi equation in curved space-time is the basic equation of particle dynamics, which reflects the inherent consistency between Lorentz symmetry theory and the Jamilton–Jacobi equation.
General relativity is a theory of gravity that cannot be renormalized, so several modified gravity theories have been proposed. Since researchers realized that Lorentz symmetry, the cornerstone of general relativity, may break at high energy cases, various gravity models based on Lorentz symmetry violation have been proposed [30–32]. In principle, Lorentz symmetry violation theory can solve the problem of irrenormalization of gravity theory. In addition, some studies on Lorentz symmetry violation suggest that the dark matter theory may be just one of the effects of theoretical models of Lorentz symmetry violation [33]. In the fields of string theory, electrodynamics and non-abel theory, Lorentz symmetry violation has attracted extensive attention [34–36]. In recent years, the Lorentz symmetry violational Dirac equation in flat space-time has been studied by introducing ether-like field terms, and the quantum correction of ether-like field terms was further studied [37, 38]. In this theory, the existence of ether-like field leads to the disappearance of Lorentz symmetry of the space-time. Therefore, properties that are inconsistent with Lorentz symmetry theory will emerge at high energy. These topics are worthy of study. On the other hand, for the dynamics of fermions in curved space-time, the modification of quantum tunneling radiation caused by ether-like field is also a subject worthy of further study when Lorentz symmetry violation is considered. At present, the quantum tunneling radiation of Dirac particles with ether-like field terms has been investigated merely in spherically symmetric black holes [39].
Due to the fact that, according to the quantum gravity theory, Lorentz relation should be corrected at high energy, we expect the dynamic equation of fermions will also be corrected to some extent in the curved Kerr anti-de-Sitter space-time. The metric of Kerr anti-de-Sitter space-time is one solution of Einstein field equation, so we can study the impact of Lorentz symmetry violation on the quantum tunneling radiation of fermions in this black hole. In this paper, the quantum tunneling radiation of fermions is corrected in the axisymmetric stationary Kerr anti-de-Sitter black hole by considering the ether-like field term. Our work is organized as follows: In section 2 , according to Lorentz symmetry violation theory, the dynamic equation of fermions with spin 1/2 is derived for Kerr–Newman black hole. Section 3 will solve this dynamic equation and obtain the corrected physical quantities such as Hawking temperature and entropy of the black hole. The last section concludes our work.
2. Lorentz symmetry violation theory and Dirac–Hamilton–Jacobi equation
Nascimento et al [38] studied the particle action and the Dirac equation in flat space-time based on Lorentz symmetry violation theory. Generalizing the ordinary derivatives in flat space-time to the covariant derivatives in curved space-time, and the commutation relation of Gamma matrices ${\bar{\gamma }}^{\mu }$ and ${\bar{\gamma }}^{\nu }$ in flat space-time to the commutation relation in curved space-time, we can get the Dirac equation with Lorentz symmetry violation for fermions with spin 1/2 and mass m in the curved space-time, namely [38]2 ) turns to ${\bar{\gamma }}^{\mu }{\bar{\gamma }}^{\nu }+{\bar{\gamma }}^{\nu }{\bar{\gamma }}^{\mu }=2{\eta }^{\mu \nu }I$ and equation (3 ) turns to ${\bar{\gamma }}^{5}{\bar{\gamma }}^{\mu }+{\bar{\gamma }}^{\mu }{\bar{\gamma }}^{5}=0$ in flat space-time. In equation (1 ), the general relativity derivative ${D}_{\mu }$ is defined as1 ), a, b and c are all small quantities. Specifically, $\tfrac{a}{{m}^{2}}\ll 1$, $\tfrac{b}{m}\ll 1$ and $\tfrac{c}{m}\ll 1$. ${u}^{\alpha }$ represents the ether-like field, which is a constant vector in the flat space-time. In the curved space-time, ${u}^{\alpha }$ is no longer a constant, but must meet the following condition:1 ) for fermions with spin 1/2, suppose4 ) and (6 ) into (1 ), and noticing ℏ is a small quantity, we can change equation (1 ) to2 ) we get8 ), equation (7 ) can be further converted to2 ), the above equation becomes1 ) to (13 ), we not only get a new dynamic equation of Dirac particles, but also a deformed Hamilton–Jacobi equation (i.e. equation (13 )) which is named as Dirac–Hamilton–Jacobi equation. S is the Hamilton principal function whose expression depends on the selected coordinate system. In a stationary space-time, $S=S(t,r,\theta ,\phi )$.
