Black Hole Radiation with Modified Dispersion Relation in Tunneling Paradigm: Static Frame

Due to the exponential high gravitational red shift near the event horizon of a black hole, it might appears that the Hawking radiation would be highly sensitive to some unknown high energy physics. To study possible deviations from the Hawking's prediction, the dispersive field theory models have been proposed, following the Unruh's hydrodynamic analogue of a black hole radiation. In the dispersive field theory models, the dispersion relations of matter fields are modified at high energies, which leads to modifications of equations of motion. In this paper, we use the Hamilton-Jacobi method to investigate the dispersive field theory models. The preferred frame is the static frame of the black hole. The dispersion relation adopted agrees with the relativistic one at low energies but is modified near the Planck mass $m_{p}$. We calculate the corrections to the Hawking temperature for massive and charged particles to $\mathcal{O}\left(m_{p}^{-2}\right) $ and massless and neutral particles to all orders. Our results suggest that the thermal spectrum of radiations near horizon is robust, e.g. corrections to the Hawking temperature are suppressed by $m_{p}$. After the spectrum of radiations near the horizon is obtained, we use the brick wall model to compute the thermal entropy of a massless scalar field near the horizon of a 4D spherically symmetric black hole. We find that the leading term of the entropy depends on how the dispersion relations of matter fields are modified, while the subleading logarithmic term does not. Finally, the luminosities of black holes are computed by using the geometric optics approximation.

The classical theory of black holes predicts that anything, including light, couldn't escape from the black holes. However, Stephen Hawking demonstrated that quantum effects could allow black holes to radiate a thermal flux of quantum particles [1]. The assumption that the ultra-high energy modes are in their ground state was used to derive the Hawking radiation in the framework of quantum field theory in curved spacetime. After this discovery, it was realized that there was the trans-Planckian problem with the calculation [2]. Due to the exponential high gravitational red shift near the horizon, the outgoing particles of the Hawking radiation originate from the extremely high (e.g., trans-Planckian) frequency modes. So the Hawking radiation relies on the validity of quantum field theory in curved spacetime to arbitrary high energies. On the other hand, quantum field theory is considered more like an effective field theory of an underlying theory whose nature remains unknown [3].
This observation poses the question of whether any unknown physics at the Planck scale could strongly influence the Hawking radiation. It is believed that the trans-Planckian physics manifests itself in certain modifications of the existing models. Thus, even though a complete theory of quantum gravity is not yet available, we can use a "bottom-to-top approach" to probe the possible effects of quantum gravity on our current theories and experiments [4]. One possible way of how such an approach works is via modifications of energy-momentum dispersion relation which has been reviewed in the framework of Lorentz violating theories in [5,6].
To study the trans-Planckian problem, the dispersive field theory models [7][8][9][10][11][12][13][14][15][16][17] have been proposed, which focused on studying the effect on the Hawking radiation due to modifications of the dispersion relations of matter fields at high energies. These models were motivated by a hydrodynamic analogue of a black hole radiation [7]. Similar to the original method for deriving the Hawking radiation, the energy fluxes for outgoing radiation were usually obtained by calculating the Bogoliubov transformations between the initial and final states of incoming and outgoing radiation. In most works, the Hawking effect could be recovered at leading order for some ranges of the black hole's temperature and the particle's frequency under some suitable assumptions although the mechanisms for recovering it were different than in the non-dispersive case. These assumptions and ranges in the models have been briefly reviewed in [18,19]. While most works used the free-fall preferred frame to modify the dispersion relation, accelerated frames has been studied in [11]. It has been found that the fluxes emitted by a black hole were not significantly affected by the choice of preferred frame as long as the acceleration of the frame was not too drastic. However, it has been numerically shown that the created particle flux dropped significantly if the acceleration became large. Thus, it was conjectured that there might be no Hawking radiation for a static frame.
There are various methods for deriving the Hawking radiation and calculating its temperature. Among them is a semiclassical method of modeling Hawking radiation as a tunneling process. This method was first proposed by Kraus and Wilczek [20,21], which is known as the null geodesic method. They employed the dynamical geometry approach to calculate the imaginary part of the action for the tunneling process of s-wave emission across the horizon and related it to the Boltzmann factor for the emission at the Hawking temperature. Later, the tunneling behaviors of particles were investigated using the Hamilton-Jacobi method [22][23][24]. In the Hamilton-Jacobi method, one ignores the self-gravitation of emitted particles and assumes that its action satisfies the relativistic Hamilton-Jacobi equation. The tunneling probability for the classically forbidden trajectory from inside to outside the horizon is obtained by using the Hamilton-Jacobi equation to calculate the imaginary part of the action for the tunneling process. Using the null geodesic method and Hamilton-Jacobi method, much fruit has been achieved [25][26][27][28][29][30][31][32][33][34][35][36]. Furthermore, the effects of quantum gravity on the Hawking radiation have been discussed in the Hamilton-Jacobi method. In fact, the minimal length deformed Hamilton-Jacobi equation for fermions in curved spacetime have been introduced and the modified Hawking temperatures have been derived [37][38][39][40][41][42]. These motivate us to use the Hamilton-Jacobi method to study the dispersive field theory models.
Note that a tunneling model has been introduced in dispersive models of analogue gravity in [17]. The Hamilton-Jacobi equations were imported to curved spacetime using the static preferred frame in [37][38][39][40][41][42], which leads us first to considering the static preferred frame for the dispersive field theory models. The models with free-fall preferred frame will be investigated in [43]. Comparisons between the results in our paper and those in [37][38][39][40][41] will be given at the end of the section II.
The remainder of our paper is organized as follows. In section II, the deformed Hamilton-Jacobi equations are derived for the dispersive models. In section IV, we solve the deformed

