Minimal Length Effects on Tunnelling from Spherically Symmetric Black Holes

In this paper, we investigate effects of the minimal length on quantum tunnelling from spherically symmetric black holes using the Hamilton-Jacobi method incorporating the minimal length. We first derive the deformed Hamilton-Jacobi equations for scalars and fermions, both of which have the same expressions. The minimal length correction to the Hawking temperature is found to depend on the black hole's mass and the mass and angular momentum of emitted particles. Finally, we calculate a Schwarzschild black hole's luminosity and find the black hole evaporates to zero mass in infinite time.


I. INTRODUCTION
The classical theory of black holes predicts that anything, including light, couldn't escape from the black holes. However, Stephen Hawking first showed that quantum effects could allow black holes to emit particles. The formula of Hawking temperature was first given in the frame of quantum field theory [1]. After that, various methods for deriving Hawking radiation have been proposed. Among them is a semiclassical method of modeling Hawking radiation as a tunneling effect proposed by Kraus and Wilczek [2,3], which is known as the null geodesic method. Later, the tunneling behaviors of particles were investigated using the Hamilton-Jacobi method [4][5][6]. Using the null geodesic method and the Hamilton-Jacobi method, much fruit has been achieved [7][8][9][10][11][12][13][14][15][16][17][18]. The key point of the Hamilton-Jacobi method is using WKB approximation to calculate the imaginary part of the action for the tunneling process.
On the other hand, various theories of quantum gravity, such as string theory, loop quantum gravity and quantum geometry, predict the existence of a minimal length [19][20][21].
The generalized uncertainty principle(GUP) [22] is a simply way to realize this minimal length.
An effective model of the GUP in one dimensional quantum mechanics is given by [23,24] where the generators of the translations in space and time are the wave vector k and the frequency ω, L f is the minimal length, and L f M f = . The quantization in position representationx = x leads to Therefore, the low energy limit p ≪ M f including order of p 3 where we neglect the factor 1 3 . From eqn. (1), it is noted that although one can increase p arbitrarily, k has an upper bound which is 1 L f . The upper bound on k implies that that particles could not possess arbitrarily small Compton wavelengths λ = 2π/k and that there exists a minimal length ∼ L f . Furthermore, the deformed Klein-Gordon/Dirac equations incorporating eqn. (4) and eqn. (5) have already be obtained in [23], which will be briefly reviewed in section II. So [23] provides a way to incorporate the minimal length with special relativity, a good starting point for studying Hawking radiation as tunnelling effect.
The black hole is a suitable venue to discuss the effects of quantum gravity. Incorporating GUP into black holes has been discussed in a lot of papers [25][26][27][28][29][30]. The thermodynamics of black holes has also been investigated in the framework of GUP [29,30]. In [31], a new form of GUP is introduced where p a is the modified four momentum, k a is the usual four momentum and α is a small  [32][33][34][35][36]. The authors consider the GUP of form where x 0i, and p 0i satisfy the canonical commutation relations. Fermions' tunnelling, black hole thermodynamics and the remnants are discussed there.
In this paper, we investigate scalars and fermions tunneling across the horizons of black holes using the deformed Hamilton-Jacobi method which incorporates the minimal length via eqn. (4) and eqn. (5). Our calculation shows that the quantum gravity correction is related not only to the black hole's mass but also to the mass and angular momentum of emitted particles.
The organization of this paper is as follows. In section II, from the modified fundamental commutation relation, we generalize the Hamilton-Jacobi in curved spacetime. In section III, incorporating GUP, we investigate the tunnelling of particles in the black holes. In section IV, we investigate how a Schwarzschild black hole evaporates in our model. Section V is devoted to our conclusion. In this paper, we take Geometrized units c = G = 1, where the Planck constant is square of the Planck Mass m p . We also assume that the emitted particles are neutral.