$\begin{eqnarray}\begin{array}{l}\left\{{\gamma }^{\mu }{D}_{\mu }[1+\displaystyle \frac{{{\hslash }}^{2}a}{{m}^{2}}{({\gamma }^{\mu }{D}_{\mu })}^{2}]\\ \,\,+\,\displaystyle \frac{b}{{\hslash }}{\gamma }^{5}+c{\hslash }{({u}^{\alpha }{D}_{\alpha })}^{2}-\displaystyle \frac{m}{{\hslash }}\right\}{\rm{\Psi }}=0,\end{array}\end{eqnarray}$
where ${\gamma }^{\mu }$ satisfies the following anti-commutation relations: $\begin{eqnarray}{\gamma }^{\mu }{\gamma }^{\nu }+{\gamma }^{\nu }{\gamma }^{\mu }=2{g}^{\mu \nu }I,\end{eqnarray}$
$\begin{eqnarray}{\gamma }^{5}{\gamma }^{\mu }+{\gamma }^{\mu }{\gamma }^{5}=0.\end{eqnarray}$
Equation ( $\begin{eqnarray}{D}_{\mu }={\partial }_{\mu }+\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Gamma }}}_{\mu }^{\alpha \beta }{{\rm{\Pi }}}_{\alpha \beta },\end{eqnarray}$
where ${{\rm{\Pi }}}_{\alpha \beta }=\tfrac{{\rm{i}}}{4}[{\gamma }^{\alpha },{\gamma }^{\beta }]$. In equation ( $\begin{eqnarray}{u}^{\alpha }{u}_{\alpha }=\mathrm{const}.\end{eqnarray}$
To solve the Dirac equation ( $\begin{eqnarray}{\rm{\Psi }}={{\rm{\Psi }}}_{{AB}}{{\rm{e}}}^{-\tfrac{{\rm{i}}S}{{\hslash }}}=\left(\begin{array}{c}A\\ B\end{array}\right){{\rm{e}}}^{-\tfrac{{\rm{i}}}{{\hslash }}S},\end{eqnarray}$
where ${\psi }_{{AB}}$ is the coefficient column matrix of ψ, A and B are column matrices of 2 × 1, and S is the Hamilton principal function. Substituting equations ( $\begin{eqnarray}\begin{array}{l}[{\rm{i}}{\gamma }^{\mu }{\partial }_{\mu }S\left(1-\displaystyle \frac{a}{{m}^{2}}{\gamma }^{\alpha }{\gamma }^{\beta }{\partial }_{\alpha }S{\partial }_{\beta }S\right)\\ \,-{{cu}}^{\alpha }{u}^{\beta }({\partial }_{\alpha }S{\partial }_{\beta }S)+b{\gamma }^{5}-m]{\rm{\Psi }}=0.\end{array}\end{eqnarray}$
Using equation ( $\begin{eqnarray}{\gamma }^{\alpha }{\gamma }^{\beta }{\partial }_{\alpha }S{\partial }_{\beta }S={g}^{\alpha \beta }{\partial }_{\alpha }S{\partial }_{\beta }S.\end{eqnarray}$
According to equation ( $\begin{eqnarray}\begin{array}{l}({\rm{i}}{\gamma }^{\mu }{\partial }_{\mu }S){\rm{\Psi }}={\left(1-\displaystyle \frac{a}{{m}^{2}}{g}^{\alpha \beta }{\partial }_{\alpha }S{\partial }_{\beta }S\right)}^{-1}\\ \,\times ({{cu}}^{\alpha }{u}^{\beta }{\partial }_{\alpha }S{\partial }_{\beta }S-b{\gamma }^{5}+m){\rm{\Psi }}\\ =\,\left(1+\displaystyle \frac{a}{{m}^{2}}{g}^{\alpha \beta }{\partial }_{\alpha }S{\partial }_{\beta }S+{ \mathcal O }({a}^{2})\right)\\ \,\times \left({{cu}}^{\alpha }{u}^{\beta }{\partial }_{\alpha }S{\partial }_{\beta }S-b{\gamma }^{5}+m\right){\rm{\Psi }}\\ =\,\left[1+\left(\displaystyle \frac{c}{m}{u}^{\alpha }{u}^{\beta }+\displaystyle \frac{a}{{m}^{2}}{g}^{\alpha \beta }\right){\partial }_{\alpha }S{\partial }_{\beta }S-\displaystyle \frac{b}{m}{\gamma }^{5}\right]m{\rm{\Psi }}\\ =\,\left[1+\left(\displaystyle \frac{c}{m}{u}^{\alpha }{u}^{\beta }+\displaystyle \frac{a}{{m}^{2}}{g}^{\alpha \beta }\right){\partial }_{\alpha }S{\partial }_{\beta }S\right]m{\rm{\Psi }}.\end{array}\end{eqnarray}$