II. DEFORMED HAMILTON-JACOBI EQUATION
In most cases, the modified dispersion relation (MDR) could take the form of where m p is Planck mass and H (x, y) = x 2 − y 2 for the unmodified dispersion relation.
Taylor expanding the right-hand side of eqn. (2) for E, m ≪ m p gives where h i,j is the coefficient of x i y j in the Taylor series of H (x, y) evaluated at (0, 0). Since eqn.
(1) when m p → ∞, we find After some manipulations, eqn. (3) can be put in the form of where α (0) = β (0) = 1 and γ (0) = 0. If the modifications to the dispersion relation are suppressed by some the scale of Lorentz violation Λ, the naturalness in effective field theories would imply that C n ∼ mp Λ n . For Λ ≪ m p , C n could become much large. To include a broader class, we consider a static black hole in the possible presence of electromagnetic potential A µ with the line element where f (r) has a simple zero at r = r h with f ′ (r h ) being finite and nonzero. The vanishing of f (r) at point r = r h indicates the presence of an event horizon. We also assume that the vector potential A µ is given by which is true for charged static black holes in most cases.
The MDR breaks the Lorentz invariance in flat spacetime. Thus, one needs to pick up a preferred frame to determine the form of the MDR. The energy and momentum in eqn. (5) are defined with respect to the preferred frame, where can be described by the unit vector u µ tangent to the observers' world lines. Explicitly, we have where p µ is the energy-momentum vector and E and p are the energy and the norm of the momentum measured in the preferred reference frame, respectively. When introducing the MDR into curved spacetime, we use the vector field u µ (x ν ). To obtain the deformed Hamilton-Jacobi equation incorporating the MDR, it is necessary to specify the profile of the preferred frame in the black hole spacetime. One of natural frames is a static frame hovering above the black hole. For such a frame, the vector field u µ (x ν ) is Plugging eqn. (9) into eqns. (8), one finds that the energy and the magnitude of the momentum becomes It can be shown that, if the classical action I is a solution of the Hamilton-Jacobi equation, then the transformation equations give where − appears since p µ = (E, − p) in our metric signature. Furthermore, since ∂ t is a Killing vector of the background spacetime, (∂ t ) µ p µ = p t is a constant. In fact, p t is the conserved energy of the particle and we define ω ≡ p t = −∂ t I, which means we can separate t from other variables when solving for I. Relating I to p µ via eqn. (11) and putting eqns. (10) into eqn. (5) give the deformed Hamilton-Jacobi equation. In the appendix, the deformed Hamilton-Jacobi equation is also derived in a more rigorous way, specifically in the language of the effective field theory. We show there that if a scalar/fermion obeys the MDR given in eqn. (5) in flat spacetime, the deformed scalar/fermionic Hamilton-Jacobi equation with respect to the preferred static frame in the black hole background spacetime can be both written as where we define A µ is the black hole's electromagnetic potential and q is the particle's charge.

III. TUNNELING RATE
In this section, we use the Hamilton-Jacobi method to investigate the particles' tunneling across the event horizon r = r h of the metric (6) by solving eqn. (12). Taking into account ∂ t I = −ω, we can employ the following ansatz for the action I where ω is the particle's energy. Plugging the ansatz into eqn. (13), we have The method of separation of variables gives the differential equation for Θ (x) where is λ is a constant and determined by h ab (x). Thus, one has and eqn. (12) becomes an ordinary differential equation for W (r). In this section, we solve eqn. (12) for ∂ r W (r), calculate its residue at r = r h and find the imaginary part of I which gives the tunneling rate Γ across the event horizon. We will calculate Im W for two cases, a massive and charged particle to O m −2 p and a neutral and massless particle to all orders.
A. Massive and Charged Particle to O m −2 p Consider a particle with the mass m and the charge q. Solving eqn. (12) for p r ≡ ∂ r W (r) gives where +/− denotes the outgoing/ingoing solutions andω (r) ≡ ω − qA t (r). Here, we have a pole at r = r h . Using the residue theory for the semi circle [1] , we get where we define κ = f ′ (r h ) /2 and The argument m mp is suppressed for α m mp , β m mp , γ m mp and C n m mp in eqns. (19) and (20) .
[1] This procedure will be discussed in detail later in this section.

B. Massless and Neutral Particle to All Orders
We now work with a particle with m = 0 and q = 0. Solving eqn. (12) for p r gives where C n ≡ C n (0) and we use α (0) = 1. To get the residue of p ± r at r = r h , we first define a few coefficients C α n ,C m,n , and η k l as follows where m is a non-negative integer and C α n are generalized binomial coefficients with C α n = n k=1 α−k+1 k . Therefore, one has where we only keep terms contributing to the residue and set k → k + 2l in the third line.
Furthermore, we denote the residue of Using the residue theory for the semi circle, one has where Note that η 0 0 = ζ 0 0 = 1.

