Lorentz Invariance Violation and Modified Hawking Fermions Tunneling Radiation

Recently the modified Dirac equation with Lorentz invariance violation has been proposed, which would be helpful to resolve some issues in quantum gravity theory and high energy physics. In this paper, the modified Dirac equation has been generalized in curved spacetime, and then fermion tunneling of black holes is researched under this correctional Dirac field theory. We also use semiclassical approximation method to get correctional Hamilton-Jacobi equation, so that the correctional Hawking temperature and correctional black hole’s entropy are derived.


Introduction
In 1974, Hawking proved black holes could radiate Hawking radiation once considering the quantum effect near the horizons of black holes [1,2].This theory indicates that black hole would be viewed as thermodynamic system, so that black hole physics can be connected closely with gravity, quantum theory, and thermodynamics physics.According to the view point of quantum tunneling theory, the virtual particles inside of black hole could cross the horizon due to the quantum tunneling effect and become real particles and then could be observed by observers as Hawking radiation.Wilczek et al. proposed a semiclassical method to study the quantum tunneling from the horizon of black hole [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21].Along with this method, the Hamilton-Jacobi method was applied to calculate the Hawking tunneling radiation.According to the Hamilton-Jacobi method, the wave function of Klein-Gordon equation can be rewritten as Φ =  exp (/ℏ) (where  is semiclassical action) and Hamilton-Jacobi equation is obtained via semiclassical approximation.Using the Hamilton-Jacobi equation, the tunneling rate could be calculated by the relationship Γ ∼ exp (−2 Im ) (where Γ is the tunneling rate at the horizon of black hole), and then the Hawking temperature can be determined.People have applied this method to research Hawking tunneling radiation of several static, stationary, and dynamical black holes.
However, since the Hamilton-Jacobi equation is derived from Klein-Gordon equation, original Hamilton-Jacobi method just can be valid for scalar particles in principle.Therefore, Kerner and Mann studied fermions tunneling of black hole by a new method [22][23][24][25][26][27][28][29][30][31][32], which assumes the wave function of Dirac equation Ψ as spin-up and spin-down and then calculates the fermions tunneling, respectively.Nevertheless, this method is still impossible to apply in arbitrary dimensional spacetime.Our work in 2009 showed that the Hamilton-Jacobi equation can also be derived from Dirac equation via semiclassical approximation, so we proved that the Hamilton-Jacobi equation can be used to study the fermions tunneling directly [33][34][35].
On the other hand, as the basis of general relativity and quantum field theory, Lorentz invariance is proposed to be spontaneously violated at higher energy scales.A possible deformed dispersion relation is given by [36][37][38][39][40][41][42][43][44]] where  0 and ⃗  are the energy and momentum of particle and  is "minimal length" with the order of the Plank length.The work of spacetime foam Liouville string models has introduced this relation with  = 1, and people also proposed quantum equation of spineless particles by using this relation.Recently, Kruglov considers the deformed dispersion relation with  = 2 and proposes modified Dirac equation [45]: where   is ordinary gamma matrix and  is space coordinate, while  is spacetime coordinate.The effect of the correctional term would be observed in higher energy experiment.
In this paper, we try to generalize the modified Dirac equation in curved spacetime and then study the correction of Hawking tunneling radiation.In Section 2, the modified Dirac equation in curved spacetime is constructed and then the modified Hamilton-Jacobi equation is derived via semiclassical approximation.We apply the modified Hamilton-Jacobi equation to the fermions tunneling radiation of 2 + 1-dimensional black string and higher dimensional BTZlike black strings in Sections 3 and 4, respectively, and Section 5 includes some conclusion and the discussion about the correction of black hole's entropy.

Modified Dirac Equation and Hamilton-Jacobi Equation in Curved Spacetime
As we all know, the gamma matrix and partial derivative should become gamma matrix in curved spacetime   and covariant D  derivative, respectively, namely, where   satisfy the relationship {  ,   } =     +     = 2  ,   is charged term of Dirac equation, and ) is spin connection.According to this transformation, we can construct the modified Dirac equation in curved spacetime as where we choose  = 1 but ℏ ̸ = 1, while  = ℏ = 1 in (1) and (2).It is assumed that  ≪ 1, so that the correctional term ℏ(  D  )(  D  ) is very small.Now let us use the modified Dirac equation to derive the modified Hamilton-Jacobi equation.Firstly, we rewrite the wave function of Dirac equation as [33][34][35] where (,   ) and Ψ are  × 1 matrices and    = −.In semiclassical approximation, we can consider that ℏ is very small, so that we can neglect the terms with ℏ after dividing by the exponential terms and multiplying by ℏ.Therefore, ( 4) is rewritten as Considering the relationship we can get where Now, multiplying both sides of ( 9) by the matrix −Γ ] ( ]  +  ] ), we can obtain The second term of the above equation could be simplified again by (8), so the above equation can be rewritten as where we can prove the relation We always ignore O( 2 ) terms because  is very small.Now, let us exchange the position of  and ] in (11) and consider the relation of gamma matrices {  ,   } = 2  ; then we can obtain Namely, where Using the idea of ( 10)-( 11) again, we can multiply both sides of ( 15) by the matrix − ] ( ]  +  ] ), so that the equation becomes The second term of the above equation could be simplified again by (15).Then, exchange  and ] and use the relationship {  ,   } = 2  , so the above equation can be rewritten as The condition that (17) has nontrivial solution required the determinant of coefficient in (17) should vanish, so we can directly get the equation Consider the square root for left side of ( 18) and ignore all O( 2 ) terms, so we can directly get the modified Hamilton-Jacobi equation: Therefore, we find that the modified Dirac equation from Lorentz invariance violation could lead to the modified Hamilton-Jacobi equation, and the correction of Hamilton-Jacobi equation depends on the energy and mass of radiation fermions.Using the modified Hamilton-Jacobi equation, we then investigate the fermions Hawking tunneling radiation of 2 + 1-dimensional black string and  + 1-dimensional BTZlike string in the following two sections.