The last step has used the condition $\tfrac{b}{m}\ll 1$, while the first order of a and c terms, coupled with the partial derivative of S, are preserved. Multiplying both sides of this equation by ${\rm{i}}{\gamma }^{\nu }{\partial }_{\nu }S$ , we get $\begin{eqnarray}\begin{array}{l}(-{\gamma }^{\mu }{\gamma }^{\nu }{\partial }_{\mu }S{\partial }_{\nu }S){\rm{\Psi }}\\ =\,[{m}^{2}+2({{cmu}}^{\alpha }{u}^{\beta }+{{ag}}^{\alpha \beta }){\partial }_{\alpha }S{\partial }_{\beta }S]{\rm{\Psi }}.\end{array}\end{eqnarray}$
With the help of equation ( $\begin{eqnarray}\begin{array}{l}[{g}^{\mu \nu }{\partial }_{\mu }S{\partial }_{\nu }S+2({{cmu}}^{\mu }{u}^{\nu }+{{ag}}^{\mu \nu })\\ \,\times \,{\partial }_{\mu }S{\partial }_{\nu }S+{m}^{2}]{{\rm{\Psi }}}_{{AB}}=0.\end{array}\end{eqnarray}$
This is a matrix equation that has a nontrivial solution only if the following equation holds: $\begin{eqnarray}\begin{array}{l}{g}^{\mu \nu }{\partial }_{\mu }S{\partial }_{\nu }S+2({{cmu}}^{\mu }{u}^{\nu }+{{ag}}^{\mu \nu })\\ \,\times \,{\partial }_{\mu }S{\partial }_{\nu }S+{m}^{2}=0.\end{array}\end{eqnarray}$
Conducting some inferences and approximations can transform the above equation into $\begin{eqnarray}[{g}^{\mu \nu }(1+2a)+2{{cmu}}^{\mu }{u}^{\nu }]{\partial }_{\mu }S{\partial }_{\nu }S+{m}^{2}=0.\end{eqnarray}$
From equations (3. Correction to tunneling radiation of spin 1/2 fermion for Kerr anti-de-Sitter black hole
The Kerr anti-de-Sitter black hole is a rotating black hole. The vacuum solution of this black hole in Boyer–Lindquist coordinate system is expressed as [40, 41]14 ) and (15 ) that the non-zero covariant components of the metric tensor are19 ) has two real solutions, denoted by ${r}_{{H}^{-}}$ and ${r}_{{H}^{+}}$, that represent inner and outer event horizon, respectively [42, 43]. Substitute equation (17 ) into (13 ), and multiply the resulting equation by ${\rho }^{2}$, then we get the Hamilton principal function S of spin 1/2 fermions in Kerr anti-de-Sitter black hole that satisfies21 ) into (20 ), and considering the fact that equation (19 ) holds at the event horizon of the black hole, we can get the dynamic equation of spin 1/2 fermions at the event horizon of the black hole, that is,24 ) by expressing the Hamilton principal function S as14 ) is stationary and axisymmetric, j is a constant. Note that $\tfrac{\partial S}{\partial t}=-\omega $ and $\tfrac{\partial S}{\partial \phi }=j$ can be easily deduced from equation (25 ). Substituting equation (25 ) into (24 ) we get