C. Calculating λ
It is easy to see that λ depends on h ab (x) . Here we consider two kinds of black holes, 4D cylindrically and spherically symmetric black holes. For a 4D cylindrically symmetric black hole, we have where −∞ < z < ∞, 0 ≤ θ ≤ 2π and α is some constant. Since ∂ θ and ∂ z are the Killing fields of the background spacetime, we can separate the variables and consider a solution for eqn. (16) of the form where J θ and J z are constant and J θ is the angular momentum along z-axis. The periodicity of θ gives J θ = n with n ∈ Z. Thus, one finds where J θ = n with n ∈ Z.
For a 4D spherically symmetric black hole, we have where 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π. ∂ φ is the Killing vector so we consider a solution for eqn.
where J φ is the angular momentum along z-axis. The periodicity of φ gives J φ = m with m ∈ Z . Since the magnitude of the angular momentum of the particle L can be expressed in terms of p θ ≡ ∂ θ Y (θ) and J φ , eqn. (16) gives λ = L 2 . Putting eqn. (31) into eqn. (16), one gets On the other hand, the Sommerfeld quantization for p θ gives where n is a non-negative integer. The integral in eqn. (34) is calculated in the classically allowed region where p θ is real, which requires that λ > m 2 2 . Integrating the quantization integral, one finds that eqn. (34) becomes Solving eqn. (35) for λ gives the WKB leading quantization of the angular momentum where l = n + |m| = 0, 1, · · · with |m| ≤ l. Note that the difference between the exact quantization of the angular momentum L 2 = l (l + 1) 2 and the WKB leading quantization

D. Tunneling Rate
When one calculates the quantum tunneling rate from Im W ± , there is so called "factortwo problem" [44]. Thus, one may have a black hole temperature which is twice the expected result. One of solutions is proposed by Mitra [45]. Mitra noted that in general, the action I could include some complex constant of integration K. In this way, the imaginary part of I becomes Im I ± = Im W ± + Im K +/− denotes the outgoing/ingoing solutions. In the semi-classical method, the absorption probability and the emission probability for a black hole are given by On the other hand, it is noted that the classical theory of black holes tells us that an incoming particle is absorbed with the probability equalling to one. Thus, one can choose K to impose the classical constraint on the absorption probability, which is Im K = − Im W − .
So eqn. (38) gives that the probability of a particle tunneling from inside to outside the horizon is Another way to circumvent this problem is considering both the contributions from spatial and temporal parts of the action to the tunneling rates.
Spatial Contribution: As pointed out in [44,46,47], Im W ± = Im p ± r dr used in eqns. (38) were not invariant under canonical transformations. Instead, one should take the closed contour integral p r dr = p + r dr − p − r dr, an invariance under canonical transformations, for the tunneling rates P emit/abs ∝ exp ± 1 Im p r dr .
Temporal Contribution: As shown in [47][48][49], the temporal part contribution came from the "rotation" which connects the interior region and the exterior region of the black hole. It was found in [49] that the direction in which the horizon was crossed did not affect the sign of the temporal contribution. However, the sign of the spatial contribution changed when the direction was reversed. Thus, the temporal contributions to P emit/abs were the same. When the horizon was crossed once, the action I got a contribution of Im (ω∆t) = πω 2κ , and for a round trip, the total contribution was Im (ω∆t) = πω κ . Taking into account the spatial and temporal contributions, one has for the absorption probability and for the emission probability where K is a constant of integration. Imposing the classical constraint on the absorption probability, one gets and Both approaches give the same expression for P emit . There is a Boltzmann factor in P emit with an effective temperature. Using eqns. (19) and (25) , we find that the effective temperature for a massive and charged particle is and that for a massless and neutral particle is where we define T 0 = κ 2π and take k B = 1.