II. DEFORMED HAMILTON-JACOBI EQUATIONS
To be generic, we will consider a spherically symmetric background metric of the form where 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. In this section, we will first derive the deformed Klein-Gordon/Dirac equations in flat spacetime and then generalize them to the curved spacetime with the metric (10).
In the (3 + 1) dimensional flat spacetime, the relations between (p i , E) and (k i , ω) can simply be generalized to where, in the spherical coordinates, one has for k Expanding eqn. (11) and eqn. (12) for small arguments to the third order gives the energy and momentum operator in position representation where we also omit the factor 1 3 . Substituting the above energy and momentum operators into the energy-momentum relation, the deformed Klein-Gordon equation satisfied by the scalar field with the mass m is where and substituting it into eqn. (16), one expands eqn. (16) in powers of and then finds that the lowest order gives the deformed scalar Hamilton-Jacobi equation in the flat spacetime Similarly, the deformed Dirac equation for a spin-1/2 fermion with the mass m takes the form as where {γ 0 , γ 0 } = 2, {γ a , γ b } = −2δ ab , and {γ 0 , γ a } = 0 with the Latin index a running over r, θ,and φ. Multiplying (γ 0 E + γ · p + m) by eqn. (19) and using the gamma matrices anticommutation relations, the deformed Dirac equation can be written as To obtain the Hamilton-Jacobi equation for the fermion, the ansatz for ψ takes the form of where v is a vector function of the spacetime. Substituting eqn. (21) into eqn. (20) and noting that the second term on RHS of eqn. (20) does not contribute to the lowest order of , we find the deformed Hamilton-Jacobi equation for a fermion is the same as the deformed one for a scalar with the same mass, namely eqn. (18). Note that one can use the deformed Maxwell's equations obtained in [23] to get the deformed Hamilton-Jacobi equation for a vector boson. However, for simplicity we just stop here.
In order to generalize the deformed Hamilton-Jacobi equation, eqn. (18), to the curved spacetime with the metric (10), we first consider the Hamilton-Jacobi equation without GUP modifications. In ref. [37], we show that the unmodified Hamilton-Jacobi equation in curved Therefore, the unmodified Hamilton-Jacobi equation in the metric (10) becomes On the other hand, the unmodified Hamilton-Jacobi equation in flat spacetime can be obtained from eqn. (18) by taking M f → ∞, Comparing eqn. (23) with eqn. (24), one finds that the Hamilton-Jacobi equation in the metric (10) can be obtained from the one in flat spacetime by making replacements in the no GUP modifications scenario. Therefore, by making , the deformed Hamilton-Jacobi equation in flat spacetime, eqn. (18), leads to the deformed Hamilton-Jacobi equation in the metric

III. QUANTUM TUNNELLING
In this section, we investigate the particles' tunneling at the event horizon r = r h of the metric (10) where GUP is taken into account. Since the metric (10) does not depend on t and φ, ∂ t and ∂ φ are killing vectors. Taking into account the Killing vectors of the background spacetime, we can employ the following ansatz for the action where ω and p φ are constants and they are the energy and the z-component of angular momentum of emitted particles, respectively. Inserting eqn. (26) into eqn. (25), we find that the deformed Hamilton-Jacobi equation becomes where Since the magnitude of the angular momentum of the particle L can be expressed in terms of p θ and p φ , one can rewrite λ as Solving eqn. (27) for where +/− represent the outgoing/ingoing solutions. In order to get the imaginary part of W ± , we need to find residue of the RHS of eqn. (30) at r = r h by expanding the RHS in a Laurent series with respect to r at r = r h . We then rewrite eqn. (30) as Using where we have Using the residue theory for semi circles, we obtain for the imaginary part of However, when one tries to calculate the tunneling rate Γ from Im W ± , there is so called "factor-two problem" [38]. One way to solve the "factor two problem" is introducing a "temporal contribution" [14-16, 39, 40]. To consider an invariance under canonical transformations, we also follow the recent work [14-16, 39, 40] and adopt the expression p r dr = p + r dr − p − r dr for the spatial contribution to Γ. The spatial and temporal contributions to Γ are given as follows.
Spatial Contribution: The spatial part contribution comes from the imaginary part of W (r). Thus, the spatial part contribution is proportional to Temporal Contribution: As pointed in Ref. [15,[38][39][40], the temporal part contribution comes from the "rotation" which connects the interior region and the exterior region of the black hole. Thus, the imaginary contribution from the temporal part when crossing the horizon is Im (ω∆t out,in ) = ω π 2κ , where κ = f ′ (r h ) 2 is the surface gravity at the event horizon.
Then the total temporal contribution for a round trip is Therefore, the tunnelling rate of the particle crossing the horizon is This is the expression of Boltzmann factor with an effective temperature where is the original Hawking temperature.