$\begin{eqnarray}\begin{array}{rcl}{\rm{d}}{s}^{2} & = & -\displaystyle \frac{{{\rm{\Delta }}}_{r}}{{\rho }^{2}}{\left({\rm{d}}t-\displaystyle \frac{{a}_{k}}{{\rm{\Xi }}}{\sin }^{2}\theta {\rm{d}}\phi \right)}^{2}+\displaystyle \frac{{\rho }^{2}}{{{\rm{\Delta }}}_{r}}{\rm{d}}{r}^{2}+\displaystyle \frac{{\rho }^{2}}{{{\rm{\Delta }}}_{\theta }}{\rm{d}}{\theta }^{2}\\ & & +\displaystyle \frac{{{\rm{\Delta }}}_{\theta }{\sin }^{2}\theta }{{\rho }^{2}}{\left({a}_{k}{\rm{d}}t-\displaystyle \frac{{r}^{2}+{a}_{k}^{2}}{{\rm{\Xi }}}{\rm{d}}\phi \right)}^{2},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}{\rho }^{2} & = & {r}^{2}+{a}_{k}^{2}{\cos }^{2}\theta ,\\ {{\rm{\Delta }}}_{r} & = & \left({r}^{2}+{a}_{k}^{2}\right)\left(\displaystyle \frac{{r}^{2}}{{{\ell }}^{2}}+1\right)-2{Mr},\\ {{\rm{\Delta }}}_{\theta } & = & 1-\displaystyle \frac{{a}_{k}^{2}}{{{\ell }}^{2}}{\cos }^{2}\theta ,\\ {\rm{\Xi }} & = & 1-\displaystyle \frac{{a}_{k}^{2}}{{{\ell }}^{2}}.\end{array}\end{eqnarray}$
Here, M is the mass of the black hole, ak is the angular momentum per unit mass $\left({a}_{k}=\tfrac{J}{M}\right)$ of the black hole, and ℓ is associated with the cosmographic constant by ${{\ell }}^{2}=-\tfrac{3}{{\rm{\Lambda }}}$. It can be seen from equations ( $\begin{eqnarray}\begin{array}{rcl}{g}_{{tt}} & = & -\displaystyle \frac{{{\rm{\Delta }}}_{r}}{{\rho }^{2}}+\displaystyle \frac{{{\rm{\Delta }}}_{\theta }{a}_{k}^{2}{\sin }^{2}\theta }{{\rho }^{2}},\\ {g}_{\phi \phi } & = & -\displaystyle \frac{{{\rm{\Delta }}}_{r}{a}_{k}^{2}{\sin }^{4}\theta }{{\rho }^{2}{{\rm{\Xi }}}^{2}}+\displaystyle \frac{{{\rm{\Delta }}}_{\theta }{({r}^{2}+{a}_{k}^{2})}^{2}{\sin }^{2}\theta }{{\rho }^{2}{{\rm{\Xi }}}^{2}},\\ {g}_{t\phi } & = & \displaystyle \frac{{{\rm{\Delta }}}_{r}{a}_{k}{\sin }^{2}\theta }{{\rho }^{2}{\rm{\Xi }}}-\displaystyle \frac{{{\rm{\Delta }}}_{\theta }{a}_{k}({r}^{2}+{a}_{k}^{2}){\sin }^{2}\theta }{{\rho }^{2}{\rm{\Xi }}},\\ {g}_{{rr}} & = & \displaystyle \frac{{\rho }^{2}}{{{\rm{\Delta }}}_{r}},\\ {g}_{\theta \theta } & = & \displaystyle \frac{{\rho }^{2}}{{{\rm{\Delta }}}_{\theta }},\end{array}\end{eqnarray}$