E. Discussion
In this section, we suppose that outgoing particles tunnel from r 1 < r h to r 2 > r h while ingoing particles from r 2 to r 1 . To to obtain the imaginary part of I for the tunneling process, we have to give an prescription for evaluating the integrals of Im W ± = r 2 r 1 p ± r dr. Following the Feynman's iǫ-prescription [50], we take the contour of the integral to be an infinitesimal semicircle below the pole at r = r h for outgoing particles. Thus, the integral becomes where we denote the semicircle centered at r = r h with the radius of R going from r h + Similarly, one has for ingoing particles To get Im W ± , we expand p ± r in powers of where p ± n (r) are some analytic functions of r around r = r h . For the non-dispersive case, only the first term in eqn. (49) appears. Thus, we can Laurent expand f (r) with respect to r at r = r h to evaluate C B/U,ε p ± r dr as ε → 0. However, these expansions for p ± r in the dispersive models look suspicious on C U/B,ε as ε → 0. In fact, can become larger than 1 if r is close enough to r h . Thus, we can not trust the expansions for p ± r any more on C U/B,ε . Nevertheless, we can assume that the singularity structure of p r in the dispersive models is the same as that in the non-dispersive case except the order of the pole at r = r h , which means the MDR effects do not introduce branch cuts or new poles for p r in the upper or lower half complex plane. Note that one may need a complete theory of quantum gravity to justify the assumption. Now consider the semicircles C U/B,R with large enough R, which lies in the region where the expansion for p r can be trusted. Under the assumption, there are no poles inside the area enclosed by (R, r h − ε), C U/B,ε , (r h + ε, R), and C U/B,R . Thus, we have where contributions from the ranges (r 1 , r h − R) and (r h + R, r 2 ) are discarded since they are always real. If the radii of the Laurent series of p ± on C B/U,R and only the coefficients a −1 of (r − r h ) −1 terms contribute to the imaginary part of the integrals. This justifies the procedure to obtain to obtain eqns. (19) and (25). Since the expansions for p ± r can be trusted on Usually, one has that the radii of the Laurent series of p ± Various theories of quantum gravity, such as string theory, loop quantum gravity and quantum geometry, predict the existence of a minimal length [51][52][53]. The generalized uncertainty principle(GUP) [54] is a simply way to realize this minimal length. To incorporate the Klein-Gordon/Dirac equation with the GUP, one usually considers the quantization in position representation. In position representation, the operators k = −i ∇ and ω = i∂ t are introduced [55]. One then can express the energy and momentum operators as functions of k and ω and obtain the deformed Klein-Gordon/Dirac equations in flat spacetime. Inserting the ansatz ϕ = exp (iEt − i p · x) in the Klein-Gordon/Dirac equation gives the dispersion relation for E and p in flat spacetime. In [37][38][39][40][41], the deformed Dirac equation was generalized to curved spacetime. The modified Hawking temperatures of various black holes were then derived via the Hamilton-Jacobi method. It turns out that the way of generalizing the Dirac equation to curved spacetime in [37][38][39][40][41] is the same as that in the appendix of the paper where we choose the static frame as the preferred one. Thus, we can use the dispersion relation for E and p obtained in flat spacetime and eqn. (20) to reproduce the modified Hawking temperatures of black holes with the metric (6) obtained in [37][38][39][40][41]. In fact, the dispersion relation in flat spacetime in [37][38][39][40][41] is given by which by comparing to eqn. (5) gives Thus, eqn. (20) becomes For example, ∆ qm for a particle with the angular momentum l = 0 in a Schwarzschild black hole with the mass M [37,41] is ∆ qm for a neutral particle with the angular momentum l = 0 in a Reissner-Nordstrom with the mass M and the charge Q [38,41] is where Here we only consider a neutral particle to make a comparison since the electromagnetic filed was included in [38,41] in a different way.
Following the argument proposed in [56], the authors in [57] obtained modified relations between the mass of a 4D Schwarzschild black hole and its entropy and temperature. The argument connecting a MDR and some modifications of the entropy of black holes is formulated in a scheme of analysis first introduced by Bekenstein [58]. In fact, the modified temperature of the black hole for the MDR in eqn. (5) with m = 0 was given by where M is the mass of the black hole and T 0 = m 2 p 8πM . On the other hand, we can use eqn. (20) to estimate the temperature of the black hole. For a massless particle in a 4D Schwarzschild black hole, eqn. (20) gives where λ is the magnitude of the angular momentum of the particle. The event horizon of the Schwarzschild black hole is r h = 2M. Near the horizon of the the black hole, one has As reported in [57], the authors obtained the relation ω δx + O 1 mp between the energy of a particle and its position uncertainty for a MDR. Near the horizon of the Schwarzschild black hole, the position uncertainty of a particle is of the order of the Schwarzschild radius of the black hole [58] δx ∼ r h = 2M. Thus, one finds for T Note that the term − mpC 3 4M in eqn. (56) could imply that the leading correction to the entropy of the black hole should have √ area dependence. Since in Loop Quantum Gravity such a √ area contribution to black-hole entropy has already been excluded, C 3 should vanish [56].
However, it has been shown in [57] that this might be evaded by combining MDR and GUP.
Our result suggests that a nonvanishing C 3 in the MDR does not lead to the presence of √ area contribution to the black hole entropy.