Fermions Tunneling of 2 + 1-Dimensional Black String
The research of gravity in 2 + 1 dimension can help people further understand the properties of gravity, and it is also important to construct the quantum gravity.Recently, Murata et al. have researched the 2 + 1-dimensional gravity with dilaton field, whose action is given by [46] where ,  3 , and  are, respectively, the dilaton field, 3dimensional Planck mass, and the parameter with mass dimension.The static black string solution is given by It is evident that the horizon of this black hole is   , but the black string is unstable as   ≲ L (where L is scale of compactification), so it is assumed that   ≫ L. Now we research the fermions tunneling of this black hole, so the modified Hamilton-Jacobi equation in this spacetime is given by where we have set  = − + () + (), yielding the radial Hamilton-Jacobi equation as where the constant  0 is from separation of variables, and (23) finally can be written as At the horizon   of the black string, the above equation is integrated via residue theorem, and we can get and the fermions tunneling rate The relationship between tunneling rate and Hawking temperature required where   and  0 are the modified and nonmodified Hawking temperature of the 2 + 1-dimensional black string, respectively, and  term is the correction.

Fermions Tunneling of Higher Dimensional BTZ-Like Black Strings
As we all know that the linear Maxwell action fails to satisfy the conformal symmetry in higher dimensional spacetime [47,48], so Hassaïne and Martínez proposed gravity theory with nonlinear Maxwell field in arbitrary dimensional spacetime: where Λ ≡ − −2 is cosmological constant.Hendi researched  + 1-dimensional static black strings solution with  = 1,  = −1, and  = /2, and it is charged BTZ-like solutions [49], whose metric is where and the electromagnetic potential is As  = 2, this solution is no other than the static charged BTZ solution.We will study the Hawking radiation and black hole temperature at the event horizon   of this black string.In (19), we can set  = − + () + (  ), where   are the space coordinates excluding the radial coordinate, so that the modified Hamilton-Jacobi equation is given by and the radial equation with constant  0 is Therefore, at the horizon   of the black string, (  ) = 0 and we finally get where  0 =   (  ).This means that the fermions tunneling rate is and the Hawking temperature is where   and  0 are the modified and nonmodified Hawking temperature of the  + 1-dimensional BTZ-like black string, respectively, and  term is the correction.

Conclusions
In this paper, we consider the deformed dispersion relation with Lorentz invariance violation and generalize the modified Dirac equation in curved spacetime.The fermions tunneling radiation of black strings is researched, and we find that the modified Dirac equation could lead to Hawking temperature's correction, which depends on the correction parameter  and particle mass  in the modified Dirac equation.Next we will discuss the correction of black hole entropy in this theory.The first law of black hole thermodynamics requires where Ξ and  are electromagnetic potential and rotating potential, so the nonmodified entropy of black hole is [1, 2, 50, 51] From the above results, we know that the relationship between modified and nonmodified Hawking temperature is   = (1 + ) 0 , since the nonmodified black hole entropy is given by where we can ignore O( 2 ) because  ≪ 1. Equation (39) shows that the correction of black hole entropy depends on , which is independent from time and space coordinates.However, it is possible that  depends on other parameters in curved spacetime, and it is very interesting that  depends on  0 .In particular, as  =  0 / 0 +⋅ ⋅ ⋅ , we can get the logarithmic correction of black hole entropy In quantum gravity theory, the logarithmic correction has been researched in detail [52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69], and according to [70], it is required that the coefficient of logarithmic correction should be −( + 1)/2( − 1) in  + 1-dimensional spacetime, so it indicates that  0 could be ( + 1)/2( − 1).
On the other hand, from the deformed dispersion relation (1) with  = 2, it is implied that the Klein-Gordon equation could be given by (− Contrasting ( 19) and ( 44) as uncharged case, we find that the correctional terms of Dirac field and scalar field are very different.The fact implies that the corrections of Hawking temperature and black hole entropy from Hawking tunneling radiation with different spin particles could be different, and this conclusion could be helpful to suggest a new idea to research the black hole information paradox.Work in these fields is currently in progress.