and the metric determinant and the non-zero entries of the inverse metric tensor are $\begin{eqnarray}\begin{array}{rcl}g & = & -\displaystyle \frac{{\rho }^{4}{\sin }^{2}\theta }{{{\rm{\Xi }}}^{2}},\\ {g}^{{tt}} & = & \displaystyle \frac{{a}_{k}^{2}{\sin }^{2}\theta }{{\rho }^{2}{{\rm{\Delta }}}_{\theta }}-\displaystyle \frac{{({r}^{2}+{a}_{k}^{2})}^{2}}{{\rho }^{2}{{\rm{\Delta }}}_{r}},\\ {g}^{\phi \phi } & = & \displaystyle \frac{{{\rm{\Xi }}}^{2}}{{\rho }^{2}{{\rm{\Delta }}}_{\theta }{\sin }^{2}\theta }-\displaystyle \frac{{{\rm{\Xi }}}^{2}{a}_{k}^{2}}{{\rho }^{2}{{\rm{\Delta }}}_{r}},\\ {g}^{t\phi } & = & \displaystyle \frac{{\rm{\Xi }}{a}_{k}({r}^{2}+{a}_{k}^{2})}{{\rho }^{2}{{\rm{\Delta }}}_{r}}-\displaystyle \frac{{a}_{k}{\rm{\Xi }}}{{\rho }^{2}{{\rm{\Delta }}}_{\theta }},\\ {g}^{{rr}} & = & {g}^{11}=\displaystyle \frac{{{\rm{\Delta }}}_{r}}{{\rho }^{2}},\\ {g}^{\theta \theta } & = & {g}^{22}=\displaystyle \frac{{{\rm{\Delta }}}_{\theta }}{{\rho }^{2}}.\end{array}\end{eqnarray}$
As to the axisymmetric and stationary Kerr anti-de-Sitter black hole, according to the null hyper-surface equation: $\begin{eqnarray}{g}^{\mu \nu }\displaystyle \frac{\partial F}{\partial {x}^{\mu }}\displaystyle \frac{\partial F}{\partial {x}^{\nu }}=0,\end{eqnarray}$
we find that the event horizon of this black hole satisfies $\begin{eqnarray}({r}_{H}^{2}+{a}_{k}^{2})\left(\displaystyle \frac{{r}_{H}^{2}}{{{\ell }}^{2}}-1\right)-2{{Mr}}_{H}=0.\end{eqnarray}$
Equation ( $\begin{eqnarray}\begin{array}{l}(1+2a)\left\{{{\rm{\Delta }}}_{r}{\left(\displaystyle \frac{\partial S}{\partial r}\right)}^{2}\right.\\ -\,\displaystyle \frac{1}{{{\rm{\Delta }}}_{r}}{\left[({r}^{2}+{a}_{k}^{2})\displaystyle \frac{\partial S}{\partial t}+{a}_{k}{\rm{\Xi }}\displaystyle \frac{\partial S}{\partial \phi }\right]}^{2}\\ +\,\displaystyle \frac{1}{{{\rm{\Delta }}}_{\theta }}{\left[{a}_{k}\sin \theta \displaystyle \frac{\partial S}{\partial t}-\displaystyle \frac{{\rm{\Xi }}}{\sin \theta }\displaystyle \frac{\partial S}{\partial \phi }\right]}^{2}\\ \left.+\,{{\rm{\Delta }}}_{\theta }{\left(\displaystyle \frac{\partial S}{\partial \theta }\right)}^{2}\right\}+2{\rho }^{2}{{cmu}}^{\mu }{u}^{\nu }{\partial }_{\mu }S{\partial }_{\nu }S+{\rho }^{2}{m}^{2}=0.\end{array}\end{eqnarray}$
To solve this equation, proper expressions should be given for ut, ur, ${u}^{\theta }$ and ${u}^{\phi }$. The principles of designing these expressions are: first, ${u}^{\alpha }$ is not a constant, but it should guarantee ${u}^{\alpha }{u}_{\alpha }=\mathrm{const};$ second, ${u}^{\alpha }$ should be related to the properties of the metric tensor, since only the metric tensor can lift or lower the indices. Therefore, we use the following expressions for ut, ur, ${u}^{\theta }$ and ${u}^{\phi }$: $\begin{eqnarray}\begin{array}{rcl}{u}^{t} & = & \displaystyle \frac{{k}_{t}}{\sqrt{{g}_{{tt}}}}=\displaystyle \frac{{k}_{t}\rho }{{({{\rm{\Delta }}}_{\theta }{a}_{k}^{2}{\sin }^{2}\theta -{{\rm{\Delta }}}_{r})}^{1/2}},\\ {u}^{r} & = & \displaystyle \frac{{k}_{r}}{\sqrt{{g}_{{rr}}}}=\displaystyle \frac{{k}_{r}{({{\rm{\Delta }}}_{r})}^{1/2}}{\rho },\\ {u}^{\theta } & = & \displaystyle \frac{{k}_{\theta }}{\sqrt{{g}_{\theta \theta }}}=\displaystyle \frac{{k}_{\theta }\rho }{{({{\rm{\Delta }}}_{\theta })}^{1/2}},\\ {u}^{\phi } & = & \displaystyle \frac{{k}_{\phi }}{\sqrt{{g}_{\phi \phi }}}=\displaystyle \frac{{k}_{\phi }\rho {\rm{\Xi }}}{\sin \theta {[{{\rm{\Delta }}}_{\theta }{({r}^{2}+{a}_{k}^{2})}^{2}-{{\rm{\Delta }}}_{r}{a}_{k}^{2}{\sin }^{2}\theta ]}^{1/2}},\end{array}\end{eqnarray}$
where all of ${k}_{t},{k}_{r},{k}_{\theta },{k}_{\phi }$ are constants. ${u}^{\alpha }$ has the following property: $\begin{eqnarray}{u}^{\alpha }{u}_{\alpha }={k}_{t}^{2}+{k}_{r}^{2}+{k}_{\theta }^{2}+{k}_{\phi }^{2}=\mathrm{const}.\end{eqnarray}$
Bringing equation ( $\begin{eqnarray}\begin{array}{l}(1+2a){{\rm{\Delta }}}_{r}^{2}\left|{}_{r\to {r}_{H}}\right.(1+2{{cmk}}_{r}^{2}){\left(\displaystyle \frac{\partial S}{\partial r}\right)}^{2}\left|{}_{r\to {r}_{H}}\right.\\ -\,{\left[({r}_{H}^{2}+{a}_{k}^{2})\displaystyle \frac{\partial S}{\partial t}+{a}_{k}{\rm{\Xi }}\displaystyle \frac{\partial S}{\partial \phi }\right]}^{2}=0.\end{array}\end{eqnarray}$
Then we get $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\partial {S}_{\pm }}{\partial r}\left|{}_{r\to {r}_{H}}\right.=\pm \displaystyle \frac{{r}_{H}^{2}+{a}_{k}^{2}}{{{\rm{\Delta }}}_{r}\left|{}_{r\to {r}_{H}}\right.(1+2a)\sqrt{1+2{{cmk}}_{r}^{2}}}\\ \times \,\left[\displaystyle \frac{\partial S}{\partial t}+\displaystyle \frac{a{\rm{\Xi }}}{{r}_{H}^{2}+{a}_{k}^{2}}\displaystyle \frac{\partial S}{\partial \phi }\right].\end{array}\end{eqnarray}$
We can do variables separation in equation ( $\begin{eqnarray}S=-\omega t+R(r)+{\rm{\Theta }}(\theta )+j\phi ,\end{eqnarray}$
where ω is the energy of fermions with mass m, and j represents the φ component of the generalized momentum of the radiated fermions. Since the curved space-time described by equation ( $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\partial {S}_{\pm }}{\partial r}\left|{}_{r\to {r}_{H}}\right.=\displaystyle \frac{{\rm{d}}{R}_{\pm }}{{\rm{d}}r}\left|{}_{r\to {r}_{H}}\right.\\ =\,\pm \displaystyle \frac{{r}_{H}^{2}+{a}_{k}^{2}}{{{\rm{\Delta }}}_{r}\left|{}_{r\to {r}_{H}}\right.(1+2a)\sqrt{1+2{{cmk}}_{r}^{2}}}(\omega -{\omega }_{0}),\end{array}\end{eqnarray}$