IV. THERMODYNAMICS OF RADIATIONS
For particles emitted in a wave mode labelled by energy ω and λ plus some other quantum numbers J i if needed, we find that (Probability for a black hole to emit a particle in this mode) = exp − ω T ef f × (Probability for a black hole to absorb a particle in the same mode), where T ef f is given by eqns. (44) or (45). The above relation for usual dispersion relation was obtained by Hartle and Hawking [59] using semiclassical analysis. If the black hole is in equilibrium, the rate of emission particles by the black hole must exactly equal the rate of absorption. Neglecting back-reaction, detailed balance condition requires that the ratio of the probability of having N particles in a particular mode with ω, λ and J i to the probability of having N −1 particles in the same mode is exp − ω T ef f . Thus, we find that the probability of having N particles P N (ω, λ, J i ) in the mode is given by where C ω,λ,J i is a normalizing constant. C ω,λ,J i is determined by the normalized condition where ǫ = 0 for bosons and ǫ = 1 for fermions. To calculate the average number n ω,λ,J i in the mode, we define where one has C ω,λ, Using eqns. (61) and (63), one can rewrite P N (ω, λ, J i ) in terms of n ω,λ,J i as where N can be any non-negative integer for bosons (ǫ = 0) but is restricted to be 0 or 1 for fermions (ǫ = 1). The von Neumann entropy for the mode is which will be calculated in the brick wall model in section V. Note that since T ef f only depends on ω and λ, the average number n ω,λ,J i and the entropy s ω,λ,J i are independent of J i . Thus, we could omit the subscript J i in n ω,λ,J i and s ω,λ,J i from now on. Defining n (x) and s (x) by we can write n ω,λ and s ω,λ with respect to n (x) and s (x) n ω,λ = n ω T ef f , When integrating over ω, we need to specify the upper limit on the energy of the emitted particle. One of the limits comes from the requirement that the energy of the particle could not exceed the mass of the black hole. Another one is from the effective field theories in the appendix. Suppose that the higher dimensional operators in the effective field theories are suppressed by some scale of Lorentz violation Λ. Usually, we can only trust the effective theories below Λ. As decoupling theorem [60] shows, the contributions above Λ in some regularized theory gets absorbed into Wilson coefficients of the effective theories, Cs and Bs in the appendix. Consequently, the energy of the particle could not exceed Λ otherwise our effective theories would break down. Note that ω Λ has also been obtained in section III. Thus, the energy of the emitted particle ω ≤ ω max ≡ min {M, Λ}. In the remaining of the paper, we would encounter the integrals like where u max = ωmax T 0 and i is a non-negative integer. For a black hole with the mass M ≫ m p , one finds that u max = 2πmp m 2 p κ ∼ 1 κmp ≫ 1. For example, κ = 1 4M and κm p ≪ 1 in the Schwarzschild metric. For such case, using n (x) ∼ e −x and s (x) ∼ xe −x for x ≫ 1, one gets which can be safely neglected for κm p ≪ 1 and hence we can let u max = ∞ in eqn. (69).
Therefore, in section V, we let u max = ∞ for integrals of the form in eqn. (69) since we are only interested in the divergent part of the entanglement entropy as κm p → 0.
Most works of the dispersive models focus on calculating the modifications of the asymptotic spectrum. There are two types of contributions to the asymptotic spectrum measured at infinity: particles generated at the black hole horizon and scattering off the background.
The first contribution depends on derivatives of f (r) at r = r h whereas the second one depends on f (r) for r > r h . As shown in section VI, the the spectrum is proportional to n ω,λ . Since the tunneling probability across the horizon is calculated in the Hamilton-Jacobi method, only the first contribution to the asymptotic spectrum is computed in our paper.
The dispersive models have been studied in 2D spacetime for most works. The higher order terms in the MDR violate conformal invariance of 2D spacetime, hence there is some scattering. Our calculations show that the spectrum of created particles near the horizon is close to a perfect thermal spectrum in the dispersive models with the static preferred frame, where it has been suggested that the asymptotic spectrum could significantly differ from the thermal one [11]. Thus, one might need the scattering off the background to explain the difference between the spectrum near the horizon and the asymptotic one. The near horizon properties of the dispersive modes has been considered in [16]. It was found that the mode mixing responsible for the Hawking effect occurred within a region of size d br ∼ m 2 3 p κ − 1 3 across the horizon and the standard Hawking spectrum has been recovered as long as d br ≪ κ −1 .
In section V, we calculate the thermal entropy near the horizon, where the effects of scattering off the background hardly play a role and hence are neglected. However, the black holes' luminosities at infinity are computed in section VI, where the effects of scattering off the background should be included. In fact, we use the geometric optics approximation to estimate these effects instead of solving the equations of motion of the fields. Under the geometric optics approximation, we find that the asymptotic spectrum is still close to a thermal one. Nevertheless, [11] might suggest that such approximation may not be robust enough and hence the asymptotic spectrum could be significantly changed.

V. ENTROPY IN BRICK WALL MODEL
In 1985 t' Hooft [61] proposed the brick wall model to calculate the entropy of a thermal gas of Hawking particles propagating just outside the black hole horizon. The entropy is calculated by methods of the WKB approximation. However, when it comes to calculate the density of states of emitted particles, t' Hooft found that they became infinite as one got closer to the horizon. To make the entropy finite, he introduced a brick wall cut-off near the horizon such that the boundary condition where Φ is the radiation's field. Moreover, another cut-off at a large distance from the horizon L ≫ r h was introduced to eliminate infrared divergences.
For simplicity, we consider in this section a 4D spherically symmetric black hole with the metric For such a black hole, the quantum numbers needed to specify a wave mode of radiation are the energy ω, the angular momentum l, the magnetic quantum number m. We also assume that the radiated particles are massless and neutral. Thus, the MDR, eqn. (2), becomes where we define H (x) = H (x, 0) and H (x) ∼ x 2 for x ≪ 1. As shown in section II, the deformed Hamilton-Jacobi equation incorporating eqn. (73) for a massless and neutral particle in the 4D spherically symmetric black hole is given by where Then, we get from eqn. (74) Define the radial wave number k (r, l, ω) by as long as p 2 r ≥ 0, and k (r, l, ω) = 0 otherwise. Taking two Dirichlet conditions at r = r h +r ε and r = L into account, one finds that the number of one-particle states not exceeding ω with fixed value of the angular momentum l is given by Thus, we obtain for the total entropy of radiation S = ω,l,m s ω,l = (2l + 1) dl dω dn (ω, l) dω s ω,l where we define a dimensionless parameter z = (l+ 1 2 ) Defining the coefficients ξ n l,k by one has for s ω,l where we define u = ω T 0 and Note that ξ 0 k,l = 0 except ξ 0 0,0 = 1, ξ n l,k<n−l = 0 and ξ n l<n−k,k = 0. Putting eqn. (82) into eqn.
(79) and integrating eqn. (79) over z gives To calculate S, the variable r may be changed by introducing . Then, eqn. (85) becomes Now the brick walls are at x = x ε and x = δ. Note that x ε → ∞ when r ε → 0 and the horizon is at x ε = ∞. Since G a (0) = 1, we can Taylor expand G a (y) at y = 0 where we find the first two coefficients of the series expansion are f a 0 = 1 and f a 1 = . Substituting eqn. (88) into eqn. (86) gives us where we use T 0 = κ 2π and = m 2 p . The entropy receives two contributions, one from the horizon and the other from the vacuum surrounding the system at large distances. The second one is irrelevant for our purposes and henceforth discarded.
For the usual scenario with H (x) = x 2 , the integrals over x in eqn. (89) become divergent for a = k = 0 as one approaches the horizon with x ε → ∞. This divergence leads to the introduction of the wall near the horizon by t' Hooft. However, the x-integrals could be finite as x ε → ∞ for some MDRs. In fact, there are two kinds of MDRs for the integrals to be finite. For the first kind of these MDRs, the high energy contributions are suppressed.
For example, the "all-order MDR" of form was given in the κ-Minkowski noncommutative spacetime in [62]. For such a MDR, one has which guaranties the convergence of the x-integrals as x ε → ∞. Another example is inspired by the all order generalized uncertainty relation considered in [63]. The MDR can be written which gives Thus, the x-integrals stay finite as x → ∞. Moreover, it was found in [63] that the entropy kept finite when the wall approached the horizon and hence the wall in the brick wall model located just outside the horizon could be avoided. For the second kind of the MDRs, the energy E in the MDRs has a maximum value and hence x ε can not go to the infinity. For example, Corley and Jacobson [9] proposed which gives So the x-integrals are finite for the Corley and Jacobson dispersion relation. It was shown in [64] that the entropy was rendered UV finite for the Corley and Jacobson dispersion relation.