IV. THERMODYNAMICS OF BLACK HOLES
For simplicity, we consider the Schwarzschild metric with f (r) = 1 − 2M r with the black hole's mass, M. The event horizon of the Schwarzschild black hole is r h = 2M. In this section, we work with massless particles. Near the horizon of the the black hole, angular momentum of the particle L ∼ pr h ∼ ωr h . Thus, one can rewrite T where T 0 = 8πM for the Schwarzschild black hole. As reported in [43], the authors obtained the relation ω δx between the energy of a particle and its position uncertainty in the framework of GUP. Near the horizon of the the Schwarzschild black hole, the position uncertainty of a particle will be of the order of the Schwarzschild radius of the black hole [44] δx ∼ r h . Thus, one finds for T where we use = m 2 p . Using the first law of the black hole thermodynamics, we find the corrected black hole entropy is where A = 4πr 2 h = 16πM 2 is the area of the horizon. The logarithmic term in eqn. (41) is the well known correction from quantum gravity to the classical Bekenstein-Hawking entropy, which have appeared in different studies of GUP modified thermodynamics of black holes [27][28][29][45][46][47][48][49][50][51]. In general, the entropy for the Schwarzschild black hole of mass M in four spacetime dimensions can be written in form of where σ =  [27][28][29][45][46][47][48][49][50][51], which is consistent with the existence of a minimal length.
3. σ > 0 : This is a subtle case. In the remaining of the section, we will investigate how the black holes evaporates in our model.
For particles emitted in a wave mode labelled by energy ω and L, we find from eqn. (37) that [52] (Probability for a black hole to emit a particle in this mode) = exp − ω T × (Probability for a black hole to absorb a particle in the same mode), where T is given by eqn. (38). Neglecting back-reaction, detailed balance condition requires that the ratio of the probability of having N particles in a particular mode with ω and L to the probability of having N − 1 particles in the same mode is exp − ω T . One then follows the standard textbook procedure to get the average number n ω,L in the mode where we define and ǫ = 0 for bosons and ǫ = 1 for fermions. In [53], counting the number of modes per frequency interval with periodic boundary conditions in a large container around the black hole, Page related the expected number emitted per mode n ω,L to the average emission rate per frequency interval Following the Page's argument, we find that in our model 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 where p r = ∂ r I is the radial wavevector. Since each particle carries off the energy ω, the total luminosity is obtained from dn ω,L dt by multiplying by the energy ω and summing up over all energy ω and L, where L 2 = (l + 1) l 2 and the degeneracy for l is (2l + 1). However, some of the radiation emitted by the horizon might not be able to reach the asymptotic region. We need to consider the greybody factor |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 angular momentum l of the particle. Taking the greybody factor into account, we find for the total luminosity Usually, one needs to solve the exact wave equations for |T i (ω)| 2 , which is 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 angular momentum l of the particle. In this approximation, the Schwarzschild black hole is just like a black sphere of radius R = 3 3/2 M [54], which puts an upper bound on l (l + 1) 2 , Note that we neglect possible modifications from GUP to the bound since we are interested in the GUP effects near the horizon. Thus, the luminosity is L = where we define u = ω T 0 , y = l(l+1) 2 M 2 ω 2 and a = T 0 , we have a ≪ 1 and hence the luminosity is which is just the Stefan's law for black holes. Therefore for large black holes, they evaporate in almost the same way as in Case 1 until M ∼ where v = au. Not worrying about exact numerical factors, one has for the evaporation rate V.

CONCLUSION
In this paper, incorporating effects of the minimal length, we derived the deformed Hamilton-Jacobi Equations for both scalars and fermions in curved spacetime based on the modified fundamental commutation relations. We investigated the particles' tunneling in the background of a spherically symmetric black holes. In this spacetime configurations, we showed that the corrected Hawking temperature is not only determined by the properties of the black holes, but also dependent on the angular momentum and mass of the emitted particles. Finally, we studied how a Schwarzschild black hole evaporates in our model. We found the black hole evaporates to zero mass in infinite time.