where $\begin{eqnarray}{\omega }_{0}=\displaystyle \frac{{a}_{k}{\rm{\Xi }}j}{{r}_{H}^{2}+{a}_{k}^{2}}\end{eqnarray}$
is the chemical potential. Integrating the above equation from the inner side to the outer side of rH with the residue theorem, we obtain $\begin{eqnarray}\begin{array}{rcl}{R}_{\pm } & = & \pm \displaystyle \int {\rm{d}}r\displaystyle \frac{{r}_{H}^{2}+{a}_{k}^{2}}{{{\rm{\Delta }}}_{r}\left|{}_{r\to {r}_{H}}\right.(1+2a)\sqrt{1+2{{cmk}}_{r}^{2}}}(\omega -{\omega }_{0})\\ & = & \pm {\rm{i}}\pi \displaystyle \frac{{r}_{H}^{2}+{a}_{k}^{2}}{{{\rm{\Delta }}}_{r}^{{\prime} }({r}_{H})(1+2a)\sqrt{1+2{{cmk}}_{r}^{2}}}(\omega -{\omega }_{0}),\end{array}\end{eqnarray}$
where $\begin{eqnarray}{{\rm{\Delta }}}_{r}^{{\prime} }({r}_{H})=2{r}_{H}\left(\displaystyle \frac{{r}_{H}^{2}}{{{\ell }}^{2}}+1\right)+2\displaystyle \frac{{r}_{H}}{{{\ell }}^{2}}({r}_{H}^{2}+{a}_{k}^{2})-2M.\end{eqnarray}$
In equation (28 ), + and − correspond to outgoing mode and incoming mode, respectively. Therefore, according to the quantum tunneling radiation theory and the semiclassical theory, we can obtain the accurate expression of the quantum tunneling rate Γ of Kerr anti-de-Sitter black hole corrected by Lorentz violation theory, namely
$\begin{eqnarray}{\rm{\Gamma }}\sim \exp [-2{\rm{Im}}({R}_{+}-{R}_{-})]=\exp (-\displaystyle \frac{\omega -{\omega }_{0}}{{T}_{H}^{{\prime} }}),\end{eqnarray}$
where ${T}_{H}^{{\prime} }$ is the Hawking temperature at the event horizon of the black hole, with expression of $\begin{eqnarray}\begin{array}{rcl}{T}_{H}^{{\prime} } & = & \displaystyle \frac{{{\rm{\Delta }}}_{r}^{{\prime} }({r}_{H})}{4\pi ({r}_{H}^{2}+{a}_{k}^{2})}(1+2a)\sqrt{1+2{{cmk}}_{r}^{2}}\\ & = & {T}_{H}(1+2a)\sqrt{1+2{{cmk}}_{r}^{2}}\\ & = & {T}_{H}(1+2a+{{cmk}}_{r}^{2}+\cdots ).\end{array}\end{eqnarray}$
Here, $\begin{eqnarray}{T}_{H}=\displaystyle \frac{{{\rm{\Delta }}}_{r}^{{\prime} }({r}_{H})}{4\pi ({r}_{H}^{2}+{a}_{k}^{2})}\end{eqnarray}$
is Hawking temperature at the event horizon of Kerr anti-de-Sitter black hole before our correction.Obviously, the corrected Hawking temperature ${T}_{H}^{{\prime} }$ depends on c and kr, which represent the coupling strength and the radial component of the ether-like field respectively, and ${T}_{H}^{{\prime} }$ increases with increasing values of them. Without the coupling parameter (c = 0), we will recover the semi-classical Hawking temperature of Kerr anti-de-Sitter black hole. Moreover, for a = 0, the temperature and its correction reduce to the case of Schwarzschild anti-de-Sitter black hole.