For the Unruh dispersion relation[7]
we have We find that the x-integrals diverge as x → 1 and a wall near the horizon is needed. However, the entropy for the Unruh dispersion relation was also found finite in [64]. This might be due to different generalization of the Hamilton-Jacobi equation in curved space in [64]. In the remaining of section, we will consider two cases, in one of which the x-integrals converge, and in the other they diverge.

A. UV Finite Case
We here assume that the x-integrals converge as x → Λ, where Λ = ∞ for the first kind of the MDRs in this case and Λ is the largest x for the second kind. Thus, we can definẽ Since H (x) ∼ x 2 for x ≪ 1, the Taylor expansion of H a+ 1 2 (x) H ′ (x) is given by where d a 0 = 2. Then one gets where θ (x) is the Heaviside step function, 0 < x 1 < Λ and c a k is a constant independent of L. Neglecting terms depending on L, one finds Plugging eqn. (101) into eqn. (89) gives us that the entropy near the horizon can be written of form For k = 0, we can choose x 1 = 0 in eqn. (101) since and where η 2 1 = 3C 2 3 16 − C 4 4 and we use ξ n l,k<n−l = 0, ξ l l,0 = (η 2 1 ζ 1 0 ) l , and ζ 1 0 = 1. Since there is no ln δ in eqn. (101) for k = 0, we have l 0 = 0. For k = 1, onlyc 0 1 contributes to l 1 and we find where we have .
Thus, we obtains for the entropy near horizon s (u) u 2 du ln m p κ+Finite terms as κm p → 0. (106)

B. Perturbative Case
In this case, a wall near the horizon is needed to regulate the x-integrals. As above, the function H (x) can be presented in the form of Taylor series where C 2 = 1. One then can have Taylor expansions for H a+ 1 2 (x) H ′ (x) where d a 0 = 2. The radial position of the brick wall near the horizon is r = r h + r ε (x = x ε ). The invariant distance of the wall from the horizon ε is defined by where we define y = T 0 u xmp = where we expand 2κ f ′ (f −1 (y 2 )) in the integral andf n are coefficients of the series. Solving eqn. (110) for y ε gives where ζ n are determined byf n . Using x ε = T 0 u yεmp , one can relate x ε to ε by where χ 0 a = 1. Focusing only on the near horizon contributions, we neglect terms involving L and use eqn. (108) to obtain where for the logarithmic term we have  gives ε 1 90π m p for a scalar field. Thus, we define α such as ε = αm p . Replacing ε by αm p in eqn. (114), we find for the entropy where we define (116)

C. Discussion
For a massless scalar field, we find the entropy near horizon in both cases can be written as S ∼ As 0 16π 3 m 2 p + s L ln m p κ + Finite terms as κm p → 0, where and A = 4πC (r 2 h ) is the horizon area. For the scenario without the MDR, the entropy near horizon [61,65,66] is where we let the proper distance ε = αm p . By comparing eqn. (117) with eqn. (118) it shows that the leading term of the entropy is affected by the effects of the MDR while the subleading logarithmic term is not. On the other hand, the first law of black hole thermodynamics dS B = dM T and eqn. (59) lead to the modified entropy of the black hole where A = 16πM 2 and κ = 1 4M . For S B , the leading term is not changed while the subleading logarithmic term is due to the MDR. These might suggest that the explanations for statistical origin of the black holes' entropy need more than the entropy of a thermal gas of Hawking particles near the horizon.
Since the deformed Hamilton-Jacobi equations and the corrections to the Hawking temperature are same for fermions and scalars with the same MDR, one may wonder if eqns.
(106) and (115) also work for fermions. In fact, it has been shown in [67] that the same argument in this section held for fermions if an appropriate boundary condition was taken instead of the too restrictive Dirichlet boundary condition.
For a MDR with H (x) in the UV finite case, we have shown that a brick wall near the horizon is not needed since the entropy is finite as one approaches the horizon. However, if one expands H (x) as a power series of x and calculates the entropy in the perturbative case, it seems that a wall near the horizon is needed to regulate the divergence. How can we reconcile the contradiction? As noted in [68], the divergence of the entropy in the perturbative case as α → 0 is more like due to the breaking down of the Taylor series.
For the typical energy ω ∼ T 0 = κ 2π , one finds that H (x) and the MDR corrections to the entropy are powers of Thus, the perturbative case is valid outside the wall at r ε = r h + 2κm 2 p . However, the perturbation would break down deep within the wall and the closed form of H (x) is needed.

VI. BLACK HOLE EVAPORATION
In [69], Page counted the number of modes per frequency interval with periodic boundary conditions in a large container around the black hole and divided it by the time it takes a particle to cross the container. He then related the expected number emitted per mode n to the average emission rate per frequency interval dn dt by for each mode and frequency interval (ω, ω + dω). Following the same argument, we find that in the MDR case dn dt = n ∂ω ∂p r dp r 2π = n dω 2π , where ∂ω ∂pr is the radial velocity of the particle and the number of modes between the wavevector interval (p r , p r + dp r ) is dpr 2π . Since each particle carries off the energy ω, the total luminosity is obtained from multiplying dn dt by the energy ω and summing up over all energy ω and quantum numbers, denoted by i, However, some of the radiation emitted by the horizon might not be able to reach the asymptotic region. Before the radiation reaches the distant observer, they must pass the curved spacetime around the black hole horizon, which plays the role of a potential barrier.
This effect on L can be described by a greybody factor from the scattering coefficients of the black hole. Actually, the greybody factor is given by |T i (ω)| 2 , where T i (ω) represents the transmission coefficient of the black hole barrier which in general can depend on the energy ω and quantum numbers i of the particle. Taking the greybody factor into account, we find for the total luminosity Since the relevant radiation usually have the energy of order M −1 , where M is the mass of the black hole, one should use the wave equations given in the appendix to compute |T i (ω)| 2 accurately. However, solving the wave equations for |T i (ω)| 2 could be very complicated. On the other hand, one can use the geometric optics approximation to estimate |T i (ω)| 2 . In the geometric optics approximation, we assume ω ≫ M and high energy waves will be absorbed unless they are aimed away from the black hole. Hence we have |T i (ω)| 2 = 1 for all the classically allowed energy ω and quantum numbers i of the particle, while |T i (E)| 2 = 0 otherwise. For the usual dispersion relations, the Stefan's law for black holes is obtained in this approximation. In the remaining of the section, we will discuss evaporations of a 4D spherically symmetric black hole with the mass M ≫ m p and a 2D black hole. For simplicity, we assume that the particles are massless and neutral.

A. 4D Spherically Symmetric Black Hole
To find the classically allowed values of angular momentum l with fixed value of energy ω, we consider eqn. (76) for a massless particle in a 4D spherically symmetric black hole, where we have λ = l + 1 2 2 2 . Since p r is always a real number in the geometric optics approximation, one has an upper bound on λ has a minimum at r min and this minimum is denoted by λ max . If the particles overcome the angular momentum barrier and get absorbed by the black hole, one must have λ ≤ λ max . Thus, the eqn. (123) becomes where g s is the number of polarization, z = (l+ 1 2 ) and we use eqn. (68) for n ω,l . Defining we find where ξ n a,k is given by eqn. (81). Substituting eqn. (127) into eqn. (125) and integrating eqn. (125) over z gives where we let u max = ∞ for M ≫ m p . Since H (x) = x 2 + n≥3 C n x n , we define h a m by where h a 0 = 1, h a 1 = aC 3 and h a 2 = C 4 a + (128) gives where [x] = max {m ∈ Z | m ≤ x} and we define and equating it to zero, one finds In the geometric optics approximation, the Schwarzschild black hole can be described as a black sphere for absorbing particles. The total luminosity are determined by the radius of the black sphere R and the temperature of the black hole T . Note that one has R = λmax ω 2 and T ef f ≈ T 0 (1 − ∆) where for massless particles Consider a sub-luminal case with C 3 = 0 and C 4 > 0, where the total luminosity decreases due to the MDR effects. In this case, the MDR effects increase the radius of the black sphere while they decrease the temperature of the black hole. The competition between the increased radius and the decreased temperature determines whether the luminosity would increase or decrease. It appears form eqn. (133) that the effects of decreased temperature wins the competition.

B. 2D Black Hole
Suppose the metric of a 2D black hole is given by where f (r) has a simple zero at r = r h . Here we consider a neutral and massless scalar particle governed by the modified dispersion relation which is the Corley and Jacobson dispersion relation for C > 0 [9]. Expressing p in terms of E gives For the 2D black hole with the event horizon at r = r h , eqn. (20) gives where we use m = 0,ω (r h ) = ω, λ = 0, α = 1, γ = 0, C 3 = 0, and C 4 = C. where 0 < α < 1. Note that |∆| < 1 for ω < Λ. Therefore, the luminosity for the black hole where T 0 = κ 2π and u max = min 2πM mpκ . For κm p ≪ 1, we can let u max = ∞ and then find (dotted lines). Note that α parameterizes the unknown quantum gravity ultraviolet cutoff Λ. In FIG. 2, it suggests that the ηC < 0 cases are highly sensitive to the physics at high energies while the ηC > 0 ones are not. If η > 0, ηC < 0/ηC > 0 implies that the particles are super-/sub-luminal. The author in [70] has shown that the Hawking radiation with sub-luminal dispersion was not sensitive to Lorentz violation at high energies due to the "mode conversion". However, the outgoing black hole modes with super-luminal dispersion emanated from some unknown quantum gravity processes.