In our scenario the tunneling radiation of fermions is used as a tool to detect the black hole’s temperature that is unknown yet to date. Obviously, the black hole’s temperature should not be affected by the radiated fermions. If one believe Lorentz violating would occur at high energy, the corrected temperature ${T}_{H}^{{\prime} }$ should be more close to the true temperature of the black hole than the original value TH. Future astronomical observations may validate our result.
4. Discussions
In the theory of black hole thermodynamics, an important physical quantity is black hole entropy. The correction of Hawking temperature will inevitably lead to the change of black hole entropy. According to the black hole thermodynamics:30 ) gives ${\rm{\Gamma }}\sim {{\rm{e}}}^{{\rm{\Delta }}{S}_{\mathrm{kBH}}^{\prime} }$, which is the tunneling rate expressed with entropy variation.
$\begin{eqnarray}{\rm{d}}M=T{\rm{d}}{S}_{k}+V{\rm{d}}J,\end{eqnarray}$
where Sk denotes the entropy of black hole, and the rotational potential V is defined as $\begin{eqnarray}V=\displaystyle \frac{{a}_{k}j}{{r}_{H}^{2}+{a}_{k}^{2}}.\end{eqnarray}$
At the event horizon rH the unmodified entropy satisfies $\begin{eqnarray}{\rm{d}}{S}_{\mathrm{kBH}}=\displaystyle \frac{{\rm{d}}M-V{\rm{d}}J}{{T}_{H}},\end{eqnarray}$
where ${S}_{\mathrm{kBH}}$ denotes the Bekenstein–Hawking of the black hole. After applying Lorentz violation, the entropy ${S}_{\mathrm{kBH}}$ is corrected as $\begin{eqnarray}\begin{array}{rcl}{S}_{\mathrm{kBH}}^{{\prime} } & = & \displaystyle \int \displaystyle \frac{{\rm{d}}M-V{\rm{d}}J}{{T}_{H}^{{\prime} }}\\ & = & {S}_{\mathrm{kBH}}(1-2a-{{cmk}}_{r}^{2}+\cdots ).\end{array}\end{eqnarray}$
In the above calculation, we ignore small quantities of higher order. We can draw a valuable conclusion from the above equation: the correction term of the entropy depends on the parameters a, c and kr, which reflects the radial behavior of Lorentz violating. We use ${\rm{\Delta }}{S}_{\mathrm{kBH}}^{\prime} $ to denote the Bekenstein–Hawking entropy variation, then equation (The main work of this paper is to correct the quantum tunneling radiation of spin 1/2 fermions in the stationary Kerr anti-de-Sitter space-time with Lorentz violation theory. It is quite complicated to correct the dynamic equation of Dirac particles in curved space-time based on Lorentz symmetry violation theory. However, the results obtained in this paper are satisfactory. By studying equations (1 ) and (13 ), we can simplify the complex calculation. The Hamilton principal function of Dirac particles is obtained from equation (13 ), then the tunneling rate of the black hole, the Hawking temperature at the event horizon of the black hole and the entropy of the black hole are obtained after correcting. These research results are valuable for further research on the quantum gravity theory and the thermodynamic evolution of black holes.
The method proposed in this paper can also be used to correct the dynamics equation of fermions with arbitrary spin, e.g. spin 3/2, and modify the tunneling rate and Hawking temperature subsequently. In short, the modification to particle dynamics equations in curved space-time based on Lorentz violation theory and related problems are topics worthy of in-depth exploration. In recent years, it is believed that Lorentz symmetry will be broken at high energy scales. Therefore, the quantum field equation affected by Lorentz violation has been successively studied. These studies, including our research on particle dynamics equations in curved space-time, can provide theoretical support for research on quantum gravity. In curved space-time, the simple but special properties of the black hole surface may become a powerful tool for verifying Lorentz violation theory in the future. We will continue our research in this area.