VII. DISCUSSION AND CONCLUSION
We used the Hamilton-Jacobi method to investigate the dispersive field theory models in this paper. Our results suggest that the thermal spectrum of radiations near horizon is robust. In fact, if the difference between the modified dispersion relation and the relativistic one was suppressed by the fundamental energy scale m p , we found that the deviation of However, the Hamilton-Jacobi method is incapable of computing them since the metric is fixed in this method. On the other hand, back-reaction appears in the the null geodesic method [20,21] to ensure energy conservation during the emission of a particle via tunneling through the horizon. These corrections lead to non-thermal corrections to the black-hole radiation spectrum. Note that there are some attempts to incorporate back-reaction effects into the the Hamilton-Jacobi method using the rainbow metric [71,72].
(b) Higher order WKB corrections. In the Hamilton-Jacobi method, we take the semiclassical limit → 0 and keep only leading order terms to calculate the Hawking temperature. Therefore, one may wonder if the Hawking temperature could receive higher order corrections in beyond the semiclassical one. The corrections has been estimated in [73] and was given by powers of case, several authors [74][75][76] argued that the tunneling method yielded no higher-order corrections to the Hawking temperature. Whether such arguments also work for the dispersive models needs to be checked.
In this paper, we used the Hamilton-Jacobi method to calculate tunneling rates of radiations across the horizon and the effective Hawking temperatures in the dispersive models with the static preferred frame. After the spectrum of radiations near the horizon was obtained, the thermal entropy of radiations near the horizon and the luminosity of the black hole were computed. Our main results are as follows: • In section II and the appendix, we used heuristic arguments and effective field theories, respectively to derive the deformed Hamilton-Jacobi equations in the dispersive models with the static preferred frame. Note that these methods can easily be generalized to any preferred frame.
• In section II, the deformed Hamilton-Jacobi equations was solved for ∂ r I and the imaginary part of I was obtained by computing the residue of ∂ r I at r = r h . The assumption for our calculation was also given, which required that the singularity structure of ∂ r I except the order of the pole at r = r h do not change after the MDR was introduced. The corrections to the Hawking temperature were calculated for massive and charged particles to O m −2 p and neutral and massless particles to all orders, respectively. It was found that corrections were suppressed by m p .
• In section IV, the average number and entropy for a mode were calculated for bosons and fermions. They could be obtained from those in the non-dispersive case by replacing the standard Hawking temperature with the modified one.
• In section V, we used the brick wall model to compute the thermal entropy of a massless scalar field near the horizon in UV finite and perturbative cases. In the UV finite case, the entropy was always finite as one approached the horizon and hence the wall near the horizon was not needed. In the perturbative case, a wall was put at r = r h + r ε to regulate the UV divergence. We assumed the proper distance between the horizon and the wall is order of m p . Thus, the entropies near the horizon in both cases were given in eqn. (117). We found that the leading term of the entropy depended on the MDR effects while the subleading logarithmic term did not.
• In section VI, we calculated luminosities of a 4D spherically symmetric black hole with the mass M ≫ m p and a 2D one. We used the geometric optics approximation to estimate the effects of scattering off the background. However, as discussed in section IV, such approximation might not be robust enough and hence the asymptotic spectrum could be significantly changed.
possibilities for breaking or modifying the Lorentz symmetry, one of which is that Lorentz invariance is spontaneously broken by extra tensor fields taking on vacuum expectation values. The most conservative approach for a framework in which to describe MDR is the effective field theory (EFT), where modifications to the dispersion relation can be described by the higher dimensional operators. Since we are only interested in modifications to the dispersion relation of the particles, we limit ourselves to the kinetic terms and neglect selfinteracting effective operators when constructing the effective field theory. We also assume that the effective theory respects U (1) gauge invariance of the charged black hole. The EFT framework can easily incorporate MDR via the introduction of extra tensors. To construct the minimal EFT in curved spacetime, we suppose that the action of the EFT contains the usual minimal gravitational couplings and the EFT coefficients are constants in the local frame [77].

Scalar Field
We work with a complex scalar field φ with the mass m and the charge q. Following guidelines we put forth, we find the effective Lagrangian for φ incorporating MDR can be written as With rotational symmetry, all extra tensors become reducible to products of a vector field

Fermionic Field
In the background spacetime with the metric g µν and the electromagnetic potential A µ , the effective Lagrangian for a spin-1/2 fermion ψ with the mass m and the charge q incorporating the MDR can be written as where extra tensors B j µ 1 ···µ k and C j µ 1 ···µn are dimensionless functions of m mp with B j µ 1 ···µ k (0) = C j µ (0) = 0, j runs over all independent operators of a given dimension, D f µ = ∂ µ + Ω µ + iq A µ , Ω µ ≡ i 2 ω ab µ Σ ab , Σ ab is the Lorentz spinor generator, ω ab µ is the spin connection and {γ µ , γ ν } = 2g µν . The Greek indices are raised and lowered by the curved metric g µν , while the Latin indices are governed by the flat metric η ab . The deformed Dirac equation is D f,µ 1 · · · D f,µ k γ µ k+1 · · · γ µn ψ = 0.